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Share Based Compensation (Tables)
 12 Months Ended
Dec. 31, 2019
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:
 
2019
 
2018
 
2017
Weighted-average estimated fair value
$
22.23

 
$
21.87

 
$
19.40

Quarterly dividend rate
$
0.46

 
$
0.44

 
$
0.42

Expected term of stock options, in years
4.2

 
4.5

 
4.7

Risk-free interest rate
2.5
%
 
2.4
%
 
1.9
%
Dividend yield
2.6
%
 
2.5
%
 
2.8
%
Expected stock price volatility
28.9
%
 
27.9
%
 
27.7
%
Present value of dividends
$
7.29

 
$
7.51

 
$
7.50
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, 2019:


 
Number of Stock Options and Stock Appreciation Rights
(in thousands)
 
Weighted-Average Exercise Price
Outstanding at beginning of the year
 
1,399
 
$
55.73

Granted
 
327
 
71.53

Exercised
 
(150)
 
51.34

Forfeited/Cancelled
 
(25)
 
66.24

Outstanding at end of the year
 
1,551
 
59.32

Vested and exercisable at end of the year
 
927
 
52.64
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 2019, 2018, and 2017, 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, 2019:
Stock Options and Stock Appreciation Rights
 
Weighted-Average Remaining Contractual Term
(Years)
 
Aggregate Intrinsic Value
(in millions)
 
 
 
 
 
Exercised in 2017
 
 
 
$
4.4

Exercised in 2018
 
 
 
17.2

Exercised in 2019
 
 
 
4.0

 
 
 
 
 
Outstanding at December 31, 2019 (a)
 
3.8
 
36.7

Vested and exercisable at December 31, 2019
 
2.7
 
28.1

_______
(a) As of December 31, 2019, 371,716 stock appreciation rights and 1,179,619 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, 2019:


Number of Share Units Outstanding (in thousands)
 
Weighted-Average Grant-Date Fair Value
Restricted Stock Units:
 
 
 
Nonvested at beginning of the year
204

 
$
57.57

Granted
43

 
71.56

Vested
(75
)
 
42.21

Forfeited
(8
)
 
67.58

Nonvested at end of the year
164

 
67.73

Performance Shares:
 
 
 
Nonvested at beginning of the year
143

 
$
62.08

Granted
58

 
71.98

Net increase due to estimated performance
16

 
66.20

Vested
(82
)
 
59.77

Forfeited
—

 
N/A

Nonvested at end of the year
135

 
68.23