# An upper bound for the firm's cost of employee stock options.

Employee stock options (ESOs) are a popular and often significant component of executive compensation. It follows that information on the firm's cost of ESOs is necessary for designing executive compensation, for evaluating the cost of management's stewardship, and for evaluating the firm's performance. Unfortunately, there are serious conceptual difficulties in applying conventional options pricing models to ESOs.The purpose of this paper is to provide an upper bound to the firm's cost of ESOs that is independent of employees' risk preferences and wealth. Such an upper bound would provide a useful point of reference for any proposed method of reporting or determining a particular firm's ESO cost, and it would help estimate the overstatement that results from applying the benchmark Black-Scholes model with the holding period equal to the contract life.(1) The upper bound is determined by identifying a price at which a portfolio exists with returns that dominate the employees' returns from holding the unhedged ESOs. This price, denoted [P.sub.s], depends on the risk and return of the underlying stock and on all other available investment opportunities but not on employees' risk aversion. Individual employees may rationally exercise their unhedged ESOs before the firm's stock exceeds [P.sub.s] depending on their level of risk aversion. However, no employees will continue holding their unhedged ESOs after [P.sub.s] is exceeded. Hence, the upper bound to the firm's ESO cost is calculated using the assumption that employees will hold their ESOs until price [P.sub.s] is exceeded. A binomial model is used to generate a probability distribution for the length of time it will take the stock price to first exceed [P.sub.s]. Using this probability distribution, the ESO costs' upper bound is computed as the sum over i of the Black-Scholes value for an option that expires at time [t.sup.i] multiplied by the probability that the stock price will first exceed [P.sub.s] at time [t.sup.i].

Under the assumption that ESOs are not hedged, the ESOs' value to risk-averse employees will be lower than the market value of tradable warrants that are otherwise identical to the ESOs. This raises the question of why the firm issues ESOs in the first place. Specifically, a firm issuing ESOs has two alternatives, 1) give employees the certainty equivalent cash payment for the ESOs and/or 2) sell warrants in the marketplace and give the proceeds to employees. If employees are risk averse and the options are not hedged, the certainty equivalent payment in alternative 1 has lower expected cost to the firm than the ESOs. Likewise, the number of ESOs the firm has to sell in alternative 2 is less than the number of ESOs the firm grants to employees. However, alternative 1 has the following disadvantages relative to issuing ESOs: a) It requires the firm to make an up-front cash outlay; b) it requires the firm to recognize an expense on its income statement; c) it is immediately taxable to the employee; and d) it does nothing to align shareholders' and employees' interests. Likewise, alternative 2 has disadvantages b, c, and d relative to issuing ESOs.

The remainder of the paper is organized as follows. Section I provides a literature review. Section II introduces a model of employees' exercise behavior and uses the model to define the upper bound for the ESO cost. A theoretical analysis based on the model shows how volatility and other parameters affect the upper bound. Section III provides computational examples for a variety of parameter values and describes how Hemmer, Matsunaga, and Shevlin's (1994) linearization method can be used to provide a closed-form approximation to the upper bound. Section IV provides implications of the model for ESO costs and incentives and for the advisability of placing restriction on ESO grants. Finally, Section V provides a summary and conclusions.

I. Motivation and Related Literature

A basic assumption underlying the valuation of American Call Options in standard options pricing models is that the options will not be exercised before the expiration date (Merton, 1990).(2) However, there are several reasons why it is unrealistic to assume that employees will hold their ESOs until the expiration date (see Hemmer et al., 1994; Huddart, 1994; Kulatilaka and Marcus, 1994; and Lambert, Larcker, and Verrecchia. 1991). First, the ESOs may represent a large proportion of employees' wealth, and employees may not have sufficient additional wealth to hedge their ESOs. Second, execution of a hedging strategy requires employees to sell the underlying stock short, and such action may conflict with insider trading restrictions. As a result, employees may exercise their ESOs before the expiration date even if they remain with the firm.(3) Furthermore, if the ESOs are not hedged, then the holding period may be related to the stock's price behavior. Moreover, Huddart and Lang (1995) provide empirical evidence that employees exercise their ESOs prior to the expiration date.(4)

If ESOs are not hedged, employees' risk preferences and wealth will affect their decision to hold or exercise the ESOs. Huddart (1994) and Kulatilaka and Marcus (1994) show that the greater the employee's risk aversion, the lower the firm's ESO cost, and the greater the employee's wealth, the greater the firm's ESO cost. Furthermore, for sufficiently risk-averse employees, volatility is negatively related to the firm's ESO cost. These results occur in their models because more risk-averse employees will hold their ESOs for a shorter period of time and more wealthy employees will hold their ESOs for a longer period of time. Huddart (1994) and Kulatilaka and Marcus' (1994) models illustrate that there is no general solution to the firm's ESO cost that uses objectively verifiable parameters.(5) One cannot even calculate an upper bound to the firm's ESO cost assuming a universal utility function because employees' risk preferences and wealth are likely to be related to the riskiness of the firm.(6) For example, employees who contract with high-risk firms are likely to be more tolerant of risk than employees who contract with low-risk firms. In summary, while Huddart (1994) and Kulatilaka and Marcus (1994) illustrate that the firm's cost of an ESO with a term of t years may be significantly less than the Black-Scholes value for an option with a contract life of t years, they do not quantify or bound the overstatement that results from applying a simple Black-Scholes model.

The motivational objectives of ESO grants suggest that the ESOs are unlikely to be hedged. ESOs align shareholders' and managers' interests in several ways. Farmer and Winter (1986) and Haugen and Senbet (1981) develop models in which ESOs reduce managers' consumption of excessive perquisites. Reitman (1993) develops a model in which ESOs reduce managers' incentive to pursue overly aggressive competitive behavior. In these models, the incentive benefits of ESOs are only obtained if the ESOs are not hedged.(7) In addition, recent empirical work linking incentive compensation to firm-specific attributes (e.g. Collins, Blackwell, and Sinkey, 1995, and Gaver and Gaver, 1995) treat ESOs as incentive payments.(8) Hence, the assumption that ESOs will not be hedged is consistent with the fact that ESOs are granted and with the notion that ESOs align shareholders' and managers' interests.

