Risk aversion combined with non-diversification drives a wedge between the company cost and the executive value of stock options. Although academics and practitioners sometimes note the shortcomings of Black-Scholes, academics routinely use Black-Scholes to measure the value, cost and pay/performance sensitivity of non-tradable executive options, and practitioners routinely use Black-Scholes (or some non-risk-adjusted modification) in order to “measure” option compensation. This standard treatment fails to differentiate between company cost and executive value. The central point of this paper is that simply decoupling the two—recognizing the divergence between the cost and value of options—produces wide-ranging implications about executive pay relevant to both research and practice.
Our results help justify, for example, why executives often claim that Black-Scholes values are too high and why they demand large premiums for accepting stock options in lieu of cash compensation. In addition, the results suggest that granting at-the-money options maximizes incentives when grants are an add-on to existing pay packages, while restricted stock is preferred when grants are accompanied by reductions in cash compensation. We also use our framework to analyze option repricings and show that cost-neutral repricings can benefit both shareholders and executives. Finally, our results are consistent with a variety of stylized facts regarding early exercise policies and behavior. We show why both executives and shareholders benefit from early-exercise provisions, why executives routinely exercise options on their vesting dates, and why (relatively) short-vesting periods are the norm.
Our results have implications for future research in executive compensation and lead to reinterpretation of existing research. Studies of time-trends in executive compensation have documented a rapid growth in executive pay levels over the past decade, driven by the dramatic growth in (Black-Scholes measures of) option compensation. However, after adjusting for the riskiness of equity-based pay, the increase in “pay” levels has plausibly been relatively modest (although there has been a dramatic growth in the cost of executive compensation). In addition, much of the observed difference in pay across industries (for example, between manufacturing and utility executives) can be explained by differences in the riskiness of pay. These findings suggest that analyses of pay levels that fail to account for risk can produce misleading results. For the same reason, studies of pay/performance sensitivities based on changes in Black- Scholes values can produce misleading results. We suggest that the derivative of executive value with respect to the stock price better captures conceptually the CEO’s pay/performance sensitivity.
Although our analysis has focused on stock options, our framework is general and applies to all risky components of pay. One of the key results of this paper is that options are a particularly expensive way to convey compensation. But the same is true for risky bonuses and other forms of contingent pay. There are important incentive benefits to both types of pay (obviously, since without such benefits, our analysis suggests that companies would never offer risky pay). Nonetheless, researchers and practitioners require a framework to understand, and quantify, the value:cost efficiency of all forms of risky compensation. Moreover, the usefulness of this framework continues to grow as companies increasingly put higher percentages of pay at risk for increasingly higher percentages of their managers and employees, a trend that also appears to be spreading internationally. We believe that there is a high return to future research that broadens our framework in precisely this way.
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Appendix A
Risk-Adjusted Pay Calculations
Our risk-adjusted pay calculations for S&P 500 executives are based on the following data available from Compustat’s ExecuComp (unless otherwise sourced):
s Shares of stock owned,
n0 Number of previously granted options held at year-end,
X0 Average exercise price for previously granted options,
Salary Base salary plus all non-variable compensation,
Bonus Bonus plus the target award from accounting-based long-term plans,
ni Number of shares in the ith
option, restricted stock, or target performance- share grant,
Xi Exercise price for the ith
option grant (equals 0 for restricted stock and performance-share grants),
T Option term,
Annualized standard deviation of continuous returns, measured over 48 months,
Equity beta, calculated from monthly data over 48 months,
rf Average yield on U.S. government securities over the fiscal year (Source: Federal Reserve System),
(rm-rf) Equity premium (assumed to be 6.5%).
In calculating risk-adjusted pay we also make assumptions regarding executive relative risk aversion, , and non-firm-related wealth, w. Our calculations are based on = 2 or = 3, assuming that w is equal to the greater of $5 million or four times cash compensation. In addition, we (somewhat arbitrarily) discount bonuses by 20% to account for the risk in bonuses. (Our results are qualitatively unchanged using discounts ranging from 0% to 50%.) Assuming that w, Salary, and Bonus is invested at the risk-free rate, rf, and that the realized
stock price at T is PT, the executive’s wealth at time T is given by
(A1) WT w(1+ rf)
T
+ sPT + n0•max(0,PT-X0) + (Salary+0.8•Bonus)(1+ rf)T
where the summation in Σni•max(0,PT-Xi) allows for multiple option and stock grants.42
If, instead of the option, he were awarded V in cash that he invested at the risk-free rate, his wealth at time T would be:
(A2) WT (w+V)(1+ rf)
T
+ sPT + n0•max(0,PT-X0)
As before, we assume the executive’s utility over wealth is U(W) with constant relative risk aversion, and define the executive’s risk-adjusted compensation as the certainty equivalent
V that equates the expected utilities of (1) and (2):
(A3) U(WTV)f(P
T)dPT U(WT)f(PT)dPT
where the distribution of stock prices in T years is lognormal with volatility and expected value equal to (rf + (rm-rf) - 2
/2)T.
