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Investment Securities | 9 Months Ended
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Sep. 30, 2022
Investments, Debt and Equity Securities [Abstract] |
Investment Securities | Investment Securities White Mountains’s portfolio of investment securities held for general investment purposes consists of fixed maturity investments, short-term investments, common equity securities, its investment in MediaAlpha and other long-term investments, which are classified as trading securities. Trading securities are reported at fair value as of the balance sheet date. Net realized and unrealized investment gains (losses) on trading securities are reported in pre-tax revenues. White Mountains’s fixed maturity investments are generally valued using industry standard pricing methodologies. Key inputs include benchmark yields, benchmark securities, reported trades, issuer spreads, bids, offers, credit ratings and prepayment speeds. Income on mortgage and asset-backed securities is recognized using an effective yield based on anticipated prepayments and the estimated economic life of the securities. When actual prepayments differ significantly from anticipated prepayments, the estimated economic life is recalculated and the remaining unamortized premium or discount is amortized prospectively over the remaining economic life. Realized investment gains (losses) resulting from sales of investment securities are accounted for using the specific identification method. Premiums and discounts on all fixed maturity investments are amortized or accreted to income over the anticipated life of the investment. Short-term investments consist of interest-bearing money market funds, certificates of deposit and other securities, which at the time of purchase, mature or become available for use within one year. Short-term investments are carried at fair value, which approximated amortized cost, as of September 30, 2022 and December 31, 2021. Other long-term investments consist primarily of unconsolidated entities, including Kudu’s Participation Contracts, private equity funds and hedge funds, a bank loan fund, Lloyd’s trust deposits, insurance-linked securities (“ILS”) funds and private debt investments. Net Investment Income White Mountains’s net investment income is comprised primarily of interest income associated with White Mountains’s fixed maturity investments and short-term investments, dividend income from common equity securities, distributions from its investment in MediaAlpha and distributions from other long-term investments. The following table presents pre-tax net investment income for the three and nine months ended September 30, 2022 and 2021:
|  | Three Months Ended |  | Nine Months Ended
|  | September 30, |  | September 30,
Millions |  | 2022 |  | 2021 |  | 2022 |  | 2021
Fixed maturity investments |  | $ | 11.1 |  |  | $ | 8.2 |  |  | $ | 28.6 |  |  | $ | 21.9 |
Short-term investments |  | 6.7 |  |  | — |  |  | 7.2 |  |  | .5 |
Common equity securities |  | .1 |  |  | .1 |  |  | .2 |  |  | .1 |
Other long-term investments |  | 17.1 |  |  | 12.0 |  |  | 45.9 |  |  | 37.1 |
Amount attributable to TPC Providers |  | (.3) |  |  | (.2) |  |  | (.8) |  |  | (.8) |
Total investment income |  | 34.7 |  |  | 20.1 |  |  | 81.1 |  |  | 58.8 |
Third-party investment expenses |  | (.8) |  |  | (.6) |  |  | (1.5) |  |  | (1.6) |
Net investment income, pre-tax |  | $ | 33.9 |  |  | $ | 19.5 |  |  | $ | 79.6 |  |  | $ | 57.2 |
Net Realized and Unrealized Investment Gains (Losses) The following table presents net realized and unrealized investment gains (losses) for the three and nine months ended September 30, 2022 and 2021:
|  | Three Months Ended September 30, |  | Nine Months Ended September 30,
Millions |  | 2022 |  | 2021 |  | 2022 |  | 2021
Realized investment gains (losses) |  |  |  |  |  |  |  |
Fixed maturity investments |  | $ | (4.9) |  |  | $ | 1.0 |  |  | $ | (10.8) |  |  | $ | 3.8 |
Short-term investments |  | (.1) |  |  | (.3) |  |  | (.4) |  |  | — |
Common equity securities |  | — |  |  | — |  |  | — |  |  | .4 |
Investment in MediaAlpha |  | — |  |  | — |  |  | — |  |  | 160.3 |
Other long-term investments |  | 3.4 |  |  | 9.2 |  |  | 24.1 |  |  | 0.9 |
Net realized investment gains (losses) |  | (1.6) |  |  | 9.9 |  |  | 12.9 |  |  | 165.4 |
Unrealized investment gains (losses) |  |  |  |  |  |  |  |
Fixed maturity investments |  | (58.8) |  |  | (7.9) |  |  | (192.9) |  |  | (27.0) |
Short-term investments |  | (2.1) |  |  | — |  |  | (2.6) |  |  | — |
Common equity securities |  | 4.3 |  |  | 1.6 |  |  | (6.3) |  |  | 7.3 |
Investment in MediaAlpha |  | (18.6) |  |  | (396.8) |  |  | (113.3) |  |  | (485.8) |
Other long-term investments |  | 27.5 |  |  | 28.4 |  |  | 40.8 |  |  | 112.3 |
Net unrealized investment gains (losses) |  | (47.7) |  |  | (374.7) |  |  | (274.3) |  |  | (393.2) |
