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UNITED STATES
SECURITIES AND EXCHANGE COMMISSION
WASHINGTON, D.C. 20549
FORM 10-Q

☒ | QUARTERLY REPORT PURSUANT TO SECTION 13 OR 15(d) OF THE SECURITIES EXCHANGE ACT OF 1934

For the quarterly period ended | September 30, 2021 |

Or
☐ | TRANSITION REPORT PURSUANT TO SECTION 13 OR 15(d) OF THE SECURITIES EXCHANGE ACT OF 1934

Commission file number: 001-16209
ARCH CAPITAL GROUP LTD.
(Exact name of registrant as specified in its charter)

Bermuda | 98-0374481
(State or other jurisdiction of incorporation or organization) | (I.R.S. Employer Identification No.)
Waterloo House, Ground Floor |
100 Pitts Bay Road, | Pembroke | HM 08, | Bermuda | (441) | 278-9250
(Address of principal executive offices) | (Registrant’s telephone number, including area code)

Securities registered pursuant to Section 12(b) of the Exchange Act:

Title of each class |  | Trading Symbol (s) |  | Name of each exchange on which registered
Common shares, $0.0011 par value per share |  | ACGL |  | NASDAQ | Stock Market
Depositary shares, each representing a 1/1000th interest in a 5.45% Series F preferred share |  | ACGLO |  | NASDAQ | Stock Market
Depositary shares, each representing a 1/1000th interest in a 4.55% Series G preferred share |  | ACGLN |  | NASDAQ | Stock Market

Indicate by check mark whether the registrant (1) has filed all reports required to be filed by Section 13 or 15(d) of the Securities Exchange Act of 1934 during the preceding 12 months (or for such shorter period that the registrant was required to file such reports), and (2) has been subject to such filing requirements for the past 90 days. Yes ☑ No ☐
Indicate by check mark whether the registrant has submitted electronically every Interactive Data File required to be submitted pursuant to Rule 405 of Regulation S-T (§232.405 of this chapter) during the preceding 12 months (or for such shorter period that the registrant was required to submit such files). Yes ☑ No ☐
Indicate by check mark whether the registrant is a large accelerated filer, an accelerated filer, a non-accelerated filer, a smaller reporting company, or an emerging growth company. See the definitions of “large accelerated filer,” “accelerated filer,” “smaller reporting company,” and “emerging growth company” in Rule 12b-2 of the Exchange Act.
Large Accelerated Filer ☑ Accelerated Filer ☐ Non-accelerated Filer ☐ Smaller reporting company ☐ Emerging growth company ☐
If an emerging growth company, indicate by check mark if the Registrant has elected not to use the extended transition period for complying with any new or revised financial accounting standards provided pursuant to Section 13(a) of the Exchange Act. ☐
Indicate by check mark whether the Registrant is a shell company (as defined in Rule 12b-2 of the Exchange Act). Yes ☐ No ☑
As of November 1, 2021, there were 386,133,743 common shares, $0.0011 par value per share, of the registrant outstanding.

Table of Contents
ARCH CAPITAL GROUP LTD.
INDEX TO FORM 10-Q

|  |  | Page No.
| PART I |  |
| Financial Information |  | 2
Item 1. | Consolidated Financial Statements |  | 4
Item 2. | Management’s Discussion and Analysis of Financial Condition and Results of Operations |  | 42
Item 3. | Quantitative and Qualitative Disclosures About Market Risk |  | 69
Item 4. | Controls and Procedures |  | 69
| PART II |  |
| Other Information |  | 70
Item 1. | Legal Proceedings |  | 70
Item 1A. | Risk Factors |  | 70
Item 2. | Unregistered Sales of Equity Securities and Use of Proceeds |  | 71
Item 3. | Defaults Upon Senior Securities |  | 71
Item 4. | Mine Safety Disclosures |  | 71
Item 5. | Other Information |  | 71
Item 6. | Exhibits |  | 72
| Signatures |  | 73

ARCH CAPITAL | 1 | 2021 THIRD QUARTER FORM 10-Q

Table of Contents
PART I. FINANCIAL INFORMATION
Cautionary Note Regarding Forward-Looking Statements
The Private Securities Litigation Reform Act of 1995 (“PSLRA”) provides a “safe harbor” for forward-looking statements. This report or any other written or oral statements made by or on behalf of us may include forward-looking statements, which reflect our current views with respect to future events and financial performance. All statements other than statements of historical fact included in or incorporated by reference in this report are forward-looking statements. Forward-looking statements, for purposes of the PSLRA or otherwise, can generally be identified by the use of forward-looking terminology such as “may,” “will,” “expect,” “intend,” “estimate,” “anticipate,” “believe” or “continue” and similar statements of a future or forward-looking nature or their negative or variations or similar terminology.
Forward-looking statements involve our current assessment of risks and uncertainties. Actual events and results may differ materially from those expressed or implied in these statements. Important factors that could cause actual events or results to differ materially from those indicated in such statements are discussed below and elsewhere in this report and in our periodic reports filed with the Securities and Exchange Commission (the “SEC”), and include:
•our ability to successfully implement our business strategy during “soft” as well as “hard” markets;
•acceptance of our business strategy, security and financial condition by rating agencies and regulators, as well as by brokers and our insureds and reinsureds;
•our ability to consummate acquisitions and integrate the business we have acquired or may acquire into our existing operations;
•our ability to maintain or improve our ratings, which may be affected by our ability to raise additional equity or debt financings, by ratings agencies’ existing or new policies and practices, as well as other factors described herein;
•general economic and market conditions (including inflation, interest rates, unemployment, housing prices, foreign currency exchange rates, prevailing credit terms and the depth and duration of a recession, including those resulting from COVID-19) and conditions specific to the reinsurance and insurance markets in which we operate;
•competition, including increased competition, on the basis of pricing, capacity (including alternative sources of capital), coverage terms, or other factors;
•developments in the world’s financial and capital markets and our access to such markets;
•our ability to successfully enhance, integrate and maintain operating procedures (including information technology) to effectively support our current and new business;
•the loss of key personnel;
•material differences between actual and expected assessments for guaranty funds and mandatory pooling arrangements;
•accuracy of those estimates and judgments utilized in the preparation of our financial statements, including those related to revenue recognition, insurance and other reserves, reinsurance recoverables, investment valuations, intangible assets, bad debts, income taxes, contingencies and litigation, and any determination to use the deposit method of accounting;
•greater than expected loss ratios on business written by us and adverse development on claim and/or claim expense liabilities related to business written by our insurance and reinsurance subsidiaries;
•the adequacy of the Company’s loss reserves;
•severity and/or frequency of losses;
•greater frequency or severity of unpredictable natural and man-made catastrophic events;
•claims for natural or man-made catastrophic events or severe economic events in our insurance, reinsurance and mortgage businesses could cause large losses and substantial volatility in our results of operations;
•the effect of climate change on our business;
•the effect of contagious disease (including COVID-19) on our business;
•acts of terrorism, political unrest and other hostilities or other unforecasted and unpredictable events;

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•availability to us of reinsurance to manage our gross and net exposures and the cost of such reinsurance;
•the failure of reinsurers, managing general agents, third party administrators or others to meet their obligations to us;
•the timing of loss payments being faster or the receipt of reinsurance recoverables being slower than anticipated by us;
•our investment performance, including legislative or regulatory developments that may adversely affect the fair value of our investments;
•changes in general economic conditions, including sovereign debt concerns or downgrades of U.S. securities by credit rating agencies, which could affect our business, financial condition and results of operations;
•changes in the method for determining the London Inter-bank Offered Rate (“LIBOR”) and the potential replacement of LIBOR;
•the volatility of our shareholders’ equity from foreign currency fluctuations, which could increase due to us not matching portions of our projected liabilities in foreign currencies with investments in the same currencies;
•changes in accounting principles or policies or in our application of such accounting principles or policies;
•changes in the political environment of certain countries in which we operate or underwrite business;
•a disruption caused by cyber-attacks or other technology breaches or failures on us or our business partners and service providers, which could negatively impact our business and/or expose us to litigation;
•statutory or regulatory developments, including as to tax matters and insurance and other regulatory matters such as the adoption of proposed legislation that would affect Bermuda-headquartered companies and/or Bermuda-based insurers or reinsurers and/or changes in regulations or tax laws applicable to us, our subsidiaries, brokers or customers, including the Tax Cuts and Jobs Act of 2017; and
•the other matters set forth under Item 1A “Risk Factors”, Item 7 “Management’s Discussion and Analysis of Financial Condition and Results of Operations” and other sections of our Annual Report on Form 10-K for the year ended December 31, 2020, as well as the other factors set forth in our other documents on file with the SEC, and management’s response to any of the aforementioned factors.
All subsequent written and oral forward-looking statements attributable to us or persons acting on our behalf are expressly qualified in their entirety by these cautionary statements. The foregoing review of important factors should not be construed as exhaustive and should be read in conjunction with other cautionary statements that are included herein or elsewhere. We undertake no obligation to publicly update or revise any forward-looking statement, whether as a result of new information, future events or otherwise.

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ITEM 1. CONSOLIDATED FINANCIAL STATEMENTS

|  | Page No.
Report of Independent Registered Public Accounting Firm |  | 5
Consolidated Balance Sheets |  |
September 30, 2021 (unaudited) and December 31, 2020 |  | 6
Consolidated Statements of Income |  |
For the three and nine month periods ended September 30, 2021 and 2020 (unaudited) |  | 7
Consolidated Statements of Comprehensive Income |  |
For the three and nine month periods ended September 30, 2021 and 2020 (unaudited) |  | 8
Consolidated Statements of Changes in Shareholders’ Equity |  |
For the three and nine month periods ended September 30, 2021 and 2020 (unaudited) |  | 9
Consolidated Statements of Cash Flows |  |
For the nine month periods ended September 30, 2021 and 2020 (unaudited) |  | 10
Notes to Consolidated Financial Statements (unaudited) |  |
Note 1 - Basis of Presentation and Recent Accounting Pronouncements |  | 11
Note 2 - A cquisitions |  | 11
Note 3 - Share Transactions |  | 12
Note 4 - Earnings Per Common Share |  | 13
Note 5 - Segment Information |  | 14
Note 6 - Reserve for Losses and Loss Adjustment Expenses |  | 19
Note 7 - Allowance for Expected Credit Losses |  | 21
Note 8 - Investment Information |  | 23
Note 9 - Fair Value |  | 30
Note 10 - Derivative Instruments |  | 36
Note 1 1 - Commitments and Contingencies |  | 37
Note 1 2 - Variable Interest Entities and Noncontrolling Interests |  | 37
Note 1 3 - Other Comprehensive Income (Loss) |  | 40
Note 1 4 - Income Taxes |  | 41
Note 1 5 - Legal Proceedings |  | 41
Note 1 6 - Transactions with Related Parties |  | 41
Note 1 7 - Subsequent Event |  | 41

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Report of Independent Registered Public Accounting Firm
To the Board of Directors and Shareholders of Arch Capital Group Ltd.
Results of Review of Interim Financial Statements
We have reviewed the accompanying consolidated balance sheet of Arch Capital Group Ltd. and its subsidiaries (the “Company”) as of September 30, 2021, and the related consolidated statements of income, comprehensive income, and changes in shareholders’ equity for the three-month and nine-month periods ended September 30, 2021 and 2020, and the consolidated statements of cash flows for the nine-month periods ended September 30, 2021 and 2020, including the related notes (collectively referred to as the “interim financial statements”). Based on our reviews, we are not aware of any material modifications that should be made to the accompanying interim financial statements for them to be in conformity with accounting principles generally accepted in the United States of America.
We have previously audited, in accordance with the standards of the Public Company Accounting Oversight Board (United States) (PCAOB), the consolidated balance sheet of the Company as of December 31, 2020, and the related consolidated statements of income, comprehensive income, changes in shareholders’ equity, and cash flows for the year then ended (not presented herein), and in our report dated February 26, 2021, we expressed an unqualified opinion on those consolidated financial statements. In our opinion, the information set forth in the accompanying consolidated balance sheet as of December 31, 2020, is fairly stated, in all material respects, in relation to the consolidated balance sheet from which it has been derived.
Basis for Review Results
These interim financial statements are the responsibility of the Company’s management. We are a public accounting firm registered with the PCAOB and are required to be independent with respect to the Company in accordance with the U.S. federal securities laws and the applicable rules and regulations of the Securities and Exchange Commission and the PCAOB. We conducted our review in accordance with the standards of the PCAOB. A review of interim financial information consists principally of applying analytical procedures and making inquiries of persons responsible for financial and accounting matters. It is substantially less in scope than an audit conducted in accordance with the standards of the PCAOB, the objective of which is the expression of an opinion regarding the financial statements taken as a whole. Accordingly, we do not express such an opinion.
/s/ PricewaterhouseCoopers LLP
New York, NY
November 4, 2021

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ARCH CAPITAL GROUP LTD. AND SUBSIDIARIES
CONSOLIDATED BALANCE SHEETS
(U.S. dollars in thousands, except share dat a)

| (Unaudited)
| September 30, 2021 |  | December 31, 2020
Assets |  |  |
Investments: |  |  |
Fixed maturities available for sale, at fair value (amortized cost: $16,629,621 and $18,143,305; net of allowance for credit losses: $2,111 and $2,397 ) | $ | 16,768,363 |  |  | $ | 18,717,825 |
Short-term investments available for sale, at fair value (amortized cost: $3,070,120 and $1,924,292; net of allowance for credit losses: $0 and $0) | 3,069,965 |  |  | 1,924,922 |
Collateral received under securities lending, at fair value (amortized cost: $— and $301,089) | — |  |  | 301,096 |
Equity securities, at fair value | 1,790,640 |  |  | 1,444,830 |
Other investments (portion measured at fair value: $2,043,970 and $3,824,796) | 2,043,970 |  |  | 4,324,796 |
Investments accounted for using the equity method | 2,741,293 |  |  | 2,047,889 |
Total investments | 26,414,231 |  |  | 28,761,358 |
Cash | 1,137,721 |  |  | 906,448 |
Accrued investment income | 75,832 |  |  | 103,299 |
Securities pledged under securities lending, at fair value (amortized cost: $— and $294,493) | — |  |  | 294,912 |
Investment in operating affiliates | 1,111,825 |  |  | 129,291 |
Premiums receivable (net of allowance for credit losses: $38,715 and $37,781) | 2,807,720 |  |  | 2,064,586 |
Reinsurance recoverable on unpaid and paid losses and loss adjustment expenses (net of allowance for credit losses: $12,831 and $11,636) | 5,358,852 |  |  | 4,500,802 |
Contractholder receivables (net of allowance for credit losses: $3,484 and $8,638) | 1,824,990 |  |  | 1,986,924 |
Ceded unearned premiums | 1,824,910 |  |  | 1,234,075 |
Deferred acquisition costs | 893,665 |  |  | 790,708 |
Receivable for securities sold | 84,019 |  |  | 92,743 |
Goodwill and intangible assets | 963,322 |  |  | 692,863 |
Other assets | 2,286,649 |  |  | 1,724,288 |
Total assets | $ | 44,783,736 |  |  | $ | 43,282,297 |
Liabilities |  |  |
Reserve for losses and loss adjustment expenses | $ | 17,331,047 |  |  | $ | 16,513,929 |
Unearned premiums | 6,165,114 |  |  | 4,838,965 |
Reinsurance balances payable | 1,403,929 |  |  | 683,263 |
Contractholder payables | 1,828,474 |  |  | 1,995,562 |
Collateral held for insured obligations | 254,259 |  |  | 215,581 |
Senior notes | 2,724,149 |  |  | 2,861,113 |
Revolving credit agreement borrowings | — |  |  | 155,687 |
Securities lending payable | — |  |  | 301,089 |
Payable for securities purchased | 357,531 |  |  | 218,779 |
Other liabilities | 1,321,470 |  |  | 1,510,888 |
Total liabilities | 31,385,973 |  |  | 29,294,856 |
Commitments and Contingencies |  |  |
Redeemable noncontrolling interests | 10,237 |  |  | 58,548 |
Shareholders' Equity |  |  |
Non-cumulative preferred shares | 830,000 |  |  | 780,000 |
Common shares ($0.0011 par, shares issued: 582,908,723 and 579,000,841) | 648 |  |  | 643 |
Additional paid-in capital | 2,061,906 |  |  | 1,977,794 |
Retained earnings | 13,842,787 |  |  | 12,362,463 |
Accumulated other comprehensive income (loss), net of deferred income tax | 49,184 |  |  | 488,895 |
Common shares held in treasury, at cost (shares: 195,650,971 and 172,280,199) | (3,396,999) |  |  | (2,503,909) |
Total shareholders' equity available to Arch | 13,387,526 |  |  | 13,105,886 |
Non-redeemable noncontrolling interests | — |  |  | 823,007 |
Total shareholders' equity | 13,387,526 |  |  | 13,928,893 |
Total liabilities, noncontrolling interests and shareholders' equity | $ | 44,783,736 |  |  | $ | 43,282,297 |

See Notes to Consolidated Financial Statements

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ARCH CAPITAL GROUP LTD. AND SUBSIDIARIES
CONSOLIDATED STATEMENTS OF INCOME
(U.S. dollars in thousands, except share data)

| (Unaudited) |  | (Unaudited)
| Three Months Ended |  | Nine Months Ended
| September 30, |  | September 30,
| 2021 |  | 2020 |  | 2021 |  | 2020
Revenues |  |  |  |  |  |  |
Net premiums earned | $ | 1,929,337 |  |  | $ | 1,771,092 |  |  | 5,998,668 |  |  | 5,180,890 |
Net investment income | 88,195 |  |  | 128,512 |  |  | 298,664 |  |  | 405,150 |
Net realized gains (losses) | (25,040) |  |  | 280,499 |  |  | 320,328 |  |  | 470,127 |
Other underwriting income | 7,274 |  |  | 5,413 |  |  | 18,913 |  |  | 18,932 |
Equity in net income (loss) of investment funds accounted for using the equity method | 105,398 |  |  | 126,735 |  |  | 299,270 |  |  | 57,407 |
Other income (loss) | (3,960) |  |  | — |  |  | 1,151 |  |  | 65 |
Total revenues | 2,101,204 |  |  | 2,312,251 |  |  | 6,936,994 |  |  | 6,132,571 |
Expenses |  |  |  |  |  |  |
Losses and loss adjustment expenses | 1,226,019 |  |  | 1,216,273 |  |  | 3,588,950 |  |  | 3,562,214 |
Acquisition expenses | 306,015 |  |  | 247,942 |  |  | 945,639 |  |  | 750,014 |
Other operating expenses | 230,832 |  |  | 215,686 |  |  | 736,808 |  |  | 659,479 |
Corporate expenses | 19,672 |  |  | 17,937 |  |  | 61,007 |  |  | 56,653 |
Amortization of intangible assets | 20,135 |  |  | 16,715 |  |  | 49,823 |  |  | 49,835 |
Interest expense | 33,176 |  |  | 41,343 |  |  | 107,222 |  |  | 105,037 |
Net foreign exchange (gains) losses | (36,078) |  |  | 44,885 |  |  | (38,366) |  |  | 11,425 |
Total expenses | 1,799,771 |  |  | 1,800,781 |  |  | 5,451,083 |  |  | 5,194,657 |
Income (loss) before income taxes and income (loss) from operating affiliates | 301,433 |  |  | 511,470 |  |  | 1,485,911 |  |  | 937,914 |
Income tax expense | (4,137) |  |  | (23,707) |  |  | (94,176) |  |  | (77,779) |
Income (loss) from operating affiliates | 124,119 |  |  | 919 |  |  | 224,052 |  |  | 6,262 |
Net income (loss) | $ | 421,415 |  |  | $ | 488,682 |  |  | $ | 1,615,787 |  |  | $ | 866,397 |
Net (income) loss attributable to noncontrolling interests | (1,473) |  |  | (69,643) |  |  | (82,203) |  |  | (4,420) |
Net income (loss) available to Arch | 419,942 |  |  | 419,039 |  |  | 1,533,584 |  |  | 861,977 |
Preferred dividends | (16,090) |  |  | (10,403) |  |  | (38,159) |  |  | (31,209) |
Loss on redemption of preferred shares | (15,101) |  |  | — |  |  | (15,101) |  |  | — |
Net income (loss) available to Arch common shareholders | $ | 388,751 |  |  | $ | 408,636 |  |  | $ | 1,480,324 |  |  | $ | 830,768 |
Net income per common share and common share equivalent |  |  |  |  |  |  |
Basic | $ | 1.00 |  |  | $ | 1.01 |  |  | $ | 3.74 |  |  | $ | 2.06 |
Diluted | $ | 0.98 |  |  | $ | 1.00 |  |  | $ | 3.66 |  |  | $ | 2.02 |
Weighted average common shares and common share equivalents outstanding |  |  |  |  |  |  |
Basic | 389,274,220 |  |  | 402,850,485 |  |  | 395,899,591 |  |  | 403,081,266 |
Diluted | 397,903,347 |  |  | 409,194,657 |  |  | 404,260,485 |  |  | 410,314,897 |

See Notes to Consolidated Financial Statements

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ARCH CAPITAL GROUP LTD. AND SUBSIDIARIES
CONSOLIDATED STATEMENTS OF COMPREHENSIVE INCOME
(U.S. dollars in thousands)

| (Unaudited) |  | (Unaudited)
| Three Months Ended |  | Nine Months Ended
| September 30, |  | September 30,
| 2021 |  | 2020 |  | 2021 |  | 2020
Comprehensive Income |  |  |  |  |  |  |
Net income (loss) | $ | 421,415 |  |  | $ | 488,682 |  |  | $ | 1,615,787 |  |  | $ | 866,397 |
Other comprehensive income (loss), net of deferred income tax |  |  |  |  |  |  |
Unrealized appreciation (decline) in value of available-for-sale investments: |  |  |  |  |  |  |
Unrealized holding gains (losses) arising during period | (95,923) |  |  | 110,782 |  |  | (279,102) |  |  | 546,291 |
Reclassification of net realized (gains) losses, included in net income (loss) | (62,654) |  |  | (79,803) |  |  | (120,504) |  |  | (368,423) |
Foreign currency translation adjustments | (31,710) |  |  | 16,709 |  |  | (54,089) |  |  | (5,729) |
Comprehensive income (loss) | 231,128 |  |  | 536,370 |  |  | 1,162,092 |  |  | 1,038,536 |
Net (income) loss attributable to noncontrolling interests | (1,473) |  |  | (69,643) |  |  | (82,203) |  |  | (4,420) |
Other comprehensive (income) loss attributable to noncontrolling interests | 9,423 |  |  | (10,820) |  |  | 13,983 |  |  | 2,127 |
Comprehensive income (loss) available to Arch | $ | 239,078 |  |  | $ | 455,907 |  |  | $ | 1,093,872 |  |  | $ | 1,036,243 |

See Notes to Consolidated Financial Statements

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ARCH CAPITAL GROUP LTD. AND SUBSIDIARIES
CONSOLIDATED STATEMENTS OF CHANGES IN SHAREHOLDERS’ EQUITY
(U.S. dollars in thousands)

| (Unaudited) |  | (Unaudited)
| Three Months Ended |  | Nine Months Ended
| September 30, |  | September 30,
| 2021 |  | 2020 |  | 2021 |  | 2020
Non-cumulative preferred shares |  |  |  |  |  |  |
Balance at beginning of period | $ | 1,280,000 |  |  | $ | 780,000 |  |  | $ | 780,000 |  |  | $ | 780,000 |
Preferred shares issued | — |  |  | — |  |  | 500,000 |  |  | — |
Preferred shares redeemed | (450,000) |  |  | — |  |  | (450,000) |  |  | — |
Balance at beginning and end of period | $ | 830,000 |  |  | $ | 780,000 |  |  | $ | 830,000 |  |  | $ | 780,000 |
Common shares |  |  |  |  |  |  |
Balance at beginning of period | 647 |  |  | 642 |  |  | 643 |  |  | 638 |
Common shares issued, net | 1 |  |  | — |  |  | 5 |  |  | 4 |
Balance at end of period | 648 |  |  | 642 |  |  | 648 |  |  | 642 |
Additional paid-in capital |  |  |  |  |  |  |
Balance at beginning of period | 2,028,919 |  |  | 1,935,514 |  |  | 1,977,794 |  |  | 1,889,683 |
Amortization of share-based compensation | 14,216 |  |  | 14,662 |  |  | 71,279 |  |  | 55,872 |
Issue costs on preferred shares | — |  |  | — |  |  | (14,179) |  |  | — |
Reversal of issue costs on preferred shares redeemed | 15,101 |  |  | — |  |  | 15,101 |  |  | — |
Other changes | 3,670 |  |  | 606 |  |  | 11,911 |  |  | 5,227 |
Balance at end of period | 2,061,906 |  |  | 1,950,782 |  |  | 2,061,906 |  |  | 1,950,782 |
Retained earnings |  |  |  |  |  |  |
Balance at beginning of period | 13,454,036 |  |  | 11,420,686 |  |  | 12,362,463 |  |  | 11,021,006 |
Cumulative effect of an accounting change (1) | — |  |  | — |  |  | — |  |  | (22,452) |
Balance at beginning of period, as adjusted | 13,454,036 |  |  | 11,420,686 |  |  | 12,362,463 |  |  | 10,998,554 |
Net income (loss) | 421,415 |  |  | 488,682 |  |  | 1,615,787 |  |  | 866,397 |
Net (income) loss attributable to noncontrolling interests | (1,473) |  |  | (69,643) |  |  | (82,203) |  |  | (4,420) |
Preferred share dividends | (16,090) |  |  | (10,403) |  |  | (38,159) |  |  | (31,209) |
Loss on redemption of preferred shares | (15,101) |  |  | — |  |  | (15,101) |  |  | — |
Balance at end of period | 13,842,787 |  |  | 11,829,322 |  |  | 13,842,787 |  |  | 11,829,322 |
Accumulated other comprehensive income (loss), net of deferred income tax |  |  |  |  |  |  |
Balance at beginning of period | 230,048 |  |  | 349,488 |  |  | 488,895 |  |  | 212,091 |
Unrealized appreciation (decline) in value of available-for-sale investments, net of deferred income tax: |  |  |  |  |  |  |
Balance at beginning of period | 264,702 |  |  | 418,487 |  |  | 501,295 |  |  | 258,486 |
Unrealized holding gains (losses) during period, net of reclassification adjustment | (158,577) |  |  | 30,979 |  |  | (399,606) |  |  | 177,868 |
Unrealized holding gains (losses) during period attributable to noncontrolling interests | 10,752 |  |  | (11,179) |  |  | 15,188 |  |  | 1,933 |
Balance at end of period | 116,877 |  |  | 438,287 |  |  | 116,877 |  |  | 438,287 |
Foreign currency translation adjustments, net of deferred income tax: |  |  |  |  |  |  |
Balance at beginning of period | (34,654) |  |  | (68,999) |  |  | (12,400) |  |  | (46,395) |
Foreign currency translation adjustments | (31,710) |  |  | 16,709 |  |  | (54,089) |  |  | (5,729) |
Foreign currency translation adjustments attributable to noncontrolling interests | (1,329) |  |  | 360 |  |  | (1,204) |  |  | 194 |
Balance at end of period | (67,693) |  |  | (51,930) |  |  | (67,693) |  |  | (51,930) |
Balance at end of period | 49,184 |  |  | 386,357 |  |  | 49,184 |  |  | 386,357 |
Common shares held in treasury, at cost |  |  |  |  |  |  |
Balance at beginning of period | (3,007,578) |  |  | (2,494,505) |  |  | (2,503,909) |  |  | (2,406,047) |
Shares repurchased for treasury | (389,421) |  |  | (601) |  |  | (893,090) |  |  | (89,059) |
Balance at end of period | (3,396,999) |  |  | (2,495,106) |  |  | (3,396,999) |  |  | (2,495,106) |
Total shareholders’ equity available to Arch | 13,387,526 |  |  | 12,451,997 |  |  | 13,387,526 |  |  | 12,451,997 |
Non-redeemable noncontrolling interests | — |  |  | 757,920 |  |  | — |  |  | 757,920 |
Total shareholders’ equity | $ | 13,387,526 |  |  | $ | 13,209,917 |  |  | $ | 13,387,526 |  |  | $ | 13,209,917 |

(1) Adoption of ASU 2016-13, “Financial Instruments - Credit Losses (Topic 326)”.
See Notes to Consolidated Financial Statements

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ARCH CAPITAL GROUP LTD. AND SUBSIDIARIES
CONSOLIDATED STATEMENTS OF CASH FLOWS
(U.S. dollars in thousands)

| (Unaudited)
| Nine Months Ended
| September 30,
| 2021 |  | 2020
Operating Activities |  |  |
Net income (loss) | $ | 1,615,787 |  |  | $ | 866,397 |
Adjustments to reconcile net income (loss) to net cash provided by operating activities: |  |  |
Net realized (gains) losses | (367,313) |  |  | (477,683) |
Equity in net (income) or loss of investment funds accounted for using the equity method and other income or loss | (372,650) |  |  | 30,306 |
Amortization of intangible assets | 49,823 |  |  | 49,835 |
Share-based compensation | 72,303 |  |  | 56,433 |
Changes in: |  |  |
Reserve for losses and loss adjustment expenses, net of unpaid losses and loss adjustment expenses recoverable | 1,548,211 |  |  | 1,668,069 |
Unearned premiums, net of ceded unearned premiums | 985,242 |  |  | 498,811 |
Premiums receivable | (847,098) |  |  | (461,766) |
Deferred acquisition costs | (247,966) |  |  | (107,238) |
Reinsurance balances payable | 618,571 |  |  | 205,620 |
Other items, net | (427,359) |  |  | 2,666 |
Net cash provided by (used for) operating activities | 2,627,551 |  |  | 2,331,450 |
Investing Activities |  |  |
Purchases of fixed maturity investments | (29,870,023) |  |  | (34,050,883) |
Purchases of equity securities | (978,951) |  |  | (1,355,848) |
Purchases of other investments | (1,350,056) |  |  | (841,886) |
Proceeds from sales of fixed maturity investments | 30,067,792 |  |  | 32,544,867 |
Proceeds from sales of equity securities | 695,633 |  |  | 731,793 |
Proceeds from sales, redemptions and maturities of other investments | 1,487,919 |  |  | 791,807 |
Proceeds from redemptions and maturities of fixed maturity investments | 1,234,412 |  |  | 645,292 |
Net settlements of derivative instruments | (67,830) |  |  | 163,290 |
Net (purchases) sales of short-term investments | (1,172,798) |  |  | (1,159,351) |
Change in cash collateral related to securities lending | — |  |  | 81,210 |
Purchase of operating affiliate | (753,916) |  |  | — |
Impact of the deconsolidation of the variable interest entity | (349,202) |  |  | — |
Purchases of fixed assets | (34,407) |  |  | (26,717) |
Other | (361,857) |  |  | (131,992) |
Net cash provided by (used for) investing activities | (1,453,284) |  |  | (2,608,418) |
Financing Activities |  |  |
Proceeds from issuance of preferred shares, net | 485,821 |  |  | — |
Redemption of preferred shares | (450,000) |  |  | — |
Purchases of common shares under share repurchase program | (872,197) |  |  | (75,486) |
Proceeds from common shares issued, net | 281 |  |  | (9,656) |
Proceeds from borrowings | — |  |  | 1,018,793 |
Repayments of borrowings | — |  |  | (304,000) |
Change in cash collateral related to securities lending | — |  |  | (81,210) |
Third party investment in non-redeemable noncontrolling interests | 15,971 |  |  | (2,867) |
Dividends paid to redeemable noncontrolling interests | (1,907) |  |  | (3,541) |
Other | (21,752) |  |  | 55,266 |
Preferred dividends paid | (38,096) |  |  | (31,209) |
Net cash provided by (used for) financing activities | (881,879) |  |  | 566,090 |
Effects of exchange rate changes on foreign currency cash and restricted cash | (34,023) |  |  | (5,847) |
Increase (decrease) in cash and restricted cash | 258,365 |  |  | 283,275 |
Cash and restricted cash, beginning of year | 1,290,544 |  |  | 903,698 |
Cash and restricted cash, end of period | $ | 1,548,909 |  |  | $ | 1,186,973 |

See Notes to Consolidated Financial Statements

ARCH CAPITAL | 10 | 2021 THIRD QUARTER FORM 10-Q

Table of Contents
ARCH CAPITAL GROUP LTD. AND SUBSIDIARIES
NOTES TO CONSOLIDATED FINANCIAL STATEMENTS (UNAUDITED)
1. Basis of Presentation and Recent Accounting Pronouncements

General
Arch Capital Group Ltd. (“Arch Capital”) is a public listed Bermuda exempted company which provides insurance, reinsurance and mortgage insurance on a worldwide basis through its wholly-owned subsidiaries. As used herein, the “Company” means Arch Capital and its subsidiaries. The Company’s consolidated financial statements included the results of Watford Holdings Ltd. and its wholly owned subsidiaries (“Watford”) through June 30, 2021. Effective July 1, 2021, Watford is wholly owned by Greysbridge Holdings Ltd., (“Greysbridge”) and Greysbridge is owned 40% by the Company, 30% by certain investment funds managed by Kelso & Company (“Kelso”) and 30% by certain investment funds managed by Warburg Pincus LLC (“Warburg”). Based on the governing documents of Greysbridge, the Company concluded that, while it retains significant influence over Watford, Watford no longer constitutes a variable interest entity. Accordingly, effective July 1, 2021, Arch no longer consolidates the results of Watford in its consolidated financial statements and footnotes. See note 12.
Basis of Presentation
The interim consolidated financial statements have been prepared in conformity with accounting principles generally accepted in the United States of America (“GAAP”). All significant intercompany transactions and balances have been eliminated in consolidation. The preparation of financial statements in conformity with GAAP requires management to make estimates and assumptions that affect the reported amounts of assets and liabilities and disclosure of contingent assets and liabilities at the date of the financial statements and the reported amounts of revenues and expenses during the reporting period. Actual results could differ materially from those estimates and assumptions. In the opinion of management, the accompanying unaudited interim consolidated financial statements reflect all adjustments (consisting of normally recurring accruals) necessary for a fair statement of results on an interim basis. The results of any interim period are not necessarily indicative of the results for a full year or any future periods.
Certain information and footnote disclosures normally included in financial statements prepared in accordance with GAAP have been condensed or omitted; however, management believes that the disclosures are adequate to make the information presented not misleading. This report should be read in conjunction with the Company’s Annual Report on Form 10-K for the year ended December 31, 2020
(“2020 Form 10-K”), including the Company’s audited consolidated financial statements and related notes.
The Company has reclassified the presentation of certain prior year information to conform to the current presentation, including the correct presentation of ‘income (loss) from operating affiliates’ on its consolidated statements of income for all periods presented to reclass such item from ‘other income (loss)’. The Company also changed its presentation of ‘investment in operating affiliates’ on its consolidated balance sheet for all periods presented to reclass such item from ‘other assets’. Such reclassifications had no effect on the Company’s net income, comprehensive income, shareholders’ equity or cash flows. Management views the impact of the prior period misclassification as not material to the financial statements on a quantitative and qualitative basis. See note 8. Tabular amounts are in U.S. Dollars in thousands, except share amounts, unless otherwise noted.
Recent Accounting Pronouncements
Recently Issued Accounting Standards Adopted
The Company adopted ASU 2019-12, “Simplifying the Accounting for Income Taxes.” This ASU eliminates certain exceptions for recognizing deferred taxes for investments, performing intraperiod tax allocations and calculating income taxes in interim periods. The ASU also clarifies the accounting for transactions that result in a step-up in the tax basis of goodwill. The adoption of this guidance did not have a material effect on the Company’s consolidated financial statements.
For information regarding additional accounting standards that the Company has not yet adopted, see note 3(r), “Significant Accounting Policies—Recent Accounting Pronouncements,” of the notes to consolidated financial statements in the Company’s 2020 Form 10-K.
2. Acquisitions

Westpac Lenders Mortgage Insurance Limited (“WLMI”)
On August 31, 2021, the Company completed the acquisition of WLMI, an Australian Prudential Regulation Authority authorized captive lenders mortgage insurance (“LMI”) provider to the Westpac Banking Corporation (“Westpac”). As part of the acquisition, WLMI will retain its existing risk in force and remain Westpac’s exclusive provider of LMI on new mortgage originations for a period of 10 years. Upon completion of this transaction, the Company renamed WLMI to Arch Lenders Mortgage Indemnity Limited (“ALMI”). ALMI will become the Company’s primary provider of LMI to the Australian market.

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ARCH CAPITAL GROUP LTD. AND SUBSIDIARIES
NOTES TO CONSOLIDATED FINANCIAL STATEMENTS (UNAUDITED)
Somerset Bridge Group Limited, Southern Rock Holdings Limited and affiliates (“Somerset”)
On August 6, 2021, the Company completed the acquisition of Somerset. The acquisition includes Somerset’s motor insurance managing general agent, distribution capabilities through direct and aggregator channels, affiliated insurer and fully integrated claims operation.
In connection with the acquisitions noted above, the Company increased its goodwill and intangible assets by $337.4 million.
3. Share Transactions

Share Repurchases
The board of directors of Arch Capital has authorized the investment in Arch Capital’s common shares through a share repurchase program. Since the inception of the share repurchase program, Arch Capital has repurchased 412.0 million common shares for an aggregate purchase price of $4.92 billion. For the nine months ended September 30, 2021, Arch Capital repurchased 22.8 million shares under the share repurchase program with an aggregate purchase price of $872.2 million. Arch Capital repurchased 2.6 million shares under the share repurchase program with an aggregate purchase price of $75.5 million during the nine months ended September 30, 2020. At September 30, 2021, $44.3 million of share repurchases were available under the program, which may be effected from time to time in open market or privately negotiated transactions. The timing and amount of the repurchase transactions under this program will depend on a variety of factors, including market conditions and corporate and regulatory considerations. See note 17.
Series G Preferred Shares
In June 2021, Arch Capital completed a $500 million underwritten public offering of 20.0 million depositary shares (the “Depositary Shares”), each of which represents a 1/1,000th interest in a share of its 4.550% Non-Cumulative Preferred Shares, Series G, $0.01 par value and $25,000 liquidation preference per share (equivalent to $25 liquidation preference per Depositary Share) (the “Series G Preferred Shares”). Each Depositary Share, evidenced by a depositary receipt, entitles the holder, through the depositary, to a proportional fractional interest in all rights and preferences of the Series G Preferred Shares represented thereby (including any dividend, liquidation, redemption and voting rights).
Holders of Series G Preferred Shares will be entitled to receive dividend payments only when, as and if declared by
the Company’s board of directors or a duly authorized committee of the board. Any such dividends will be payable from, and including, the date of original issue on a non-cumulative basis, quarterly in arrears on the last day of March, June, September and December of each year, at an annual rate of 4.550%. Dividends on the Series G Preferred Shares are not cumulative. The Company will be restricted from paying dividends on or repurchasing its common shares unless certain dividend payments are made on the Series G Preferred Shares. The Company may not declare or pay a dividend on the Series G Preferred Shares under certain circumstances, including if the Company is or, after giving effect to such payment, would be in breach of applicable individual or group solvency and liquidity requirements or applicable individual or group enhanced capital requirements ("ECR"). The Series G Preferred Shares may not be redeemed at any time if the ECR would be breached immediately before or after giving effect to such redemption, unless the Company replaces the capital represented by preference shares to be redeemed with capital having equal or better capital treatment.
Except in specified circumstances relating to certain tax or corporate events, the Series G Preferred Shares are not redeemable prior to June 11, 2026. On and after that date, the Series G Preferred Shares will be redeemable at the Company’s option, in whole or in part, at a redemption price of $25,000 per share of the Series G Preferred Shares (equivalent to $25 per depositary share), plus any declared and unpaid dividends, without accumulation of any undeclared dividends to, but excluding, the redemption date. The Depositary Shares will be redeemed if and to the extent the related Series G Preferred Shares are redeemed by the Company. Neither the Depositary Shares nor the Series G Preferred Shares have a stated maturity, nor will they be subject to any sinking fund or mandatory redemption. The Series G Preferred Shares are not convertible into any other securities. The Series G Preferred Shares do not have voting rights, except under limited circumstances.
The net proceeds from the Series G Preferred Shares offering of approximately $485.8 million were primarily used to redeem the Company’s issued and outstanding 5.25% Series E Non-Cumulative Preferred Shares in September 2021. The preferred shares were redeemed at a redemption price equal to $25 per share, plus all declared and unpaid dividends to (but excluding) the redemption date. In accordance with GAAP, following the redemption, original issuance costs related to such shares have been removed from additional paid-in capital and recorded as a “loss on redemption of preferred shares.” Such adjustment had no impact on total shareholders’ equity or cash flows.