II. The Employee's Exercise Behavior

This section develops a model and explores its implications concerning the employees' option exercise behavior.

A. The Model

For ease of exposition, the analysis begins with Non-Qualified Stock Options (NQOs) and is later extended, under certain conditions, to Incentive Stock Options (ISOs). The difference between ISOs and NQOs is as follows. With NQOs, the employee reports the difference between the exercise price and the market price on the exercise date as ordinary income for the year in which exercise occurs. Price appreciation subsequent to the exercise date is treated as capital gain income, subject to holding period requirements for capital gains. With ISOs, the employee can defer tax until the stock is sold (except possibly the Alternative Minimum Tax). If the stock is held for longer than the qualifying holding period for ISOs, the difference between the exercise price and the market price on the date the stock is sold is treated as a capital gain and taxed in the year the stock is sold.(9) This capital gain will never be taxed if the employee dies before selling the stock.

The model uses the following assumptions:

1. Employees do not hedge their ESOs.

2. Equilibrium market prices are characterized by a security market line, where expected stock returns have a positive linear relation with systematic risk.

3. Stock returns distributions are completely described by the expected value and variance.

4. Returns from all perfectly diversified portfolios are taxed as ordinary income in the year they are earned.

5. Employees' deviations from the optimal exercise strategy are expected to be random, and these random deviations have a zero mean effect on the firm's expected ESO cost.(10)

There are two types of risk from holding a security: unsystematic risk and systematic risk. Unsystematic risk is defined as the risk that can be removed by holding a sufficiently diversified portfolio, while systematic risk is defined as the risk that cannot be diversified away. The total risk of the security is the systematic risk plus the unsystematic risk. Two principles of finance relating to risk are that the expected rate of return on a security is positively and linearly related to its systematic risk and that unsystematic risk is not rewarded by the market with a higher expected rate of return. Hence, when employees hold ESOs that are not hedged, they bear a significant amount of unsystematic risk that is not compensated by the market.

Employees can always exercise in-the-money ESOs, sell the stock, and invest the after-tax proceeds in a perfectly diversified portfolio, where the systematic and total risk of the diversified portfolio equals the total risk of the underlying stock. After switching out of the ESOs, employees will be compensated by the market for all the risk borne because all the risk will be systematic. At the same time, the employees will bear no more total risk than if the ESOs were still held. As a result, the employees may earn a higher expected profit than the one earned by continuing to hold the ESOs, without bearing additional risk.

Let E(R) represent the expected rate of return on the underlying stock, let E(G) represent the expected rate of stock price appreciation on the underlying stock (E(R) [greater than or equal to] E(G)), let E(DP) represent the expected rate of return on the diversified portfolio, let [Tau] represent the employees' marginal tax rate, let X represent the exercise price, and let P represent the stock price. Appendix A formally provides the proof that, for any price in excess of [P.sub.s] and for any holding period, the expected pre-tax profit from switching into the diversified portfolio exceeds the expected pre-tax profit from continuing to hold the ESOs.

[P.sub.s]E(G) = ([P.sub.s] - X)(1 - [Tau])E(DP) or: (1)

[P.sub.s] = X / [1 - E(G)/(E(DP)(1 - [Tau]))]

The intuition of Equation (1) is as follows. ([P.sub.s] - X)(1 - [Tau]) represents the amount the employees will have available for investment if the options are exercised and the underlying stock is sold. This amount is simply the options' intrinsic value less the tax liability from the exercise. If [P.sub.s]E(G) = ([P.sub.s]X)(1 - [Tau])E(DP), then for any holding period, the employees' expected pre-tax profit from continuing to hold the options at most equals the employees' expected pre-tax profit from the strategy of exercising the ESOs, selling the stock, and investing the after-tax proceeds in a perfectly diversified portfolio with systematic and total risk equal to the total risk of the underlying stock.

Assumption (4) ensures that the statutory tax rate on income from the two strategies is the same. At prices in excess of [P.sub.s], the switching strategy of exercising the ESOs and investing the proceeds in the diversified portfolio will generate greater pre-tax profits for each period until the ESOs are exercised than the strategy of continuing to hold the ESOs.(11) Since all taxes on the ESOs will eventually be paid by the ESOs' expiration date, the switching strategy dominates at all prices in excess of price [P.sub.s](12,13) While each firm has a single unique [P.sub.s], a given employee's optimal decision whether to exercise the ESOs at a price below [P.sub.s] is the solution to his or her personal portfolio optimization problem.(14) However, it will never be rational for employees to continue holding their unhedged ESOs after the stock price first exceeds [P.sub.s].

B. The Maximum ESO Cost

Using Equation (1), the upper bound for the firm's ESO cost can now be derived. Let the grant date be denoted by time 0. If the firm does not pay dividends, a necessary condition for employees to hold their ESOs until time [t.sup.i] is that the stock price first exceeds [P.sub.s] (the switching price) at time [t.sup.i]. If the firm pays dividends, the necessary condition for employees to hold their ESOs until time [t.sup.i] is that time [t.sup.i] is the earlier of 1) when price [P.sub.s] is first exceeded or 2) when the remaining time-value of the option first becomes less than the present value of the dividends through the ESOs' expiration date. The probability that the necessary conditions hold for a given time [t.sup.i], denoted [P.sup.i], is computed using a binomial process to model the stock price path. For each possible [t.sup.i] between the vesting and expiration dates, the Black-Scholes value for an option that expires at time [t.sup.i], denoted [V.sup.i], is multiplied by the probability that the necessary condition for time [t.sup.i] holds. The sum, over i, of the product [P.sup.i][V.sup.i], is the maximum ESO cost.(15) Intuitively, this cost is the expected Black-Scholes value for the ESOs, where the Black-Scholes value is a function of employees' holding period (as well as of volatility and the risk-free rate), and employees are assumed to switch out of their ESOs when it is impossible to rationally continue holding the ESOs. A related construct is the maximum expected holding period. It is defined as the sum, over [t.sup.i], of the products of [t.sup.i] and [P.sup.i], and it is computed using the same binomial model that is used for the maximum ESO cost.(16) Intuitively, the maximum expected holding period is the probability-weighted average time until it is impossible for employees to rationally continue holding the ESOs.(17)

To calculate the maximum ESO cost, estimates of [R.sub.m], [R.sub.f], [[Beta].sub.us], [[Sigma].sub.Rm], [[Sigma].sup.[Epsilon]us], E(G), E(DP), and volatility are required. These parameters can be estimated with past returns data, in the same manner as volatility is estimated for the Black-Scholes model. Using a CAPM framework, E(DP) can be derived as follows.(18) Let [V.sub.us] and [V.sub.dp] represent the total risk of the underlying stock and the diversified portfolio, respectively. (To simplify the discussion, the remainder of the paper deals with the case where the firm does not pay dividends.)