Net realized and unrealized investment gains (losses), before amount attributable to TPC providers(1) |  | (49.3) |  |  | (364.8) |  |  | (261.4) |  |  | (227.8) |
Amount attributable to TPC Providers |  | 1.3 |  |  | (1.7) |  |  | 6.2 |  |  | (6.5) |
Net realized and unrealized investment gains (losses) |  | $ | (48.0) |  |  | $ | (366.5) |  |  | $ | (255.2) |  |  | $ | (234.3) |
Fixed maturity and short-term investments |  |  |  |  |  |  |  |
Net realized and unrealized investment gains (losses) |  | $ | (65.9) |  |  | $ | (7.2) |  |  | $ | (206.7) |  |  | $ | (23.2) |
Less: net realized and unrealized gains (losses) on investment securities sold during the period |  | .1 |  |  | (5.7) |  |  | (3.0) |  |  | (6.9) |
Net unrealized investment gains (losses) on investment securities held at the end of the period |  | $ | (66.0) |  |  | $ | (1.5) |  |  | $ | (203.7) |  |  | $ | (16.3) |
Common equity securities and investment in MediaAlpha |  |  |  |  |  |  |  |
Net realized and unrealized investment gains (losses) on common equity securities |  | $ | 4.3 |  |  | $ | 1.6 |  |  | $ | (6.3) |  |  | $ | 7.7 |
Net realized and unrealized investment gains (losses) from investment in MediaAlpha |  | (18.6) |  |  | (396.8) |  |  | (113.3) |  |  | (325.5) |
Total net realized and unrealized investment gains (losses) |  | (14.3) |  |  | (395.2) |  |  | (119.6) |  |  | (317.8) |
Less: net realized and unrealized gains (losses) on investment securities sold during the period |  | — |  |  | .4 |  |  | — |  |  | 20.3 |
Net unrealized investment gains (losses) on investment securities held at the end of the period |  | $ | (14.3) |  |  | $ | (395.6) |  |  | $ | (119.6) |  |  | $ | (338.1) |
(1) For the three months ended September 30, 2022 and 2021, includes $(22.6) and $(7.0) of realized and unrealized investment gains (losses) related to foreign currency exchange. For the nine months ended September 30, 2022 and 2021, includes $(44.4) and $(7.4) of realized and unrealized investment gains (losses) related to foreign currency exchange. The following table presents total gains included in earnings attributable to net unrealized investment gains for Level 3 investments for the three and nine months ended September 30, 2022 and 2021 for investments still held at the end of the period:
|  | Three Months Ended |  | Nine Months Ended
|  | September 30, |  | September 30,
Millions |  | 2022 |  | 2021 |  | 2022 |  | 2021
Other long-term investments |  | $ | 40.8 |  |  | $ | 26.5 |  |  | $ | 57.7 |  |  | $ | 75.5 |
Total net unrealized investment gains, pre-tax - Level 3 investments |  | $ | 40.8 |  |  | $ | 26.5 |  |  | $ | 57.7 |  |  | $ | 75.5 |
Investment Holdings The following tables present the cost or amortized cost, gross unrealized investment gains (losses), net foreign currency gains (losses) and carrying values of White Mountains’s fixed maturity investments as of September 30, 2022 and December 31, 2021:
|  | September 30, 2022
Millions |  | Cost or Amortized Cost |  | Gross Unrealized Gains |  | Gross Unrealized Losses |  | Net Foreign Currency Gains (Losses) |  | Carrying Value
U.S. Government and agency obligations |  | $ | 227.9 |  |  | $ | — |  |  | $ | (11.0) |  |  | $ | — |  |  | $ | 216.9 |
Debt securities issued by corporations |  | 1,068.8 |  |  | — |  |  | (91.2) |  |  | (3.1) |  |  | 974.5 |
Municipal obligations |  | 289.3 |  |  | — |  |  | (24.4) |  |  | — |  |  | 264.9 |
Mortgage and asset-backed securities |  | 278.3 |  |  | — |  |  | (37.6) |  |  | — |  |  | 240.7 |
Collateralized loan obligations |  | 169.7 |  |  | — |  |  | (7.6) |  |  | (4.8) |  |  | 157.3 |
Total fixed maturity investments |  | $ | 2,034.0 |  |  | $ | — |  |  | $ | (171.8) |  |  | $ | (7.9) |  |  | $ | 1,854.3 |
|  | December 31, 2021
Millions |  | Cost or Amortized Cost |  | Gross Unrealized Gains |  | Gross Unrealized Losses |  | Net Foreign Currency Gains (Losses) |  | Carrying Value
U.S. Government and agency obligations |  | $ | 212.1 |  |  | $ | .5 |  |  | $ | (1.1) |  |  | $ | — |  |  | $ | 211.5 |
Debt securities issued by corporations |  | 993.3 |  |  | 8.7 |  |  | (8.7) |  |  | (.4) |  |  | 992.9 |
Municipal obligations |  | 276.4 |  |  | 16.8 |  |  | (1.3) |  |  | — |  |  | 291.9 |
Mortgage and asset-backed securities |  | 277.2 |  |  | 2.9 |  |  | (2.5) |  |  | — |  |  | 277.6 |
Collateralized loan obligations |  | 136.5 |  |  | — |  |  | (.4) |  |  | (1.1) |  |  | 135.0 |
Total fixed maturity investments |  | $ | 1,895.5 |  |  | $ | 28.9 |  |  | $ | (14.0) |  |  | $ | (1.5) |  |  | $ | 1,908.9 |
The following table presents the cost or amortized cost and carrying values of White Mountains’s fixed maturity investments by contractual maturity as of September 30, 2022. Actual maturities could differ from contractual maturities because borrowers may have the right to call or prepay certain obligations with or without call or prepayment penalties.