42
Our methodology requires that the grant-date market price and the expiration term be the same for all grants made to a single executive during the year. When grants were made at different dates (and different prices), we “normalized” the price by adjusting the exercise price and the shares granted. When grants were made for various terms, we used the term for the largest grant. This restriction was seldom required: 98% of the sample executives had only single grants or multiple grants with a common expiration term.
Appendix B
A Binomial Framework for Non-tradable Options
We begin by building a traditional binomial tree (Cox, Ross, and Rubinstein, 1979), starting with an initial stock price of P0. The stock prices in the second period are either P0 (with probability ) or P0/ (probability 1- ); subsequent prices are determined similarly for h
periods per year and Y years (the total number of periods is T Yh). In the traditional model with risk-neutral growth, the probabilities π are determined so that the expected return is the risk-free rate. We depart from the traditional model by assuming an expected continuously compounded annual return of ln(1+rf+ (rm-rf)) and (therefore) a per-period return of m e /h. The “uptick” and probability are chosen to solve:
+ (1- )(1/ ) = m, 2
+ (1- )(1/ )2 - m2= ,
where m and ≡m2 e / h−1 are (respectively) the mean and variance of the assumed lognormal distribution of stock prices. Applying the quadratic formula,
= 1 2m (m 2+ + 1)− (m2+ +1)2−4m2
(
)
, =m− 1 − 1 .In order to determine executive option values and early exercise decisions, we employ a backward induction algorithm combined with same utility function (and parameter assumptions) used in our previous analysis. As before, we assume the executive holds non- firm-related wealth, w, invested at the risk-free rate, rf, holds s shares of company stock, and is
given a grant of n options at exercise price X. If the executive decides to exercise his options early, the profits from the exercise are invested in the riskless asset, which is then held until the final period.
The executive’s exercise decision rule is: exercise at any period t if the expected utility from exercise is greater than the expected utility from holding the option to the next period. Specifically, utility based on final period T wealth is calculated at each final period node. (The final period exercise decision is trivial since, at in-the-money nodes, all options will be exercised, and at out-of-the-money nodes, all options expire worthless.) Then, in the period prior to the final period (T-1), the executive solves:
{
E}
T T T U U U Max( (1 ) ), −1 − ++ −where UT+ is the expected utility in T if the stock price increases by λ (an uptick), UT− is the expected utility in T if the stock price increases by 1/λ (a downtick), and UTE−1 is the expected utility in T-1 if the executive exercises in T-1. Evaluating utility in the event of exercise is straightforward since no further decisions are made by the executive. Expected utility under early exercise is based on post-exercise holdings of safe wealth and stock evaluated at all possible (given the current node) final-period stock prices. Following evaluation at T-1, the same process is repeated at T-2 and backward induction is then used until the root node is reached and the tree is fully grown. At completion, each of the nY nodes of the final tree contains a stock price Pt,i, an expected utility, and an indicator for whether the option has been
exercised.
The expected utilities in each node of the binomial tree assume cash, stock, and options are held until period T, and therefore denote final-period utilities. We estimate the executive value of the option grant by finding the grant-date cash award (invested until T at the risk-free rate) that yields the expected utility at the first node of the binomial tree.
Table 1
Ratio of Executive Value to Black-Scholes Cost for Option with $30 Exercise Price, for Various Pairs of Relative Risk Aversion ( ) and Diversification
Stock Price B-S Cost = 2 50% Stock = 2 67% Stock = 3 50% Stock = 3 67% Stock Payout Probability $5 $0.39 24.5% 13.4% 5.8% 2.2% 13.1% $15 $4.95 49.7% 31.6% 22.3% 10.7% 51.5% $30 $16.55 63.5% 44.8% 36.7% 21.1% 77.9% $45 $30.11 69.1% 51.2% 44.0% 27.8% 88.4% $60 $44.40 71.9% 54.9% 48.0% 32.0% 93.3%
Note: B-S cost is the Black-Scholes value of one option with an exercise price of $30. Executive values are estimated numerically assuming that the executive has constant relative risk aversion, = 2 or = 3, and assuming (using the Capital Asset Pricing Model, CAPM) that the distribution of stock prices in T = 10 years is lognormal with volatility = .30 and expected value (rf + (rm-rf) -
2
/2)T , where = 1 is the firm’s systematic risk, rf = 6% is the risk-free rate, and rm- rf = 6.5% is the equity premium. The payout probability that the market price at expiration, PT, exceeds the exercise price, X, is calculated under the standard CAPM assumption that Ln(PT/P0) is normally distributed with mean Ln(rf + (rm-rf) -
2
/2)T and variance 2 T.
