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Maintaining Optimal CEO Incentives Through Equity Grants and CEO Portfolio Rebalancing

This paper examines the joint hypotheses that firms set optimal levels for CEO incentives, and that firms and CEOs jointly correct deviations from these optimal levels through equity grants and CEO portfolio rebalancing. I investigate two equity-based CEO incentives, pay-for-performance sensitivity and risk-taking incentive. Pay-for-performance sensitivity is defined as the change in CEO wealth for a given change in the firm's stock price, while risk-taking incentive the sensitivity of CEO wealth to equity risk. I find that firms' and CEOs' combined annual adjustment to pay-for-performance sensitivity or risk-taking incentive is negatively related to the degree that each incentive deviates from its target level at the beginning of the year, consistent with firms and CEOs jointly correcting the incentive deviations. Overall, the findings suggest that firms and CEOs coordinate their equity-granting and portfolio-rebalancing decisions to manage optimal CEO incentive levels consistent with economic theory.

of costs of adjustment, such as information gathering and processing costs, deviations from the optimal PPS and RI levels may not be immediately corrected.
Correcting such incentive deviations is essential to ensure the efficient management of the firm. Core and Guay (1999) find evidence consistent with firms setting optimal PPS targets and using equity grants to correct PPS deviations. However, there exists no evidence on the adjustment of RI. I extend prior work by addressing two unanswered questions. First, I directly study how excessive PPS is reduced. Equity grants cannot decrease PPS. Second, I examine how optimal RI is maintained concurrent with PPS. This question is particularly relevant when the two incentives deviate in opposite directions. Equity grants, usually including at-the-money options, cannot simultaneously correct both deviations because they increase both PPS and RI.
Based on principal-agent theory, I argue that optimal PPS and RI arise endogenously from a second-best contract between shareholders and their CEO. Although it is in CEOs' interest to reduce their risk exposure through diversification, empirical evidence is consistent with firms successfully enforcing the adoption of an optimal portfolio through various mechanisms, which include restricted stock grants, option grants with a vesting schedule, target ownership plans that require minimum stockholdings (Core and Larcker, 2000), and possibly the threat of termination.
When CEO incentives exceed the target levels, the optimal contract allows CEOs to reduce the excess. In reality, most CEOs have considerable flexibility in adjusting their equity holdings.
For example, GE's former CEO Jack Welch exercised 500,000 options and sold about 627,700 shares of stock in 1999. Likewise, when the incentives fall below target levels, CEOs should be required to increase their equity holdings. It is more likely for firms to grant CEOs new stock and options, because requiring CEOs to purchase equity on the open market is more costly due to transaction costs and potential capital constraints. A large purchase can also exert upward stock-price pressure. Accordingly, I allow for both equity grants and CEO portfolio rebalancing as incentive-adjusting mechanisms. I hypothesize that firms set optimal equity incentive targets and that, when PPS and RI deviate from their targets, firms alter equity grants and CEOs rebalance their firm-specific stock and option holdings to correct both deviations.
Under the assumption that, on average, CEOs' stock and option portfolios provide optimal PPS and RI, I estimate optimal PPS and RI levels cross-sectionally based on firm characteristics.
Residuals from these models are proxies for incentive deviations, and I use them to predict equity (stock and option) grants and CEO portfolio rebalancing (option exercise, stock sales, and purchases). I find evidence consistent with my predictions. Specifically, when both incentives are above their optimal levels, I find that CEOs exercise options and sell stock to reduce their excess incentives. If one incentive is insufficient and the other excessive, my findings are consistent with firms making equity grants to increase the insufficient incentive and CEOs reducing their stock or option holdings to decrease the excessive one. When both incentives are below their optimal levels, I find firms grant equity and CEOs do not fully undo the increase.
My results are robust to various sensitivity analyses, including alternative performance measures and rank or firm fixed-effects specifications. This paper makes three contributions. First, my evidence is consistent with firms setting optimal PPS and RI targets and with firms and CEOs jointly correcting PPS deviations and RI deviations. Prior research focuses exclusively on how firms use equity grants to manage optimal PPS levels. Second, simultaneously considering PPS and RI leads to different predictions on equity grants and CEOs' portfolio rebalancing from those based on the independent analysis of either incentive. When PPS is excessive and RI insufficient, focusing on PPS alone leads to a prediction of no equity grants, assuming that no other granting reasons such as compensation exist. I predict that the firm grants stock options to increase RI, and that the CEO sells stock to decrease PPS. My findings support my prediction.
Finally, I find that CEOs rebalance their portfolios to adjust their own incentives in a manner consistent with shareholders' interests. My findings indicate that, when firms grant equity to increase insufficient CEO incentives, CEOs do not fully undo the increase through portfolio rebalancing. On the other hand, when faced with excessive incentives, CEOs exercise options and sell stock to reduce their incentives. Prior research offers little evidence on whether shareholders implicitly control the portfolio-rebalancing behavior of their CEOs. Although Ofek and Yermack (2000) investigate how CEOs rebalance their firm-specific equity portfolios following equity grants, they do not examine how that rebalancing affects CEO incentives. Nor do they discuss the reasons for the equity grants.
Section II of this paper develops four hypotheses about CEO and firm actions in the face of incentive deviations. Section III describes the research design. I report empirical results in Section IV and present sensitivity analyses in Section V. Section VI concludes with a discussion of contributions and some suggestions for future research.

