Managerial incentives to increase firm volatility provided by debt, stock, and options. Joshua D. Anderson

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1 Managerial incentives to increase firm volatility provided by debt, stock, and options Joshua D. Anderson (617) John E. Core* (617) Abstract We measure a manager s risk-taking incentives as the total sensitivity of the manager s debt, stock, and option holdings to firm volatility. We compare this measure to the option vega and to relative measures used by the prior literature. Vega does not capture risk incentives from managers stock and debt holdings, and does not reflect the fact that employee options are warrants. The relative measures do not incorporate the sensitivity of options to volatility. The new measure explains risk choices better than vega and the relative measures. Our measure should be useful for future research on managers risk choices. First draft: October 2011 This draft: May 30, 2013 * Corresponding author. We gratefully acknowledge comments from Ana Albuquerque (discussant), Wayne Guay, Mitchell Petersen, Eric So, Daniel Taylor, Anand Venkateswaran, Jerry Zimmerman, and seminar participants at the American Accounting Association 2012 Annual Meeting, Columbia University, MIT Sloan School of Management, Northeastern University, Pennsylvania State University, Temple University, and the University of Technology Sydney. We thank Ingolf Dittmann for his estimates of CEO non-firm wealth. We appreciate the financial support of the MIT Sloan School of Management.

2 1. Introduction A large literature uses the sensitivity of stock options to an increase in stock volatility ( vega ) to study whether managers equity portfolios provide incentives to increase risk. Studies on early samples show a strong positive association between vega and risk-taking (Guay, 1999; Coles et al., 2006), whereas studies on later samples show mixed results (e.g., Hayes et al., 2012). We re-examine vega and show that it has two shortcomings: (1) it does not capture potential risk incentives from managers stock and inside debt (unsecured pensions and deferred compensation), and (2) it does not reflect the fact that employee options are warrants. We derive and calculate an overall measure of a manager s risk-taking incentives using the total sensitivity of the manager s debt, stock, and option holdings to firm volatility. Limited liability implies that equity is an option on firm value with a strike price equal to the face value of debt. Consequently, an increase in firm volatility increases equity value by reducing debt value (Black and Scholes 1973; Merton, 1974). When a firm has options, this increase in equity value is shared between the stock and options. This implies that the option sensitivity to volatility is larger than vega. Because options are warrants, an increase in volatility that increases option value comes in part from a decrease in stock value. If the firm has no debt, all of the increase in option value comes from a decrease in stock value. This implies a stock sensitivity to volatility that goes from being negative to positive as leverage increases. A manager s attitude toward risk will be affected by the sensitivities of the managers holdings of debt, stock, and options to firm volatility. 1

3 To estimate these sensitivities, we follow Merton (1974) and value total firm equity (stock and stock options) as an option on the value of firm assets. The model gives an estimate of the decrease in debt value for an increase in firm volatility. This decrease in debt value implies an equal increase in equity value. In turn, the increase in equity value is shared between the stock and stock options. We estimate the CEO s sensitivities by applying the CEO s ownership of debt, stock, and options to the firm s sensitivities. We estimate these sensitivities for a sample of 5,967 Execucomp CEO-years from 2006 to The typical CEO in our sample owns roughly 2% of the debt, 2% of the stock, and 16% of the options. In terms of incentives to increase volatility, this CEO has small negative incentives from debt, small positive incentives from stock, and large positive incentives from options. A one standard deviation increase in firm volatility increases the average CEO s wealth by $3 million, or 7% of total wealth. The total sensitivity increases as leverage increases, but the vega roughly remains constant with leverage. As leverage increases, the debt sensitivity becomes more negative (making the CEO averse to risk increases), but the equity sensitivity (the sum of stock and option sensitivities) increases more rapidly. This occurs because the stock sensitivity changes from being negative to strongly positive. Because vega ignores the debt and stock sensitivities, it can be a noisy and biased measure of risk-taking incentives. If the total sensitivity better reflects CEO incentives, we expect it to be more highly associated with CEOs risk-taking choices. To test this conjecture, we examine the association between the total sensitivity and vega and three proxies for future firm 2

4 risk: stock volatility, research and development expense, and leverage. We specify regression models following Coles et al. (2006) and Hayes et al. (2012). Our results suggest that the total sensitivity is more highly associated with risk-taking than is vega. Theory suggests that incentives relative to wealth determine risk-taking. For this reason, we examine modified specifications in which we scale the CEO s risk-taking incentives with a proxy for the CEO s total wealth. We also scale the CEO s incentives to increase stock price ( delta ) by total wealth. Prior research (e.g., Coles et al., 2006) controls for wealth effects by using tenure and cash compensation as proxies for CEO wealth. We find that the scaled measures explain firm risk better than the unscaled measures, and that the scaled total sensitivity explains firm risk better than the scaled vega. The total sensitivity measure requires data on CEOs inside debt, which data became available only in To avoid this limitation, we also examine the equity sensitivity, which is equal to the sum of the stock sensitivity and the option sensitivity (or the total sensitivity minus the debt sensitivity). The equity sensitivity is very highly correlated with the total sensitivity because the debt sensitivity is small and has low variance. We compute the equity sensitivity from and compare it with vega. In this sample, we also find that equity sensitivity explains risk-taking better than vega, and that the scaled equity sensitivity is superior to the unscaled measures. 1 1 Our finding that the equity sensitivity is superior to vega suggests that the stock sensitivity provides important incentives. Guay (1999) also examines the stock sensitivity, but finds that it does not have a large effect on incentives. Potential reasons for the difference in our findings include: (1) we value options as warrants, (2) we use a different asset volatility calculation, and (3) we use a different sample. 3