|  | September 30, 2022
Millions |  | Cost or Amortized Cost |  | Carrying Value
Due in one year or less |  | $ | 207.5 |  |  | $ | 203.8 |
Due after one year through five years |  | 917.9 |  |  | 849.4 |
Due after five years through ten years |  | 354.8 |  |  | 309.3 |
Due after ten years |  | 105.8 |  |  | 93.8 |
Mortgage and asset-backed securities and collateralized loan obligations |  | 448.0 |  |  | 398.0 |
Total fixed maturity investments |  | $ | 2,034.0 |  |  | $ | 1,854.3 |
The following tables present the cost or amortized cost, gross unrealized investment gains (losses), net foreign currency gains (losses), and carrying values of common equity securities, White Mountains’s investment in MediaAlpha and other long-term investments as of September 30, 2022 and December 31, 2021:
|  | September 30, 2022
Millions |  | Cost or Amortized Cost |  | Gross Unrealized Gains |  | Gross Unrealized Losses |  | Net Foreign Currency Gains (Losses) |  | Carrying Value
Common equity securities |  | $ | 324.2 |  |  | $ | 32.7 |  |  | $ | (8.7) |  |  | $ | (15.4) |  |  | $ | 332.8 |
Investment in MediaAlpha |  | $ | — |  |  | $ | 148.2 |  |  | $ | — |  |  | $ | — |  |  | $ | 148.2 |
Other long-term investments |  | $ | 1,390.7 |  |  | $ | 345.0 |  |  | $ | (90.0) |  |  | $ | (23.0) |  |  | $ | 1,622.7 |
|  | December 31, 2021
Millions |  | Cost or Amortized Cost |  | Gross Unrealized Gains |  | Gross Unrealized Losses |  | Net Foreign Currency Gains (Losses) |  | Carrying Value
Common equity securities |  | $ | 236.3 |  |  | $ | 16.1 |  |  | $ | — |  |  | $ | (1.3) |  |  | $ | 251.1 |
Investment in MediaAlpha |  | $ | — |  |  | $ | 261.6 |  |  | $ | — |  |  | $ | — |  |  | $ | 261.6 |
Other long-term investments |  | $ | 1,186.7 |  |  | $ | 239.0 |  |  | $ | (44.1) |  |  | $ | (3.8) |  |  | $ | 1,377.8 |
Fair Value Measurements Fair value measurements are categorized into a hierarchy that distinguishes between inputs based on market data from independent sources (observable inputs) and a reporting entity’s internal assumptions based upon the best information available when external market data is limited or unavailable (unobservable inputs). Quoted prices in active markets for identical assets or liabilities have the highest priority (Level 1), followed by observable inputs other than quoted prices, including prices for similar but not identical assets or liabilities (Level 2) and unobservable inputs, including the reporting entity’s estimates of the assumptions that market participants would use, having the lowest priority (Level 3). See Note 17 — “Fair Value of Financial Instruments.” As of September 30, 2022 and December 31, 2021, White Mountains used quoted market prices or other observable inputs to determine fair value for approximately 69% and 68% of the investment portfolio. Fair Value Measurements by Level The following tables present White Mountains’s fair value measurements for investments as of September 30, 2022 and December 31, 2021 by level. The major security types were based on the legal form of the securities. White Mountains has disaggregated its fixed maturity investments based on the issuing entity type, which impacts credit quality, with debt securities issued by U.S. government entities carrying minimal credit risk, while the credit and other risks associated with other issuers, such as corporations, municipalities or entities issuing mortgage and asset-backed securities vary depending on the nature of the issuing entity type. White Mountains further disaggregates debt securities issued by corporations by industry sector because investors often reference commonly used benchmarks and their subsectors to monitor risk and performance. Accordingly, White Mountains has further disaggregated this asset class into subclasses based on the similar sectors and industry classifications it uses to evaluate investment risk and performance against commonly used benchmarks, such as the Bloomberg Barclays U.S. Intermediate Aggregate.
