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Valuation of Stock Options of Non-Publicly Traded Companies

William R. Cron, Central Michigan University
Randall B. Hayes, Central Michigan University

Introduction

Companies use stock options primarily for two purposes. Many companies, large and small, use stock options as a part of employee compensation packages. A second use, often employed by smaller companies to conserve cash, is as a means of payment for goods and services. But regardless of the purpose, both the issuer and the recipient need to have some idea of the value of the options being exchanged. The recipient needs to know how much he or she has been paid for the work done or the goods furnished. The issuer needs to know how much value has been given up in the transaction. Other parties frequently have a need to estimate the value of options. In a divorce case, courts often require a valuation of options held as part of a marital estate in order to arrive at an equitable distribution. In a similar fashion, probate courts often require a valuation of options held in the name of a decedent to accurately value the estate for an equitable distribution to heirs, and to properly determine the estate tax liability. Recent developments in financial accounting also require a valuation of stock options. If a firm seeks audited financial statements, Financial Accounting Statement 123 Revised (FAS 123R) mandates the firm account for its options using their fair value. The FASB requires this accounting of all firms, large or small.

Accepted techniques exist for valuing the options of publicly traded companies. In contrast, and despite the widespread use of stock options to compensate managers and to pay for services, almost nothing has been written on how to value the options of non-publicly traded companies. This paper addresses this deficiency. The paper first examines the history of accounting for options and discusses the Black-Scholes model often used to value the options of publicly traded companies. We then suggest how a useful model of firm valuation, the Gordon Growth model, can be used to estimate the stock price and volatility variables necessary to apply the Black-Scholes model to non-publicly traded companies. A final section provides an example of how to use the model in practice.

Our aim is to suggest a reasonable way to estimate the value of the options of non-publicly traded companies, and by doing so, significantly improve the measurement techniques for the options of non-publicly traded companies. If we can increase the credibility of option valuations, this will reduce the uncertainty accountants, analysts, and companies encounter in this area. A better measurement technique will improve the reliability of the financial statements of non-publicly traded companies that currently issue options. Better financial reports will enable these companies to better compete with publicly traded companies for debt and equity capital.

In addition, a better valuation methodology will make options a more attractive element in the executive compensation packages of companies that do not currently use stock options. These companies will have an additional mechanism to compensate managers that does not consume their cash resources. They would also be able to develop compensation packages that enable them to better compete with the packages offered by publicly traded companies. As a number of authors note (Todd 1991; Hackett and McDermott 1999; Geer 1997), a chronic problem confronting non-publicly traded companies is developing executive pay packages that enable them to compete with publicly traded companies in attracting and retaining managerial talent.

A History of Accounting for Options

For many years, the accounting for employee stock options followed the intrinsic value rules of Opinion 25 of the now-defunct Accounting Principles Board. The intrinsic value approach valued options by calculating the difference between the market price of the stock and the exercise price of the option at the date of issue. The usual result was that no value was assigned to a companys options since most option contracts set the exercise price either at or slightly above the market price at the date of the grant. With no value assigned to the option, no compensation cost was recorded when options were awarded to employees, or if the options were issued for goods and services, no cost was associated with the transaction. The intrinsic value method systematically understated the values that were changing hands. The method effectively ignored the real value of an option, which lies in the holders ability to buy the issuing companys stock at a fixed price for some period of time in the future. This ability is the latent forward contract component of the option, and it is what makes the option valuable to the holder (Balsam 1994; Cron and Hayes 2004). Throughout the 1980s and 1990s, options became a more important part of many compensation packages, and the systematic bias in their valuation caused a crescendoof criticism from the financial analyst community. In response, the Financial Accounting Standards Board (FASB) issued an exposure draft that proposed that the value of the options should be measured using a fair value approach. The FASB believed that valuation theory and practice had progressed far enough to enable accountants to reach reasonable estimates of the value of options issued by publicly held companies. The FASB proposed that auditors estimate the value of any options issued by a company and record a commensurate compensation expense. While the FASB did not endorse any particular valuation method that auditors should use, the board did mention the widespread acceptance of the Black-Scholes method by the financial community (FAS 123R).