5 A concern about regressions of incentives on risk-taking is reverse causality (that is, when risk is expected to be high, firms use high risk-taking incentives). To explore the robustness of our results, we follow Hayes et al. (2012) and use the introduction of option expensing in 2005 as an exogenous change to incentives. Consistent with Hayes et al., we find no significant association between changes in vega and changes in firm risk for our sample. However, the change in scaled equity sensitivity is significantly positively associated with both the change in stock volatility and the change in leverage. We find no association, however, with the change in R&D expense. Our derivation of the sensitivity of debt, stock, and options also implies that the relative risk-taking measures used in the recent literature (e.g., Cassell et al., 2012; Sundaram and Yermack, 2007; Wei and Yermack, 2011) are noisy and can be biased. These measures do not correctly incorporate the sensitivity of option value to firm volatility. We calculate a measure that correctly weights the manager s debt, stock, and option sensitivities. The prior measures suggest that CEOs on average are highly aligned with debt holders: the average CEO has debt incentives to reduce volatility that are three times his or her equity incentives to increase volatility. By contrast, the corrected measure, which explicitly takes into account the incentives to increase firm volatility from options, is an order of magnitude smaller, suggesting that CEOs have little alignment with debt holders: the average CEO has incentives to reduce volatility that are equal to 0.4 times his or her equity incentives to increase volatility. Consistent with prior literature, we find that these ratios are negatively associated with risk choices. However, our 4

6 scaled total sensitivity measure is more highly associated with risk-taking choices than the relative ratios. We contribute to the literature in several ways. We calculate a measure of risk-taking incentives that includes the sensitivity of managers debt and stock holdings. In addition, our measure better calculates the sensitivity of the manager s stock options to firm volatility. We compare this measure to vega and to the relative measures used by the prior literature. We find that the new measure is more highly associated with risk choices than vega and the relative measures. Our measure should be useful for future research on managers risk choices. The remainder of the paper proceeds as follows. In the next section, we define the sensitivity of firm debt, stock, and options to firm volatility. We then define the corresponding measures of the sensitivity of the CEO s portfolio to firm volatility. In the third section, we describe how we select a sample of CEOs, and compare various measures of incentives. In the fourth section, we compare regressions using the measures to explain various firm outcomes and provide robustness tests. In the fifth section, we conclude. 2. Definition of incentive measures In this section we first show how firm debt, equity, and option values change with changes in firm volatility, and then we relate these changes to measures of managerial incentives. 2.1 Sensitivity of firm capital structure to firm volatility In general, firms are financed with debt, equity, and employee options: (1) 5

7 Debt is the market value of the debt, Stock is the market value of stock, and Options is the market value of options. It is convenient to express stock and option values in per share amounts, and we assume the firm has n shares of stock outstanding with stock price P. The firm has qn stock options outstanding with option price W. For simplicity in our notation, we assume for the moment that all options have the same exercise price and time to maturity so that each option is worth W. To begin, suppose that there are no stock options outstanding, so that (1) becomes: (2) Black and Scholes (1973) and Merton (1973) show that equity can be valued as a call with a strike price equal to the face value of debt. Under the assumption that changes in firm volatility do not change the value of the firm: 0 (3) Therefore, any loss in debt value due to volatility increases is offset by an equal gain in equity value: (4) More volatile returns increase the value of equity holders call option, which reduces the value of debt. The interests of debt and equity conflict. Equity prefers higher firm volatility, which raises the value of its call; debt prefers lower firm volatility, which increases the value of its short call. Now consider a firm with no debt financed with stock and employee stock options: (5) 6

8 Employee stock options are warrants (W) because exercising the options results in the firm issuing new shares of stock and receiving the strike price. Analogous to (4), an increase in firm volatility has the following effect on the stock price and the option price: (6) Equation (6) shows that the price of a share of stock in a firm with only stock and employee stock options decreases when firm volatility increases (Galai and Schneller, 1978). The share price decreases because the increased volatility makes it more likely that the option will be in the money and that the current value of a share outstanding will be diluted. This result for options on stock is similar to the result when stock is an option on the value of the levered firm. Increases in volatility do not change the value of the firm: Therefore, any gains to the options are offset by losses to the stock. Now we combine the results for debt and options. An increase in firm volatility affects debt, stock, and option value according to the following relation: (7) In firms with both debt and options, shareholders have purchased a call on the assets, and they have sold a call on the equity to employees. They are in a position with respect to the equity similar to the position of the debt holders with respect to the assets. When the firm is levered, increasing firm volatility causes shareholders to gain from the call on the asset but to lose on the call on the equity. Since the change in stockholders value is a combination of these two 7

