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Earnings Per Common Share (Details) - USD ($)
$ / shares in Units, $ in Thousands
 3 Months Ended 12 Months Ended
Dec. 31, 2020
Sep. 30, 2020
Jun. 30, 2020
Mar. 31, 2020
Dec. 31, 2019
Sep. 30, 2019
Jun. 30, 2019
Mar. 31, 2019
Dec. 31, 2020
Dec. 31, 2019
Dec. 31, 2018
Numerator:                      
Net income                 $ 1,465,711 $ 1,693,300 $ 727,821
Net (income) loss attributable to noncontrolling interests                 (60,190) (56,981) 30,150
Net income available to Arch $ 543,544 $ 419,039 $ 298,821 $ 144,117 $ 326,384 $ 392,453 $ 468,954 $ 448,528 1,405,521 1,636,319 757,971
Preferred share dividends (10,403) (10,403) (10,403) (10,403) (10,403) (10,403) (10,403) (10,403) (41,612) (41,612) (41,645)
Loss on redemption of preferred shares                 0 0 (2,710)
Net income available to Arch common shareholders $ 533,141 $ 408,636 $ 288,418 $ 133,714 $ 315,981 $ 382,050 $ 458,551 $ 438,125 $ 1,363,909 $ 1,594,707 $ 713,616
Denominator:                      
Weighted average common shares outstanding                 403,062,179 401,802,815 401,036,376
Series D preferred securities [1]                 0 0 3,311,245
Weighted average common shares outstanding — basic                 403,062,179 401,802,815 404,347,621
Effect of dilutive common share equivalents:                      
Nonvested restricted shares                 1,682,309 1,673,770 1,474,207
Stock options [2]                 5,514,967 8,132,893 7,084,650
Weighted average common shares and common share equivalents outstanding — diluted                 410,259,455 411,609,478 412,906,478
Earnings per common share:                      
Basic (per share) $ 1.32 $ 1.01 $ 0.72 $ 0.33 $ 0.78 $ 0.95 $ 1.14 $ 1.09 $ 3.38 $ 3.97 $ 1.76
Diluted (per share) $ 1.30 $ 1.00 $ 0.71 $ 0.32 $ 0.76 $ 0.92 $ 1.12 $ 1.07 $ 3.32 $ 3.87 $ 1.73
Stock Options                      
Antidilutive Securities Excluded from Computation of Earnings Per Share [Line Items]                      
Antidilutive securities excluded from computation of earnings per common share (shares)                 2,249,821 1,302,017 5,673,821
[1] The company has determined that, based on a review of the terms, features and rights of the Company’s non-voting common equivalent preferred shares compared to the rights of the Company’s common shareholders, the underlying common shares that the convertible securities convert to were common share equivalents at the time of their issuance.
[2] 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 2020, 2019 and 2018, the number of stock options excluded were 2,249,821, 1,302,017 and 5,673,821, respectively.