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Equity Incentive Program | 6 Months Ended
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Jun. 30, 2023
Share-Based Payment Arrangement [Abstract] |
Equity Incentive Program | 12. 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 six months ended June 30, 2023, the Company issued stock-settled appreciation rights ("SARs") covering 359,715 shares, performance share awards ("PSAs") of 43,656 and restricted stock units ("RSUs") of 82,055. During the six months ended June 30, 2022, the Company issued SARs covering 335,285 shares, PSAs of 40,087 and RSUs of 76,509. 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
| 2023 |  | 2022
Risk-free interest rate | 3.91 | % |  | 1.86 | %
Dividend yield | 1.32 | % |  | 1.25 | %
Expected life (years) | 5.4 |  | 5.4
Volatility | 30.65 | % |  | 29.46 | %
Grant price | $153.25 |  | $160.21
Fair value per share at date of grant | $47.27 |  | $42.07
The PSAs granted in 2023 and 2022 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 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 PSAs granted in the respective periods were as follows:
| PSAs
| 2023 |  | 2022
Risk-free interest rate | 4.28 | % |  | 1.68 | %
Dividend yield | 1.32 | % |  | 1.25 | %
Expected life (years) | 2.9 |  | 2.9
Volatility | 27.30 | % |  | 31.10 | %
Grant price | $153.25 |  | $160.21
Fair value per share at date of grant | $249.48 |  | $196.40
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 $153.25 and $160.21 for RSUs granted in 2023 and 2022, 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 June 30, |  | Six Months Ended June 30,
| 2023 |  | 2022 |  | 2023 |  | 2022
Pre-tax stock-based compensation expense | $ | 6,441 |  |  | $ | 7,218 |  |  | $ | 18,723 |  |  | $ | 18,331 |
Tax benefit | (587) |  |  | (731) |  |  | (1,951) |  |  | (1,846) |
Total stock-based compensation expense, net of tax | $ | 5,854 |  |  | $ | 6,487 |  |  | $ | 16,772 |  |  | $ | 16,485 |

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