9 opposing effects, whether stockholders prefer more volatility depends on the number of options outstanding and firm leverage, as we illustrate next Estimation of firm sensitivities To estimate the sensitivities described above, we calculate the value of debt and options using standard pricing models. We then increase firm volatility by 1%, hold firm value fixed, and recalculate the values of debt, stock, and options. We estimate the sensitivities to a one percent change in firm volatility as the difference between these values. Appendix A describes the details. We first price employee options as warrants using the Black-Scholes model, as modified to account for dividend payouts by Merton (1973), and modified to reflect warrant pricing by Schulz and Trautmann (1994). Calculating option value this way gives a value for total firm equity. Second, we model firm equity as an option on the levered firm following Merton (1974) using the Black-Scholes formula. This model allows us to calculate total firm value and firm volatility following the approach of Eberhart (2005). With these values in hand, we calculate the value of the debt as a put on the firm s assets with strike price equal to the face value of debt. To calculate the sensitivities, we increase firm volatility by 1%, which implies a 1% increase in stock volatility. We use this new firm volatility to determine a new debt value. The sensitivity of the debt to a change in firm volatility is the difference between this value and the value at the lower firm volatility. From (7), equity increases by the magnitude of the decrease in the debt value. Finally, we use the higher equity value and higher stock volatility to compute a new value for stock and stock options following Schulz and Trautmann (1994). The difference 8

10 between these stock and option values and those calculated in the first step is the sensitivity to firm volatility for the stock and options Example of firm sensitivities To give intuition for the foregoing relations, in Panel A of Table 1, we show the sensitivities for an example firm. We use values that are approximately the median values of our sample described below. The market value of assets is $2.5 billion and firm volatility is 35%. Options are 7% of shares outstanding, and have a price-to-strike ratio of The options and the debt have a maturity of four years. Leverage is the face value of debt divided by the sum of the book value of debt and market value of equity. To calculate the values and sensitivities, we assume a risk-free rate of 2.25%, that the interest rate on debt is equal to the risk-free rate, and that the firm pays no dividends. The first set of rows shows the change in the value of firm debt, stock, and options for a 1% change in the standard deviation of the assets at various levels of leverage. An increase in volatility reduces debt value, and this reduction is greater for greater leverage. This reduction in debt value is shared between the stock and options. Options always benefit from increases in volatility. When leverage is low, the sensitivity of debt to firm volatility is very low. Since there is little debt to transfer value from, option holders gain at the expense of stockholders when volatility increases. As leverage increases, the sensitivity of debt to firm volatility decreases. As this happens, the stock sensitivity becomes positive as the stock offsets losses to options with gains against the debt. 2.2 Managers incentives from the sensitivity of firm capital structure to firm volatility 9

11 2.2.1 Total incentives to increase firm volatility We now use the above results to derive measures of managerial incentives. A manager s (risk-neutral) incentives to increase volatility from a given security are equal to the security s sensitivity to firm volatility multiplied by the fraction owned by the manager. If the manager owns α of the outstanding stock, β of the outstanding debt, and options, the manager s total incentives to increase firm volatility are: where (8) is the manager s average per option sensitivity to firm volatility, computed to reflect that employee options are warrants Vega incentives to increase stock volatility Prior literature uses the vega, the sensitivity of managers option holdings to a change in stock volatility, as a proxy for incentives to increase volatility (Guay, 1999; Core and Guay, 2002; Coles et al., 2006; Hayes et al., 2012). The vega is the change in the Black-Scholes option value for a change in stock volatility: (9) (Here, we use the notation O to indicate that the option is valued using Black-Scholes, in contrast to the notation W to indicate that the option is valued as a warrant.) Comparing the vega with the total sensitivity in (8), one can see that the vega is a subset of total risk-taking incentives. In particular, it does not include incentives from debt and stock and does not account for the fact that employee stock options are warrants. Inspection of the difference between (8) and (9) 10

12 reveals that for the vega to be similar to total risk-taking incentives, the firm must have low or no leverage (so that the volatility increase causes little re-distribution from debt value to equity value) and the firm must have low amounts of options (so that the volatility increase causes little re-distribution from stock value to option value) Relative incentives to increase volatility Jensen and Meckling (1976) suggest a scaled measure of incentives: the ratio of riskreducing incentives to risk-increasing incentives. The ratio of risk-reducing to risk-increasing incentives in (8) is equal to the ratio of debt incentives (multiplied by -1) to stock and option incentives: (10a) From Panel A of Table 1, the sensitivity of stock to volatility can be negative when the firm has options but little leverage. In this case the ratio of risk-reducing to risk-increasing incentives is: (10b) We term this ratio the relative sensitivity ratio. As in Jensen and Meckling, the ratio is informative about whether the manager has net incentives to increase or decrease firm risk. It can be useful to know whether risk-reducing incentives are greater than risk-increasing incentives (that is, whether Eq. (8) is negative or positive, or equivalently whether the ratio in Eq. (10) is greater or less than one). If the ratio in Eq. (10) is less than one, then the manager has more riskincreasing incentives than risk-reducing incentives and vice versa if the ratio is greater than one. 11

