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Stock-Based Compensation | 12 Months Ended
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Dec. 31, 2023
Stock-Based Compensation |
Stock-Based Compensation | 15. Stock-Based Compensation Grants of restricted stock and restricted stock units and performance share units have been determined and administered by the compensation committee of the Board of Directors. Total stock-based compensation expense was $12.9 million, $10.5 million and $10.6 million for the years ended December 31, 2023, 2022 and 2021, respectively. Stock-based compensation expense is recognized using the straight-line method over the vesting period and generally vests over a three-year vesting period. Certain awards provide for accelerated vesting when the sum of an employee's age and years of service is at least 75. We recognize forfeitures as they occur. Total income tax benefit recognized for stock-based compensation arrangements was $2.7 million, $2.2 million and $2.2 million for each of the years ended December 31, 2023, 2022 and 2021. We generally issue treasury shares for stock options and restricted stock, unless treasury shares are not available. Upon the vesting of restricted shares, we have allowed the holder to elect to surrender an amount of shares to meet their statutory tax withholding requirements. These shares are accounted for as treasury stock based upon the value of the stock on the date of vesting. ​ Restricted Stock and Restricted Stock Units The following table summarizes activity under our restricted stock plans (shares in thousands):
​ | ​ | ​ | ​ | ​ | ​ | ​
​ | ​ | Year Ended | ​
​ | ​ | December 31, 2023 | ​
​ | ​ | ​ | ​ | Weighted- | ​
​ | ​ | ​ | ​ | Average Grant | ​
Restricted Stock and Restricted Stock Units |  | Shares |  | Date Fair Value |
Unvested at beginning of year |  | 104 | ​ | $ | 76.39 | ​
Granted |  | 72 | ​ | $ | 164.47 | ​
Vested |  | (63) | ​ | $ | 79.32 | ​
Forfeited |  | — | ​ | $ | — | ​
Unvested at end of year |  | 113 | ​ | $ | 130.83 | ​
​ Approximately $7.5 million of compensation expense related to restricted stock and restricted stock units will be recognized over a weighted-average period of 2.6 years. We determine the fair value of restricted stock and restricted stock units based on the quoted price of our stock at the date of grant. The weighted-average grant date fair value per share of restricted stock shares and units awarded during 2023, 2022 and 2021 was $164.47, $90.17 and $76.73, respectively. The fair value of restricted stock vested during the years ended December 31, 2023, 2022 and 2021 was $9.3 million, $6.6 million and $5.5 million, respectively. Performance Stock Units Under the 2017 Plan, we granted dollar-denominated performance vesting restricted stock units (“PSUs”), which cliff vest at the end of a three-year performance period. The PSUs are subject to two performance measures; 50% of the PSUs are based on the annual performance of our stock price relative to a group of our peers (total shareholder return) and 50% of the PSUs are measured based on meeting or exceeding a pre-determined annual earnings per share target as set by our Board of Directors (EPS). Depending on the Company’s performance in relation to the established performance measures, the awards may vest at zero to a maximum of 2.0 times the dollar-denominated award granted at target. Upon achievement of the necessary performance metrics, the award will be determined in dollars and may be settled in cash or stock based on the market price of the Company’s common stock at the end of the performance period, at our discretion. Compensation expense for dollar-denominated performance units will ultimately be equal to the final dollar value awarded to the grantee upon vesting, settled either in cash or stock. However, throughout the performance period we must record and accrue expense based on an estimate of that future payout. For units determined by EPS performance, the awards are evaluated quarterly against established targets in order to estimate the liability throughout the vesting period. For units determined by total shareholder return performance, a Monte Carlo simulation model is used to estimate accruals throughout the vesting period. The model simulates our total shareholder return and compares it against our peer group over the three-year performance period to produce a predicted distribution of relative share performance. This is applied to the reward criteria to give an expected value of the total shareholder return element. The calculated fair market value as of December 31, 2023 was $15.0 million. Of this amount, $5.4 million relates to the PSUs granted in 2021 whose performance period ended December 31, 2023. These awards will be settled within the upcoming year either in cash or stock. The fair value of performance stock units vested during the years ended December 31, 2023, 2022, and 2021 was $4.5 million, $3.5 million and $2.2 million, respectively. The expense related to performance stock units for the years ended December 31, 2023, 2022 and 2021 was $6.6 million, $5.3 million and $5.7 million, respectively. At the December 31, 2023 calculated fair market value, approximately $2.1 million of compensation expense related to performance stock units will be recognized over a weighted-average period of 1.4 years. ​ We estimated the fair value of the total shareholder return portion of the PSUs as of December 31, 2023, 2022, and 2021 using a Monte Carlo simulation model with the following assumptions: ​
​ | ​ | ​ | ​ | ​ | ​ | ​
December 31, 2023 | ​ | 2023 PSU Grant | ​ | 2022 PSU Grant
Risk-free interest rate | ​ | 4.2 | % | ​ | 4.7 | %
Dividend yield | ​ | 0.5 | % | ​ | 0.5 | %
Volatility | ​ | 35.0 | % | ​ | 35.1 | %
Look-back period (in years) | ​ | 2.0 | ​ | ​ | 1.0 | ​
​ | ​ | ​ | ​ | ​ | ​ | ​
December 31, 2022 | ​ | 2022 PSU Grant | ​ | 2021 PSU Grant
Risk-free interest rate | ​ | 4.4 | % | ​ | 4.7 | %
Dividend yield | ​ | 0.5 | % | ​ | 0.5 | %
Volatility | ​ | 33.5 | % | ​ | 35.1 | %
Look-back period (in years) | ​ | 2.0 | ​ | ​ | 1.0 | ​
​ | ​ | ​ | ​ | ​ | ​ | ​
December 31, 2021 | ​ | 2021 PSU Grant | ​ | 2020 PSU Grant
Risk-free interest rate | ​ | 0.7 | % | ​ | 0.4 | %
Dividend yield | ​ | 0.5 | % | ​ | 0.5 | %
Volatility | ​ | 50.1 | % | ​ | 31.8 | %
Look-back period (in years) | ​ | 2.0 | ​ | ​ | 1.0 | ​
​ The look-back period reflects the remaining performance period as of the respective year-end dates. The risk-free interest rate for the remaining performance period is based on U.S. Treasury rates as of the respective year-end dates. The assumption for the expected volatility reflects the daily annualized historical volatility on the Company’s dividend adjusted close stock prices measured over the look-back period. The dividend yield assumption is based on the annualized most recent quarterly dividend divided by the stock price on the respective year-end dates.

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