Stock Options for Undiversified Executives* Brian J. Hall Harvard Business School Kevin J. Murphy Marshall School of Business University of Southern California June 2000 Abstract We develop a certainty-equivalence framework to analyze the cost and value of, and incentives provided by, non-tradable options held by undiversified risk-averse executives. We derive “Executive Value” lines, the risk-adjusted analogues to Black-Scholes lines, and distinguish between “executive value” and “company cost.” We show that this framework explains (or sheds light on) virtually all of the major stylized facts about stock option practices, including: 1. Why executives often argue that Black-Scholes values are too high, 2. Why executives typically demand large premiums to exchange options for cash (suggesting that options are an expensive way to convey pay), 3. Why executive reported pay levels have increased substantially over the past decade (while risk-adjusted increases have been far more modest), 4. Why virtually all options are granted at-the-money, 5. Why “indexed options” and “premium options” are rarely observed in practice despite their strong advocates and seemingly attractive features, 6. Why companies often re-set the exercise price on underwater options, 7. Why executives often exercise stock options well before expiration (even for non-dividend paying stocks), forfeiting substantial option value, 8. Why companies allow executives to exercise options prior to expiration (and why this policy is advantageous to both executives and shareholders), 9. Why short vesting periods (which have modest effects on executive value and company cost) are the norm (and why long vesting periods are almost never observed in practice). *We would like to thank Thomas Knox for excellent research assistance. Stock Options for Undiversified Executives by Brian J. Hall and Kevin J. Murphy Stock options, which give the recipient the right to buy a share of stock at a pre- specified exercise price for a pre-specified term, have emerged as the single largest component of compensation for US executives (Hall and Liebman, 1998; Murphy, 1999). In fiscal 1998, 97% of S&P 500 companies granted options to their top executives, compared to only 82% in 1992. Moreover, the grant-date value of stock options accounted for 40% of total pay for S&P 500 CEOs in 1998, up from only 25% of total pay in 1992.1 Stock-options have become increasingly important for rank-and-file workers as well: 45% of US companies awarded options to their exempt salaried employees in 1998, while 12% and 10% awarded options to their non-exempt and hourly employees, respectively.2 In addition to conveying compensation, executive stock options provide a direct link between executive wealth and company stock-price performance, and therefore mitigate agency problems between shareholders and managers (Jensen and Meckling, 1976). Indeed, as documented by Jensen and Murphy (1990) and Hall and Liebman (1998), virtually all of the sensitivity of pay to corporate performance for the typical CEO is attributable to his or 1 Data extracted from Compustat’s ExecuComp database, using grant-date option values based on ExecuComp’s Black-Scholes’ calculations. 2 Prevalence data are based on the American Compensation Association’s 1997-1998 Salary Budget Survey, and include survey results from 735 publicly traded corporations. JUNE 2000 EXECUTIVE STOCK OPTIONS PAGE 2 her holdings of company stock and stock options, and not to accounting-based bonuses or year-to-year changes in other components of compensation. Given the emerging dominance of stock options in the provision of both compensation and incentives, it is important to find ways to measure both the value and the incentives provided by executive stock options. Virtually all research in the area, including papers written by both authors and the “facts” noted in the previous two paragraphs, have relied on option pricing formulas such as Black-Scholes (1973) to evaluate executive options. But, executive stock options differ in critical ways from the options “priced” in the valuation formulas. In particular, while option pricing theory assumes that options are freely tradable and that option holders can hedge the risk of options by short-selling stock, executive options are non-tradable and held by risk-averse undiversified executives who cannot easily hedge. While researchers and practitioners have routinely noted the shortcomings of using standard option formulas to value executive options, there have been few attempts to “correct” for the non-tradability of executive options, or to explore the implications of such a correction.3 The purpose of this paper is to analyze the valuation of, and incentives provided by, non-tradable stock options held by risk-averse undiversified executives. Section 2 develops our framework, which provides an operationally useful alternative to Black-Scholes for both valuing stock options and measuring the incentives created by such options. We distinguish between the cost to the company and the value to the executive-recipient. Ignoring (for the moment) complications related to early exercise, potential forfeiture, and executive inside information, option-pricing methodologies such as Black-Scholes are appropriate for 3 Noteworthy exceptions include Lambert, Larcker, and Verrecchia (1991), Kulatilaka and Marcus (1994), Rubenstein (1995), Carpenter (1998), Murphy (1999), DeTemple and Sundaresan (1999), Muelbroek (2000) and Hall and Murphy (2000). JUNE 2000 EXECUTIVE STOCK OPTIONS PAGE 3 measuring the amount outside investors would pay for an option, and therefore provide an estimate of the company’s cost of granting an option. We estimate the value of non-tradable options to an undiversified risk-averse executive using the “certainty equivalence” approach, following Lambert, Larcker, and Verrecchia (1991), and show that option-pricing formulas such as Black-Scholes systematically overstate the value an executive places on a non- tradable option. In addition to introducing our framework, Section 2 explores several implications of our comparison of the cost and value of executive stock options. Our analysis supports frequent claims by executives that Black-Scholes values are “too high,” and also explains why executives often demand large premiums to accept options in lieu of cash payments. We show that these premiums depend on whether the option is in-the-money or out-of-the- money, which in turn depends on the probability that the option will expire unexercised. Finally, our results also have implications for academic research that adds options values to other risky components of pay to establish “total compensation.” At best, this aggregate calculation is a measure of the company’s cost of the compensation package, and not an estimate of the executive’s value of the compensation package. We show that, while the reported level of total compensation for S&P 500 executives has risen dramatically from 1992 to 1998, the growth in “risk-adjusted pay” has been relatively modest. Section 3 analyzes the incentives created by non-tradable executive options. We define incentives as the derivative of the executive’s value with respect to the stock price, and consider the incentive effects of setting the exercise price above the grant-date market price (premium options) or below the grant-date price (discount options), “indexing” the exercise price to the performance of the market, and repricing options following declines in stock JUNE 2000 EXECUTIVE STOCK OPTIONS PAGE 4 prices. Although academics and the business press have routinely called on companies to issue premium or indexed options, and to refrain from repricing, we show that such policies are not generally in the interest of shareholders. First, we show that, holding constant the company’s cost of making