II. Hypothesis Development
I define PPS as the change in the dollar value of a CEO's stock and option portfolio for a 1% change in the stock price, and RI as the dollar change in portfolio value for a 0.01 change in annualized standard deviation of stock returns. 2 I discuss alternative CEO incentive measures in Section V, Sensitivity Analysis. 2 I ignore the incentives from potential CEO turnover and from year-to-year adjustments in cash pay, as well as the variation in incentives from other performance measures. It is reasonable to assume that the majority of CEO incentives are from variation in the value of their equity holdings (Hall and Liebman, 1998;Core, Guay, and Verrecchia, 2000).
Assuming that optimal incentive levels exist for both PPS and RI, I predict that firms and CEOs jointly correct deviations from these target levels to ensure incentive alignment.
Deviations from optimal incentive levels can occur for several reasons. First, target levels shift due to changes in CEO or firm characteristics. Second, incentives from a given equity portfolio vary with stock price, stock-return volatility, and the time remaining until option expiration. 3 Finally, CEOs periodically buy and sell stock and exercise options to realize compensation.
Based on the direction of CEOs' incentive deviations, I divide these deviations into four types and summarize my predictions in the first three columns of Table 1. The first type of deviations, both PPS and RI are below their optimal levels, can occur if, after an exogenous shock, the firm is faced with more growth opportunities. Since more growth opportunities aggravate information asymmetry, the firm needs to provide its CEO with a higher PPS to ensure the adoption of profitable projects (Yermack, 1995). Also, the problem of underinvestment in risky projects is particularly damaging for high-growth firms because their prospects depend crucially on such projects, so RI needs to be higher too. Since firms can structure CEO compensation to provide reservation utility and replenish insufficient incentives at the same time, the response is to grant options to increase both incentives. 4 If the options do not exactly correct 3 The value of one share of common stock increases dollar-for-dollar with stock price. PPS from stock options is determined by an option's delta. If option holders can fully diversify the riskiness in option payoff, the Black-Scholes formula shows that delta increases monotonically with stock price. For risk-averse and undiversified executives, delta is smaller but also increases with stock price (Hall and Murphy, 2002).
Common stock provides little RI for most firms (Guay, 1999). RI from options is often computed as the partial derivative of their Black-Scholes value to stock-return volatility and has a "bell-shaped" relation to stock price. RI is very small when stock price is close to zero. It increases with stock price because, at a higher stock price, an upsurge in volatility makes it more likely that the option will finish in the money. Since a small increase in volatility can cause the option to go in the money, RI is maximized at a stock price lower than the exercise price. For in-themoney options, increases in stock-return volatility are less desirable because the CEO's payoff is exposed to risk. Therefore, RI falls as options go further in the money and gradually approaches zero. Lambert et al. (1991) show that, if the probability of the option finishing in the money is sufficiently high, an undiversified executive's stock options can increase aversion to risk; i.e., provide negative RI. the deviations, restricted stock awards can serve as a fine-tuning device. For the incentive increase to be effective, the CEO must not fully undo the equity grants by portfolio rebalancing.
I state the hypothesis that applies to Deviation Type 1 as follows: H1: When PPS and RI are both too low, firms grant new equity (including options) to increase the incentives. CEOs do not fully undo the increased incentives.
Next, consider the type of deviations where PPS is low and RI is excessive. This scenario can occur after a dramatic stock-price decline that leaves the CEO with many moderately-out-ofthe-money and/or moderately-in-the-money options. PPS always declines with stock price, but RI can increase because near-the-money options provide high RI. 5 Assuming that the CEO remains with the firm after the dismal stock performance, it is in the shareholders' interest to replenish insufficient PPS. Granting restricted stock increases PPS, and has little impact on RI.
Excessive RI, on the other hand, may not be easily reduced. CEOs would not exercise out-ofthe-money and cannot exercise unvested options. Although CEOs can reduce RI by exercising slightly-in-the-money options, they forfeit the time value of the options by doing so. Huddart (1994) shows that risk-averse employees exercise stock options at a threshold price-to-strike ratio, i.e., the ratio of stock price to exercise price, that is a function of their level of risk tolerance. He further shows that very few exercises take place when the stock price is close to the exercise price because most employees' thresholds are not exceeded. Huddart and Lang (1996) find supporting evidence that more exercises occur when options are deeper in-themoney. A possible measure to correct both deviations simultaneously is for the firm to cancel CEOs' base salaries. Executive employment contracts typically guarantee minimum salary increases for the subsequent three years. Murphy (1999) finds that 85% of the 1,000 large companies he studies grant options to their CEOs in 1998. Since equity compensation was prevalent in the 1990s, it is likely that a large incentive shortfall can be gradually corrected by requiring the CEO to hold the granted equity.
out-of-the-money options and award new at-the-money options, or to reprice underwater options to set them at-the-money. Gilson and Vetsuypens (1993) (1999) find that firms make more than one equity grant to their CEOs in less than 10% of the fiscal years examined. 6 If one incentive is off from its target but the other is right on the target, I classify this scenario as if the ontarget incentive were excessive because the predictions are similar. For example, if PPS is insufficient and RI is at

