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Equity Incentive Program | 9 Months Ended
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Sep. 30, 2020
Share-based Payment Arrangement, Noncash Expense [Abstract] |
Share-based Compensation | 13. Equity Incentive Program The Company typically grants equity awards annually at its regularly scheduled first quarter meeting of the Compensation Committee of the Board of Directors. During the nine months ended September 30, 2020, the Company issued stock-settled appreciation rights ("SARs") covering 390,780 shares, performance share awards of 49,056 and restricted stock units ("RSUs") of 83,512. The Company uses the Black-Scholes option pricing model to determine the fair value of each SAR on the date of grant. Expected volatilities are based on Dover's stock price history, including implied volatilities from traded options on Dover stock. The Company uses historical data to estimate SAR exercise and employee termination patterns within the valuation model. The expected life of SARs granted is derived from the output of the option valuation model and represents the average period of time that SARs granted are expected to be outstanding. The interest rate for periods within the contractual life of the SARs is based on the U.S. Treasury yield curve in effect at the time of grant. The assumptions used in determining the fair value of the SARs awarded during the respective periods were as follows:
| SARs |  |
| 2020 |  | 2019
Risk-free interest rate | 1.44 | % |  | 2.51 | %
Dividend yield | 1.65 | % |  | 2.13 | %
Expected life (years) | 5.5 |  | 5.6
Volatility | 22.76 | % |  | 22.35 | %
Grant price | $119.86 |  | $91.20
Fair value per share at date of grant | $22.54 |  | $17.55
The performance share awards granted in 2020 are market condition awards as attainment is based on Dover's performance relative to its peer group (companies listed under the S&P 500 Industrials sector) for the relevant performance period. The performance period and vesting period for these awards is approximately three years. These awards were valued on the date of grant using the Monte Carlo simulation model (a binomial lattice-based valuation model), and are generally recognized ratably over the vesting period, and the fair value is not subject to change based on future market conditions. The assumptions used in determining the fair value of the performance shares granted in 2020 were as follows:
| Performance Shares
| 2020
Risk-free interest rate | 1.40 | %
Dividend yield | 1.65 | %
Expected life (years) | 2.9
Volatility | 23.30 | %
Grant price | $119.86
Fair value per share at date of grant | $165.71
The performance share awards granted in 2019 are considered performance condition awards as attainment is based on Dover's performance relative to established internal metrics. The fair value of these awards was determined using Dover's closing stock price on the date of grant. The expected attainment of the internal metrics for these awards is analyzed each reporting period, and the related expense is adjusted based on expected attainment, if that attainment differs from previous estimates. The cumulative effect on current and prior periods of a change in attainment is recognized in selling, general and administrative expenses in the Condensed Consolidated Statements of Earnings in the period of change. The fair value and average attainment used in determining stock-based compensation cost for the performance shares issued in 2019 is as follows for the nine months ended September 30, 2020:
| Performance Shares
| 2019
Fair value per share at date of grant | $91.20
Average attainment rate reflected in expense | 168.96%
The Company also has granted RSUs, and the fair value of these awards was determined using Dover's closing stock price on the date of grant. Stock-based compensation is reported within selling, general and administrative expenses in the Condensed Consolidated Statements of Earnings. The following table summarizes the Company’s compensation expense relating to all stock-based incentive plans:
| Three Months Ended September 30, |  |  |  | Nine Months Ended September 30, |  |
| 2020 |  | 2019 |  | 2020 |  | 2019
Pre-tax stock-based compensation expense | $ | 7,371 |  |  | $ | 7,876 |  |  | $ | 15,591 |  |  | $ | 24,493 |
Tax benefit | (854) |  |  | (489) |  |  | (1,822) |  |  | (2,035) |
Total stock-based compensation expense, net of tax | $ | 6,517 |  |  | $ | 7,387 |  |  | $ | 13,769 |  |  | $ | 22,458 |

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