an option grant, incentives are maximized by setting exercise prices at or below grant-date market prices. Next, we show that indexed options are less efficient than traditional (fixed exercise price) options in providing incentives to undiversified executives. Finally, we show that repricings structured to hold constant the total cost of the options can simultaneously benefit shareholders and executives. Section 4 relaxes our maintained assumption that options are held for their full term, and analyzes the timing of exercise decisions by risk-averse executives. Several researchers have documented that executive (and employee) options are exercised relatively early in their term, even when the underlying stock pays no dividends. We show that such behavior is entirely consistent with our framework: risk-averse executives will exercise early following price run-ups to “lock in” a gain. More importantly, we demonstrate that allowing early exercise is advantageous to both executives and shareholders. Allowing early exercise increases the executive’s value from receiving an option, while also reducing the company’s cost of granting the option (precisely because the executive will exercise “too early” from an outside investor’s standpoint). Overall, our framework demonstrates the importance of incorporating risk into analyses of executive compensation, and explains or gives insights into virtually all stock option pay practices and behaviors. In addition, our framework offers some general implications for the design of employee bonus plans. Section 5 summarizes and generalizes these findings. JUNE 2000 EXECUTIVE STOCK OPTIONS PAGE 5 2. Option Values for Undiversified Executives Central to modern option theory is the idea that option holders can fully hedge the risk associated with holding options by short-selling the underlying stock or taking similar actions that achieve the same purpose. Black and Scholes (1973) and Merton (1973) demonstrated that, since investors can hedge, options can be valued as if investors were risk neutral and all assets appreciate at the risk-free rate. Under these assumptions, the value of options can be estimated by computing the expected value of the option upon exercise assuming that the expected return on the stock is equal to the risk-free rate, and then discounting the expected value to the grant date using the risk-free rate. This risk-neutrality assumption is central to all option pricing models and methodologies, including the Black- Scholes model (as amended by Merton to account for dividends), binomial models, arbitrage pricing models and Monte Carlo methodologies.4 Compensation consultants, practitioners, and academic researchers have routinely used Black-Scholes or similar methodologies to value executive stock options. However, while the assumptions underlying option-pricing methodologies reasonably describe the situation faced by sophisticated outside investors holding freely traded options, they do not describe the situation faced by executives (and other employees) holding options on their own company’s stock. In contrast to outside investors, company executives cannot trade or sell their options, and are also forbidden from hedging the risks by short-selling company stock; such actions would obviously defeat a primary purpose of the option grants, which is to align 4 See Hull (1997) and Merton (1997) for a comprehensive treatment of these issues. JUNE 2000 EXECUTIVE STOCK OPTIONS PAGE 6 the financial interests of the managers with those of the shareholders.5 In addition, while outside investors tend to be well-diversified (holding small amounts of stock in a large number of companies), company executives are inherently undiversified, with their physical as well as human capital invested disproportionately in their company.6 These substantive violations of the underlying assumptions suggest that traditional methodologies are not appropriate in determining the value of executive stock options. When defining the “value” of executive stock options, it is critical to distinguish between two fundamentally distinct option valuation concepts: the economic cost to the company, and the economic value to the executive-recipient. The economic or opportunity cost of granting an option is the amount the company could have received if it were to sell the option to an outside investor rather than giving it to the executive. Ignoring for the moment complications related to potential forfeiture, early exercise, and executive inside information, traditional valuation formulas provide an estimate of how much outside investors would pay for an option, and therefore represent an estimate of the company’s cost of granting an option. However, traditional valuation methodologies do not measure the value of a non-tradable option to an undiversified, risk-averse executive. We estimate the value of a non-tradable option to an undiversified risk-averse executive as the amount of riskless cash compensation the executive would exchange for the option, using a “certainty equivalence” approach similar to that adopted by Lambert, 5 Although executives cannot explicitly short-sell company stock they may engage in related transactions (such as zero-cost collars) that reduce their risk. Existing evidence suggests that such transactions are observed but are not widespread (Bettis, Bizjak, and Lemmon, 1999). 6 Indeed, in addition to being “forced” not to trade or hedge their options, executives are routinely required (through ownership guidelines imposed by the board) or pressured (by informal board requirements or through the desire to signal to markets) to hold more company stock than dictated from an optimal- portfolio standpoint. JUNE 2000 EXECUTIVE STOCK OPTIONS PAGE 7 Larcker, and Verricchia (1991). In particular, we suppose that an executive has non-firm- related wealth of w, holds s shares of company stock, and is granted n options to buy n shares of stock at exercise price X in T years. Assuming that w 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 (1) WT w(1+ rf)T + sPT + n•max(0,PT-X). 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: (2) WV T (w+V)(1+ rf)T + sPT. We assume the executive’s utility over wealth is U(W), and define the executive’s value of n options as the certainty equivalent V that equates expected utilities (1) and (2): (3) U(W V )f(PT)dPT T U(WT)f(PT)dPT Solving (3) numerically requires assumptions about the form of the utility function, U(W), and the distribution of future stock prices, f(PT). We assume that the executive has 1 constant relative risk aversion , so that U(W) ln(W) when =1, and U(W) 1− W 1− when 1. We adopt the Capital Asset Pricing Model (CAPM) and assume that the distribution of stock prices in T years is lognormal with volatility σ and expected value equal to (rf + (rm- 2 r f) - /2)T, where is the firm’s systematic risk and rm is the return on the market portfolio.7 Unless otherwise noted, the figures and numerical calculations in this article are derived assuming no dividends, = .30 (the median volatility for S&P 500 firms), = 1 (the market 7 For tractability, we assume that the distribution of future stock prices is the same whether the executive receives options or cash. If the grant provides incentives that shift the distribution, and if the shift is not already incorporated into stock prices as of the grant date, we will underestimate both the cost and value of the option. Since most of our results hinge on the difference between (rather than the levels of) cost and value, this assumption does not affect our main qualitative results. JUNE 2000 EXECUTIVE STOCK OPTIONS PAGE 8 average beta, by definition), rf = 6%, and an equity premium of rm- rf = 6.5%. The qualitative results in this paper are not sensitive to reasonable changes in these parameters. 