Fig. 1. Pay for luck relationship: Panel A depicts pay for luck with fixed influence costs; Panel B depicts pay for luck with marginal influence costs.

Fig. 2. Asymmetric performance: Panel A depicts the realizations of luck and skill; while Panel B depicts the implied pay for luck relationship that would be estimated.

Descriptive statistics of chief executive officers (CEOs) and firms. The data are collected for every CEO in the ExecuComp database as defined by the CEOANN field for each year 1992–2001. Salary and bonus represent the CEO's yearly salary and bonus values. Cash compensation is the sum of salary, bonus, long-term incentive payouts and all other cash compensation paid. Option grants represent the Black and Scholes value of the options granted to the CEO in the year. Total direct compensation is the sum of salary, bonus, other annual compensation, long-term incentive payouts, other cash payouts, and the value of restricted stock and stock option awards. CEO age is the CEO's age in the data year, and CEO tenure is calculated as the difference between the fiscal year-end of the current year and the date at which the CEO became CEO as given by Became_CEO. Stock return is the one-year percentage return for the firm over its fiscal year. Market cap of equity is the firm's market capitalization at the end of the firm's fiscal year in millions of dollars. The standard deviation of stock returns (stock volatility) is computed using the five years of monthly data preceding the data year. Compensation data are in thousands, and market values are in millions of yearly dollars. Panel A contains the full ExecuComp sample, and Panel B, which contains only CEOs with at least two consecutive years of coverage, contains the subsample upon which we perform our analysis.

Descriptive statistics of performance benchmarks. The data are collected for every firm in which a chief executive officer (CEO) in the ExecuComp database is identified as defined by the CEOANN field for each year 1992–2001. The equal-weighted and value-weighted industry returns on are based on the firm's two-digit standard industrial classification (SIC) code. Summary statistics for returns are in decimal form. The percent positive represents the proportion of the sample for which the relative benchmark return is positive.

Simple correlations among chief executive officers (CEO) and firm variables. Pairwise correlations are carried out for each data item as collected for every CEO in the ExecuComp database as defined by the CEOANN field for each year 1992–2001. Option grants represent the Black and Scholes value of the options granted to the CEO in the year. Total direct compensation (total comp) is the sum of salary, bonus, other annual compensation, long-term incentive payouts, other cash payouts, and the value of restricted stock and stock option awards. CEO age is the CEO's age in the data year, and CEO tenure is calculated as the difference between the fiscal year-end of the current year and the date at which the CEO became CEO as given by Became_CEO. Stock return is the percentage return for the firm over its fiscal year. Market cap of equity is the firm's market capitalization at the end of the firm's fiscal year. The standard deviation of stock returns (stock volatility) is computed using the five years of monthly data preceding the data year. Significance levels are given in parentheses below the correlations. * Indicates different from zero at the 1% level and ** at the 5% level.

Column 1 of Panel A contains an ordinary least squares regression of changes in total direct chief executive officer compensation on the contribution of exogenous factors (luck) on the performance of the firm's dollar returns, the contribution of firm-specific performance, the cdf of the dollar variance of firm returns, tenure, and an interaction of luck with the cdf of the variance of luck and skill with the variance of skill, plus executive fixed effects and year effects. Column 2 replaces total compensation with bonus, and Column 3 uses the Black and Scholes value of options granted. Total direct compensation is the sum of salary, bonus, other annual compensation, long-term incentive payouts, other cash payouts, and the value of restricted stock and stock option awards. In the row labeled Pr(b=a), we provide the p-value from the test that b-a=0. Pr(b=a) is evaluated at the median for both variance of luck and skill. Robust standard errors are reported in parentheses, and the coefficients on the intercept, the cdf of the dollar variance tenure, and the year and executive fixed effects are suppressed for convenience. * Indicates different from zero at the 1% level, ** at the 5% level, and *** at the 10% level. In each case, there are 6,263 observations. Panel B contains summary statistics on the predicted dollar values of luck (λ) and the residual skill () from our first-stage regression of firm returns on equally-weighted and value-weighted average industry returns, where industry is given by the firm's two-digit standard industrial classification code.

Testing for asymmetry around zero. Column 1 contains an ordinary least squares regression of changes in total direct chief executive officer compensation on the contribution of exogenous factors (luck) on the performance of the firm's dollar returns, the contribution of firm-specific performance, and interactions of both luck and skill with dummy variables indicating that luck or skill are negative. We also control for the cdf of the dollar variance of firm returns, tenure, and an interaction of luck with the cdf of the variance of luck and skill with the variance of skill, plus executive fixed effects and year effects. Column 2 estimates the same model, but drops the interaction of skill and the dummy variable indicating that skill is negative. Column 3 estimates the same model as Column 1, but relies on a median regression. Using the same specification as in Column 1, Column 4 replaces total compensation with bonus and Column 5 uses the Black and Scholes value of options granted. Total direct compensation is the sum of salary, bonus, other annual compensation, long-term incentive payouts, other cash payouts, and the value of restricted stock and stock option awards. In the row labeled Pr(b_D=a_D), we provide the p-value from the test that b_D-a_D=0. Median bad luck removed for median firm is equal to the opposite of the coefficient on the luck×luck is down (b_D), divided by the sensitivity of pay to luck (coefficient b on λ) for a firm with the median riskiness of luck. Robust standard errors are reported in parentheses, and the coefficients on the intercept, the cdf of the dollar variance, tenure, and the year and executive fixed effects are suppressed for convenience. *Indicates different from zero at the 1% level and ** at the 5% level. N/A stands for not applicable. In each case, there are 6,263 observations.