Sample Selection and Incentive Measurement
I began by extracting a sample of CEOs from the ExecuComp database. For firms not identifying their CEOs, I select the executive with the highest cash compensation. The initial sample includes 11,608 CEO-year observations from 1992 to 1999. I then eliminate financial firms and firms with incomplete Compustat data. Financial firms are excluded because they have few investment expenditures, which I later use as proxies for growth opportunities. CEOs holding more than 25% of their company's common stock are also excluded because their compensation may not be designed for contracting purposes. The sample used to estimate optimal incentives consists of 9,045 CEO-year observations. When examining incentive adjustments, I retain only those CEOs remaining in their positions the second year and those with complete data on incentive adjustments. The selection procedures result in a final total of 7,054 CEO-year observations. Guay (1999) finds that, except for firms in severe financial distress, CEO option holdings predominantly drive RI. 7 Accordingly, when calculating CEO incentives, I use RI from stockoption holdings as a proxy for total RI (Guay, 1999), and likewise use RI from stock-option grants as a proxy for RI grants.
Calculating firms' adjustments to PPS and RI through equity (stock and option) grants is straightforward. The ExecuComp database provides the dollar amount of restricted-stock grants.
Multiplying this value by 1% produces the PPS from restricted stock grants. I estimate PPS and target level, restricted stock grants are predicted to increase PPS. RI remains constant as a result, and the CEO does not subsequently reduce RI by option exercise. 7 Common stock also provides RI, because the payoff to common stock in a levered firm can be viewed as a call option. The option's underlying asset is the firm value and the exercise price is the face value of debt. Guay (1999) RI from option grants using the Black-Scholes (1973)

Empirical Models for Estimating Optimal Incentives
The empirical models to estimate optimal incentives are based on previous studies (Core and Guay, 1999;Guay, 1999;Himmelberg, 1999). I modify their approach by simultaneously estimating optimal PPS and RI in a two-stage least squares model to capture incentive endogeneity.
These studies identify firm size, investment opportunity set, CEO risk aversion, and industry differences as the prominent common determinants of PPS and RI. They also use CEO tenure and firms' free-cash flow problem as determinants that directly affect PPS alone (see Core and Guay, 1999;Guay, 1999;Himmelberg et al., 1999 for in-depth discussions). Based on Guay (1999), I include PPS as a determinant of optimal RI. Higher PPS ties CEO wealth more closely to stock price, and makes the CEO less diversified, implying a need for higher RI. However, I do not use RI as a determinant of PPS, because how RI affects PPS is not obvious.
Among the common determinants of both incentives, I measure firm size as the log of market value of assets, calculated as the sum of the book value of liabilities plus market value of equity. Similar to Himmelberg et al. (1999), firms' growth opportunities are proxied by their asset composition, i.e., property, plant and equipment (PP&E), changes in working capital, capital expenditures and acquisitions (CAPEX) and R&D expenditures (R&D), all scaled by market value of assets. I set missing R&D expenses to zero, and use a dummy variable (RDUM) to indicate whether the firm reports R&D expense. CEO cash compensation serves as a proxy for both CEO wealth and the degree of risk aversion. 10 I include industry dummies, corresponding to firms' two-digit SIC codes, to control for industry differences.
The free-cash-flow problem directly affects PPS alone. Following Lang et al. (1991), I measure it as [(operating cash flow -common and preferred dividends)/market value of assets)] if the firm has a market-to-book ratio lower than one, and zero if the market-to-book ratio is higher than or equal to one. In PPS estimation, I also use the log of a firm's five-year stockreturn volatility as a proxy for the uncertainty in the firm's operating environment. 11 I do not 10 My results are robust when CEO age is included as an additional explanatory variable in Eqs. (1) and (2). Its exclusion in the reported results is because about half of the observations have missing values for CEO age. 11 Firms' five-year stock-return volatility has a skewness of 2.5. The logarithmic transformation reduces the skewness to 0.79. Stock-return volatility is positively related to both the noisiness in stock price and the firms' include this variable in RI estimation, because RI encourages CEO risk-taking, so the causality is reversed.
All above variables are measured at the end of each fiscal year, and I summarize their acronyms in Appendix C. The following first-stage model estimates PPS: I estimate Eq. (1) each year using only annual observations. I then use the predicted value of log(1+ PPS) it to estimate RI annually in the second-stage model:

