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Stock compensation plans (Tables)
 12 Months Ended
Dec. 31, 2019
Dec. 31, 2018
Share-based Payment Arrangement, Noncash Expense [Abstract]    
Summary of Share-based Compensation Activity
The following table summarizes the share-based compensation activity:
201920182017
(Dollars in millions)
Share-based compensation expense$26.9  $22.4  $19.4  
Total income tax benefit recognized for share-based compensation arrangements$21.1  $20.7  $12.8  
Net excess tax benefit$15.4  $15.9  $6.6  
  
Weighted-Average Assumptions used to Estimate Fair Value of Options Granted  
The fair value of options granted in 2019, 2018 and 2017 was estimated at the date of grant using a Black-Scholes option pricing model. The following weighted-average assumptions were used:
 201920182017
Risk-free interest rate2.44 %2.67 %1.88 %
Expected life of option4.99 years4.98 years4.94 years
Expected dividend yield0.47 %0.54 %0.71 %
Expected volatility23.92 %22.65 %21.74 %
Summary of Stock Option Activity
The following table summarizes the option activity during 2019:
Shares Subject to OptionsWeighted Average Exercise PriceWeighted Average Remaining Contractual Life In Years
Aggregate
Intrinsic
Value
(Dollars in thousands)
Outstanding, beginning of the year1,471,449  $136.62  
Granted162,087  288.45  
Exercised(291,696) 99.96  
Forfeited or expired(16,308) 245.65  
Outstanding, end of the year1,325,532  161.91  6.0284,371  
Exercisable, end of the year1,018,703  $133.71  5.3247,275  
  
Weighted-Average Assumptions used to Estimate Fair Value of Non-Vested Shares Granted The following weighted-average assumptions were used:
201920182017
Risk-free interest rate2.41 %2.41 %1.47 %
Expected dividend yield0.46 %0.53 %0.71 %
  
Summary of Non-Vested Restricted Stock Unit Activity
The following table summarizes the non-vested restricted stock unit activity during 2019:
Number of
Non-Vested
Shares
Weighted
Average
Grant-Date
Fair Value
Weighted
Average
Remaining
Contractual
Life
Aggregate
Intrinsic
Value
(In years)(Dollars in thousands)
Outstanding, beginning of the year201,812  $188.10  
Granted69,799  286.51  
Vested(80,835) 150.83  
Forfeited(13,428) 236.31  
Outstanding, end of the year177,348  $240.17  1.2$66,761