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Segment Information (Tables)
 9 Months Ended
Sep. 30, 2021
Segment Reporting [Abstract]  
Analysis of underwriting income or loss by segment and reconciliation to net income available to common shareholders
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
 InsuranceReinsuranceMortgageSub-TotalOtherTotal
Gross premiums written (1)$1,596,619 $1,251,760 $360,934 $3,207,415 $— $3,207,415 
Premiums ceded(442,806)(630,371)(60,207)(1,131,486)— (1,131,486)
Net premiums written1,153,813 621,389 300,727 2,075,929 — 2,075,929 
Change in unearned premiums(215,143)57,313 11,238 (146,592)— (146,592)
Net premiums earned938,670 678,702 311,965 1,929,337 — 1,929,337 
Other underwriting income (loss)— 3,293 3,981 7,274 — 7,274 
Losses and loss adjustment expenses(668,630)(545,846)(11,543)(1,226,019)— (1,226,019)
Acquisition expenses(152,467)(129,450)(24,098)(306,015)— (306,015)
Other operating expenses(138,931)(45,647)(46,254)(230,832)— (230,832)
Underwriting income (loss)$(21,358)$(38,948)$234,051 173,745 — 173,745 
Net investment income88,195 — 88,195 
Net realized gains (losses)(25,040)— (25,040)
Equity in net income (loss) of investment funds accounted for using the equity method105,398 — 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 affiliates301,433 — 301,433 
Income tax (expense) benefit(4,137)— (4,137)
Income (loss) from operating affiliates124,119 — 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 Arch419,942 — 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 ratio71.2 %80.4 %3.7 %63.5 %— %63.5 %
Acquisition expense ratio16.2 %19.1 %7.7 %15.9 %— %15.9 %
Other operating expense ratio14.8 %6.7 %14.8 %12.0 %— %12.0 %
Combined ratio102.2 %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.’
Three Months Ended
September 30, 2020
 InsuranceReinsuranceMortgageSub-TotalOtherTotal
Gross premiums written (1)$1,206,328 $1,004,590 $346,248 $2,556,914 $197,480 $2,681,032 
Premiums ceded(382,167)(400,388)(47,783)(830,086)(50,164)(806,888)
Net premiums written824,161 604,202 298,465 1,726,828 147,316 1,874,144 
Change in unearned premiums(105,007)(49,704)52,944 (101,767)(1,285)(103,052)
Net premiums earned719,154 554,498 351,409 1,625,061 146,031 1,771,092 
Other underwriting income (loss)(31)298 4,600 4,867 546 5,413 
Losses and loss adjustment expenses(525,321)(422,084)(153,055)(1,100,460)(115,813)(1,216,273)
Acquisition expenses(102,420)(85,388)(35,716)(223,524)(24,418)(247,942)
Other operating expenses(122,541)(41,818)(36,708)(201,067)(14,619)(215,686)
Underwriting income (loss)$(31,159)$5,506 $130,530 104,877 (8,273)96,604 
Net investment income99,857 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 method126,735 — 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 affiliates432,896 78,574 511,470 
Income tax (expense) benefit(23,638)(69)(23,707)
Income (loss) from operating affiliates919 — 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 Arch409,295 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 ratio73.0 %76.1 %43.6 %67.7 %79.3 %68.7 %
Acquisition expense ratio14.2 %15.4 %10.2 %13.8 %16.7 %14.0 %
Other operating expense ratio17.0 %7.5 %10.4 %12.4 %10.0 %12.2 %
Combined ratio104.2 %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.’
Nine Months Ended
September 30, 2021
 InsuranceReinsuranceMortgageSub-TotalOtherTotal
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 written3,112,207 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 earned2,623,571 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 income256,354 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 method299,270 — 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 affiliates1,396,476 89,435 1,485,911 
Income tax (expense) benefit(93,942)(234)(94,176)
Income (loss) from operating affiliates224,052 — 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 Arch1,524,650 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 ratio66.7 %72.5 %8.7 %58.8 %78.0 %59.8 %
Acquisition expense ratio15.9 %18.5 %8.6 %15.6 %18.9 %15.8 %
Other operating expense ratio15.6 %7.3 %14.6 %12.4 %9.9 %12.3 %
Combined ratio98.2 %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.’
Nine Months Ended
September 30, 2020
 InsuranceReinsuranceMortgageSub-TotalOtherTotal
Gross premiums written (1)$3,444,335 $2,934,174 $1,084,337 $7,461,860 $590,309 $7,831,554 
Premiums ceded(1,119,165)(967,698)(136,154)(2,222,031)(150,437)(2,151,853)
Net premiums written2,325,170 1,966,476 948,183 5,239,829 439,872 5,679,701 
Change in unearned premiums(202,188)(388,321)113,965 (476,544)(22,267)(498,811)
Net premiums earned2,122,982 1,578,155 1,062,148 4,763,285 417,605 5,180,890 
Other underwriting income (loss)(31)1,767 15,649 17,385 1,547 18,932 
Losses and loss adjustment expenses(1,550,632)(1,235,586)(444,721)(3,230,939)(331,275)(3,562,214)
Acquisition expenses(317,428)(255,516)(108,304)(681,248)(68,766)(750,014)
Other operating expenses(370,947)(125,831)(120,178)(616,956)(42,523)(659,479)
Underwriting income (loss)$(116,056)$(37,011)$404,594 $251,527 $(23,412)$228,115 
Net investment income313,916 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 method57,407 — 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 affiliates935,980 1,934 937,914 
Income tax (expense) benefit(78,112)333 (77,779)
Income (loss) from operating affiliates6,262 — 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 Arch862,257 (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 ratio73.0 %78.3 %41.9 %67.8 %79.3 %68.8 %
Acquisition expense ratio15.0 %16.2 %10.2 %14.3 %16.5 %14.5 %
Other operating expense ratio17.5 %8.0 %11.3 %13.0 %10.2 %12.7 %
Combined ratio105.5 %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.’