Design of Tests for the Hypotheses
Since I assume that Eqs. (1) and (2) approximate optimal levels of PPS and RI at the end of year t, positive (negative) estimated residuals, e PPS, it or e RI, it , indicate that the incentives are above (below) their optima. To test my hypotheses that firms and CEOs jointly correct incentive growth opportunities. If the growth-opportunity proxies capture CEO discretion well, stock-return volatility reflects the noise in stock price, and a negative coefficient ensues. If the proxies for the growth opportunities do not fully capture CEO discretion, the coefficient on stock-return volatility may be positive. deviations, I conduct two tests. The first investigates the relation between incentive deviations and combined adjustments by firms and CEOs. The second examines firms' and CEOs' separate adjustments. I expect my first test to be more powerful because it allows a direct test on whether net adjustments increase an insufficient incentive or decrease an excessive incentive.
My tests differ in scope from prior research. For example, Core and Guay (1999) examines firms' PPS adjustments through equity grants, analogous to part of my second test. In addition, I proxy for the degree of incentive deviation by relative, as opposed to absolute deviations in prior work. To obtain the relative measures, I scale PPS deviations, e PPS, it , by one plus log of total PPS, i.e., (1+log(1+PPS t )), and scale RI deviations, e RI, it , by one plus log of total RI, i.e., (1+log(1+RI t )). In both denominators, one is added inside the log because some CEOs have zero incentives. Another one is then added to the log so that the denominators are nonzero. 12 I argue that, compared with absolute deviations, relative incentive deviations better capture the degree of deviation, and therefore the benefits of incentive alignment. These two types of deviation measures should yield the same results when correction of deviations is immediate and complete. However, when adjusting costs exist, relative deviations may be a superior measure.
Since I measure the degree of deviation by relative deviations, I also express incentive adjustments in relative terms. To do so, I first transform incentive adjustments logarithmically, because Eqs. (1) and (2) estimate the log of optimal incentives. Then, I scale the logarithmically transformed PPS adjustments by one plus log of beginning-of-the-year PPS, i.e., (1+log(1+PPS t )), and scale RI adjustments analogously. For example, I define logarithmically transformed combined incentive adjustments as LN∆PPS and LN∆RI. Since combined adjustments (∆PPS and ∆RI) can be negative, I measure LN∆PPS as the logarithm of 1 plus the absolute value of ∆PPS, and LN∆PPS takes on the sign of ∆PPS. In an analogous manner, I measure LN∆RI. The following two equations examine combined PPS and RI adjustments:   (4) where, In both equations, incentive adjustments and incentive deviations are in their relative forms. I use three dummy variables, D 1 , D 2 , and D 3 , to differentiate among the four deviation types discussed in Section II, Hypothesis Development. For example, D 1 equals one when PPS is insufficient and RI excessive, and zero otherwise. In Eq.
(3), the sum of coefficients, λ 1 +λ 2 , reflects the relation between combined PPS adjustments and PPS deviations in this scenario. In Eq.
(3), I also control for the size of RI deviations and RI adjustments as well as other possible determinants of PPS adjustments. Likewise, in Eq. (4), the relation between combined RI adjustments and RI deviations is tested after controlling for PPS deviations, PPS adjustments, and other determinants. Other determinants of equity grants include firms' financing constraints, the level of CEO pay, and stock-price performance. In addition, recent stock price performance, the ratio of stock price to option exercise price, and stock-return volatility, among other factors, are among the determinants of CEO portfolio rebalancing. I discuss these variables in section IV, empirical results. In Eqs. (3) and (4), I do not include variables that potentially control for CEO risk-aversion, such as CEO cash compensation or CEO age. According to Core and Guay (2000), if CEOs can rebalance their portfolio of total wealth when portfolio risk deviates from the contracted level, no risk adjustments are needed to value their portfolio changes.
After examining combined adjustments in Eqs. (3) and (4) Eq. (5) estimates firms' or CEOs' adjustment to PPS; Eq. (6) estimates firms' or CEOs' adjustments to RI. Since CEOs' adjustment to PPS or RI can be negative, I use as dependent variables log of the absolute value of the adjustments plus one, and the variables take on the sign of the raw adjustments. Same as in Eqs. (3) and (4)