2.1 Executive Value Lines Figure 1 illustrates our methodology by showing how the value of a ten-year non- tradable stock option with an exercise price of $30 varies with changes in stock prices. The “intrinsic value” is defined as the (positive) spread between the market price and exercise price, and the Black-Scholes value, C(P), approximates the company’s cost of granting an option. These two lines, showing option payouts and values, are of course standard in option analyses. What is new in Figure 1 are the “executive value” lines, which plot the certainty- equivalent values as a function of stock prices. The figure depicts the per-share values of non-tradable options to undiversified executives, assuming that executives have $5 million in initial wealth split between company stock and safe cash, and assuming that executives receive a grant of options to purchase 5,000 share of stock at an exercise price of $30. Four executive-value lines are drawn, for different pairs of risk aversion and diversification, representing executives with relative risk aversion of =2 or =3 and holding either 50% or 66% of their wealth in company stock. Under traditional valuation methodologies, option values depend on six factors: the exercise price, stock price, dividend yield, stock-return volatility, risk-free rate, and the time until expiration. Our numerical analysis of equation (3) shows that the certainty-equivalent value (that is, the value of a non-tradable option to an executive recipient) depends on these six parameters, but also depends on the executive’s risk-aversion, initial wealth, and diversification. Figure 1 illustrates these new comparative static results. For example, executive values are strictly decreasing in risk-aversion: the executive value lines for JUNE 2000 EXECUTIVE STOCK OPTIONS PAGE 9 executives with relative risk aversion of =3 lie strictly below those for executives with =2. In addition, executives with large holdings of company stock (relative to their wealth) place lower values on options: the executive value lines for executives holding 66% of their initial wealth in company stock (rather than safe cash) lie strictly below those for executives holding 50% of their wealth in stock. The executive value lines in Figure 1 lie below the Black-Scholes line, V(P)0 is only 35% (and not 50%) because the distribution of stock prices is skewed: the mean relative return is zero, but the median relative return is less than zero. JUNE 2000 EXECUTIVE STOCK OPTIONS PAGE 27 the scarcity of indexed option plans in part reflects accounting considerations,23 we believe that our analysis helps show the benefit of traditional option plans over indexed plans. Although index options are less costly than traditional options to grant, they are disproportionately less valuable to the executive-recipients. Moreover, the incentives provided by indexed options are weaker than the incentives provided by an equal-dollar cost of traditional options. 3.1.3 Option Repricing One of the most controversial executive pay practices involves resetting the exercise price on outstanding options following a decline in the company’s stock price (Saly, 1994). As shown in Figure 5, options lose incentive value once the stock price falls sufficiently below the exercise price that the executive perceives little chance of exercising: this “loss of incentives” is a common justification for option “repricings” following share-price declines. Since repricing effectively “forgives” executives for dismal performance, companies adopting repricing policies create perverse incentives for executives holding options. A recent twist on standard share-for-share repricing practices is “Black-Scholes repricing,” in which the executives exchange their options with a high exercise price for a smaller number of options with a lower exercise price. The exchange is structured so that the total Black-Scholes value of the option is the same immediately before and after the exchange. From the executive’s perspective, the exchange is beneficial since both options have the same expected value but the lower-priced options are less risky (i.e., have a higher 23 Under US accounting rules, as long as stock options have a pre-specified exercise price and expiration date, companies incur an accounting charge equal to the grant-date “spread” between the market price and the exercise price (amortized over the life of the option). Indexed options do not have a fixed exercise price, and therefore do not qualify for favorable accounting treatment. JUNE 2000 EXECUTIVE STOCK OPTIONS PAGE 28 probability of ultimately being exercised). From the shareholder’s perspective, the cost is the same with or without the exchange. Executive incentives can either increase or decrease, depending on the specific parameters involved. One of the first companies to adopt Black-Scholes repricing was General Dynamics in 1991. As described in Dial and Murphy (1995), the CEO of General Dynamics exchanged 105,000 options with an exercise price of $45 (the market price on the original grant date) for 51,500 options with an exercise price of $25.50 (the market price on the exchange date). The number of options offered in the exchange was determined to maintain the pre-exchange Black-Scholes cost of approximately $388,000. Assuming that CEO Anders had risk aversion of =2 and $4 million in wealth (67% in stock), he valued the 105,000 out-of-the- money options at $190,000 and the 51,500 at-the-money options at $255,000.24 Clearly participating in the exchange made sense from Anders’ perspective, while imposing little cost on the shareholders. According to our model, Anders’ option incentives fell slightly from $18,500 (the change in Anders’ value of the old options for a $1 change in the stock price) to $17,500 (the change in his value of the new options for each $1 change in stock prices). 4. Early Exercise Our valuation analysis in Sections 2 and 3 assumed that executives hold options until their expiration date. However, in practice, executive options vest and become exercisable within a few years from grant, and executive (and employee) options are routinely exercised 24 Details of our calculation are available upon request, but are based on = .2425, dividend yield of 4%, = 1, rf = 8%, and rm- rf = 6.5%. The $4 million in total wealth is an assumption, but the 67% in stock is JUNE 2000 EXECUTIVE STOCK OPTIONS PAGE 29 relatively early in their term (Huddart and Lang, 1996; Huddart, 1998). In this section, we relax our maintained assumption that executives hold their options until the expiration date and analyze the decision to exercise early.25 In particular, we analyze how the company’s cost, the executive’s value, and the incentive strength are affected by allowing early exercise. In order to isolate how risk aversion and diversification drive executive exercise decisions, we continue to ignore dividends. While investors holding freely tradable American call options on non-dividend paying stocks will never exercise early (Merton, 1973), we show that risk-averse executives holding non-tradable options will exercise early to “lock in” an option gain. 4.1 Methodology We analyze executive option values and early exercise decisions using a modified binomial approach, described in detail in Appendix B. Our approach is similar to traditional binomial option valuation (Cox, Ross, and Rubinstein, 1979) with two major differences. First, while binomial price “trees” under the traditional model are based on expected returns equal to the risk-free rate (reflecting that option holders perfectly hedge the risk of options), price trees under our modified approach are based on CAPM expected returns, E(r)=rf+ (rm-rf). Second, while under the traditional approach the payout from exercising is compared to the expected value of holding for another period, under our modified approach we compare the expected utility from exercising (and holding cash until the final period) to consistent with Anders’ stock holdings (relative to the $4 million). The new options were granted with a ten year term; the old options had nine years remaining. 