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ARCH CAPITAL GROUP LTD. AND SUBSIDIARIES
NOTES TO CONSOLIDATED FINANCIAL STATEMENTS (UNAUDITED)
4. Earnings Per Common Share

The following table sets forth the computation of basic and diluted earnings per common share:

| Three Months Ended |  | Nine Months Ended
| September 30, |  | September 30,
| 2021 |  | 2020 |  | 2021 |  | 2020
Numerator: |  |  |  |  |  |  |
Net income (loss) | $ | 421,415 |  |  | $ | 488,682 |  |  | $ | 1,615,787 |  |  | $ | 866,397 |
Amounts attributable to noncontrolling interests | (1,473) |  |  | (69,643) |  |  | (82,203) |  |  | (4,420) |
Net income (loss) available to Arch | 419,942 |  |  | 419,039 |  |  | 1,533,584 |  |  | 861,977 |
Preferred dividends | (16,090) |  |  | (10,403) |  |  | (38,159) |  |  | (31,209) |
Loss on redemption of preferred shares | (15,101) |  |  | — |  |  | (15,101) |  |  | — |
Net income (loss) available to Arch common shareholders | $ | 388,751 |  |  | $ | 408,636 |  |  | $ | 1,480,324 |  |  | $ | 830,768 |
Denominator: |  |  |  |  |  |  |
Weighted average common shares and common share equivalents outstanding — basic | 389,274,220 |  |  | 402,850,485 |  |  | 395,899,591 |  |  | 403,081,266 |
Effect of dilutive common share equivalents: |  |  |  |  |  |  |
Nonvested restricted shares | 2,131,915 |  |  | 1,580,791 |  |  | 1,877,930 |  |  | 1,690,447 |
Stock options (1) | 6,497,212 |  |  | 4,763,381 |  |  | 6,482,964 |  |  | 5,543,184 |
Weighted average common shares and common share equivalents outstanding — diluted | 397,903,347 |  |  | 409,194,657 |  |  | 404,260,485 |  |  | 410,314,897 |
Earnings per common share: |  |  |  |  |  |  |
Basic | $ | 1.00 |  |  | $ | 1.01 |  |  | $ | 3.74 |  |  | $ | 2.06 |
Diluted | $ | 0.98 |  |  | $ | 1.00 |  |  | $ | 3.66 |  |  | $ | 2.02 |

(1) Certain stock options were not included in the computation of diluted earnings per share where the exercise price of the stock options exceeded the average market price and would have been anti-dilutive or where, when applying the treasury stock method to in-the-money options, the sum of the proceeds, including unrecognized compensation, exceeded the average market price and would have been anti-dilutive. For the 2021 third quarter and 2020 third quarter, the number of stock options excluded were 1,948,006 and 4,713,241, respectively. For the nine months ended September 30, 2021 and 2020 period, the number of stock options excluded were 2,397,507 and 2,361,413, respectively.

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ARCH CAPITAL GROUP LTD. AND SUBSIDIARIES
NOTES TO CONSOLIDATED FINANCIAL STATEMENTS (UNAUDITED)
5. Segment Information

The Company classifies its businesses into three underwriting segments — insurance, reinsurance and mortgage — and two other operating segments — ‘other’ and corporate (non-underwriting). The Company determined its reportable segments using the management approach described in accounting guidance regarding disclosures about segments of an enterprise and related information. The accounting policies of the segments are the same as those used for the preparation of the Company’s consolidated financial statements. Intersegment business is allocated to the segment accountable for the underwriting results.
The Company’s insurance, reinsurance and mortgage segments each have managers who are responsible for the overall profitability of their respective segments and who are directly accountable to the Company’s chief operating decision makers, the Chief Executive Officer of Arch Capital, the Chief Financial Officer and Treasurer of Arch Capital and the President and Chief Underwriting Officer of Arch Capital. The chief operating decision makers do not assess performance, measure return on equity or make resource allocation decisions on a line of business basis. Management measures segment performance for its three underwriting segments based on underwriting income or loss. The Company does not manage its assets by underwriting segment, with the exception of goodwill and intangible assets, and, accordingly, investment income is not allocated to each underwriting segment.
The insurance segment consists of the Company’s insurance underwriting units which offer specialty product lines on a worldwide basis. Product lines include: construction and national accounts; excess and surplus casualty; lenders products; professional lines; programs; property, energy, marine and aviation; travel, accident and health; and other (consisting of alternative markets, excess workers' compensation and surety business).
The reinsurance segment consists of the Company’s reinsurance underwriting units which offer specialty product lines on a worldwide basis. Product lines include: casualty; marine and aviation; other specialty; property catastrophe; property excluding property catastrophe (losses on a single risk, both excess of loss and pro rata); and other (consisting of life reinsurance, casualty clash and other).
The mortgage segment includes the Company’s U.S. primary mortgage insurance, U.S. credit-risk transfer (“CRT”) which are predominately with government sponsored enterprises (“GSE’s”) and international mortgage insurance and reinsurance operations. Arch Mortgage Insurance Company and United Guaranty Residential Insurance Company (combined “Arch MI U.S.”) are approved as eligible mortgage insurers by Federal National Mortgage Association (“Fannie Mae”) and Federal Home Loan Mortgage Corporation (“Freddie Mac”), each a GSE. Arch MI U.S. also includes Arch Mortgage Guaranty Company, which is not a GSE-approved entity.
The corporate (non-underwriting) segment results include net investment income, other income (loss), corporate expenses, transaction costs and other, interest expense, items related to the Company’s non-cumulative preferred shares, net realized gains or losses (which includes changes in the allowance for credit losses on financial assets and net impairment losses recognized in earnings), equity in net income or loss of investment funds accounted for using the equity method, net foreign exchange gains or losses, income or loss from operating affiliates and income taxes. Such amounts exclude the results of the ‘other’ segment.
Through June 30, 2021, the ‘other’ segment included the results of Watford. In July 2021, the Company announced the completion of the previously disclosed acquisition of Watford by Greysbridge. Based on the governing documents of Greysbridge, the Company has concluded that, while it retains significant influence over Watford, Watford no longer constitutes a variable interest entity. Accordingly, effective July 1, 2021, Arch no longer consolidates the results of Watford in its consolidated financial statements. S ee note 12.

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ARCH CAPITAL GROUP LTD. AND SUBSIDIARIES
NOTES TO CONSOLIDATED FINANCIAL STATEMENTS (UNAUDITED)
The following tables summarize the Company’s underwriting income or loss by segment, together with a reconciliation of underwriting income or loss to net income available to Arch common shareholders:

| Three Months Ended
| September 30, 2021
| Insurance |  | Reinsurance |  | Mortgage |  | Sub-Total |  | Other |  | Total
Gross premiums written (1) | $ | 1,596,619 |  |  | $ | 1,251,760 |  |  | $ | 360,934 |  |  | $ | 3,207,415 |  |  | $ | — |  |  | $ | 3,207,415 |
Premiums ceded | (442,806) |  |  | (630,371) |  |  | (60,207) |  |  | (1,131,486) |  |  | — |  |  | (1,131,486) |
Net premiums written | 1,153,813 |  |  | 621,389 |  |  | 300,727 |  |  | 2,075,929 |  |  | — |  |  | 2,075,929 |
Change in unearned premiums | (215,143) |  |  | 57,313 |  |  | 11,238 |  |  | (146,592) |  |  | — |  |  | (146,592) |
Net premiums earned | 938,670 |  |  | 678,702 |  |  | 311,965 |  |  | 1,929,337 |  |  | — |  |  | 1,929,337 |
Other underwriting income (loss) | — |  |  | 3,293 |  |  | 3,981 |  |  | 7,274 |  |  | — |  |  | 7,274 |
Losses and loss adjustment expenses | (668,630) |  |  | (545,846) |  |  | (11,543) |  |  | (1,226,019) |  |  | — |  |  | (1,226,019) |
Acquisition expenses | (152,467) |  |  | (129,450) |  |  | (24,098) |  |  | (306,015) |  |  | — |  |  | (306,015) |
Other operating expenses | (138,931) |  |  | (45,647) |  |  | (46,254) |  |  | (230,832) |  |  | — |  |  | (230,832) |
Underwriting income (loss) | $ | (21,358) |  |  | $ | (38,948) |  |  | $ | 234,051 |  |  | 173,745 |  |  | — |  |  | 173,745 |
Net investment income |  |  |  |  |  |  | 88,195 |  |  | — |  |  | 88,195 |
Net realized gains (losses) |  |  |  |  |  |  | (25,040) |  |  | — |  |  | (25,040) |
Equity in net income (loss) of investment funds accounted for using the equity method |  |  |  |  |  |  | 105,398 |  |  | — |  |  | 105,398 |
Other income (loss) |  |  |  |  |  |  | (3,960) |  |  | — |  |  | (3,960) |
Corporate expenses (2) |  |  |  |  |  |  | (18,636) |  |  | — |  |  | (18,636) |
Transaction costs and other (2) |  |  |  |  |  |  | (1,036) |  |  | — |  |  | (1,036) |
Amortization of intangible assets |  |  |  |  |  |  | (20,135) |  |  | — |  |  | (20,135) |
Interest expense |  |  |  |  |  |  | (33,176) |  |  | — |  |  | (33,176) |
Net foreign exchange gains (losses) |  |  |  |  |  |  | 36,078 |  |  | — |  |  | 36,078 |
Income (loss) before income taxes and income (loss) from operating affiliates |  |  |  |  |  |  | 301,433 |  |  | — |  |  | 301,433 |
Income tax (expense) benefit |  |  |  |  |  |  | (4,137) |  |  | — |  |  | (4,137) |
Income (loss) from operating affiliates |  |  |  |  |  |  | 124,119 |  |  | — |  |  | 124,119 |
Net income (loss) |  |  |  |  |  |  | 421,415 |  |  | — |  |  | 421,415 |
Amounts attributable to redeemable noncontrolling interests |  |  |  |  |  |  | (1,473) |  |  | — |  |  | (1,473) |
Amounts attributable to nonredeemable noncontrolling interests |  |  |  |  |  |  | — |  |  | — |  |  | — |
Net income (loss) available to Arch |  |  |  |  |  |  | 419,942 |  |  | — |  |  | 419,942 |
Preferred dividends |  |  |  |  |  |  | (16,090) |  |  | — |  |  | (16,090) |
Loss on redemption of preferred shares |  |  |  |  |  |  | (15,101) |  |  | — |  |  | (15,101) |
Net income (loss) available to Arch common shareholders |  |  |  |  |  |  | $ | 388,751 |  |  | $ | — |  |  | $ | 388,751 |
Underwriting Ratios |  |  |  |  |  |  |  |  |  |  |
Loss ratio | 71.2 | % |  | 80.4 | % |  | 3.7 | % |  | 63.5 | % |  | — | % |  | 63.5 | %
Acquisition expense ratio | 16.2 | % |  | 19.1 | % |  | 7.7 | % |  | 15.9 | % |  | — | % |  | 15.9 | %
Other operating expense ratio | 14.8 | % |  | 6.7 | % |  | 14.8 | % |  | 12.0 | % |  | — | % |  | 12.0 | %
Combined ratio | 102.2 | % |  | 106.2 | % |  | 26.2 | % |  | 91.4 | % |  | — | % |  | 91.4 | %
Goodwill and intangible assets | $ | 261,103 |  |  | $ | 176,128 |  |  | $ | 526,091 |  |  | $ | 963,322 |  |  | $ | — |  |  | $ | 963,322 |

(1) Certain amounts included in the gross premiums written of each segment are related to intersegment transactions. Accordingly, the sum of gross premiums written for each segment does not agree to the total gross premiums written as shown in the table above due to the elimination of intersegment transactions in the total.
(2) Certain expenses have been excluded from ‘corporate expenses’ and reflected in ‘transaction costs and other.’

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ARCH CAPITAL GROUP LTD. AND SUBSIDIARIES
NOTES TO CONSOLIDATED FINANCIAL STATEMENTS (UNAUDITED)

| Three Months Ended
| September 30, 2020
| Insurance |  | Reinsurance |  | Mortgage |  | Sub-Total |  | Other |  | Total
Gross premiums written (1) | $ | 1,206,328 |  |  | $ | 1,004,590 |  |  | $ | 346,248 |  |  | $ | 2,556,914 |  |  | $ | 197,480 |  |  | $ | 2,681,032 |
Premiums ceded | (382,167) |  |  | (400,388) |  |  | (47,783) |  |  | (830,086) |  |  | (50,164) |  |  | (806,888) |
Net premiums written | 824,161 |  |  | 604,202 |  |  | 298,465 |  |  | 1,726,828 |  |  | 147,316 |  |  | 1,874,144 |
Change in unearned premiums | (105,007) |  |  | (49,704) |  |  | 52,944 |  |  | (101,767) |  |  | (1,285) |  |  | (103,052) |
Net premiums earned | 719,154 |  |  | 554,498 |  |  | 351,409 |  |  | 1,625,061 |  |  | 146,031 |  |  | 1,771,092 |
Other underwriting income (loss) | (31) |  |  | 298 |  |  | 4,600 |  |  | 4,867 |  |  | 546 |  |  | 5,413 |
Losses and loss adjustment expenses | (525,321) |  |  | (422,084) |  |  | (153,055) |  |  | (1,100,460) |  |  | (115,813) |  |  | (1,216,273) |
Acquisition expenses | (102,420) |  |  | (85,388) |  |  | (35,716) |  |  | (223,524) |  |  | (24,418) |  |  | (247,942) |
Other operating expenses | (122,541) |  |  | (41,818) |  |  | (36,708) |  |  | (201,067) |  |  | (14,619) |  |  | (215,686) |
Underwriting income (loss) | $ | (31,159) |  |  | $ | 5,506 |  |  | $ | 130,530 |  |  | 104,877 |  |  | (8,273) |  |  | 96,604 |
Net investment income |  |  |  |  |  |  | 99,857 |  |  | 28,655 |  |  | 128,512 |
Net realized gains (losses) |  |  |  |  |  |  | 210,984 |  |  | 69,515 |  |  | 280,499 |
Equity in net income (loss) of investment funds accounted for using the equity method |  |  |  |  |  |  | 126,735 |  |  | — |  |  | 126,735 |
Other income (loss) |  |  |  |  |  |  | — |  |  | — |  |  | — |
Corporate expenses (2) |  |  |  |  |  |  | (16,263) |  |  | — |  |  | (16,263) |
Transaction costs and other (2) |  |  |  |  |  |  | (1,674) |  |  | — |  |  | (1,674) |
Amortization of intangible assets |  |  |  |  |  |  | (16,715) |  |  | — |  |  | (16,715) |
Interest expense |  |  |  |  |  |  | (36,224) |  |  | (5,119) |  |  | (41,343) |
Net foreign exchange gains (losses) |  |  |  |  |  |  | (38,681) |  |  | (6,204) |  |  | (44,885) |
Income (loss) before income taxes and income (loss) from operating affiliates |  |  |  |  |  |  | 432,896 |  |  | 78,574 |  |  | 511,470 |
Income tax (expense) benefit |  |  |  |  |  |  | (23,638) |  |  | (69) |  |  | (23,707) |
Income (loss) from operating affiliates |  |  |  |  |  |  | 919 |  |  | — |  |  | 919 |
Net income (loss) |  |  |  |  |  |  | 410,177 |  |  | 78,505 |  |  | 488,682 |
Amounts attributable to redeemable noncontrolling interests |  |  |  |  |  |  | (882) |  |  | (993) |  |  | (1,875) |
Amounts attributable to nonredeemable noncontrolling interests |  |  |  |  |  |  | — |  |  | (67,768) |  |  | (67,768) |
Net income (loss) available to Arch |  |  |  |  |  |  | 409,295 |  |  | 9,744 |  |  | 419,039 |
Preferred dividends |  |  |  |  |  |  | (10,403) |  |  | — |  |  | (10,403) |
Net income (loss) available to Arch common shareholders |  |  |  |  |  |  | $ | 398,892 |  |  | $ | 9,744 |  |  | $ | 408,636 |
Underwriting Ratios |  |  |  |  |  |  |  |  |  |  |
Loss ratio | 73.0 | % |  | 76.1 | % |  | 43.6 | % |  | 67.7 | % |  | 79.3 | % |  | 68.7 | %
Acquisition expense ratio | 14.2 | % |  | 15.4 | % |  | 10.2 | % |  | 13.8 | % |  | 16.7 | % |  | 14.0 | %
Other operating expense ratio | 17.0 | % |  | 7.5 | % |  | 10.4 | % |  | 12.4 | % |  | 10.0 | % |  | 12.2 | %
Combined ratio | 104.2 | % |  | 99.0 | % |  | 64.2 | % |  | 93.9 | % |  | 106.0 | % |  | 94.9 | %
Goodwill and intangible assets | $ | 282,146 |  |  | $ | 20,319 |  |  | $ | 403,662 |  |  | $ | 706,127 |  |  | $ | 7,650 |  |  | $ | 713,777 |

(1) Certain amounts included in the gross premiums written of each segment are related to intersegment transactions. Accordingly, the sum of gross premiums written for each segment does not agree to the total gross premiums written as shown in the table above due to the elimination of intersegment transactions in the total.
(2) Certain expenses have been excluded from ‘corporate expenses’ and reflected in ‘transaction costs and other.’

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ARCH CAPITAL GROUP LTD. AND SUBSIDIARIES
NOTES TO CONSOLIDATED FINANCIAL STATEMENTS (UNAUDITED)

| Nine Months Ended
| September 30, 2021
| Insurance |  | Reinsurance |  | Mortgage |  | Sub-Total |  | Other |  | Total
Gross premiums written (1) | $ | 4,381,372 |  |  | $ | 4,080,840 |  |  | $ | 1,143,691 |  |  | $ | 9,602,213 |  |  | $ | 457,465 |  |  | $ | 9,890,912 |
Premiums ceded | (1,269,165) |  |  | (1,535,607) |  |  | (171,923) |  |  | (2,973,005) |  |  | (102,763) |  |  | (2,907,002) |
Net premiums written | 3,112,207 |  |  | 2,545,233 |  |  | 971,768 |  |  | 6,629,208 |  |  | 354,702 |  |  | 6,983,910 |
Change in unearned premiums | (488,636) |  |  | (484,607) |  |  | 10,735 |  |  | (962,508) |  |  | (22,734) |  |  | (985,242) |
Net premiums earned | 2,623,571 |  |  | 2,060,626 |  |  | 982,503 |  |  | 5,666,700 |  |  | 331,968 |  |  | 5,998,668 |
Other underwriting income (loss) | — |  |  | 3,148 |  |  | 15,026 |  |  | 18,174 |  |  | 739 |  |  | 18,913 |
Losses and loss adjustment expenses | (1,750,257) |  |  | (1,494,539) |  |  | (85,112) |  |  | (3,329,908) |  |  | (259,042) |  |  | (3,588,950) |
Acquisition expenses | (417,541) |  |  | (381,060) |  |  | (84,297) |  |  | (882,898) |  |  | (62,741) |  |  | (945,639) |
Other operating expenses | (409,386) |  |  | (150,856) |  |  | (143,697) |  |  | (703,939) |  |  | (32,869) |  |  | (736,808) |
Underwriting income (loss) | $ | 46,387 |  |  | $ | 37,319 |  |  | $ | 684,423 |  |  | $ | 768,129 |  |  | $ | (21,945) |  |  | $ | 746,184 |
Net investment income |  |  |  |  |  |  | 256,354 |  |  | 42,310 |  |  | 298,664 |
Net realized gains (losses) |  |  |  |  |  |  | 239,690 |  |  | 80,638 |  |  | 320,328 |
Equity in net income (loss) of investment funds accounted for using the equity method |  |  |  |  |  |  | 299,270 |  |  | — |  |  | 299,270 |
Other income (loss) |  |  |  |  |  |  | 1,151 |  |  | — |  |  | 1,151 |
Corporate expenses (2) |  |  |  |  |  |  | (59,279) |  |  | — |  |  | (59,279) |
Transaction costs and other (2) |  |  |  |  |  |  | (793) |  |  | (935) |  |  | (1,728) |
Amortization of intangible assets |  |  |  |  |  |  | (48,925) |  |  | (898) |  |  | (49,823) |
Interest expense |  |  |  |  |  |  | (98,812) |  |  | (8,410) |  |  | (107,222) |
Net foreign exchange gains (losses) |  |  |  |  |  |  | 39,691 |  |  | (1,325) |  |  | 38,366 |
Income (loss) before income taxes and income (loss) from operating affiliates |  |  |  |  |  |  | 1,396,476 |  |  | 89,435 |  |  | 1,485,911 |
Income tax (expense) benefit |  |  |  |  |  |  | (93,942) |  |  | (234) |  |  | (94,176) |
Income (loss) from operating affiliates |  |  |  |  |  |  | 224,052 |  |  | — |  |  | 224,052 |
Net income (loss) |  |  |  |  |  |  | 1,526,586 |  |  | 89,201 |  |  | 1,615,787 |
Amounts attributable to redeemable noncontrolling interests |  |  |  |  |  |  | (1,936) |  |  | (1,953) |  |  | (3,889) |
Amounts attributable to nonredeemable noncontrolling interests |  |  |  |  |  |  | — |  |  | (78,314) |  |  | (78,314) |
Net income (loss) available to Arch |  |  |  |  |  |  | 1,524,650 |  |  | 8,934 |  |  | 1,533,584 |
Preferred dividends |  |  |  |  |  |  | (38,159) |  |  | — |  |  | (38,159) |
Loss on redemption of preferred shares |  |  |  |  |  |  | (15,101) |  |  | — |  |  | (15,101) |
Net income (loss) available to Arch common shareholders |  |  |  |  |  |  | $ | 1,471,390 |  |  | $ | 8,934 |  |  | $ | 1,480,324 |
Underwriting Ratios |  |  |  |  |  |  |  |  |  |  |
Loss ratio | 66.7 | % |  | 72.5 | % |  | 8.7 | % |  | 58.8 | % |  | 78.0 | % |  | 59.8 | %
Acquisition expense ratio | 15.9 | % |  | 18.5 | % |  | 8.6 | % |  | 15.6 | % |  | 18.9 | % |  | 15.8 | %
Other operating expense ratio | 15.6 | % |  | 7.3 | % |  | 14.6 | % |  | 12.4 | % |  | 9.9 | % |  | 12.3 | %
Combined ratio | 98.2 | % |  | 98.3 | % |  | 31.9 | % |  | 86.8 | % |  | 106.8 | % |  | 87.9 | %

(1) Certain amounts included in the gross premiums written of each segment are related to intersegment transactions. Accordingly, the sum of gross premiums written for each segment does not agree to the total gross premiums written as shown in the table above due to the elimination of intersegment transactions in the total.
(2) Certain expenses have been excluded from ‘corporate expenses’ and reflected in ‘transaction costs and other.’

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ARCH CAPITAL GROUP LTD. AND SUBSIDIARIES
NOTES TO CONSOLIDATED FINANCIAL STATEMENTS (UNAUDITED)

| Nine Months Ended
| September 30, 2020
| Insurance |  | Reinsurance |  | Mortgage |  | Sub-Total |  | Other |  | Total
Gross premiums written (1) | $ | 3,444,335 |  |  | $ | 2,934,174 |  |  | $ | 1,084,337 |  |  | $ | 7,461,860 |  |  | $ | 590,309 |  |  | $ | 7,831,554 |
Premiums ceded | (1,119,165) |  |  | (967,698) |  |  | (136,154) |  |  | (2,222,031) |  |  | (150,437) |  |  | (2,151,853) |
Net premiums written | 2,325,170 |  |  | 1,966,476 |  |  | 948,183 |  |  | 5,239,829 |  |  | 439,872 |  |  | 5,679,701 |
Change in unearned premiums | (202,188) |  |  | (388,321) |  |  | 113,965 |  |  | (476,544) |  |  | (22,267) |  |  | (498,811) |
Net premiums earned | 2,122,982 |  |  | 1,578,155 |  |  | 1,062,148 |  |  | 4,763,285 |  |  | 417,605 |  |  | 5,180,890 |
Other underwriting income (loss) | (31) |  |  | 1,767 |  |  | 15,649 |  |  | 17,385 |  |  | 1,547 |  |  | 18,932 |
Losses and loss adjustment expenses | (1,550,632) |  |  | (1,235,586) |  |  | (444,721) |  |  | (3,230,939) |  |  | (331,275) |  |  | (3,562,214) |
Acquisition expenses | (317,428) |  |  | (255,516) |  |  | (108,304) |  |  | (681,248) |  |  | (68,766) |  |  | (750,014) |
Other operating expenses | (370,947) |  |  | (125,831) |  |  | (120,178) |  |  | (616,956) |  |  | (42,523) |  |  | (659,479) |
Underwriting income (loss) | $ | (116,056) |  |  | $ | (37,011) |  |  | $ | 404,594 |  |  | $ | 251,527 |  |  | $ | (23,412) |  |  | $ | 228,115 |
Net investment income |  |  |  |  |  |  | 313,916 |  |  | 91,234 |  |  | 405,150 |
Net realized gains (losses) |  |  |  |  |  |  | 523,964 |  |  | (53,837) |  |  | 470,127 |
Equity in net income (loss) of investment funds accounted for using the equity method |  |  |  |  |  |  | 57,407 |  |  | — |  |  | 57,407 |
Other income (loss) |  |  |  |  |  |  | 65 |  |  | — |  |  | 65 |
Corporate expenses (2) |  |  |  |  |  |  | (51,407) |  |  | — |  |  | (51,407) |
Transaction costs and other (2) |  |  |  |  |  |  | (5,246) |  |  | — |  |  | (5,246) |
Amortization of intangible assets |  |  |  |  |  |  | (49,835) |  |  | — |  |  | (49,835) |
Interest expense |  |  |  |  |  |  | (86,599) |  |  | (18,438) |  |  | (105,037) |
Net foreign exchange gains (losses) |  |  |  |  |  |  | (17,812) |  |  | 6,387 |  |  | (11,425) |
Income (loss) before income taxes and income (loss) from operating affiliates |  |  |  |  |  |  | 935,980 |  |  | 1,934 |  |  | 937,914 |
Income tax (expense) benefit |  |  |  |  |  |  | (78,112) |  |  | 333 |  |  | (77,779) |
Income (loss) from operating affiliates |  |  |  |  |  |  | 6,262 |  |  | — |  |  | 6,262 |
Net income (loss) |  |  |  |  |  |  | 864,130 |  |  | 2,267 |  |  | 866,397 |
Amounts attributable to redeemable noncontrolling interests |  |  |  |  |  |  | (1,873) |  |  | (3,125) |  |  | (4,998) |
Amounts attributable to nonredeemable noncontrolling interests |  |  |  |  |  |  | — |  |  | 578 |  |  | 578 |
Net income (loss) available to Arch |  |  |  |  |  |  | 862,257 |  |  | (280) |  |  | 861,977 |
Preferred dividends |  |  |  |  |  |  | (31,209) |  |  | — |  |  | (31,209) |
Net income (loss) available to Arch common shareholders |  |  |  |  |  |  | $ | 831,048 |  |  | $ | (280) |  |  | $ | 830,768 |
Underwriting Ratios |  |  |  |  |  |  |  |  |  |  |
Loss ratio | 73.0 | % |  | 78.3 | % |  | 41.9 | % |  | 67.8 | % |  | 79.3 | % |  | 68.8 | %
Acquisition expense ratio | 15.0 | % |  | 16.2 | % |  | 10.2 | % |  | 14.3 | % |  | 16.5 | % |  | 14.5 | %
Other operating expense ratio | 17.5 | % |  | 8.0 | % |  | 11.3 | % |  | 13.0 | % |  | 10.2 | % |  | 12.7 | %
Combined ratio | 105.5 | % |  | 102.5 | % |  | 63.4 | % |  | 95.1 | % |  | 106.0 | % |  | 96.0 | %

(1) Certain amounts included in the gross premiums written of each segment are related to intersegment transactions. Accordingly, the sum of gross premiums written for each segment does not agree to the total gross premiums written as shown in the table above due to the elimination of intersegment transactions in the total.
(2) Certain expenses have been excluded from ‘corporate expenses’ and reflected in ‘transaction costs and other.’

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ARCH CAPITAL GROUP LTD. AND SUBSIDIARIES
NOTES TO CONSOLIDATED FINANCIAL STATEMENTS (UNAUDITED)
6. Reserve for Losses and Loss Adjustment Expenses

The following table represents an analysis of losses and loss adjustment expenses and a reconciliation of the beginning and ending reserve for losses and loss adjustment expenses:

| Three Months Ended |  | Nine Months Ended
| September 30, |  | September 30,
| 2021 |  | 2020 |  | 2021 |  | 2020
Reserve for losses and loss adjustment expenses at beginning of period | $ | 17,196,648 |  |  | $ | 15,044,874 |  |  | $ | 16,513,929 |  |  | $ | 13,891,842 |
Unpaid losses and loss adjustment expenses recoverable | 4,146,020 |  |  | 4,156,157 |  |  | 4,314,855 |  |  | 4,082,650 |
Net reserve for losses and loss adjustment expenses at beginning of period | 13,050,628 |  |  | 10,888,717 |  |  | 12,199,074 |  |  | 9,809,192 |
Net incurred losses and loss adjustment expenses relating to losses occurring in: |  |  |  |  |  |  |
Current year | 1,348,528 |  |  | 1,264,315 |  |  | 3,812,381 |  |  | 3,673,346 |
Prior years | (122,509) |  |  | (48,042) |  |  | (223,431) |  |  | (111,132) |
Total net incurred losses and loss adjustment expenses | 1,226,019 |  |  | 1,216,273 |  |  | 3,588,950 |  |  | 3,562,214 |
Net losses and loss adjustment expense reserves of acquired business (2) | 104,307 |  |  | — |  |  | 104,307 |  |  | $ | — |
Retroactive reinsurance transactions (1) | — |  |  | — |  |  | (183,893) |  |  | 60,635 |
Impact of deconsolidation of Watford (3) | (1,460,611) |  |  | — |  |  | (1,460,611) |  |  | — |
Net foreign exchange (gains) losses | (78,152) |  |  | 114,122 |  |  | 10,818 |  |  | 22,706 |
Net paid losses and loss adjustment expenses relating to losses occurring in: |  |  |  |  |  |  |
Current year | (208,923) |  |  | (189,961) |  |  | (432,348) |  |  | (359,395) |
Prior years | (417,368) |  |  | (512,263) |  |  | (1,610,397) |  |  | (1,578,464) |
Total net paid losses and loss adjustment expenses | (626,291) |  |  | (702,224) |  |  | (2,042,745) |  |  | (1,937,859) |
Net reserve for losses and loss adjustment expenses at end of period | 12,215,900 |  |  | 11,516,888 |  |  | 12,215,900 |  |  | 11,516,888 |
Unpaid losses and loss adjustment expenses recoverable | 5,115,147 |  |  | 4,383,638 |  |  | 5,115,147 |  |  | 4,383,638 |
Reserve for losses and loss adjustment expenses at end of period | $ | 17,331,047 |  |  | $ | 15,900,526 |  |  | $ | 17,331,047 |  |  | $ | 15,900,526 |

(1) During the 2021 first quarter, the Company entered into a reinsurance to close and other related agreements with Premia Managing Agency Limited (“Premia”), in connection with the 2018 and prior years of account related to the acquisition of Barbican Group Holdings Limited (“Barbican”). During the 2020 first quarter, the Company entered into a reinsurance to close agreement of the 2017 and prior years of account previously covered by a third party arrangement.
(2) Represents activity related to the Company’s acquisitions in the 2021 period. See note 2.
(3) See note 12.
Development on Prior Year Loss Reserves
2021 Third Quarter
During the 2021 third quarter, the Company recorded net favorable development on prior year loss reserves of $122.5 million, which consisted of $5.1 million from the insurance segment, $72.3 million from the reinsurance segment and $45.1 million from the mortgage segment.
The insurance segment’s net favorable development of $5.1 million, or 0.5 loss ratio points, for the 2021 third quarter consisted of $49.0 million of net favorable development in short-tailed and long-tailed lines and $43.9 million of net adverse development in medium-tailed lines. Net favorable development in short-tailed lines reflected $5.4 million of
favorable development in lenders products, primarily from the 2020 accident year (i.e., the year in which a loss occurred), $5.4 million of favorable development from property (excluding marine), primarily from the 2020 accident year, and $5.1 million of favorable development in travel and accident, across most accident years. Net favorable development in long-tailed lines reflected $26.3 million of favorable development related to construction and national accounts, across most accident years, and $6.7 million of favorable development related to other business, including alternative markets, primarily from the 2015 to 2018 accident years. Net adverse development in medium-tailed lines included $37.0 million of adverse development in contract binding business, across most accident years, partially offset by favorable development in marine and programs, primarily from more recent accident years.

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ARCH CAPITAL GROUP LTD. AND SUBSIDIARIES
NOTES TO CONSOLIDATED FINANCIAL STATEMENTS (UNAUDITED)
The reinsurance segment’s net favorable development of $72.3 million, or 10.7 loss ratio points, for the 2021 third quarter consisted of $65.4 million of net favorable development in short-tailed lines and $6.9 million in medium-tailed and long-tailed lines. Net favorable development in short-tailed lines reflected $46.1 million of favorable development related to property catastrophe and property other than property catastrophe business, primarily from the 2017 to 2020 underwriting years (i.e., all premiums and losses attributable to contracts having an inception or renewal date within the given twelve-month period) and $18.9 million of favorable development related to other specialty, primarily from the 2012 to 2017 underwriting years. Net favorable development in medium-tailed and long-tailed lines included $4.7 million of favorable development in casualty, primarily from the 2013 and 2014 underwriting years.
The mortgage segment’s net favorable development was $45.1 million, or 14.5 loss ratio points, for the 2021 third quarter, about half of which came from U.S. primary mortgage insurance, from better than expected cure activity in pre-pandemic delinquencies and recoveries on second lien and student loans, and the other half from our CRT portfolio and international mortgage insurance.
2020 Third Quarter
During the 2020 third quarter, the Company recorded net favorable development on prior year loss reserves of $48.0 million, which consisted of $2.3 million from the insurance segment, $42.0 million from the reinsurance segment and $4.5 million from the mortgage segment, partially offset by $0.7 million unfavorable from the ‘other’ segment.
The insurance segment’s net favorable development of $2.3 million, or 0.3 loss ratio points, for the 2020 third quarter consisted of $12.9 million of net favorable development in short-tailed and long-tailed lines and $10.6 million of net adverse development in medium-tailed lines. Net favorable development of $11.8 million in short-tailed lines reflected $8.0 million of favorable development from property (excluding marine), primarily from the 2015 to 2018 accident years and $3.4 million of favorable development in travel and accident, primarily from the 2019 accident year. Net favorable development of $1.1 million in long-tailed lines reflected $8.7 million of favorable development in construction and national accounts, primarily from the 2018 accident year, and $4.2 million of favorable development related to other business, including alternative markets and excess workers’ compensation, primarily from the 2013 to 2017 accident years, partially offset by $11.7 million of adverse development in executive assurance and casualty, primarily from the 2015 and 2019 accident year. Net adverse development in medium-tailed lines included $7.1 million of adverse development in program business, primarily from
2015 to 2018 accident years and $3.7 million of adverse development in contract binding, across all accident years.
The reinsurance segment’s net favorable development of $42.0 million, or 7.6 loss ratio points, for the 2020 third quarter consisted of $45.6 million of net favorable development in short-tailed and medium-tailed lines and net adverse development of $3.6 million from long-tailed lines. Net favorable development in short-tailed lines reflected $27.6 million of favorable development related to property catastrophe and property other than property catastrophe business, primarily from the 2016 to 2019 underwriting years and $7.8 million of favorable development from other specialty, primarily from the 2016 to 2019 underwriting years. Net favorable development of $9.4 million in medium-tailed lines reflected favorable development in marine and aviation across most underwriting years. Adverse development of $3.6 million in long-tailed lines reflected an increase in reserves from casualty, primarily from the 2012 to 2019 underwriting years.
The mortgage segment’s net favorable development was $4.5 million, or 1.3 loss ratio points, for the 2020 third quarter, primarily driven by subrogation recoveries on second lien and student loan business.
Nine Months Ended September 30, 2021
During the nine months ended September 30, 2021, the Company recorded net favorable development on prior year loss reserves of $223.4 million, which consisted of $13.1 million from the insurance segment, $119.6 million from the reinsurance segment and $99.1 million from the mortgage segment, partially offset by $8.4 million of adverse development from the ‘other’ segment (activity for the six months ended June 30, 2021 prior to deconsolidation of Watford).
The insurance segment’s net favorable development of $13.1 million, or 0.5 loss ratio points, for the 2021 period consisted of $102.5 million of net favorable development in short-tailed and long-tailed lines, partially offset by $89.4 million of net adverse development in medium-tailed lines. Net favorable development of $65.3 million in short-tailed lines reflected $27.0 million of favorable development from property (excluding marine), primarily from the 2019 and 2020 accident years, $24.0 million of favorable development in lenders products, primarily from the 2020 accident year, and $14.4 million of favorable development in travel and accident, primarily from the 2017 to 2020 accident years. Net favorable development of $37.1 million in long-tailed lines included favorable development primarily related to construction, national accounts and alternative markets, primarily from the 2016 to 2019 accident years. Net adverse development in medium-tailed lines reflected $57.1 million of adverse development in contract binding business, primarily from the 2014 to 2019 accident years, $26.2 million of adverse development in professional liability business,

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ARCH CAPITAL GROUP LTD. AND SUBSIDIARIES
NOTES TO CONSOLIDATED FINANCIAL STATEMENTS (UNAUDITED)
primarily from the 2019 and 2020 accident years, and $6.9 million of adverse development in programs business, primarily from the 2019 accident year.
The reinsurance segment’s net favorable development of $119.6 million, or 5.8 loss ratio points, for the 2021 period consisted of $139.7 million of net favorable development from short-tailed and medium-tailed lines, partially offset by $20.1 million of net adverse development from long-tailed lines. Net favorable development of $132.6 million in short-tailed lines reflected $97.1 million of favorable development from other specialty lines, primarily from the 2016 to 2019 underwriting years, and $71.7 million of favorable development from property other than property catastrophe business, primarily from the 2017 to 2020 underwriting years. Such amounts were partially offset by adverse development of $36.3 million from property catastrophe, primarily from the 2020 underwriting year. Adverse development in long-tailed lines reflected an increase in reserves from casualty, primarily from the 2018 underwriting year.
The mortgage segment’s net favorable development was $99.1 million, or 10.1 loss ratio points, for the 2021 period, which included reserve releases associated with various vintage credit risk transfer contracts that were called by the GSEs, favorable development on U.S. and international business and subrogation recoveries on second lien and student loan business.
Nine Months Ended September 30, 2020
During the nine months ended September 30, 2020, the Company recorded net favorable development on prior year loss reserves of $111.1 million, which consisted of $5.9 million from the insurance segment, $93.8 million from the reinsurance segment, $10.8 million from the mortgage segment and $0.6 million from the ‘other’ segment.
The insurance segment’s net favorable development of $5.9 million, or 0.3 loss ratio points, for the 2020 period consisted of $41.6 million of net favorable development in short-tailed and long-tailed lines, partially offset by $35.7 million of net adverse development in medium-tailed lines. Net favorable development of $27.2 million in short-tailed lines reflected $17.5 million of favorable development from property (excluding marine), primarily from the 2015 to 2018 accident years, $6.2 million of favorable development on travel and accident, primarily from 2019 accident year, and $3.5 million of favorable development in lenders products, primarily from the 2018 and 2019 accident years. Net favorable development of $14.4 million in long-tailed lines included $11.7 million of favorable development related to other business, including alternative markets and excess workers’ compensation, primarily from the 2013 to 2017 accident years. Net adverse development in medium-tailed lines reflected $23.0 million of adverse development in contract binding business, across all accident years, and $13.5 million of adverse development
in program business, primarily from the 2016 to 2018 accident years.
The reinsurance segment’s net favorable development of $93.8 million, or 5.9 loss ratio points, for the 2020 period consisted of $113.0 million of net favorable development from short-tailed and medium-tailed lines, partially offset by $19.2 million of net adverse development from long-tailed lines. Net favorable development of $101.8 million in short-tailed lines reflected $52.1 million related to property catastrophe and property other than property catastrophe business, primarily from the 2016 to 2019 underwriting years, and $47.1 million from other specialty lines, across most underwriting years. Adverse development in long-tailed lines of $19.2 million reflected an increase in reserves from casualty, primarily from the 2012 to 2015 underwriting years.
The mortgage segment’s net favorable development was $10.8 million, or 1.0 loss ratio points, for the 2020 period, primarily driven by subrogation recoveries on second lien and student loan business.
7. Allowance for Expected Credit Losses

Premiums Receivable
The following table provides a roll forward of the allowance for expected credit losses of the Company’s premium receivables:

| Premium Receivables, Net of Allowance | Allowance for Expected Credit Losses |
Three Months Ended September 30, 2021 |  |
Balance at beginning of period | $ | 2,866,578 |  | $ | 35,979 |  |
Change for provision of expected credit losses (1) |  | 2,736 |  |
Balance at end of period | $ | 2,807,720 |  | $ | 38,715 |  |
Three Months Ended September 30, 2020 |  |
Balance at beginning of period | $ | 2,203,753 |  | $ | 36,054 |  |
Change for provision of expected credit losses (1) |  | 1,046 |  |
Balance at end of period | $ | 2,225,311 |  | $ | 37,100 |  |
Nine Months Ended September 30, 2021 |  |  |
Balance at beginning of period | $ | 2,064,586 |  | $ | 37,781 |  |
Change for provision of expected credit losses (1) |  | 934 |  |
Balance at end of period | $ | 2,807,720 |  | $ | 38,715 |  |
Nine Months Ended September 30, 2020 |  |  |
Balance at beginning of period | $ | 1,778,717 |  | $ | 21,003 |  |
Cumulative effect of accounting change (2) |  | 6,539 |  |
Change for provision of expected credit losses (1) |  | 9,558 |  |
Balance at end of period | $ | 2,225,311 |  | $ | 37,100 |  |

(1)Amounts deemed uncollectible are written-off in operating expenses. For the 2021 third quarter and 2020 third quarter, amounts written off were $1.2 million and nil, respectively. For the nine months ended September 30, 2021 and 2020 period, amounts written off were were $2.4 million and $2.3 million, respectively.
(2)Adoption of ASU 2016-13, “Financial Instruments - Credit Losses (Topic 326)”.

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Table of Contents
ARCH CAPITAL GROUP LTD. AND SUBSIDIARIES
NOTES TO CONSOLIDATED FINANCIAL STATEMENTS (UNAUDITED)
Reinsurance Recoverables
The following table provides a roll forward of the allowance for expected credit losses of the Company’s reinsurance recoverables:

| Reinsurance Recoverables, Net of Allowance |  | Allowance for Expected Credit Losses |
Three Months Ended September 30, 2021 |  |
Balance at beginning of period | $ | 4,314,515 |  |  | $ | 11,029 |  |
Change for provision of expected credit losses |  |  | 1,802 |  |
Balance at end of period | $ | 5,358,852 |  |  | $ | 12,831 |  |
Three Months Ended September 30, 2020 |  |
Balance at beginning of period | $ | 4,363,507 |  |  | $ | 13,595 |  |
Change for provision of expected credit losses |  |  | 399 |  |
Balance at end of period | $ | 4,621,937 |  |  | $ | 13,994 |  |
Nine Months Ended September 30, 2021 |  |  |  |
Balance at beginning of period | $ | 4,500,802 |  |  | $ | 11,636 |  |
Change for provision of expected credit losses |  |  | 1,195 |  |
Balance at end of period | $ | 5,358,852 |  |  | $ | 12,831 |  |
Nine Months Ended September 30, 2020 |  |  |  |
Balance at beginning of period | $ | 4,346,816 |  |  | $ | 1,364 |  |
Cumulative effect of accounting change (1) |  |  | 12,010 |  |
Change for provision of expected credit losses |  |  | 620 |  |
Balance at end of period | $ | 4,621,937 |  |  | $ | 13,994 |  |

(1) Adoption of ASU 2016-13, “Financial Instruments - Credit Losses (Topic 326)”.
The following table summarizes the Company’s reinsurance recoverables on paid and unpaid losses (not including ceded unearned premiums):

| September 30, |  | December 31
| 2021 |  | 2020
Reinsurance recoverable on unpaid and paid losses and loss adjustment expenses | $ | 5,358,852 |  | $ | 4,500,802
% due from carriers with A.M. Best rating of “A-” or better | 69.1 | % |  | 63.9 | %
% due from all other carriers with no A.M. Best rating (1) | 30.9 | % |  | 36.1 | %
Largest balance due from any one carrier as % of total shareholders’ equity | 6.5 | % |  | 1.8 | %

(1) At September 30, 2021 and December 31, 2020 over 93% and 94% of such amount were collateralized through reinsurance trusts, funds withheld arrangements, letters of credit or other, respectively.
Contractholder Receivables
The following table provides a roll forward of the allowance for expected credit losses of the Company’s contractholder receivables:

| Contract-holder Receivables, Net of Allowance |  | Allowance for Expected Credit Losses |
Three Months Ended September 30, 2021 |  |
Balance at beginning of period | $ | 1,882,948 |  |  | $ | 4,471 |  |
Change for provision of expected credit losses |  |  | (987) |  |
Balance at end of period | $ | 1,824,990 |  |  | $ | 3,484 |  |
Three Months Ended September 30, 2020 |  |
Balance at beginning of period | $ | 2,179,124 |  |  | $ | 6,290 |  |
Change for provision of expected credit losses |  |  | (389) |  |
Balance at end of period | 2,185,614 |  |  | $ | 5,901 |  |
Nine Months Ended September 30, 2021 |  |  |  |
Balance at beginning of period | $ | 1,986,924 |  |  | $ | 8,638 |  |
Change for provision of expected credit losses |  |  | (5,154) |  |
Balance at end of period | $ | 1,824,990 |  |  | $ | 3,484 |  |
Nine Months Ended September 30, 2020 |  |  |  |
Balance at beginning of period | $ | 2,119,460 |  |  | $ | — |  |
Cumulative effect of accounting change (1) |  |  | 6,663 |  |
Change for provision of expected credit losses |  |  | (762) |  |
Balance at end of period | $ | 2,185,614 |  |  | $ | 5,901 |  |

(1) Adoption of ASU 2016-13, “Financial Instruments - Credit Losses (Topic 326)”.

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ARCH CAPITAL GROUP LTD. AND SUBSIDIARIES
NOTES TO CONSOLIDATED FINANCIAL STATEMENTS (UNAUDITED)
8. Investment Information

Available For Sale Investments
The following table summarizes the fair value and cost or amortized cost of the Company’s securities classified as available for sale:

| Estimated Fair Value |  | Gross Unrealized Gains |  | Gross Unrealized Losses |  | Allowance for Expected Credit Losses (2) |  | Cost or Amortized Cost |  |
September 30, 2021 |  |  |  |  |  |  |  |  |  |  |
Fixed maturities (1): |  |  |  |  |  |  |  |  |  |  |
Corporate bonds | $ | 6,403,617 |  |  | $ | 155,246 |  |  | $ | (40,649) |  |  | $ | (1,440) |  |  | $ | 6,290,460 |  |  |
Mortgage backed securities | 405,797 |  |  | 3,384 |  |  | (3,966) |  |  | (24) |  |  | 406,403 |  |  |
Municipal bonds | 382,722 |  |  | 20,096 |  |  | (1,206) |  |  | (2) |  |  | 363,834 |  |  |
Commercial mortgage backed securities | 579,424 |  |  | 3,598 |  |  | (548) |  |  | (3) |  |  | 576,377 |  |  |
U.S. government and government agencies | 4,460,515 |  |  | 12,912 |  |  | (30,320) |  |  | — |  |  | 4,477,923 |  |  |
Non-U.S. government securities | 1,863,734 |  |  | 46,378 |  |  | (28,948) |  |  | (82) |  |  | 1,846,386 |  |  |
Asset backed securities | 2,672,554 |  |  | 12,576 |  |  | (7,700) |  |  | (560) |  |  | 2,668,238 |  |  |
Total | 16,768,363 |  |  | 254,190 |  |  | (113,337) |  |  | (2,111) |  |  | 16,629,621 |  |  |
Short-term investments | 3,069,965 |  |  | 1,625 |  |  | (1,780) |  |  | — |  |  | 3,070,120 |  |  |
Total | $ | 19,838,328 |  |  | $ | 255,815 |  |  | $ | (115,117) |  |  | $ | (2,111) |  |  | $ | 19,699,741 |  |  |
December 31, 2020 |  |  |  |  |  |  |  |  |  |  |
Fixed maturities (1): |  |  |  |  |  |  |  |  |  |  |
Corporate bonds | $ | 7,856,571 |  |  | $ | 414,247 |  |  | $ | (34,388) |  |  | $ | (896) |  |  | $ | 7,477,608 |  |  |
Mortgage backed securities | 630,001 |  |  | 8,939 |  |  | (5,028) |  |  | (278) |  |  | 626,368 |  |  |
Municipal bonds | 494,522 |  |  | 27,291 |  |  | (3,835) |  |  | (11) |  |  | 471,077 |  |  |
Commercial mortgage backed securities | 389,900 |  |  | 8,722 |  |  | (2,954) |  |  | (122) |  |  | 384,254 |  |  |
U.S. government and government agencies | 5,557,077 |  |  | 22,612 |  |  | (12,611) |  |  | — |  |  | 5,547,076 |  |  |
Non-U.S. government securities | 2,433,733 |  |  | 153,891 |  |  | (8,060) |  |  | — |  |  | 2,287,902 |  |  |
Asset backed securities | 1,634,804 |  |  | 19,225 |  |  | (10,715) |  |  | (1,090) |  |  | 1,627,384 |  |  |
Total | 18,996,608 |  |  | 654,927 |  |  | (77,591) |  |  | (2,397) |  |  | 18,421,669 |  |  |
Short-term investments | 1,924,922 |  |  | 2,693 |  |  | (2,063) |  |  | — |  |  | 1,924,292 |  |  |
Total | $ | 20,921,530 |  |  | $ | 657,620 |  |  | $ | (79,654) |  |  | $ | (2,397) |  |  | $ | 20,345,961 |  |  |

(1) In securities lending transactions, the Company receives collateral in excess of the fair value of the fixed maturities pledged. For purposes of this table, the Company has excluded the collateral received under securities lending, at fair value and included the securities pledged under securities lending, at fair value. See “—Securities Lending Agreements.” In September 2021, the Company terminated its securities lending program and no longer enters into securities lending agreements with financial institutions.
(2) Effective January 1, 2020, the Company adopted ASU 2016-13 and as a result any credit impairment losses on the Company’s available-for-sale investments are recorded as an allowance, subject to reversal.

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ARCH CAPITAL GROUP LTD. AND SUBSIDIARIES
NOTES TO CONSOLIDATED FINANCIAL STATEMENTS (UNAUDITED)
The following table summarizes, for all available for sale securities in an unrealized loss position, the fair value and gross unrealized loss by length of time the security has been in a continual unrealized loss position:

| Less than 12 Months |  | 12 Months or More |  | Total
| Estimated Fair Value |  | Gross Unrealized Losses |  | Estimated Fair Value |  | Gross Unrealized Losses |  | Estimated Fair Value |  | Gross Unrealized Losses
September 30, 2021 |  |  |  |  |  |  |  |  |  |  |
Fixed maturities (1): |  |  |  |  |  |  |  |  |  |  |
Corporate bonds | $ | 2,744,615 |  |  | $ | (36,595) |  |  | $ | 68,450 |  |  | $ | (4,054) |  |  | $ | 2,813,065 |  |  | $ | (40,649) |
Mortgage backed securities | 257,789 |  |  | (3,113) |  |  | 22,769 |  |  | (853) |  |  | 280,558 |  |  | (3,966) |
Municipal bonds | 46,450 |  |  | (731) |  |  | 7,148 |  |  | (475) |  |  | 53,598 |  |  | (1,206) |
Commercial mortgage backed securities | 76,518 |  |  | (256) |  |  | 6,578 |  |  | (292) |  |  | 83,096 |  |  | (548) |
U.S. government and government agencies | 3,621,719 |  |  | (29,797) |  |  | 9,755 |  |  | (523) |  |  | 3,631,474 |  |  | (30,320) |
Non-U.S. government securities | 1,283,266 |  |  | (27,523) |  |  | 22,304 |  |  | (1,425) |  |  | 1,305,570 |  |  | (28,948) |
Asset backed securities | 1,119,638 |  |  | (6,524) |  |  | 37,078 |  |  | (1,176) |  |  | 1,156,716 |  |  | (7,700) |
Total | 9,149,995 |  |  | (104,539) |  |  | 174,082 |  |  | (8,798) |  |  | 9,324,077 |  |  | (113,337) |
Short-term investments | 265,011 |  |  | (1,780) |  |  | — |  |  | — |  |  | 265,011 |  |  | (1,780) |
Total | $ | 9,415,006 |  |  | $ | (106,319) |  |  | $ | 174,082 |  |  | $ | (8,798) |  |  | $ | 9,589,088 |  |  | $ | (115,117) |
December 31, 2020 |  |  |  |  |  |  |  |  |  |  |
Fixed maturities (1): |  |  |  |  |  |  |  |  |  |  |
Corporate bonds | $ | 747,442 |  |  | $ | (33,086) |  |  | $ | 3,934 |  |  | $ | (1,302) |  |  | $ | 751,376 |  |  | $ | (34,388) |
Mortgage backed securities | 284,619 |  |  | (4,788) |  |  | 3,637 |  |  | (240) |  |  | 288,256 |  |  | (5,028) |
Municipal bonds | 67,937 |  |  | (3,835) |  |  | — |  |  | — |  |  | 67,937 |  |  | (3,835) |
Commercial mortgage backed securities | 126,624 |  |  | (2,916) |  |  | 2,655 |  |  | (38) |  |  | 129,279 |  |  | (2,954) |
U.S. government and government agencies | 1,285,907 |  |  | (12,611) |  |  | — |  |  | — |  |  | 1,285,907 |  |  | (12,611) |
Non-U.S. government securities | 543,844 |  |  | (7,658) |  |  | 2,441 |  |  | (402) |  |  | 546,285 |  |  | (8,060) |
Asset backed securities | 634,470 |  |  | (9,110) |  |  | 57,737 |  |  | (1,605) |  |  | 692,207 |  |  | (10,715) |
Total | 3,690,843 |  |  | (74,004) |  |  | 70,404 |  |  | (3,587) |  |  | 3,761,247 |  |  | (77,591) |
Short-term investments | 97,920 |  |  | (2,063) |  |  | — |  |  | — |  |  | 97,920 |  |  | (2,063) |
Total | $ | 3,788,763 |  |  | $ | (76,067) |  |  | $ | 70,404 |  |  | $ | (3,587) |  |  | $ | 3,859,167 |  |  | $ | (79,654) |

(1) In securities lending transactions, the Company receives collateral in excess of the fair value of the fixed maturities pledged. For purposes of this table, the Company has excluded the collateral received under securities lending, at fair value and included the securities pledged under securities lending, at fair value. See “—Securities Lending Agreements.” In September 2021, the Company terminated its securities lending program and no longer enters into securities lending agreements with financial institutions.
At September 30, 2021, on a lot level basis, approximately 3,910 security lots out of a total of approximately 10,020 security lots were in an unrealized loss position and the largest single unrealized loss from a single lot in the Company’s fixed maturity portfolio was $2.5 million. At December 31, 2020, on a lot level basis, approximately 2,320 security lots out of a total of approximately 11,180 security lots were in an unrealized loss position and the largest single unrealized loss from a single lot in the Company’s fixed maturity portfolio was $0.9 million.

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ARCH CAPITAL GROUP LTD. AND SUBSIDIARIES
NOTES TO CONSOLIDATED FINANCIAL STATEMENTS (UNAUDITED)
The contractual maturities of the Company’s fixed maturities are shown in the following table. Expected maturities, which are management’s best estimates, will differ from contractual maturities because borrowers may have the right to call or prepay obligations with or without call or prepayment penalties.

|  | September 30, 2021 |  | December 31, 2020
Maturity |  | Estimated Fair Value |  | Amortized Cost |  | Estimated Fair Value |  | Amortized Cost
Due in one year or less |  | $ | 416,601 |  |  | $ | 410,373 |  |  | $ | 348,200 |  |  | $ | 339,951 |
Due after one year through five years |  | 7,862,472 |  |  | 7,772,773 |  |  | 10,629,959 |  |  | 10,340,819 |
Due after five years through 10 years |  | 4,483,025 |  |  | 4,453,526 |  |  | 4,881,564 |  |  | 4,654,754 |
Due after 10 years |  | 348,490 |  |  | 341,931 |  |  | 482,180 |  |  | 448,139 |
|  | 13,110,588 |  |  | 12,978,603 |  |  | 16,341,903 |  |  | 15,783,663 |
Mortgage backed securities |  | 405,797 |  |  | 406,403 |  |  | 630,001 |  |  | 626,368 |
Commercial mortgage backed securities |  | 579,424 |  |  | 576,377 |  |  | 389,900 |  |  | 384,254 |
Asset backed securities |  | 2,672,554 |  |  | 2,668,238 |  |  | 1,634,804 |  |  | 1,627,384 |
Total (1) |  | $ | 16,768,363 |  |  | $ | 16,629,621 |  |  | $ | 18,996,608 |  |  | $ | 18,421,669 |

(1) In securities lending transactions, the Company receives collateral in excess of the fair value of the fixed maturities pledged. For purposes of this table, the Company has excluded the collateral received under securities lending, at fair value and included the securities pledged under securities lending, at fair value. See “—Securities Lending Agreements.” In September 2021, the Company terminated its securities lending program and no longer enters into securities lending agreements with financial institutions.
Securities Lending Agreements
In September 2021, the Company terminated its securities lending program and no longer enters into securities lending agreements with financial institutions to enhance investment income. Prior to the termination of this program, the Company loaned certain of its securities to third parties, primarily major brokerage firms, for short periods of time through a lending agent. The Company maintained legal control over the securities it lent (shown as ‘Securities pledged under securities lending, at fair value’ on the Company’s balance sheet), retained the earnings and cash flows associated with the loaned securities and received a fee from the borrower for the temporary use of the securities. An indemnification agreement with the lending agent protected the Company in the event a borrower became insolvent or failed to return any of the securities on loan from the Company.
The Company received collateral (shown as ‘Collateral received under securities lending, at fair value’ on the Company’s balance sheet) in the form of cash or U.S. government and government agency securities. At September 30, 2021, the Company had no cash collateral or security collateral due to the termination of the program. At December 31, 2020, the fair value of the cash collateral received on securities lending was nil, and the fair value of security collateral received was $301.1 million.
The carrying value of collateral held under the Company’s securities lending transactions by significant investment category and remaining contractual maturity of the underlying agreements was as follows at December 31, 2020 (no balances at September 30, 2021 due to the termination of the program):

|  | Remaining Contractual Maturity of the Agreements
|  | Overnight and Continuous |  | Less than 30 Days |  | 30-90 Days |  | 90 Days or More |  | Total
December 31, 2020 |  |  |  |  |  |  |  |  |  |
U.S. government and government agencies |  | $ | 142,317 |  |  | $ | — |  |  | $ | 139,290 |  |  | $ | — |  |  | $ | 281,607 |
Corporate bonds |  | 3,021 |  |  | — |  |  | — |  |  | — |  |  | 3,021 |
Equity securities |  | 16,461 |  |  | — |  |  | — |  |  | — |  |  | 16,461 |
Total |  | $ | 161,799 |  |  | $ | — |  |  | $ | 139,290 |  |  | $ | — |  |  | $ | 301,089 |
Gross amount of recognized liabilities for securities lending in offsetting disclosure in note 10 |  | $ | — |
Amounts related to securities lending not included in offsetting disclosure in note 10 |  | $ | 301,089 |

Equity Securities, at Fair Value
At September 30, 2021, the Company held $1.8 billion of equity securities, at fair value, compared to $1.4 billion at
December 31, 2020. Such holdings include publicly traded common stocks primarily in the consumer cyclical and non-cyclical, technology, communication and financial sectors

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ARCH CAPITAL GROUP LTD. AND SUBSIDIARIES
NOTES TO CONSOLIDATED FINANCIAL STATEMENTS (UNAUDITED)
and exchange-traded funds in fixed income, equity and other sectors.
Other Investments
The following table summarizes the Company’s other investments and other investable assets:

| September 30, 2021 |  | December 31, 2020
Fixed maturities | $ | 414,007 |  |  | $ | 843,354 |
Other investments | 1,489,759 |  |  | 2,331,885 |
Short-term investments | 115,681 |  |  | 557,008 |
Equity securities | 24,523 |  |  | 92,549 |
Investments accounted for using the fair value option | $ | 2,043,970 |  |  | $ | 3,824,796 |
Other investable assets (1) | — |  |  | 500,000 |
Total other investments | $ | 2,043,970 |  |  | $ | 4,324,796 |

(1) Participation interests in a receivable of a reverse repurchase agreement.
The following table summarizes the Company’s other investments, as detailed in the previous table, by strategy:

| September 30, 2021 |  | December 31, 2020
Term loan investments | $ | 528,585 |  |  | $ | 1,231,731 |
Lending | 579,563 |  |  | 572,636 |
Credit related funds | 56,997 |  |  | 90,780 |
Energy | 84,880 |  |  | 65,813 |
Investment grade fixed income | 131,910 |  |  | 138,646 |
Infrastructure | 26,359 |  |  | 165,516 |
Private equity | 81,465 |  |  | 48,750 |
Real estate | — |  |  | 18,013 |
Total | $ | 1,489,759 |  |  | $ | 2,331,885 |

Investments Accounted For Using the Equity Method
The following table summarizes the Company’s investments accounted for using the equity method, by strategy:

| September 30, 2021 |  | December 31, 2020
Credit related funds | $ | 952,794 |  |  | $ | 740,060 |
Equities | 414,322 |  |  | 343,058 |
Real estate | 340,510 |  |  | 258,518 |
Lending | 330,368 |  |  | 179,629 |
Private equity | 365,840 |  |  | 235,289 |
Infrastructure | 222,484 |  |  | 175,882 |
Energy | 114,975 |  |  | 115,453 |
Total | $ | 2,741,293 |  |  | $ | 2,047,889 |

Certain of the Company’s other investments are in investment funds for which the Company has the option to redeem at agreed upon values as described in each investment fund’s subscription agreement. Depending on the terms of the various subscription agreements, investments in investment funds may be redeemed daily, monthly, quarterly or on other terms. Two common redemption restrictions
which may impact the Company’s ability to redeem these investment funds are gates and lockups. A gate is a suspension of redemptions which may be implemented by the general partner or investment manager of the fund in order to defer, in whole or in part, the redemption request in the event the aggregate amount of redemption requests exceeds a predetermined percentage of the investment fund’s net assets which may otherwise hinder the general partner or investment manager’s ability to liquidate holdings in an orderly fashion in order to generate the cash necessary to fund extraordinarily large redemption payouts. A lockup period is the initial amount of time an investor is contractually required to hold the security before having the ability to redeem. If the investment funds are eligible to be redeemed, the time to redeem such fund can take weeks or months following the notification.
Limited Partnership Interests
In the normal course of its activities, the Company invests in limited partnerships as part of its overall investment strategy. Such amounts are included in ‘investments accounted for using the equity method’ and ‘investments accounted for using the fair value option.’ The Company has determined that it is not required to consolidate these investments because it is not the primary beneficiary of the funds. The Company’s maximum exposure to loss with respect to these investments is limited to the investment carrying amounts reported in the Company’s consolidated balance sheet and any unfunded commitment.
The following table summarizes investments in limited partnership interests where the Company has a variable interest by balance sheet line item:

| September 30, 2021 |  | December 31, 2020
Investments accounted for using the equity method (1) | 2,741,293 |  |  | 2,047,889 |
Investments accounted for using the fair value option (2) | 176,884 |  |  | 184,720 |
Total | $ | 2,918,177 |  |  | $ | 2,232,609 |

(1) Aggregate unfunded commitments were $2.3 billion at September 30, 2021, compared to $1.8 billion at December 31, 2020.
(2) Aggregate unfunded commitments were $22.9 million at September 30, 2021, compared to $35.6 million at December 31, 2020.

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ARCH CAPITAL GROUP LTD. AND SUBSIDIARIES
NOTES TO CONSOLIDATED FINANCIAL STATEMENTS (UNAUDITED)
Net Investment Income
The components of net investment income were derived from the following sources:

| September 30,
| 2021 |  | 2020
Three Months Ended |  |  |
Fixed maturities | $ | 75,964 |  |  | $ | 98,344 |
Term loans | 1,736 |  |  | 22,459 |
Equity securities | 9,867 |  |  | 6,659 |
Short-term investments | 1,858 |  |  | 1,332 |
Other (1) | 17,378 |  |  | 22,060 |
Gross investment income | 106,803 |  |  | 150,854 |
Investment expenses | (18,608) |  |  | (22,342) |
Net investment income | $ | 88,195 |  |  | $ | 128,512 |
Nine Months Ended |  |  |
Fixed maturities | $ | 255,215 |  |  | $ | 318,582 |
Term loans | 33,343 |  |  | 66,141 |
Equity securities | 24,101 |  |  | 18,885 |
Short-term investments | 3,603 |  |  | 9,611 |
Other (1) | 51,683 |  |  | 57,926 |
Gross investment income | 367,945 |  |  | 471,145 |
Investment expenses | (69,281) |  |  | (65,995) |
Net investment income | $ | 298,664 |  |  | $ | 405,150 |

(1) Includes income distributions from investment funds and other items.
Net Realized Gains (Losses)
Net realized gains (losses), which include changes in the allowance for credit losses on financial assets and net impairment losses recognized in earnings were as follows:

| September 30,
| 2021 |  | 2020
Three Months Ended |  |  |
Available for sale securities: |  |  |
Gross gains on investment sales | $ | 86,819 |  |  | $ | 104,733 |
Gross losses on investment sales | (18,446) |  |  | (16,862) |
Change in fair value of assets and liabilities accounted for using the fair value option: |  |  |
Fixed maturities | (7,492) |  |  | 34,115 |
Other investments | 3,811 |  |  | 61,622 |
Equity securities | 3,042 |  |  | 4,048 |
Short-term investments | 16 |  |  | 3,377 |
Equity securities, at fair value: |  |  |
Net realized gains (losses) on sales during the period | 14,736 |  |  | 26,549 |
Net unrealized gains (losses) on equity securities still held at reporting date | (40,155) |  |  | 33,562 |
Allowance for credit losses: |  |  |
Investments related | (456) |  |  | 1,332 |
Underwriting related | (3,985) |  |  | 351 |
Derivative instruments (1) | (21,435) |  |  | 20,369 |
Other (2) | (41,495) |  |  | 7,303 |
Net realized gains (losses) | $ | (25,040) |  |  | $ | 280,499 |
Nine Months Ended |  |  |
Available for sale securities: |  |  |
Gross gains on investment sales | $ | 267,362 |  |  | $ | 515,086 |
Gross losses on investment sales | (132,071) |  |  | (98,654) |
Change in fair value of assets and liabilities accounted for using the fair value option: |  |  |
Fixed maturities | 19,973 |  |  | (25,370) |
Other investments | 111,550 |  |  | (67,608) |
Equity securities | 10,599 |  |  | 5,803 |
Short-term investments | 648 |  |  | (1,936) |
Equity securities, at fair value: |  |  |
Net realized gains (losses) on sales during the period | 86,155 |  |  | 7,760 |
Net unrealized gains (losses) on equity securities still held at reporting date | 45,400 |  |  | 3,682 |
Allowance for credit losses: |  |  |
Investments related | (1,208) |  |  | (4,763) |
Underwriting related | 2,664 |  |  | (8,753) |
Net impairments losses | — |  |  | (533) |
Derivative instruments (1) | (36,428) |  |  | 146,722 |
Other (2) | (54,316) |  |  | (1,309) |
Net realized gains (losses) | $ | 320,328 |  |  | $ | 470,127 |

(1) See note 10 for information on the Company’s derivative instruments.
(2) 2021 periods reflected $33.1 million of losses related to the Company’s deconsolidation of Watford.

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ARCH CAPITAL GROUP LTD. AND SUBSIDIARIES
NOTES TO CONSOLIDATED FINANCIAL STATEMENTS (UNAUDITED)
Equity in Net Income (Loss) of Investment Funds Accounted for Using the Equity Method
The Company recorded $105.4 million of equity in net income related to investment funds accounted for using the equity method in the 2021 third quarter, compared to income of $126.7 million for the 2020 third quarter, and an income of $299.3 million for the nine months ended September 30, 2021, compared to income of $57.4 million for nine months ended September 30, 2020. In applying the equity method, investments are initially recorded at cost and are subsequently adjusted based on the Company’s proportionate share of the net income or loss of the funds (which include changes in the market value of the underlying securities in the funds). Such investments are generally recorded on a one to three month lag based on the availability of reports from the investment funds.
Investments in Operating Affiliates
Investments in which the Company has significant influence over the operating and financial policies are classified as ‘investments in operating affiliates’ on the Company’s balance sheets and are accounted for under the equity method. Such investments primarily include the Company’s investment in Coface, Greysbridge and Premia. Investments in Coface and Premia are generally recorded on a three month lag, while the Company’s investment in Greysbridge is not recorded on a lag.
In 2021, the Company completed the share purchase agreement with Natixis to purchase 29.5% of the common equity of Coface, a France-based leader in the global trade credit insurance market. The consideration paid was €9.95 per share, or an aggregate €453 million (approximately $546 million) including related fees. Income (loss) from
operating affiliates reflected a one-time gain of $74.5 million realized from the acquisition. As a result of equity method accounting rules, approximately $36 million of additional gain was deferred and will generally be recognized over the next five years. As of September 30, 2021, the Company owned approximately 29.86% of the issued shares of Coface, or 30.10% excluding treasury shares, with a carrying value of $615.9 million.
In July 2021, the Company announced the completion of the previously disclosed acquisition of Watford by Greysbridge for a cash purchase price of $35.00 per common share. Effective July 1, 2021, Watford is wholly owned by Greysbridge, and Greysbridge is owned 40% by the Company, 30% by certain investment funds managed by Kelso and 30% by certain investment funds managed by Warburg. At September 30, 2021 the Company’s carrying value in Greysbridge was $363.3 million, which reflected the Company’s aggregate purchase price of $278.9 million along with income (loss) from operating affiliates, which included a one-time gain of $95.7 million recognized from the acquisition. In addition, the ‘net realized gains (losses)’ line on the Company’s consolidated statements of income included a $33.1 million loss as a result of deconsolidation of Watford in the Company’s financial statements following the close of the transaction. See n ote 12.
Income from operating affiliates for the 2021 third quarter was $124.1 million, compared to an income of $0.9 million, for the 2020 third quarter, and income of $224.1 million for the nine months ended September 30, 2021, compared to an income of $6.3 million for the nine months ended September 30, 2020. The income from operating affiliates for the 2021 period, primarily related to the Company’s recent acquisitions of Coface and Greysbridge.

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ARCH CAPITAL GROUP LTD. AND SUBSIDIARIES
NOTES TO CONSOLIDATED FINANCIAL STATEMENTS (UNAUDITED)
Allowance for Expected Credit Losses
The following table provides a roll forward of the allowance for expected credit losses of the Company’s securities classified as available for sale:

|  | Structured Securities (1) |  | Municipal Bonds |  | Corporate Bonds |  |  |  | Total
Three Months Ended September 30, 2021 |  |  |  |  |  |  |  |
Balance at beginning of period |  | $ | 759 |  |  | $ | 6 |  |  | $ | 1,359 |  |  |  |  | $ | 2,124 |
Additions for current-period provision for expected credit losses |  | 48 |  |  | — |  |  | — |  |  |  |  | 48 |
Additions (reductions) for previously recognized expected credit losses |  | 14 |  |  | (4) |  |  | 395 |  |  |  |  | 405 |
Reductions due to disposals (3) |  | (234) |  |  | — |  |  | (232) |  |  |  |  | (466) |
Balance at end of period |  | $ | 587 |  |  | $ | 2 |  |  | $ | 1,522 |  |  |  |  | $ | 2,111 |
Three Months Ended September 30, 2020 |  |  |  |  |  |  |  |
Balance at beginning of period |  | $ | 1,726 |  |  | $ | 28 |  |  | $ | 4,115 |  |  |  |  | $ | 5,869 |
Additions for current-period provision for expected credit losses |  | 27 |  |  | — |  |  | 202 |  |  |  |  | 229 |
Additions (reductions) for previously recognized expected credit losses |  | 403 |  |  | 33 |  |  | (1,996) |  |  |  |  | (1,560) |
Reductions due to disposals |  | (28) |  |  | — |  |  | (577) |  |  |  |  | (605) |
Balance at end of period |  | $ | 2,128 |  |  | $ | 61 |  |  | $ | 1,744 |  |  |  |  | $ | 3,933 |
Nine Months Ended September 30, 2021 |  |  |  |  |  |  |  |  |  |
Balance at beginning of period |  | $ | 1,490 |  |  | $ | 11 |  |  | $ | 896 |  |  |  |  | $ | 2,397 |
Additions for current-period provision for expected credit losses |  | 282 |  |  | — |  |  | 2,428 |  |  |  |  | 2,710 |
Additions (reductions) for previously recognized expected credit losses |  | (751) |  |  | (9) |  |  | (557) |  |  |  |  | (1,317) |
Reductions due to disposals (3) |  | (434) |  |  | — |  |  | (1,245) |  |  |  |  | (1,679) |
Balance at end of period |  | $ | 587 |  |  | $ | 2 |  |  | $ | 1,522 |  |  |  |  | $ | 2,111 |
Nine Months Ended September 30, 2020 |  |  |  |  |  |  |  |  |  |
Balance at beginning of period |  | $ | — |  |  | $ | — |  |  | $ | — |  |  |  |  | $ | — |
Cumulative effect of accounting change (2) |  | 517 |  |  | — |  |  | 117 |  |  |  |  | 634 |
Additions for current-period provision for expected credit losses |  | 2,868 |  |  | 67 |  |  | 7,643 |  |  |  |  | 10,578 |
Additions (reductions) for previously recognized expected credit losses |  | (903) |  |  | 8 |  |  | (4,920) |  |  |  |  | (5,815) |
Reductions due to disposals |  | (354) |  |  | (14) |  |  | (1,096) |  |  |  |  | (1,464) |
Balance at end of period |  | $ | 2,128 |  |  | $ | 61 |  |  | $ | 1,744 |  |  |  |  | $ | 3,933 |

(1) Includes asset backed securities, mortgage backed securities and commercial mortgage backed securities.
(2) Adoption of ASU 2016-13, “Financial Instruments - Credit Losses (Topic 326)”.
(3) Reduction for the 2021 periods primarily related to the Company’s deconsolidation of Watford.