13 When the manager s portfolio of debt, stock, and options mirrors the firm s capital structure, the ratio in (10) is one. Jensen and Meckling (1976) posit that a manager with such a portfolio would have no incentives whatsoever to reallocate wealth between capital providers (p. 352) by increasing the risk of the underlying assets. If the firm has no employee options, the stock sensitivity is always positive and the relative sensitivity ratio (10) becomes: (11) The first expression follows from (4):, and the sensitivity of total debt value to volatility divides off. The second equality follows from the definition of β and α as the manager s fractional holdings of debt and stock. An advantage of this ratio is that, if in fact the firm has no employee options, one does not have to estimate the sensitivity of debt to volatility to compute the ratio. Much prior literature (e.g., Anantharaman et al., 2011; Cassell et al., 2012; Sundaram and Yermack, 2007; Tung and Wang, 2011; Wang et al., 2010, 2011) uses this measure, and terms it the relative leverage ratio, as it compares the manager s leverage to the firm s leverage. Since most firms have options in their capital structure, to operationalize the relative leverage ratio, researchers make an ad hoc adjustment by adding the Black-Scholes value of the options (O for the firm and for the CEO) to the value of the firm s stock and CEO s stock: (12) 12

14 Alternatively, Wei and Yermack (2011) make a different ad hoc adjustment for options by converting the options into equivalent units of stock by multiplying the options by their Black- Scholes delta for the irm and for the CEO : (13) That these adjustments for options are not correct may be seen by comparing the measures to the correct relative sensitivity measure shown in (10), which uses the option sensitivity to firm volatility. Equations (12) and (13), which instead use the option value and delta implicitly assume that options are less sensitive to firm volatility then stock or debt. Only when the firm has no employee options are the relative leverage and incentive ratios equal to the relative sensitivity ratio. However, it is important to note that scaling away the levels information contained in (8) can lead to incorrect inference, even when calculated correctly. For example, imagine two CEOs who both have $1 million total wealth and both have a relative sensitivity ratio of 0.9. Although they are otherwise identical, CEO A has risk-reducing incentives of -$900 and has risk-increasing incentives of $1,000, while CEO B has risk-reducing incentives of -$90,000 and has risk-increasing incentives of $100,000. The relative measure (0.9) scales away the sensitivities and suggests that both CEOs make the same risk choices. However, CEO B is much more likely to take risks: his wealth increases by $10,000 (1% of wealth) for each 1% increase in firm volatility, while CEO A s increases by only $100 (0.01% of wealth). 13

15 2.2.4 Empirical estimation of CEO sensitivities We calculate CEO sensitivities as weighted functions of the firm sensitivities. The CEO s debt and stock sensitivities are the CEO s percentage ownership of debt and stock multiplied by the firm sensitivities. We calculate the average strike price and maturity of the manager s options following Core and Guay (2002). We calculate the value of the CEO s options following Schulz and Trautmann (1994). Appendix A.5 provides details and notes the necessary Execucomp, Compustat, and CRSP variable names. Calculating the sensitivities requires a normalization for the partial derivatives. Throughout this paper we report results using a 1% increase in firm volatility, which is equivalent to a 1% increase in stock volatility. In other words, to calculate a sensitivity, we first calculate a value using current volatility, then increase volatility by 1% and re-calculate the value. The sensitivity is the difference in these values. Prior literature (e.g., Guay, 1999) calculates vega using a 0.01 increase in stock volatility. The disadvantage of using a 0.01 increase in stock volatility for our calculations is that it implies an increase in firm volatility that grows smaller than 0.01 as firm leverage increases. So that the measures are directly comparable, we therefore use a 1% increase in stock volatility to compute vega. The 1% vega is highly correlated (0.91) with the 0.01 increase vega used in the prior literature, and all of our inferences in Tables 4, 6, 7, and 8 below with the 1% vega are identical to those with the 0.01 increase vega Example of CEO sensitivities In Panel B of Table 1, we illustrate how incentives to take risk vary with firm leverage for an example CEO (of the example firm introduced above). The example CEO owns 2% of the 14

16 firm s debt, 2% of the firm s stock, and 16% of the firm s options. These percentages are similar to the averages for our main sample described below. Columns (2) to (4) show the sensitivities of the CEO s debt, stock, and options to a 1% change in firm volatility for various levels of leverage. As with the firm sensitivities, the example CEO s debt sensitivity decreases monotonically with leverage, while the sensitivities of stock and options increase monotonically with leverage. Column (5) shows that the total equity sensitivity, which is the sum of the stock and option sensitivities, increases sharply as the stock sensitivity goes from being negative to positive. Column (6) shows the total sensitivity, which is the sum of the debt, stock, and option sensitivities. These total risk-taking incentives increase monotonically with leverage as the decrease in the debt sensitivity is outweighed by the increase in the equity sensitivity. Column (7) shows the vega for the example CEO. In contrast to the equity sensitivity and the total sensitivity which both increase in leverage, the vega first increases and then decreases with leverage in this example. Part of the reason is that the vega does not capture the debt and stock sensitivities. Holding this aside, the vega does not measure well the sensitivity of the option to firm volatility. It captures the fact that the option price is sensitive to stock volatility, but it misses the fact that equity value benefits from decreases in debt value. As leverage increases, the sensitivity of stock price to firm volatility increases dramatically (as shown by the increasingly negative debt sensitivity), but this effect is omitted from the vega calculations. Consequently, the vega is likely to be a good approximation of the CEO s incentives to increase risk when leverage is very low, but the approximation is much noisier as leverage increases. 15