|  | September 30, 2022
Millions |  | Fair Value |  | Level 1 |  | Level 2 |  | Level 3
Fixed maturity investments: |  |  |  |  |  |  |  |
U.S. Government and agency obligations |  | $ | 216.9 |  |  | $ | 216.9 |  |  | $ | — |  |  | $ | — |
Debt securities issued by corporations: |  |  |  |  |  |  |  |
Financials |  | 269.0 |  |  | — |  |  | 269.0 |  |  | — |
Consumer |  | 181.4 |  |  | — |  |  | 181.4 |  |  | — |
Industrial |  | 115.8 |  |  | — |  |  | 115.8 |  |  | — |
Healthcare |  | 114.3 |  |  | — |  |  | 114.3 |  |  | — |
Technology |  | 110.8 |  |  | — |  |  | 110.8 |  |  | — |
Utilities |  | 75.9 |  |  | — |  |  | 75.9 |  |  | — |
Communications |  | 49.1 |  |  | — |  |  | 49.1 |  |  | — |
Energy |  | 36.1 |  |  | — |  |  | 36.1 |  |  | — |
Materials |  | 22.1 |  |  | — |  |  | 22.1 |  |  | — |
Total debt securities issued by corporations |  | 974.5 |  |  | — |  |  | 974.5 |  |  | — |
Municipal obligations |  | 264.9 |  |  | — |  |  | 264.9 |  |  | — |
Mortgage and asset-backed securities |  | 240.7 |  |  | — |  |  | 240.7 |  |  | — |
Collateralized loan obligations |  | 157.3 |  |  | — |  |  | 157.3 |  |  | — |
Total fixed maturity investments |  | 1,854.3 |  |  | 216.9 |  |  | 1,637.4 |  |  | — |
Short-term investments |  | 1,196.4 |  |  | 1,196.4 |  |  | — |  |  | — |
Common equity securities (1) |  | 332.8 |  |  | 46.4 |  |  | 286.4 |  |  | — |
Investment in MediaAlpha |  | 148.2 |  |  | 148.2 |  |  | — |  |  | — |
Other long-term investments |  | 1,058.0 |  |  | — |  |  | 13.9 |  |  | 1,044.1 |
Other long-term investments — NAV (2) |  | 564.7 |  |  | — |  |  | — |  |  | — |
Total other long-term investments |  | 1,622.7 |  |  | — |  |  | 13.9 |  |  | 1,044.1 |
Total investments |  | $ | 5,154.4 |  |  | $ | 1,607.9 |  |  | $ | 1,937.7 |  |  | $ | 1,044.1 |
(1) Consist of investments in exchange traded funds (“ETFs”) and listed funds that predominantly invest in international equities. (2) Consists of private equity funds and hedge funds, a bank loan fund, Lloyd’s trust deposits and ILS funds for which fair value is measured at net asset value (“NAV”) using the practical expedient. Investments for which fair value is measured at NAV are not classified within the fair value hierarchy.
|  | December 31, 2021
Millions |  | Fair Value |  | Level 1 |  | Level 2 |  | Level 3
Fixed maturity investments: |  |  |  |  |  |  |  |
U.S. Government and agency obligations |  | $ | 211.5 |  |  | $ | 211.5 |  |  | $ | — |  |  | $ | — |
Debt securities issued by corporations: |  |  |  |  |  |  |  |
Financials |  | 264.2 |  |  | — |  |  | 264.2 |  |  | — |
Consumer |  | 178.1 |  |  | — |  |  | 178.1 |  |  | — |
Technology |  | 117.9 |  |  | — |  |  | 117.9 |  |  | — |
Industrial |  | 112.9 |  |  | — |  |  | 112.9 |  |  | — |
Healthcare |  | 112.8 |  |  | — |  |  | 112.8 |  |  | — |
Utilities |  | 70.9 |  |  | — |  |  | 70.9 |  |  | — |
Communications |  | 56.0 |  |  | — |  |  | 56.0 |  |  | — |
Energy |  | 48.0 |  |  | — |  |  | 48.0 |  |  | — |
Materials |  | 32.1 |  |  | — |  |  | 32.1 |  |  | — |
Total debt securities issued by corporations |  | 992.9 |  |  | — |  |  | 992.9 |  |  | — |
Municipal obligations |  | 291.9 |  |  | — |  |  | 291.9 |  |  | — |
Mortgage and asset-backed securities |  | 277.6 |  |  | — |  |  | 277.6 |  |  | — |
Collateralized loan obligations |  | 135.0 |  |  | — |  |  | 135.0 |  |  | — |
Total fixed maturity investments |  | 1,908.9 |  |  | 211.5 |  |  | 1,697.4 |  |  | — |
Short-term investments |  | 465.9 |  |  | 465.9 |  |  | — |  |  | — |
Common equity securities (1) |  | 251.1 |  |  | — |  |  | 251.1 |  |  | — |
Investment in MediaAlpha |  | 261.6 |  |  | 261.6 |  |  | — |  |  | — |
Other long-term investments |  | 895.3 |  |  | — |  |  | 4.7 |  |  | 890.6 |
Other long-term investments — NAV (2) |  | 482.5 |  |  | — |  |  | — |  |  | — |
Total other long-term investments |  | 1,377.8 |  |  | — |  |  | 4.7 |  |  | 890.6 |
Total investments |  | $ | 4,265.3 |  |  | $ | 939.0 |  |  | $ | 1,953.2 |  |  | $ | 890.6 |
(1) Consist of investments in listed funds that predominantly invest in international equities. (2) Consists of private equity funds and hedge funds, a bank loan fund, Lloyd’s trust deposits and ILS funds for which fair value is measured at NAV using the practical expedient. Investments for which fair value is measured at NAV are not classified within the fair value hierarchy. Investments Held on Deposit or as Collateral As of September 30, 2022 and December 31, 2021, investments of $455.1 million and $479.5 million were held in trusts required to be maintained in relation to HG Global’s reinsurance agreements with BAM. BAM is required to maintain deposits with certain insurance regulatory agencies in order to maintain their insurance licenses. The fair value of such deposits, which represent state deposits and are included within the investment portfolio, totaled $4.6 million and $4.8 million as of September 30, 2022 and December 31, 2021. Lloyd’s trust deposits are generally required of Lloyd's syndicates to protect policyholders in non-U.K. markets and are pledged into Lloyd’s trust accounts to provide a portion of the capital needed to