25 Related analyses of early-exercise decisions include Huddart (1994), Carpenter (1998), and DeTemple and Sundaresan (1999). Our framework for analyzing early exercise is closest to Huddart’s (1994), who focuses primarily on measuring the cost of options to the firm. Relative to his model, our framework allows the executive to hold a richer set of assets prior to exercise—cash, stock and options rather than options only—and employs a more flexible utility framework. JUNE 2000 EXECUTIVE STOCK OPTIONS PAGE 30 the “expected” expected utility from holding the option for another period. 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. We assume that stock acquired through exercise is sold immediately, with the cash proceeds invested at the risk-free rate.26 In addition, we assume that options are exercisable immediately upon grant (alternative vesting schedules are analyzed below in Section 4.4) We maintain our other assumptions from Sections 2 and 3. In particular, we assume the option is granted for T=10 years at an exercise price of X=$30 (the grant-date market price), and assume = .30, = 1 rf = 6%, and rm- rf = 6.5%. We compute executive values for a grant of 5,000 options for executives with constant relative risk aversion of =2 or =3 and holding either 50% or 66% of their $5 million initial wealth in company stock. Our binomial tree is calculated for h=50 periods per year, or a total of hT=500 periods and 125,250 individual nodes. For each node, we record information on the stock price, the expected utility, and an indicator for whether the option has been exercised. 4.2 Early-Exercise Decisions Undiversified executives with exercisable options face a tradeoff. If they exercise early, they can invest the “spread” between the market and exercise price at the risk-free rate, thereby locking in the gain. However, by exercising early (and immediately selling the acquired shares), they sacrifice both upside potential in stock prices and the ability to defer payment of the exercise price. The exercise decision in each period will naturally depend on the realized stock price (relative to the exercise price). If the price is sufficiently high, the 26 Ignoring taxes, which we do throughout, it would be irrational for an executive to exercise an option early and to hold the stock. JUNE 2000 EXECUTIVE STOCK OPTIONS PAGE 31 expected utility from locking in the gain will exceed the utility from holding the option for another period. But, at sufficiently low stock prices, even risk-averse, undiversified executives will not exercise options early. We define the “threshold price” as the stock price where the executive is just indifferent between exercising early or holding the option for another period. Figure 8 shows how the threshold price varies over the 10-year term of the option for managers with the same pairs of risk aversion and diversification as before.27 Several results emerge. First, in any period, more risk-averse and less diversified executives have lower threshold prices (that is, they will choose to exercise early at lower realized stock prices). For example, an executive with = 2 and half of his wealth in company stock has a first-year price threshold of more than $90. The price would have to more than triple (from its $30 initial value) in the first year for such an executive to exercise early. However, a more risk averse ( = 3) and less diversified (66% of his wealth in stock) executive has a first-year exercise price boundary of less than $60. Second, Figure 8 reveals that threshold prices decrease over time. As time passes, the benefits of holding the option (deferring the payment of the exercise price, and missing out on future price increases) fall, and an executive’s threshold for locking in option gains is lower. On the expiration date, of course, the threshold price falls to the exercise price (in this case, $30). But, the day before exercise, the threshold price is above $50 for an executive 27 Within our binomial framework, we estimate the threshold price for each period as the average of the lowest price that induces early exercise and the highest price that induces holding the option for another period. Because of the discreteness in the binomial model (which considers only “n” possible stock prices in the n th period), our estimated threshold prices vary within a range from period to period (depending, for example, on whether “n” is even or odd in a period). As an expositional simplification, the threshold prices in Figure 7 have been smoothed somewhat by using linear interpolation and a larger number of periods per year (h=75 rather than 50) than in the rest of the analysis. JUNE 2000 EXECUTIVE STOCK OPTIONS PAGE 32 with = 2 and 50% of his initial wealth in stock, and about $35 for an executive with =3 and 66% of his initial wealth in stock. Figure 9 shows the cumulative probabilities of early exercise for executives.28 The figure indicates that there is a relatively high probability of early exercise after the first two years, although the probabilities differ greatly (consistent with the very different threshold prices in Figure 8) with differences in risk aversion and diversification. For example, the likelihood that a (relatively) high-risk-aversion ( = 3) and low-diversification (wealth in stock at 66%) executive will exercise within the first 4 years is about 50%, and within 8 years is 75%. Conversely, an executive with = 2 and 50% of his wealth in stock has only an 18% chance of exercising within the first 4 years, and 50% chance of exercising within the first eight years. As of the day before expiration, the cumulative exercise probabilities range from 65% (for a = 2, 50% executive) to 84% (for a = 3, 66% executive). Since all in-the-money options are exercised in the last period, the cumulative probabilities of exercise by the expiration day are quite close for executives with different characteristics, ranging from 78 percent to 85 percent.29 Early exercise is therefore entirely consistent with our framework: risk-averse executives will exercise early following price run-ups to “lock in” a gain. This result has important implications for the value, cost, and incentive strength of executive stock options, the subject to which we now turn. 28 The probabilities are based on a simulation with 100,000 sample price paths (through the tree). At each period t, the cumulative probability of early exercise is calculated as the percentage of price paths (out of 100,000) that surpassed the threshold price at t or earlier. 