17 The final Columns (8) to (10) illustrate the various relative incentive measures. The relative sensitivity measure in Column (8) is calculated following Eq. (10) as the negative of the sum of debt and stock sensitivities divided by the option sensitivity when the stock sensitivity is negative (as for the three lower leverage values) and as the negative of debt sensitivity divided by the sum of the stock and option sensitivity otherwise. Thus, risk-reducing incentives are -$6 for the low-leverage firms and -$57 for the high-leverage firms. The risk-increasing incentives are $46 for the low-leverage firms and $115 for the high-leverage firms. Accordingly, as leverage increases, the relative sensitivity measure increases from 0.13 (= 6/46) to 0.50 (= 57/115), indicating that the CEO is more identified with debt holders (has fewer relative risktaking incentives). This inference that risk-taking incentives decline is the opposite of the increase in risk-taking incentives shown in Column (6) for the total sensitivity and total sensitivity as a percentage of total wealth. This example illustrates the point above that scaling away the levels information contained in Eq. (8) can lead to incorrect inference even when the relative ratio is calculated correctly. In columns (9) and (10) of the final rows, we illustrate how the relative leverage and relative incentive ratios for our example CEO. The relative leverage ratio is computed by dividing the CEO s percentage debt ownership (2%) by the CEO s ownership of total stock and option value (roughly 2.4%). Because these value ratios do not change much with leverage, the relative leverage ratio stays about 0.8, suggesting that the CEO is highly identified with debt holders. The relative incentive ratio, which is similarly computed by dividing the CEO s percentage debt ownership (2%) by the CEO s ownership of total stock and option delta (roughly 16

18 2.8%), also shows high identification with debt holders and little change with leverage. Again, this is inconsistent with the substantial increase in risk-taking incentives illustrated in Column (6) for the total sensitivity. As noted above, these ratios only measure relative incentives correctly when the firm has little or no options, and even a correctly calculated relative risk-taking ratio can lead to incorrect inference. The incentive ratios are not informative about the magnitude of the sensitivity of the manager s wealth to an increase in firm volatility. 3. Sample and Variable Construction 3.1 Sample Selection We use two samples of Execucomp CEO data. Our main sample contains Execucomp CEOs from 2006 to 2010, and our secondary sample, described in more detail in Section 4.4 below, contains Execucomp CEOs from 1994 to The total incentive measures described above require information on CEO inside debt (pensions and deferred compensation) and on firm options outstanding. Execucomp provides information on inside debt only beginning in 2006 (when the SEC began to require detailed disclosures). Our main sample therefore begins in The sample ends in 2010 because our tests require one-year ahead data that is only available through Following Coles et al. (2006) and Hayes et al. (2012), we remove financial firms (firms with SIC codes between 6000 and 6999) and utility firms (firms with SIC codes between 4900 and 4999). We identify an executive as CEO if we can calculate CEO tenure from Execucomp data and if the CEO is in office at the end of the year. If the firm has more than one CEO during the year, we choose the 17

19 individual with the higher total pay. We merge the Execucomp data with data from Compustat and CRSP. The resulting sample contains 5,967 CEO-year observations that have complete data. 3.2 Descriptive statistics firm size, volatility, and leverage Table 2, Panel A shows descriptive statistics for volatility, the market value of firm debt, stock, and options, and leverage for the firms in our sample. We describe in Appendix A.3 how we estimate firm market values following Eberhart (2005). To mitigate the effect of outliers, we winsorize all variables each year at the 1 st and 99 th percentiles. Because our sample consists of S&P 1500 firms, the firms are large and have moderate volatility. Most firms in the sample have low leverage. The median value of leverage is 14%, and the mean is 19%. These low amounts of leverage suggest low agency costs of asset substitution for most sample firms (Jensen and Meckling, 1976). 3.3 Descriptive statistics CEO incentive measures Table 3, Panel A shows full sample descriptive statistics for the CEO incentive measures. We detail in Appendix A how we calculate these sensitivities. As with the firm variables described above, we winsorize all incentive variables each year at the 1 st and 99 th percentiles. 2 The average CEO in our sample has some incentives from debt to decrease risk, but the amount of these incentives is low. This is consistent with low leverage in the typical sample firm. Nearly half of the CEOs have no debt incentives. The magnitude of the incentives from stock to increase firm risk is also small for most managers, but there is substantial variation in these 2 Consequently, the averages in the table do not add, i.e., the average total sensitivity is not equal the sum of the average debt sensitivity and average equity sensitivity. 18