Evidence on the Estimated Optimal Levels of PPS and RI
I estimate Eq. (1) and Eq. (2)   observations), and (IV) both incentives are higher than their optimal levels (3,320 observations).
For the two deviation-types where RI is excessive, the mean price-to-strike ratio is 1.49 or 1.72.
When RI is insufficient, mean price-to-strike ratios are above two. This finding is consistent with deep-in-the-money options providing insufficient RI. Firms are also more likely to reprice CEO options that offer excessive RI. The percentage of firms that reprice CEO options are higher in the categories that RI is too high (2.37% or 2.05%), compared with cases when RI is too low (1.54% or 1.12%). Further, about 10% of the CEOs in my sample hold no options.
These CEOs are almost exclusively categorized in type (I) and (III) where RI is insufficient. The above evidence shows that my estimation models largely capture the conjectured characteristics of the CEO portfolio holdings for each deviation types.

Evidence on the Four Hypotheses
I examine both combined and separate incentive adjustments by firms and CEOs. Results on combined adjustments, i.e., Eqs. (3) and (4), are reported in table 5, and those on separate adjustments, i.e., Eqs. (5) and (6) deviations are -1.35 and -0.21 respectively, both significant at a p-value less than 0.001. The -0.21 coefficient on RI deviation indicate that a CEO whose RI is 10% below average receives a positive RI adjustment that is 2.1% higher than a CEO with an optimal level of RI.
The following four subsections present specific findings for each of the four deviation types discussed in Section II, Hypothesis Development. 15 For each deviation type, I start with evidence on firms' and CEOs' combined adjustments; then I analyze how the combined effects are achieved by separately examining equity grants and CEO portfolio rebalancing.

4.2a Evidence on H1 (both incentives are below their optimal levels)
When both incentives are below their optimal levels, evidence in deviations are -0.65 (λ 1 ) and -0.25 (γ 1 ) respectively, both significant at p-values less than 0.005.

4.2b Evidence on H2 (PPS is insufficient and RI excessive relative to their target levels)
When PPS is insufficient and RI excessive relative to their target levels, my findings are consistent with that combined adjustments increase PPS but do not reduce excessive RI.
Referring once again to Table 5, note that firms' and CEOs' combined PPS adjustments are significantly negatively related to beginning-of-the-year PPS deviations (λ 1 + λ 2 of -1.10).
Combined RI adjustments are positively related to RI deviations (γ 1 + γ 2 of 0.30), but the relation is not significant.
When investigating separate adjustments by firms and CEOs in Table 6, I find that the observed combined PPS increase is due to firms granting equity to increase PPS and CEOs not fully undoing the increase through portfolio rebalancing. Specifically, in the two versions of Eq.

4.2c Evidence on H3 (RI is below the optimal level but PPS excessive)
When PPS is excessive and RI is below the optimal level, combined annual PPS adjustments are significantly negatively related to beginning-of-the-year PPS deviations, and RI adjustments bear the same relationship to RI deviations. In Table 5, the coefficients on PPS and RI deviations are -2.82 (λ 1 + λ 3 ) and -0.20 (γ 1 + γ 3 ) respectively. My findings are consistent with PPS being reduced and RI increased.
Separate analysis of firms' and CEOs' incentive adjustments is consistent with firms using option grants to increase RI.

4.2d Evidence on H4 (both incentives are excessive)
When both incentives are above their optimal levels, the evidence in Table 5  year)/the prior year's cash dividends and stock repurchases] is less than two in the current or any of the previous two years. If the denominator is zero for all three years, the firm is also categorized as dividend constrained. Yermack (1995) and Dechow, Hutton, and Sloan (1996) find that the use of stock options is greater when firms have higher net operating loss carryforwards. I use a dummy variable to indicate nonzero tax loss carry-forwards. In both tables 5 and 6, the signs on the coefficients of most of these variables are consistent with my predictions.
Firms may pay part of the CEO compensation with equity (Baber et al., 1996;Murphy, 1999). Similar to Core and Guay (1999), I control for the potential association between total CEO compensation and firm performance by including both the current year's and the next year's stock returns. These returns also control for CEOs' psychological biases in option exercise attributed to stock-price performance (e.g., Heath et al., 1999). As illustrated in Table 6, equity grants increase with stock returns (Columns 1 and 4). CEOs also reduce more PPS and RI through option exercises and stock sales when the stock price increases (Columns 3 and 6).  Table 6), consistent with CEOs' exercising more options and selling more stock when options are deeper-in-the-money.

