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Stock compensation plans (Tables)
 12 Months Ended
Dec. 31, 2020
Share-based Payment Arrangement, Noncash Expense [Abstract]  
Summary of Share-based Compensation Activity
The following table summarizes the share-based compensation activity:
202020192018
(Dollars in millions)
Share-based compensation expense$20.7 $26.9 $22.4 
Total income tax benefit recognized for share-based compensation arrangements22.0 21.1 20.7 
Net excess tax benefit17.5 15.4 15.9 
Weighted-Average Assumptions used to Estimate Fair Value of Options Granted
The fair value of options granted in 2020, 2019 and 2018 was estimated at the date of grant using a Black-Scholes option pricing model. The following weighted-average assumptions were used:
 202020192018
Risk-free interest rate1.16 %2.44 %2.67 %
Expected life of option5.00 years4.99 years4.98 years
Expected dividend yield0.39 %0.47 %0.54 %
Expected volatility23.98 %23.92 %22.65 %
Summary of Stock Option Activity
The following table summarizes the option activity during 2020:
Shares Subject to OptionsWeighted Average Exercise PriceWeighted Average Remaining Contractual Life In Years
Aggregate
Intrinsic
Value
Outstanding, beginning of the year1,325,532 $161.91 
Granted130,206 347.64 
Exercised(289,324)106.79 
Forfeited or expired(9,099)290.49 
Outstanding, end of the year1,157,315 195.57 5.71$249,979 
Exercisable, end of the year903,680 $163.27 4.94$224,388 
Weighted-Average Assumptions used to Estimate Fair Value of Non-Vested Shares Granted The following weighted-average assumptions were used:
202020192018
Risk-free interest rate1.07 %2.41 %2.41 %
Expected dividend yield0.38 %0.46 %0.53 %
Summary of Non-Vested Restricted Stock Unit Activity
The following table summarizes the non-vested restricted stock unit activity during 2020:
Number of
Non-Vested
Shares
Weighted
Average
Grant-Date
Fair Value
Weighted
Average
Remaining
Contractual
Life
Aggregate
Intrinsic
Value
Outstanding, beginning of the year177,348 $240.17 
Granted52,464 344.70 
Vested(67,851)195.95 
Forfeited(10,718)285.37 
Outstanding, end of the year151,243 $293.06 1.2$62,236