20 incentives, with a standard deviation of approximately $47 thousand as compared to $16 thousand for debt incentives. The average sensitivity of the CEO s options to firm volatility is much larger. The mean value of the total (debt, stock, and option) sensitivity is $65 thousand, which indicates that a 1% increase in firm volatility provides the average CEO in our sample with $65 thousand in additional wealth. The vega is smaller than the total sensitivity, and has strictly positive values as compared to the total sensitivity which has about 8% negative values. 3 While the level measures of the CEO s incentives are useful, they are difficult to interpret in cross-sectional comparisons of CEOs who have different amounts of wealth. Wealthier CEOs will respond less to the same incentives if wealthier CEOs are less risk-averse. 4 In this case, a direct way to generate a measure of the strength of incentives across CEOs is to scale the level of incentives by the CEO s wealth. We estimate CEO total wealth as the sum of the value of the CEO s debt, stock, and option portfolio and wealth outside the firm. 5 We use the measure of CEO outside wealth developed by Dittmann and Maug (2007). 6,7 The average scaled total sensitivity is 0.14% of wealth. The value is low because the sensitivities are calculated with 3 This vega is calculated for a 1% increase in stock volatility rather than the 0.01 increase used in prior literature to make it comparable to the total sensitivity. 4 It is frequently assumed in the literature (e.g., Hall and Murphy, 2002; Lewellen, 2006; Conyon et al., 2011) that CEOs have decreasing absolute risk aversion. 5 We value the options as warrants following Schulz and Trautmann (1994). This is consistent with how we calculate the sensitivities of the CEOs portfolios. 6 To develop the proxy, Dittmann and Maug assume that the CEO enters the Execucomp database with no wealth, and then accumulates outside wealth from cash compensation and selling shares. Dittmann and Maug assume that the CEO does not consume any of his outside wealth. The only reduction in outside wealth comes from using cash to exercise his stock options and paying U.S. federal taxes. Dittmann and Maug claim that their proxy is the best available given that managers preferences for saving and consumption are unobservable. We follow Dittmann and Maug (2007) and set negative estimates of outside wealth to missing. 7 The wealth proxy is missing for approximately 13% of CEOs. For those CEOs, we impute outside wealth using a model that predicts outside wealth as a function of CEO and firm characteristics. If we instead discard observations with missing wealth, our inference below is the same. 19

21 respect to a 1% increase in firm volatility. If the average CEO increases firm volatility by one standard deviation (19.9%), that CEO s wealth increases by 7%. While some CEOs have net incentives to decrease risk, these incentives are small. For the CEO at the first percentile of the distribution who has large risk-reducing incentives, a one standard deviation decrease in firm volatility increases the CEO s wealth by 2%. We present sample descriptive statistics sorted by leverage in Table 3, Panel B. We rank the sample by leverage and divide it into five groups of 1,193 firm-years. The first set of rows shows the sensitivity of the value of CEO s debt, stock, and options for a 1% increase in firm volatility at various levels of leverage. The second set of rows show the mean sensitivity of the CEO s debt, stock, and options as a percentage of the CEO s wealth. Since CEO wealth varies greatly, the percentage values are more interpretable, and we concentrate our discussion on the values in these rows. Column (2) shows that as leverage increases the mean of the CEOs debt sensitivity to firm volatility decreases. Intuitively, the sensitivity of the firm s debt decreases in leverage, so the sensitivity of the CEO s inside debt also decreases, holding percentage ownership constant. The mean stock and option sensitivities in Columns (3) and (4) both increase with leverage. While the CEO s stock sensitivity is negative for low levels of leverage, the mean incentives from stock are very small as a fraction of wealth. When leverage is high, the magnitude of the stock sensitivity is much larger. CEOs option sensitivity also increases with leverage. Given the large increase in equity sensitivity shown in Column (5), the CEOs total sensitivity to firm volatility in Column (6) increases across leverage bins. By contrast, the vega in Column (7) has no relation with 20

22 leverage. The vega excludes one of the two channels that affect option sensitivity to firm volatility: the stock-sensitivity channel. When leverage is high, the stock price responds strongly to increases in firm volatility, changing the value of the stock options. Since the vega excludes this channel, it is biased downwards when leverage is high. 3.4 Descriptive statistics CEO relative incentive measures Panel A shows that the mean (median) relative leverage ratio is 3.09 (0.18), and the mean (median) relative incentive ratio is 2.31 (0.15). These values are similar to those in Cassell et al. (2012), who also use an Execucomp sample. These ratios are skewed, and are approximately one at the third quartile, suggesting that 25% of our sample CEOs have incentives to decrease risk. This fraction is much larger than the 8% of CEOs with net incentives to reduce risk based on the total sensitivity measure, and suggests a bias in the relative leverage and incentive measures. By contrast, the mean (median) relative sensitivity ratio is 0.42 (0.03), suggesting low incentives to decrease risk. Panel B shows that the relative ratios (Columns (8) through (10) in the second set of rows) all decrease in leverage, consistent with the increase in the total sensitivity in Column (6). The mean relative sensitivity ratio in Column (8) is below one for all of the leverage bins. This is consistent with the intuition that firms must provide risk-averse CEOs with incentives to increase risk. For the 40% of firms with the lowest leverage, the relative leverage and incentive ratios are much greater than one. These measures suggest that low leverage firms choose to strongly identify their CEOs with their debt holders. However, these are the firms that have few agency problems from debt, so there is little need to provide strong identification with debt holders. This 21

