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Share Based Compensation (Tables)
12 Months Ended
Dec. 31, 2020
Share-based Payment Arrangement [Abstract]  
Weighted Average Fair Value and Assumptions
The following table shows the weighted-average fair value for our stock options and the assumptions we used to estimate fair value:
202020192018
Weighted-average estimated fair value$22.50 $22.23 $21.87 
Quarterly dividend rate$0.48 $0.46 $0.44 
Expected term of stock options, in years4.24.24.5
Risk-free interest rate1.3 %2.5 %2.4 %
Dividend yield2.5 %2.6 %2.5 %
Expected stock price volatility28.5 %28.9 %27.9 %
Present value of dividends
$7.89 $7.29 $7.51 
Data With Respect to Stock Options SARs Activity
The following table shows information about outstanding stock options and stock appreciation rights for the year ended December 31, 2020:

Number of Stock Options and Stock Appreciation Rights
(in thousands)
Weighted-Average Exercise Price
Outstanding at beginning of the year1,55159.32 
Granted28877.07 
Exercised(337)54.69 
Forfeited/Cancelled(33)72.28 
Expired(2)69.24 
Outstanding at end of the year1,46763.56 
Vested and exercisable at end of the year90757.10 
Schedule of Share-Based Compensation, Aggregate Intrinsic Value and Weighted Average Remaining Contractual Term
The following table shows the aggregate intrinsic value of stock options and stock appreciation rights exercised in 2020, 2019, and 2018, and the weighted-average remaining contractual term and aggregate intrinsic value of stock options and stock appreciation rights outstanding and vested as of December 31, 2020:
Stock Options and Stock Appreciation RightsWeighted-Average Remaining Contractual Term
(Years)
Aggregate Intrinsic Value
(in millions)
Exercised in 2018$17.2
Exercised in 20194.0 
Exercised in 20207.6 
Outstanding at December 31, 2020 (a)3.828.8
Vested and exercisable at December 31, 20202.823.6 
_______
(a) As of December 31, 2020, 153,366 stock appreciation rights and 1,313,577 stock options were outstanding.
Schedule of Share-Based Compensation, Restricted Stock Units and Performance Shares Award Activity
The following table shows information about restricted stock units and performance shares for the year ended December 31, 2020:

Number of Share Units Outstanding (in thousands)Weighted-Average Grant-Date Fair Value
Restricted Stock Units:
Nonvested at beginning of the year164 $67.73 
Granted34 77.08 
Vested(56)61.19 
Forfeited(4)71.82 
Nonvested at end of the year138 72.59 
Performance Shares:
Nonvested at beginning of the year135 $68.23 
Granted54 80.63 
Net increase due to estimated performance71.71 
Vested(70)65.16 
Forfeited(11)70.54 
Nonvested at end of the year112 76.03