This time the criticism came from the companies that issued options (Mayer 2002). Many of these companies believed that a requirement to recognize an expense in connection with the issuance of stock options, and thereby report lower profits, would penalize them in their efforts to raise capital. The intense political pressure exerted on the FASB, and the lack of support by the Securities and Exchange Commission, caused the FASB to relent. The board issued FAS 123, which allowed the use of either the fair value or the intrinsic value method. When a company elects the intrinsic value method, however, FAS 123 required companies to present pro-forma income amounts calculated using the fair value method. FAS 123 was a compromise, but few people were pleased. Recent accounting scandals, fueled by overly generous stock option grants, have re-focused attention on the issue. As a result, the FASB issued a revision of FAS 123 (FAS 123R), which mandates the use of the fair value method for the valuation of stock options.

Options of Non-Publicly Traded Companies

The unhappy experience of the FASB indicates that options occupy an important role in the financial evaluation of companies. Virtually all the attention, however, has been focused on the valuation of options issued by companies with publicly traded stock. What about the valuation of options from companies whose stock is not publicly traded? Options often occupy an important role in the compensation packages of non-publicly traded firms. Often, non-publicly traded companies are chronically short of cash as they try to finance the growth in capital assets, inventories, and employee rolls. One way to conserve precious cash is to pay employees with stock options. Indeed, stock options often play a central role in the business plans of start-up companies where they constitute the principal means to acquire employee services without paying cash (Geer 1997; Hackett and McDermott 1999; Todd 1991). In addition, non-publicly traded companies often begin to include options in their compensation packages after they become established and foresee a public offering of their stock on the horizon (Todd 1991).

Banks, venture capitalists, and potential equity investors all have a need for information in regard to options in order to perform reliable financial analyses. As mentioned earlier, probate and family courts have similar needs. The models that the FASB relies on for fair value option valuations, however, require information about the behavior of stock prices on major exchanges. The Black-Sholes method, for example, not only requires information about common stock prices, it also requires estimates of the volatility of stock prices. No such information exists for non-publicly traded firms. So what is an accountant of a non-publicly traded company to do? Non-publicly traded companies still need to issue certified financial statements to satisfy their creditors and other parties entitled to reliable financial information. Reliable accounting information is also required by employees who already have equity stakes and by those employees who may be offered equity stakes. Unfortunately for the accountant in these circumstances, the valuation literature provides little guidance. Early research focused on an econometric approach to value options (Shelton 1967; Samuelson 1965; Giguere 1958; Kassouf 1968). This approach attempts to determine an options value based on a statistical analysis of the behavior of the prices of publicly traded options relative to several financial attributes of the companies issuing the securities. The statistical technique uses multiple regression on several companies at various points in time. Later research focused on developing a more theoretical approach. The Black-Scholes model and the binomial model are examples of these efforts. Like the econometric approaches, these models focus on publicly traded companies.

Our thesis is that since accepted valuation techniques exist for the options of publicly traded firms, it makes sense to try to estimate the terms in these models when valuing the options of non-publicly traded companies. In the following sections, we discuss ways to estimate the terms of the Black-Sholes model, a model that has achieved legitimacy in the financial literature and is extensively utilized by publicly traded firms. Our technique uses an income capitalization approach to estimate the value and price volatility of a non-publicly traded companys stock. The estimated terms are then adjusted for factors specific to non-publicly traded companies. Specifically, we incorporate discounts for lack of marketability and lack of control into both the stock price and volatility estimates.

The Black-Scholes Model

The Black-Scholes model, developed in1972, was one of the first attempts to value stock options using the behavior of stock prices in the equity markets. The model has been rigorously tested over the thirty years since its development, and while not perfect, the model has achieved widespread acceptance by both financial analysts and accounting regulators. In the accounting arena, FAS 123 and 123R identifies Black-Scholes as one of the accepted methods for valuing stock options. Recent discussions between the FASB, the International Accounting Standards Board, and a group of valuation experts also specifically identify this model as an acceptable approach.

While the mathematical form appears complex, the variables that comprise the terms of the model are not (Damodaran 2003). The model holds that the value of an option is a function of only six variables: the current price of the stock, the exercise price of the option, the time to maturity, the standard deviation of stock price, the stocks dividend yield, and the risk-free rate of return. That these six variables would govern the value of an option makes intuitive sense. The risk-free rate specifies the rate offered by securities with a zero risk of default, and this rate specifies a baseline rate of return that all risky financial assets (like a stock option) must exceed. The current stock price and its dividend yield represent measures of the cash cost to buy the stock and the periodic rate of return the stockholder can expect. The standard deviation of the stock price measures its volatility, and when considered in conjunction with the current stock price, helps evaluate the likelihood that the stock price will exceed the exercise price of the option. The Black-Scholes formula for the valuation of the option (V) is:

V = SeytN(d1) PertN(d2)(1)

Once d1 and d2 are calculated, the cumulative probabilities of achieving these values, N(d1) and N(d2), are determined from a standardized normal distribution. An analyst can derive these probabilities using almost any spreadsheet program.