23 puzzling observation is a result of these ratios incorrectly weighting the sensitivity of the CEOs options to firm volatility. 3.5 Correlations CEO incentive measures Panel C of Table 3 shows Pearson correlations between the incentive measures. Focusing first on the levels, the total sensitivity and vega are highly correlated (0.69). The total sensitivity is almost perfectly correlated with the equity sensitivity (0.99). Since CEOs inside debt sensitivity to firm volatility has a low variance, including debt sensitivity does not provide much incremental information about CEOs incentives. The scaled total sensitivity and scaled vega are also highly correlated (0.79), and the scaled total sensitivity is almost perfectly correlated with the scaled equity sensitivity (0.98). The relative leverage and relative incentive ratios are almost perfectly correlated (0.99). Because the correlation is so high, we do not include the relative incentive ratio in our subsequent analyses. 4. Associations of incentive measures with firm risk choices 4.1 Research Design Unscaled incentive measures We examine how the CEO s incentives at time t are related to firm risk choices at time t+1 using regressions of the following form: Firm Risk Choice Risk-taking Incentives Delta Control (14) The form of the regression is similar to those in Guay (1999), Coles et al. (2006) and Hayes et al. (2012). 22

24 Guay (1999, p. 46) shows that manager s incentives to increase risk are positively related to the sensitivity of wealth to volatility, but negatively related to the increase in the manager s risk premium that occurs when firm risk increases. Prior researchers examining vega (e.g. Armstrong et al., 2013, p. 7) argue that vega provides managers with an unambiguous incentive to adopt risky projects, and that this relation should manifest empirically so long as the regression adequately controls for differences in the risk premiums. Delta (incentives to increase stock price) is an important determinant of the manager s risk premium. When a manager s wealth is more concentrated in firm stock, he or she is less diversified, and requires a greater risk premium when firm risk increases. We control for the delta of the CEO s equity portfolio measured following Core and Guay (2002). We also control for cash compensation and CEO tenure, which prior literature (Guay, 1999; Coles et al. 2006) uses as proxies for the CEO s outside wealth and risk aversion. We use three proxies for firm risk choices: (1) ln(stock Volatility t+1 ) measured using daily stock volatility over year t+1, (2) R&D Expense t+1 measured as the ratio of R&D expense to total assets, and (3) Book Leverage t+1 measured as the book value of long-term debt to the book value of assets. Like the prior literature, we consider ln(stock Volatility) to be a summary measure of the outcome of firm risk choices, R&D Expense to be a major input to increased risk through investment risk, and Book Leverage to be a major input to increased risk through capital structure risk. We measure all control variables at t and all risk choice variables at t+1. By doing this, we hope to mitigate concerns about reverse causality. 23

25 Other control variables in these regressions follow Coles et al. (2006) and Hayes et al. (2012). We control for firm size using ln(sales), and for growth opportunities using Market-to- Book. All regressions include year and 2-digit SIC industry fixed effects. In the regression with ln(stock Volatility t+1 ), we also control for risk from past R&D Expense and CAPEX and Book Leverage. In the regression with R&D Expense, we also control for ln(sales Growth) and Surplus Cash. In the regression with Book Leverage as the dependent variable, we control for ROA, and follow Hayes et al. (2012) by controlling for PPE, the quartile rank of a modified version of the Altman (1968) Z-score, and whether the firm has a long-term issuer credit rating Scaled incentive measures Prior literature identifies CEO wealth as an important determinant of CEO s attitudes toward risk. As noted above, the larger delta is relative to wealth, the greater the risk premium the CEO demands. Likewise, the larger risk-taking incentives are relative to wealth, the more a given risk increase will change the CEO s wealth, and the greater the CEO s motivation to increase risk. To capture these effects more directly, we scale risk-taking incentives and delta by wealth and control for wealth in the following alternative specification: Firm Risk Choice Risk-taking Incentives/wealth Delta/wealth + wealth + Control (15) 24

26 To enable comparison across the models, we include the same control variables in (15) as in (14) above. 4.2 Association of level and scaled incentive measures with firm risk choices Table 4, Panel A contains the Stock Volatility regressions. Vega in Column (1) has an unexpected significant negative coefficient. This result is inconsistent with findings in Coles at al. (2006) for When we examine data from in Section 4.4 below, however, we find a positive coefficient on vega. This finding and findings in Hayes et al. (2012) are consistent with changes in the cross-sectional relation between vega and risk-taking over time. The coefficient on total sensitivity in Column (2) is positive but not significant. As noted above, scaling the level of incentives by total wealth can provide a better cross-sectional measure of CEOs incentives. The coefficients on both the scaled vega in Column (3) and the total sensitivity in Column (4) are both positive, and the coefficient on scaled total sensitivity is significant. To evaluate the nonnested hypothesis that the scaled total sensitivity in Column (4) better explains stock volatility than the scaled vega in Column (3), we test whether the adjusted R 2 is significantly greater using a Vuong test. This test compares the explanatory power of the regressions (Vuong, 1989). 8 We cluster the standard errors by firm and year (Barth, Gow, and Taylor, 2012). This test (labeled Col. 3 = Col. 4 at the bottom of Panel A) rejects the scaled vega 8 The J-test is another widely-used test of nonnested hypotheses. In contrast to the J-test, the Vuong test is preferable because it compares the specifications directly without embedding one model in the other (Greene, 2008, p. 139). 25

