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Segment Information (Tables)
 9 Months Ended
Sep. 30, 2020
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, 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)919 — 919 
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 taxes433,815 78,574 512,389 
Income tax (expense) benefit(23,638)(69)(23,707)
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.’
Three Months Ended
September 30, 2019
 InsuranceReinsuranceMortgageSub-TotalOtherTotal
Gross premiums written (1)$1,005,874 $662,572 $375,092 $2,043,292 $249,960 $2,181,121 
Premiums ceded(302,034)(226,096)(57,703)(585,587)(94,208)(567,664)
Net premiums written703,840 436,476 317,389 1,457,705 155,752 1,613,457 
Change in unearned premiums(98,504)(72,621)25,611 (145,514)(29,920)(175,434)
Net premiums earned605,336 363,855 343,000 1,312,191 125,832 1,438,023 
Other underwriting income (loss)— (1,208)3,955 2,747 579 3,326 
Losses and loss adjustment expenses(422,782)(270,379)(13,080)(706,241)(96,214)(802,455)
Acquisition expenses(91,259)(62,393)(34,396)(188,048)(23,072)(211,120)
Other operating expenses(115,408)(32,533)(37,003)(184,944)(11,568)(196,512)
Underwriting income (loss)$(24,113)$(2,658)$262,476 235,705 (4,443)231,262 
Net investment income126,874 34,614 161,488 
Net realized gains (losses)80,014 (18,659)61,355 
Equity in net income (loss) of investment funds accounted for using the equity method17,130 — 17,130 
Other income (loss)1,338 — 1,338 
Corporate expenses (2)(15,066)— (15,066)
Transaction costs and other (2)(1,995)— (1,995)
Amortization of intangible assets(20,003)— (20,003)
Interest expense(23,237)(8,091)(31,328)
Net foreign exchange gains (losses)29,794 3,330 33,124 
Income (loss) before income taxes430,554 6,751 437,305 
Income tax (expense) benefit(38,116)— (38,116)
Net income (loss)392,438 6,751 399,189 
Amounts attributable to redeemable noncontrolling interests— (6,600)(6,600)
Amounts attributable to nonredeemable noncontrolling interests— (136)(136)
Net income (loss) available to Arch392,438 15 392,453 
Preferred dividends(10,403)— (10,403)
Net income (loss) available to Arch common shareholders$382,035 $15 $382,050 
Underwriting Ratios     
Loss ratio69.8 %74.3 %3.8 %53.8 %76.5 %55.8 %
Acquisition expense ratio15.1 %17.1 %10.0 %14.3 %18.3 %14.7 %
Other operating expense ratio19.1 %8.9 %10.8 %14.1 %9.2 %13.7 %
Combined ratio104.0 %100.3 %24.6 %82.2 %104.0 %84.2 %
Goodwill and intangible assets$158,990 $— $457,860 $616,850 $7,650 $624,500 

(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)6,327 — 6,327 
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 taxes942,242 1,934 944,176 
Income tax (expense) benefit(78,112)333 (77,779)
Net income (loss)864,130 2,267 866,397 
Dividends 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.’
Nine Months Ended
September 30, 2019
 InsuranceReinsuranceMortgageSub-TotalOtherTotal
Gross premiums written (1)$2,867,753 $1,890,974 $1,095,607 $5,853,574 $598,627 $6,196,809 
Premiums ceded(914,751)(627,120)(149,358)(1,690,469)(178,118)(1,613,195)
Net premiums written1,953,002 1,263,854 946,249 4,163,105 420,509 4,583,614 
Change in unearned premiums(201,719)(186,450)72,436 (315,733)2,735 (312,998)
Net premiums earned1,751,283 1,077,404 1,018,685 3,847,372 423,244 4,270,616 
Other underwriting income (loss)— 4,393 11,867 16,260 1,844 18,104 
Losses and loss adjustment expenses(1,168,677)(751,147)(50,226)(1,970,050)(318,480)(2,288,530)
Acquisition expenses(265,177)(173,504)(98,722)(537,403)(81,654)(619,057)
Other operating expenses(338,327)(102,197)(116,697)(557,221)(39,368)(596,589)
Underwriting income (loss)$(20,898)$54,949 $764,907 798,958 (14,414)784,544 
Net investment income371,161 102,314 473,475 
Net realized gains (losses)316,201 6,167 322,368 
Equity in net income (loss) of investment funds accounted for using the equity method96,533 — 96,533 
Other income (loss)3,550 — 3,550 
Corporate expenses (2)(47,911)— (47,911)
Transaction costs and other (2)(5,363)— (5,363)
Amortization of intangible assets(60,214)— (60,214)
Interest expense(70,094)(19,579)(89,673)
Net foreign exchange gains (losses)28,779 2,918 31,697 
Income (loss) before income taxes1,431,600 77,406 1,509,006 
Income tax (expense) benefit(128,454)(20)(128,474)
Net income (loss)1,303,146 77,386 1,380,532 
Dividends attributable to redeemable noncontrolling interests— (15,778)(15,778)
Amounts attributable to nonredeemable noncontrolling interests— (54,819)(54,819)
Net income (loss) available to Arch1,303,146 6,789 1,309,935 
Preferred dividends(31,209)— (31,209)
Net income (loss) available to Arch common shareholders$1,271,937 $6,789 $1,278,726 
Underwriting Ratios     
Loss ratio66.7 %69.7 %4.9 %51.2 %75.2 %53.6 %
Acquisition expense ratio15.1 %16.1 %9.7 %14.0 %19.3 %14.5 %
Other operating expense ratio19.3 %9.5 %11.5 %14.5 %9.3 %14.0 %
Combined ratio101.1 %95.3 %26.1 %79.7 %103.8 %82.1 %

(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.’