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Equity Incentive Program | 9 Months Ended
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Sep. 30, 2024
Share-Based Payment Arrangement [Abstract] |
Equity Incentive Program | 13. Equity Incentive Program The Company typically makes its annual grants of equity awards pursuant to actions taken by the Compensation Committee of the Board of Directors at its regularly scheduled first quarter meeting. During the nine months ended September 30, 2024, the Company issued stock-settled appreciation rights ("SARs") covering 345,354 shares, performance share awards ("PSAs") of 43,602 and restricted stock units ("RSUs") of 87,755. During the nine months ended September 30, 2023, the Company issued SARs covering 349,491 shares, PSAs of 43,656 and RSUs of 85,053. 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 awards 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
| 2024 |  | 2023
Risk-free interest rate | 4.13 | % |  | 3.91 | %
Dividend yield | 1.28 | % |  | 1.32 | %
Expected life (years) | 5.5 |  | 5.4
Volatility | 31.32 | % |  | 30.65 | %
Grant price | $160.11 |  | $153.25
Fair value per share at date of grant | $51.17 |  | $47.27
The PSAs granted in 2024 vest based on the attainment of two equally weighted measures: (i) Dover’s performance relative to established internal metrics (performance condition) and (ii) Dover's performance relative to its peer group (companies listed under the S&P 500 Industrials sector; market condition). The grant date fair value of the performance condition portion is determined using Dover’s closing stock price at the date of grant and the amount of expense recognized over the vesting period is subject to adjustment based on the expected attainment of the performance condition. The fair value per share at the date of grant for the 2024 performance condition portion is $177.19. The grant date fair value of the 2024 market condition portion, and all 2023 PSAs, is determined using the Monte Carlo simulation model. The amount of expense recognized over the vesting period is not subject to change based on future market conditions. The assumptions used in the Monte Carlo model to determine the fair value of the PSAs granted in the respective periods were as follows:
| PSAs
| 2024 |  | 2023
Risk-free interest rate | 4.37 | % |  | 4.28 | %
Dividend yield | 1.15 | % |  | 1.32 | %
Expected life (years) | 2.8 |  | 2.9
Volatility | 23.30 | % |  | 27.30 | %
Grant price | $177.19 |  | $153.25
Fair value per share at date of grant | $287.62 |  | $249.48
The performance and vesting periods for all 2024 and 2023 PSAs is three years. 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, which was $160.11 and $153.25 for RSUs granted in 2024 and 2023, respectively. 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,
| 2024 |  | 2023 |  | 2024 |  | 2023
Pre-tax stock-based compensation expense | $ | 8,187 |  |  | $ | 6,687 |  |  | $ | 32,297 |  |  | $ | 24,836 |
Tax benefit | (729) |  |  | (692) |  |  | (3,141) |  |  | (2,536) |
Total stock-based compensation expense, net of tax | $ | 7,458 |  |  | $ | 5,995 |  |  | $ | 29,156 |  |  | $ | 22,300 |
Pre-tax stock-based compensation expense attributable to discontinued operations was $103 and $58 for the three months ended September 30, 2024 and 2023, respectively, and $672 and $632 for the nine months ended September 30, 2024 and 2023, respectively. These expenses are included within stock-based compensation expense in the condensed consolidated statements of stockholders' equity. See Note 4 — Discontinued and Disposed Operations for further details.

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