27 in favor of the scaled total sensitivity (p-value < 0.01). A similar test (labeled Col. 2 = Col. 4) rejects the level of total sensitivity in favor of the scaled total sensitivity (p-value < 0.01). In contrast, in Panel B all four measures have positive and significant relations with R&D Expense, consistent with the prediction that greater risk-taking incentives lead to more R&D. In this instance, vega has significantly higher explanatory power than the total sensitivity (p-values < 0.01). Again, the scaled measures do a better job of explaining R&D Expense than the level measures (p-values < 0.02). In Panel C, vega is unexpectedly negatively related to Book Leverage. The total sensitivity, however, is positively related to leverage. We note that the total sensitivity is a noisy measure of incentives to increase leverage. An increase in leverage does not affect asset volatility, but does increase stock volatility. The total sensitivity therefore is only correlated with a leverage increase through components sensitive to stock volatility (options and the warrant effect of options on stock), but not through components sensitive to asset volatility (debt and the debt effect on equity). 9 The scaled measures are both positively and significantly associated with leverage. Consistent with Panel A, using the scaled total sensitivity rather than the scaled vega improves the explanatory power (p-value < 0.01). 9 Similar to Lewellen (2006), we also calculate a direct measure of the sensitivity of the manager s portfolio to a leverage increase. To do this, we assume that leverage increases because 1% of the asset value is used to repurchase equity. The firm repurchases shares and options pro rata so that option holders and shareholders benefit equally from the repurchase. The CEO does not sell stock or options. The sensitivity of the manager s portfolio to the increase in leverage has a 0.67 correlation with the total sensitivity. The sensitivity to increases in leverage has a significantly higher association with book leverage than the total sensitivity. However, there is not a significant difference in the association when both measures are scaled by wealth. 26

28 Based on Table 4, the total sensitivity scaled by wealth is positively related to each of the risk variables. The specification that includes scaled total sensitivity also provides the most explanatory power for most of the firm risk variables. In summary, scaling the incentive variables and including wealth significantly improves the specification, and using the scaled total sensitivity rather than the scaled vega improves the explanatory power further. 4.3 Association of relative ratios with firm risk choices The preceding section compares the total sensitivity to vega. In this section, we compare the scaled total sensitivity to the relative leverage ratio. 10 The regression specifications are identical to Table 4. These specifications are similar to, but not identical to, those of Cassell et al. (2012). 11 Because our main interest is the incentive variables, the only controls we tabulate are wealth and delta scaled by wealth. The relative leverage ratio has two shortcomings as a regressor. First, it is not defined for firms with no debt or for CEOs with no equity incentives, so our largest sample in Table 5 is 4,994 firm-years as opposed to 5,967 in Table 4. Second, as noted above and in Cassell et al. (2012), when the CEOs inside debt is large relative to firm debt, the ratio takes on very large values. As one way of addressing this problem, we trim extremely large values by winsorizing 10 Again, because the relative incentive ratio is almost perfectly correlated with the relative leverage ratio, results with the relative incentive ratio are virtually identical, and therefore we do not tabulate those results. 11 An important difference is in the control for delta. We include delta scaled by wealth in our regressions as a proxy for risk aversion, and find it to be highly negatively associated with risk-taking as predicted. Cassell et al. (2012) include delta as part of a composite variable that combines delta, vega, and the CEO s debt equity ratio. They find inconsistent signs and little significance with the variable. 27

29 the ratio at the 90 th percentile. 12 We present results for the subsample where the relative leverage ratio is defined in Columns (1) and (2) of each panel. As another way of addressing this problem, Cassell et al. (2012) use the natural logarithm of the ratio; this solution (which eliminates CEOs with no inside debt) results in a further reduction in sample size to a maximum of 3,329. We present results for the subsample where ln(relative Leverage) is defined in Columns (3) and (4) of each panel. Note that the total sensitivity measure is defined more often than the relative leverage ratio or its natural logarithm. The total sensitivity measure could be used to study manager s incentives in a broader sample of firms than the relative ratios. In Panel A, the relative leverage ratio is negatively related to stock volatility, which is consistent with CEOs talking less risk when they are more identified with debt holders, but the relation is not significant. Scaled total sensitivity is significantly related to stock volatility in this subsample. In addition, this regression has a higher adjusted R 2 (p-value: 0.01). In this subsample, both the logarithm of the relative leverage ratio and the scaled total sensitivity are significantly related to stock volatility. However, in the restricted subsample, the explanatory power of the logarithm of the relative leverage ratio and the scaled total sensitivity are not significantly different. In Panel B when R&D expense is the dependent variable, the relative leverage ratio is again insignificant in Column (1), while the scaled total sensitivity is significantly related to R&D expense in the larger subsample in Column (2). In addition, this regression has a higher 12 If instead we winsorize the relative leverage ratio at the 99 th percentile, it is not significant in any specification. 28