The Black-Scholes model represented a breakthrough because now an option could be valued directly from readily available data. The only measurement that represented a challenge to the analyst was the standard deviation. Computing this value required a time series of price data, and prior to the advent of databases from the Center for Research in Security Prices (the CRSP tapes), analysts had to compute the measurement manually. The CRSP tapes made the calculations easy, and in the mid-1970s analysts began to follow option prices just like they followed stock prices.

With a way of estimating their value, options became a popular form of compensation to employees. Companies could provide an estimated value of an option grant, thereby giving employees a better idea of the amount of their compensation.

Of course, all this was true for publicly traded companies. The Black-Scholes model requires measures of the stock price and its standard deviation through time. Such data does not exist for non-publicly traded companies. Does that mean that Black-Sholes cannot be used for the stock options of non-publicly traded companies? Or can the missing factors be estimated using other sources of information? We know that a minority interest in the shares of non-publicly traded companies has value in spite of the lack of a ready market for these shares. Business appraisers have developed techniques for the valuation of the shares of these companies and, by applying discounts for the lack of marketability and control, have arrived at an estimated value for a minority interest in these shares. Our hypothesis is that the inputs used in the Black-Scholes model can be estimated with enough precision to provide a reliable estimate for options written on these companies. The following section describes the process. We first describe how to estimate the time to maturity and the risk-free rate. We then use an accepted earnings capitalization model to develop an estimation procedure for both the stock price and the volatility of the stock price. We then show how adjustments for lack of marketability and for lack of control can be incorporated into the analysis. After this discussion, we present an example of how the estimation procedure can be used in practice to value the options of non-publicly traded companies.

Estimating Variables in the Black-Scholes Model

Estimating the Time to Maturity and the Risk-Free Rate

For publicly traded stock options, the expectation is that the options will not be exercised until just prior to their expiration. When employees hold the options, however, experience indicates that the options are often exercised early. For this reason, it makes sense to have the variable for time (t) represent the time to expected exercise and not the time to expiration. With this adjustment, the risk-free rate of return® should represent the rate on Treasury securities with a maturity date equal to the date of expected exercise, i.e., if the option is expected to be exercised in five years, the rate on five-year Treasury notes should be used for the risk-free rate.

Estimating the Stock Price

Analysts frequently use the earnings capitalization model, sometimes referred to as the Gordon Growth Model, in the valuation of firms. The basic single stage version of the model is applicable to mature firms that have already transitioned from initially high and sporadic annual growth rates to growth rates that are reasonably constant (Pratt, Reilly, and Schweihs 2000). The analyst can also apply the model to other types of firms by simply expanding it. The expanded model is applicable to firms that transition to lower, more stable growth rates following an initial high growth rate stage. Scenarios that call for the expanded two-stage model include situations where patent rights or barriers to entry allow for high initial growth rates and more normal growth rates when the patents expire or new firms enter the industry. These are situations typical for start-up companies (Damodaran 2003). The analyst can construct a three-stage version of the model for more complicated situations. While our analysis focuses on the single stage version of the model, the methodology we utilize is equally applicable to multiple stage models.

The basic single stage version of the Gordon Growth Model posits that the firms estimated value per share (Se) is the product of next periods earnings per share (E1) times the firms earnings capitalization rate©:

Se = E1 x C (2)

Studying the components of the earnings capitalization rate reveals that it is the reciprocal of the difference between the firms cost of equity capital (k) and its expected future growth rate (g):

C = 1 / (k g)

The expected growth rate is driven by the firms reinvestment rate of earnings and its return on capital or:

g = reinvestment rate x return on capital

Both the reinvestment rate and the return on capital can be calculated from the firms financial statements. Since we are dealing with a mature firm, a reasonable assumption is that these measures will continue unchanged into the future. With this assumption, next years earnings per share, E1, is estimated by simply multiplying the current years earnings, E0, by (1 + g).

The other component of the capitalization rate, the cost of equity (k), is driven by several factors. Specifically, the cost of equity consists of a risk free rate and an equity risk premium that compensates stockholders for the uncertainty of future returns. Analysts frequently use the rate on 20-year treasury obligations to represent the risk-free rate. The interest rate on these securities has been relatively constant in recent years; the time series has varied by only a couple of percentage points over the past decade.

The other component of the cost of equity, the equity risk premium, is composed of three parts: a basic equity risk premium, a size premium, and a premium for risk factors unique to the individual firm. Analysts often measure the basic equity risk premium by the historical difference between the market rate of return on equities and the risk-free rate. Ibbotson Associates (2005) estimates this difference to be historically about 7 to 8 percent. Valuation texts suggest the equity risk premium should be multiplied by the firms beta. For non-publicly traded companies, of course, the beta is unavailable but that does not mean the beta adjustment should be ignored. A reasonable starting position is to assume a beta of one, and then adjust this value for firm-specific factors that would increase or decrease the measure. For example, a high debt to equity ratio would indicate that the beta adjustment should be higher than one. In addition, membership in a stable industry where the betas of the publicly traded members are less than one would indicate a lower beta adjustment.

The second component of the cost of equity, the size premium, is an attempt to incorporate a factor for the increased risk of small companies. Ibbotson Associates (2005) estimates a factor for each of three size categories, ranging from .5 for mid-sized companies to 2.6 for smaller companies. Like long-term interest rates, the size factors appear to have been relatively constant over the last ten years.

The final component is the unsystematic risk factor. There is no accepted model to estimate the factor, but there is wide agreement that analysts should add some amount to the equity risk premium to reflect risk characteristics specific to the company (Pratt, Reilly and Schweihs 2000). The factor can be either positive or negative. For example, a high turnover of management for a firm would indicate a positive unsystematic risk adjustment to its cost of capital. A company with large purchase contracts that provide a stable source of revenue would indicate a negative adjustment. The analyst can determine with some reliability whether the adjustment is positive or negative through a benchmarking analysis of comparable firms. The absolute size of the adjustment, however, is something that will involve a subjective evaluation by the analyst. In most cases, the absolute size of the adjustment will be small, but larger adjustments may be justified in unusual situations where the level of uncertainty is a good deal higher or lower than comparable firms in the companys industry.

Adjusting for Lack of Marketability and Control

The classic income capitalization model, expressed in equation (2), provides a reasonable estimate of the stock price of a publicly traded company when the stock is held by a controlling interest. In an effort to value the stock of a non-publicly traded company held by a non-controlling interest, the valuation formula needs to be adjusted for two additional factors.

Lack of Marketability. The expense and time involved in finding buyers, and the risk of not finding a buyer at all, reduces the value of non-publicly traded stock. For an analyst, the major issue is estimating the size of the discount to apply to the firms valuation. Evidence on the size of the discount comes primarily from three sources. Transactions in restricted shares of stock are one source. Letter stocks represent a typical type of restricted stock. Companies often issue letter stocks to avoid the registration costs of a public offering. The purchaser of these shares is restricted from selling them for a specified period, often twenty-four months. A number of studies analyzing the size of discount applicable to letter stocks have been conducted, and most studies estimate the lack-of-marketability penalty to be somewhere between 30 and 40 percent (Pratt, Reilly and Schweihs 2000).

A second source of evidence comes from analyses of private transactions occurring prior to an initial public offering. These studies focus on a comparison between the prices for freely tradable shares and restricted, but otherwise identical, shares of stock. These studies indicate a discount approximately 10 percent higher than the letter stock studies, or somewhere between 33 and 45 percent.

Finally, Trout (2003) investigates the differences in put prices on long-term equity anticipation securities (sometimes called LEAPS). Trout concludes that the minimum discount for marketability is approximately 24 percent.

These studies suggest a range to begin with in the estimation of the discount for lack of marketability. The selection of a particular value for the discount will depend on whether any evidence exists regarding possible external buyers for the shares, such as the existence of an ESOP or the potential for a public offering in the future. The number of shares involved in the option contract will also affect the exact size of the discount (small blocks are easier to sell than large blocks), as will any restrictions on transfer of shares (e.g., the shares may only be sold to certain family members).

Lack of Control. A discount for lack of control also needs to be determined. The income capitalization model of equation (2) produces an estimated value for a controlling interest. Control of a company entitles the controlling shareholder to a number of valuable prerogatives, such as the ability to appoint members of the board of directors, declare dividends, and determine management compensation. Minority interests lack these prerogatives, and this deficiency makes their stock worth less. Again, determining the exact amount of the discount is problematical. Contrary to the lack of marketability discount, there is not a body of empirical evidence available to provide an initial starting point for a lack of control discount. The available evidence concerns discounts from proportionate shares of book value in sales of minority interests in limited partnerships. The evidence suggests that the discount for a minority interest is between 20 and 30 percent (Pratt, Reilly and Schweihs 2000).

Adding discounts for marketability (dm) and control (dc) to Equation (2) yields: Se = E1 x C x (1 – dm) x (1 – dc) (3)

This formula, we believe, represents a reasonable approach to estimating a surrogate for the share price term (S) in the Black-Scholes model of equation (1).

Estimating Volatility

The FASB Approach. The volatility, or standard deviation, of the stock price is a key variable in the Black-Sholes model. In FAS 123 and FAS 123R, the FASB suggests three ways to estimate volatility for non-publicly traded stock. One alternative they propose is to find a company that is similar to the non-publicly traded company, and then use the volatility measure of the similar companys stock as the volatility measure for the stock of the non-publicly traded firm. Their second proposal is to compute the volatility of an index of stock prices for a similar industry, and then use this as the volatility measure of the stock of the non-publicly traded company. The third alternative, suggested in FAS 123, is simply to assume zero volatility for the stock price of the non-publicly traded company.

Each of the alternatives has deficiencies. Identifying similar publicly traded companies is something that sounds easy in theory but is difficult in practice. While the analyst may find it possible to identify companies that face similar risk factors (i.e., operate in the same markets), it is almost impossible to identify companies whose operating dynamics and financing practices, the very characteristics that determine value, are the same. There always will be differences, and they usually are substantial. Moreover, the publicly traded company normally will be larger in sales, assets, and capitalization. As discussed earlier, a firms size reduces its cost of capital, and that will affect not only the valuation of the firm but also the volatility of that value. As such, differences in size alone mean the volatility measure of the publicly traded company cannot be readily assigned to the non-publicly traded company.

The second alternative proposed by the FASB suggests using the variance of an index of stock prices. The comparability factor is even more difficult here. The index for a similar industry will represent a mix of different dynamics, and determining what this mix is will be hard enough. For example, if some companies in the index rely on capital leases, while other companies rely on operating leases, does the index of stock prices represent the stock of a mythical company that relies on both operating and capital leases? Or is the effect of lease financing on the stock price washed out? Moreover, without going into the mathematics of the issue, determining the average variance for an index from a distribution of variances from a number of different companies is fraught with problems. Should the variances be weighted by the size of the companies, or should some other weighting be used? Because the variances are bounded by one and have no upward limit, the values will be distributed in a lognormal fashion. This being the case, should we use the arithmetic or geometric mean as the measure of central tendency? Or should we use the median, or some other measure? Finally, even if the analyst can find some index with which he or she is reasonably comfortable, the stock prices of the index, or the index itself, should be adjusted for the lack of marketability and the lack of control of the non-publicly traded stock. How should these discounts be applied? Should it be applied to the individual stock prices of the companies in the index, or to the index as a whole? The choice will make a difference, but it isnt clear which way it should be done.

The third suggestion of FAS 123 is to assume a zero volatility measure in the option valuation formula. From a valuation perspective, this is perhaps the worst alternative. An important component of the value of an option comes from the variance of the stock price, i.e., the variance helps determine how likely it will be that the stock price will at sometime exceed the exercise price. With a zero variance assumed, the statistical likelihood of the stock price exceeding the exercise price will also be zero. Using the assume a zero variance approach will seriously, and consistently, understate the true value of an option. The estimated value of the option will always be biased below its true value, and usually not by a small amount.

An Alternative Approach to Estimating Volatility

None of the approaches suggested by the FASB represents a good solution to the problem. We believe a better solution is to use equation (3) to derive an estimate of the standard deviation of the stock price (?Se). Assuming the capitalization rate and the discounts for marketability and control are constant through time, the formula for the standard deviation of the stock price is:

?Se = ?E x C x (1 – dm) x (1 – dc) (4)

With the exception of the estimate of the unsystematic risk component of the cost of capital, all the terms in equations (3) and (4) can be estimated using the financial statements of the company or published sources. The data are essentially specific to the company. No data are borrowed from other companies or indexes. We believe this is an important attribute of the procedure. By using data either derived or specific to the non-publicly traded firm, the uncertainty of the valuation is reduced. There is no worry about the comparability of other publicly traded firms or the applicability of external indexes of volatility. There will still be questions about valuation. For example: Has the growth rate been correctly estimated? Do the reported earnings fairly represent a measure of income? Has the equity risk premium been reasonably stable over time? These questions, however, concern the company itself and not whether the right data have been borrowed from other sources.