E(R) = Rf + [[Beta].sub.us]([R.sub.m] - [R.sub.f])

[V.sub.us] = [[Beta].sub.[us.sup.2]] [[Sigma].sup.[2.sub.Rm]] + [[[Sigma].sup.2].sub.[Epsilon]us]

where [[Beta].sub.us] ([[Beta].sub.dp]) is the beta on the underlying stock (diversified portfolio), [R.sub.m] is the expected return on the market portfolio, [[Sigma].sub.[Epsilon]us] is the unsystematic risk on the underlying stock, [[Sigma].sub.Rm] is the risk on the market portfolio, and [R.sub.f] is the risk-free rate of return.

[Mathematical Expression Omitted]

[V.sub.dp] = [[Beta].sub.[dp.sup.2]][[Sigma].sup.[2.sub.Rm]]

Given that [V.sub.dp] = [V.sub.us], [[Beta].sub.[dp.sup.2]][[Sigma].sup.[2.sub.Rm]] = [[Beta].sub.[us.sup.2]][[Sigma].sup.[2.sub.Rm]] + [[Sigma].sup.[2.sub.[Epsilon]us]], or:

[[Beta].sub.dp] = [[Beta].sub.[us.sup.2]] + [[Sigma].sup.[2.sub.[Epsilon]us]] / [[Sigma].sup.[2.sub.[Rm.sup.1/2]]] (3)

Combining Equations (2) and (3)

[Mathematical Expression Omitted]

[Mathematical Expression Omitted]

By design, the switching price is determined using tax assumptions and other computational simplifications that favor the strategy of continuing to hold the ESOs. The consequence of relaxing these assumptions and simplifications is that the solution for the switching price will be less than [P.sub.s]. If assumption (4) is relaxed, the effective tax rate on profits from the diversified portfolio will be lower than the effective tax rate on profits from the ESOs. As a result, the price at which the after-tax profits from the two strategies are equal will be less than [P.sub.s], and all employees will rationally exercise their ESOs even before the stock price first exceeds [P.sub.s]. Further, at [P.sub.s], the expected end-of-period after-tax wealth is actually greater with the switching strategy. This occurs because if the cumulative return on the underlying stock through the ESOs' expiration date is less than (X - [P.sub.s])/[P.sub.s] percent (with [P.sub.s] [greater than] X) the employees will have zero end-of-period wealth if they hold the ESOs.(20) However, even if the cumulative return on the diversified portfolio over this period is less than (X- [P.sub.s])/[P.sub.s] percent, the employees will have some remaining wealth at the ESOs expiration date. As a result, the highest price at which it is even possible for rational employees to continue holding their ESOs is, in fact, less than [P.sub.s]. The exact price at which all employees will have exercised their ESOs is a function of the holding period, the time remaining until the expiration date, the difference between the stock and exercise prices, and the tax regimes for ESOs and other investments. However, unlike [P.sub.s], this price cannot be determined with a closed-form expression. Consequently, the difference between the maximum ESO cost and the benchmark cost represents the minimum overstatement that results from applying standard options pricing models to ESOs.

Finally, while the previous discussion deals with NQOs, the maximum ESO cost for NQOs may be applicable to ISOs. Halperin, Mozes, and Balsam (1994) show that the following two conditions are necessary for NQOs and ISOs to have the same maximum ESO cost.

1. Employees will not hold the underlying stock until their death. That is, employees will realize all capital gains at some point in their lifetimes.

2. At least one of the following hold: a) There is no difference between ordinary income and capital gain tax rates and b) returns on the diversified portfolio are treated as capital gain income.

If the necessary conditions hold, the statutory tax rate that will be applied to profits on the diversified portfolio will be equal to the statutory tax rate that will be applied to profits from the ISOs. Hence, the [P.sub.s] for NQOs applies to ISOs as well. However, if the necessary conditions do not hold, the switching price for ISOs will be greater than the [P.sub.s] for NQOs because the statutory tax rate will be greater for profits on the diversified portfolio.(21)

C. Parameters That Affect the Maximum ESO Cost

Equation (1) is useful for developing intuitions about the parameters that are related to the maximum ESO cost. Ceteris paribus, the higher [P.sub.s], the longer employees will hold their ESOs and the greater the firm's maximum ESO cost. In Equation (1), [P.sub.s] is an increasing function of [Tau], the employees' marginal tax rate. Therefore, the higher [Tau], the higher the firm's maximum ESO cost.(22) The intuition is that the higher the individual's marginal tax rate, the longer employees are likely to hold their ESOs, and the higher the firm's ESO cost is likely to be. An additional insight is that the maximum ESO cost is negatively related to the market risk premium. Equation (4) implies that E(R)/E(DP) is a decreasing function of [Mathematical Expression Omitted], which represents the market risk premium. Hence, the greater the market risk premium, the lower E(R)/E(DP), the lower [P.sub.s], and the lower the maximum ESO cost. Conversely, [P.sub.s] is independent of the contract life of the ESO grant. As a result, an increase in the ESO's term from n years to n + j years will have little or no effect on the maximum ESO cost if the stock price is very likely to exceed [P.sub.s] within a period of n years. This will occur if n is long, [P.sub.s] is low, or the stock is very volatile.(23)

In the case of marketable stock options, there is a well-established positive relation between volatility and the option value, and that relation is the same regardless of the relative amounts of systematic and unsystematic risk in the underlying stock. It is natural to examine whether a similar relation can be established between volatility and the maximum ESO cost. While the specific analysis of the relation between volatility and the maximum ESO cost is presented in Appendix B, the results of the analysis can be summarized as follows. Unlike the case of marketable stock options, the relation between volatility and the maximum ESO cost crucially depends on the individual amounts of systematic and unsystematic risk in the underlying stock. If two firms have equal volatility (total risk) but one firm has greater unsystematic risk, the two firms' maximum ESO costs will differ. Furthermore, depending on whether the increase in volatility is generated by systematic risk or unsystematic risk, an increase in volatility may either increase or decrease the maximum ESO cost.