support obligations at Lloyd’s. As of September 30, 2022 and December 31, 2021, Ark held Lloyd’s trust deposits with a fair value of $117.8 million and $113.8 million The underwriting capacity of a member of Lloyd’s must be supported by providing a deposit (“Funds at Lloyd’s”) in the form of cash, securities or letters of credit in an amount determined by Lloyd’s. The amount of such deposit is calculated for each member through the completion of an annual capital adequacy exercise. These requirements allow Lloyd’s to evaluate that each member has sufficient assets to meet its underwriting liabilities plus a required solvency margin. As of September 30, 2022 and December 31, 2021, the fair value of Ark’s Funds at Lloyd’s investment deposits totaled $311.4 million and $342.8 million. As of September 30, 2022 and December 31, 2021, Ark had $49.8 million and $50.0 million of short-term investments pledged as collateral under an uncommitted standby letter of credit. See Note 7 — “Debt.” Debt Securities Issued by Corporations The following table presents the credit ratings of debt securities issued by corporations held in White Mountains’s investment portfolio as of September 30, 2022 and December 31, 2021:
|  | Fair Value at
Millions |  | September 30, 2022 |  | December 31, 2021
AAA |  | $ | 11.2 |  |  | $ | 12.0 |
AA |  | 90.5 |  |  | 85.0 |
A |  | 524.0 |  |  | 490.4 |
BBB |  | 343.0 |  |  | 396.8 |
Other |  | 5.8 |  |  | 8.7 |
Debt securities issued by corporations (1) |  | $ | 974.5 |  |  | $ | 992.9 |
(1) Credit ratings are based upon issuer credit ratings provided by Standard & Poor’s Financial Services LLC (“Standard & Poor’s”), or if unrated by Standard & Poor’s, long-term obligation ratings provided by Moody’s Investor Service, Inc. Mortgage and Asset-backed Securities and Collateralized Loan Obligations The following table presents the fair value of White Mountains’s mortgage and asset-backed securities and collateralized loan obligations as of September 30, 2022 and December 31, 2021:
|  | September 30, 2022 |  | December 31, 2021
Millions |  | Fair Value |  | Level 2 |  | Level 3 |  | Fair Value |  | Level 2 |  | Level 3
Mortgage-backed securities: |  |  |  |  |  |  |  |  |  |  |  |
Agency: |  |  |  |  |  |  |  |  |  |  |  |
FNMA |  | $ | 114.1 |  |  | $ | 114.1 |  |  | $ | — |  |  | $ | 125.4 |  |  | $ | 125.4 |  |  | $ | — |
FHLMC |  | 74.5 |  |  | 74.5 |  |  | — |  |  | 90.5 |  |  | 90.5 |  |  | — |
GNMA |  | 28.9 |  |  | 28.9 |  |  | — |  |  | 40.1 |  |  | 40.1 |  |  | — |
Total agency (1) |  | 217.5 |  |  | 217.5 |  |  | — |  |  | 256.0 |  |  | 256.0 |  |  | — |
Non-agency: Residential |  | .4 |  |  | .4 |  |  | — |  |  | .5 |  |  | .5 |  |  | — |
Total non-agency |  | .4 |  |  | .4 |  |  | — |  |  | .5 |  |  | .5 |  |  | — |
Total mortgage-backed securities |  | 217.9 |  |  | 217.9 |  |  | — |  |  | 256.5 |  |  | 256.5 |  |  | — |
Other asset-backed securities: |  |  |  |  |  |  |  |  |  |  |  |
Credit card receivables |  | 11.9 |  |  | 11.9 |  |  | — |  |  | 12.3 |  |  | 12.3 |  |  | — |
Vehicle receivables |  | 10.9 |  |  | 10.9 |  |  | — |  |  | 8.8 |  |  | 8.8 |  |  | — |
Total other asset-backed securities |  | 22.8 |  |  | 22.8 |  |  | — |  |  | 21.1 |  |  | 21.1 |  |  | — |
Total mortgage and asset-backed securities |  | 240.7 |  | 240.7 |  | — |  |  | 277.6 |  |  | 277.6 |  |  | — |
Collateralized loan obligations |  | 157.3 |  |  | 157.3 |  |  | — |  |  | 135.0 |  |  | 135.0 |  |  | — |
Total mortgage and asset-backed securities and collateralized loan obligations |  | $ | 398.0 |  |  | $ | 398.0 |  |  | $ | — |  |  | $ | 412.6 |  |  | $ | 412.6 |  |  | $ | — |
(1) Represents publicly traded mortgage-backed securities which carry the full faith and credit guaranty of the U.S. Government (i.e., GNMA) or are guaranteed by a government sponsored entity (i.e., FNMA, FHLMC). As of September 30, 2022, White Mountains’s investment portfolio included $157.3 million of collateralized loan obligations that are within the senior tranches of their respective fund securitization structures. All of White Mountains’s collateral loan obligations were rated AAA or AA as of September 30, 2022. Investment in MediaAlpha Following the MediaAlpha IPO, White Mountains’s investment in MediaAlpha is accounted for at fair value based on the publicly traded share price of MediaAlpha’s common stock and is presented as a separate line item on the balance sheet. At the September 30, 2022 closing price of $8.75 per share, the fair value of White Mountains’s investment in MediaAlpha was $148.2 million. See Note 2 — “Significant Transactions.” Other Long-Term Investments The following table presents the carrying values of White Mountains’s other long-term investments as of September 30, 2022 and December 31, 2021:
|  | Fair Value at
Millions |  | September 30, 2022 |  | December 31, 2021
Kudu’s Participation Contracts |  | $ | 813.2 |  |  | $ | 669.5 |
PassportCard/DavidShield |  | 132.0 |  |  | 120.0 |
Elementum Holdings, L.P. |  | 45.0 |  |  | 45.0 |
Other unconsolidated entities (1) |  | 37.0 |  |  | 34.4 |
Total unconsolidated entities |  | 1,027.2 |  |  | 868.9 |