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ARCH CAPITAL GROUP LTD. AND SUBSIDIARIES
NOTES TO CONSOLIDATED FINANCIAL STATEMENTS (UNAUDITED)
Restricted Assets
The Company is required to maintain assets on deposit, which primarily consist of fixed maturities, with various regulatory authorities to support its underwriting operations. The Company’s subsidiaries maintain assets in trust accounts as collateral for transactions with affiliated companies and also have investments in segregated portfolios primarily to provide collateral or guarantees for letters of credit to third parties. See note 18, “Commitments and Contingencies,” of the notes to consolidated financial statements in the Company’s 2020 Form 10-K.
The following table details the value of the Company’s restricted assets:

| September 30, 2021 |  | December 31, 2020
Assets used for collateral or guarantees: |  |  |
Affiliated transactions | $ | 4,271,208 |  |  | $ | 4,643,334 |
Third party agreements | 2,594,053 |  |  | 3,083,324 |
Deposits with U.S. regulatory authorities | 803,878 |  |  | 827,552 |
Deposits with non-U.S. regulatory authorities | 469,497 |  |  | 179,099 |
Total restricted assets | $ | 8,138,636 |  |  | $ | 8,733,309 |

Reconciliation of Cash and Restricted Cash
The following table details reconciliation of cash and restricted cash within the Consolidated Balance Sheets:

| September 30, 2021 |  | December 31, 2020
Cash | $ | 1,137,721 |  |  | $ | 906,448 |
Restricted cash (included in ‘other assets’) | $ | 411,188 |  |  | $ | 384,096 |
Cash and restricted cash | $ | 1,548,909 |  |  | $ | 1,290,544 |

9. Fair Value

Accounting guidance regarding fair value measurements addresses how companies should measure fair value when they are required to use a fair value measure for recognition or disclosure purposes under GAAP and provides a common definition of fair value to be used throughout GAAP. It defines fair value as the price that would be received to sell an asset or paid to transfer a liability in an orderly fashion between market participants at the measurement date. In addition, it establishes a three-level valuation hierarchy for the disclosure of fair value measurements. The valuation hierarchy is based upon the transparency of inputs to the valuation of an asset or liability as of the measurement date. The level in the hierarchy within which a given fair value measurement falls is determined based on the lowest level input that is significant to the measurement (Level 1 being the highest priority and Level 3 being the lowest priority).
The levels in the hierarchy are defined as follows:

Level 1: |  | Inputs to the valuation methodology are observable inputs that reflect quoted prices (unadjusted) for identical assets or liabilities in active markets
Level 2: |  | Inputs to the valuation methodology include quoted prices for similar assets and liabilities in active markets, and inputs that are observable for the asset or liability, either directly or indirectly, for substantially the full term of the financial instrument
Level 3: |  | Inputs to the valuation methodology are unobservable and significant to the fair value measurement

Following is a description of the valuation methodologies used for securities measured at fair value, as well as the general classification of such securities pursuant to the valuation hierarchy. The Company reviews its securities measured at fair value and discusses the proper classification of such investments with investment advisers and others.
The Company determines the existence of an active market based on its judgment as to whether transactions for the financial instrument occur in such market with sufficient frequency and volume to provide reliable pricing information. The independent pricing sources obtain market quotations and actual transaction prices for securities that have quoted prices in active markets. The Company uses quoted values and other data provided by nationally recognized independent pricing sources as inputs into its process for determining fair values of its fixed maturity investments. To validate the techniques or models used by pricing sources, the Company's review process includes, but is not limited to: (i) quantitative analysis (e.g., comparing the quarterly return for each managed portfolio to its target benchmark, with significant differences identified and investigated); (ii) a review of the average number of prices obtained in the pricing process and the range of resulting fair values; (iii) initial and ongoing evaluation of methodologies used by outside parties to calculate fair value; (iv) a comparison of the fair value estimates to the Company’s knowledge of the current market; (v) a comparison of the pricing services' fair values to other pricing services' fair values for the same investments; and (vi) periodic back-testing, which includes randomly selecting purchased or sold securities and comparing the executed prices to the fair value estimates from the pricing service. A price source hierarchy was maintained in order to determine which price source would be used (i.e., a price obtained from a pricing service with more seniority in the hierarchy will be used over a less senior one in all cases). The hierarchy prioritizes pricing services based on availability and reliability and assigns the highest priority to index providers. Based on the above

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ARCH CAPITAL GROUP LTD. AND SUBSIDIARIES
NOTES TO CONSOLIDATED FINANCIAL STATEMENTS (UNAUDITED)
review, the Company will challenge any prices for a security or portfolio which are considered not to be representative of fair value. The Company did not adjust any of the prices obtained from the independent pricing sources at September 30, 2021.
In certain circumstances, when fair values are unavailable from these independent pricing sources, quotes are obtained directly from broker-dealers who are active in the corresponding markets. Such quotes are subject to the validation procedures noted above. Where quotes are unavailable, fair value is determined by the Investment Manager using quantitative and qualitative assessments such as internally modeled values. Of the $23.8 billion of financial assets and liabilities measured at fair value at September 30, 2021, approximately $9.0 million, or 0.0%, were priced using non-binding broker-dealer quotes or modeled valuations. Of the $26.5 billion of financial assets and liabilities measured at fair value at December 31, 2020, approximately $150.1 million, or 0.6%, were priced using non-binding broker-dealer quotes or modeled valuations.
Fixed maturities
The Company uses the market approach valuation technique to estimate the fair value of its fixed maturity securities, when possible. The market approach includes obtaining prices from independent pricing services, such as index providers and pricing vendors, as well as to a lesser extent quotes from broker-dealers. The independent pricing sources obtain market quotations and actual transaction prices for securities that have quoted prices in active markets. Each source has its own proprietary method for determining the fair value of securities that are not actively traded. In general, these methods involve the use of “matrix pricing” in which the independent pricing source uses observable market inputs including, but not limited to, investment yields, credit risks and spreads, benchmarking of like securities, broker-dealer quotes, reported trades and sector groupings to determine a reasonable fair value.
The following describes the significant inputs generally used to determine the fair value of the Company’s fixed maturity securities by asset class:
•U.S. government and government agencies — valuations provided by independent pricing services, with all prices provided through index providers and pricing vendors. The Company determined that all U.S. Treasuries would be classified as Level 1 securities due to observed levels of trading activity, the high number of strongly correlated pricing quotes received on U.S. Treasuries and other factors. The fair values of U.S. government agency securities are generally determined using the spread above the risk-free yield curve. As the yields for the risk-free yield curve and the spreads for these securities are observable market inputs, the
fair values of U.S. government agency securities are classified within Level 2.
•Corporate bonds — valuations provided by independent pricing services, substantially all through index providers and pricing vendors with a small amount through broker-dealers. The fair values of these securities are generally determined using the spread above the risk-free yield curve. These spreads are generally obtained from the new issue market, secondary trading and from broker-dealers who trade in the relevant security market. As the significant inputs used in the pricing process for corporate bonds are observable market inputs, the fair value of these securities are classified within Level 2. A small number of securities are included in Level 3 due to a low level of transparency on the inputs used in the pricing process.
•Mortgage-backed securities — valuations provided by independent pricing services, substantially all through pricing vendors and index providers with a small amount through broker-dealers. The fair values of these securities are generally determined through the use of pricing models (including Option Adjusted Spread) which use spreads to determine the expected average life of the securities. These spreads are generally obtained from the new issue market, secondary trading and from broker-dealers who trade in the relevant security market. The pricing services also review prepayment speeds and other indicators, when applicable. As the significant inputs used in the pricing process for mortgage-backed securities are observable market inputs, the fair value of these securities are classified within Level 2. A small number of securities are included in Level 3 due to a low level of transparency on the inputs used in the pricing process.
•Municipal bonds — valuations provided by independent pricing services, with all prices provided through index providers and pricing vendors. The fair values of these securities are generally determined using spreads obtained from broker-dealers who trade in the relevant security market, trade prices and the new issue market. As the significant inputs used in the pricing process for municipal bonds are observable market inputs, the fair value of these securities are classified within Level 2.
•Commercial mortgage-backed securities — valuations provided by independent pricing services, substantially all through index providers and pricing vendors with a small amount through broker-dealers. The fair values of these securities are generally determined through the use of pricing models which use spreads to determine the appropriate average life of the securities. These spreads are generally obtained from the new issue market, secondary trading and from broker-dealers who trade in the relevant security market. The pricing services also review prepayment speeds and other indicators, when applicable. As the significant inputs used in the pricing process for commercial

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ARCH CAPITAL GROUP LTD. AND SUBSIDIARIES
NOTES TO CONSOLIDATED FINANCIAL STATEMENTS (UNAUDITED)
mortgage-backed securities are observable market inputs, the fair value of these securities are classified within Level 2.
•Non-U.S. government securities — valuations provided by independent pricing services, with all prices provided through index providers and pricing vendors. The fair values of these securities are generally based on international indices or valuation models which include daily observed yield curves, cross-currency basis index spreads and country credit spreads. As the significant inputs used in the pricing process for non-U.S. government securities are observable market inputs, the fair value of these securities are classified within Level 2.
•Asset-backed securities — valuations provided by independent pricing services, substantially all through index providers and pricing vendors with a small amount through broker-dealers. The fair values of these securities are generally determined through the use of pricing models (including Option Adjusted Spread) which use spreads to determine the appropriate average life of the securities. These spreads are generally obtained from the new issue market, secondary trading and from broker-dealers who trade in the relevant security market. The pricing services also review prepayment speeds and other indicators, when applicable. As the significant inputs used in the pricing process for asset-backed securities are observable market inputs, the fair value of these securities are classified within Level 2. A small number of securities are included in Level 3 due to a low level of transparency on the inputs used in the pricing process.
Equity securities
The Company determined that exchange-traded equity securities would be included in Level 1 as their fair values are based on quoted market prices in active markets. Certain equity securities are included in Level 2 of the valuation hierarchy as the significant inputs used in the pricing process for such securities are observable market inputs. Other equity securities are included in Level 3 due to the lack of an available independent price source for such securities. As the significant inputs used to price these securities are unobservable, the fair value of such securities are classified as Level 3.
Other investments
The Company’s other investments include term loan investments for which fair values are estimated by using quoted prices of term loan investments with similar characteristics, pricing models or matrix pricing. Such investments are generally classified within Level 2. The fair values for certain of the Company’s other investments are determined using net asset values as advised by external fund managers. The net asset value is based on the fund manager’s valuation of the underlying holdings in accordance with the fund’s governing documents. In accordance with applicable
accounting guidance, certain investments that are measured at fair value using the net asset value per share (or its equivalent) practical expedient have not been classified in the fair value hierarchy. A small number of securities are included in Level 3 due to the lack of an available independent price source for such securities.
Derivative instruments
The Company’s futures contracts, foreign currency forward contracts, interest rate swaps and other derivatives trade in the over-the-counter derivative market. The Company uses the market approach valuation technique to estimate the fair value for these derivatives based on significant observable market inputs from third party pricing vendors, non-binding broker-dealer quotes and/or recent trading activity. As the significant inputs used in the pricing process for these derivative instruments are observable market inputs, the fair value of these securities are classified within Level 2.
Short-term investments
The Company determined that certain of its short-term investments held in highly liquid money market-type funds, Treasury bills and commercial paper would be included in Level 1 as their fair values are based on quoted market prices in active markets. The fair values of other short-term investments are generally determined using the spread above the risk-free yield curve and are classified within Level 2.
Residential mortgage loans
The Company’s residential mortgage loans (included in ‘other assets’ in the consolidated balance sheets) include amounts related to the Company’s whole mortgage loan purchase and sell program. Fair values of residential mortgage loans are generally determined based on market prices. As significant inputs used in pricing process for these residential mortgage loans are observable market inputs, the fair value of these securities are classified within Level 2.
Contingent consideration liabilities
Contingent consideration liabilities (included in ‘other liabilities’ in the consolidated balance sheets) include amounts related to various Company’s acquisitions. Such amounts are remeasured at fair value at each balance sheet date with changes in fair value recognized in ‘net realized gains (losses).’ To determine the fair value of contingent consideration liabilities, the Company estimates future payments using an income approach based on modeled inputs which include a weighted average cost of capital. The Company determined that contingent consideration liabilities would be included within Level 3.

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ARCH CAPITAL GROUP LTD. AND SUBSIDIARIES
NOTES TO CONSOLIDATED FINANCIAL STATEMENTS (UNAUDITED)
The following table presents the Company’s financial assets and liabilities measured at fair value by level at September 30, 2021:

|  |  | Estimated Fair Value Measurements Using:
| Estimated Fair Value |  | Quoted Prices in Active Markets for Identical Assets (Level 1) |  | Significant Other Observable Inputs (Level 2) |  | Significant Unobservable Inputs (Level 3)
Assets measured at fair value (1): |  |  |  |  |  |  |
Available for sale securities: |  |  |  |  |  |  |
Fixed maturities: |  |  |  |  |  |  |
Corporate bonds | $ | 6,403,617 |  |  | $ | — |  |  | $ | 6,403,604 |  |  | $ | 13 |
Mortgage backed securities | 405,797 |  |  | — |  |  | 405,797 |  |  | — |
Municipal bonds | 382,722 |  |  | — |  |  | 382,722 |  |  | — |
Commercial mortgage backed securities | 579,424 |  |  | — |  |  | 579,424 |  |  | — |
U.S. government and government agencies | 4,460,515 |  |  | 4,431,935 |  |  | 28,580 |  |  | — |
Non-U.S. government securities | 1,863,734 |  |  | — |  |  | 1,863,734 |  |  | — |
Asset backed securities | 2,672,554 |  |  | — |  |  | 2,669,110 |  |  | 3,444 |
Total | 16,768,363 |  |  | 4,431,935 |  |  | 12,332,971 |  |  | 3,457 |
Short-term investments | 3,069,965 |  |  | 2,046,391 |  |  | 1,023,574 |  |  | — |
Equity securities, at fair value | 1,790,640 |  |  | 1,756,462 |  |  | 31,596 |  |  | 2,582 |
Derivative instruments (4) | 89,958 |  |  | — |  |  | 89,958 |  |  | — |
Residential mortgage loans | 7,701 |  |  | — |  |  | 7,701 |  |  | — |
Fair value option: |  |  |  |  |  |  |
Corporate bonds | 374,326 |  |  | — |  |  | 374,326 |  |  | — |
Non-U.S. government bonds | 19,829 |  |  | — |  |  | 19,829 |  |  | — |
Mortgage backed securities | — |  |  | — |  |  | — |  |  | — |
Commercial mortgage backed securities | — |  |  | — |  |  | — |  |  | — |
Asset backed securities | 19,852 |  |  | — |  |  | 19,852 |  |  | — |
U.S. government and government agencies | — |  |  | — |  |  | — |  |  | — |
Short-term investments | 115,681 |  |  | 1,190 |  |  | 114,491 |  |  | — |
Equity securities | 24,522 |  |  | 19,987 |  |  | — |  |  | 4,535 |
Other investments | 359,011 |  |  | 17,861 |  |  | 311,393 |  |  | 29,757 |
Other investments measured at net asset value (2) | 1,130,748 |  |  |  |  |  |  |
Total | 2,043,969 |  |  | 39,038 |  |  | 839,891 |  |  | 34,292 |
Total assets measured at fair value | $ | 23,770,596 |  |  | $ | 8,273,826 |  |  | $ | 14,325,691 |  |  | $ | 40,331 |
Liabilities measured at fair value: |  |  |  |  |  |  |
Contingent consideration liabilities | $ | (17,811) |  |  | $ | — |  |  | $ | — |  |  | $ | (17,811) |
Securities sold but not yet purchased (3) | — |  |  | — |  |  | — |  |  | — |
Derivative instruments (4) | (43,526) |  |  | — |  |  | (43,526) |  |  | — |
Total liabilities measured at fair value | $ | (61,337) |  |  | $ | — |  |  | $ | (43,526) |  |  | $ | (17,811) |

(1) In securities lending transactions, the Company receives collateral in excess of the fair value of the securities pledged. For purposes of this table, the Company has excluded the collateral received under securities lending, at fair value and included the securities pledged under securities lending, at fair value. See note 8, “—Securities Lending Agreements.” In September 2021, the Company terminated its securities lending program and no longer enters into securities lending agreements with financial institutions.
(2) In accordance with applicable accounting guidance, certain investments that are measured at fair value using the net asset value per share (or its equivalent) practical expedient have not been classified in the fair value hierarchy. The fair value amounts presented in this table are intended to permit reconciliation of the fair value hierarchy to the amounts presented in the consolidated balance sheets.
(3) Represents the Company’s obligations to deliver securities that it did not own at the time of sale. Such amounts are included in “other liabilities” on the Company’s consolidated balance sheets.
(4) See note 10.

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ARCH CAPITAL GROUP LTD. AND SUBSIDIARIES
NOTES TO CONSOLIDATED FINANCIAL STATEMENTS (UNAUDITED)
The following table presents the Company’s financial assets and liabilities measured at fair value by level at December 31, 2020:

|  |  | Estimated Fair Value Measurements Using:
| Estimated Fair Value |  | Quoted Prices in Active Markets for Identical Assets (Level 1) |  | Significant Other Observable Inputs (Level 2) |  | Significant Unobservable Inputs (Level 3)
Assets measured at fair value (1): |  |  |  |  |  |  |
Available for sale securities: |  |  |  |  |  |  |
Fixed maturities: |  |  |  |  |  |  |
Corporate bonds | $ | 7,856,571 |  |  | $ | — |  |  | $ | 7,856,558 |  |  | $ | 13 |
Mortgage backed securities | 630,001 |  |  | — |  |  | 630,001 |  |  | — |
Municipal bonds | 494,522 |  |  | — |  |  | 494,522 |  |  | — |
Commercial mortgage backed securities | 389,900 |  |  | — |  |  | 389,900 |  |  | — |
U.S. government and government agencies | 5,557,077 |  |  | 5,463,356 |  |  | 93,721 |  |  | — |
Non-U.S. government securities | 2,433,733 |  |  | — |  |  | 2,433,733 |  |  | — |
Asset backed securities | 1,634,804 |  |  | — |  |  | 1,631,378 |  |  | 3,426 |
Total | 18,996,608 |  |  | 5,463,356 |  |  | 13,529,813 |  |  | 3,439 |
Short-term investments | 1,924,922 |  |  | 1,920,565 |  |  | 4,357 |  |  | — |
Equity securities, at fair value | 1,460,959 |  |  | 1,401,653 |  |  | 17,291 |  |  | 42,015 |
Derivative instruments (4) | 177,383 |  |  | — |  |  | 177,383 |  |  | — |
Fair value option: |  |  |  |  |  |  |
Corporate bonds | 651,294 |  |  | — |  |  | 650,309 |  |  | 985 |
Non-U.S. government bonds | 35,263 |  |  | — |  |  | 35,263 |  |  | — |
Mortgage backed securities | 3,282 |  |  | — |  |  | 3,282 |  |  | — |
Commercial mortgage backed securities | 1,090 |  |  | — |  |  | 1,090 |  |  | — |
Asset backed securities | 152,151 |  |  | — |  |  | 152,151 |  |  | — |
U.S. government and government agencies | 274 |  |  | 164 |  |  | 110 |  |  | — |
Short-term investments | 557,008 |  |  | 420,131 |  |  | 136,877 |  |  | — |
Equity securities | 92,549 |  |  | 23,373 |  |  | 188 |  |  | 68,988 |
Other investments | 1,134,229 |  |  | 51,149 |  |  | 1,015,977 |  |  | 67,103 |
Other investments measured at net asset value (2) | 1,197,656 |  |  |  |  |  |  |
Total | 3,824,796 |  |  | 494,817 |  |  | 1,995,247 |  |  | 137,076 |
Total assets measured at fair value | $ | 26,384,668 |  |  | $ | 9,280,391 |  |  | $ | 15,724,091 |  |  | $ | 182,530 |
Liabilities measured at fair value: |  |  |  |  |  |  |
Contingent consideration liabilities | $ | (461) |  |  | $ | — |  |  | $ | — |  |  | $ | (461) |
Securities sold but not yet purchased (3) | (21,679) |  |  | — |  |  | (21,679) |  |  | — |
Derivative instruments (4) | (108,705) |  |  | — |  |  | (108,705) |  |  | — |
Total liabilities measured at fair value | $ | (130,845) |  |  | $ | — |  |  | $ | (130,384) |  |  | $ | (461) |

(1) In securities lending transactions, the Company receives collateral in excess of the fair value of the securities pledged. For purposes of this table, the Company has excluded the collateral received under securities lending, at fair value and included the securities pledged under securities lending, at fair value. See note 8, “—Securities Lending Agreements.” In September 2021, the Company terminated its securities lending program and no longer enters into securities lending agreements with financial institutions.
(2) In accordance with applicable accounting guidance, certain investments that are measured at fair value using the net asset value per share (or its equivalent) practical expedient have not been classified in the fair value hierarchy. The fair value amounts presented in this table are intended to permit reconciliation of the fair value hierarchy to the amounts presented in the consolidated balance sheets.
(3) Represents the Company’s obligations to deliver securities that it did not own at the time of sale. Such amounts are included in “other liabilities” on the Company’s consolidated balance sheets.
(4) See note 10.

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ARCH CAPITAL GROUP LTD. AND SUBSIDIARIES
NOTES TO CONSOLIDATED FINANCIAL STATEMENTS (UNAUDITED)
The following table presents a reconciliation of the beginning and ending balances for all financial assets and liabilities measured at fair value on a recurring basis using Level 3 inputs:

| Assets | Liabilities
s | Available For Sale |  | Fair Value Option |  | Fair Value |  |
| Structured Securities (1) |  | Corporate Bonds |  | Corporate Bonds |  | Other Investments |  | Equity Securities |  | Equity Securities |  | Contingent Consideration Liabilities
Three Months Ended September 30, 2021 |  |  |  |  |  |  |  |  |  |  |  |  |
Balance at beginning of period | $ | 3,424 |  |  | $ | 13 |  |  | $ | 998 |  |  | $ | 73,900 |  |  | $ | 73,678 |  |  | $ | 49,136 |  |  | $ | (466) |
Total gains or (losses) (realized/unrealized) |  |  |  |  |  |  |  |  |  |  |  |  |
Included in earnings (2) | 10 |  |  | — |  |  | — |  |  | — |  |  | 38 |  |  | 11 |  |  | — |
Included in other comprehensive income | 10 |  |  | — |  |  | — |  |  | — |  |  | — |  |  | — |  |  | — |
Purchases, issuances, sales and settlements |  |  |  |  |  |  |  |  |  |  |  |  |
Purchases | — |  |  | — |  |  | — |  |  | — |  |  | — |  |  | 208 |  |  | (17,345) |
Issuances | — |  |  | — |  |  | — |  |  | — |  |  | — |  |  | — |  |  | — |
Sales (3) | — |  |  | — |  |  | (998) |  |  | (44,143) |  |  | (69,181) |  |  | (46,773) |  |  | — |
Settlements | — |  |  | — |  |  | — |  |  | — |  |  | — |  |  | — |  |  | — |
Transfers in and/or out of Level 3 | — |  |  | — |  |  | — |  |  | — |  |  | — |  |  | — |  |  | — |
Balance at end of period | $ | 3,444 |  |  | $ | 13 |  |  | $ | — |  |  | $ | 29,757 |  |  | $ | 4,535 |  |  | $ | 2,582 |  |  | $ | (17,811) |
Three Months Ended September 30, 2020 |  |  |  |  |  |  |  |  |  |  |  |  |
Balance at beginning of period | $ | 3,450 |  |  | $ | 857 |  |  | $ | 998 |  |  | $ | 46,453 |  |  | $ | 61,447 |  |  | $ | 51,981 |  |  | $ | (1,250) |
Total gains or (losses) (realized/unrealized) |  |  |  |  |  |  |  |  |  |  |  |  |
Included in earnings (2) | (75) |  |  | (5,872) |  |  | (34) |  |  | 885 |  |  | 2,076 |  |  | (946) |  |  | — |
Included in other comprehensive income | 191 |  |  | 6,936 |  |  | — |  |  | — |  |  | — |  |  | — |  |  | — |
Purchases, issuances, sales and settlements |  |  |  |  |  |  |  |  |  |  |  |  |
Purchases | — |  |  | — |  |  | — |  |  | 22,436 |  |  | — |  |  | — |  |  | — |
Issuances | — |  |  | — |  |  | — |  |  | — |  |  | — |  |  | — |  |  | — |
Sales | — |  |  | — |  |  | — |  |  | (3,588) |  |  | — |  |  | (8,349) |  |  | — |
Settlements | (11) |  |  | — |  |  | — |  |  | — |  |  | — |  |  | — |  |  | 620 |
Transfers in and/or out of Level 3 | — |  |  | (1,908) |  |  | — |  |  | — |  |  | — |  |  | — |  |  | — |
Balance at end of period | $ | 3,555 |  |  | $ | 13 |  |  | $ | 964 |  |  | $ | 66,186 |  |  | $ | 63,523 |  |  | $ | 42,686 |  |  | $ | (630) |
Nine Months Ended September 30, 2021 |  |  |  |  |  |  |  |  |  |  |  |  |
Balance at beginning of year | $ | 3,426 |  |  | $ | 13 |  |  | $ | 985 |  |  | $ | 67,103 |  |  | $ | 68,988 |  |  | $ | 42,015 |  |  | $ | (461) |
Total gains or (losses) (realized/unrealized) |  |  |  |  |  |  |  |  |  |  |  |  |
Included in earnings (2) | (46) |  |  | — |  |  | 13 |  |  | 881 |  |  | 4,728 |  |  | 1,837 |  |  | — |
Included in other comprehensive income | 67 |  |  | — |  |  | — |  |  | — |  |  | — |  |  | — |  |  | — |
Purchases, issuances, sales and settlements |  |  |  |  |  |  |  |  |  |  |  |  |
Purchases | — |  |  | — |  |  | — |  |  | 13,003 |  |  | — |  |  | 5,503 |  |  | (17,345) |
Issuances | — |  |  | — |  |  | — |  |  | — |  |  | — |  |  | — |  |  | — |
Sales (3) | — |  |  | — |  |  | (998) |  |  | (51,230) |  |  | (69,181) |  |  | (46,773) |  |  | — |
Settlements | (3) |  |  | — |  |  | — |  |  | — |  |  | — |  |  | — |  |  | (5) |
Transfers in and/or out of Level 3 | — |  |  | — |  |  | — |  |  | — |  |  | — |  |  | — |  |  | — |
Balance at end of period | $ | 3,444 |  |  | $ | 13 |  |  | $ | — |  |  | $ | 29,757 |  |  | $ | 4,535 |  |  | $ | 2,582 |  |  | $ | (17,811) |
Nine Months Ended September 30, 2020 |  |  |  |  |  |  |  |  |  |  |  |  |
Balance at beginning of year | $ | 5,216 |  |  | $ | 8,851 |  |  | $ | 932 |  |  | $ | 68,817 |  |  | $ | 58,094 |  |  | $ | 55,889 |  |  | $ | (7,998) |
Total gains or (losses) (realized/unrealized) |  |  |  |  |  |  |  |  |  |  |  |  |
Included in earnings (2) | (130) |  |  | (5,865) |  |  | (34) |  |  | (129) |  |  | 5,429 |  |  | 7,132 |  |  | (72) |
Included in other comprehensive income | (118) |  |  | 397 |  |  | — |  |  | — |  |  | — |  |  | — |  |  | — |
Purchases, issuances, sales and settlements |  |  |  |  |  |  |  |  |  |  |  |  |
Purchases | — |  |  | — |  |  | 66 |  |  | 22,460 |  |  | — |  |  | 3,464 |  |  | — |
Issuances | — |  |  | — |  |  | — |  |  | — |  |  | — |  |  | — |  |  | — |
Sales | — |  |  | — |  |  | — |  |  | (27,946) |  |  | — |  |  | (23,799) |  |  | — |
Settlements | (1,413) |  |  | (1,462) |  |  | — |  |  | — |  |  | — |  |  | — |  |  | 7,440 |
Transfers in and/or out of Level 3 | — |  |  | (1,908) |  |  | — |  |  | 2,984 |  |  | — |  |  | — |  |  | — |
Balance at end of period | $ | 3,555 |  |  | $ | 13 |  |  | $ | 964 |  |  | $ | 66,186 |  |  | $ | 63,523 |  |  | $ | 42,686 |  |  | $ | (630) |

(1) Includes asset backed securities, mortgage backed securities and commercial mortgage backed securities.
(2) Gains or losses were included in net realized gains (losses).
(3) Sales for the 2021 periods primarily related to the Company’s deconsolidation of Watford.

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ARCH CAPITAL GROUP LTD. AND SUBSIDIARIES
NOTES TO CONSOLIDATED FINANCIAL STATEMENTS (UNAUDITED)
Financial Instruments Disclosed, But Not Carried, At Fair Value
The Company uses various financial instruments in the normal course of its business. The carrying values of cash, accrued investment income, receivable for securities sold, certain other assets, payable for securities purchased and certain other liabilities approximated their fair values at September 30, 2021, due to their respective short maturities. As these financial instruments are not actively traded, their respective fair values are classified within Level 2.
At September 30, 2021, the Company’s senior notes were carried at their cost, net of debt issuance costs, of $2.7 billion and had a fair value of $3.3 billion. At December 31, 2020, the Company’s senior notes were carried at their cost, net of debt issuance costs, of $2.9 billion and had a fair value of $3.7 billion. The fair values of the senior notes were obtained from a third party pricing service and are based on observable market inputs. As such, the fair values of the senior notes are classified within Level 2.
10. Derivative Instruments

The Company’s investment strategy allows for the use of derivative instruments. The Company’s derivative instruments are recorded on its consolidated balance sheets at fair value. The Company utilizes exchange traded U.S. Treasury note, Eurodollar and other futures contracts and commodity futures to manage portfolio duration or replicate investment positions in its portfolios and the Company routinely utilizes foreign currency forward contracts, currency options, index futures contracts and other derivatives as part of its total return objective. In addition, certain of the Company’s investments are managed in portfolios which incorporate the use of foreign currency forward contracts which are intended to provide an economic hedge against foreign currency movements.
In addition, the Company purchases to-be-announced mortgage backed securities (“TBAs”) as part of its investment strategy. TBAs represent commitments to purchase a future issuance of agency mortgage backed securities. For the period between purchase of a TBA and issuance of the underlying security, the Company’s position is accounted for as a derivative. The Company purchases TBAs in both long and short positions to enhance investment performance and as part of its overall investment strategy.
The following table summarizes information on the fair values and notional values of the Company’s derivative instruments:

| Estimated Fair Value |  |
| Asset Derivatives |  | Liability Derivatives |  | Notional Value (1)
September 30, 2021 |  |  |  |  |
Futures contracts (2) | $ | 36,010 |  |  | $ | (17,562) |  |  | $ | 2,796,442 |
Foreign currency forward contracts (2) | 7,380 |  |  | (13,914) |  |  | 1,234,665 |
TBAs (3) | 49,227 |  |  | — |  |  | 47,603 |
Other (2) | 46,568 |  |  | (12,050) |  |  | 4,184,225 |
Total | $ | 139,185 |  |  | $ | (43,526) |  |  |
December 31, 2020 |  |  |  |  |
Futures contracts (2) | $ | 11,046 |  |  | $ | (4,496) |  |  | $ | 3,099,796 |
Foreign currency forward contracts (2) | 52,716 |  |  | (6,202) |  |  | 1,656,729 |
TBAs (3) | — |  |  | — |  |  | — |
Other (2) | 113,621 |  |  | (98,007) |  |  | 5,763,919 |
Total | $ | 177,383 |  |  | $ | (108,705) |  |  |

(1) Represents the absolute notional value of all outstanding contracts, consisting of long and short positions.
(2) The fair value of asset derivatives are included in ‘other assets’ and the fair value of liability derivatives are included in ‘other liabilities.’
(3) The fair value of TBAs are included in ‘fixed maturities available for
sale, at fair value.’
The Company did not hold any derivatives which were designated as hedging instruments at September 30, 2021 or December 31, 2020.
The Company’s derivative instruments can be traded under master netting agreements, which establish terms that apply to all derivative transactions with a counterparty. In the event of a bankruptcy or other stipulated event of default, such agreements provide that the non-defaulting party may elect to terminate all outstanding derivative transactions, in which case all individual derivative positions (loss or gain) with a counterparty are closed out and netted and replaced with a single amount, usually referred to as the termination amount, which is expressed in a single currency. The resulting single net amount, where positive, is payable to the party “in-the-money” regardless of whether or not it is the defaulting party, unless the parties have agreed that only the non-defaulting party is entitled to receive a termination payment where the net amount is positive and is in its favor. Contractual close-out netting reduces derivatives credit exposure from gross to net exposure.
At September 30, 2021, asset derivatives and liability derivatives of $133.1 million and $42.2 million, respectively, were subject to a master netting agreement, compared to $138.8 million and $93.0 million, respectively, at December 31, 2020. The remaining derivatives included in the preceding table were not subject to a master netting agreement.

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ARCH CAPITAL GROUP LTD. AND SUBSIDIARIES
NOTES TO CONSOLIDATED FINANCIAL STATEMENTS (UNAUDITED)
Realized and unrealized contract gains and losses on the Company’s derivative instruments are reflected in ‘net realized gains (losses)’ in the consolidated statements of income, as summarized in the following table:

Derivatives not designated as |  | September 30,
hedging instruments: |  | 2021 |  | 2020
Three Months Ended |  |  |  |
Net realized gains (losses): |  |  |  |
Futures contracts |  | $ | (10,073) |  |  | $ | 10,945 |
Foreign currency forward contracts |  | (16,146) |  |  | 10,813 |
TBAs |  | (46) |  |  | 120 |
Other (1) |  | 4,830 |  |  | (1,509) |
Total |  | $ | (21,435) |  |  | $ | 20,369 |
Nine Months Ended |  |  |  |
Net realized gains (losses): |  |  |  |
Futures contracts |  | $ | (17,394) |  |  | $ | 105,282 |
Foreign currency forward contracts |  | (36,922) |  |  | 3,466 |
TBAs |  | (46) |  |  | 1,129 |
Other (1) |  | 17,934 |  |  | 36,845 |
Total |  | $ | (36,428) |  |  | $ | 146,722 |

(1) Includes realized gains and losses on swaps, options and other derivatives contracts.
11. Commitments and Contingencies

Investment Commitments
The Company’s investment commitments, which are primarily related to agreements entered into by the Company to invest in funds and separately managed accounts when called upon, were approximately $2.6 billion at September 30, 2021, compared to $2.1 billion at December 31, 2020.
Interest Paid
Interest paid on the Company’s senior notes and other borrowings were $75.8 million for the nine months ended September 30, 2021, compared to $60.6 million for the 2020 period.
12. Variable Interest Entities and Noncontrolling Interests

Watford
In March 2014, the Company invested $100.0 million and acquired approximately 11% of Watford’s outstanding common equity. Watford was considered a VIE and the Company concluded that it was the primary beneficiary of Watford, through June 30, 2021. As such, the results of Watford were included in the Company’s consolidated
financial statements as of and for the periods ended June 30, 2021.
In the 2020 fourth quarter, Arch Capital, Watford and Greysbridge, a wholly-owned subsidiary of Arch Capital, entered into an Agreement and Plan of Merger (as amended, the “Merger Agreement”). The merger and the related Greysbridge equity financing closed on July 1, 2021. Effective July 1, 2021, Watford is wholly owned by Greysbridge, and Greysbridge is owned 40% by the Company, 30% by certain investment funds managed by Kelso and 30% by certain investment funds managed by Warburg. Based on the governing documents of Greysbridge, the Company concluded that, while it retains significant influence over Watford, Watford no longer constitutes a variable interest entity. Accordingly, effective July 1, 2021, the Company no longer consolidates the results of Watford in its consolidated financial statements and footnotes. Beginning in the 2021 third quarter, the Company classifies its investment as ‘investments in operating affiliates’ on the Company’s balance sheets and is accounted for under the equity method.
The following table provides the carrying amount and balance sheet caption in which the assets and liabilities of Watford were reported at December 31, 2020:

|  |  | December 31,
|  |  | 2020
Assets |  |  |
Investments accounted for using the fair value option (1) |  |  | $ | 1,790,385 |
Fixed maturities available for sale, at fair value |  |  | 655,249 |
Equity securities, at fair value |  |  | 52,410 |
Cash |  |  | 211,451 |
Accrued investment income |  |  | 14,679 |
Premiums receivable |  |  | 224,377 |
Reinsurance recoverable on unpaid and paid losses and loss adjustment expenses |  |  | 286,590 |
Ceded unearned premiums |  |  | 122,339 |
Deferred acquisition costs |  |  | 53,705 |
Receivable for securities sold |  |  | 37,423 |
Goodwill and intangible assets |  |  | 7,650 |
Other assets |  |  | 75,801 |
Total assets of consolidated VIE |  |  | $ | 3,532,059 |
Liabilities |  |  |
Reserve for losses and loss adjustment expenses |  |  | $ | 1,519,583 |
Unearned premiums |  |  | 407,714 |
Reinsurance balances payable |  |  | 63,269 |
Revolving credit agreement borrowings |  |  | 155,687 |
Senior notes |  |  | 172,689 |
Payable for securities purchased |  |  | 25,881 |
Other liabilities |  |  | 193,494 |
Total liabilities of consolidated VIE |  |  | $ | 2,538,317 |
Redeemable noncontrolling interests |  |  | $ | 52,398 |

(1) Included in “other investments” on the Company’s balance sheet.

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ARCH CAPITAL GROUP LTD. AND SUBSIDIARIES
NOTES TO CONSOLIDATED FINANCIAL STATEMENTS (UNAUDITED)
Through June 30, 2021, Watford generated $47.0 million of cash provided by operating activities, $96.3 million of cash provided by investing activities and $2.0 million of cash used for financing activities, compared to $133.6 million of cash provided by operating activities, $242.0 million of cash provided by investing activities and $279.7 million of cash used for financing activities for the nine months ended September 30, 2020.
Non-redeemable noncontrolling interests
Through June 30, 2021, the Company accounted for the portion of Watford’s common equity attributable to third party investors in the shareholders’ equity section of its consolidated balance sheets. The portion of Watford’s income or loss attributable to third party investors was recorded in the consolidated statements of income in ‘net (income) loss attributable to noncontrolling interests.’
The following table sets forth activity in the non-redeemable noncontrolling interests:

| September 30,
| 2021 |  | 2020
Three Months Ended |  |  |
Balance, beginning of period | $ | 918,874 |  |  | $ | 679,089 |
Impact of deconsolidation of Watford | (918,874) |  |  | — |
Additional paid in capital attributable to noncontrolling interests | — |  |  | 243 |
Amounts attributable to noncontrolling interests | — |  |  | 67,768 |
Other comprehensive income (loss) attributable to noncontrolling interests | — |  |  | 10,820 |
Balance, end of period | $ | — |  |  | $ | 757,920 |
Nine Months Ended |  |  |
Balance, beginning of year | $ | 823,007 |  |  | $ | 762,777 |
Impact of deconsolidation of Watford | (918,874) |  |  |
Additional paid in capital attributable to noncontrolling interests | 22,113 |  |  | 715 |
Repurchases attributable to non-redeemable noncontrolling interests (1) | — |  |  | (2,867) |
Amounts attributable to noncontrolling interests | 78,314 |  |  | (578) |
Other comprehensive income (loss) attributable to noncontrolling interests | (4,560) |  |  | (2,127) |
Balance, end of period | $ | — |  |  | $ | 757,920 |

(1) During 2020, Watford’s board of directors authorized the investment in Watford’s common shares through a share repurchase program.
Redeemable noncontrolling interests
Through June 30, 2021, the Company accounted for redeemable noncontrolling interests in the mezzanine section of its consolidated balance sheets in accordance with applicable accounting guidance. Such redeemable noncontrolling interests primarily related to the Watford Preference Shares issued in late March 2014 with a par value of $0.01 per share and a liquidation preference of $25.00 per
share. The Watford Preference Shares were issued at a discounted amount of $24.50 per share. Through June 30, 2021 preferred dividends, including the accretion of the discount and issuance costs, were included in ‘net (income) loss attributable to noncontrolling interests’ in the Company’s consolidated statements of income.
The following table sets forth activity in the redeemable non-controlling interests:

| September 30,
| 2021 |  | 2020
Three Months Ended |  |  |
Balance, beginning of period | $ | 57,533 |  |  | $ | 55,986 |
Impact of deconsolidation of Watford | (48,919) |  |  | — |
Accretion of preference share issuance costs | — |  |  | 23 |
Other | 1,623 |  |  | 1,826 |
Balance, end of period | $ | 10,237 |  |  | $ | 57,835 |
Nine Months Ended |  |  |
Balance, beginning of year | $ | 58,548 |  |  | $ | 55,404 |
Impact of deconsolidation of Watford | (48,919) |  |  | — |
Accretion of preference share issuance costs | — |  |  | 70 |
Other | 608 |  |  | 2,361 |
Balance, end of period | $ | 10,237 |  |  | $ | 57,835 |

The portion of income or loss attributable to third party investors, recorded in the Company’s consolidated statements of income in ‘net (income) loss attributable to noncontrolling interests,’ are summarized in the table below:

| September 30,
| 2021 |  | 2020
Three Months Ended |  |  |
Amounts attributable to non-redeemable noncontrolling interests | $ | — |  |  | $ | (67,768) |
Amounts attributable to redeemable noncontrolling interests | (1,473) |  |  | (1,875) |
Net (income) loss attributable to noncontrolling interests | $ | (1,473) |  |  | $ | (69,643) |
Nine Months Ended |  |  |
Amounts attributable to non-redeemable noncontrolling interests | $ | (78,314) |  |  | $ | 578 |
Amounts attributable to redeemable noncontrolling interests | (3,889) |  |  | (4,998) |
Net (income) loss attributable to noncontrolling interests | $ | (82,203) |  |  | $ | (4,420) |

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ARCH CAPITAL GROUP LTD. AND SUBSIDIARIES
NOTES TO CONSOLIDATED FINANCIAL STATEMENTS (UNAUDITED)
Bellemeade Re
The Company has entered into aggregate excess of loss mortgage reinsurance agreements with various special purpose reinsurance companies domiciled in Bermuda (the “Bellemeade Agreements”). At the time the Bellemeade Agreements were entered into, the applicability of the accounting guidance that addresses VIEs was evaluated. As a result of the evaluation of the Bellemeade Agreements, the Company concluded that these entities are VIEs. However, given that the ceding insurers do not have the unilateral power to direct those activities that are significant to their economic performance, the Company does not consolidate such entities in its consolidated financial statements.
The following table presents the total assets of the Bellemeade entities, as well as the Company’s maximum exposure to loss associated with these VIEs, calculated as the maximum historical observable spread between the benchmark index for each respective transaction and short term invested trust asset yields. The benchmark index for agreements effective prior to 2021 is based on one-month LIBOR, while the 2021 agreements benchmark index is based on the Secured Overnight Financing Rate (“SOFR”). SOFR is a measure of the cost of borrowing cash overnight, collateralized by U.S. Treasury securities, and is based on directly observable U.S. Treasury-backed repurchase transactions.

|  |  | September 30, 2021 |  |  |  | December 31, 2020
|  |  | Maximum Exposure to Loss |  |  |  | Maximum Exposure to Loss
Bellemeade Entities (Issue Date) | Total VIE Assets |  | On-Balance Sheet (Asset) Liability |  | Off-Balance Sheet |  | Total |  | Total VIE Assets |  | On-Balance Sheet (Asset) Liability |  | Off-Balance Sheet |  | Total
Bellemeade 2017-1 Ltd. (Oct-17) | $ | 145,573 |  |  | $ | (214) |  |  | $ | 585 |  |  | $ | 371 |  |  | $ | 145,573 |  |  | $ | (245) |  |  | $ | 844 |  |  | $ | 599 |
Bellemeade 2018-1 Ltd. (Apr-18) | 228,938 |  |  | (764) |  |  | 1,683 |  |  | 919 |  |  | 250,095 |  |  | (903) |  |  | 2,245 |  |  | 1,342 |
Bellemeade 2018-2 Ltd. (Aug-18) | — |  |  | — |  |  | — |  |  | — |  |  | 108,395 |  |  | (138) |  |  | 280 |  |  | 142 |
Bellemeade 2018-3 Ltd. (Oct-18) | 302,563 |  |  | (1,049) |  |  | 2,328 |  |  | 1,279 |  |  | 302,563 |  |  | (1,320) |  |  | 3,262 |  |  | 1,942 |
Bellemeade 2019-1 Ltd. (Mar-19) | 210,529 |  |  | (931) |  |  | 8,142 |  |  | 7,211 |  |  | 219,256 |  |  | (1,361) |  |  | 8,461 |  |  | 7,100 |
Bellemeade 2019-2 Ltd. (Apr-19) | 398,316 |  |  | (787) |  |  | 5,658 |  |  | 4,871 |  |  | 398,316 |  |  | (730) |  |  | 5,201 |  |  | 4,471 |
Bellemeade 2019-3 Ltd. (Jul-19) | 491,634 |  |  | (826) |  |  | 3,971 |  |  | 3,145 |  |  | 528,084 |  |  | (861) |  |  | 5,079 |  |  | 4,218 |
Bellemeade 2019-4 Ltd. (Oct-19) | 468,737 |  |  | (682) |  |  | 4,761 |  |  | 4,079 |  |  | 468,737 |  |  | (890) |  |  | 6,676 |  |  | 5,786 |
Bellemeade 2020-1 Ltd. (Jun-20) (1) | — |  |  | — |  |  | — |  |  | — |  |  | 275,068 |  |  | (178) |  |  | 1,012 |  |  | 834 |
Bellemeade 2020-2 Ltd. (Sep-20) (2) | 266,704 |  |  | (279) |  |  | 2,629 |  |  | 2,350 |  |  | 423,420 |  |  | (556) |  |  | 6,839 |  |  | 6,283 |
Bellemeade 2020-3 Ltd. (Nov-20) (3) | 381,410 |  |  | (395) |  |  | 6,646 |  |  | 6,251 |  |  | 418,158 |  |  | (631) |  |  | 9,605 |  |  | 8,974 |
Bellemeade 2020-4 Ltd. (Dec-20) (4) | 226,916 |  |  | (100) |  |  | 2,190 |  |  | 2,090 |  |  | 321,393 |  |  | (156) |  |  | 6,816 |  |  | 6,660 |
Bellemeade 2021-1 Ltd. (Mar-21) (5) | 579,717 |  |  | 229 |  |  | 4,217 |  |  | 4,446 |  |  | — |  |  | — |  |  | — |  |  | — |
Bellemeade 2021-2 Ltd. (Jun-21) (6) | 522,807 |  |  | 906 |  |  | 5,090 |  |  | 5,996 |  |  | — |  |  | — |  |  | — |  |  | — |
Bellemeade 2021-3 Ltd. (Sep-21) (7) | 507,873 |  |  | 182 |  |  | 4,561 |  |  | 4,743 |  |  | — |  |  | — |  |  | — |  |  | — |
Total | $ | 4,731,717 |  |  | $ | (4,710) |  |  | $ | 52,461 |  |  | $ | 47,751 |  |  | $ | 3,859,058 |  |  | $ | (7,969) |  |  | $ | 56,320 |  |  | $ | 48,351 |

(1) An additional $79 million capacity was provided directly to Arch MI U.S. by a separate panel of reinsurers and is not reflected in this table.
(2) An additional $26 million capacity was provided directly to Arch MI U.S. by a separate panel of reinsurers and is not reflected in this table.
(3) An additional $34 million capacity was provided directly to Arch MI U.S. by a separate panel of reinsurers and is not reflected in this table.
(4) An additional $16 million capacity was provided directly to Arch MI U.S. by a separate panel of reinsurers and is not reflected in this table.
(5) An additional $64 million capacity was provided directly to Arch MI U.S. by a separate panel of reinsurers and is not reflected in this table.
(6) An additional $93 million capacity was provided directly to Arch MI U.S. by a separate panel of reinsurers and is not reflected in this table.
(7) An additional $131 million capacity was provided directly to Arch MI U.S. by a separate panel of reinsurers and is not reflected in this table.

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ARCH CAPITAL GROUP LTD. AND SUBSIDIARIES
NOTES TO CONSOLIDATED FINANCIAL STATEMENTS (UNAUDITED)
13. Other Comprehensive Income (Loss)

The following tables present details about amounts reclassified from accumulated other comprehensive income and the tax effects allocated to each component of other comprehensive income (loss):

|  |  |  | Amounts Reclassified from AOCI
|  | Consolidated Statement of Income |  | Three Months Ended |  | Nine Months Ended
Details About |  | Line Item That Includes |  | September 30, |  | September 30,
AOCI Components |  | Reclassification |  | 2021 |  | 2020 |  | 2021 |  | 2020
Unrealized appreciation on available-for-sale investments |  |  |  |  |  |  |  |
|  | Net realized gains (losses) |  | $ | 68,373 |  |  | $ | 87,871 |  |  | $ | 135,291 |  |  | $ | 416,432 |
|  | Provision for credit losses |  | (457) |  |  | 1,333 |  |  | (1,208) |  |  | (4,762) |
|  | Other-than-temporary impairment losses |  | — |  |  | — |  |  | — |  |  | (533) |
|  | Total before tax |  | 67,916 |  |  | 89,204 |  |  | 134,083 |  |  | 411,137 |
|  | Income tax (expense) benefit |  | (5,262) |  |  | (9,401) |  |  | (13,579) |  |  | (42,714) |
|  | Net of tax |  | $ | 62,654 |  |  | $ | 79,803 |  |  | $ | 120,504 |  |  | $ | 368,423 |

| Before Tax Amount |  | Tax Expense (Benefit) |  | Net of Tax Amount
Three Months Ended September 30, 2021 |  |  |  |  |
Unrealized appreciation (decline) in value of investments: |  |  |  |  |
Unrealized holding gains (losses) arising during period | $ | (104,607) |  |  | $ | (8,684) |  |  | $ | (95,923) |
Less reclassification of net realized gains (losses) included in net income | 67,916 |  |  | 5,262 |  |  | 62,654 |
Foreign currency translation adjustments | (32,060) |  |  | (350) |  |  | (31,710) |
Other comprehensive income (loss) | $ | (204,583) |  |  | $ | (14,296) |  |  | $ | (190,287) |
Three Months Ended September 30, 2020 |  |  |  |  |
Unrealized appreciation (decline) in value of investments: |  |  |  |  |
Unrealized holding gains (losses) arising during period | $ | 119,265 |  |  | $ | 8,483 |  |  | $ | 110,782 |
Less reclassification of net realized gains (losses) included in net income | 89,204 |  |  | 9,401 |  |  | 79,803 |
Foreign currency translation adjustments | 16,918 |  |  | 209 |  |  | 16,709 |
Other comprehensive income (loss) | $ | 46,979 |  |  | $ | (709) |  |  | $ | 47,688 |
Nine Months Ended September 30, 2021 |  |  |  |  |
Unrealized appreciation (decline) in value of investments: |  |  |  |  |
Unrealized holding gains (losses) arising during period | $ | (307,910) |  |  | $ | (28,808) |  |  | $ | (279,102) |
Less reclassification of net realized gains (losses) included in net income | 134,083 |  |  | 13,579 |  |  | 120,504 |
Foreign currency translation adjustments | (54,083) |  |  | 6 |  |  | (54,089) |
Other comprehensive income (loss) | $ | (496,076) |  |  | $ | (42,381) |  |  | $ | (453,695) |
Nine Months Ended September 30, 2020 |  |  |  |  |
Unrealized appreciation (decline) in value of investments: |  |  |  |  |
Unrealized holding gains (losses) arising during period | $ | 611,390 |  |  | $ | 65,099 |  |  | $ | 546,291 |
Less reclassification of net realized gains (losses) included in net income | 411,137 |  |  | 42,714 |  |  | 368,423 |
Foreign currency translation adjustments | (5,911) |  |  | (182) |  |  | (5,729) |
Other comprehensive income (loss) | $ | 194,342 |  |  | $ | 22,203 |  |  | $ | 172,139 |

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ARCH CAPITAL GROUP LTD. AND SUBSIDIARIES
NOTES TO CONSOLIDATED FINANCIAL STATEMENTS (UNAUDITED)
14. Income Taxes

The Company’s income tax provision on income before income taxes, including income (loss) from operating affiliates, resulted in an effective tax rate of 5.5% for the nine months ended September 30, 2021, compared to 8.2% for the nine months ended September 30, 2020. The effective tax rate for the 2021 period included discrete income tax benefits of $28.7 million which had the effect of decreasing the effective tax rate on net income available to Arch common shareholders by 1.7%. The discrete tax items in the 2021 period primarily related to the partial release of a valuation allowance on certain U.K. deferred tax assets.
The Company’s effective tax rate, which is based upon the expected annual effective tax rate, may fluctuate from period to period based on the relative mix of income or loss reported by jurisdiction and the varying tax rates in each jurisdiction.
The Company had a net deferred tax asset of $162.5 million at September 30, 2021, compared to a net deferred tax asset of $15.7 million at December 31, 2020. The change is primarily a result of mortgage contingency reserves activity, fixed asset capitalization and market value fluctuations in the investment portfolio. In addition, the Company paid $202.4 million and $146.8 million of income taxes for the nine months ended September 30, 2021 and 2020, respectively.
15. Legal Proceedings

The Company, in common with the insurance industry in general, is subject to litigation and arbitration in the normal course of its business. As of September 30, 2021, the Company was not a party to any litigation or arbitration which is expected by management to have a material adverse effect on the Company’s results of operations and financial condition and liquidity.
16. Transactions with Related Parties

In the 2021 first quarter, as part of the Company’s acquisition of Barbican, the Company entered into an agreement with Premia Managing Agency Limited for the reinsurance to close of Syndicate 1955’s 2018 underwriting year of account into Premia Syndicate 1884’s 2021 underwriting year of account. The reinsurance to close covers legacy business underwritten by Syndicate 1955 on the underwriting 2018 and prior years of account and under the agreement, approximately $380 million of net liabilities was transferred to Syndicate 1884, with an effective date of January 1, 2021. Barbican recorded reinsurance recoverable on unpaid and paid losses and funds held liability of nil and $8.8 million, respectively, at September 30, 2021, compared to
$199.8 million and $149.6 million, respectively, at December 31, 2020.
In July 2021, following consummation of the Merger Agreement and the related Greysbridge equity financing, pursuant to which Watford is wholly owned by Greysbridge, and Greysbridge is owned 40% by the Company, 30% by certain funds managed by Kelso and 30% by certain funds managed by Warburg, the Company entered into certain reinsurance transactions with Watford. For the three months ended September 30, 2021, the Company ceded premiums written related to such transactions of $316.2 million (which includes reinsurance transactions in force as well as those entered into in conjunction with the Merger Agreement). In addition, Watford paid certain acquisition costs and administrative fees to the Company. At September 30, 2021, the Company recorded a reinsurance recoverable on unpaid and paid losses from Watford of $874.9 million and a reinsurance balance payable to Watford of $281.2 million. See note 12, “Variable Interest Entities and Noncontrolling Interests,” for information about Watford.
The Company has a put/call option that was entered into in connection with the Greysbridge equity financing, whereby beginning January 1, 2024 the Company will have a call right (but not the obligation) and Warburg and Kelso will each have a put right (but not the obligation) to buy/sell one third of their initial shares annually at the tangible book value per share of Greysbridge for the most recently ended fiscal quarter.
As of September 30, 2021, the Company owns $35.0 million in aggregate principal amount of Watford Holdings Ltd’s 6.5% senior notes, due July 2, 2029 and approximately 6.6% of Watford’s preference shares.
17. Subsequent Event

Share Repurchases
In October 2021, the Company announced that its Board of Directors has increased its share repurchase program to an aggregate of up to $1.5 billion, which may be effected from time to time in open market or privately negotiated transactions through December 31, 2022. The timing and amount of the repurchase transactions under this program will depend on a variety of factors, including market conditions and corporate and regulatory considerations.
From October 1 to October 13, 2021, the Company repurchased approximately 1.2 million common shares for an aggregate purchase price of $45.5 million. At October 27, 2021 approximately $1.5 billion of repurchases were available under the share repurchase program.

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ITEM 2. MANAGEMENT’S DISCUSSION AND ANALYSIS OF FINANCIAL CONDITION AND RESULTS OF OPERATIONS

The following is a discussion and analysis of our financial condition and results of operations. This should be read in conjunction with our consolidated financial statements included in Item 1 of this report and also our Management’s Discussion and Analysis of Financial Condition and Results of Operations contained in our Annual Report on Form 10-K for the year ended December 31, 2020 (“2020 Form 10-K”). In addition, readers should review “Risk Factors” set forth in Item 1A of Part I of our 2020 Form 10-K and “ITEM 1A—Risk Factors” of this Form 10-Q. Tabular amounts are in U.S. Dollars in thousands, except share amounts, unless otherwise noted.
Arch Capital Group Ltd. (“Arch Capital” and, together with its subsidiaries, “Arch”, “we” or “us”) is a publicly listed Bermuda exempted company with approximately $16.1 billion in capital at September 30, 2021 and, through operations in Bermuda, the United States, Europe, Canada, Australia and Hong Kong, writes insurance, reinsurance and mortgage insurance on a worldwide basis.
CURRENT OUTLOOK

As we approach the end of 2021, our three areas of focus during the year have remained constant. In our property and casualty segments of insurance and reinsurance, we continue our growth in the sectors where rates allow for returns that are substantially higher than our cost of capital. Our mortgage insurance segment has transitioned, for the most part, from forbearance to recovery and is producing results that made a significant contribution to our underwriting income in the third quarter. We have also been keenly focused on actively managing our investments and capital to enhance our returns over the long run.
The 2021 third quarter reflected the benefits of attractive pricing in almost all of our insurance markets. As a result, we currently expect the next several quarters to continue to show improved underwriting margins, partially due to the compounding of rate-on-rate increases and the rebalancing of our mix of business. We believe that this time-tested strategy of protecting capital through soft markets and increasing our writings in hard markets gives us the best chance to generate superior risk adjusted returns over time. As long as rate increases support returns above our required thresholds, we expect to continue to grow our writings.
The trajectory and market acceptance of rate increases reinforce why we remain optimistic that improved economics in the property casualty market will be sustainable for some time. The property casualty industry is facing many degrees
of uncertainty, including heightened catastrophe activity, rising inflation, COVID’s ongoing influence on the global economy and perennially low interest rates.
Rate improvements have enabled us to continue to expand writings in our property casualty segments as we have been for two years now. Rate increases remain well above the long-term loss cost trends and have spread to more lines than last year. Overall, 2021 rates are up around 10% compared to 2020 and we currently expect that the benefit of higher premium levels will be reflected well into 2022 and beyond. Positive rate increases have accelerated in lower limit accounts which, until now, had lagged the increases in larger accounts. Our early focus on Lloyd’s and business in the UK has improved our scale and our economics in this market. Some of our business lines that were most impacted by COVID, like travel, are recapturing some of the lost volume as both business and consumer travel increases.
In reinsurance, strong growth was observed across most of our lines of business, but especially in our casualty and other specialty lines where strong rates increases and growth in new accounts helped increase the top line. At less than 6% of our tangible equity, we remained underweight in property catastrophe exposure and we will deploy more capital to the line if expected returns improve meaningfully above our target. Consistent with our insurance segment, we expect the ongoing rate improvements to be reflected in our underwriting results over the next several quarters.
For our U.S. primary mortgage operations, delinquencies continue to be better than our expectations at the beginning of the COVID-19 pandemic, as notices of default have declined to pre-pandemic levels at September 30, 2021, which is another indicator of improved conditions. Additionally, loans in forbearance continue to decline as federal programs conclude and we remain optimistic that most of these loans will ultimately cure.
In the 2021 third quarter, insurance in force for our U.S. primary mortgage operations remained steady at $280.4 billion, while insurance in force for the total mortgage segment was $457.7 billion. Overall, the market remains competitive but rational and our mortgage business continues to generate returns on capital in the mid teens. Mortgage originations continue at a pace similar to last year’s record origination volume and credit quality remains excellent. Outside of the U.S., we increased our writings in Australia as a result of the housing market remaining strong and due to our acquisition of Westpac’s LMI business.

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We remain committed to providing solutions across many offerings as the marketplace evolves, including the mortgage credit risk transfer programs initiated by government sponsored enterprises, or “GSEs.” In addition, we enter into aggregate excess of loss mortgage reinsurance agreements with various special purpose reinsurance companies domiciled in Bermuda and issue mortgage insurance linked notes, increasing our protection for mortgage tail risk. The Bellemeade structures provide approximately $5.1 billion of aggregate reinsurance coverage at September 30, 2021.
FINANCIAL MEASURES

Management uses the following three key financial indicators in evaluating our performance and measuring the overall growth in value generated for Arch Capital’s common shareholders:
Book Value per Share
Book value per share represents total common shareholders’ equity available to Arch divided by the number of common shares outstanding. Management uses growth in book value per share as a key measure of the value generated for our common shareholders each period and believes that book value per share is the key driver of Arch Capital’s share price over time. Book value per share is impacted by, among other factors, our underwriting results, investment returns and share repurchase activity, which has an accretive or dilutive impact on book value per share depending on the purchase price.
Book value per share was $32.43 at September 30, 2021, compared to $32.02 at June 30, 2021 and $28.75 at September 30, 2020. The 1.3% increase in book value per share for the 2021 third quarter reflected strong underwriting returns, which were impacted by a high level of losses from catastrophic events, along with flat total return on investments for the period. The 12.8% increase in book value per share over the trailing twelve months primarily reflected strong underwriting results.
Operating Return on Average Common Equity
Operating return on average common equity (“Operating ROAE”) represents annualized after-tax operating income available to Arch common shareholders divided by the average of beginning and ending common shareholders’ equity available to Arch during the period. After-tax operating income available to Arch common shareholders, a non-GAAP financial measure as defined in Regulation G, represents net income available to Arch common shareholders, excluding net realized gains or losses (which includes changes in the allowance for credit losses on financial assets and net impairment losses recognized in earnings) equity in net income or loss of investment funds
accounted for using the equity method, net foreign exchange gains or losses, transaction costs and other, loss on redemption of preferred shares and income taxes. Management uses Operating ROAE as a key measure of the return generated to common shareholders. See “Comment on Non-GAAP Financial Measures.”
Our Operating ROAE was 9.3% for the 2021 third quarter, compared to 4.2% for the 2020 third quarter, and 10.1% for the nine months ended September 30, 2021, compared to 3.9% for the 2020 period. Results for the 2021 third quarter reflected a one-time gain of $95.7 million recognized from the Company’s previously disclosed acquisition of a 40% share of Greysbridge. Returns for the 2021 periods reflected strong underwriting returns and income from operating affiliates, while the 2020 period reflected the impact of COVID-19 on underwriting results.
Total Return on Investments
Total return on investments includes investment income, equity in net income or loss of investment funds accounted for using the equity method, net realized gains and losses (excluding changes in the allowance for credit losses on non-investment related financial assets) and the change in unrealized gains and losses generated by Arch’s investment portfolio. Total return is calculated on a pre-tax basis and before investment expenses and reflects the effect of financial market conditions along with foreign currency fluctuations. In addition, total return incorporates the timing of investment returns during the periods. The following table summarizes our total return compared to the benchmark return against which we measured our portfolio during the periods. See “Comment on Non-GAAP Financial Measures.”

| Arch Portfolio |  | Benchmark Return
Pre-tax total return (before investment expenses):
2021 Third Quarter | 0.01 | % |  | (0.40) | %
2020 Third Quarter | 2.30 | % |  | 2.41 | %
Nine Months Ended September 30, 2021 | 1.50 | % |  | 0.87 | %
Nine Months Ended September 30, 2020 | 5.19 | % |  | 3.67 | %

Total return for the 2021 third quarter reflected movements in interest rates and credit spreads on our fixed income portfolio. We continue to maintain a short duration on our portfolio of 2.68 years at September 30, 2021.
The benchmark return index is a customized combination of indices intended to approximate a target portfolio by asset mix and average credit quality while also matching the approximate estimated duration and currency mix of our insurance and reinsurance liabilities. Although the estimated duration and average credit quality of this index will move as the duration and rating of its constituent securities change,

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generally we do not adjust the composition of the benchmark return index except to incorporate changes to the mix of liability currencies and durations noted above. The benchmark return index should not be interpreted as expressing a preference for or aversion to any particular sector or sector weight. The index is intended solely to provide, unlike many master indices that change based on the size of their constituent indices, a relatively stable basket of investable indices. At September 30, 2021, the benchmark return index had an average credit quality of “Aa2” by Moody’s Investors Service (“Moody’s”), and an estimated duration of 3.18 years.
The benchmark return index included weightings to the following indices:

| %
ICE BoAML 1-5 Year A - AAA U.S. Corporate Index | 13.00 | %
ICE BoAML 5-10 Year A - AAA U.S. Corporate Index | 11.00 |
ICE BoAML 1-5 Year U.S. Treasury Index | 11.00 |
MSCI ACWI Net Total Return USD Index | 9.30 |
ICE BoAML 1-10 Year BBB U.S. Corporate Index | 5.00 |
JPM CLOIE Investment Grade | 5.00 |
S&P/LSTA Leveraged Loan Total Return Index | 4.965 |
ICE BoAML U.S. Mortgage Backed Securities Index | 4.00 |
ICE BoAML AAA US Fixed Rate CMBS | 4.00 |
ICE BoAML 1-5 Year U.K. Gilt Index | 4.00 |
ICE BoAML German Government 1-10 Year Index | 3.50 |
ICE BoAML 0-3 Year U.S. Treasury Index | 3.25 |
ICE BoAML 5-10 Year U.S. Treasury Index | 3.00 |
ICE BoAML 1-10 Year U.S. Municipal Securities Index | 3.00 |
Bloomberg Barclays ABS Aaa Index | 3.00 |
ICE BoAML 1-5 Year Australia Government Index | 2.75 |
ICE BoAML U.S. High Yield Constrained Index | 2.50 |
ICE BoAML 1-5 Year Canada Government Index | 2.00 |
ICE BofA CCC and Lower US High Yield Constrained Index | 1.38 |
Bloomberg Barclays Global High Yield Index | 1.38 |
S&P DJ Global ex-US Select Real Estate Securities Net Index | 0.825 |
FTSE Nareit All Mortgage Capped Index Total Return USD | 0.825 |
Bloomberg Barclays CMBS: Erisa Eligible Unhedged USD | 0.825 |
ICE BoAML 15+ Year Canada Government Index | 0.50 |
Total | 100.00 | %

COMMENT ON NON-GAAP FINANCIAL MEASURES

Throughout this filing, we present our operations in the way we believe will be the most meaningful and useful to investors, analysts, rating agencies and others who use our financial information in evaluating the performance of our company. This presentation includes the use of after-tax operating income available to Arch common shareholders, which is defined as net income available to Arch common shareholders, excluding net realized gains or losses (which includes changes in the allowance for credit losses on financial assets and net impairment losses recognized in earnings), equity in net income or loss of investment funds accounted for using the equity method, net foreign exchange gains or losses, transaction costs and other, loss on redemption of preferred shares and income taxes, and the use of annualized operating return on average common equity. The presentation of after-tax operating income available to Arch common shareholders and annualized operating return on average common equity are non-GAAP financial measures as defined in Regulation G. The reconciliation of such measures to net income available to Arch common shareholders and annualized net income return on average common equity (the most directly comparable GAAP financial measures) in accordance with Regulation G is included under “Results of Operations” below.
We believe that net realized gains or losses, equity in net income or loss of investment funds accounted for using the equity method, net foreign exchange gains or losses, transaction costs and other and loss on redemption of preferred shares in any particular period are not indicative of the performance of, or trends in, our business. Although net realized gains or losses, equity in net income or loss of investment funds accounted for using the equity method and net foreign exchange gains or losses are an integral part of our operations, the decision to realize investment gains or losses, the recognition of the change in the carrying value of investments accounted for using the fair value option in net realized gains or losses, the recognition of net impairment losses, the recognition of equity in net income or loss of investment funds accounted for using the equity method and the recognition of foreign exchange gains or losses are independent of the insurance underwriting process and result, in large part, from general economic and financial market conditions. Furthermore, certain users of our financial information believe that, for many companies, the timing of the realization of investment gains or losses is largely opportunistic. In addition, changes in the allowance for credit losses and net impairment losses recognized in earnings on the Company’s investments represent other-than-temporary declines in expected recovery values on securities without actual realization. The use of the equity method on certain of our investments in certain funds that invest in fixed maturity securities is driven by the ownership structure of such funds

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(either limited partnerships or limited liability companies). In applying the equity method, these investments are initially recorded at cost and are subsequently adjusted based on our proportionate share of the net income or loss of the funds (which include changes in the market value of the underlying securities in the funds). This method of accounting is different from the way we account for our other fixed maturity securities and the timing of the recognition of equity in net income or loss of investment funds accounted for using the equity method may differ from gains or losses in the future upon sale or maturity of such investments. Transaction costs and other include advisory, financing, legal, severance, incentive compensation and other transaction costs related to acquisitions. We believe that transaction costs and other, due to their non-recurring nature, are not indicative of the performance of, or trends in, our business performance. The loss on redemption of preferred shares related to the redemption of the Company's Series E preferred shares in September 2021 and had no impact on shareholders' equity or cash flows. Due to these reasons, we exclude net realized gains or losses, equity in net income or loss of investment funds accounted for using the equity method, net foreign exchange gains or losses, transaction costs and other and loss on redemption of preferred shares from the calculation of after-tax operating income available to Arch common shareholders.
We believe that showing net income available to Arch common shareholders exclusive of the items referred to above reflects the underlying fundamentals of our business since we evaluate the performance of and manage our business to produce an underwriting profit. In addition to presenting net income available to Arch common shareholders, we believe that this presentation enables investors and other users of our financial information to analyze our performance in a manner similar to how management analyzes performance. We also believe that this measure follows industry practice and, therefore, allows the users of financial information to compare our performance with our industry peer group. We believe that the equity analysts and certain rating agencies which follow us and the insurance industry as a whole generally exclude these items from their analyses for the same reasons.
Our segment information includes the presentation of consolidated underwriting income or loss and a subtotal of underwriting income or loss before the contribution from the ‘other’ segment, through June 30, 2021. Such measures represent the pre-tax profitability of our underwriting operations and include net premiums earned plus other underwriting income, less losses and loss adjustment expenses, acquisition expenses and other operating expenses. Other operating expenses include those operating expenses that are incremental and/or directly attributable to our individual underwriting operations. Underwriting income or loss does not incorporate items included in our corporate
(non-underwriting) segment. While these measures are presented in note 5, “Segment Information,” of the notes accompanying our consolidated financial statements, they are considered non-GAAP financial measures when presented elsewhere on a consolidated basis. The reconciliations of underwriting income or loss to income before income taxes (the most directly comparable GAAP financial measure) on a consolidated basis and a subtotal before the contribution from the ‘other’ segment through June 30, 2021, in accordance with Regulation G, is shown in note 5, “Segment Information” to our consolidated financial statements.
We measure segment performance for our three underwriting segments based on underwriting income or loss. We do not manage our assets by underwriting segment, with the exception of goodwill and intangibles and, accordingly, investment income and other non-underwriting related items are not allocated to each underwriting segment. The ‘other’ segment includes the results of Watford through June 30, 2021.
Along with consolidated underwriting income, we provide a subtotal of underwriting income or loss before the contribution from the ‘other’ segment. Through June 30, 2021, the ‘other’ segment included the results of Watford Holdings Ltd. Watford Holdings Ltd. is the parent of Watford Re Ltd., a multi-line Bermuda reinsurance company (together with Watford Holdings Ltd., “Watford”). Pursuant to GAAP, Watford was considered a variable interest entity and we concluded that we were the primary beneficiary of Watford. As such, we consolidated the results of Watford in our consolidated financial statements through June 30, 2021. In the 2020 fourth quarter, Arch Capital, Watford, and Greysbridge Ltd., a wholly-owned subsidiary of Arch Capital, entered into an Agreement and Plan of Merger (as amended, the “Merger Agreement”). Arch Capital assigned its rights under the Merger Agreement to Greysbridge Holdings Ltd. (“Greysbridge”). The merger and the related Greysbridge equity financing closed on July 1, 2021. Effective July 1, 2021, Watford is wholly owned by Greysbridge, and Greysbridge is owned 40% by Arch and 30% by certain funds managed by Kelso and 30% by certain funds managed by Warburg. Based on the governing documents of Greysbridge, we concluded that, while we retain significant influence over Greysbridge, Greysbridge does not constitute a variable interest entity. Accordingly, effective July 1, 2021, we no longer consolidate the results of Watford in our consolidated financial statements and footnotes. See note 12, “Variable Interest Entities and Noncontrolling Interests” and note 5, “Segment Information,” to our consolidated financial statements for additional information on Watford.
Our presentation of segment information includes the use of a current year loss ratio which excludes favorable or adverse

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development in prior year loss reserves. This ratio is a non-GAAP financial measure as defined in Regulation G. The reconciliation of such measure to the loss ratio (the most directly comparable GAAP financial measure) in accordance with Regulation G is shown on the individual segment pages. Management utilizes the current year loss ratio in its analysis of the underwriting performance of each of our underwriting segments.
Total return on investments includes investment income, equity in net income or loss of investment funds accounted for using the equity method, net realized gains and losses (excluding changes in the allowance for credit losses on non-investment related financial assets) and the change in unrealized gains and losses generated by Arch’s investment portfolio. Total return is calculated on a pre-tax basis and before investment expenses, excludes amounts reflected in the ‘other’ segment, and reflects the effect of financial market conditions along with foreign currency fluctuations. In addition, total return incorporates the timing of investment returns during the periods. There is no directly comparable GAAP financial measure for total return. Management uses total return on investments as a key measure of the return generated to Arch common shareholders, and compares the return generated by our investment portfolio against benchmark returns during the periods.
RESULTS OF OPERATIONS

The following table summarizes our consolidated financial data, including a reconciliation of net income or loss available to Arch common shareholders to after-tax operating income or loss available to Arch common shareholders.

| Three Months Ended |  | Nine Months Ended
| September 30, |  | September 30,
| 2021 |  | 2020 |  | 2021 |  | 2020
Net income available to Arch common shareholders | $ | 388,751 |  |  | $ | 408,636 |  |  | $ | 1,480,324 |  |  | $ | 830,768 |
Net realized (gains) losses | 25,040 |  |  | (219,726) |  |  | (247,949) |  |  | (517,007) |
Equity in net (income) loss of investment funds accounted for using the equity method | (105,398) |  |  | (126,735) |  |  | (299,270) |  |  | (57,407) |
Net foreign exchange (gains) losses | (36,078) |  |  | 39,462 |  |  | (39,522) |  |  | 17,003 |
Transaction costs and other | 1,036 |  |  | 1,674 |  |  | 889 |  |  | 5,246 |
Loss on redemption of preferred shares | 15,101 |  |  | — |  |  | 15,101 |  |  | — |
Income tax expense (1) | 6,236 |  |  | 17,010 |  |  | 32,100 |  |  | 48,088 |
After-tax operating income available to Arch common shareholders | $ | 294,688 |  |  | $ | 120,321 |  |  | $ | 941,673 |  |  | $ | 326,691 |
Beginning common shareholders’ equity | $ | 12,706,072 |  |  | $ | 11,211,825 |  |  | $ | 12,325,886 |  |  | $ | 10,717,371 |
Ending common shareholders’ equity | $ | 12,557,526 |  |  | $ | 11,671,997 |  |  | $ | 12,557,526 |  |  | $ | 11,671,997 |
Average common shareholders’ equity | $ | 12,631,799 |  |  | $ | 11,441,911 |  |  | $ | 12,441,706 |  |  | $ | 11,194,684 |
Annualized net income return on average common equity % | 12.3 |  |  | 14.3 |  |  | 15.9 |  |  | 9.9 |
Annualized operating return on average common equity % | 9.3 |  |  | 4.2 |  |  | 10.1 |  |  | 3.9 |

(1) Income tax expense on net realized gains or losses, equity in net income or loss of investment funds accounted for using the equity method, net foreign exchange gains or losses and transaction costs and other reflects the relative mix reported by jurisdiction and the varying tax rates in each jurisdiction.

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Segment Information
We classify our businesses into three underwriting segments — insurance, reinsurance and mortgage — and two other operating segments — corporate (non-underwriting) and ‘other.’ Our insurance, reinsurance and mortgage segments each have managers who are responsible for the overall profitability of their respective segments and who are directly accountable to our chief operating decision makers, the Chief Executive Officer of Arch Capital, the Chief Financial Officer and Treasurer of Arch Capital and the President and Chief Underwriting Officer of Arch Capital. The chief operating decision makers do not assess performance, measure return on equity or make resource allocation decisions on a line of business basis. Management measures segment performance for our three underwriting segments based on underwriting income or loss. We do not manage our assets by underwriting segment, with the exception of goodwill and intangible assets, and, accordingly, investment income is not allocated to each underwriting segment.
We determined our reportable segments using the management approach described in accounting guidance regarding disclosures about segments of an enterprise and related information. The accounting policies of the segments are the same as those used for the preparation of our consolidated financial statements. Intersegment business is allocated to the segment accountable for the underwriting results.
Insurance Segment
The following tables set forth our insurance segment’s underwriting results:

| Three Months Ended September 30,
| 2021 |  | 2020 |  | % Change
Gross premiums written | $ | 1,596,619 |  |  | $ | 1,206,328 |  |  | 32.4 |
Premiums ceded | (442,806) |  |  | (382,167) |  |  |
Net premiums written | 1,153,813 |  |  | 824,161 |  |  | 40.0 |
Change in unearned premiums | (215,143) |  |  | (105,007) |  |  |
Net premiums earned | 938,670 |  |  | 719,154 |  |  | 30.5 |
Other underwriting income (loss) | — |  |  | (31) |  |  |
Losses and loss adjustment expenses | (668,630) |  |  | (525,321) |  |  |
Acquisition expenses | (152,467) |  |  | (102,420) |  |  |
Other operating expenses | (138,931) |  |  | (122,541) |  |  |
Underwriting income (loss) | $ | (21,358) |  |  | $ | (31,159) |  |  | 31.5 |
Underwriting Ratios |  |  |  |  | % Point Change
Loss ratio | 71.2 | % |  | 73.0 | % |  | (1.8) |
Acquisition expense ratio | 16.2 | % |  | 14.2 | % |  | 2.0 |
Other operating expense ratio | 14.8 | % |  | 17.0 | % |  | (2.2) |
Combined ratio | 102.2 | % |  | 104.2 | % |  | (2.0) |

| Nine Months Ended September 30,
| 2021 |  | 2020 |  | % Change
Gross premiums written | $ | 4,381,372 |  |  | $ | 3,444,335 |  |  | 27.2 |
Premiums ceded | (1,269,165) |  |  | (1,119,165) |  |  |
Net premiums written | 3,112,207 |  |  | 2,325,170 |  |  | 33.8 |
Change in unearned premiums | (488,636) |  |  | (202,188) |  |  |
Net premiums earned | 2,623,571 |  |  | 2,122,982 |  |  | 23.6 |
Other underwriting income | — |  |  | (31) |  |  |
Losses and loss adjustment expenses | (1,750,257) |  |  | (1,550,632) |  |  |
Acquisition expenses | (417,541) |  |  | (317,428) |  |  |
Other operating expenses | (409,386) |  |  | (370,947) |  |  |
Underwriting income (loss) | $ | 46,387 |  |  | $ | (116,056) |  |  | 140.0 |
Underwriting Ratios |  |  |  |  | % Point Change
Loss ratio | 66.7 | % |  | 73.0 | % |  | (6.3) |
Acquisition expense ratio | 15.9 | % |  | 15.0 | % |  | 0.9 |
Other operating expense ratio | 15.6 | % |  | 17.5 | % |  | (1.9) |
Combined ratio | 98.2 | % |  | 105.5 | % |  | (7.3) |

The insurance segment consists of our insurance underwriting units which offer specialty product lines on a worldwide basis. Product lines include:
•Construction and national accounts: primary and excess casualty coverages to middle and large accounts in the construction industry and a wide range of products for middle and large national accounts, specializing in loss sensitive primary casualty insurance programs (including large deductible, self-insured retention and retrospectively rated programs).
•Excess and surplus casualty: primary and excess casualty insurance coverages, including middle market energy business, and contract binding, which primarily provides casualty coverage through a network of appointed agents to small and medium risks.
•Lenders products: collateral protection, debt cancellation and service contract reimbursement products to banks, credit unions, automotive dealerships and original equipment manufacturers and other specialty programs that pertain to automotive lending and leasing.
•Professional lines: directors’ and officers’ liability, errors and omissions liability, employment practices liability, fiduciary liability, crime, professional indemnity and other financial related coverages for corporate, private equity, venture capital, real estate investment trust, limited partnership, financial institution and not-for-profit clients of all sizes and medical professional and general liability insurance coverages for the healthcare industry. The business is predominately written on a claims-made basis.
•Programs: primarily package policies, underwriting workers’ compensation and umbrella liability business in support of desirable package programs, targeting program

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managers with unique expertise and niche products offering general liability, commercial automobile, inland marine and property business with minimal catastrophe exposure.
•Property, energy, marine and aviation: primary and excess general property insurance coverages, including catastrophe-exposed property coverage, for commercial clients. Coverages for marine include hull, war, specie and liability. Aviation and standalone terrorism are also offered.
•Travel, accident and health: specialty travel and accident and related insurance products for individual, group travelers, travel agents and suppliers, as well as accident and health, which provides accident, disability and medical plan insurance coverages for employer groups, medical plan members, students and other participant groups.
•Other: includes alternative market risks (including captive insurance programs), excess workers’ compensation and employer’s liability insurance coverages for qualified self-insured groups, associations and trusts, and contract and commercial surety coverages, including contract bonds (payment and performance bonds) primarily for medium and large contractors and commercial surety bonds for Fortune 1,000 companies and smaller transaction business programs.
Premiums Written.
The following tables set forth our insurance segment’s net premiums written by major line of business:

| Three Months Ended September 30, |  |
| 2021 |  | 2020 |  |  |  |
| Amount |  | % |  | Amount |  | % |  |  |  |  |  |  |  |
Property, energy, marine and aviation | $ | 215,062 |  |  | 18.6 |  |  | $ | 152,193 |  |  | 18.5 |  |  |  |  |  |  |  |  |
Professional lines | 310,185 |  |  | 26.9 |  |  | 199,163 |  |  | 24.2 |  |  |  |  |  |  |  |  |
Programs | 196,048 |  |  | 17.0 |  |  | 123,768 |  |  | 15.0 |  |  |  |  |  |  |  |  |
Construction and national accounts | 92,253 |  |  | 8.0 |  |  | 88,790 |  |  | 10.8 |  |  |  |  |  |  |  |  |
Excess and surplus casualty | 98,320 |  |  | 8.5 |  |  | 78,889 |  |  | 9.6 |  |  |  |  |  |  |  |  |
Travel, accident and health | 62,837 |  |  | 5.4 |  |  | 28,972 |  |  | 3.5 |  |  |  |  |  |  |  |  |
Lenders products | 38,905 |  |  | 3.4 |  |  | 60,830 |  |  | 7.4 |  |  |  |  |  |  |  |  |
Other | 140,203 |  |  | 12.2 |  |  | 91,556 |  |  | 11.1 |  |  |  |  |  |  |  |  |
Total | $ | 1,153,813 |  |  | 100.0 |  |  | $ | 824,161 |  |  | 100.0 |  |  |  |  |  |  |  |  |

2021 Third Quarter versus 2020 Period. Gross premiums written by the insurance segment in the 2021 third quarter were 32.4% higher than in the 2020 third quarter, while net premiums written were 40.0% higher. The higher level of net premiums written reflected increases in most lines of business, due in part to rate increases, new business opportunities and growth in existing accounts.

| Nine Months Ended September 30, |  |
| 2021 |  | 2020 |  |  |  |
| Amount |  | % |  | Amount |  | % |  |  |  |  |  |  |  |
Property, energy, marine and aviation | $ | 593,322 |  |  | 19.1 |  |  | $ | 439,579 |  |  | 18.9 |  |  |  |  |  |  |  |  |
Professional Lines | 803,392 |  |  | 25.8 |  |  | 526,180 |  |  | 22.6 |  |  |  |  |  |  |  |  |
Programs | 503,822 |  |  | 16.2 |  |  | 341,230 |  |  | 14.7 |  |  |  |  |  |  |  |  |
Construction and national accounts | 304,624 |  |  | 9.8 |  |  | 261,933 |  |  | 11.3 |  |  |  |  |  |  |  |  |
Excess and surplus casualty | 258,259 |  |  | 8.3 |  |  | 209,011 |  |  | 9.0 |  |  |  |  |  |  |  |  |
Travel, accident and health | 226,214 |  |  | 7.3 |  |  | 183,015 |  |  | 7.9 |  |  |  |  |  |  |  |  |
Lenders products | 114,151 |  |  | 3.7 |  |  | 117,812 |  |  | 5.1 |  |  |  |  |  |  |  |  |
Other | 308,423 |  |  | 9.9 |  |  | 246,410 |  |  | 10.6 |  |  |  |  |  |  |  |  |
Total | $ | 3,112,207 |  |  | 100.0 |  |  | $ | 2,325,170 |  |  | 100.0 |  |  |  |  |  |  |  |  |

Nine Months Ended September 30, 2021 versus 2020 Period. Gross premiums written by the insurance segment for the nine months ended September 30, 2021 were 27.2% higher than in the 2020 period, while net premiums written were 33.8% higher than in the 2020 period. The increase in net premiums written reflected growth across most lines of business, due in part to rate increases, new business opportunities and growth in existing accounts.
Net Premiums Earned.
The following tables set forth our insurance segment’s net premiums earned by major line of business:

| Three Months Ended September 30, |  |
| 2021 |  | 2020 |  |  |  |
| Amount |  | % |  | Amount |  | % |  |  |  |  |  |  |  |
Property, energy, marine and aviation | $ | 187,905 |  |  | 20.0 |  |  | $ | 133,827 |  |  | 18.6 |  |  |  |  |  |  |  |  |
Professional lines | 249,007 |  |  | 26.5 |  |  | 168,502 |  |  | 23.4 |  |  |  |  |  |  |  |  |
Programs | 137,299 |  |  | 14.6 |  |  | 104,861 |  |  | 14.6 |  |  |  |  |  |  |  |  |
Construction and national accounts | 94,523 |  |  | 10.1 |  |  | 95,386 |  |  | 13.3 |  |  |  |  |  |  |  |  |
Excess and surplus casualty | 84,048 |  |  | 9.0 |  |  | 69,978 |  |  | 9.7 |  |  |  |  |  |  |  |  |
Travel, accident and health | 56,102 |  |  | 6.0 |  |  | 36,726 |  |  | 5.1 |  |  |  |  |  |  |  |  |
Lenders products | 33,030 |  |  | 3.5 |  |  | 33,401 |  |  | 4.6 |  |  |  |  |  |  |  |  |
Other | 96,756 |  |  | 10.3 |  |  | 76,473 |  |  | 10.6 |  |  |  |  |  |  |  |  |
Total | $ | 938,670 |  |  | 100.0 |  |  | $ | 719,154 |  |  | 100.0 |  |  |  |  |  |  |  |  |

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| Nine Months Ended September 30, |  |
| 2021 |  | 2020 |  |  |  |
| Amount |  | % |  | Amount |  | % |  |  |  |  |  |  |  |
Property, energy, marine and aviation | $ | 512,880 |  |  | 19.5 |  |  | $ | 365,791 |  |  | 17.2 |  |  |  |  |  |  |  |  |
Professional Lines | 662,776 |  |  | 25.3 |  |  | 475,014 |  |  | 22.4 |  |  |  |  |  |  |  |  |
Programs | 369,113 |  |  | 14.1 |  |  | 322,203 |  |  | 15.2 |  |  |  |  |  |  |  |  |
Construction and national accounts | 293,043 |  |  | 11.2 |  |  | 286,691 |  |  | 13.5 |  |  |  |  |  |  |  |  |
Excess and surplus casualty | 232,314 |  |  | 8.9 |  |  | 196,041 |  |  | 9.2 |  |  |  |  |  |  |  |  |
Travel, accident and health | 168,378 |  |  | 6.4 |  |  | 166,218 |  |  | 7.8 |  |  |  |  |  |  |  |  |
Lenders products | 119,507 |  |  | 4.6 |  |  | 81,855 |  |  | 3.9 |  |  |  |  |  |  |  |  |
Other | 265,560 |  |  | 10.1 |  |  | 229,169 |  |  | 10.8 |  |  |  |  |  |  |  |  |
Total | $ | 2,623,571 |  |  | 100.0 |  |  | $ | 2,122,982 |  |  | 100.0 |  |  |  |  |  |  |  |  |

Net premiums written are primarily earned on a pro rata basis over the terms of the policies for all products, usually 12 months. Net premiums earned reflect changes in net premiums written over the previous five quarters. Net premiums earned in the 2021 third quarter were 30.5% higher than in the 2020 third quarter. Net premiums earned for the nine months ended September 30, 2021 were 23.6% higher than in the 2020 period.
Losses and Loss Adjustment Expenses.
The table below shows the components of the insurance segment’s loss ratio:

| Three Months Ended |  | Nine Months Ended
| September 30, |  | September 30,
| 2021 |  | 2020 |  | 2021 |  | 2020
Current year | 71.7 | % |  | 73.3 | % |  | 67.2 | % |  | 73.3 | %
Prior period reserve development | (0.5) | % |  | (0.3) | % |  | (0.5) | % |  | (0.3) | %
Loss ratio | 71.2 | % |  | 73.0 | % |  | 66.7 | % |  | 73.0 | %

Current Year Loss Ratio.
2021 Third Quarter versus 2020 Period. The insurance segment’s current year loss ratio in the 2021 third quarter was 1.6 points lower than in the 2020 third quarter. The 2021 third quarter loss ratio reflected 12.2 points of current year catastrophic activity, primarily related to Hurricane Ida and other global events, compared to 10.3 points of catastrophic activity for the 2020 third quarter, which included exposure to the COVID-19 global pandemic. The insurance segment’s current year loss ratio for the nine months ended September 30, 2021 was 6.1 points lower than in the 2020 period and reflected 7.0 points of current year catastrophic activity, compared to 9.9 points in the 2020 period. The balance of the change in the 2021 loss ratios resulted, in part, from changes in mix of business and the level of large attritional losses.
Prior Period Reserve Development.
The insurance segment’s net favorable development was $5.1 million, or 0.5 points, for the 2021 third quarter, compared to $2.3 million, or 0.3 points, for the 2020 third quarter, and $13.1 million, or 0.5 points for the nine months ended September 30, 2021, compared to $5.9 million, or 0.3 points, for the 2020 period. See note 6, “Reserve for Losses and Loss Adjustment Expenses,” to our consolidated financial statements for information about the insurance segment’s prior year reserve development.
Underwriting Expenses.
2021 Third Quarter versus 2020 Period. The insurance segment’s underwriting expense ratio was 31.0% in the 2021 third quarter, consistent with 31.2% in the 2020 third quarter.
Nine Months Ended September 30, 2021 versus 2020 Period. The insurance segment’s underwriting expense ratio was 31.5% for the nine months ended September 30, 2021, compared to 32.5% for the 2020 period, with the decrease primarily due to growth in net premiums earned.
Reinsurance Segment
The following tables set forth our reinsurance segment’s underwriting results:

| Three Months Ended September 30,
| 2021 |  | 2020 |  | % Change
Gross premiums written | $ | 1,251,760 |  |  | $ | 1,004,590 |  |  | 24.6 |
Premiums ceded | (630,371) |  |  | (400,388) |  |  |
Net premiums written | 621,389 |  |  | 604,202 |  |  | 2.8 |
Change in unearned premiums | 57,313 |  |  | (49,704) |  |  |
Net premiums earned | 678,702 |  |  | 554,498 |  |  | 22.4 |
Other underwriting income (loss) | 3,293 |  |  | 298 |  |  |
Losses and loss adjustment expenses | (545,846) |  |  | (422,084) |  |  |
Acquisition expenses | (129,450) |  |  | (85,388) |  |  |
Other operating expenses | (45,647) |  |  | (41,818) |  |  |
Underwriting income (loss) | $ | (38,948) |  |  | $ | 5,506 |  |  | (807.4) |
Underwriting Ratios |  |  |  |  | % Point Change
Loss ratio | 80.4 | % |  | 76.1 | % |  | 4.3 |
Acquisition expense ratio | 19.1 | % |  | 15.4 | % |  | 3.7 |
Other operating expense ratio | 6.7 | % |  | 7.5 | % |  | (0.8) |
Combined ratio | 106.2 | % |  | 99.0 | % |  | 7.2 |

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| Nine Months Ended September 30,
| 2021 |  | 2020 |  | % Change
Gross premiums written | $ | 4,080,840 |  |  | $ | 2,934,174 |  |  | 39.1 |
Premiums ceded | (1,535,607) |  |  | (967,698) |  |  |
Net premiums written | 2,545,233 |  |  | 1,966,476 |  |  | 29.4 |
Change in unearned premiums | (484,607) |  |  | (388,321) |  |  |
Net premiums earned | 2,060,626 |  |  | 1,578,155 |  |  | 30.6 |
Other underwriting income | 3,148 |  |  | 1,767 |  |  |
Losses and loss adjustment expenses | (1,494,539) |  |  | (1,235,586) |  |  |
Acquisition expenses | (381,060) |  |  | (255,516) |  |  |
Other operating expenses | (150,856) |  |  | (125,831) |  |  |
Underwriting income (loss) | $ | 37,319 |  |  | $ | (37,011) |  |  | 200.8 |
Underwriting Ratios |  |  |  |  | % Point Change
Loss ratio | 72.5 | % |  | 78.3 | % |  | (5.8) |
Acquisition expense ratio | 18.5 | % |  | 16.2 | % |  | 2.3 |
Other operating expense ratio | 7.3 | % |  | 8.0 | % |  | (0.7) |
Combined ratio | 98.3 | % |  | 102.5 | % |  | (4.2) |

The reinsurance segment consists of our reinsurance underwriting units which offer specialty product lines on a worldwide basis. Product lines include:
•Casualty: provides coverage to ceding company clients on third party liability and workers’ compensation exposures from ceding company clients, primarily on a treaty basis. Exposures include, among others, executive assurance, professional liability, workers’ compensation, excess and umbrella liability, excess motor and healthcare business.
•Marine and aviation: provides coverage for energy, hull, cargo, specie, liability and transit, and aviation business, including airline and general aviation risks. Business written may also include space business, which includes coverages for satellite assembly, launch and operation for commercial space programs.
•Other specialty: provides coverage to ceding company clients for proportional motor and other lines, including surety, accident and health, workers’ compensation catastrophe, agriculture, trade credit and political risk.
•Property catastrophe: provides protection for most catastrophic losses that are covered in the underlying policies written by reinsureds, including hurricane, earthquake, flood, tornado, hail and fire, and coverage for other perils on a case-by-case basis. Property catastrophe reinsurance provides coverage on an excess of loss basis when aggregate losses and loss adjustment expense from a single occurrence or aggregation of losses from a covered peril exceed the retention specified in the contract.
•Property excluding property catastrophe: provides coverage for both personal lines and commercial property exposures and principally covers buildings, structures, equipment and contents. The primary perils in this business
include fire, explosion, collapse, riot, vandalism, wind, tornado, flood and earthquake. Business is assumed on both a proportional and excess of loss treaty basis and on a facultative basis. In addition, facultative business is written which focuses on commercial property risks on an excess of loss basis.
•Other: includes life reinsurance business on both a proportional and non-proportional basis, casualty clash business and, in limited instances, non-traditional business which is intended to provide insurers with risk management solutions that complement traditional reinsurance.
Premiums Written.
The following tables set forth our reinsurance segment’s net premiums written by major line of business:

| Three Months Ended September 30, |  |
| 2021 |  | 2020 |  |  |  |
| Amount |  | % |  | Amount |  | % |  |  |  |  |  |  |  |
Property excluding property catastrophe | $ | 237,025 |  |  | 38.1 |  |  | $ | 223,880 |  |  | 37.1 |  |  |  |  |  |  |  |  |
Property catastrophe | (7,125) |  |  | (1.1) |  |  | 42,125 |  |  | 7.0 |  |  |  |  |  |  |  |  |
Other specialty | 167,006 |  |  | 26.9 |  |  | 159,969 |  |  | 26.5 |  |  |  |  |  |  |  |  |
Casualty | 187,066 |  |  | 30.1 |  |  | 142,401 |  |  | 23.6 |  |  |  |  |  |  |  |  |
Marine and aviation | 19,159 |  |  | 3.1 |  |  | 27,839 |  |  | 4.6 |  |  |  |  |  |  |  |  |
Other | 18,258 |  |  | 2.9 |  |  | 7,988 |  |  | 1.3 |  |  |  |  |  |  |  |  |
Total | $ | 621,389 |  |  | 100.0 |  |  | $ | 604,202 |  |  | 100.0 |  |  |  |  |  |  |  |  |

2021 Third Quarter versus 2020 Period. Gross premiums written by the reinsurance segment in the 2021 third quarter were 24.6% higher than in the 2020 third quarter, while net premiums written were 2.8% higher. The lower level of growth in net premiums written compared to gross premiums written primarily reflected a higher level of premiums ceded due to a one-time $161.2 million adjustment, resulting from retrocessions to Watford following its ownership change on July 1, 2021. Absent this item, the growth in net premiums written would have been 29.5%, consistent with the level of growth in gross premiums written, reflecting increases in most lines of business, due in part to new business opportunities and rate increases.

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| Nine Months Ended September 30, |  |
| 2021 |  | 2020 |  |  |  |
| Amount |  | % |  | Amount |  | % |  |  |  |  |  |  |  |
Property excluding property catastrophe | $ | 778,959 |  |  | 30.6 |  |  | $ | 546,443 |  |  | 27.8 |  |  |  |  |  |  |  |  |
Property catastrophe | 197,724 |  |  | 7.8 |  |  | 248,893 |  |  | 12.7 |  |  |  |  |  |  |  |  |
Other Specialty | 747,662 |  |  | 29.4 |  |  | 562,296 |  |  | 28.6 |  |  |  |  |  |  |  |  |
Casualty | 631,212 |  |  | 24.8 |  |  | 438,330 |  |  | 22.3 |  |  |  |  |  |  |  |  |
Marine and aviation | 131,045 |  |  | 5.1 |  |  | 109,996 |  |  | 5.6 |  |  |  |  |  |  |  |  |
Other | 58,631 |  |  | 2.3 |  |  | 60,518 |  |  | 3.1 |  |  |  |  |  |  |  |  |
Total | $ | 2,545,233 |  |  | 100.0 |  |  | $ | 1,966,476 |  |  | 100.0 |  |  |  |  |  |  |  |  |

Nine Months Ended September 30, 2021 versus 2020 Period. Gross premiums written by the reinsurance segment for the nine months ended September 30, 2021 were 39.1% higher than in the 2020 period, while net premiums written were 29.4% higher than in the 2020 period. The increase in net premiums written reflected growth in property excluding property catastrophe, other specialty and casualty primarily due to new business and rate increases.
Net Premiums Earned.
The following tables set forth our reinsurance segment’s net premiums earned by major line of business:

| Three Months Ended September 30, |  |
| 2021 |  | 2020 |  |  |  |
| Amount |  | % |  | Amount |  | % |  |  |  |  |  |  |  |
Property excluding property catastrophe | $ | 210,280 |  |  | 31.0 |  |  | $ | 163,081 |  |  | 29.4 |  |  |  |  |  |  |  |  |
Property catastrophe | 61,107 |  |  | 9.0 |  |  | 69,524 |  |  | 12.5 |  |  |  |  |  |  |  |  |
Other specialty | 195,649 |  |  | 28.8 |  |  | 141,201 |  |  | 25.5 |  |  |  |  |  |  |  |  |
Casualty | 159,697 |  |  | 23.5 |  |  | 136,421 |  |  | 24.6 |  |  |  |  |  |  |  |  |
Marine and aviation | 29,818 |  |  | 4.4 |  |  | 26,744 |  |  | 4.8 |  |  |  |  |  |  |  |  |
Other | 22,151 |  |  | 3.3 |  |  | 17,527 |  |  | 3.2 |  |  |  |  |  |  |  |  |
Total | $ | 678,702 |  |  | 100.0 |  |  | $ | 554,498 |  |  | 100.0 |  |  |  |  |  |  |  |  |

| Nine Months Ended September 30, |  |
| 2021 |  | 2020 |  |  |  |
| Amount |  | % |  | Amount |  | % |  |  |  |  |  |  |  |
Property excluding property catastrophe | $ | 600,842 |  |  | 29.2 |  |  | $ | 399,752 |  |  | 25.3 |  |  |  |  |  |  |  |  |
Property catastrophe | 225,285 |  |  | 10.9 |  |  | 177,750 |  |  | 11.3 |  |  |  |  |  |  |  |  |
Other Specialty | 571,364 |  |  | 27.7 |  |  | 467,592 |  |  | 29.6 |  |  |  |  |  |  |  |  |
Casualty | 492,574 |  |  | 23.9 |  |  | 404,248 |  |  | 25.6 |  |  |  |  |  |  |  |  |
Marine and aviation | 112,699 |  |  | 5.5 |  |  | 76,562 |  |  | 4.9 |  |  |  |  |  |  |  |  |
Other | 57,862 |  |  | 2.8 |  |  | 52,251 |  |  | 3.3 |  |  |  |  |  |  |  |  |
Total | $ | 2,060,626 |  |  | 100.0 |  |  | $ | 1,578,155 |  |  | 100.0 |  |  |  |  |  |  |  |  |

Net premiums written, irrespective of the class of business, are generally earned on a pro rata basis over the terms of the underlying policies or reinsurance contracts. Net premiums earned by the reinsurance segment in the 2021 third quarter were 22.4% higher than in the 2020 third quarter, and reflect changes in net premiums written over the previous five quarters. For the nine months ended September 30, 2021, net premiums earned were 30.6% higher than in the 2020 period.
Other Underwriting Income (Loss).
Other underwriting income for the 2021 third quarter was $3.3 million, compared to an income of $0.3 million for the 2020 third quarter, and an income of $3.1 million for the nine months ended September 30, 2021, compared to an income of $1.8 million for the 2020 period.
Losses and Loss Adjustment Expenses.
The table below shows the components of the reinsurance segment’s loss ratio:

| Three Months Ended |  | Nine Months Ended
| September 30, |  | September 30,
| 2021 |  | 2020 |  | 2021 |  | 2020
Current year | 91.1 | % |  | 83.7 | % |  | 78.3 | % |  | 84.2 | %
Prior period reserve development | (10.7) | % |  | (7.6) | % |  | (5.8) | % |  | (5.9) | %
Loss ratio | 80.4 | % |  | 76.1 | % |  | 72.5 | % |  | 78.3 | %

Current Year Loss Ratio.
2021 Third Quarter versus 2020 Period. The reinsurance segment’s current year loss ratio in the 2021 third quarter was 7.4 points higher than in the 2020 third quarter. The 2021 third quarter loss ratio reflected 34.6 points of current year catastrophic activity, primarily related to Hurricane Ida, European floods and other global events. The 2020 third quarter included 26.1 points of catastrophic activity, which included exposure to the COVID-19 pandemic.
Nine Months Ended September 30, 2021 versus 2020 Period. The reinsurance segment’s current year loss ratio for the nine months ended September 30, 2021 was 5.9 points lower than in the 2020 period and reflected 20.1 points of current year catastrophic activity, compared to 21.6 points in the 2020 period. The 2020 period loss ratio included exposure to the COVID-19 pandemic.
Prior Period Reserve Development.
The reinsurance segment’s net favorable development was $72.3 million, or 10.7 points, for the 2021 third quarter, compared to $42.0 million, or 7.6 points, for the 2020 third quarter, and $119.6 million, or 5.8 points, for the nine months ended September 30, 2021, compared to $93.8 million, or 5.9 points, for the 2020 period. See note 6, “Reserve for Losses and Loss Adjustment Expenses,” to our consolidated financial statements for information about the reinsurance segment’s prior year reserve development.
Underwriting Expenses.
2021 Third Quarter versus 2020 Period. The underwriting expense ratio for the reinsurance segment was 25.8% in the

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2021 third quarter, compared to 22.9% in the 2020 third quarter, with the increase primarily resulting from changes in mix of business to lines with higher acquisition costs and a higher level of expenses related to favorable development of prior year loss reserves.
Nine Months Ended September 30, 2021 versus 2020 Period. The underwriting expense ratio for the reinsurance segment was 25.8% for the nine months ended September 30, 2021, compared to 24.2% for the 2020 period. The comparison of the underwriting expense ratios also reflected changes in the mix and type of business and a higher level of net premiums earned for the 2021 period.
Mortgage Segment
Our mortgage operations include U.S. and international mortgage insurance and reinsurance operations as well as participation in GSE credit risk-sharing transactions. Our mortgage group includes direct mortgage insurance in the U.S. primarily through Arch Mortgage Insurance Company, United Guaranty Residential Insurance Company and Arch Mortgage Guaranty Company (together, “Arch MI U.S.”); mortgage reinsurance by Arch Reinsurance Ltd. (“Arch Re Bermuda”) to mortgage insurers on both a proportional and non-proportional basis globally; direct mortgage insurance in Europe through Arch Insurance (EU) Designated Activity Company (“Arch Insurance EU”); in Hong Kong through Arch MI Asia Limited (“Arch MI Asia”); in Australia through Arch Lenders Mortgage Indemnity Limited (“ALMI”) and participation in various GSE credit risk-sharing products primarily through Arch Re Bermuda.
The following tables set forth our mortgage segment’s underwriting results.

| Three Months Ended September 30,
| 2021 |  | 2020 |  | % Change
Gross premiums written | $ | 360,934 |  |  | $ | 346,248 |  |  | 4.2 |
Premiums ceded | (60,207) |  |  | (47,783) |  |  |
Net premiums written | 300,727 |  |  | 298,465 |  |  | 0.8 |
Change in unearned premiums | 11,238 |  |  | 52,944 |  |  |
Net premiums earned | 311,965 |  |  | 351,409 |  |  | (11.2) |
Other underwriting income | 3,981 |  |  | 4,600 |  |  |
Losses and loss adjustment expenses | (11,543) |  |  | (153,055) |  |  |
Acquisition expenses | (24,098) |  |  | (35,716) |  |  |
Other operating expenses | (46,254) |  |  | (36,708) |  |  |
Underwriting income | $ | 234,051 |  |  | $ | 130,530 |  |  | 79.3 |
Underwriting Ratios |  |  |  |  | % Point Change
Loss ratio | 3.7 | % |  | 43.6 | % |  | (39.9) |
Acquisition expense ratio | 7.7 | % |  | 10.2 | % |  | (2.5) |
Other operating expense ratio | 14.8 | % |  | 10.4 | % |  | 4.4 |
Combined ratio | 26.2 | % |  | 64.2 | % |  | (38.0) |

| Nine Months Ended September 30,
| 2021 |  | 2020 |  | % Change
Gross premiums written | $ | 1,143,691 |  |  | $ | 1,084,337 |  |  | 5.5 |
Premiums ceded | (171,923) |  |  | (136,154) |  |  |
Net premiums written | 971,768 |  |  | 948,183 |  |  | 2.5 |
Change in unearned premiums | 10,735 |  |  | 113,965 |  |  |
Net premiums earned | 982,503 |  |  | 1,062,148 |  |  | (7.5) |
Other underwriting income | 15,026 |  |  | 15,649 |  |  |
Losses and loss adjustment expenses | (85,112) |  |  | (444,721) |  |  |
Acquisition expenses | (84,297) |  |  | (108,304) |  |  |
Other operating expenses | (143,697) |  |  | (120,178) |  |  |
Underwriting income | $ | 684,423 |  |  | $ | 404,594 |  |  | 69.2 |
Underwriting Ratios |  |  |  |  | % Point Change
Loss ratio | 8.7 | % |  | 41.9 | % |  | (33.2) |
Acquisition expense ratio | 8.6 | % |  | 10.2 | % |  | (1.6) |
Other operating expense ratio | 14.6 | % |  | 11.3 | % |  | 3.3 |
Combined ratio | 31.9 | % |  | 63.4 | % |  | (31.5) |

Premiums Written.
The following tables set forth our mortgage segment’s net premiums written by underwriting location (i.e., where the business is underwritten):

| Three Months Ended September 30, |  |
| 2021 |  | 2020 |  |  |  |
| Amount |  | % |  | Amount |  | % |  |  |  |  |  |  |  |
Underwriting location: |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
United States | $ | 221,315 |  |  | 73.6 |  |  | $ | 245,971 |  |  | 82.4 |  |  |  |  |  |  |  |  |
Other | 79,412 |  |  | 26.4 |  |  | 52,494 |  |  | 17.6 |  |  |  |  |  |  |  |  |
Total | $ | 300,727 |  |  | 100.0 |  |  | $ | 298,465 |  |  | 100.0 |  |  |  |  |  |  |  |  |

2021 Third Quarter versus 2020 Period. Gross premiums written by the mortgage segment in the 2021 third quarter were 4.2% higher than in the 2020 third quarter, while net premiums written were 0.8% higher. The increase in gross premiums written reflected growth in Australian single premium mortgage insurance partially as a result of the previously disclosed acquisition of Westpac Lenders Mortgage Insurance Limited. The lower increase in net premiums written reflected a higher level of premiums ceded on U.S. primary mortgage insurance.

| Nine Months Ended September 30, |  |
| 2021 |  | 2020 |  |  |  |
| Amount |  | % |  | Amount |  | % |  |  |  |  |  |  |  |
Underwriting location: |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
United States | $ | 703,489 |  |  | 72.4 |  |  | $ | 771,203 |  |  | 81.3 |  |  |  |  |  |  |  |  |
Other | 268,279 |  |  | 27.6 |  |  | 176,980 |  |  | 18.7 |  |  |  |  |  |  |  |  |
Total | $ | 971,768 |  |  | 100.0 |  |  | $ | 948,183 |  |  | 100.0 |  |  |  |  |  |  |  |  |

Nine Months Ended September 30, 2021 versus 2020 Period. Gross premiums written by the mortgage segment for the nine months ended September 30, 2021 were 5.5% higher than in the 2020 period, while net premiums written for the nine months ended September 30, 2021 were 2.5% higher

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than in the 2020 period, primarily reflecting growth in Australian single premium mortgage insurance and the benefit of premiums received related to the exercise of early redemption options by GSEs for certain seasoned callable credit risk transfer contracts. This growth was partially offset by a lower level of U.S. primary mortgage insurance in force on monthly premium policies, which resulted from the continued high level of refinancing activity.
The persistency rate, which represents the percentage of mortgage insurance in force at the beginning of a 12-month period that remains in force at the end of such period, was 57.7% for the Arch MI U.S. portfolio of mortgage insurance policies at September 30, 2021, reflecting the higher level of mortgage refinancing activity, compared to 58.7% at December 31, 2020.
The following tables provide details on the new insurance written (“NIW”) generated by Arch MI U.S. NIW represents the original principal balance of all loans that received coverage during the period.

(U.S. Dollars in millions) | Three Months Ended September 30, |  |
2021 |  | 2020 |  |
| Amount |  | % |  | Amount |  | % |  |  |  |
Total new insurance written (NIW) (1) | $ | 27,841 |  |  |  |  | $ | 32,787 |  |  |  |  |  |  |
Credit quality (FICO): |  |  |  |  |  |  |  |  |  |  |
>=740 | $ | 17,514 |  |  | 62.9 |  |  | $ | 21,160 |  |  | 64.5 |  |  |  |  |
680-739 | 9,012 |  |  | 32.4 |  |  | 10,562 |  |  | 32.2 |  |  |  |  |
620-679 | 1,315 |  |  | 4.7 |  |  | 1,065 |  |  | 3.2 |  |  |  |  |
Total | $ | 27,841 |  |  | 100.0 |  |  | $ | 32,787 |  |  | 100.0 |  |  |  |  |
Loan-to-value (LTV): |  |  |  |  |  |  |  |  |  |  |
95.01% and above | $ | 1,554 |  |  | 5.6 |  |  | $ | 2,561 |  |  | 7.8 |  |  |  |  |
90.01% to 95.00% | 14,240 |  |  | 51.1 |  |  | 13,967 |  |  | 42.6 |  |  |  |  |
85.01% to 90.00% | 8,394 |  |  | 30.1 |  |  | 10,052 |  |  | 30.7 |  |  |  |  |
85.00% and below | 3,653 |  |  | 13.1 |  |  | 6,207 |  |  | 18.9 |  |  |  |  |
Total | $ | 27,841 |  |  | 100.0 |  |  | $ | 32,787 |  |  | 100.0 |  |  |  |  |
Monthly vs. single: |  |  |  |  |  |  |  |  |  |  |
Monthly | $ | 26,515 |  |  | 95.2 |  |  | $ | 31,928 |  |  | 97.4 |  |  |  |  |
Single | 1,326 |  |  | 4.8 |  |  | 859 |  |  | 2.6 |  |  |  |  |
Total | $ | 27,841 |  |  | 100.0 |  |  | $ | 32,787 |  |  | 100.0 |  |  |  |  |
Purchase vs. refinance: |  |  |  |  |  |  |  |  |  |  |
Purchase | $ | 25,711 |  |  | 92.3 |  |  | $ | 24,256 |  |  | 74.0 |  |  |  |  |
Refinance | 2,130 |  |  | 7.7 |  |  | 8,531 |  |  | 26.0 |  |  |  |  |
Total | $ | 27,841 |  |  | 100.0 |  |  | $ | 32,787 |  |  | 100.0 |  |  |  |  |

(1)Represents the original principal balance of all loans that received coverage during the period.

(U.S. Dollars in millions) | Nine Months Ended September 30, |  |
2021 |  | 2020 |  |
| Amount |  | % |  | Amount |  | % |  |  |  |
Total new insurance written (NIW) (1) | $ | 83,232 |  |  |  |  | $ | 74,116 |  |  |  |  |  |  |
Credit quality (FICO): |  |  |  |  |  |  |  |  |  |  |
>=740 | $ | 54,572 |  |  | 65.6 |  |  | $ | 47,080 |  |  | 63.5 |  |  |  |  |
680-739 | 25,543 |  |  | 30.7 |  |  | 24,130 |  |  | 32.6 |  |  |  |  |
620-679 | 3,117 |  |  | 3.7 |  |  | 2,906 |  |  | 3.9 |  |  |  |  |
Total | $ | 83,232 |  |  | 100.0 |  |  | $ | 74,116 |  |  | 100.0 |  |  |  |  |
Loan-to-value (LTV): |  |  |  |  |  |  |  |  |  |  |
95.01% and above | $ | 4,646 |  |  | 5.6 |  |  | $ | 6,177 |  |  | 8.3 |  |  |  |  |
90.01% to 95.00% | 40,464 |  |  | 48.6 |  |  | 30,569 |  |  | 41.2 |  |  |  |  |
85.01% to 90.00% | 25,381 |  |  | 30.5 |  |  | 23,521 |  |  | 31.7 |  |  |  |  |
85.01% and below | 12,741 |  |  | 15.3 |  |  | 13,849 |  |  | 18.7 |  |  |  |  |
Total | $ | 83,232 |  |  | 100.0 |  |  | $ | 74,116 |  |  | 100.0 |  |  |  |  |
Monthly vs. single: |  |  |  |  |  |  |  |  |  |  |
Monthly | $ | 78,229 |  |  | 94.0 |  |  | $ | 71,011 |  |  | 95.8 |  |  |  |  |
Single | 5,003 |  |  | 6.0 |  |  | 3,105 |  |  | 4.2 |  |  |  |  |
Total | $ | 83,232 |  |  | 100.0 |  |  | $ | 74,116 |  |  | 100.0 |  |  |  |  |
Purchase vs. refinance: |  |  |  |  |  |  |  |  |  |  |
Purchase | $ | 71,226 |  |  | 85.6 |  |  | $ | 51,511 |  |  | 69.5 |  |  |  |  |
Refinance | 12,006 |  |  | 14.4 |  |  | 22,605 |  |  | 30.5 |  |  |  |  |
Total | $ | 83,232 |  |  | 100.0 |  |  | $ | 74,116 |  |  | 100.0 |  |  |  |  |

(1)Represents the original principal balance of all loans that received coverage during the period.
Net Premiums Earned.
The following tables set forth our mortgage segment’s net premiums earned by underwriting location:

| Three Months Ended September 30, |  |
| 2021 |  | 2020 |  |  |  |
| Amount |  | % |  | Amount |  | % |  |  |  |  |  |  |  |
Underwriting location: |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
United States | $ | 236,892 |  |  | 75.9 |  |  | $ | 290,451 |  |  | 82.7 |  |  |  |  |  |  |  |  |
Other | 75,073 |  |  | 24.1 |  |  | 60,958 |  |  | 17.3 |  |  |  |  |  |  |  |  |
Total | $ | 311,965 |  |  | 100.0 |  |  | $ | 351,409 |  |  | 100.0 |  |  |  |  |  |  |  |  |

2021 Third Quarter versus 2020 Period. Net premiums earned for the 2021 third quarter were 11.2% lower than in the 2020 third quarter, and reflected a lower level of single premium policy terminations.

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| Nine Months Ended September 30, |  |
| 2021 |  | 2020 |  |  |  |
| Amount |  | % |  | Amount |  | % |  |  |  |  |  |  |  |
Underwriting location: |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
United States | $ | 747,830 |  |  | 76.1 |  |  | $ | 884,265 |  |  | 83.3 |  |  |  |  |  |  |  |  |
Other | 234,673 |  |  | 23.9 |  |  | 177,883 |  |  | 16.7 |  |  |  |  |  |  |  |  |
Total | $ | 982,503 |  |  | 100.0 |  |  | $ | 1,062,148 |  |  | 100.0 |  |  |  |  |  |  |  |  |

Nine Months Ended September 30, 2021 versus 2020 Period. Net premiums earned for the nine months ended September 30, 2021 were 7.5% lower than in the 2020 period, primarily reflecting a lower level of single premiums earned, partially offset by an increase in earnings from Australian single premium policy terminations.
Other Underwriting Income.
Other underwriting income, which is primarily related to GSE credit risk-sharing transactions was $4.0 million for the 2021 third quarter, compared to $4.6 million for the 2020 third quarter.
Losses and Loss Adjustment Expenses.
The table below shows the components of the mortgage segment’s loss ratio:

| Three Months Ended |  | Nine Months Ended
| September 30, |  | September 30,
| 2021 |  | 2020 |  | 2021 |  | 2020
Current year | 18.2 | % |  | 44.9 | % |  | 18.8 | % |  | 42.9 | %
Prior period reserve development | (14.5) | % |  | (1.3) | % |  | (10.1) | % |  | (1.0) | %
Loss ratio | 3.7 | % |  | 43.6 | % |  | 8.7 | % |  | 41.9 | %

Current Year Loss Ratio.
2021 Third Quarter versus 2020 Period. The mortgage segment’s current year loss ratio was 26.7 points lower in the 2021 third quarter than in the 2020 third quarter. The mortgage segment’s current year loss ratio was 24.1 points lower for the nine months ended September 30, 2021 than for the 2020 period. The lower current year loss ratios for the 2021 period reflect decrease in loss assumptions related to COVID-19 pandemic, primarily driven by lower delinquencies.
For the 2020 periods, the increase in incurred losses was primarily due to, the financial stress related to the COVID-19 pandemic. Segregating estimated losses due to COVID-19 from the overall mortgage segment estimated losses would require the number of delinquencies specifically attributable to COVID-19. As this analysis cannot be performed accurately, the Company is not reporting COVID-19 provisions separately from its overall loss provisions.
Prior Period Reserve Development.
The mortgage segment’s net favorable development was $45.1 million, or 14.5 points, for the 2021 third quarter, compared to $4.5 million, or 1.3 points, for the 2020 third quarter, and $99.1 million, or 10.1 points, for the nine months ended September 30, 2021, compared to $10.8 million, or 1.0 points, for the 2020 period. See note 6, “Reserve for Losses and Loss Adjustment Expenses,” to our consolidated financial statements for information about the mortgage segment’s prior year reserve development.
Underwriting Expenses.
2021 Third Quarter versus 2020 Period. The underwriting expense ratio for the mortgage segment was 22.5% in the 2021 third quarter, compared to 20.6% in the 2020 third quarter, with the increase primarily due to a lower level in net premiums earned on U.S. primary mortgage insurance business.
Nine Months Ended September 30, 2021 versus 2020 Period. The underwriting expense ratio for the mortgage segment was 23.2% for the nine months ended September 30, 2021, compared to 21.5% for the 2020 period, with the increase primarily due to a lower level in net premiums earned on U.S. primary mortgage insurance business.
Corporate (Non-Underwriting) Segment
The corporate (non-underwriting) segment results include net investment income, other income (loss), corporate expenses, transaction costs and other, amortization of intangible assets, interest expense, items related to our non-cumulative preferred shares, net realized gains or losses (which includes changes in the allowance for credit losses on financial assets and net impairment losses recognized in earnings), equity in net income or loss of investment funds accounted for using the equity method, net foreign exchange gains or losses, income or loss from operating affiliates and income taxes. Such amounts exclude the results of the ‘other’ segment. See note 1, “Basis of Presentation and Recent Accounting Pronouncements,” to our consolidated financial statements for information about the change in presentation of income or loss from operating affiliates.

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Net Investment Income.
The components of net investment income were derived from the following sources:

| Three Months Ended |  | Nine Months Ended
| September 30, |  | September 30,
| 2021 |  | 2020 |  | 2021 |  | 2020
Fixed maturities | $ | 75,964 |  |  | $ | 84,608 |  |  | $ | 232,690 |  |  | $ | 277,862 |
Equity securities | 9,867 |  |  | 6,659 |  |  | 23,799 |  |  | 18,312 |
Short-term investments | 1,858 |  |  | 1,162 |  |  | 3,474 |  |  | 5,444 |
Other (1) | 19,114 |  |  | 24,594 |  |  | 55,699 |  |  | 62,898 |
Gross investment income | 106,803 |  |  | 117,023 |  |  | 315,662 |  |  | 364,516 |
Investment expenses (2) | (18,608) |  |  | (17,166) |  |  | (59,308) |  |  | (50,600) |
Net investment income | $ | 88,195 |  |  | $ | 99,857 |  |  | $ | 256,354 |  |  | 313,916 |

(1) Amounts include dividends and other distributions on investment funds, term loan investments, funds held balances, cash balances and other items.
(2) Investment expenses were approximately 0.32% of average invested assets for the 2021 third quarter, compared to 0.32% for the 2020 third quarter, and 0.32% for the nine months ended September 30, 2021, compared to 0.31% for the 2020 period.
The lower level of net investment income for the 2021 third quarter primarily related to lower yields available in the financial market. The pre-tax investment income yield, calculated based on amortized cost and on an annualized basis, was 1.41% for the 2021 third quarter, compared to 1.76% for the 2020 third quarter, and 1.40% for the nine months ended September 30, 2021, compared to 1.94% for the 2020 period.
Corporate Expenses.
Corporate expenses were $18.6 million for the 2021 third quarter, compared to $16.3 million for the 2020 third quarter, and $59.3 million for the nine months ended September 30, 2021, compared to $51.4 million for the 2020 period. The increase in corporate expenses was primarily due to higher incentive compensation costs.
Transaction Costs and Other.
Transaction costs and other were $1.0 million for the 2021 third quarter, compared to $1.7 million for the 2020 third quarter, and $0.8 million for the nine months ended September 30, 2021, compared to $5.2 million for the 2020 period. Amounts in the 2021 and 2020 periods are primarily related to acquisitions activity for the respective period.
Amortization of Intangible Assets.
Amortization of intangible assets for the 2021 third quarter was $20.1 million, compared to $16.7 million for the 2020 third quarter, and $48.9 million for the nine months ended
September 30, 2021, compared to $49.8 million for the 2020 period. Amounts in 2021 and 2020 primarily related to amortization of finite-lived intangible assets. The increase in amortization of intangible assets expense was a result of acquisitions closed during the 2021 third quarter.
Interest Expense.
Interest expense was $33.2 million for the 2021 third quarter, compared to the $36.2 million for the 2020 third quarter, and $98.8 million for the nine months ended September 30, 2021, compared to $86.6 million for the 2020 period.The higher level of interest expense in 2021 period mainly resulted from the issuance of $1.0 billion of 3.635% senior notes on June 30, 2020.
Loss on Redemption of Preferred Shares.
In September 2021, we redeemed all 5.25% Series E preferred shares and, in accordance with GAAP, we recorded a loss of $15.1 million to remove original issuance costs related to the redeemed shares from additional paid-in capital. Such adjustment had no impact on total shareholders’ equity or cash flows.
Net Realized Gains or Losses.
We recorded net realized losses of $25.0 million for the 2021 third quarter, compared to net realized gains of $211.0 million for the 2020 third quarter, and net realized gains of $239.7 million for the nine months ended September 30, 2021, compared to net realized gains of $524.0 million for the 2020 period. In addition, 2021 third quarter included $33.1 million loss as a result of deconsolidation of Watford in our financial statements following the close of the transaction. Currently, our portfolio is actively managed to maximize total return within certain guidelines. The effect of financial market movements on the investment portfolio will directly impact net realized gains and losses as the portfolio is adjusted and rebalanced. Net realized gains or losses from the sale of fixed maturities primarily results from our decisions to reduce credit exposure, to change duration targets, to rebalance our portfolios or due to relative value determinations.
Net realized gains or losses also include realized and unrealized contract gains and losses on our derivative instruments, changes in the fair value of assets accounted for using the fair value option and in the fair value of equities, along with changes in the allowance for credit losses on financial assets and net impairment losses recognized in earnings. See note 8, “Investment Information—Net Realized Gains (Losses),” to our consolidated financial statements for additional information. See note 8, “Investment Information—Allowance for Credit Losses,” to our consolidated financial statements for additional information.

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Equity in Net Income (Loss) of Investment Funds Accounted for Using the Equity Method.
We recorded $105.4 million of equity in net income related to investment funds accounted for using the equity method in the 2021 third quarter, compared to income of $126.7 million for the 2020 third quarter, and $299.3 million of income for the nine months ended September 30, 2021, compared to income of $57.4 million for the 2020 period. Such investments are generally recorded on a one to three month lag based on the availability of reports from the investment funds. Investment funds accounted for using the equity method totaled $2.7 billion at September 30, 2021, compared to $2.0 billion at December 31, 2020. See note 8, “Investment Information—Investments Accounted For Using the Equity Method,” to our consolidated financial statements for additional information.
Net Foreign Exchange Gains or Losses.
Net foreign exchange gains for the 2021 third quarter were $36.1 million, compared to net foreign exchange losses for the 2020 third quarter of $38.7 million. Net foreign exchange gains for the nine months ended September 30, 2021 were $39.7 million, compared to net foreign exchange losses for the 2020 period of $17.8 million. Amounts in both periods were primarily unrealized and resulted from the effects of revaluing our net insurance liabilities required to be settled in foreign currencies at each balance sheet date.
Income Tax Expense.
Our income tax provision on income (loss) before income taxes, including income (loss) from operating affiliates, resulted in an expense of 1.0% for the 2021 third quarter, compared to 5.4% for the 2020 third quarter, and 5.8% for the nine months ended September 30, 2021, compared to 8.3% for the 2020 period. The effective tax rates for the 2021 third quarter and nine months ended September 30, 2021 included discrete income tax benefits of $25.3 million and $28.7 million, respectively. The discrete tax items primarily related to the partial release of a valuation allowance on certain U.K. deferred tax assets in the third quarter. Our effective tax rate, which is based upon the expected annual effective tax rate, may fluctuate from period to period based on the relative mix of income or loss reported by jurisdiction and the varying tax rates in each jurisdiction.
Income (loss) from operating affiliates.
We recorded $124.1 million of net income from our operating affiliates in the 2021 third quarter, compared to income of $0.9 million for the 2020 third quarter, and $224.1 million of income for the nine months ended September 30, 2021, compared to $6.3 million for the 2020 period. Results for the 2021 third quarter reflected a one-time gain of $95.7 million recognized from the Company’s previously disclosed
acquisition of a 40% share of Greysbridge. Results for 2021 period, primarily include income from our investment in Coface, Greysbridge and Premia.
Other Segment
Through June 30, 2021, the ‘other’ segment included the results of Watford. Pursuant to GAAP, Watford was considered a variable interest entity and we concluded that we were the primary beneficiary of Watford. As such, we consolidated the results of Watford in our consolidated financial statements through June 30, 2021. In July 2021, we announced the completion of the previously disclosed acquisition of Watford by Greysbridge. Based on the governing documents of Greysbridge, the Company has concluded that, while it retains significant influence over Watford, Watford no longer constitutes a variable interest entity. Accordingly, effective July 1, 2021, Arch no longer consolidates the results of Watford in its consolidated financial statements. See note 12, “Variable Interest Entities and Noncontrolling Interests” and note 5, “Segment Information” to our consolidated financial statements for additional information on Watford.
CRITICAL ACCOUNTING POLICIES,
ESTIMATES AND RECENT ACCOUNTING PRONOUNCEMENTS

Critical accounting policies, estimates and recent accounting pronouncements are discussed in Management’s Discussion and Analysis of Financial Condition and Results of Operations contained in our 2020 Form 10-K, updated where applicable in the notes accompanying our consolidated financial statements, including note 1, “Basis of Presentation and Recent Accounting Pronouncements.”

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FINANCIAL CONDITION

Investable Assets Held by Arch
The following table summarizes the fair value of the investable assets held by Arch:

Investable assets (1): | Estimated Fair Value |  | % of Total |  |
September 30, 2021 |  |  |  |  |
Fixed maturities (2) | $ | 17,182,370 |  |  | 63.0 |  |  |
Short-term investments (2) | 3,185,646 |  |  | 11.7 |  |  |
Cash | 1,137,721 |  |  | 4.2 |  |  |
Equity securities (2) | 1,815,163 |  |  | 6.7 |  |  |
Other investments (2) | 1,489,759 |  |  | 5.5 |  |  |
Other investable assets (3) | — |  |  | — |  |  |
Investments accounted for using the equity method | 2,741,293 |  |  | 10.0 |  |  |
Securities transactions entered into but not settled at the balance sheet date | (273,512) |  |  | (1.0) |  |  |
Total investable assets held by Arch | $ | 27,278,440 |  |  | 100.0 |  |  |
Average effective duration (in years) | 2.68 |  |  |  |  |
Average S&P/Moody’s credit ratings (4) | AA-/Aa3 |  |  |  |
Embedded book yield (5) | 1.54 | % |  |  |  |
December 31, 2020 |  |  |  |  |
Fixed maturities (2) | $ | 18,771,296 |  |  | 69.9 |  |  |
Short-term investments (2) | 2,063,240 |  |  | 7.7 |  |  |
Cash | 694,997 |  |  | 2.6 |  |  |
Equity securities (2) | 1,436,104 |  |  | 5.3 |  |  |
Other investments (2) | 1,480,347 |  |  | 5.5 |  |  |
Other investable assets (3) | 500,000 |  |  | 1.9 |  |  |
Investments accounted for using the equity method | 2,047,889 |  |  | 7.6 |  |  |
Securities transactions entered into but not settled at the balance sheet date | (137,578) |  |  | (0.5) |  |  |
Total investable assets held by Arch | $ | 26,856,295 |  |  | 100.0 |  |  |
Average effective duration (in years) | 3.01 |  |  |  |  |
Average S&P/Moody’s credit ratings (4) | AA/Aa2 |  |  |  |
Embedded book yield (5) | 1.56 | % |  |  |  |

(1)In securities lending transactions, we receive collateral in excess of the fair value of the securities pledged. For purposes of this table, we have excluded the collateral received under securities lending, at fair value and included the securities pledged under securities lending, at fair value. In September 2021, the Company terminated its securities lending program.
(2)Includes investments carried as available for sale, at fair value and at fair value under the fair value option.
(3)Represents participation interests in a receivable of a reverse repurchase agreement.
(4)Average credit ratings on our investment portfolio on securities with ratings by Standard & Poor’s Rating Services (“S&P”) and Moody’s Investors Service (“Moody’s”).
(5)Before investment expenses.
At September 30, 2021, approximately $18.4 billion, or 67.3%, of total investable assets held by Arch were internally managed, compared to $19.2 billion, or 71.4%, at December 31, 2020.
The following table summarizes our fixed maturities and fixed maturities pledged under securities lending agreements (“Fixed Maturities”) by type:

| Estimated Fair Value |  | % of Total
September 30, 2021 |  |  |
Corporate bonds | $ | 6,777,943 |  |  | 39.4 |
Residential mortgage backed securities | 405,797 |  |  | 2.4 |
Municipal bonds | 382,722 |  |  | 2.2 |
Commercial mortgage backed securities | 579,424 |  |  | 3.4 |
U.S. government and government agencies | 4,460,515 |  |  | 26.0 |
Non-U.S. government securities | 1,883,563 |  |  | 11.0 |
Asset backed securities | 2,692,406 |  |  | 15.7 |
Total | $ | 17,182,370 |  |  | 100.0 |
December 31, 2020 |  |  |
Corporate bonds | $ | 8,039,745 |  |  | 42.8 |
Residential mortgage backed securities | 616,619 |  |  | 3.3 |
Municipal bonds | 492,734 |  |  | 2.6 |
Commercial mortgage backed securities | 390,990 |  |  | 2.1 |
U.S. government and government agencies | 5,354,863 |  |  | 28.5 |
Non-U.S. government securities | 2,310,157 |  |  | 12.3 |
Asset backed securities | 1,566,188 |  |  | 8.3 |
Total | $ | 18,771,296 |  |  | 100.0 |

The following table provides the credit quality distribution of our Fixed Maturities. For individual fixed maturities, S&P ratings are used. In the absence of an S&P rating, ratings from Moody’s are used, followed by ratings from Fitch Ratings.

| Estimated Fair Value |  | % of Total
September 30, 2021 |  |  |
U.S. government and gov’t agencies (1) | $ | 4,830,467 |  |  | 28.1 |
AAA | 3,257,679 |  |  | 19.0 |
AA | 2,217,452 |  |  | 12.9 |
A | 2,773,104 |  |  | 16.1 |
BBB | 2,807,788 |  |  | 16.3 |
BB | 522,357 |  |  | 3.0 |
B | 348,036 |  |  | 2.0 |
Lower than B | 43,751 |  |  | 0.3 |
Not rated | 381,736 |  |  | 2.2 |
Total | $ | 17,182,370 |  |  | 100.0 |
December 31, 2020 |  |  |
U.S. government and gov’t agencies (1) | $ | 5,963,758 |  |  | 31.8 |
AAA | 3,117,046 |  |  | 16.6 |
AA | 2,063,738 |  |  | 11.0 |
A | 3,760,280 |  |  | 20.0 |
BBB | 2,699,201 |  |  | 14.4 |
BB | 574,189 |  |  | 3.1 |
B | 268,095 |  |  | 1.4 |
Lower than B | 54,795 |  |  | 0.3 |
Not rated | 270,194 |  |  | 1.4 |
Total | $ | 18,771,296 |  |  | 100.0 |

(1)Includes U.S. government-sponsored agency residential mortgage-backed securities and agency commercial mortgage-backed securities.

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The following table provides information on the severity of the unrealized loss position as a percentage of amortized cost for all Fixed Maturities which were in an unrealized loss position:

Severity of gross unrealized losses: | Estimated Fair Value |  | Gross Unrealized Losses |  | % of Total Gross Unrealized Losses
September 30, 2021 |  |  |  |  |
0-10% | $ | 9,287,536 |  |  | $ | (106,042) |  |  | 93.6 |
10-20% | 27,315 |  |  | (4,173) |  |  | 3.7 |
20-30% | 8,233 |  |  | (2,034) |  |  | 1.8 |
Greater than 30% | 993 |  |  | (1,088) |  |  | 1.0 |
Total | $ | 9,324,077 |  |  | $ | (113,337) |  |  | 100.0 |
December 31, 2020 |  |  |  |  |
0-10% | $ | 3,583,981 |  |  | $ | (55,542) |  |  | 79.4 |
10-20% | 95,495 |  |  | (12,183) |  |  | 17.4 |
20-30% | 1,061 |  |  | (406) |  |  | 0.6 |
Greater than 30% | 1,249 |  |  | (1,785) |  |  | 2.6 |
Total | $ | 3,681,786 |  |  | $ | (69,916) |  |  | 100.0 |

The following table summarizes our top ten exposures to fixed income corporate issuers by fair value at September 30, 2021, excluding guaranteed amounts and covered bonds:

| Estimated Fair Value |  | Credit Rating (1)
Bank of America Corporation | $ | 359,893 |  |  | A-/A2
JPMorgan Chase & Co. | 294,411 |  |  | A-/A2
Citigroup Inc. | 247,892 |  |  | BBB+/A3
Wells Fargo & Company | 231,542 |  |  | BBB+/A1
Morgan Stanley | 215,424 |  |  | BBB+/A1
The Goldman Sachs Group, Inc. | 180,324 |  |  | BBB+/A2
Dai-ichi Life Holdings, Inc. | 111,418 |  |  | AA-/A1
Apple Inc. | 111,280 |  |  | AA+/Aa1
Westpac Banking Corporation | 107,389 |  |  | AA-/Aa3
Nestlé S.A. | 83,838 |  |  | AA-/Aa3
Total | $ | 1,943,411 |  |  |

(1)Average credit ratings as assigned by S&P and Moody’s, respectively.
The following table provides information on our structured securities, which includes residential mortgage-backed securities (“RMBS”), commercial mortgage-backed securities (“CMBS”) and asset-backed securities (“ABS”):

| Agencies |  | Investment Grade |  | Below Investment Grade |  | Total
Sep 30, 2021 |  |  |  |  |  |  |
RMBS | $ | 345,175 |  |  | $ | 48,301 |  |  | $ | 12,321 |  |  | $ | 405,797 |
CMBS | 24,779 |  |  | 502,218 |  |  | 52,427 |  |  | 579,424 |
ABS | — |  |  | 2,374,981 |  |  | 317,425 |  |  | 2,692,406 |
Total | $ | 369,954 |  |  | $ | 2,925,500 |  |  | $ | 382,173 |  |  | $ | 3,677,627 |
Dec 31, 2020 |  |  |  |  |  |  |
RMBS | $ | 584,499 |  |  | $ | 4,102 |  |  | $ | 28,018 |  |  | $ | 616,619 |
CMBS | 24,396 |  |  | 342,491 |  |  | 24,103 |  |  | 390,990 |
ABS | — |  |  | 1,403,137 |  |  | 163,051 |  |  | 1,566,188 |
Total | $ | 608,895 |  |  | $ | 1,749,730 |  |  | $ | 215,172 |  |  | $ | 2,573,797 |

The following table summarizes our equity securities, which include investments in exchange traded funds:

| September 30, 2021 |  | December 31, 2020
Equities (1) | $ | 856,837 |  |  | $ | 676,437 |
Exchange traded funds |  |  |
Fixed income (2) | 348,613 |  |  | 341,139 |
Equity and other (3) | 609,713 |  |  | 418,528 |
Total | $ | 1,815,163 |  |  | $ | 1,436,104 |

(1)Primarily in consumer non-cyclical, technology, communications financial and consumer cyclical at September 30, 2021.
(2)Primarily in corporate at September 30, 2021.
(3)Primarily in large cap stocks, foreign equities, technology, financial and utilities at September 30, 2021.
The following table summarizes our other investments and other investable assets:

| September 30, 2021 |  | December 31, 2020
Lending | $ | 579,563 |  |  | $ | 572,636 |
Term loan investments | 528,585 |  |  | 380,193 |
Energy | 84,880 |  |  | 65,813 |
Credit related funds | 56,997 |  |  | 90,780 |
Investment grade fixed income | 131,910 |  |  | 138,646 |
Infrastructure | 26,359 |  |  | 165,516 |
Private equity | 81,465 |  |  | 48,750 |
Real estate | — |  |  | 18,013 |
Total fair value option | $ | 1,489,759 |  |  | $ | 1,480,347 |
Other investable assets | — |  |  | 500,000 |
Total other investments | $ | 1,489,759 |  |  | $ | 1,980,347 |

For details on our investments accounted for using the equity method, see note 8, “Investment Information—Investments Accounted For Using the Equity Method,” to our consolidated financial statements.

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Our investment strategy allows for the use of derivative instruments. We utilize various derivative instruments such as futures contracts to enhance investment performance, replicate investment positions or manage market exposures and duration risk that would be allowed under our investment guidelines if implemented in other ways. See note 10, “Derivative Instruments,” to our consolidated financial statements for additional disclosures related to derivatives.
Accounting guidance regarding fair value measurements addresses how companies should measure fair value when they are required to use a fair value measure for recognition or disclosure purposes under GAAP and provides a common definition of fair value to be used throughout GAAP. See note 9, “Fair Value,” to our consolidated financial statements for a summary of our financial assets and liabilities measured at fair value, segregated by level in the fair value hierarchy.
Reinsurance
The effects of reinsurance on written and earned premiums and losses and loss adjustment expenses (“LAE”) with unaffiliated reinsurers were as follows:

| Three Months Ended |  | Nine Months Ended
| September 30, |  | September 30,
| 2021 |  | 2020 |  | 2021 |  | 2020
Premiums written: |  |  |  |  |  |  |
Direct | $ | 1,993,098 |  |  | $ | 1,667,449 |  |  | $ | 5,795,236 |  |  | $ | 4,841,874 |
Assumed | 1,214,317 |  |  | 1,013,583 |  |  | 4,095,676 |  |  | 2,989,680 |
Ceded | (1,131,486) |  |  | (806,888) |  |  | (2,907,002) |  |  | (2,151,853) |
Net | $ | 2,075,929 |  |  | $ | 1,874,144 |  |  | $ | 6,983,910 |  |  | $ | 5,679,701 |
Premiums earned: |  |  |  |  |  |  |
Direct | $ | 1,754,462 |  |  | $ | 1,618,583 |  |  | $ | 5,263,286 |  |  | $ | 4,723,630 |
Assumed | 1,129,434 |  |  | 880,024 |  |  | 3,169,016 |  |  | 2,387,286 |
Ceded | (954,559) |  |  | (727,515) |  |  | (2,433,634) |  |  | (1,930,026) |
Net | $ | 1,929,337 |  |  | $ | 1,771,092 |  |  | $ | 5,998,668 |  |  | $ | 5,180,890 |
Losses and LAE: |  |  |  |  |  |  |
Direct | $ | 1,101,793 |  |  | $ | 1,131,696 |  |  | $ | 3,134,305 |  |  | $ | 3,279,737 |
Assumed | 961,285 |  |  | 635,199 |  |  | 2,182,852 |  |  | 1,689,176 |
Ceded | (837,059) |  |  | (550,622) |  |  | (1,728,207) |  |  | (1,406,699) |
Net | $ | 1,226,019 |  |  | $ | 1,216,273 |  |  | $ | 3,588,950 |  |  | $ | 3,562,214 |

See note 7, “Allowance for Expected Credit Losses,” to our consolidated financial statements for information about our reinsurance recoverables and related allowance for credit losses.
Bellemeade Re
We have entered into aggregate excess of loss mortgage reinsurance agreements with various special purpose reinsurance companies domiciled in Bermuda (the “Bellemeade Agreements”). For the respective coverage periods, we will retain the first layer of the respective aggregate losses and the special purpose reinsurance companies will provide second layer coverage up to the outstanding coverage amount. We will then retain losses in excess of the outstanding coverage limit. The aggregate excess of loss reinsurance coverage generally decreases over a ten-year period as the underlying covered mortgages amortize, unless provisional call options embedded within certain of the Bellemeade Agreements are executed or if pre-defined delinquency triggering events occur.
The following table summarizes the respective coverages and retentions at September 30, 2021:

|  |  | September 30, 2021
| Initial Coverage at Issuance |  | Current Coverage |  | Remaining Retention, Net
Bellemeade 2017-1 Ltd. (1) | $ | 368,114 |  |  | $ | 145,573 |  |  | $ | 124,777 |
Bellemeade 2018-1 Ltd. (2) | 374,460 |  |  | 228,938 |  |  | 121,411 |
Bellemeade 2018-3 Ltd. (3) | 506,110 |  |  | 302,563 |  |  | 126,578 |
Bellemeade 2019-1 Ltd. (4) | 341,790 |  |  | 210,529 |  |  | 98,340 |
Bellemeade 2019-2 Ltd. (5) | 621,022 |  |  | 398,316 |  |  | 156,085 |
Bellemeade 2019-3 Ltd. (6) | 700,920 |  |  | 491,634 |  |  | 178,759 |
Bellemeade 2019-4 Ltd. (7) | 577,267 |  |  | 468,737 |  |  | 113,674 |
Bellemeade 2020-2 Ltd. (8) | 449,167 |  |  | 268,468 |  |  | 227,140 |
Bellemeade 2020-3 Ltd. (9) | 451,816 |  |  | 405,881 |  |  | 159,615 |
Bellemeade 2020-4 Ltd. (10) | 337,013 |  |  | 238,480 |  |  | 133,872 |
Bellemeade 2021-1 Ltd. (11) | 643,577 |  |  | 643,577 |  |  | 157,685 |
Bellemeade 2021-2 Ltd. (12) | 616,017 |  |  | 616,017 |  |  | 146,956 |
Bellemeade 2021-3 Ltd. (13) | 639,391 |  |  | 639,391 |  |  | 145,790 |
Total | $ | 6,626,664 |  |  | $ | 5,058,104 |  |  | $ | 1,890,682 |

(1) Issued in October 2017, covering in-force policies issued between January 1, 2017 and June 30, 2017.
(2) Issued in April 2018, covering in-force policies issued between July 1, 2017 and December 31, 2017.
(3) Issued in October 2018, covering in-force policies issued between January 1, 2018 and June 30, 2018.
(4) Issued in March 2019, covering in-force policies primarily issued between 2005-2008 under United Guaranty Residential Insurance Company (“UGRIC”); as well as policies issued through 2015 under both UGRIC and Arch Mortgage Insurance Company.
(5) Issued in April 2019, covering in-force policies issued between July 1, 2018 and December 31, 2018.
(6) Issued in July 2019, covering in-force policies issued in 2016.
(7) Issued in October 2019, covering in-force policies issued between January 1, 2019 and June 30, 2019.
(8) Issued in September 2020, covering in-force policies issued between January 1, 2020 and May 31, 2020. $423 million was directly funded by Bellemeade 2020-2 Ltd. with an additional $26 million of capacity provided directly to Arch MI U.S. by a separate panel of reinsurers.

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(9) Issued in November 2020, covering in-force policies issued between June 1, 2020 and August 31, 2020. $418 million was directly funded by Bellemeade 2020-3 Ltd. with an additional $34 million of capacity provided directly to Arch MI U.S. by a separate panel of reinsurers.
(10) Issued in December 2020, covering in-force policies issued between July 1, 2019 and December 31, 2019. $321 million was directly funded by Bellemeade 2020-4 Ltd. with an additional $16 million of capacity provided directly to Arch MI U.S. by a separate panel of reinsurers.
(11) Issued in March 2021, covering in-force policies issued between September 1, 2020 and November 30, 2020. $580 million was directly funded by Bellemeade Re 2021-1 Ltd. with an additional $64 million capacity provided directly to Arch MI U.S. by a separate panel of reinsurers.
(12) Issued in June 2021, covering in-force policies issued between December 1, 2020 and March 31, 2021. $523 million was directly funded by Bellemeade Re 2021-2 Ltd. via insurance-linked notes, with an additional $93 million capacity provided directly to Arch MI U.S. by a separate panel of reinsurers.
(13) Issued in September 2021, covering in-force policies issued between April 1, 2021 and June 30, 2021. $508 million was directly funded by Bellemeade Re 2021-3 Ltd. via insurance-linked notes, with an additional $131 million capacity provided directly to Arch MI U.S. by a separate panel of reinsurers.
Reserve for Losses and Loss Adjustment Expenses
We establish reserve for losses and loss adjustment expenses (“Loss Reserves”) which represent estimates involving actuarial and statistical projections, at a given point in time, of our expectations of the ultimate settlement and administration costs of losses incurred. Estimating Loss Reserves is inherently difficult. We utilize actuarial models as well as available historical insurance industry loss ratio experience and loss development patterns to assist in the establishment of Loss Reserves. Actual losses and loss adjustment expenses paid will deviate, perhaps substantially, from the reserve estimates reflected in our financial statements.
At September 30, 2021 and December 31, 2020, our Loss Reserves, net of unpaid losses and loss adjustment expenses recoverable, by type and by operating segment were as follows:

| September 30, 2021 |  | December 31, 2020
Insurance segment: |  |  |
Case reserves | $ | 2,147,930 |  |  | $ | 2,051,640 |
IBNR reserves | 4,341,016 |  |  | 3,889,823 |
Total net reserves | 6,488,946 |  |  | 5,941,463 |
Reinsurance segment: |  |  |
Case reserves | 1,562,072 |  |  | 1,560,523 |
Additional case reserves | 549,476 |  |  | 280,472 |
IBNR reserves | 2,617,728 |  |  | 2,253,953 |
Total net reserves | 4,729,276 |  |  | 4,094,948 |
Mortgage segment: |  |  |
Case reserves | 731,671 |  |  | 631,921 |
IBNR reserves | 266,007 |  |  | 271,702 |
Total net reserves | 997,678 |  |  | 903,623 |
Other segment: |  |  |
Case reserves | — |  |  | 566,587 |
Additional case reserves | — |  |  | 32,321 |
IBNR reserves | — |  |  | 660,132 |
Total net reserves | — |  |  | 1,259,040 |
Total: |  |  |
Case reserves | 4,441,673 |  |  | 4,810,671 |
Additional case reserves | 549,476 |  |  | 312,793 |
IBNR reserves | 7,224,751 |  |  | 7,075,610 |
Total net reserves | $ | 12,215,900 |  |  | $ | 12,199,074 |

At September 30, 2021 and December 31, 2020, the insurance segment’s Loss Reserves by major line of business, net of unpaid losses and loss adjustment expenses recoverable, were as follows:

| September 30, 2021 |  | December 31, 2020
Insurance segment: |  |  |
Professional lines (1) | $ | 1,587,042 |  |  | $ | 1,482,820 |
Construction and national accounts | 1,468,668 |  |  | 1,395,067 |
Excess and surplus casualty (2) | 903,966 |  |  | 816,495 |
Programs | 760,273 |  |  | 699,354 |
Property, energy, marine and aviation | 599,431 |  |  | 517,692 |
Travel, accident and health | 93,632 |  |  | 98,910 |
Lenders products | 70,520 |  |  | 48,946 |
Other (3) | 1,005,414 |  |  | 882,179 |
Total net reserves | $ | 6,488,946 |  |  | $ | 5,941,463 |

(1)Includes professional liability, executive assurance and healthcare business.
(2)Includes casualty and contract binding business.
(3)Includes alternative markets, excess workers’ compensation and surety business.

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At September 30, 2021 and December 31, 2020, the reinsurance segment’s Loss Reserves by major line of business, net of unpaid losses and loss adjustment expenses recoverable, were as follows:

| September 30, 2021 |  | December 31, 2020
Reinsurance segment: |  |  |
Casualty (1) | $ | 2,079,990 |  |  | $ | 1,995,849 |
Other specialty (2) | 1,093,611 |  |  | 917,178 |
Property excluding property catastrophe | 669,922 |  |  | 594,033 |
Marine and aviation | 231,203 |  |  | 204,205 |
Property catastrophe | 526,356 |  |  | 268,858 |
Other (3) | 128,194 |  |  | 114,825 |
Total net reserves | $ | 4,729,276 |  |  | $ | 4,094,948 |

(1)Includes executive assurance, professional liability, workers’ compensation, excess motor, healthcare and other.
(2)Includes non-excess motor, surety, accident and health, workers’ compensation catastrophe, agriculture, trade credit and other.
(3)Includes life, casualty clash and other.
At September 30, 2021 and December 31, 2020, the mortgage segment’s Loss Reserves by major line of business, net of unpaid losses and loss adjustment expenses recoverable, were as follows:

| September 30, 2021 |  | December 31, 2020
U.S. primary mortgage insurance (1) | $ | 726,636 |  |  | $ | 649,748 |
Other | 271,042 |  |  | 253,875 |
Total net reserves | $ | 997,678 |  |  | $ | 903,623 |

(1) At September 30, 2021, 27.9% represents policy years 2011 and prior and the remainder from later policy years. At December 31, 2020, 28.3% of total net reserves represent policy years 2011 and prior and the remainder from later policy years.
Mortgage Operations Supplemental Information
The mortgage segment’s insurance in force (“IIF”) and risk in force (“RIF”) were as follows at September 30, 2021 and December 31, 2020:

(U.S. Dollars in millions) | September 30, 2021 |  | December 31, 2020
Amount |  | % |  | Amount |  | %
Insurance In Force (IIF) (1): |  |  |  |  |  |  |
U.S. primary mortgage insurance | $ | 280,379 |  |  | 61.3 |  |  | $ | 280,579 |  |  | 66.2 |
U.S. credit risk transfer (CRT) and other (2) | 108,203 |  |  | 23.6 |  |  | 103,535 |  |  | 24.4 |
International mortgage insurance/reinsurance (3) | 69,127 |  |  | 15.1 |  |  | 39,425 |  |  | 9.3 |
Total | $ | 457,709 |  |  | 100.0 |  |  | $ | 423,539 |  |  | 100.0 |
Risk In Force (RIF) (4): |  |  |  |  |  |  |
U.S. primary mortgage insurance | $ | 70,320 |  |  | 84.8 |  |  | $ | 70,522 |  |  | 90.5 |
U.S. credit risk transfer (CRT) and other (2) | 4,817 |  |  | 5.8 |  |  | 4,699 |  |  | 6.0 |
International mortgage insurance/reinsurance (3) | 7,803 |  |  | 9.4 |  |  | 2,673 |  |  | 3.4 |
Total | $ | 82,940 |  |  | 100.0 |  |  | $ | 77,894 |  |  | 100.0 |

(1)Represents the aggregate dollar amount of each insured mortgage loan’s current principal balance.
(2)Includes all CRT transactions, which are predominantly with GSEs, and other U.S. reinsurance transactions.
(3)International mortgage insurance and reinsurance with risk primarily located in Australia, which reflects WLMI acquisition in the 2021 third quarter and to lesser extent Europe and Asia.
(4)The aggregate dollar amount of each insured mortgage loan’s current principal balance multiplied by the insurance coverage percentage specified in the policy for insurance policies issued and after contract limits and/or loss ratio caps for risk-sharing or reinsurance.
The IIF and RIF for our U.S. primary mortgage insurance business by policy year were as follows at September 30, 2021:

(U.S. Dollars in millions) | IIF |  | RIF |  | Delinquency
Amount |  | % |  | Amount |  | % |  | Rate (1)
Policy year: |  |  |  |  |  |  |  |  |
2011 and prior | $ | 11,888 |  |  | 4.2 |  |  | $ | 2,672 |  |  | 3.8 |  |  | 9.51 | %
2012 | 1,974 |  |  | 0.7 |  |  | 505 |  |  | 0.7 |  |  | 2.76 | %
2013 | 4,876 |  |  | 1.7 |  |  | 1,342 |  |  | 1.9 |  |  | 2.72 | %
2014 | 5,454 |  |  | 1.9 |  |  | 1,501 |  |  | 2.1 |  |  | 3.37 | %
2015 | 9,810 |  |  | 3.5 |  |  | 2,637 |  |  | 3.8 |  |  | 2.92 | %
2016 | 16,150 |  |  | 5.8 |  |  | 4,324 |  |  | 6.1 |  |  | 3.64 | %
2017 | 15,001 |  |  | 5.4 |  |  | 3,910 |  |  | 5.6 |  |  | 4.50 | %
2018 | 16,193 |  |  | 5.8 |  |  | 4,105 |  |  | 5.8 |  |  | 5.75 | %
2019 | 29,860 |  |  | 10.6 |  |  | 7,463 |  |  | 10.6 |  |  | 3.58 | %
2020 | 88,737 |  |  | 31.6 |  |  | 21,777 |  |  | 31.0 |  |  | 0.91 | %
2021 | 80,436 |  |  | 28.7 |  |  | 20,084 |  |  | 28.6 |  |  | 0.19 | %
Total | $ | 280,379 |  |  | 100.0 |  |  | $ | 70,320 |  |  | 100.0 |  |  | 2.67 | %

(1)Represents the ending percentage of loans in default.

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The IIF and RIF for our U.S. primary mortgage insurance business by policy year were as follows at December 31, 2020:

(U.S. Dollars in millions) | IIF |  | RIF |  | Delinquency
Amount |  | % |  | Amount |  | % |  | Rate (1)
Policy year: |  |  |  |  |  |  |  |  |
2011 and prior | $ | 14,588 |  |  | 5.2 |  |  | $ | 3,327 |  |  | 4.7 |  |  | 11.36 | %
2012 | 3,651 |  |  | 1.3 |  |  | 992 |  |  | 1.4 |  |  | 2.98 | %
2013 | 7,546 |  |  | 2.7 |  |  | 2,107 |  |  | 3.0 |  |  | 3.30 | %
2014 | 8,261 |  |  | 2.9 |  |  | 2,273 |  |  | 3.2 |  |  | 4.06 | %
2015 | 15,032 |  |  | 5.4 |  |  | 4,048 |  |  | 5.7 |  |  | 3.72 | %
2016 | 24,958 |  |  | 8.9 |  |  | 6,648 |  |  | 9.4 |  |  | 4.77 | %
2017 | 24,748 |  |  | 8.8 |  |  | 6,413 |  |  | 9.1 |  |  | 5.52 | %
2018 | 27,304 |  |  | 9.7 |  |  | 6,918 |  |  | 9.8 |  |  | 6.76 | %
2019 | 48,304 |  |  | 17.2 |  |  | 12,001 |  |  | 17.0 |  |  | 4.61 | %
2020 | 106,187 |  |  | 37.8 |  |  | 25,795 |  |  | 36.6 |  |  | 0.76 | %
Total | $ | 280,579 |  |  | 100.0 |  |  | $ | 70,522 |  |  | 100.0 |  |  | 4.19 | %

(1)Represents the ending percentage of loans in default.
The following tables provide supplemental disclosures on risk in force for our U.S. primary mortgage insurance business at September 30, 2021 and December 31, 2020:

(U.S. Dollars in millions) | September 30, 2021 |  | December 31, 2020
Amount |  | % |  | Amount |  | %
Credit quality (FICO): |  |  |  |  |  |  |
>=740 | $ | 41,927 |  |  | 59.6 |  |  | $ | 40,774 |  |  | 57.8 |
680-739 | 23,732 |  |  | 33.7 |  |  | 24,498 |  |  | 34.7 |
620-679 | 4,323 |  |  | 6.1 |  |  | 4,837 |  |  | 6.9 |
<620 | 338 |  |  | 0.5 |  |  | 413 |  |  | 0.6 |
Total | $ | 70,320 |  |  | 100.0 |  |  | $ | 70,522 |  |  | 100.0 |
Weighted average FICO score | 745 |  |  |  |  | 743 |  |  |
Loan-to-value (LTV): |  |  |  |  |  |  |
95.01% and above | $ | 7,708 |  |  | 11.0 |  |  | $ | 8,643 |  |  | 12.3 |
90.01% to 95.00% | 38,378 |  |  | 54.6 |  |  | 37,877 |  |  | 53.7 |
85.01% to 90.00% | 19,980 |  |  | 28.4 |  |  | 20,013 |  |  | 28.4 |
85.00% and below | 4,254 |  |  | 6.0 |  |  | 3,989 |  |  | 5.7 |
Total | $ | 70,320 |  |  | 100.0 |  |  | $ | 70,522 |  |  | 100.0 |
Weighted average LTV | 92.8 | % |  |  |  | 92.8 | % |  |
Total RIF, net of external reinsurance | $ | 54,847 |  |  |  |  | $ | 56,658 |  |  |

(U.S. Dollars in millions) | September 30, 2021 |  | December 31, 2020
Amount |  | % |  | Amount |  | %
Total RIF by State: |  |  |  |  |  |  |
Texas | $ | 5,590 |  |  | 7.9 |  |  | $ | 5,636 |  |  | 8.0 |
California | 5,451 |  |  | 7.8 |  |  | 5,261 |  |  | 7.5 |
Florida | 3,344 |  |  | 4.8 |  |  | 3,632 |  |  | 5.2 |
Minnesota | 2,936 |  |  | 4.2 |  |  | 2,520 |  |  | 3.6 |
North Carolina | 2,921 |  |  | 4.2 |  |  | 2,622 |  |  | 3.7 |
Illinois | 2,920 |  |  | 4.2 |  |  | 2,762 |  |  | 3.9 |
Georgia | 2,908 |  |  | 4.1 |  |  | 2,959 |  |  | 4.2 |
Massachusetts | 2,519 |  |  | 3.6 |  |  | 2,464 |  |  | 3.5 |
Virginia | 2,412 |  |  | 3.4 |  |  | 2,526 |  |  | 3.6 |
Ohio | 2,263 |  |  | 3.2 |  |  | 2,264 |  |  | 3.2 |
Other | 37,056 |  |  | 52.7 |  |  | 37,876 |  |  | 53.7 |
Total | $ | 70,320 |  |  | 100.0 |  |  | $ | 70,522 |  |  | 100.0 |

The following table provides supplemental disclosures for our U.S. primary mortgage insurance business related to insured loans and loss metrics:

(U.S. Dollars in thousands, except policy, loan and claim count) | Nine Months Ended
September 30,
2021 |  | 2020
Roll-forward of insured loans in default: |  |  |
Beginning delinquent number of loans | 52,234 |  |  | 20,163 |
New notices | 26,483 |  |  | 87,760 |
Cures | (46,334) |  |  | (48,234) |
Paid claims | (613) |  |  | (1,327) |
Ending delinquent number of loans (1) | 31,770 |  |  | 58,362 |
Ending number of policies in force (1) | 1,188,768 |  |  | 1,245,408 |
Delinquency rate (1) | 2.67 | % |  | 4.69 | %
Losses: |  |  |
Number of claims paid | 613 |  |  | 1,327 |
Total paid claims | $ | 22,848 |  |  | $ | 55,559 |
Average per claim | $ | 37.3 |  |  | $ | 41.9 |
Severity (2) | 80.2 | % |  | 93.3 | %
Average case reserve per default (in thousands) | $ | 23.5 |  |  | $ | 10.1 |

(1)Includes first lien primary and pool policies.
(2)Represents total paid claims divided by RIF of loans for which claims were paid.
The risk to capital ratio, which represents total current (non-delinquent) risk in force, net of reinsurance, divided by total statutory capital, for Arch MI U.S. was approximately 8.6 to 1 at September 30, 2021, compared to 9.3 to 1 at December 31, 2020.