III. Computing the Maximum ESO Cost

This section provides computations of the maximum ESO cost using the method introduced in Section II. In addition, Hemmer et al.'s (1994) linearization method is adapted to provide a closed-form approximation to the maximum ESO cost.

A. Computations for the Maximum ESO Cost

Computations of the maximum ESO cost are presented in Table 1 for the following ESO: a 10-year grant life, no exercise restrictions, a $50 stock price on the grant date, a $50 exercise price (X), [R.sub.f] = 0.06, and the firm does not pay dividends (i.e., E(R) = E(G)). Panels A, B, and C of Table 1 present results for E(R) = 0.08, E(R) = 0.12, and E(R) = 0.16, respectively.(24) E(R) is a function of the stock's systematic risk and of the market risk premium. For each value of E(R), E(DP) ranges from E(R)*1.4 to E(R)*3.6, and for each combination of E(R) and E(DP), volatility varies from 0.1 to 0.6. The higher E(DP) relative to E(R), the greater the amount of unsystematic risk in the underlying stock relative to its systematic risk and/or the greater the market risk premium (see Equation (4)). A relation such as E(DP) = E(R)*3.6 would arise where almost all (i.e., approximately 80 to 85%) of the total risk in the underlying stock is unsystematic. If E(DP) = E(R)*3.6 and E(R) = 0.16, the value for E(DP) is 0.544, and the interpretation of that value is that if all the risk in the underlying stock were priced in the same way as the stock's systematic risk is priced, the expected rate of return on the underlying stock would be 0.544 rather than 0.16.(25)

Table 1 also presents the maximum expected holding period, which provides the intuition for the difference between the maximum ESO cost and the benchmark Black-Scholes cost with the holding period equal to ten years (the contract period). The lower the maximum ESO holding period, the greater the difference between the ESO contract period and the average time until the switching price is first exceeded. Hence, the lower the maximum ESO holding period, the lower the maximum ESO cost and the greater the minimum overstatement of the ESO cost that results from using options pricing models with the holding period equal to ten years. It follows that for any volatility level, the Black-Scholes cost for the ESOs with a holding period of ten years is identical to the maximum ESO cost where the maximum expected holding period is ten years.

The results in Table 1 show that for a given level of volatility and E(R), the maximum ESO holding period is a decreasing function of E(DP). In cases where E(DP) is high relative to E(R), the maximum holding period may be less than half the ESOs' term. For example, if volatility = 0.4, E(R) = 0.12 (see Panel B), and E(DP) = 0.168, the maximum ESO holding period is 10 years. However, if E(DP) = 0.408, the maximum ESO holding period is 5.24 years. As another example of a set of parameters for which early exercise is rational, if E(R) = 0.16 (see Panel C), volatility = 0.4, and E(DP) = 0.512, the maximum ESO holding period is only [TABULAR DATA FOR TABLE 1 OMITTED] 4.86 years. The intuition of these results is that the higher E(DP) relative to E(R), the less attractive the strategy of holding the ESOs relative to the switching strategy, the lower the switching price, and the shorter the ESO holding period. In addition, the results in Table 1 show that for a given E(R) and E(DP), the maximum ESO holding period is a decreasing function of volatility, consistent with the analysis of volatility provided in Appendix B. For example, if volatility = 0.1, E(R) = 0.08 (see Panel A), and E(DP) = 0.192, the maximum expected holding period is 9.61 years. However, if volatility = 0.4, the maximum expected holding period is 7.91 years.

Regarding the ESO cost, results in Table 1 illustrate that for any given level of volatility and E(R), the minimum overstatement from using options-pricing models to value ESOs is an increasing function of E(DP). In cases where E(DP) is high relative to E(R), the minimum overstatement from applying options-pricing models to ESOs often exceeds 50%. For example, if E(R) = 0.12 (see Panel B), volatility = 0.4, and E(DP) = 0.168, the maximum ESO cost is $31.28 and the Black-Scholes cost is also $31.28. However, if E(R) = 0.432, the maximum ESO cost is $19.84 and the Black-Scholes cost is $31.28. In the latter case, the Black-Scholes cost overstates the firm's ESO cost by a minimum of 57.7%. Likewise, if E(R) = 0.16 (see Panel C), volatility = 0.6, and E(DP) = 0.288, the maximum ESO cost is $35.19 and the Black-Scholes cost is $37.61. However, if E(R) = 0.544, the maximum ESO cost is $23.54 and the Black-Scholes cost is $37.61. In this case, the Black-Scholes cost overstates the firm's ESO cost by a minimum of 59.8%. The overstatement is an increasing function of E(DP) because the higher E(DP) relative to E(R), the lower the period employees will hold their ESOs.

The results in Table 1 also illustrate that, ceteris paribus (e.g., E(R) and E(DP)), the maximum ESO cost is an increasing function of volatility. For example, if E(R) = 0.08 (see Panel A), E(DP) = 0.144, and volatility = 0.2, the maximum ESO cost is $24.59, while if volatility is 0.5, the maximum ESO cost is $33.50. Likewise, if E(R) = 0.12 (see Panel B), E(DP) = 0.288, and volatility = 0.1, the maximum ESO cost is $19.78, while if volatility is 0.4, the maximum ESO cost is $25.81. However, the analysis presented in Section II and Appendix B shows that the ceteris paribus condition is unlikely to hold.