Table 2
Explanatory Regressions for Risk-Adjusted and Unadjusted Pay for S & P 5 0 0 C E O s , 1 9 9 2 - 1 9 9 9
Dependent Variables: Ln(CEO Total Compensation)
Independent Variable No Risk Adjustment Risk-Adjusted with = 2 Risk-Adjusted with = 3 (1) (2) (3) Intercept 6.09 (50.8) 5.09 (45.0) 4.99 (46.4) Ln(Sales) 0.294 (23.6) 0.350 (29.8) 0.338 (30.2) Return on Assets 0.406 (1.7) .145 (0.6) 0.109 (0.5) Finance (Dummy) 0.263 (6.2) 0.323 (8.0) 0.316 (8.2) Utility (Dummy) -0.629 (-11.6) -0.427 (-8.3) -0.342 (-7.0) Year 1992 (Dummy) -0.921 (-15.1) -0.518 (-9.0) -0.414 (-7.6) Year 1993 (Dummy) -0.772 (-12.8) -0.427 (-7.5) -0.337 (-6.2) Year 1994 (Dummy) -0.664 (-11.1) -0.302 (-5.3) -0.225 (-4.2) Year 1995 (Dummy) -0.565 (-9.5) -0.206 (-3.7) -0.142 (-2.7) Year 1996 (Dummy) -0.360 (-6.1) -0.041 (-0.7) 0.012 (-0.2) Year 1997 (Dummy) -0.193 (-3.3) 0.050 (0.9) 0.061 (1.2) Year 1998 (Dummy) -0.117 (-2.0) -.023 (-0.4) -0.030 (-0.6) R2 0.268 0.275 0.266
Note: t-statistics in parentheses. Sample size is 3,765 for all regressions. Total compensation defined as the sum of salaries, bonuses, benefits, share options (valued on date of grant using the Black-Scholes formula), LTIP-related stock grants (valued at 80% of face value for performance-contingent awards), and other compensation. Risk- adjusted compensation is computed assuming constant relative risk aversion and assuming that CEO has “safe wealth” equal to the greater of $5 million or four times total compensation. All monetary variables in 1999-constant dollars.
Table 3
Cost, Value, and Incentives from Executive Options with and without Early Exercise
Options Exercisable at Expiration Options Exercisable at Grant Risk Aversion % of Wealth in Stock Company Cost Executive Value Value Cost Incentives ∂V∂/P Company Cost Executive Value Value Cost Incentives ∂V∂/P 2 50% $16.55 $10.51 63.5% $0.45 $14.76 $12.40 84.0% $0.61 2 67% $16.55 $7.41 44.8% $0.35 $13.60 $9.96 73.2% $0.56 3 50% $16.55 $6.07 36.7% $0.26 $13.06 $9.42 72.1% $0.53 3 67% $16.55 $3.49 21.1% $0.17 $11.57 $7.33 63.4% $0.49
Note: The data depict the per-share costs, value, and incentives from a grant of 5,000 ten-year stock options with an exercise price of $30 (the market price on the date of grant). The cost is estimated as the binomial valuation of an American option (under risk-neutral pricing), where the exercise decision is determined not by the investor but rather by the threshold prices in Figure 7. The value is estimated as the grant-date cash award that yields the same expected utility as receiving an option, and incentives are measured as the derivative of this value with respect to an incremental change in the stock price. The executive is assumed to have initial wealth of $5 million, split between riskless cash and company stock.
Table 4
Likelihood of Exercising on Vesting Date
Likelihood of Exercising on Vesting Date Risk Aversion Diversification (% of Wealth in stock) Vesting at 2 Years Vesting at 3 Years Vesting at 4 Years 2 50% 1.4% 5.8% 10.0% 2 67% 5.9% 12.9% 19.8% 3 50% 5.9% 16.7% 23.7% 3 67% 17.0% 26.3% 33.3%
Note: The executive, with initial wealth of $5 million, split between riskless cash and company stock, is assumed to receive 5,000 ten-year stock options with an exercise price of $30 (the market price on the date of grant). Exercise probabilities are based on a simulation with 100,000 sample price paths (through the tree) for each risk/diversification/vesting group. The likelihood of exercising at vesting date is calculated as the percentage of price paths (out of 100,000) that exceed the threshold price at the executive’s first opportunity to exercise.
Figure 1
Executive Value Lines: Option Values for Undiversified Executives
$0 $15 $30 $45 $60 $0 $5 $10 $15 $20 $25 $30 $35 $40 $45 Stock Price "Intrinisic Value"
Option Values for Executive with: