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Dec 1994

Stock-based compensation: estimating option values. (Accounting)

by Franz, Diana R.

    Abstract- The controversial, FASB-issued exposure draft called 'Accounting for Stock-Based Compensation' requires firms to value stock options at market value. These would have to be valued with the use of a mathematical model because there is actually no actual market for these options. The exposure draft allows appraisers to value the options through theoretically acceptable valuation models, specifically the Black-Scholes model and the Cox-Ross-Rubinstein binomial model. Although preference for any of these two valuation models is found to have no material effect on the estimate of the option's value, the models become problematic when one considers that they both use variables that require estimates of the future, particularly the volatility of the stock price and the expected dividend yield. There is a risk that firms will be tempted to compute for these variables to manipulate the estimated value of the options.

Originally, the exposure draft mandated disclosure of stock-based compensation for 1994 financial statements. Recognition of expense would be required beginning with 1997 financial statements. In June of this year, the FASB agreed to delay the disclosure requirements for one year. However, they did not delay the recognition provisions. The final statement is expected to be issued by March 31, 1995.

One criticism of the proposed statement, the one endorsed by AcSEC, is that the value of employee stock options cannot be measured with any reasonable degree of accuracy. According to this view, the FASB's reliance on sophisticated option-pricing models is misplaced. The models were developed for publicly traded options that expire in a matter of months. Employee options typically run for years. Critics argue that the estimates needed for the models may be nearly on target for a few months but could be extremely inaccurate over several years.

A discussion of the following three topics is presented to assist in the evaluation of the merits of this criticism:

* A review of option-pricing models in use today.

* A hypothetical example that examines the sensitivity of the models to error ratio in the underlying variables.

* A real-world example to see how the proposed rules would work in practice.

Current Option-Pricing Models

The FASB's exposure draft permits options to be valued using any theoretically sound valuation model. Two models are specifically mentioned: the Black-Scholes model and the Cox-Ross-Rubinstein binomial model. For a stock that pays dividends, the original Black-Scholes model must be modified to include dividend yield. The Cox-Ross-Rubinstein model achieves a closer approximation to the option's true value when the stock pays dividends.

Both models estimate an option's value using these variables:

* The current price of the stock,

* The volatility of the stock price,

* The option's exercise price,

* The length of time until the option expires,

* The risk-free interest rate, and

* The stock's expected dividend yield.

The following example uses data from Appendix B of the exposure draft. It illustrates that both models produce similar estimates of an option's value. In paragraph 197 of the exposure draft, an option with the following characteristics is presented:

Current stock price $50 Option exercise price $50 Risk-free interest rate 6.5% Time to expiration (in years) 6 Expected volatility of stock 30% Expected dividend yield 1.5%

Using the Black-Scholes model, the FASB calculated the value of the option to be $18.02. We used the Cox-Ross-Rubinstein model and calculated a value of $17.99. The three-cent difference illustrates that the valuation models, while not identical, give very similar results.

The two methods diverge significantly only for options in which the current stock price is much greater than, or much less than, the exercise price, i.e., deep in-the-money and deep out-of-money options. As a practical matter, most employee stock options are at-the-money options, where the exercise price equals the stock price when the options are issued. Therefore, the choice of valuation models does not materially affect the estimate of an option's value. We used the Cox- Ross-Rubinstein model to compute all of the option values shown in the following examples.

Most of the variables included in option-pricing models are known at the time the options are issued. However, two of the variables, volatility and dividend yield, require estimates of the future. How do errors in either of these estimates affect the computed value of an option?

What if the Estimate of Volatility Is Wrong?

Volatility is a measure of the propensity of a stock price to undergo large changes. Some stocks experience wide swings, while others trade in a fairly narrow range. A stock's volatility is measured by the standard deviation of its return. If a stock's current price is $100 and its annualized standard deviation of return is 20%, then there is a two- thirds probability that the price a year from now will be within the range $120 ($100 x 120%) to $83 ($100/120%). The larger the standard deviation, the greater the range of likely prices.

Although each stock is different, some generalizations about volatility can be made. First, volatility can change over time, but the changes tend to be temporary. A stock that has recently experienced a change in volatility is likely to revert to its long-run average. This will be true as long as the company has not made fundamental changes, such as its lines of business or capital structure. Second, if volatility is estimated over a year or longer, large changes are unlikely. Estimates 10% above or below the long-run average are common, but estimates as high as double or as low as half of the long-run average are rare. Third, volatility of an individual stock is affected by overall market forces, such as interest rates and general economic conditions. The volatilities of stocks tend to move up and down together, although firm- specific factors (such as the firm's asset mix or its level of debt) can counteract or accentuate market forces.

Theoretically, the measure of volatility needed for an option-pricing model is the volatility that the stock will experience over the life of the option. Of course, this is unknown at the time the option is issued. The best that can be done is to measure past volatility and adjust the measurement to account for expected future changes in circumstances. One of the foremost criticisms of the FASB's exposure draft is that volatility cannot be reliably estimated over the decade-long life of a typical employee stock option.

Exhibit 1 shows that an option's estimated value increases as the volatility of the stock increases. This relation holds true for all options. The reason is that volatility affects the upside potential of a stock as well as downside risk. Option holders participate fully in the stock's upside potential. If the option is in the money at the end of its term, further increases in price are enjoyed as much by option holders as by stockholders. But option holders do not participate fully in the downside risk of a stock. If the option is out of the money at the end of its term, it is unimportant to the option holder how far out of the money it is.

For purposes of this example, we have assumed that the "true" level of volatility is 30%. Therefore, the correct value of the option is $17.99. But, what if the estimate of volatility is 10% too high, or 10% too low? What if the estimate is double its true value, or half its true value?

Exhibit 1 shows that if the option period is six years and the estimate of volatility is 10% too low or too high, the estimate of the option's value will be correspondingly 5.6% too low or too high. If volatility is estimated at half or twice what it should be, the estimate of option value will be correspondingly 27% too low or 53% too high.

The example continues with a 10-year option and demonstrates that the relative effect of mismeasuring volatility is less for options with longer terms to expiration. For the 10-year option, a 10% error in the estimate of volatility results in only a 4.5% error in the value of the option, compared with the 5.6% valuation error for the six-year option.

In summary, errors caused by the mis-estimation of volatility are not trivial; but they are not as large as the FASB's most strident critics have feared. The proportional error in the estimated value of an option is smaller than the proportional error in projected volatility that caused it. Moreover, the importance of an error in the estimate of volatility diminishes with longer-lived options.

What if the Measure of Dividend Yield Is Wrong?

Dividend yield is defined as the annual dividend payout divided by the average price of the stock. Management has more control over dividend yield than volatility; at least the numerator (dividend payout) is controllable even though the denominator (stock price) is not. A constant dividend yield is assumed by both the modified Black-Scholes model used in the FASB's exposure draft and the Cox-Ross-Rubinstein model we used.

Exhibit 2 shoves that an option's value decreases as dividend yield increases. The reason is that option holders do not receive dividends on their options. They only benefit from the stock's appreciation. If more of the total return on the stock takes the form of current dividends, less is left over for appreciation.

In Exhibit 2, we set the "true" dividend yield at 1.5%, varied it from half to double the "true" yield, and calculated option values. With an option with a six-year period, if the estimate of dividend yield is 10% too low, the option value will be 1.7% too high. If it is 10% too high, the option value will be 1.7% too low.

The relative effect of mismeasuring dividend yield grows larger with increases in the life of the option. For a 10-year option, a 10% error in dividend yield results in a 2.4% error in the value of the option.

A Real-World Example

It is useful to consider a real-world example to illustrate how the FASB's proposal would work in practice. We have chosen Chrysler Corporation's grant of stock options to Lee Iacocca in November 1978. This example has a number of characteristics that make it interesting:

* It would have been especially difficult to project Chrysler's volatility because of the troubles the company was facing at the time.

* The company's dividend policy was not constant during the term of the options. In fact, the dividend was eliminated entirely during a portion of the option's term.

* The full ten-year term of the options has elapsed, making it possible to compare estimates of volatility and dividend yield with actual experience.

Lee Iacocca was named president of Chrysler Corporation on November 2, 1978. At that time, he received options on 400,000 shares of Chrysler stock. The options had a ten-year term and an exercise price of $11.07. The options did not vest immediately, but for purposes of this illustration, we have assumed that they did. The closing price of Chrysler's stock on November 2, 1978 was $10.875. Ten-year Treasury Notes auctioned on November 8, 1978 yielded 8.85%.

Both the volatility and dividend yield of Chrysler's stock varied significantly during the 15-year period covered, five before the options were granted and 10 after. Volatility was quite high in the early 80's, but eventually settled back to its long-run average of about 40%. Dividend yield fluctuated erratically until dividends were eliminated in 1980. When dividends resumed in 1983, the yield was relatively stable.













































The variables used in the option-pricing model are--

Current stock price $10.875 Option exercise price $11.07 Risk-free interest rate 8.85% Time to expiration 10 years

We estimated volatility and dividend yield over three periods: five years, two years, and one year before the options were issued. We also calculated the actual average volatility and dividend yield over the term of the options. The results were:











If the FASB's proposal had been in effect in 1978, an accountant who knew Chrysler's volatility and dividend yield in advance would have recorded compensation expense in the amount of $2,540,000 (400,000 x $6.35). Compensation, based on five-year, two-year, and one-year estimates of volatility and dividend yield, would have been $1,496,000, $1,008,000, and $880,000 respectively.

There is, of course, no reason volatility and dividend yield must be estimated in the same way. An accountant in 1978 might have reasoned as follows: Volatility tends to revert over time to its long-run average. Therefore, the five-year estimate of 41.13% is preferable to the estimates based on shorter periods, so long as the company has not fundamentally changed. As for dividend yield, the company cut its dividend from $0.25 per quarter to $0.10 per quarter beginning with the fourth quarter of 1978. Annualizing that dividend payout, the stock's dividend yield is $0.40 / $10.875 = 3.68%. Given volatility of 41.13% and a dividend yield of 3.68%, the value of one option is $4.51 and the charge to compensation expense would have been $1,804,000.

As a percentage of Chrysler's 1978 pre-tax loss of $286 million, these amounts do not appear to be material. However, compensation is often a sensitive issue to shareholders and other financial statement users, so thresholds of materiality for compensation may be at lower levels than for other operating expenses. Also, we have examined options granted to a single individual. In many firms, the use of stock options is so prevalent that these differences in estimates would be considered material by any standard.

Estimation is Nothing New

It is nothing new for the application of accounting standards to involve estimation. Loan loss reserves and post-employment benefit accruals contain a potential for much larger estimation errors. There is a risk, however, that estimates of volatility and dividend yield will be selected, not for their intrinsic validity, but to manipulate the estimated value of the options. There will generally be a downward bias, since most managements would prefer to understate their compensation rather than overstate it. But the bias will never be as great as it is under current rules, where the options are valued at zero.

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