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By Wanda A. Wallace

The disclosure requirements of SFAS No. 123, Accounting for Stock-Based Compensation, are effective for financial statements for fiscal years beginning after December 15, 1995. Directions for tracking information will be needed as entities apply that statement. An option pricing model choice will, at all times, require specification of a set of key inputs for valuation. These include--

* the risk-free rate,

* the volatility of future stock movements,

* the expected life of the option,

* the expected dividend yield,

* the exercise price, and

* the stock price.

Risk Free Rate

SFAS No. 123 has made selection of the risk-free rate simple, as it has prescribed that a zero coupon bond rate should be used that matches the term of the expected life of the option. While the selection is clearly dependent on the life specification, once that choice is made, reference to The Wall Street Journal (WSJ) or similar data sources at the date of grant can be made to select the input. To be specific, the Money & Investing section of the WSJ has a listing termed "Treasury Bonds, Notes & Bills" that includes a subheading of "U.S. Treasury Strips." The type denoted "bp" for the maturity that matches the expected life is the risk-free rate required by SFAS
No. 123.


The volatility of the future stock movements is to be measured as the annualized standard deviation of stock returns, appropriately adjusted for dividends, stock dividends, and stock splits (so that variability is not inflated by periodic payout practices or price adjustment due to changes in the denomination of the underlying stock instrument). You likely recall from your basic statistics course the concept of standard deviation, which is the square root of the variance. The variance is defined as the sum of squared deviations from the mean, adjusted for the number of observations, less one. Specifically, to find the standard deviation of four numbers: $4, $7, $8, $5, which generates a mean or average value of $6 ($24 divided by 4), first the differences of each value from the mean are determined, which would be: -2, 1, 2, -1. These differences are then squared to generate: 4, 1, 4, and 1. Then the sum of squared differences is 10, with the denominator equal to 4­1, leading to a variance of 10/3 or 3.33. The standard deviation is the square root of this measure or 1.82. This is a simple example of the computation that is to be applied to returns.

Returns simply means that stock prices are compared to compute the current price divided by the former price observation. The result of this ratio is then transformed to a natural logarithm before the standard deviation is computed. Natural logarithms are designated typically as Ln; such a computation is available on many calculators and within most spreadsheet software. The reasoning for the Ln transformation ties to empirical evidence regarding the distribution of stock returns and is considered the generally accepted form for data on which the standard deviation of returns computation is to be based. Note that the standard deviation formula is available in most spreadsheet programs, commonly short-handed in formula form as @stdev or some similar abbreviation.

Daily, Weekly, Monthly, or Annual? The question still remains as to which stock prices are to be input into the analysis of volatility, i.e., should we use daily, weekly, monthly, or annual prices? The prescription found in SFAS No. 123 is twofold. One prescription recognizes the advisability of having at least 30 data points on which to base an estimate, and the other prescription requires that the time frame over which historical volatility is measured be matched to the expected life of the option. This suggests that annual prices are likely to be of limited use, since 30 years of data, while fulfilling the first prescription, will fail to match the second. Hence, the choices will tend to be among daily, weekly, and monthly, and this choice is up to the estimator. One consideration discussed in SFAS No. 123 relates to whether the stock is thinly traded. If that is the case, it is unlikely that daily data is of use, and a choice of weekly or monthly would be advisable. Whichever data interval is selected, it is important that such observations be contiguous, in equal intervals, and that when the standard deviation is computed that it be appropriately adjusted to an annualized standard deviation. This simply means that the standard deviation computed is multiplied by the square root of 12 for monthly data, the square root of 52 for weekly data, or the square root of the number of trading days in the year for daily data, which is likely to approximate 260 observations.

Stock Dividends, Splits, and Dividend Payout Adjustments to Returns. For stock dividends, stock splits, and dividends, the approach is to effectively adjust the raw return data to make adjacent observations in apples-to-apples form. For example, if two adjacent observations were $70 and $30 due in part to a stock split's influence, the rate of return derived by comparing the former price to the latter price would set the $70 to equivalent terms of $35 before it is compared to the $30 value. It is apparent that the large volatility induced if the $70 and $30 value are compared would inflate the variability of the stock prices and lead to improper reflection of expected stock price movements.

Interdependence with Expected Life. The interesting facet of both the risk-free rate and the volatility estimates is that they are explicitly linked to the expected life of the option. This is the variable that perhaps is most distinctive from traditional applications of option pricing models. This is because employee stock options are commonly issued for longer terms than those of traded options, are subject to a vesting period, and are observed to be exercised earlier than traded options. Moreover, the rights of employees are not tradable, and the approach of SFAS No. 123 is to acknowledge such a difference by prescribing the use of expected life rather than the term to maturity or the actual life of the option. Keep in mind that the shorter the expected life of the option, the lower the value of an option, all things held constant.

Expected Life

By definition, the expected life must be at least the vesting term. However, this is an element of employee stock options on which entities need to begin to collect data to support their selection of inputs into future estimations. SFAS No. 123 expects that actual experience in the past will be considered and suggests that groups of homogeneous exercise experience be formed as a step in option valuation. The example provided suggests that middle managers may have a tendency to exercise quicker than top managers and, as a result, two pools of option grants could be formed, one of which is valued using the expected life for middle management grants and the second being valued using the longer life historically experienced for top management grants.

Expected Dividend Yield

The expected dividend yield is likewise expected to be computed from historical data by and large and for a period matching the expected life of the option. It is acknowledged that the theory calls for expected dividends, meaning that SFAS No. 123 expects publicly available information be considered as to future dividend policies and for such considerations to be applied to adjust historical practices. SFAS No. 123 describes some parties' concerns as to proprietary and legal considerations as estimates are based on expectations and the latter inputs are disclosed. However the role of such measures in the estimation process has already been made transparent in certain SEC associated disclosures and is described as no different from other disclosures that are common in such areas as expected returns and discount rates applied to pensions and other accounts involving estimations.

Exercise and Stock Prices

The exercise price and stock price are explicit in the instrument issued in most cases, which means the designers of the instrument will carefully select when they grant an option and what spread is permitted between the exercise price and stock price. One choice available is to index the exercise price or to set performance thresholds that must be achieved for the instrument to become active. Any uncertainties embedded in the instrument are largely subject to SFAS No. 5 on contingencies in determining the valuation inputs for SFAS No. 123.

SFAS No. 123 requires estimates of compensation cost for periods during which it is not possible to determine the fair value be based on the current intrinsic value of the award, determined in accordance with the terms that would apply if the option or similar instrument had been currently exercised--intrinsic value refers to the difference in the stock price and exercise price.

Limited Information Settings

All of the inputs prescribed are expected to largely reflect historical experiences; however, SFAS No. 123 emphasizes that conditions may arise where such historical quantities may need adjustment or may be unavailable. An obvious example is a newly traded entity that lacks stock price information for a period equivalent to the expected life of the option. SFAS No. 123 recognizes the propriety of forming a peer group on which to base estimates of each input. Research has also suggested that a combination of a peer group estimate and an historical entity-specific estimate, each weighing 50-50 can enhance the overall performance of the volatility estimate. This type of approach is referred to as a shrinkage forecast that achieves some control for a known phenomenon in volatility measures over time: mean reversion. This merely means that high volatility tends to decrease and low volatility tends to increase over time, meaning that some tempering of historical measures through weighing of peer group experiences can be helpful. The peer groups suggested by existing research would be based on industry and size.

Consideration of Critical Events

The possibility of a critical event influencing the applicability of historical information is likewise recognized in SFAS No. 123. Examples include the presence of a takeover bid or a large restructuring of an entity that could well be expected to influence price volatility and expectations for future price movements. SFAS No. 123 permits these events to be "set aside" in a manner that removes their unusual effects from historical patterns in formulating future projections.

This suggests that an information search of historical price movements could be warranted on an exception basis. In other words, all price movements of a single day that exceed an appropriate percentage threshold (e.g., 20%) should perhaps be examined
to consider whether a demonstrable unusual event is associated with such changes. Care must be taken in the approach to time-series-based computations when a contiguous set of observations is precluded due to such "key events." One approach frequently used in statistical software programs is to substitute for missing observations the mean value of all other observations. Of course, this does not solve the notion of cyclical, seasonal, or weekend effects known to arise in stock price patterns. An alternative that may be available if a sufficient number of observations are available pre and post the event is to form two estimates, pre and post, and combine them as a weighted average, based on the proportion of the option's expected life represented in each volatility

Nonpublic Entities

One other consideration merits mention and it relates to nonpublic entities. A nonpublic entity is defined in SFAS No. 123 as "An entity other than one (a) whose equity securities trade in a public market either on a stock exchange (domestic or foreign) or in the over-the-counter market, including securities quoted only locally or regionally, (b) that makes a filing with a regulatory agency in preparation for the sale of any class of equity securities in a public market, or (c) that is controlled by an entity covered by (a) or (b)." For this type of entity, SFAS No. 123 recognizes an added difficulty in computing volatility and therefore permits the omission of that input from the computation of option value. When an option pricing model computation is made with zero volatility, it is termed minimum value. Note that if a nonpublic entity believes it can derive an estimate of volatility, it is
permitted to report fair value in a parallel manner to that applied by a
public entity. *

Wanda A. Wallace, PhD, CMA, CIA, CPA, is The John N. Dalton Professor of Business Administration of The College of William and Mary.

Douglas R. Carmichael, PhD, CPA
Baruch College

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