Private equity funds and hedge funds |  | 239.8 |  |  | 153.8 |
Bank loan fund |  | 155.7 |  |  | 163.0 |
Lloyd’s trust deposits |  | 117.8 |  |  | 113.8 |
ILS funds |  | 49.4 |  |  | 51.9 |
Private debt investments |  | 13.0 |  |  | 14.1 |
Other |  | 19.8 |  |  | 12.3 |
Total other long-term investments |  | $ | 1,622.7 |  |  | $ | 1,377.8 |
(1) Includes White Mountains’s non-controlling equity interests in certain private common equity securities, convertible preferred securities, limited liability company units and Simple Agreement for Future Equity (“SAFE”) investments. Private Equity Funds and Hedge Funds White Mountains invests in private equity funds and hedge funds, which are included in other long-term investments. The fair value of these investments is generally estimated using the NAV of the funds. As of September 30, 2022, White Mountains held investments in 16 private equity funds and two hedge funds. The largest investment in a single private equity fund or hedge fund was $49.3 million and $31.3 million as of September 30, 2022 and December 31, 2021. The following table presents the fair value of investments and unfunded commitments in private equity funds and hedge funds by investment objective and sector as of September 30, 2022 and December 31, 2021:
|  | September 30, 2022 |  | December 31, 2021
Millions |  | Fair Value |  | Unfunded Commitments |  | Fair Value |  | Unfunded Commitments
Private equity funds |  |  |  |  |  |  |  |
Aerospace/Defense/Government |  | $ | 102.9 |  |  | $ | 45.1 |  |  | $ | 69.8 |  |  | $ | 11.8 |
Financial services |  | 77.9 |  |  | 54.9 |  |  | 67.7 |  |  | 29.3 |
Real estate |  | 4.3 |  |  | 2.5 |  |  | 4.3 |  |  | 2.9 |
Total private equity funds |  | 185.1 |  |  | 102.5 |  |  | 141.8 |  |  | 44.0 |
Hedge funds |  |  |  |  |  |  |  |
Long/short equity financials and business services |  | 49.3 |  |  | — |  |  | — |  |  | — |
European small/mid cap |  | 5.4 |  |  | — |  |  | 12.0 |  |  | — |
Total hedge funds |  | 54.7 |  |  | — |  |  | 12.0 |  |  | — |
Total private equity funds and hedge funds included in other long-term investments |  | $ | 239.8 |  |  | $ | 102.5 |  |  | $ | 153.8 |  |  | $ | 44.0 |
Investments in private equity funds are generally subject to a lock-up period during which investors may not request a redemption. Distributions prior to the expected termination date of the fund may be limited to dividends or proceeds arising from the liquidation of the fund’s underlying investments. In addition, certain private equity funds have the option to extend the lock-up period. The following table presents investments in private equity funds that were subject to lock-up periods as of September 30, 2022:
Millions |  | 1 – 3 years |  | 3 – 5 years |  | 5 – 10 years |  | >10 years |  | Total
Private equity funds — expected lock-up period remaining |  | $27.5 |  | $19.0 |  | $135.4 |  | $3.2 |  | $185.1
Investors in private equity funds are generally subject to indemnification obligations outside of the capital commitment period and prior to the winding up of the fund. As of September 30, 2022 and December 31, 2021, White Mountains is not aware of any indemnification claims relating to its investments in private equity funds. Redemption of investments in most hedge funds is subject to restrictions, including lock-up periods where no redemptions or withdrawals are allowed, restrictions on redemption frequency and advance notice periods for redemptions. Amounts requested for redemptions remain subject to market fluctuations until the redemption effective date, which generally falls at the end of the defined redemption period. Advance notice requirements for redemptions from White Mountains’s hedge funds investments range from 45 to 90 days. One of White Mountains’s hedge fund investments also limits redemptions to every second anniversary following the date of the initial investment. Bank Loan Fund White Mountains’s other long-term investments include a bank loan fund with a fair value of $155.7 million as of September 30, 2022. The fair value of this investment is estimated using the NAV of the fund. The bank loan fund’s investment objective is to provide, on an unleveraged basis, high current income consistent with preservation of capital and low duration. The bank loan fund primarily invests in a broad portfolio of U.S. dollar-denominated, non-investment grade, floating-rate senior secured loans and may invest in other financial instruments, such as secured and unsecured corporate debt, credit default swaps, reverse repurchase agreements, synthetic indices and cash and cash equivalents. The investment in the bank loan fund is subject to restrictions on redemption frequency and advance notice periods for redemptions. Amounts requested for redemptions remain subject to market fluctuations until the redemption effective date, which generally falls at the end of the defined redemption period. White Mountains may redeem all or a portion of its bank loan fund investment as of any calendar month-end upon 15 calendar days advanced written notice. Lloyd’s Trust Deposits White Mountains’s other long-term investments include Lloyd’s trust deposits, which