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Segment Information (Tables)
 3 Months Ended
Mar. 31, 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
March 31, 2021
 InsuranceReinsuranceMortgageSub-TotalOtherTotal
Gross premiums written (1)$1,415,886 $1,471,060 $391,246 $3,277,293 $216,523 $3,397,206 
Premiums ceded(421,047)(471,948)(56,051)(948,147)(37,212)(888,749)
Net premiums written994,839 999,112 335,195 2,329,146 179,311 2,508,457 
Change in unearned premiums(175,365)(354,212)1,122 (528,455)(31,580)(560,035)
Net premiums earned819,474 644,900 336,317 1,800,691 147,731 1,948,422 
Other underwriting income (loss)— (1,198)6,897 5,699 411 6,110 
Losses and loss adjustment expenses(535,747)(484,870)(63,689)(1,084,306)(118,794)(1,203,100)
Acquisition expenses(128,222)(118,025)(30,082)(276,329)(28,152)(304,481)
Other operating expenses(137,113)(60,514)(49,131)(246,758)(14,275)(261,033)
Underwriting income (loss)$18,392 $(19,707)$200,312 198,997 (13,079)185,918 
Net investment income78,729 20,127 98,856 
Net realized gains (losses)101,336 41,125 142,461 
Equity in net income (loss) of investment funds accounted for using the equity method71,686 — 71,686 
Other income (loss)(1,741)— (1,741)
Corporate expenses (2)(23,468)— (23,468)
Transaction costs and other (2)(1,201)(715)(1,916)
Amortization of intangible assets(14,402)— (14,402)
Interest expense(34,197)(4,149)(38,346)
Net foreign exchange gains (losses)21,505 (1,442)20,063 
Income (loss) before income taxes and income (loss) from operating affiliates397,244 41,867 439,111 
Income tax (expense) benefit(38,852)(8)(38,860)
Income (loss) from operating affiliates75,457 — 75,457 
Net income (loss)433,849 41,859 475,708 
Amounts attributable to redeemable noncontrolling interests117 (972)(855)
Amounts attributable to nonredeemable noncontrolling interests— (36,697)(36,697)
Net income (loss) available to Arch433,966 4,190 438,156 
Preferred dividends(10,403)— (10,403)
Net income (loss) available to Arch common shareholders$423,563 $4,190 $427,753 
Underwriting Ratios
Loss ratio65.4 %75.2 %18.9 %60.2 %80.4 %61.7 %
Acquisition expense ratio15.6 %18.3 %8.9 %15.3 %19.1 %15.6 %
Other operating expense ratio16.7 %9.4 %14.6 %13.7 %9.7 %13.4 %
Combined ratio97.7 %102.9 %42.4 %89.2 %109.2 %90.7 %
Goodwill and intangible assets$276,211 $17,807 $377,841 $671,859 $7,650 $679,509 
(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
March 31, 2020
 InsuranceReinsuranceMortgageSub-TotalOtherTotal
Gross premiums written (1)$1,207,645 $1,122,519 $368,945 $2,698,537 $234,902 $2,832,830 
Premiums ceded(378,897)(325,339)(44,327)(747,991)(48,202)(695,584)
Net premiums written828,748 797,180 324,618 1,950,546 186,700 2,137,246 
Change in unearned premiums(112,829)(253,720)20,408 (346,141)(46,661)(392,802)
Net premiums earned715,919 543,460 345,026 1,604,405 140,039 1,744,444 
Other underwriting income (loss)— 2,120 4,599 6,719 133 6,852 
Losses and loss adjustment expenses(507,108)(430,069)(67,566)(1,004,743)(110,676)(1,115,419)
Acquisition expenses(107,337)(79,606)(38,536)(225,479)(21,804)(247,283)
Other operating expenses(129,649)(45,297)(45,896)(220,842)(13,702)(234,544)
Underwriting income (loss)$(28,175)$(9,392)$197,627 160,060 (6,010)154,050 
Net investment income113,028 32,125 145,153 
Net realized gains (losses)(72,109)(294,851)(366,960)
Equity in net income (loss) of investment funds accounted for using the equity method(4,209)— (4,209)
Other income (loss)32 — 32 
Corporate expenses (2)(18,201)— (18,201)
Transaction costs and other (2)(2,595)— (2,595)
Amortization of intangible assets(16,631)— (16,631)
Interest expense(25,245)(7,310)(32,555)
Net foreign exchange gains (losses)63,307 9,364 72,671 
Income (loss) before income taxes and income (loss) from operating affiliates197,437 (266,682)(69,245)
Income tax (expense) benefit(27,945)— (27,945)
Income (loss) from operating affiliates8,516 — 8,516 
Net income (loss)178,008 (266,682)(88,674)
Amounts attributable to redeemable noncontrolling interests(57)(1,096)(1,153)
Amounts attributable to nonredeemable noncontrolling interests— 233,944 233,944 
Net income (loss) available to Arch177,951 (33,834)144,117 
Preferred dividends(10,403)— (10,403)
Net income (loss) available to Arch common shareholders$167,548 $(33,834)$133,714 
Underwriting Ratios     
Loss ratio70.8 %79.1 %19.6 %62.6 %79.0 %63.9 %
Acquisition expense ratio15.0 %14.6 %11.2 %14.1 %15.6 %14.2 %
Other operating expense ratio18.1 %8.3 %13.3 %13.8 %9.8 %13.4 %
Combined ratio103.9 %102.0 %44.1 %90.5 %104.4 %91.5 %
Goodwill and intangible assets$268,296 $2,516 $426,988 $697,800 $7,650 $705,450 

(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.’