B. Approximating the Maximum ESO Cost with a Closed-Form Expression

While the maximum ESO cost can be computed using a probability distribution for the date at which the stock price first exceeds [P.sub.s], a simpler computation can be used to obtain a quick closed-form estimate of the upper bound to the firm's ESO cost. Let the estimate of the maximum ESO holding period be denoted by t. Given t, one can use Hemmer et al.'s (1994) linearization method to approximate the maximum ESO cost. Let the Black-Scholes cost assuming the options will be held until the expiration date (vesting date) be denoted by [C.sub.exp. date] ([C.sub.vesting]), and let the maximum ESO cost be denoted by [C.sub.max]. Let the time from the grant date until the expiration date (vesting date) be denoted by [t.sub.exp. date] ([t.sub.vesting]). Hemmer et al. illustrate how their approximation mitigates the problem that arises from the fact that the Black-Scholes cost is a concave function of the options' term.(26, 27) Their method is as follows:

x = ([C.sub.exp. date] - [C.sub.vesting]) / ([t.sub.exp. date] - [t.sub.vesting])

[C.sub.max] = [C.sub.vesting] + [x.sup.*](t - [t.sub.vesting])

A closed-form method of approximating t is as follows.(28) Let the stock price on the grant date be denoted by C, and let Y represent the future value factor required to compound C into [P.sub.s] such that [e.sup.Y] = [P.sub.s]/C. Given [P.sub.s] and C, one can compute Y. Y can be written as Y = E(G) * t. Given Y and E(G), one can then solve for t in the expression Y = E(G) * t. For example, assume that the solution to [P.sub.s] in Equation (1) is $120, C = $50 and E(G)= 0.10. Y is computed as 0.875 because the natural log of 2.4 (120/50) is 0.875. Given Y = 0.875 and E(G) = 0.10, it follows that t = 8.75, or that it will take approximately 8 and 3/4 years for the stock price to move from $50 to $120. If [P.sub.s] does not exist or if t is greater than the ESO contract life, the maximum ESO cost equals the benchmark value.

IV. Implications of the Model

One implication of the model and results in Table 1 is that the overstatement from using options-pricing models for ESOs is likely to be greatest for those firms that have the greatest need to issue ESOs. Small, high-tech firms in the start-up or growth stages often offer large numbers of ESOs in lieu of cash compensation to attract talented managers. Because these firms typically have high unsystematic risk, E(R)/E(DP) and [P.sub.s] will be low, and the maximum ESO cost will be significantly below the benchmark Black-Scholes value. A second implication is that restrictions on early exercise may have a significant effect on the firm's ESO cost. For example, if E(R) = 0.16 (Panel C), E(DP) = 0.384, volatility = 0.3, and there are no restrictions on when the ESOs can be exercised, the maximum ESO cost is $21.67, and the maximum ESO holding period is 6.71 years. However, if employees are restricted from exercising their ESOs prior to the expiration date (ten years), the ESO cost (using either the maximum ESO cost or the Black-Scholes value) is $27.84. Conversely, the ESOs must be more valuable to employees if there are no exercise restrictions because employees can always choose to hold their ESOs until the expiration date. Hence, the advisability of placing restrictions on early exercise depends on the firm's motivation for issuing the ESOs. If the ESOs are intended as a form of deferred compensation or tax planning, it would not make sense for the firm to increase its expected ESO cost by placing significant restrictions on early exercise. However, if the ESOs are intended to align managers' interests with shareowners' interests and to reduce managerial turnover, the expected cost of restrictions on early exercise may be exceeded by the expected motivational benefits of these restrictions. A related implication is that if the firm has high unsystematic risk relative to total risk (high E(DP)/E(R)), the firm will need to place lengthy exercise restrictions on the ESOs in order to ensure that employees will hold their ESOs until the expiration date. Otherwise, employees will rationally exercise their ESOs considerably prior to the expiration date.

A third implication of the model is that the relation between volatility and the firm's ESO cost differs from the relation between volatility and the value of marketable options. With ESOs that are not hedged, the employee's return depends on the actual return on the underlying stock, and the employee must consider the relative attractiveness of switching into other investments. As a result, an increase in volatility due to an increase in unsystematic risk (with no change in systematic risk) may actually decrease the firm's ESO cost. This will occur if the attractiveness of continuing to hold the ESOs sufficiently decreases relative to the attractiveness of alternative investments and there is a substantial decrease in the amount of time employees hold their ESOs.

A fourth implication of the model relates to the incentives provided by ESOs. Specifically, investment decisions may have different effects on the firm's value and on the ESOs' expected value to employees. For example, if employees take actions that increase E(R) and E(G), the effect on [P.sub.s] will depend on the way both systematic and unsystematic risk change. If, for example, E(G) increases and unsystematic risk stays constant, [P.sub.s] will increase, employees will hold their ESOs longer, and the expected value of the ESOs to employees will increase. Hence, ESOs may align employees' and owners' interests with respect to an investment opportunity that increases the firm's value without increasing its unsystematic risk. On the other hand, if the investment opportunity contains sufficient unsystematic risk, [P.sub.s] will decrease, and employees will exercise their ESOs at a lower stock price. It is possible that the expected value of the ESOs to employees may decrease due to the lower [P.sub.s]. Hence, employees holding unhedged ESOs may choose to forego investments that increase the firm's value but also contain high unsystematic risk.

Additional implications of the model relate to financial accounting issues. The Financial Accounting Standards Board's Statement of Financial Accounting Standards No. 123, Accounting for Stock Based Compensation (FASB, 1995), requires firms to estimate their employees' holding period, but the Standard does not provide much guidance on how this holding period should be estimated. This paper suggests two methods of estimating employees' holding period. One method is the expected ESO holding period discussed in Section II, and the other method the closed-form approximation discussed in Section III.

A second accounting issue relates to the possibility that employees may depart from the firm before they vest in the options or prior to the options' expiration date.(29) Jennergren and Naslund (1993) deal with this issue by assuming that the early departure event follows a Poisson distribution and is independent of the firm's stock price. They then derive an expression linking the Black-Scholes cost to the annual departure probability. Cuny and Jorian (1995) derive another result using the assumption that the early departure event is related to the firm's stock price. However, both Jennergren and Naslund and Cuny and Jorian's results are based on the assumption that the ESOs would be held until the expiration date in the absence of the early departure event. Reflecting this assumption, SFAS 123 requires firms to calculate their expected ESO cost by using an exogenously specified early departure probability.

However, if ESO grants are intended to elicit the desired behavior from employees and to bond employees to the firm, employees are arguably more likely to depart after exercising their ESOs, when the motivational and bonding mechanism no longer exists. As a result, early exercise and early departure may coincide, and models of early exercise may also model early departure. Hence, employee departure between the vesting and expiration dates is, in effect, built into the model and captured by [P.sub.s], and it does not need to be separately considered through exogenous, ad hoc adjustments.

V. Summary and Suggestions for Future Research

This paper provides an upper bound to the firm's ESO cost that is independent of employees' risk preferences and wealth. The upper bound is a useful point of reference for any proposed method of reporting or determining a particular firm's ESO cost and gives the minimum overstatement that results from applying options-pricing models to that firm's ESO cost. The upper bound is determined by identifying a price for which a portfolio exists with returns that dominate the employee's returns from continuing to hold the ESOs. This price depends on the risk and return of the underlying stock and on all other available investment opportunities, but it does not depend on employees' risk preferences or wealth.

There are several suggestions for future research. First, the model's characterization that employees will never hold their ESOs after price [P.sub.s] is exceeded can be empirically tested with data on actual exercise behavior. Such research might provide supporting evidence for the validity of the model used to determine the ESO costs' upper bound. Second, the model of employees' exercise behavior can be extended into a utility framework. This might provide additional insights into the incentives provided by ESOs, for firms with different amounts of systematic and unsystematic risk. Third, future work can attempt to derive formal conditions under which increases in volatility will either increase or decrease the firm's maximum ESO cost. Such work would further contribute to the understanding of how volatility affects the firm's ESO cost and of how the firm's cost of ESOs differs from the value of tradable options.

1 This benchmark is used because there is no model for determining the expected holding period other than the model introduced in this paper.

2 Or, if the stock pays dividends, on some determinable date prior to the expiration date.

3 These issues do not arise with transferable options, as long as at least one investor has the ability to hedge the options and earn risk-free arbitrage profits.

4 Huddart and Lang analyze the exercise behavior of over 50,000 employees at eight firms. These firms shared their proprietary data with the authors with the stipulation that they remain anonymous.

5 Specifically, the expected ESO cost is a function of employees' degree of risk aversion, which is not observable.

6 Smith and Zimmerman (1976) propose a lower bound for the firm's ESO cost that is essentially the minimum-value method. In the minimum-value method, the exercise price is first discounted from the option's expiration date back to the option's grant date using the riskless rate. Then, this figure is subtracted from the market price of the stock on the grant date. However, Smith and Zimmerman's proposed lower bound is not derived from any theory or model concerning the employee's exercise strategy for ESOs.

7 Hemmer (1993) develops a model in which the firm grants options in a manner such that the employees' wealth is hedged through the ESOs and employees' wealth remains hedged only if they select the action desired by the firm. However, the extent to which Hemmer's model is descriptive of practice is an empirical question.

8 Further, Gaver and Gaver's (1995) measurement of incentive compensation assumes that employees holding ESOs will realize the entire difference between the stock price on the exercise date and the exercise price. This measurement technique clearly requires that the ESOs are not hedged.

9 On the other hand, if the ISOs are disqualified (i.e., the stock is held for less than the qualifying ISO holding period), the entire difference between the exercise price and the market price on the date the stock is sold is reported as ordinary income.

10 Assumption 4 allows one to ignore tax issues in constructing a closed-form expression for the maximum price at which employees will continue to hold their ESOs. Assumption 5 allows the possible occurrence of random events, such as liquidity needs, to be ignored.

11 To illustrate this point, note that if the stock price, P, is higher than [P.sub.s], PE(G) [less than] (P - X)(1 - [Tau])E(DP). This condition implies that the pre-tax profits will be greater with the switching strategy in the first period after the ESOs after exercised. The same result will hold every subsequent period because the condition PE(G) [less than] (P - X)(1 - [Tau])E(DP) implies that P(1 + E(G))E(G) [less than] (P - X)(1 - [Tau])(1 + E(DP)(1 - [Tau]))E(DP). The latter condition implies that the expected pre-tax profits will be greater with the switching strategy in the second period after the ESOs are exercised. This exercise can be repeated for every period until the ESO expiration date.

12 More formally, the employees' expected utility from the strategy of switching into the diversified portfolio when the stock price first exceeds [P.sub.s] will be greater than the expected utility from the strategy of continuing to hold the options, with first-order stochastic dominance.

13 Risk is ignored in comparing the switching strategy with the strategy of continuing to hold the ESOs because, by construction, the risk in the former strategy is lower than the risk in the latter strategy.

14 Given a particular holding period, employees have a unique expected utility from continuing to hold the ESOs for every price less than [P.sub.s]. Hence, one can sketch an ESO curve in the space defined by expected utility and price. Employees will optimally hold all their ESOs until price [P.sub.s] is first exceeded, if and only if the following condition holds: For every point on the ESO Curve, the employees' indifference curve on which that point lies never intersects any feasible portfolio consisting of a combination of the ESOs and other securities.

15 In cases with vesting restrictions, [P.sup.i] for time [t.sup.i] is modified as follows. If the firm does not pay dividends, [P.sup.i] is the probability that [t.sup.i] is the first time after the vesting restrictions lapse that the stock price exceeds [P.sub.s]. If the firm pays dividends, [P.sup.i] is the probability that time [t.sup.i] is the earlier of 1) the first time after the vesting restrictions lapse that price [P.sub.s] is exceeded and 2) when the remaining time-value of the option first becomes less than the present value of the dividends through the ESOs' expiration date. The summation of the products [P.sup.i][V.sup.i] only includes those [t.sup.i] after the vesting restrictions lapse.

16 Or, if the firm pays dividends, the maximum holding period is the probability-weighted average amount of time until the earlier of 1) the first time the stock price exceeds [P.sub.s] and 2) when the remaining time-value of the option first becomes less than the present value of the dividends through the ESOs' expiration date.

17 The maximum ESO cost can also be calculated using closed-form expressions developed for capped call options, if the firm does not pay dividends. In a capped-call option, the call writer pays the intrinsic value of the option when the underlying instrument first exceeds some threshold value (the cap). However, the binomial approach is necessary for computing the maximum ESO cost in the case where the firm pays dividends and for computing the maximum expected holding period. In addition, a binomial model can be modified to incorporate the effects of anticipated parameter changes. This is especially important for long-lived ESOs, where the options' life may extend from the firm's start-up phase through its growth and maturity phases.

18 The derivation for E(DP) is adapted from Mozes (1994).

19 This final expression is useful because Equation (1) shows that [P.sub.s] is a function of E(G)/E(DP). In cases where the firm does not pay dividends, E(G) = E(R), and [P.sub.s] is an increasing function of E(R)/E(DP).

20 To illustrate this point, note that [P.sub.s][(X - [P.sub.s])/[P.sub.s]] = X - [P.sub.s]. Therefore, if the cumulative return on the underlying stock is equal to or less than (X - [P.sub.s])/[P.sub.s], the value of the underlying stock will be equal to or less than [P.sub.s] + X - [P.sub.s] = X, at the expiration date. In this case, the ESOs will expire worthless.

21 For Alternative Minimum Tax (AMT) purposes, the difference between the current stock price and the exercise price is a tax preference item during the year that the ISOs are exercised. If employees plan to sell the stock immediately upon exercise, the AMT is not relevant. Likewise, if employees plan to sell the stock after holding it for the period necessary to qualify for the capital gain tax rate, the AMT is a fairly minor issue. The AMT is a more serious tax planning issue for ISOs when employees plan to continue holding the stock after exercising the ISOs.

However, if the two necessary conditions hold, the AMT will not affect the upper bound of the firm's ESO cost. The reason is that, when the stock price exceeds the [P.sub.s] for NQOs, all employees holding ISOs will exercise their ISOs and sell the stock after holding it for the qualifying period.

22 Thus, an increase in individual tax rates may simultaneously make ESOs more costly to the firm and less valuable to the employee.

23 The entire analysis can also be equivalently presented in terms of the maximum ESO holding period. All parameters that are negatively (not) related to the maximum ESO cost are negatively (not) related to the maximum ESO holding period.

24 Alternative sets of parameters were also examined, with similar results to those reported in Table 1.

25 There are few individual investments with an expected rate of return of 0.544. However, if the security market line is linear in risk-expected return space and (riskless) unlimited borrowing is allowed, there will exist a leveraged portfolio with an expected rate of return of 0.544 and systematic risk equal to the sum of systematic and unsystematic risk in the underlying stock. If there are restrictions on borrowing, there may be a ceiling on the expected rate of return that can be achieved. The model ignores the possibility that such a ceiling exists because the value assumed for the ceiling would be completely arbitrary.

26 The concavity problem can be illustrated with the following example. Assume that the Black-Scholes costs are $20, $27, and $30 for holding periods of 6, 8, and 10 years, respectively. If there is a 50% chance that the options will be held 6 years and a 50% chance that the options will he held 10 years, the correct computation of the expected cost is 0.5*$20 + 0.5*$30 = $25. However, if the expected cost is computed using the average holding period of 8 years, the cost will be incorrectly calculated as $27. The overstatement of the grant cost from using the average holding period is referred to as the concavity problem.

27 In contrast, the maximum ESO cost does not suffer from the concavity or early exercise problems, because it does not use an average t.

28 The discussion in Section II shows that when calculating the maximum ESO holding period, there is virtually zero additional cost in calculating the maximum ESO cost. Therefore, the method of Hemmer et al. (1994) is not beneficial in conjunction with using the maximum ESO holding period to estimate t.

29 There are also estimation issues that apply equally to ESOs and other long-lived options. These issues, summarized by Rubinstein (1994), deal with the fact that parameters such as volatility and dividend yield are unlikely to remain stable over long periods of time. For example, Lauterbach and Schultz (1990) show the difficulty in estimating volatility for long-lived options and warrants. However, Noreen and Wolfson (1981) find that market prices of publicly traded long-lived warrants are within 15%-20% of the Black-Scholes prices.

References

Collins, M.C., D.W. Blackwell, and J.F. Sinkey, 1995, "The Relationship Between Corporate Compensation Policies and Investment Opportunities: Empirical Evidence for Large Bank Holding Companies," Financial Management, (Autumn), 40-53.

Cuny, C. and P. Jorian, 1995, "Valuing Executive Stock Options with a Departure Decision," Journal of Accounting and Economics, Forthcoming.

Farmer, R.E.A. and R.A. Winter, 1986, "The Role of Options in the Resolution of Agency Problems: A Comment," Journal of Finance (December), 1157-1170.

Financial Accounting Standards Board, 1995, "Statement of Financial Accounting Standards No. 123, Accounting for Stock Based Compensation," Norwalk, CT (October).

Gaver, J.J. and K. Gaver, 1995, "Compensation Policy and the Investment Opportunity Set," Financial Management (Spring), 19-32.

Halperin, R., H. Mozes, and S. Balsam, 1994, "A Role for Incentive Stock Options After The Tax Reform Act of 1986," Fordham University Working Paper.

Haugen, R.A. and L.W. Senbet, 1981, "Resolving the Agency Problems of External Capital Through Options," Journal of Finance (June), 629-647.

Hemmer, T., 1993, "Risk-Free Incentive Contracts: Eliminating Agency Costs Using Option-Based Compensation Schemes," Journal of Accounting and Economics (October), 447-473.

Hemmer, T., S. Matsunaga, and T. Shevlin, 1994, "Estimating the 'Fair Value' of Employee Stock Options with Expected Early Exercise," Accounting Horizons (December), 23-42.

Huddart, S., 1994, "Employee Stock Options," Journal of Accounting and Economics (September), 207-231.

Huddart, S. and M. Lang, 1995, "Employee Stock Options: An Empirical Analysis," Duke University Working Paper.

Jennergren, L.P. and B. Naslund, 1993, "A Comment on Valuation of Executive Stock Options and the FASB Proposal," Accounting Review (January), 179-183.

Kulatilaka, N. and A.J. Marcus, 1994, "Valuing Employee Stock Options," Financial Analysts Journal (November-December), 46-56.

Lambert, R.A., D.F. Larcker, and R.A. Verrecchia, 1991, "Portfolio Considerations in Valuing Executive Compensation," Journal of Accounting Research (Spring), 129-149.

Lauterbach, B. and P. Schultz, 1990, "Pricing Warrants: An Empirical Study of the Black-Scholes Model and Its Alternatives," Journal of Finance (September), 1181-1209.

