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Stock-based Compensation | 3 Months Ended
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Mar. 31, 2020
Stock-based compensation [Abstract] |
Stock-based Compensation | 4. Stock-Based Compensation We have several stock-based compensation plans under which employees and non-employee directors receive stock options, nonvested retention shares, and nonvested stock units. We refer to the nonvested shares and stock units collectively as “retention awards”. We have elected to issue treasury shares to cover option exercises and stock unit vestings, while new shares are issued when retention shares are granted. Information regarding stock-based compensation appears in the table below:
Millions, |  |  |  |
for the Three Months Ended March 31, | 2020 | 2019
Stock-based compensation, before tax: |  |  |  |
Stock options | $ | 4 | $ | 5
Retention awards |  | 18 |  | 22
Total stock-based compensation, before tax | $ | 22 | $ | 27
Excess tax benefits from equity compensation plans | $ | 33 | $ | 39
Stock Options – We estimate the fair value of our stock option awards using the Black-Scholes option pricing model. The table below shows the annual weighted-average assumptions used for valuation purposes:
Weighted-Average Assumptions | 2020 | 2019
Risk-free interest rate |  | 1.5% |  | 2.5%
Dividend yield |  | 2.1% |  | 2.2%
Expected life (years) |  | 4.9 |  | 5.2
Volatility |  | 23.4% |  | 22.7%
Weighted-average grant-date fair value of options granted | $ | 32.20 | $ | 30.37
The risk-free rate is based on the U.S. Treasury yield curve in effect at the time of grant; the expected dividend yield is calculated as the ratio of dividends paid per share of common stock to the stock price on the date of grant; the expected life is based on historical and expected exercise behavior; and expected volatility is based on the historical volatility of our stock price over the expected life of the option. A summary of stock option activity during the three months ended March 31, 2020, is presented below:
| Options (thous.) | Weighted-Average ‎Exercise Price | Weighted-Average Remaining Contractual Term | Aggregate Intrinsic Value (millions)
Outstanding at January 1, 2020 | 3,502 | $ | 113.38 | 6.1 | yrs. | $ | 236
Granted | 558 |  | 176.63 |  | N/A |  | N/A
Exercised | (602) |  | 101.15 |  | N/A |  | N/A
Forfeited or expired | (6) |  | 142.54 |  | N/A |  | N/A
Outstanding at March 31, 2020 | 3,452 | $ | 125.69 | 6.6 | yrs. | $ | 85
Vested or expected to vest at March 31, 2020 | 3,418 | $ | 125.31 | 6.6 | yrs. | $ | 85
Options exercisable at March 31, 2020 | 2,306 | $ | 108.27 | 5.5 | yrs. | $ | 79
Stock options are granted at the closing price on the date of grant, have 10 year contractual terms, and vest no later than 3 years from the date of grant. None of the stock options outstanding at March 31, 2020, are subject to performance or market-based vesting conditions. At March 31, 2020, there was $28 million of unrecognized compensation expense related to nonvested stock options, which is expected to be recognized over a weighted-average period of 1.7 years. Additional information regarding stock option exercises appears in the following table:
Millions, |  |  |  |
for the Three Months Ended March 31, | 2020 | 2019
Intrinsic value of stock options exercised | $ | 48 | $ | 138
Cash received from option exercises |  | 33 |  | 72
Treasury shares repurchased for employee taxes |  | (8) |  | (22)
Tax benefit realized from option exercises |  | 11 |  | 34
Aggregate grant-date fair value of stock options vested |  | 14 |  | 15
Retention Awards – The fair value of retention awards is based on the closing price of the stock on the grant date. Dividends and dividend equivalents are paid to participants during the vesting periods. Changes in our retention awards during the three months ended March 31, 2020, were as follows:
| Shares ‎(thous.) | Weighted-Average ‎Grant-Date Fair Value
Nonvested at January 1, 2020 | 1,898 | $ | 112.12
Granted | 304 |  | 185.97
Vested | (608) |  | 75.89
Forfeited | (22) |  | 134.81
Nonvested at March 31, 2020 | 1,572 | $ | 140.10
Retention awards are granted at no cost to the employee or non-employee director and vest over periods lasting up to 4 years. At March 31, 2020, there was $131 million of total unrecognized compensation expense related to nonvested retention awards, which is expected to be recognized over a weighted-average period of 2.2 years. Performance Retention Awards – In February 2020, our Board of Directors approved performance stock unit grants. The basic terms of these performance stock units are identical to those granted in February 2019, except for different annual return on invested capital (ROIC) performance targets. The plan also includes relative operating income growth (OIG) as a modifier compared to the companies included in the S&P 500 Industrials Index. We define ROIC as net operating profit adjusted for interest expense (including interest on average operating lease liabilities) and taxes on interest divided by average invested capital adjusted for average operating lease liabilities. The modifier can be up to +/- 25% of the award earned based on the ROIC achieved. Stock units awarded to selected employees under these grants are subject to continued employment for 37 months and the attainment of certain levels of ROIC, modified for the relative OIG. We expense the fair value of the units that are probable of being earned based on our forecasted ROIC over the 3-year performance period, and with respect to the third year of the plan, the relative OIG modifier. We measure the fair value of these performance stock units based upon the closing price of the underlying common stock as of the date of grant, reduced by the present value of estimated future dividends. Dividend equivalents are paid to participants only after the units are earned. The assumptions used to calculate the present value of estimated future dividends related to the February 2020 grant were as follows:
| 2020
Dividend per share per quarter | $ | 0.97
Risk-free interest rate at date of grant |  | 1.4%
Changes in our performance retention awards during the three months ended March 31, 2020, were as follows:
| Shares ‎(thous.) | Weighted-Average ‎Grant-Date Fair Value
Nonvested at January 1, 2020 | 929 | $ | 123.32
Granted | 287 |  | 166.63
Vested | (336) |  | 102.58
Unearned | (8) |  | 150.42
Forfeited | (32) |  | 138.71
Nonvested at March 31, 2020 | 840 | $ | 145.57
At March 31, 2020, there was $29 million of total unrecognized compensation expense related to nonvested performance retention awards, which is expected to be recognized over a weighted-average period of 1.6 years. This expense is subject to achievement of the performance measures established for the performance stock unit grants.

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