V. Sensitivity Analysis
This section provides several sensitivity analyses designed to evaluate the robustness of my results. The results reported in section IV remain qualitatively unchanged.  (table 5), and R 2 improves. The results are consistent with incentive deviations being further adjusted to the optimal levels in the second year. I report results from Eqs. (5) and (6), i.e., separate adjustments by the firm and the CEO in table 7B.

Tests over a Two-year
They are similar to those from the annual estimation except for the following. When both incentives fall short of the optimal levels, my findings are consistent with firms granting more PPS (RI) over the next two-years to CEOs with bigger PPS (RI) shortfalls.   incentives from their stock and option holdings using a certainty-equivalent approach, same as that in Lambert et al. (1991). Under the approach, I make specific assumptions of CEOs' coefficient of relative risk aversion. Appendix B describes details of this estimation procedure.

Alternative Specifications -Rank Regressions and Firm-Effects
The inference based on this procedure is similar to my reported main results, i.e., firms and CEOs jointly correct deviations from optimal PPS and RI levels. In addition, I find that, compared with Black-Scholes model (1973), this alternative procedure leads to minor changes in the ranking of PPS, but it changes the ranking of RI significantly. The rank correlation coefficient between RI estimated by Black-Scholes model (1973) and that based on the certaintyequivalent approach is only 0.82, while that between the two types of PPS measures is 0.98.

VI. Conclusions
This paper examines how firms and CEOs jointly maintain optimal levels of two portfolio equity incentives, pay-for-performance sensitivity (PPS) and risk-taking incentive (RI). I include both equity grants and CEO portfolio rebalancing decisions as incentive-adjusting mechanisms, and hypothesize that the firm and the CEO jointly correct both PPS and RI incentive deviations.
Overall, the empirical findings support my hypotheses.
I provide evidence on four types of incentive deviations based on a broad sample of 9,045 This paper provides more comprehensive evidence than prior research on the maintenance of optimal incentive. First, I ask a broader question that captures the simultaneous maintenance of both optimal PPS and RI levels. Second, I examine two incentive-adjusting mechanisms, equity grants and CEO portfolio rebalancing decisions. Prior research only focuses on equity grants. Third, I use relative incentive deviations to measure the degree of departure from optimal levels, as opposed to the absolute deviations used in prior research. I argue that the relative measure better captures the severity of incentive misalignment. As a final contribution, this paper extends our understanding of CEO incentives by analyzing the dynamics and imperfections in the incentive-adjusting process. I find that incentive adjustments are more complete over a two-year horizon than in an annual window. When CEOs hold a larger proportion of vested options, they tend to reduce more of their excess RI. Based on my findings, a fruitful avenue for future research is to explore how the (temporary) misalignment of incentives influences firms' investment decisions and performance.

Appendix A. Calculation of PPS and RI
I first discuss the calculation of PPS and RI from CEOs' equity (stock and options) holdings at the end of each fiscal year. Then I describe the estimation of adjustments to PPS and RI through equity grants and CEO portfolio rebalancing in the next year.

Estimating year-end PPS and RI from CEOs' stock and option holdings
At each fiscal year-end, pay-for-Performance Sensitivity (PPS) is measured as the change in the dollar value of a CEO's firm-specific equity (stock and option) portfolio for a 1% change in the firm's stock price. Risk-taking incentive (RI) is calculated as the change in the dollar value of a CEO's firm-specific equity portfolio for a 0.01 change in firm's stock-return volatility. Formally, let the CEO's firm-specific wealth be W = n s S + n c C where, n s = Number of shares of common stock owned; n c = Number of stock options owned; S = Stock price; C = Value of a call option on one share of stock.
Based on the Black-Scholes formula (1973) for valuing European call options, as modified by Merton (1973) to account for dividend payouts, PPS and RI can be calculated as: I proxy for expected stock-return volatility and expected dividend yield by ExecuComp's historical volatility and dividend yield. Risk-free rate is estimated by the treasury bond yield corresponding to the option's remaining time-to-maturity. I assume that the maturity of unexercisable (exercisable) options is one (three) years less than the maturity of options granted in the current year. When no options are granted to the CEO in the current year, I use the average maturity of options granted to other executives in the same firm. When none of the executives receive options, I use nine (six) years as the maturity of unexercisable (exercisable) options.

Estimating PPS and RI provided by new equity (stock and option) grants
To calculate PPS and RI from new option grants, the following parameters to the Black-Scholes formula are readily available from ExecuComp database: (1) number of options granted, (2) grant-day price, (3) exercise price, and (4) maturity. In addition, I estimate expected stock-return volatility, expected dividend yield, and the risk-free rate the same as in the above section.
PPS from the restricted-stock grants is the dollar value granted multiplied by 1%, and the dollar value is provided by the ExecuComp database. I set RI from restricted stock grants to zero, because they provide little risk-taking incentives.

Estimating PPS and RI adjustments made by the CEOs in the second year
Decrease (increase) in PPS or RI is codes negative (positive). I separately calculate incentive adjustments due to option exercise and due to stock ownership changes. I assume that the options exercised are from the previous year-end's exercisable options. To calculate changes to PPS and RI due to the options exercised, the only missing parameter to the Black-Scholes formula is the market price at which these options are exercised. The ExecuComp provides the number (NUMBER) and realized value (VALUE) of the options exercised. If K1 is the average exercise price of those exercisable options, the average market price at which options are exercised can be estimated as VALUE / NUMBER + K1; when the number of option exercise exceeds exercisable options, I assume the excess come from unexercisable options. Let K2 be the average exercise price of unexercisable options, and the average market price at which the options are excised is (VALUE + K1* (number of exercisable options) + K2 * (VALUE -number of exercisable options)) /NUMBER; CEOs may retain or sell the stock acquired upon option exercise, or sell or purchase stock independent of option exercise. The change in the number of stock owned during the year can be calculated as (Shares owned) t+1 -(Shares owned) t -(number of restricted stock granted) t+1 .
ExecuComp only provides the value of restricted stock granted as of the date of the grant (RSTKGRNT), but not the number of grants. Since restricted stock is usually granted with options, I divide the granted dollar value by the market price on the option-grant day to estimate the number of restricted stock granted. When no options are granted or when the grant-day market price is missing, I use the average of stock prices at the beginning and the end of the year. To calculate PPS adjustments through the change in CEOs' stock holdings, I multiply the change in the CEO's stock holding by the estimated average price at which options are exercised.

Combining the CEO's and the firm's adjustments.
Let ∆PPS and ∆RI be the combined changes to PPS and RI by shareholders and the CEO over the course of the second year ∆PPS t+1 = (PPS from equity grants) t+1 + (CEO's Adjustments to PPS) t+1 , ∆RI t+1 = (RI from equity grants) t+1 + (CEO's Adjustments to RI) t+1

Appendix B. A Certainty-equivalent Approach to Estimate CEO Incentives
I only use the certainty-equivalent approach to re-estimate incentives from CEOs' portfolio holdings, not their incentive adjustments. Under the certainty-equivalent approach, I make the following assumptions: (1) A CEO's utility function is U(X) = 1/(1-a)X 1-a . The CEO's coefficient of relative risk aversion, a, is constant. I consider a ranging between 0.5 and 4.0.
(2) S represents stock price in T years. Stock return in T years is lognormally distributed with volatility σ and expected value (µ -σ 2 /2)T. I assume µ to be 0.10 and σ to be 0.30.
(3) The CEO's entire wealth consists of three components, one unrelated to firm performance with a value W, m shares of stock with total value of mS, and n stock options with payoff function nZ(S).
I define the value of n stock options to the CEO as C, so that the CEO is indifferent between this amount and the uncertain payoff nZ(S), given the composition of the remainder of his wealth. Formally, C is the solution to the equation,
Since risk-averse CEOs also value their stock holdings lower than the stock's market value, I define H, the value of both stock and option holdings to CEOs, as,

∫ U(W + mS + nZ(S)) f(S) dS = ∫ (W + H) f(S) dS
The following equations calculate incentives provided by CEOs' stock and option holdings.

PPS
Pay-for-performance sensitivity, defined as the change in the dollar value of a CEO's firm-specific equity portfolio for a 1 percent change in the firm's stock price.

RI
Risk-taking incentive, defined as the change in the dollar value of a CEO's firm-specific equity portfolio for a 0.01 change in the firm's stock-return volatility. ∆PPS = Firm's adjustment to PPS (through equity grants) + CEO's adjustments to PPS. ∆RI = Firm's adjustment to RI (through equity grants) + CEO's adjustments to RI ..

LN∆PPS
= ± log(1+ | ∆PPS |). The variable takes on the sign of ∆PPS, which can be positive, negative, or zero.

LN∆RI
= ± log(1+ | ∆RI |). The variable takes on the sign of ∆RI, which can be positive, negative, or zero.

FIRM CHARACTERISTICS
Market Value of Assets = book value of debt + market value of equity. Free-cash-flow problem = [(Operating cash flow -common and preferred dividends)/Market value of assets], when MTB < 1; or 0, when MTB >=1.

Interest coverage
= Interest expense/Operating income before depreciation.

Net Operating Loss
= 1 if net operating loss carry forward is non-zero in the current or any of the two previous years, and 0 otherwise.