There inevitably will be situations where the valuation analyst is uncertain whether he or she has reasonably accurate answers to the above questions. In those instances, it would be wise to use the estimating procedure of FAS 123R as a reality check. If the valuation using our procedure is reasonably close to the valuation obtained by following FAS 123R, it seems reasonable to conclude that an appropriate valuation has been obtained. If the valuations are different, however, it will be incumbent on the analyst to delve deeper and find out why. Wisely, the FASB gave accountants wide latitude in their selection of valuation techniques for options. The analyst has the freedom to choose which method he or she believes yields the best estimate, or the analyst can elect to yse an average of the two valuations. All FASB requires is a well-considered approach to the valuation questions.

An Example

Table 1 presents the data to calculate the estimated stock price (Se) and the standard deviation of the estimated stock price (?Se) for an example company. The table also lists where the analyst can obtain the data to make the estimates. Inserting the values in Table 1 into equations (4) and (5) yields the following estimated values:

S = $10.62

?s = 2.83

Assuming the option contract specifies that the exercise price equals the estimated current stock price (P = $10.62), the time to exercise is five years (t = 5), and the dividend yield is 3 percent (y = .03), inserting Se and ?Se into the Black-Scholes model of equation (1) yields an estimate the value of the option of:

C = $9.13

As mentioned above, most of the data to derive this estimate came from the companys financial statements or readily available reference material. The only estimates the analyst would be required to make are those for the companys beta, the appropriate discounts for marketability and control, and the value for the unsystematic risk appropriate for the circumstances. Analysts commonly make all these estimates in their valuation practices.

Conclusion

Many companies have decided that it makes sense to share equity ownership with their managers. A favored way to accomplish this is through stock options. Unlike the years before FAS 123R, however, companies now must estimate and disclose the value of their options in their audited financial statements. This has left many smaller, non-publicly traded companies that have issued stock options in a difficult situation. The FASB requires a valuation for their options, but existing valuation models require extensive stock price data, something non-publicly traded companies definitely lack. What are these companies to do? For other non-publicly traded companies that have not issued options, the requirements of FAS 123R act as an impediment to using options to compensate and reward managers. Issuing options means disclosing suspect valuations, and perhaps penalizing the perceived integrity of their financial statements. With no well-accepted alternative valuation approach, many non-publicly traded companies sensibly shy away from using options in their management compensation packages.

This paper presents the needed alternative. We borrow the Black-Scholes model and show how to develop surrogates for its variables. Developing the surrogates isnt difficult. For the most part, the necessary data comes from publicly available sources and the companys own financial statements. The approach isnt perfect but it is good enough to yield reasonable estimates of an options value. With a reasonable valuation approach, companies should find it easier to use stock options as a means of rewarding managers with equity ownership.

Notes

The Black-Scholes model is the most widely used approach to value options. While it is rarely used, FAS 123R suggests that using the binomial model might be a more flexible approach.

FAS 123R suggests both of these adjustments, although the FASB uses the re-definition of t to also adjust the employee stock options value for its lack of marketability. We believe the adjustment for lack of marketability should be done in a direct fashion, and a later section of the paper describes our adjustment procedure.

The firms estimated value can be expressed as follows:

m

Se = E0 ? (1+g1)i / (1+k)i +

i=1

[E0(1+g1)m(1+g2) / (k-g2)]/(1+k)m

The first term is the discounted value of the earnings during the first m years, when earnings are growing at the rate of g1. The next term represents the discounted value of the earnings for period m+1 to infinity. Once earnings have grown at the rate of g1 for m periods, they will continue to grow at the rate of g2 from period m+1 in perpetuity. Factoring out E0 yields:

m

Se = E0 { ? (1+g1)i / (1+k)i +

i=1

[(1+g1)m(1+g2) / (k-g2)]/(1+k)m }

The fundamental point is that the value of a firm is still the product of the initial income (the only variable) and the constant expression within the braces. Examining this equation also demonstrates that the only difference between a single stage model and a two-stage model is the complexity of the constant expression.

In FAS 123R, the FASB specifically states that a discount for lack of marketability should not be applied. The resolution of the apparent conflict of this prohibition with a discount for lack of marketability we are using lies in the level at which it is applied. The prohibition that the FASB is referring to applies to the reduction in the value of the options because they cannot be sold or transferred to another individual. The discount we are applying is at the stock valuation level. Our model directly values the firms shares. Once the firms stock price is adjusted for their lack of marketability, the adjusted price is used to calculate the value of an option. The option value would not need to be discounted further as specified in FAS 123R.

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