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Shareholders’ Equity and Book Value per Share
The following table presents the calculation of book value per share:

(U.S. dollars in thousands, except share data) | September 30, 2021 |  | December 31, 2020
Total shareholders’ equity available to Arch | $ | 13,387,526 |  |  | $ | 13,105,886 |
Less preferred shareholders’ equity | 830,000 |  |  | 780,000 |
Common shareholders’ equity available to Arch | $ | 12,557,526 |  |  | $ | 12,325,886 |
Common shares and common share equivalents outstanding, net of treasury shares (1) | 387,257,752 |  |  | 406,720,642 |
Book value per share | $ | 32.43 |  |  | $ | 30.31 |

(1)Excludes the effects of 17,419,530 and 17,839,333 stock options and 739,407 and 1,153,784 restricted stock units outstanding at September 30, 2021 and December 31, 2020, respectively.
LIQUIDITY

Liquidity is a measure of our ability to access sufficient cash flows to meet the short-term and long-term cash requirements of our business operations.
Arch Capital is a holding company whose assets primarily consist of the shares in its subsidiaries. Generally, Arch Capital depends on its available cash resources, liquid investments and dividends or other distributions from its subsidiaries to make payments, including the payment of debt service obligations and operating expenses it may incur and any dividends or liquidation amounts with respect to our preferred and common shares.
For the nine months ended September 30, 2021, Arch Capital received dividends of $1.4 billion from Arch Re Bermuda, our Bermuda-based reinsurer and insurer, which can pay approximately $2.3 billion to Arch Capital during the remainder of 2021 without providing an affidavit to the Bermuda Monetary Authority (“BMA”).
For the nine months ended September 30, 2021, Arch-U.S. received $200.0 million of dividends from Arch U.S. MI Holdings Inc., a subsidiary of Arch-U.S., which received a total of $300.0 million of ordinary and extraordinary dividends, $140 million from United Guaranty Residential Insurance Company (“UGRIC”) and $160 million from Arch Mortgage Insurance Company (“AMIC”). UGRIC and AMIC have minimal ordinary dividend capacity remaining for the remainder of 2021.
In June 2021, Arch Capital completed a $500.0 million underwritten public offering of 20.0 million depositary shares, each of which represents a 1/1,000th interest in a
share of its 4.550% Non-Cumulative Preferred Shares. See note 3, “Share Transactions.”
We expect that our liquidity needs, including our anticipated (re)insurance obligations and operating and capital expenditure needs, for the next twelve months, will be met by funds generated from underwriting activities and investment income, as well as by our balance of cash, short-term investments, proceeds on the sale or maturity of our investments, and our credit facilities.
Cash Flows
The following table summarizes our cash flows from operating, investing and financing activities.

| Nine Months Ended
| September 30,
| 2021 |  | 2020
Total cash provided by (used for): |  |  |
Operating activities | $ | 2,580,697 |  |  | $ | 2,198,037 |
Investing activities | (1,184,436) |  |  | (2,850,392) |
Financing activities | (895,943) |  |  | 845,612 |
Effects of exchange rate changes on foreign currency cash | (30,501) |  |  | (2,878) |
Increase (decrease) in cash and restricted cash | $ | 469,817 |  |  | $ | 190,379 |

•Cash provided by operating activities for the nine months ended September 30, 2021 reflected a higher level of premiums collected than in the 2020 period.
•Cash used for investing activities for the nine months ended September 30, 2021 was lower than in the 2020 period. Activity for the nine months ended September 30, 2021 reflected cash used to purchase our non-controlling interest in Coface and Watford, while the 2020 period reflected a higher level of securities purchased, and the investing of proceeds from our issuance of the senior notes.
•Cash used for financing activities for the nine months ended September 30, 2021 reflected $485.8 million inflow from issuance of preferred shares, $450.0 million related to redemption of our Series E preferred shares and $872.2 million of repurchases under our share repurchase program. Activity for the 2020 period primarily reflected the issuance of $1.0 billion of our senior notes and $75.5 million of repurchases under our share repurchase program.

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CAPITAL RESOURCES

The following table provides an analysis of our capital structure:

(U.S. dollars in thousands, except share data) | Sep 30, 2021 |  | Dec 31, 2020
Senior notes | $ | 2,724,149 |  |  | $ | 2,723,423 |
Shareholders’ equity available to Arch: |  |  |
Series E non-cumulative preferred shares | $ | — |  |  | $ | 450,000 |
Series F non-cumulative preferred shares | 330,000 |  |  | 330,000 |
Series G non-cumulative preferred shares | 500,000 |  |  | — |
Common shareholders’ equity | 12,557,526 |  |  | 12,325,886 |
Total | $ | 13,387,526 |  |  | $ | 13,105,886 |
Total capital available to Arch | $ | 16,111,675 |  |  | $ | 15,829,309 |
Debt to total capital (%) | 16.9 |  |  | 17.2 |
Preferred to total capital (%) | 5.2 |  |  | 4.9 |
Debt and preferred to total capital (%) | 22.1 |  |  | 22.1 |

Arch MI U.S. is required to maintain compliance with the GSEs requirements, known as the Private Mortgage Insurer Eligibility Requirements or “PMIERs.” The financial requirements require an eligible mortgage insurer’s available assets, which generally include only the most liquid assets of an insurer, to meet or exceed “minimum required assets” as of each quarter end. Minimum required assets are calculated from PMIERs tables with several risk dimensions (including origination year, original loan-to-value and original credit score of performing loans, and the delinquency status of non-performing loans) and are subject to a minimum amount. Arch MI U.S. satisfied the PMIERs’ financial requirements as of September 30, 2021 with an estimated PMIER sufficiency ratio of 195%, compared to 173% at December 31, 2020.
Arch Capital, through its subsidiaries, provides financial support to certain of its insurance subsidiaries and affiliates, through certain reinsurance arrangements beneficial to the ratings of such subsidiaries. Historically, our insurance, reinsurance and mortgage insurance subsidiaries have entered into separate reinsurance arrangements with Arch Re Bermuda covering individual lines of business. The reinsurance agreements between our U.S.-based property casualty insurance and reinsurance subsidiaries and Arch Re
Bermuda were canceled on a cutoff basis as of January 1, 2018. In 2019, certain reinsurance agreements between our insurance subsidiaries and Arch Re Bermuda were reinstated.
GUARANTOR INFORMATION

The below table provides a description of our senior notes payable at September 30, 2021:

|  | Interest |  | Principal |  | Carrying
Issuer/Due |  | (Fixed) |  | Amount |  | Amount
Arch Capital: |  |  |  |  |  |
May 1, 2034 |  | 7.350 | % |  | $ | 300,000 |  |  | $ | 297,457 |
June 30, 2050 |  | 3.635 | % |  | 1,000,000 |  | 988,665
Arch-U.S.: |  |  |  |  |  |
Nov. 1, 2043 (1) |  | 5.144 | % |  | 500,000 |  | 495,033
Arch Finance: |  |  |  |  |  |
Dec. 15, 2026 (1) |  | 4.011 | % |  | 500,000 |  | 497,527
Dec. 15, 2046 (1) |  | 5.031 | % |  | 450,000 |  | 445,467
Total |  |  |  | $ | 2,750,000 |  |  | $ | 2,724,149 |

(1)Fully and unconditionally guaranteed by Arch Capital.
Our senior notes were issued by Arch Capital, Arch Capital Group (U.S.) Inc. (“Arch-U.S.”) and Arch Capital Finance LLC (“Arch Finance”). Arch-U.S. is a wholly-owned subsidiary of Arch Capital and Arch Finance is a wholly-owned finance subsidiary of Arch-U.S. Our 2034 senior notes and 2050 senior notes issued by Arch Capital are unsecured and unsubordinated obligations of Arch Capital and ranked equally with all of its existing and future unsecured and unsubordinated indebtedness. The 2043 senior notes issued by Arch-U.S. are unsecured and unsubordinated obligations of Arch-U.S. and Arch Capital and rank equally and ratably with the other unsecured and unsubordinated indebtedness of Arch-U.S. and Arch Capital. The 2026 senior notes and 2046 senior notes issued by Arch Finance are unsecured and unsubordinated obligations of Arch Finance and Arch Capital and rank equally and ratably with the other unsecured and unsubordinated indebtedness of Arch Finance and Arch Capital.
Arch-U.S. and Arch Finance depend on their available cash resources, liquid investments and dividends or other distributions from their subsidiaries or affiliates to make payments, including the payment of debt service obligations and operating expenses they may incur.

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The following tables present condensed financial information for Arch Capital (parent guarantor) and Arch-U.S. (subsidiary issuer):

| September 30, 2021 |  | December 31, 2020 |  |  |  |  |  |
| Arch Capital |  | Arch-U.S. |  | Arch Capital |  | Arch-U.S. |  |  |  |  |  |
Assets |  |  |  |  |  |  |  |  |  |  |  |  |
Total investments | $ | 135 |  |  | $ | 565,314 |  |  | $ | 172 |  |  | $ | 396,547 |  |  |  |  |  |  |
Cash | 10,796 |  |  | 10,386 |  |  | 18,932 |  |  | 11,368 |  |  |  |  |  |  |
Investment in operating affiliates | 7,099 |  |  | — |  |  | 7,731 |  |  | — |  |  |  |  |  |  |
Due from subsidiaries and affiliates | — |  |  | 200,786 |  |  | — |  |  | 201,515 |  |  |  |  |  |  |
Other assets | 10,281 |  |  | 36,601 |  |  | 10,659 |  |  | 34,405 |  |  |  |  |  |  |
Total assets | $ | 28,311 |  |  | $ | 813,087 |  |  | $ | 37,494 |  |  | $ | 643,835 |  |  |  |  |  |  |
Liabilities |  |  |  |  |  |  |  |  |  |  |  |  |
Senior notes | 1,286,122 |  |  | 495,033 |  |  | 1,285,867 |  |  | 494,944 |  |  |  |  |  |  |
Due to subsidiaries and affiliates | 4 |  |  | 507,103 |  |  | — |  |  | 586,805 |  |  |  |  |  |  |
Other liabilities | 37,226 |  |  | 55,132 |  |  | 23,270 |  |  | 41,876 |  |  |  |  |  |  |
Total liabilities | $ | 1,323,352 |  |  | $ | 1,057,268 |  |  | $ | 1,309,137 |  |  | $ | 1,123,625 |  |  |  |  |  |  |
Non-cumulative preferred shares | $ | 830,000 |  |  | — |  |  | $ | 780,000 |  |  | — |  |  |  |  |  |  |

|  |  | Nine Months Ended |  | Year Ended |  |  |  |  |  |
|  |  | September 30, 2021 |  | December 31, 2020 |  |  |  |  |  |
|  |  |  |  | Arch Capital |  | Arch-U.S. |  | Arch Capital |  | Arch-U.S. |  |  |  |  |  |
Revenues |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
Net investment income |  |  |  |  | $ | 1,160 |  |  | $ | 9,392 |  |  | $ | 53 |  |  | $ | 18,084 |  |  |  |  |  |  |
Net realized gains (losses) |  |  |  |  | — |  |  | 66,147 |  |  | (2,110) |  |  | 26,096 |  |  |  |  |  |  |
Equity in net income (loss) of investments accounted for using the equity method |  |  |  |  | — |  |  | 13,726 |  |  | — |  |  | 2,507 |  |  |  |  |  |  |
Total revenues |  |  |  |  | 1,160 |  |  | 89,265 |  |  | (2,057) |  |  | 46,687 |  |  |  |  |  |  |
Expenses |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
Corporate expenses |  |  |  |  | 54,726 |  |  | 4,438 |  |  | 65,566 |  |  | 7,227 |  |  |  |  |  |  |
Interest expense |  |  |  |  | 44,055 |  |  | 35,455 |  |  | 40,445 |  |  | 47,566 |  |  |  |  |  |  |
Net foreign exchange (gains) losses |  |  |  |  | 7 |  |  | — |  |  | 3 |  |  | — |  |  |  |  |  |  |
Total expenses |  |  |  |  | 98,788 |  |  | 39,893 |  |  | 106,014 |  |  | 54,793 |  |  |  |  |  |  |
Income (loss) before income taxes and income (loss) from operating affiliates |  |  |  |  | (97,628) |  |  | 49,372 |  |  | (108,071) |  |  | (8,106) |  |  |  |  |  |  |
Income tax (expense) benefit |  |  |  |  | — |  |  | (12,843) |  |  | — |  |  | 2,689 |  |  |  |  |  |  |
Income (loss) from operating affiliates |  |  |  |  | (443) |  |  | — |  |  | (437) |  |  | — |  |  |  |  |  |  |
Net income available to Arch |  |  |  |  | (98,071) |  |  | 36,529 |  |  | (108,508) |  |  | (5,417) |  |  |  |  |  |  |
Preferred dividends |  |  |  |  | (38,159) |  |  | — |  |  | (41,612) |  |  | — |  |  |  |  |  |  |
Loss on redemption of preferred shares |  |  |  |  | (15,101) |  |  | — |  |  | — |  |  | — |  |  |  |  |  |  |
Net income (loss) available to Arch common shareholders |  |  |  |  | $ | (151,331) |  |  | $ | 36,529 |  |  | $ | (150,120) |  |  | $ | (5,417) |  |  |  |  |  |  |

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SHARE REPURCHASE PROGRAM

The board of directors of Arch Capital has authorized the investment in Arch Capital’s common shares through a share repurchase program. For the nine months ended September 30, 2021, Arch Capital repurchased 22.8 million shares under the share repurchase program with an aggregate purchase price of $872.2 million. Since the inception of the share repurchase program through September 30, 2021, Arch Capital has repurchased 412.0 million common shares for an aggregate purchase price of $4.92 billion. At September 30, 2021, approximately $44.3 million of share repurchases were available under the program. On October 8, 2021, the board of directors of ACGL increased the aggregate purchase amount authorized under the share repurchase program to $1.5 billion, which may be effected from time to time in open market or privately negotiated transactions through December 31, 2022. The timing and amount of the repurchase transactions under this program will depend on a variety of factors, including market conditions and corporate and regulatory considerations. We will continue to monitor our share price and, depending upon results of operations, market conditions and the development of the economy, as well as other factors, we will consider share repurchases on an opportunistic basis. See note 17, “Subsequent Event.”
CATASTROPHIC EVENTS AND SEVERE ECONOMIC EVENTS

We have large aggregate exposures to natural and man-made catastrophic events, pandemic events like COVID-19 and severe economic events. Natural catastrophes can be caused by various events, including hurricanes, floods, windstorms, earthquakes, hailstorms, tornadoes, explosions, severe winter weather, fires, droughts and other natural disasters. Man-made catastrophic events may include acts of war, acts of terrorism and political instability. Catastrophes can also cause losses in non-property business such as mortgage insurance, workers’ compensation or general liability. In addition to the nature of property business, we believe that economic and geographic trends affecting insured property, including inflation, property value appreciation and geographic concentration, tend to generally increase the size of losses from catastrophic events over time.
Our models employ both proprietary and vendor-based systems and include cross-line correlations for property, marine, offshore energy, aviation, workers compensation and personal accident. We seek to limit the probable maximum pre-tax loss to a specific level for severe catastrophic events. Currently, we seek to limit our 1-in-250 year return period net probable maximum loss from a severe catastrophic event in any geographic zone to approximately 25% of tangible shareholders’ equity available to Arch (total shareholders’
equity available to Arch less goodwill and intangible assets). We reserve the right to change this threshold at any time.
Based on in-force exposure estimated as of October 1, 2021, our modeled peak zone catastrophe exposure was a windstorm affecting the Florida Tri-County, with a net probable maximum pre-tax loss of $671 million, followed by windstorms affecting the Northeastern U.S. and the Gulf of Mexico regions with net probable maximum pre-tax losses of $650 and $661 million, respectively. Our exposures to other perils, such as U.S. earthquake and international events, were less than the exposures arising from U.S. windstorms and hurricanes. As of October 1, 2021, our modeled peak zone earthquake exposure (San Francisco earthquake) represented approximately 75% of our peak zone catastrophe exposure, and our modeled peak zone international exposure (UK windstorm) was substantially less than both our peak zone windstorm and earthquake exposures.
Effective July 1, 2021, our insurance operations had in effect a reinsurance program which provided coverage for certain property-catastrophe related losses equal to $276 million in excess of various retentions per occurrence.
We also have significant exposure to losses due to mortgage defaults resulting from severe economic events in the future. For our U.S. mortgage insurance business, we have developed a proprietary risk model (“Realistic Disaster Scenario” or “RDS”) that simulates the maximum loss resulting from a severe economic downturn impacting the housing market. The RDS models the collective impact of adverse conditions for key economic indicators, the most significant of which is a decline in home prices. The RDS model projects paths of future home prices, unemployment rates, income levels and interest rates and assumes correlation across states and geographic regions. The resulting future performance of our in-force portfolio is then estimated under the economic stress scenario, reflecting loan and borrower information.
Currently, we seek to limit our modeled RDS loss from a severe economic event to approximately 25% of tangible shareholders’ equity available to Arch. We reserve the right to change this threshold at any time. Based on in-force exposure estimated as of October 1, 2021, our modeled RDS loss was approximately 6% of tangible shareholders’ equity available to Arch.
Net probable maximum loss estimates are net of expected reinsurance recoveries, before income tax and before excess reinsurance reinstatement premiums. RDS loss estimates are net of expected reinsurance recoveries and before income tax. Catastrophe loss estimates are reflective of the zone indicated and not the entire portfolio. Since hurricanes and windstorms can affect more than one zone and make multiple landfalls, our catastrophe loss estimates include clash estimates from

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other zones. Our catastrophe loss estimates and RDS loss estimates do not represent our maximum exposures and it is highly likely that our actual incurred losses would vary materially from the modeled estimates. There can be no assurances that we will not suffer pre-tax losses greater than 25% of our tangible shareholders’ equity from one or more catastrophic events or severe economic events due to several factors. These factors include the inherent uncertainties in estimating the frequency and severity of such events and the margin of error in making such determinations resulting from potential inaccuracies and inadequacies in the data provided by clients and brokers, the modeling techniques and the application of such techniques or as a result of a decision to change the percentage of shareholders' equity exposed to a single catastrophic event or severe economic event. In addition, actual losses may increase if our reinsurers fail to meet their obligations to us or the reinsurance protections purchased by us are exhausted or are otherwise unavailable. See “Risk Factors—Risks Relating to Our Industry” and “Management’s Discussion and Analysis of Financial Condition and Results of Operations—Catastrophic Events and Severe Economic Events” in our 2020 Form 10-K.
OFF-BALANCE SHEET ARRANGEMENTS

Off-balance sheet arrangements are discussed in Management’s Discussion and Analysis of Financial Condition and Results of Operations contained in our 2020 Form 10-K.
MARKET SENSITIVE INSTRUMENTS AND RISK MANAGEMENT

In accordance with the SEC’s Financial Reporting Release No. 48, we performed a sensitivity analysis to determine the effects that market risk exposures could have on the future earnings, fair values or cash flows of our financial instruments as of September 30, 2021. Market risk represents the risk of changes in the fair value of a financial instrument and is comprised of several components, including liquidity, basis and price risks.
An analysis of material changes in market risk exposures at September 30, 2021 that affect the quantitative and qualitative disclosures presented in our 2020 Form 10-K (see section captioned “Management’s Discussion and Analysis of Financial Condition and Results of Operations—Market Sensitive Instruments and Risk Management”) were as follows:
Investment Market Risk
Fixed Income Securities. We invest in interest rate sensitive securities, primarily debt securities. We consider the effect of interest rate movements on the fair value of our fixed
maturities, fixed maturities pledged under securities lending agreements, short-term investments and certain of our other investments, equity securities and investment funds accounted for using the equity method which invest in fixed income securities (collectively, “Fixed Income Securities”) and the corresponding change in unrealized appreciation. As interest rates rise, the fair value of our Fixed Income Securities falls, and the converse is also true. Based on historical observations, there is a low probability that all interest rate yield curves would shift in the same direction at the same time. Furthermore, at times interest rate movements in certain credit sectors exhibit a much lower correlation to changes in U.S. Treasury yields. Accordingly, the actual effect of interest rate movements may differ materially from the amounts set forth in the following tables.
The following table summarizes the effect that an immediate, parallel shift in the interest rate yield curve would have had on our Fixed Income Securities:

(U.S. dollars in billions) | Interest Rate Shift in Basis Points
-100 |  | -50 |  | — |  | +50 |  | +100
Sept. 30, 2021 |  |  |  |  |  |  |  |  |
Total fair value | $ | 25.29 |  |  | $ | 25.00 |  |  | $ | 24.67 |  |  | $ | 24.35 |  |  | $ | 24.06 |
Change from base | 2.5 | % |  | 1.3 | % |  |  |  | (1.3) | % |  | (2.5) | %
Change in unrealized value | $ | 0.62 |  |  | $ | 0.32 |  |  |  |  | $ | (0.32) |  |  | $ | (0.62) |
Dec. 31, 2020 |  |  |  |  |  |  |  |  |
Total fair value | $ | 25.82 |  |  | $ | 25.44 |  |  | $ | 25.07 |  |  | $ | 24.69 |  |  | $ | 24.31 |
Change from base | 3.0 | % |  | 1.5 | % |  |  |  | (1.5) | % |  | (3.0) | %
Change in unrealized value | $ | 0.75 |  |  | $ | 0.38 |  |  |  |  | $ | (0.38) |  |  | $ | (0.75) |

In addition, we consider the effect of credit spread movements on the market value of our Fixed Income Securities and the corresponding change in unrealized value. As credit spreads widen, the fair value of our Fixed Income Securities falls, and the converse is also true. In periods where the spreads on our Fixed Income Securities are much higher than their historical average due to short-term market dislocations, a parallel shift in credit spread levels would result in a much more pronounced change in unrealized value.

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The following table summarizes the effect that an immediate, parallel shift in credit spreads in a static interest rate environment would have had on our Fixed Income Securities:

(U.S. dollars in billions) | Credit Spread Shift in Percentage Points
-100 |  | -50 |  | — |  | +50 |  | +100
Sept. 30, 2021 |  |  |  |  |  |  |  |  |
Total fair value | $ | 25.24 |  |  | $ | 24.97 |  |  | $ | 24.67 |  |  | $ | 24.38 |  |  | $ | 24.11 |
Change from base | 2.3 | % |  | 1.2 | % |  |  |  | (1.2) | % |  | (2.3) | %
Change in unrealized value | $ | 0.57 |  |  | $ | 0.30 |  |  |  |  | $ | (0.30) |  |  | $ | (0.57) |
Dec. 31, 2020 |  |  |  |  |  |  |  |  |
Total fair value | $ | 25.54 |  |  | $ | 25.32 |  |  | $ | 25.07 |  |  | $ | 24.82 |  |  | $ | 24.59 |
Change from base | 1.9 | % |  | 1.0 | % |  |  |  | (1.0) | % |  | (1.9) | %
Change in unrealized value | $ | 0.48 |  |  | $ | 0.25 |  |  |  |  | $ | (0.25) |  |  | $ | (0.48) |

Another method that attempts to measure portfolio risk is Value-at-Risk (“VaR”). VaR measures the worst expected loss under normal market conditions over a specific time interval at a given confidence level. The 1-year 95th percentile parametric VaR reported herein estimates that 95% of the time, the portfolio loss in a one-year horizon would be less than or equal to the calculated number, stated as a percentage of the measured portfolio’s initial value. The VaR is a variance-covariance based estimate, based on linear sensitivities of a portfolio to a broad set of systematic market risk factors and idiosyncratic risk factors mapped to the portfolio exposures. The relationships between the risk factors are estimated using historical data, and the most recent data points are generally given more weight. As of September 30, 2021, our portfolio’s VaR was estimated to be 3.2% compared to an estimated 4.3% at December 31, 2020. In periods where the volatility of the risk factors mapped to our portfolio’s exposures is higher due to market conditions, the resulting VaR is higher than in other periods.
Equity Securities. At September 30, 2021 and December 31, 2020, the fair value of our investments in equity securities (excluding securities included in Fixed Income Securities above) totaled $1.5 billion and $1.1 billion, respectively. These investments are exposed to price risk, which is the potential loss arising from decreases in fair value. An immediate hypothetical 10% decline in the value of each position would reduce the fair value of such investments by approximately $146.7 million and $109.5 million at September 30, 2021 and December 31, 2020, respectively, and would have decreased book value per share by approximately $0.38 and $0.27, respectively. An immediate hypothetical 10% increase in the value of each position would increase the fair value of such investments by approximately $146.7 million and $109.5 million at September 30, 2021 and December 31, 2020, respectively, and would have increased book value per share by approximately $0.38 and $0.27, respectively.
Investment-Related Derivatives. At September 30, 2021, the notional value of all derivative instruments (excluding to-be-announced mortgage backed securities which are included in the fixed income securities analysis above and foreign currency forward contracts which are included in the foreign currency exchange risk analysis below) was $6.8 billion, compared to $8.6 billion at December 31, 2020. If the underlying exposure of each investment-related derivative held at September 30, 2021 depreciated by 100 basis points, it would have resulted in a reduction in net income of approximately $68.2 million, and a decrease in book value per share of approximately $0.18 per share, compared to $85.7 million and $0.21 per share, respectively, on investment-related derivatives held at December 31, 2020. If the underlying exposure of each investment-related derivative held at September 30, 2021 appreciated by 100 basis points, it would have resulted in an increase in net income of approximately $68.2 million, and an increase in book value per share of approximately $0.18 per share, compared to $85.7 million and $0.21 per share, respectively, on investment-related derivatives held at December 31, 2020. See note 10, “Derivative Instruments,” to our consolidated financial statements for additional disclosures concerning derivatives.
For further discussion on investment activity, please refer to “Financial Condition—Investable Assets.”
Foreign Currency Exchange Risk
Foreign currency rate risk is the potential change in value, income and cash flow arising from adverse changes in foreign currency exchange rates. Through our subsidiaries and branches located in various foreign countries, we conduct our insurance and reinsurance operations in a variety of local currencies other than the U.S. Dollar. We generally hold investments in foreign currencies which are intended to mitigate our exposure to foreign currency fluctuations in our net insurance liabilities. We may also utilize foreign currency forward contracts and currency options as part of our investment strategy. See note 10, “Derivative Instruments,” to our consolidated financial statements for additional information.

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The following table provides a summary of our net foreign currency exchange exposures, as well as foreign currency derivatives in place to manage these exposures:

(U.S. dollars in thousands, except per share data) | September 30, 2021 |  | December 31, 2020
Net assets (liabilities), denominated in foreign currencies, excluding shareholders’ equity and derivatives | $ | (813,451) |  |  | $ | (309,968) |
Shareholders’ equity denominated in foreign currencies (1) | 1,088,134 |  |  | 695,355 |
Net foreign currency forward contracts outstanding (2) | 39,171 |  |  | 1,108,161 |
Net exposures denominated in foreign currencies | $ | 313,854 |  |  | $ | 1,493,548 |
Pre-tax impact of a hypothetical 10% appreciation of the U.S. Dollar against foreign currencies: |  |  |
Shareholders’ equity | $ | (31,385) |  |  | $ | (149,355) |
Book value per share | $ | (0.08) |  |  | $ | (0.37) |
Pre-tax impact of a hypothetical 10% decline of the U.S. Dollar against foreign currencies: |  |  |
Shareholders’ equity | $ | 31,385 |  |  | $ | 149,355 |
Book value per share | $ | 0.08 |  |  | $ | 0.37 |

(1) Represents capital contributions held in the foreign currencies of our operating units.
(2) Represents the net notional value of outstanding foreign currency forward contracts.
Although we generally attempt to match the currency of our projected liabilities with investments in the same currencies, from time to time we may elect to over or underweight one or more currencies, which could increase our exposure to foreign currency fluctuations and increase the volatility of our shareholders’ equity. Historical observations indicate a low probability that all foreign currency exchange rates would shift against the U.S. Dollar in the same direction and at the same time and, accordingly, the actual effect of foreign currency rate movements may differ materially from the amounts set forth above. For further discussion on foreign exchange activity, please refer to “Results of Operations.”
Effects of Inflation
We do not believe that inflation has had a material effect on our consolidated results of operations, except insofar as inflation may affect our reserve for losses and loss adjustment expenses and interest rates. The potential exists, after a catastrophe loss or pandemic events like COVID-19, for the development of inflationary pressures in a local economy. The anticipated effects of inflation on us are considered in our catastrophe loss models. The actual effects of inflation on our results cannot be accurately known until claims are ultimately settled.
OTHER FINANCIAL INFORMATION

The consolidated financial statements as of September 30, 2021 have been reviewed by PricewaterhouseCoopers LLP, the registrant's independent public accountants, whose report is included as an exhibit to this filing. The report of PricewaterhouseCoopers LLP states that they did not audit and they do not express an opinion on that unaudited financial information. Accordingly, the degree of reliance on their report on such information should be restricted in light of the limited nature of the review procedures applied. PricewaterhouseCoopers LLP is not subject to the liability provisions of Section 11 of the Securities Act of 1933 for their report on the unaudited financial information because that report is not a "report" or a "part" of the registration statement prepared or certified by PricewaterhouseCoopers LLP within the meaning of Sections 7 and 11 of the Securities Act of 1933.

ITEM 3. QUANTITATIVE AND QUALITATIVE DISCLOSURES ABOUT MARKET RISK

Reference is made to the information appearing above under the subheading “Market Sensitive Instruments and Risk Management” under the caption “Management’s Discussion and Analysis of Financial Condition and Results of Operations,” which information is hereby incorporated by reference.

ITEM 4. CONTROLS AND PROCEDURES

Evaluation of Disclosure Controls and Procedures
In connection with the filing of this Form 10-Q, our management, including the Chief Executive Officer and Chief Financial Officer, conducted an evaluation of the effectiveness of disclosure controls and procedures pursuant to applicable Exchange Act Rules as of the end of the period covered by this report. Based on that evaluation, the Chief Executive Officer and Chief Financial Officer concluded that the disclosure controls and procedures were effective as of the end of and during the period covered by this report with respect to information being recorded, processed, summarized and reported within time periods specified in the SEC’s rules and forms and with respect to timely communication to them and other members of management responsible for preparing periodic reports of all material information required to be disclosed in this report as it relates to Arch Capital and its consolidated subsidiaries.
We continue to enhance our operating procedures and internal controls to effectively support our business and our regulatory and reporting requirements. Our management does

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not expect that our disclosure controls or our internal controls will prevent all errors and all fraud. A control system, no matter how well conceived and operated, can provide only reasonable, not absolute, assurance that the objectives of the control system are met. Further, the design of a control system must reflect the fact that there are resource constraints, and the benefits of controls must be considered relative to their costs. As a result of the inherent limitations in all control systems, no evaluation of controls can provide absolute assurance that all control issues and instances of fraud, if any, within the company have been detected. These inherent limitations include the realities that judgments in decision-making can be faulty, and that breakdowns can occur because of simple error or mistake. Additionally, controls can be circumvented by the individual acts of some persons or by collusion of two or more people. The design of any system of controls also is based in part upon certain assumptions about the likelihood of future events, and there can be no assurance that any design will succeed in achieving its stated goals under all potential future conditions; over time, controls may become inadequate because of changes in conditions, or the degree of compliance with the policies or procedures may deteriorate. As a result of the inherent limitations in a cost-effective control system, misstatement due to error or fraud may occur and not be detected. Accordingly, our disclosure controls and procedures are designed to provide reasonable, not absolute, assurance that the disclosure controls and procedures are met.
Changes in Internal Controls Over Financial Reporting
There have been no changes in internal control over financial reporting that occurred during the quarter ended September 30, 2021 that have materially affected, or are reasonably likely to materially affect, internal control over financial reporting. We have not experienced any material impact to our internal controls over financial reporting despite the fact that most of our employees are working remotely due to the COVID-19. We are continually monitoring and assessing COVID-19 situation on our internal controls to minimize the impact on their design and operating effectivenes s.
PART II. OTHER INFORMATION

ITEM 1. LEGAL PROCEEDINGS

We, in common with the insurance industry in general, are subject to litigation and arbitration in the normal course of our business. As of September 30, 2021, we were not a party to any litigation or arbitration which is expected by management to have a material adverse effect on our results of operations and financial condition and liquidity.

ITEM 1A. RISK FACTORS

There were no material changes from the risk factors previously disclosed in our Annual Report on Form 10-K for the year ended December 31, 2020.

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ITEM 2. UNREGISTERED SALES OF EQUITY SECURITIES AND USE OF PROCEEDS

The following table summarizes our purchases of common shares for the 2021 third quarter:

|  | Issuer Purchases of Equity Securities
Period |  | Total Number of Shares Purchased (1) |  | Average Price Paid per Share |  | Total Number of Shares Purchased as Part of Publicly Announced Plans or Programs |  | Approximate Dollar Value of Shares that May Yet be Purchased Under the Plan or Programs (2)
7/1/2021-7/31/2021 |  | 729,925 |  |  | $ | 39.01 |  |  | 727,778 |  |  | $ | 402,822 |
8/1/2021-8/31/2021 |  | 4,173,006 |  |  | 40.66 |  |  | 4,130,776 |  |  | $ | 234,886 |
9/1/2021-9/30/2021 |  | 4,864,398 |  |  | 39.32 |  |  | 4,846,402 |  |  | $ | 44,331 |
Total |  | 9,767,329 |  |  | $ | 39.87 |  |  | 9,704,956 |  |  |

(1)Represents repurchases by Arch Capital of shares, from time to time, from employees in order to facilitate the payment of withholding taxes on restricted shares granted and the exercise of stock appreciation rights. We purchased these shares at their fair value, as determined by reference to the closing price of our common shares on the day the restricted shares vested or the stock appreciation rights were exercised.
(2)Remaining amount available at September 30, 2021 under Arch Capital’s share repurchase authorization, under which repurchases may be effected from time to time in open market or privately negotiated transactions. On October 8, 2021, the board of directors of ACGL increased the aggregate purchase amount authorized under the share repurchase program to $1.5 billion. Repurchases under this authorization may be effected from time to time in open market or privately negotiated transactions through December 31, 2022.

ITEM 3. DEFAULTS UPON SENIOR SECURITIES

None.

ITEM 4. MINE SAFETY DISCLOSURES

Not applicable.

ITEM 5. OTHER INFORMATION

In accordance with Section 10a(i)(2) of the Securities Exchange Act of 1934, as amended, we are responsible for disclosing non-audit services to be provided by our independent auditor, PricewaterhouseCoopers LLP, which are approved by the Audit Committee of our board of directors. During the 2021 third quarter, the Audit Committee approved engagements of PricewaterhouseCoopers LLP for permitted non-audit services, which consisted of tax consulting services, tax compliance services and other accounting consulting services.

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ITEM 6. EXHIBITS

|  |  |  | Incorporated by Reference |  |
Exhibit Number |  | Exhibit Description |  | Form |  | Original Number |  | Date Filed |  | Filed Herewith
10.1 |  | First Amendment to Third Amended and Restated Credit Agreement, dated as of August 12, 2020 by and among Arch Capital Group Ltd. (“ACGL”), the other Loan Parties party hereto, the Lenders party hereto, and Bank of America, N.A., as Administrative Agent (“Third Amended and Restated Credit Agreement”). |  |  |  |  |  |  |  | X
10.2 |  | The LIBOR Transition Amendment to the Third Amended and Restated Credit Agreement, dated as of September 29, 2021. |  |  |  |  |  |  |  | X
15 |  | Accountants’ Awareness Letter (regarding unaudited interim financial information) |  |  |  |  |  |  |  | X
31.1 |  | Certification of Chief Executive Officer pursuant to Section 302 of the Sarbanes-Oxley Act of 2002 |  |  |  |  |  |  |  | X
31.2 |  | Certification of Chief Financial Officer pursuant to Section 302 of the Sarbanes-Oxley Act of 2002 |  |  |  |  |  |  |  | X
32.1 |  | Certification of Chief Executive Officer pursuant to Section 906 of the Sarbanes-Oxley Act of 2002 |  |  |  |  |  |  |  | X
32.2 |  | Certification of Chief Financial Officer pursuant to Section 906 of the Sarbanes-Oxley Act of 2002 |  |  |  |  |  |  |  | X
101.INS |  | XBRL Instance Document |  |  |  |  |  |  |  |
101.SCH |  | XBRL Taxonomy Extension Schema Document |  |  |  |  |  |  |  |
101.CAL |  | XBRL Taxonomy Extension Calculation Linkbase Document |  |  |  |  |  |  |  |
101.DEF |  | XBRL Taxonomy Extension Definition Linkbase Document |  |  |  |  |  |  |  |
101.LAB |  | XBRL Taxonomy Extension Label Linkbase Document |  |  |  |  |  |  |  |
101.PRE |  | XBRL Taxonomy Extension Presentation Linkbase Document |  |  |  |  |  |  |  |

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SIGNATURES
Pursuant to the requirements of the Securities Exchange Act of 1934, the registrant has duly caused this report to be signed on its behalf by the undersigned thereunto duly authorized.

|  | ARCH CAPITAL GROUP LTD.
|  | (REGISTRANT)
|  | /s/ Marc Grandisson
Date: November 4, 2021 |  | Marc Grandisson
|  | Chief Executive Officer (Principal Executive Officer)
|  | /s/ François Morin
Date: November 4, 2021 |  | François Morin
|  | Executive Vice President, Chief Financial Officer (Principal Financial and Accounting Officer) and Treasurer

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