29 The difference reflects options that were in-the-money earlier in the term (and exercised by high risk- averse low diversified executives), but out-of-the-money at expiration. JUNE 2000 EXECUTIVE STOCK OPTIONS PAGE 33 4.3 Cost, Value, and Incentives Allowing executives to exercise prior to the full term affects both the cost and value of executive stock options. Allowing early exercise unambiguously increases the value of an option to an undiversified executive, since executives could always hold the option to full term but in practice choose not to. In contrast, allowing early exercise actually reduces the company’s cost of granting an option. As discussed in Section 2, the economic cost of granting an option is the amount the company could have raised if it were to sell the option to an outside investor instead of giving it to the executive. If outside investors made the exercise decision, then the company’s cost would be the usual binomial valuation of an American option (which, for non-dividend-paying stocks, is simply the Black-Scholes value). But, in this case, the exercise decision is not made by the investor but rather by an executive who for a variety of reasons is not expected to make the same exercise decisions as an unrestricted outside investor. Since the executive exercise decisions are suboptimal from the standpoint of the outside investor, the amount the investor is willing to pay for the option is clearly reduced when exercise decisions are made by the executive, rather than by the investor. The cost reduction comes from the fact that early exercise essentially removes the right-hand tail of payoffs to executive option holders, who truncate the huge payoffs that would otherwise come from dramatic increases in the company’s stock price. For example, consider the executives who prematurely exercised their options in the mid-1990s at Cisco, Microsoft, General Electric and other high-flying companies of the decade. Such early exercises led to far lower company costs than would have been the case if these executives had not exercised their options early. JUNE 2000 EXECUTIVE STOCK OPTIONS PAGE 34 As before, we measure the value of an immediately exercisable executive option as the grant-date cash award that yields the expected utility as receiving an option, and measure the incentives from the option as the slope of the executive-value line, V / P , which defines how the option value changes with an incremental change in the stock price. We measure the cost of the executive option as the usual binomial valuation of an American option (under risk-neutral pricing) but with a catch: the exercise decision is made by an undiversified executive rather than by the investor. In particular, we measure the cost of the option under the assumption that the investor is “forced” to exercise when even the stock price exceeds the threshold prices in Figure 8.30 Table 4 shows how the company’s cost, the executive’s value, and incentives (defined as the derivative of value with respect to changes in stock prices) are affected by the possibility of early exercise for various pairs of risk aversion and diversification. 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 columns under “Options Exercisable at Expiration” basically replicate the results in Table 1 (for an at-the- money grant): the cost is the Black-Scholes cost of $16.55, executive values range from $3.49 to $10.51, and the incentives range from $0.17 to $0.45. The calculations under “Options Exercisable at Grant” allow executives to exercise options anytime during their term. Compared to the case without early exercise, the company’s cost is lower, and the executive’s value and incentives are higher, for grants that allow early exercise. For example, for an executive with = 2 and 66% of his wealth in company stock, allowing early exercise increases his value from $7.41 to $9.96, while reducing the company’s cost of granting the 30 Formally, we compute the value of a path-dependent barrier option (Hull, 1997), where the barrier (the JUNE 2000 EXECUTIVE STOCK OPTIONS PAGE 35 option from $16.55 to $13.60. In addition to increasing executive value while reducing company cost, Table 4 shows that allowing early exercise also increases executive incentives. For example, for an executive with = 2 and 50% of his wealth in stock, a one dollar increase in the stock price will increase the value of an option without an early exercise provision by only 45¢, but will increase the value of an otherwise identical exercisable option by 61¢. Allowing early exercise doubles the option incentives for a = 3 and 50% executive (from 26¢ to 53¢), and nearly triples the incentives for a = 3 and 66% executive. Overall, our results help explain not only why executive options are so often exercised early, but why it is in the interest of shareholders to allow early exercise. Early exercise reduces the company’s cost of granting options, while increasing the value and incentives to the executive recipient. 4.4 Vesting Schedules The analysis so far in this section has assumed that options are immediately exercisable upon grant, and can be freely exercised at anytime during the term of the option. In practice, however, options typically become exercisable only when they “vest” (that is, when the options are no longer subject to forfeiture if the executive leaves the firm). Although there is a range of observed practices, the most common schedules vest options 33% annually over three years, 25% annually over four years, or 20% annually over five years. Huddart and Lang (1996) have noted that executives (and employees) exercise a disproportionate number of options immediately upon vesting, creating spikes in option- price path above which the executive exercises) varies over time as shown in Figure 8. JUNE 2000 EXECUTIVE STOCK OPTIONS PAGE 36 exercise patterns. Table 5 replicates this empirical result for our hypothetical risk-averse, undiversified executives. The table shows the likelihood of exercising on the vesting date, for options with cliff-vesting in two, three, or four years. For example, the table shows that an executive with = 3 and 50% of his wealth in company stock has a 17% chance of exercising his options on the vesting date if they vest in two years, and a 33% chance if they vest in four years. The spike is naturally smaller for executives who are more risk tolerant and more diversified: the likelihood of exercising on the vesting date for a = 2, 50% executive is only 1.4% for options vesting in two years, and 10% for options vesting in four years. Table 4 showed that allowing early exercise reduced the company’s cost of granting the option and increasing the perceived value of the option to the executive. Therefore, the ratio of the executive’s value to the company’s cost is higher for companies with immediately exercisable options than for companies not allowing early exercise. Figure 10 plots the intermediate cases, showing value:cost ratios for option grants with vesting periods that range from 0 years (complete early exercise is allowed) to 10 years (no early exercise is allowed). Two primary results emerge from Figure 10. First, the value:cost ratios fall more dramatically in the cases in which executives are more risk-averse and less diversified. Second, the value:cost ratios are very flat at low vesting durations, especially in cases in which risk-aversion is relatively low and diversification is relatively high. This results suggests that reasonably short vesting periods do not decrease the value:cost efficiency of options very much. To the extent that short vesting periods create benefits to companies—in terms of retention and ensuring that options are held and provide positive incentives for some JUNE 2000 EXECUTIVE STOCK OPTIONS PAGE 37 minimum period of time—our analysis suggests that the counterbalancing “efficiency” costs of short vesting may not be too large. Thus, our analysis helps explain why short vesting periods are so common while long vesting periods (of more than five years) are virtually non-existent. 5. Conclusion Stock options have become an increasingly important source of compensation and incentives for top-level executives. We develop a framework for analyzing the cost and value of, and the incentives provided by, non-tradable options held by undiversified and risk-averse executives (and other employees). Our framework enables us to derive “Executive Value” lines (the risk-adjusted analogues to the Black-Scholes lines) and provides an operationally useful alternative to standard option pricing models for understanding and analyzing a very wide range of stock option pay practices—and indeed, compensation and incentives design issues more broadly. Our methodology allows us to adjust option values, and total compensation packages, for executive risk-aversion. Executive risk aversion is at the heart of the agency problem (Jensen and Meckling, 1976; Holmstrom, 1979) and it creates the tradeoff between incentives and risk that has been explored extensively by economists (Aggarwal and Samwick, 1999). Conversely, the implications of adjusting compensation for risk have not been explored extensively by economists and are not well understood. Indeed, neither academics nor practitioners make risk-adjustments to options. Economists routinely use Black-Scholes to measure the value and pay-performance sensitivity of options. And practitioners continue to use Black-Scholes (or some non-risk adjusted modification of JUNE 2000 EXECUTIVE STOCK OPTIONS PAGE 38 Black-Scholes) in order to measure option compensation. Our analysis “explains” or sheds light on virtually every major stylized fact about stock option pay practices and leads to reinterpretation of major trends in, and facts about, the level of top executive pay. For example, our analysis demonstrates that, while Black-Scholes represents a reasonable measure of the company cost of stock options, it is not a reasonable measure of executive value. Indeed, for plausible parameter estimates, Black-Scholes overestimates executive value by a substantial margin. This helps explain 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. Although the complexity of stock options may contribute to low executive value somewhat, our analysis suggests that these “low” executive valuations are perfectly consistent with rationality on the part of undiversified executives. The Black-Scholes “benchmark” is simply not appropriate for measuring executive value. Our analysis has implications for the central design issue with regard to stock options: the setting of the exercise price. One of the most striking facts about option design is that nearly all executive stock options are issued “at the money.” Premium options and indexed options are exceedingly rare, despite their strong advocates among academics and the business press. Although accounting rules may explain some of the paucity of these plans, our analysis suggests an economic justification that does not rely on accounting considerations. In particular, we find that increasing the exercise price or indexing the exercise price to the market reduces the executive’s value far more than it reduces the company’s cost of granting the option, thus making such options a bad deal for executives and a costly proposition for shareholders. Moreover, options granted with premium or JUNE 2000 EXECUTIVE STOCK OPTIONS PAGE 39 indexed exercise prices provide poor incentives for risk-averse executives, because of the high probability that they will expire out of the money. We find instead that, for a given “company cost” of options, option incentives are maximized by setting exercise prices at or near the grant-date market price. Our analysis helps explain why options are often repriced when they fall out of the money. Canceling and reissuing an outstanding option at a lower exercise price increases the executive’s perceived value of the option by a greater amount than it increases the company’s cost. It is therefore easy to see why boards often succumb to repricing pressures from executives. In addition, we show that repricings can actually be in shareholders’ interest when they are structured to hold constant the total cost of the options (i.e. executives are given a smaller number of options in return for a reduction in the exercise price). In this case, the repricing is cost-neutral for the company, but is value-increasing and often incentive- increasing to the executive recipient. Therefore, although a formal policy of option repricings creates a set of new problems, our analysis suggests that one-time repricings are not only understandable, but also typically optimal. Executives often exercise their options well before expiration, sometimes sacrificing substantial (Black-Scholes) option value in doing so. This behavior is entirely consistent with our framework: risk-averse executives exercise their options when the price rises above a “price threshold” in order to lock-in early gains. The most striking fact about early exercise, however, it not that it happens, but rather that allowing early exercise represents a “win-win” design policy. Allowing early exercise substantially raises the value (and by extension, the incentives) of options: risk-averse executives prefer to lock-in gains. It also substantially lowers the cost to shareholders: by exercising early, executives forfeit the very large option JUNE 2000 EXECUTIVE STOCK OPTIONS PAGE 40 gains that come with very large stock price increases. Our analysis also enables us to analyze vesting rules. For the same reason that American options are preferable to European options, vesting (which restricts executives from selling early for a specified period) lowers executive value while raising company cost. However, we find that the relationship between vesting periods and the value:cost “efficiency” ratio of options is highly non-linear: for short vesting periods (one to three years), the value:cost ratio is fairly flat. Thus, if there are (unmodeled) benefits associated with short vesting (such as retention), companies can reap those benefits with only modest effects on the value:cost efficiency of their options. Consistent with our analysis, long vesting periods, which significantly reduce the value:cost efficiency of options, are almost never observed in practice. Our framework is also useful for examining and understanding longitudinal and cross- sectional trends in pay levels. For example, one of the most striking features about executive pay over the past two decades is the dramatic increase in the level of CEO pay. We show, however, that the results are far less dramatic when the level of CEO pay is risk-adjusted. Between 1992 and 1998, reported CEO pay rose by 160 percent. However, on a risk-adjusted basis, the increase was a third as large. Moreover, the most dramatic increase in average CEO pay during the period was between 1997 and 1998. On a risk-adjusted basis, CEO pay actually declined over the same period. The framework we have advanced is completely general. That is, it applies to all risky components of pay, not just options. 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 JUNE 2000 EXECUTIVE STOCK OPTIONS PAGE 41 pay (obviously, since without such benefits, our analysis suggests that companies would never offer risky pay), but there is a strong need for a framework for researchers and practitioners to understand, and quantify, the value:cost efficiency of risky compensation. Moreover, this need 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 beyond the borders of the US. We believe that there is a high return to future research that broadens our framework in precisely this way. JUNE 2000 EXECUTIVE STOCK OPTIONS PAGE 42 References Aggarwal, Rajesh K. and Andrew A. Samwick (1999a), “The Other Side of the Tradeoff: The Impact of Risk on Executive Compensation,” Journal of Political Economy, 107, 65-105. Aggarwal, Rajesh K. and Andrew A. Samwick (1999b), “Executive Compensation, Strategic Competition, and Relative Performance Evaluation: Theory and Evidence,” forthcoming in the Journal of Finance. Antle, R. and A. Smith (1986), “An Empirical Investigation of the Relative Performance Evaluation of Corporate Executives,” Journal of Accounting Research, 24(1), 1-39. Bettis, J. Carr, John M. Bizjak, and Michael L. Lemmon (1999) “Insider Trading in Derivative Securities: An Empirical Examination of the Use of Zero-Cost Collars and Equity Swaps by Corporate Insiders.” Arizona State University. Black, F. and M. Scholes (1973), “The Pricing of Options and Corporate Liabilities,” Journal of Political Economy, 81, 637-59. Carpenter, Jennifer (1998), “The Exercise and Valuation of Executive Stock Options,” Journal of Financial Economics, 48(2): 127-158. Cox, John C., Steven R. Ross, and Mark Rubinstein (1979), Option Pricing: A Simplified Approach, Journal of Financial Economics, 7, 229-263. DeTemple, Jerome and Suresh Sundaresan (1999), Nontraded Asset Valuation with Portfolio Constraints: A Binomial Approach, The Review of Financial Studies, 12 (4), 835-872. Dial, Jay and Kevin. J. Murphy (1995), “Incentives, Downsizing, and Value Creation at General Dynamics,” Journal of Financial Economics, 37(3), 261-314. Gibbons, Robert and Kevin. J. Murphy (1990), “Relative Performance Evaluation for Chief Executive Officers,” Industrial and Labor Relations Review, 43(3), 30s-51s. Hall, Brian J. and Jeffrey B. Liebman (1988), “Are CEOs Really Paid Like Bureaucrats?” Quarterly Journal of Economics, 113, August, 653-691. Hall, Brian J. and Kevin J. Murphy (2000), “Optimal Exercise Prices for Risk Averse Executives,” American Economic Review, May, 209-214. Holmstrom, Bengt. (1982), “Moral Hazard in Teams,” The Bell Journal of Economics, 13(2), 324-40. Holmstrom, Bengt. (1979), “Moral Hazard and Observability,” The Bell Journal of Economics, 10, 74-91. JUNE 2000 EXECUTIVE STOCK OPTIONS PAGE 43 Huddart, Steven (1994), “Employee Stock Options,” Journal of Accounting and Economics, 18, 207-231. Huddart, Steven and Mark Lang (1996), “Employee Stock Options Exercises: An Empirical Analysis, Journal of Accounting and Economics, 21, 5-43. Huddart, S. (1998), “Patterns of Stock Option Exercise in the United States,” in J. Carpenter and D. Yermack, eds., Executive Compensation and Shareholder Value, NYU and Kluwer Academic Publishers, Norwell, MA, 118-140. Hull, John (1997), Options, Futures, and Other Derivative Securities, 3rd Edition, Prentice Hall: New Jersey. Jensen, Michael C. and William M. Meckling (1976), “Theory of the Firm: Managerial Behavior, Agency Costs and Ownership Structure,” Journal of Financial Economics, III, 305-360. Jensen, M. and K. J. Murphy (1990), “Performance Pay and Top-Management Incentives,” Journal of Political Economy, 98(2), 225-64. Joskow, Paul, Nancy Rose and Catherine Wolfram (1996), “Political Constraints on Executive Compensation: Evidence from the Electric Utility Industry,” RAND Journal of Economics, 27 (1), 165-182. Kulatilaka, Nalin and Alan J. Marcus (1994), “Valuing Employee Stock Options,” Financial Analysts Journal, November- December, 46-56. Lambert, R., W. Lanen, and D. Larcker (1989), “Executive Stock Option Plans and Corporate Dividend Policy,” Journal of Financial and Quantitative Analysis, 24(4), 409-425. Lambert, Richard A.; Larcker, David F. and Verrecchia, Robert E. (1991), “Portfolio Considerations in Valuing Executive Compensation,” Journal of Accounting Research, 29(1), 129-49. Margrabe, W. (1978), “The Value of an Option to Exchange One Asset for Another,” Journal of Finance, 33, 177-186. Merton, Robert C. (1997), Continuous-Time Finance, Blackwell Publishers. Merton, Robert C. (1973) “Theory of Rational Option Pricing,” Bell Journal of Economics and Management Science, IV, 141-183. Murphy, Kevin J.. (1999) “Executive Compensation,” in Orley Ashenfelter and David Card, eds., Handbook of Labor Economics, Vol. III, North Holland. Ofek, E. and D. Yermack (1998), “Taking Stock: Does Equity-Based Compensation Increase Managers’ Ownership?” Stern School of Business, New York University. JUNE 2000 EXECUTIVE STOCK OPTIONS PAGE 44 Rappaport, Alfred (1999) “New Thinking on How to Link Executive Pay with Performance,” Harvard Business Review, March/April, 91-101. Rubenstein (1995), “On the Accounting Valuation of Employee Stock Options,” Journal of Derivatives, Fall. Saly, J. (1994), “Repricing Executive Stock Options in a Down Market,” Journal of Accounting and Economics, 18(3), 325-356. JUNE 2000 EXECUTIVE STOCK OPTIONS PAGE 45 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 JUNE 2000 EXECUTIVE STOCK OPTIONS PAGE 46 bonuses. 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+.8*Bonus)(1+ rf)T + ni•max(0,PT-Xi), where the summation in ni•max(0,PT-Xi) allows for multiple option and stock grants.31 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(W V )f(PT)dPT T U(WT)f(PT)dPT where the distribution of stock prices in T years is lognormal with volatility σ and expected 2 value equal to (rf + (rm-rf) - /2)T. 31 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. JUNE 2000 EXECUTIVE STOCK OPTIONS PAGE 47 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= , ≡ m2  e − 1 are (respectively) the mean and variance of the assumed 2 /h where m and   lognormal distibution of stock prices. Applying the quadratic formula, = 1 2m ((m 2 + + 1) − (m 2 + ) + 1) 2 − 4m 2 , 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 JUNE 2000 EXECUTIVE STOCK OPTIONS PAGE 48 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: { + − Max UT + (1 − )UT ),UT −1 E } + − where UT is the expected utility in T if the stock price increases by λ (an uptick), UT is the E expected utility in T if the stock price increases by 1/λ (a downtick), and UT −1 is the expected if 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. JUNE 2000 EXECUTIVE STOCK OPTIONS PAGE 49 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 B-S =2 =2 =3 =3 Payout Price Cost 50% Stock 66% Stock 50% Stock 66% Stock 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 2T. JUNE 2000 EXECUTIVE STOCK OPTIONS PAGE 50 Table 2 Explanatory Regressions for Risk-Adjusted and Unadjusted Pay for S&P 500 CEOs Dependent Variables: CEO Total Compensation No Risk Risk-Adjusted Risk-Adjusted Independent Variable Adjustment with =2 with =3 (1) (2) (3) 5.90 4.97 4.96 Intercept (49.7) (49.4) (51.2) 0.302 0.332 0.311 Ln(Sales) (23.2) (30.2) (29.4) 0.226 0.310 0.306 Finance (Dummy) (5.4) (8.8) (9.0) -0.650 -0.364 -0.276 Utility (Dummy) (-12.4) (-8.2) (-6.5) -0.766 -0.327 -0.243 Year 1992 (Dummy) (-13.8) (-6.9) (-5.4) -0.621 -0.238 -0.161 Year 1993 (Dummy) (-11.2) (-5.1) (-3.6) -0.519 -0.139 -0.079 Year 1994 (Dummy) (-9.5) (-3.0) (-1.8) -0.400 -0.046 -0.005 Year 1995 (Dummy) (-7.4) (-1.0) (-0.1) -0.221 0.058 0.055 Year 1996 (Dummy) (-4.1) (1.3) (1.3) -0.076 0.127 0.113 Year 1997 (Dummy) (-1.4) (2.8) (2.6) R2 0.266 0.287 0.263 Note: t-statistics in parentheses. Sample size is 3,351 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 cash compensation. All monetary variables in 1988-constant dollars. JUNE 2000 EXECUTIVE STOCK OPTIONS PAGE 51 Table 3 Ratio of Executive Value to Cost for an Indexed Option with $30 Exercise Price, for Various Pairs of Relative Risk Aversion ( ) and Diversification Stock B-S =2 =2 =3 =3 Payout Price Cost 50% Stock 66% Stock 50% Stock 66% Stock Probability $5 $0.04 2.7% 1.3% 0.4% 0.1% 0.4% $15 $1.68 12.2% 7.0% 4.3% 1.8% 10.2% $30 $9.22 24.1% 15.6% 12.2% 6.1% 34.6% $45 $20.41 32.2% 22.4% 19.1% 10.8% 54.7% $60 $33.35 37.8% 27.5% 24.4% 14.9% 68.5% Note: B-S cost is the Black-Scholes value of one indexed 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 volatility =.25 (the median volatility of market-adjusted returns), rf =0, (rm-rf) =0. JUNE 2000 EXECUTIVE STOCK OPTIONS PAGE 52 Table 4 Cost, Value, and Incentives from Executive Options with and without Early Exercise Options Exercisable at Expiration Options Exercisable at Grant % of Risk Wealth Company Executive Value Incentives Company Executive Value Incentives Aversion in Stock Cost Value Cost ∂V∂/P Cost Value Cost ∂V∂/P 2 50% $16.55 $10.51 63.5% $0.45 $14.76 $12.40 84.0% $0.61 2 66% $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 66% $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. JUNE 2000 EXECUTIVE STOCK OPTIONS PAGE 53 Table 5 Likelihood of Exercising on Vesting Date Likelihood of Exercising on Vesting Date Diversification Risk (% of Wealth Vesting at Vesting at Vesting at Aversion in stock) 2 Years 3 Years 4 Years 2 50% 1.4% 5.8% 10.0% 2 66% 5.9% 12.9% 19.8% 3 50% 5.9% 16.7% 23.7% 3 66% 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. JUNE 2000 EXECUTIVE STOCK OPTIONS PAGE 54 Figure 1 Executive Value Lines: Option Values for Undiversified Executives $45 Black-Scholes "Cost" $40 $35 "Intrinisic Value" $30 Option Values for Executive with: $25 =2, 50% of Wealth in Stock =2, 66% of Wealth in Stock $20 =3, 50% of Wealth in Stock =3, 66% of Wealth in Stock $15 $10 $5 $0 $0 $15 $30 $45 $60 Out-of-the-Money At-the-Money In-the-Money Stock Price Note: Executive values for ten-year options with an exercise price of $30 are estimated using the “certainty equivalence” approach, and are defined as the amount of riskless cash compensation the executive would exchange for the option. Certainty equivalents 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. JUNE 2000 EXECUTIVE STOCK OPTIONS PAGE 55 Figure 2 Executive Indifference Curves: Amount of Stock-Based Pay Required to Offset $300,000 in Cash Compensation $3.0 "Cost" of Stock-Based Pay for Executives with: $2.5 =3, 66% in Stock =3, 50% in Stock $2.0 $1.5 =2, 66% in Stock =2, 50% in Stock $1.0 $0.5 $0.0 Res Stock FMV Option Cash Discount Options Premium Options $-0.5 0% 0% 50% 100% 150% 200% 250% 300% [100%] [93%] [78%] [63%] [51%] [42%] [35%] Exercise Price as % of Stock Price [Payout Probability] Note: 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 JUNE 2000 EXECUTIVE STOCK OPTIONS PAGE 56 Figure 3 CEO Pay and Risk-Adjusted Pay in S&P 500 Industrials, 1992-1998 $5,129 $5,000 Unadjusted Pay in Columns 7% $4,208 Risk-Adjusted Pay for Executive with: $4,000 7% Risk Aversion = 2 $3,754 Risk Aversion = 3 7% $3,221 53% 8% $3,000 $2,714 48% 8% 46% $2,375 8% 42% $1,986 $2,000 36% 7% 34% Other 18% 30% Stock- 21% 21% Based 24% 23% 21% $1,000 22% Bonus 22% 36% 32% 27% 26% 23% Salary 41% $0 1992 1993 1994 1995 1996 1997 1998 Note: Median pay levels (in 1998-constant dollars) based on ExecuComp data for S&P 500 CEOs (financial firms and utilities excluded). Total compensation (in columns) defined as the sum of salaries, bonuses, benefits, stock options (valued on date of grant using the Black-Scholes formula), stock grants, and other compensation. Executive values are estimated using the “certainty equivalence” approach, and are defined as the amount of riskless cash compensation the executive would exchange for his stock and option grants, conditional on his current stock and option holdings. The risk-adjusted value of accounting-based bonuses are assumed to be worth 80% of actual bonuses. The CEO's safe wealth is assumed to be the greater of $5 million or four times cash compensation. Certainty equivalents are estimated numerically assuming that the executive has constant relative risk aversion of 2 or 3, and assuming (using the Capital Asset Pricing Model, CAPM) that the distribution of stock prices over the actual term of the options granted is lognormal with volatility and expected value (rf + (rm-rf) - 2/2)T, where and are determined using monthly stock-return data over 48 months, rf is the average yield on US government securities during the year of grant, and rm- rf = 6% is the equity premium. JUNE 2000 EXECUTIVE STOCK OPTIONS PAGE 57 Figure 4 Value of Stock and Options Held by S&P 500 CEOs, 1992-1998 $60 $50 S&P 500 Financial Firms $40 $30 S&P 500 Industrials $20 $10 S&P 500 Utilities $0 1992 1993 1994 1995 1996 1997 1998 Note: Data from Compustat's ExecuComp database, and are in 1998-constant dollars. The value of option holdings is defined as the in-the-money value of all exercisable and non-exercisable options held at the end of the fiscal year. JUNE 2000 EXECUTIVE STOCK OPTIONS PAGE 58 Figure 5 Incentives (per share) from 5,000 Options Granted to Undiversified Executives $1.00 Slope of Black-Scholes Line $0.90 $0.80 $0.70 Slope of Executive Value Lines =2, 50% of Wealth in Stock $0.60 =2, 66% of Wealth in Stock $0.50 =3, 50% of Wealth in Stock $0.40 =3, 66% of Wealth in Stock $0.30 $0.20 $0.10 $0.00 $0 $15 $30 $45 $60 Out-of-the-Money At-the-Money In-the-Money Stock Price Note: The figure shows the “slopes” of the Black-Scholes and executive valuations in Figure 1, which in turn depict the per-share cost and value of a grant of 5,000 ten-year options with an exercise price of $30. We define “incentives” as the change in the certainty-equivalent option value for each $1 change in the stock price. JUNE 2000 EXECUTIVE STOCK OPTIONS PAGE 59 Figure 6 Incentives from $300,000 “worth” of Options Granted to Undiversified Executives $25 Change in Black-Scholes Value $20 $15 $10 Change in Executive Value: =2, 50% in Stock =2, 66% in Stock $5 =3, 50% in Stock =3, 66% in Stock Max Max Max Max 40% 65% 85% 130% $0 Res Stock FMV Option Discount Options Premium Options 0% 50% 100% 150% 200% 250% 300% [100%] [93%] [78%] [63%] [51%] [42%] [35%] Exercise Price as % of Stock Price [Payout Probability] Note: The figure assumes that executives with $5 million in initial wealth are granted stock options with a Black-Scholes value of $300,000; the number of options granted naturally increases as the exercise price increases. We define “incentives” as the change in the certainty-equivalent option value for each $1 change in the stock price. JUNE 2000 EXECUTIVE STOCK OPTIONS PAGE 60 Figure 7 Incentives from $300,000 “worth” of Indexed Options Granted to Undiversified Executives $25 Change in Black-Scholes Value $20 $15 Change in Executive Value: =2, 50% in Stock $10 =2, 66% in Stock =3, 50% in Stock =3, 66% in Stock $5 Max Max 20% 40% $0 Res Stock FMV Option Discount Options Premium Options 0% 50% 100% 150% 200% 250% 300% [100%] [68%] [35%] [18%] [10%] [6%] [4%] Exercise Price as % of Stock Price [Payout Probability] Note: The figure assumes that executives with $5 million in initial wealth are granted ten-year indexed stock options with a Black-Scholes value of $300,000; the number of options granted naturally increases as the exercise price increases. We define “incentives” as the change in the certainty-equivalent option value for each $1 change in the stock price. Indexed options are calculated assuming volatility =.25 (the median volatility of market-adjusted returns), rf =0, (rm-rf) =0. JUNE 2000 EXECUTIVE STOCK OPTIONS PAGE 61 Figure 8 “Threshold Prices” for Early Exercise Decisions on Immediately Exercisable Options $100 $90 $80 =2, 50% in Stock =2, 66% in Stock $70 $60 =3, 50% in Stock =3, 66% in Stock $50 $40 Maximum price for each period in Binomial Tree $30 1 2 3 4 5 6 7 8 9 10 Years Since Option Grant Note: The threshold price is the stock price where the executive is just indifferent between exercising early or holding the option for another period. The figure assumes that executives with $5 million in initial wealth are granted 5,000 immediately exercisable stock options. At exactly T=10, the threshold price falls to the exercise price of $30. JUNE 2000 EXECUTIVE STOCK OPTIONS PAGE 62 Figure 9 Cumulative Exercise Probabilities for Immediately Exercisable Options 100% 80% =3, 66% in Stock =3, 50% in Stock 60% 40% =2, 66% in Stock =2, 50% in Stock 20% 0% 1 2 3 4 5 6 7 8 9 10 Years Since Option Grant Note: The threshold price is the stock price where the executive is just indifferent between exercising early or holding the option for another period. The figure assumes that executives with $5 million in initial wealth are granted 5,000 stock options. JUNE 2000 EXECUTIVE STOCK OPTIONS PAGE 63 Figure 10 The Ratio of the Executive Value to the Company’s Cost, by Vesting Date 100% =2, 50% in Stock 80% =2, 66% in Stock 60% =3, 50% in Stock =3, 66% in Stock 40% 20% 0% 0 1 2 3 4 5 6 7 8 9 10 Immediately Number of Years Until Options are Exercisable No Early Exercisable Exercise Note: The threshold price is the stock price where the executive is just indifferent between exercising early or holding the option for another period. The figure assumes that executives with $5 million in initial wealth are granted 5,000 stock options.