consist of non-U.K. deposits and Canadian comingled pooled funds. The Lloyd’s trust deposits invest primarily in short-term government securities, agency securities and corporate bonds held in trusts that are managed by Lloyd's of London. These investments are generally required of Lloyd's syndicates to protect policyholders in non-U.K. markets and are pledged into Lloyd’s trust accounts to provide a portion of the capital needed to support obligations at Lloyd’s. The fair value of the Lloyd’s trust deposits is generally estimated using the NAV of the funds. As of September 30, 2022, White Mountains held Lloyd’s trust deposits with a fair value of $117.8 million. Insurance-Linked Securities Funds White Mountains’s other long-term investments include ILS fund investments. The fair value of these investments is generally estimated using the NAV of the funds. As of September 30, 2022, White Mountains held investments in ILS funds with a fair value of $49.4 million. Investments in ILS funds are generally subject to restrictions, including lock-up periods where no redemptions or withdrawals are allowed, non-renewal clauses, restrictions on redemption frequency and advance notice periods for redemptions. From time to time, natural catastrophe, liquidity, market or other events will occur that make the determination of fair value for underlying investments in ILS funds less certain due to the potential for loss development. In such circumstances, the impacted investments may be subject to additional lock-up provisions. ILS funds are typically subject to monthly and annual restrictions on redemptions and advance redemption notice period requirements that range between 30 and 90 days. Amounts requested for redemption remain subject to market fluctuations until the redemption effective date, which generally falls at the end of the defined redemption period. One of the ILS funds in White Mountains’s portfolio requires shareholders to provide advance redemption notice on or before September 15 of each calendar year. Amounts requested for redemption in this fund remain subject to market fluctuation until the underlying investment has fully matured or been commuted, which may be up to a period of three years from the start of each calendar year. Rollforward of Level 3 Investments Level 3 measurements as of September 30, 2022 and 2021 consist of securities for which the estimated fair value has not been determined based upon quoted market price inputs for identical or similar securities. The following table presents the changes in White Mountains’s fair value measurements for Level 3 investments for the nine months ended September 30, 2022 and 2021:
Level 3 Investments |  |  |  |
Millions |  |  |  |  |  |  |  |  |  |  | Other Long-term Investments |  |  |  | Other Long-term Investments |  |
Balance at December 31, 2021 |  |  |  |  |  |  |  |  |  |  | $ | 890.6 |  |  | Balance at December 31, 2020 |  | $ | 614.2 |  |  |  |  |
Net realized and unrealized gains |  |  |  |  |  |  |  |  |  |  | 57.3 |  |  | Net realized and unrealized gains |  | 79.0 |  |  |  |  |
Amortization/accretion |  |  |  |  |  |  |  |  |  |  | — |  |  | Amortization/accretion |  | — |  |  |  |  |
Purchases |  |  |  |  |  |  |  |  |  |  | 127.9 |  |  | Purchases |  | 143.6 |  |  |  |  |
Sales |  |  |  |  |  |  |  |  |  |  | (31.7) |  |  | Sales |  | (31.9) |  |  |  |  |
Effect of Ark Transaction |  |  |  |  |  |  |  |  |  |  | — |  |  | Effect of Ark Transaction |  | 9.6 |  |  |  |  |
Transfers in |  |  |  |  |  |  |  |  |  |  | — |  |  | Transfers in |  | — |  |  |  |  |
Transfers out |  |  |  |  |  |  |  |  |  |  | — |  |  | Transfers out |  | — |  |  |  |  |
Balance at September 30, 2022 |  |  |  |  |  |  |  |  |  |  | $ | 1,044.1 |  |  | Balance at September 30, 2021 |  | $ | 814.5 |  |  |  |  |
Fair Value Measurements — Transfers Between Levels - Nine months ended September 30, 2022 and 2021 Transfers between levels are recorded using the fair value measurement as of the end of the quarterly period in which the event or change in circumstance giving rise to the transfer occurred. During the nine months ended September 30, 2022 and 2021, there were no fixed maturity investments or other long-term investments classified as Level 3 measurements in the prior period that were transferred to Level 2 measurements. During the nine months ended September 30, 2022 and 2021, there were no fixed maturity investments or other long-term investments classified as Level 2 measurements in the prior period that were transferred to Level 3 measurements. Significant Unobservable Inputs The following tables present significant unobservable inputs used in estimating the fair value of White Mountains’s other long-term investments, classified within Level 3 as of September 30, 2022 and December 31, 2021. The tables below exclude $42.9 million and $46.7 million of Level 3 other long-term investments generally valued based on recent or expected transaction prices. The fair value of investments in private equity funds and hedge funds, bank loan funds, Lloyd’s trust deposits and ILS funds are generally estimated using the NAV of the funds.