Merton, R.C., 1990, Continuous-Time Finance, Basil Blackwell, Cambridge, MA.

Mozes, H.A., 1994, "Determining the Substitution Rate Between Incentive Stock Options and Non-Qualified Stock Options," Journal of the American Taxation Association (Spring), 138-157.

Noreen, E. and M. Wolfson, 1981, "Equilibrium Warrant Pricing Models and Accounting for Executive Stock Options," Journal of Accounting Research (Autumn), 384-398.

Reitman, D., 1993, "Stock Options and the Strategic Use of Managerial Incentives," American Economic Review (June), 513-524.

Rubinstein, M., 1994, "On the Evaluating of Employee Stock Options," FASB Roundtable Discussion on ESOs, Norwalk, CT (April 18).

Smith, C. and J. Zimmerman, 1976, "Valuing Employee Stock Option Plans Using the Option Pricing Models," Journal of Accounting Research (Autumn), 357-64.

Appendix A: Proof of Equation 1

The proof shows that the employees' expected utility will be greater if they exercise the option immediately after price [P.sub.s] is first exceeded than if they continue to hold the options, regardless of the time horizon.

Let i represent an arbitrary holding period of time i. Further, let E([R.sup.i]) represent E[(1 + R).sup.i] - 1, and let E([DP.sup.i]) represent E[(1 + DP).sup.i] - 1. EU[ ] represents the expected utility of a random variable. The utility function need only have a positive first derivative with respect to wealth.

It follows that if ([P.sub.s] - X)(1 - [Tau])E(DP) [greater than] [P.sub.s]E(R) then ([P.sub.s] - X)(1 - [Tau])E([DP.sup.i]) [greater than] [P.sub.s]E([R.sup.i]). Adding ([P.sub.s] - X) to both sides of the inequality and then multiplying both sides by (1 - [Tau]) gives the following inequality:

(1 - [Tau])([P.sub.s] - X) + (1- [Tau])([P.sub.s] - X)E([DP.sup.i])

[greater than] (1 - [Tau]) [[P.sub.s](1 + E([U.sup.i]) - X]

Given that the risk of the diversified portfolio is no greater than the risk of continuing to hold the options, it immediately follows from first order stochastic dominance that:

EU [(1 - [Tau])([P.sub.s] - X) + (1 - [Tau])([P.sub.s] X)([DP.sup.i])]

[greater than] EU [ max [0, (1 - [Tau])[[P.sub.s](1 + [U.sup.i]) - X] ] ].

QED

APPENDIX B: Analysis of the Relation Between Volatility and the Maximum ESO Cost

1. Equation (1) implies that for a given E(G), [P.sub.s] is a decreasing function of E(DP), and Equation (4) implies that E(DP) is an increasing function of the underlying stock's unsystematic risk. It follows that if the unsystematic risk in the underlying stock increases without a concurrent decrease in systematic risk, [P.sub.s] and the value of the ESOs to employees will decrease. However, the increase in unsystematic risk increases volatility, and the Black-Scholes (or binomial) cost of an ESO with a given holding period is an increasing function of volatility. As a result, the decrease in the expected ESO cost from the shorter holding period (due to the lower [P.sub.s]) may be offset by the higher volatility used to compute the Black-Scholes (or binomial) cost for each holding period [t.sup.i].(a) Hence, the effect on the maximum ESO cost from an increase in volatility due to an increase in unsystematic risk is case-specific and cannot be generalized.(b)

2. The effects of an increase in volatility due to an increase in the underlying stock's systematic risk are also ambiguous. For ease of exposition, assume that the firm does not pay dividends, so that E(R) = E(G) and E(G)/E(DP) = E(R)/E(DP). Equation (4) implies that E(R)/E(DP) is an increasing function of the systematic risk in the underlying stock ([[Beta].sub.us]), and Equation (1) implies that [P.sub.s] is an increasing function of E(G)/E(DP). As a result, an increase in systematic risk will increase [P.sub.s] and volatility. However, given the higher expected rate of return, employees' holding period may decrease with an increase in systematic risk because it may take less time for the stock price to first exceed the higher [P.sub.s]. If the holding period decreases, the negative effect on the maximum ESO cost from the shorter holding period may be greater than the positive effect on the ESO cost from the increase in volatility. Hence, the effect on the maximum ESO cost from an increase in volatility due to an increase in systematic risk is case-specific and cannot be generalized.

3. Finally, to analyze the case where an increase in volatility represents an increase in both systematic and unsystematic risk, again assume that the firm does not pay dividends, so that E(R) = E(G). If E(R)/E(DP) is unchanged, the increase in the maximum ESO cost due to the increase in volatility may be offset by the shorter holding period that results from the increase in E(R). If E(R)/E(DP) is positively related to the increase in volatility, the increase in the maximum ESO cost due to the increases in volatility and [P.sub.s] may be offset by the shorter holding period resulting from the increase in E(R). Similarly, if E(R)/E(DP) (and [P.sub.s]) is negatively related to the increase in volatility, the decrease in the expected ESO cost that results from the decrease in the holding period due to the increase in E(R) may be offset by the increased volatility. Hence, there is no analytical relation between volatility per se, and the firm's ESO cost.

a Likewise, holding total risk (volatility) constant, an increase in unsystematic risk will lower [P.sub.s]. However, the accompanying decrease in systematic risk will also lower E(G), and it may take longer, on average, for the stock price to reach the lower [P.sub.s]. As a result, when holding total risk constant, an increase in unsystematic risk may either increase or decrease the length of time employees will hold their ESOs.

b Huddart (1994) and Kulatilaka and Marcus (1994) provide numerical examples of cases where volatility is negatively related to the firm's ESO cost. In their examples, the negative effect on the firm's ESO cost from the decrease in the holding period due to employees' risk-aversion is greater than positive effect from the higher volatility. However, their numerical examples are not generalizable.

Haim A. Mozes is Associate Professor of Accounting at Fordham University, Graduate School of Business Administration, New York, NY.

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Title Annotation: | includes appendices |
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Author: | Mozes, Haim A. |

Publication: | Financial Management |

Date: | Dec 22, 1995 |

Words: | 9579 |

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