Piecewise linear regressions. This table contains median regressions of changes in total direct chief executive officer compensation on the contributions of exogenous factors (luck) and firm-specific performance (skill) on the performance of the firm's dollar returns over various regions (based on percentiles) of these two variables. Specifically, in Columns 1 and 2, we define low luck as the region over which luck took a value less than the 10th percentile, high luck as the region over which luck took a value greater than the 90th percentile, and medium luck as the region in between. We do this similarly for the values of skill in the second column. In Column 3, we alter the breakpoints to the 20th/80th percentiles, respectively. The regressions also include as control variables the cdf of the dollar variance of firm returns, an interaction of luck with the cdf of the variance of luck and skill with the variance of skill, plus firm fixed effects and year effects. Total direct compensation is the sum of salary, bonus, other annual compensation, long-term incentive payouts, other cash payouts, and the value of restricted stock and stock option awards. In the row labeled Pr(LowLuck=MedLuck), we provide the p-value from the test that LowLuck-MedLuck=0. Bootstrapped standard errors with 100 repetitions are reported in parentheses, and the coefficients on the intercept, the cdf of the dollar variance, tenure, and the year and executive fixed effects are suppressed for convenience. * Indicates different from zero at the 1% level, ** at the 5% level, and *** at the 10% level. In each case, there are 6,263 observations.

Pay-for-performance and executive skill. This table contains ordinary least squares regressions of changes in total compensation on dollar returns (luck plus skill and measures of each firm-CEO (chief executive officer) pair's median skill measure. Robust standard errors are in parentheses, and the coefficients on the intercept, the cdf of the dollar variance tenure, and the year and executive fixed effects are suppressed for convenience. * Indicates different from zero at the 1% level and ** at the 5% level.

Luck versus skill and CEO turnover. This table contains probit regressions of the probability the CEO leaves her firm in a given year. Age is a dummy variable that takes on the value one if the CEO is 65 years or older. Column 1 uses continuous measures of the contribution of exogenous factors (luck) on the performance of the firm's dollar returns and the contribution of firm-specific performance (skill). Column 2 uses dummy variables taking on the value one if the relevant measure is negative. In the row labeled Pr(b=a), we provide the p-value from the test that b-a=0. The estimated intercept term and coefficients on controls for market capitalization, asset value, and year dummies are suppressed for convenience. * Indicates different from zero at the 1% level and ** at the 5% level. In each case, there are 6,809 observations.

Differential asymmetry based on governance index. Columns 1–5 of this table contain ordinary least squares regressions of changes in total direct chief executive officer compensation on the contribution of exogenous factors (luck) on the performance of the firm's dollar returns, the contribution of firm-specific performance, the cdf of the dollar variance of firm returns, tenure, and interactions of luck with the cdf of the variance of luck and skill with the variance of skill, plus industry fixed effects and year effects. Columns 1 and 3 contain coefficient estimates for the subsample of firms in the bottom quintile (G6) of the Gompers et al. (2003) Corporate Governance Index (strong governance). Columns 2 and 4 contain coefficient estimates for the subsample of firms in the top quintile (G12) of the Corporate Governance Index (weak governance). Columns 3 and 4 include the interaction of luck with a dummy variable indicating whether luck is negative. Column 5 contains estimates from the same model for the full sample but includes two additional interactions of luck with a dummy variable indicating whether luck is negative with another dummy variable indicating whether corporate governance is good (G6) or poor (G12). Here, we can interpret the coefficient on luck with an indicator of whether luck is negative as capturing the firms of intermediate governance quality. Total direct compensation is the sum of salary, bonus, other annual compensation, long-term incentive payouts, other cash payouts, and the value of restricted stock and stock option awards. Robust standard errors are reported in parentheses, and the coefficients on the intercept, the cdf of the dollar variance, tenure, and the year and industry fixed effects are suppressed for convenience. * Indicates different from zero at the 1% level, ** at the 5% level, and *** at the 10% level.

Option granting policies and luck. The dependent variable in all cases is the Black and Scholes value of options granted. Lagged option grants is the value of options granted in the previous year. Robust standard errors are in parentheses, and the coefficients on the control variables (risk, luck, skill, luck and skill interacted with the cdf of risk, asset value, market capitalization, debt/assets, tenure, and year and two-digit standard industrial classification dummies) are suppressed for convenience. * Indicates different from zero at the 1% level, ** at the 5% level, and *** at the 10% level. N/A stands for not applicable. In each case, there are 6,263 observations.