Cash constraint
= 3 years' average of [(common and preferred dividends + cash flow used in investing activities -cash flow from operations)]/Market value of assets.

Dividend Constraint
= 1, if (Retained earnings + cash dividend and stock repurchases)/(prior year's cash dividend and stock repurchases) <=2 in the current or any of the two previous years or if the denominator is zero for all three years. Dividend Constraint t equals 0 otherwise.

PROXIES FOR GROWTH OPPORTUNITIES ∆(Working Capital)
Changes in working capital, scaled by market value of assets.

CAPEX
The sum of capital expenditure and acquisitions, scaled by market value of assets. R&D R&D expenditures, scaled by market value of assets.

RDUM
Dummy variable equal to 1 when R&D expense is missing, and 0 otherwise.

PP&E
Property, plant, and equipment, scaled by market value of assets.

Fraction of Black-Scholes Value Realizable
[(year-end stock price -Average exercise price of options held at year-end)/year-end Black-Scholes value per option held] Table 1 Summary of Hypotheses and The firm grants restricted stock to increase PPS.

•
The CEO does not undo the PPS increase by option exercise and stock sales.

•
The CEO may or may not be able to reduce RI by option exercise. The firm does not further increase the incentives by equity grants.
• The CEO reduces PPS by stock sales.

•
The CEO may or may not be able to reduce RI by option exercise.   Variables are defined in Appendix C. Table 3 Estimating Optimal Levels of PPS and RI Optimal PPS and RI are estimated with 9,045 CEO-year observations from 1992 to 1999. Equations (1) and (2) are estimated annually in a two-stage least squares model, with the estimation of PPS being the first stage. For parsimony, I report results a pooled cross-sectional and time-series model with seven year dummies. Fifty-four dummy variables corresponding to the two-digit SIC codes are included. Intercept terms and coefficients on dummy variables are not reported. T-statistics are reported in the parentheses. Predicted signs on the coefficients are in Columns 1 and 3. Residuals from the two equations, e PPS and e RI , are the estimated incentive deviations at the end of each fiscal year.

Table 5 Determinants of Firms' and CEOs' Combined Incentive Adjustments
The sample consists of 7,054 CEO-year observations from 1992 to 1999. T-statistics are presented in the parentheses, and calculated using White's heteroskedasticity-consistent standard errors. Intercept terms and coefficients on fifty-four industry dummies and seven year-dummies are not reported.

Determinants of Separate Incentive Adjustments by Firms or by CEOs
The sample consists of 7,054 CEO-year observations. Eq. (5A) and Eq. (5B) separately estimate firms' or CEOs' adjustments to PPS; Eq. (6A) and Eq. (6B) separately estimate firms' or CEOs' adjustments to RI. I run Eq. (5B) in an OLS model, and t-statistics are calculated using White's heteroskedasticity-consistent standard errors. I run other equations using TOBIT analysis, and t-statistics are based on maximum likelihood standard errors. Intercept terms and coefficients on fifty-four industry dummies and seven year-dummies are not shown. I report t-statistics in parentheses. Predicted signs on the coefficients are presented in the columns to their left.    I scale.(e PPS, it ), the dependent variables in both Eq. (5A) and Eq. (5B), and (LN∆PPS) it+1 by (1+log(1+PPS it )). I scale (e RI, it ), the dependent variables in both Eq. (6A) and Eq. (6B), and (LN∆RI) it+1 by (1+log(1+RI it )).

Control Variables for Firm Performance and CEO Option-Exercising Behavior
Other variables are defined in Appendix C.

Table 7A Sensitivity Analysis on Firms' and CEOs' Combined Adjustments
I report results on three sensitivity analyses: (1) two-year's combined incentive adjustments (Columns 1 and 3), (2) restrictions on excessive-incentive reduction (Columns 2 and 4), and (3) on the subsample only including CEOs with option holdings (Column 5). In the third test, I re-estimate Eqs. (1)-(4), and only report results from Eq. (4). t-statistics are reported in the parentheses, and are calculated using White's heteroskedasticity-consistent standard errors. Coefficients on the hypothesis variables and on some selected control variables are shown.

Determinants of PPS and RI from two-year's Equity Grants
The sample consists of 4,746 CEO-year observations from 1992 to 1999. Eq. (5) estimates the sum of PPS from two-year's equity grants, and Eq. (6) estimates the sum of RI from two-year's equity grants. I use TOBIT models to analyze both equations. t-statistics are reported in parentheses, and are based on maximum likelihood standard errors. Intercept terms and coefficients on fifty-four industry dummies and seven year-dummies are not shown. I only report coefficients on the hypothesis variables and on some selected control variables.