$ in Millions |  | September 30, 2022
Description |  | Valuation Technique(s) (1) |  | Fair Value (2) |  | Unobservable Inputs
|  |  |  |  |  | Discount Rate (6) |  | Terminal Cash Flow Exit Multiple (x) or Terminal Revenue Growth Rate (%) (6)
Kudu’s Participation Contracts (3)(4)(5) |  | Discounted cash flow |  | $813.2 |  | 17% - 24% |  | 7x - 16x
PassportCard/DavidShield |  | Discounted cash flow |  | $132.0 |  | 24% |  | 4%
Elementum Holdings, L.P. |  | Discounted cash flow |  | $45.0 |  | 18% |  | 4%
Private debt investments |  | Discounted cash flow |  | $11.0 |  | 10% |  | N/A
(1) Key inputs to the discounted cash flow analysis generally include projections of future revenue and earnings, discount rates and terminal exit multiples or growth rates. (2) Includes the net unrealized investment gains (losses) associated with foreign currency; foreign currency effects based on observable inputs. (3) Since Kudu’s Participation Contracts are not subject to corporate taxes within Kudu Investment Management, LLC, pre-tax discount rates are applied to pre-tax cash flows in determining fair values. The weighted average discount rate and weighted average terminal cash flow exit multiple applied to Kudu’s Participation Contracts is 20% and 9.7x. (4) As of September 30, 2022, three of Kudu’s Participation Contracts with a total fair value of $312.4 were valued using a probability weighted expected return method, which was based on a discounted cash flow analysis and an expected sale transaction. (5) In 2022, Kudu deployed a total of $97.9 into new and existing Kudu Participation Contracts, including Gramercy Funds Management, GenTrust, EC Management Services, and TK Partners. (6) Increases (decreases) to the discount rates in isolation would result in lower (higher) fair value measurements, while increases (decreases) to the terminal cash flow exit multiples or terminal revenue growth rates in isolation would result in higher (lower) fair value measurements.
$ in Millions |  | December 31, 2021
Description |  | Valuation Technique(s) (1) |  | Fair Value (2) |  | Unobservable Inputs
|  |  |  |  |  | Discount Rate (6) |  | Terminal Cash Flow Exit Multiple (x) or Terminal Revenue Growth Rate (%) (6)
Kudu’s Participation Contracts (3)(4)(5) |  | Discounted cash flow |  | $669.5 |  | 18% - 23% |  | 7x - 13x
PassportCard/DavidShield |  | Discounted cash flow |  | $120.0 |  | 23% |  | 4%
Elementum Holdings, L.P. |  | Discounted cash flow |  | $45.0 |  | 17% |  | 4%
Private debt investments |  | Discounted cash flow |  | $9.4 |  | 8% |  | N/A
(1) Key inputs to the discounted cash flow analysis generally include projections of future revenue and earnings, discount rates and terminal exit multiples or growth rates. (2) Includes the net unrealized investment gains (losses) associated with foreign currency; foreign currency effects based on observable inputs. (3) Since Kudu’s Participation Contracts are not subject to corporate taxes within Kudu Investment Management, LLC, pre-tax discount rates are applied to pre-tax cash flows in determining fair values. The weighted average discount rate and weighted average terminal cash flow exit multiple applied to Kudu’s Participation Contracts is 20% and 10x. (4) In 2021, Kudu deployed a total of $223.4 into new and existing Kudu Participation Contracts, including TIG Advisors, TK Partners, Third Eye Capital Management, Douglass Winthrop Advisors, Granahan Investment Management and Radcliffe Capital Management. (5) As of December 31, 2021, one of Kudu’s Participation Contracts with a total fair value of $78.8 was valued using a probability weighted expected return method, which was based on a discounted cash flow analysis and an expected sale transaction. (6) Increases (decreases) to the discount rates in isolation would result in lower (higher) fair value measurements, while increases (decreases) to the terminal cash flow exit multiples or terminal revenue growth rates in isolation would result in higher (lower) fair value measurements.

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