Encyclopedia of Financial Models, Volume II

By Frank J. Fabozzi

Volume 2 of the Encyclopedia of Financial Models

The need for serious coverage of financial modeling has never been greater, especially with the size, diversity, and efficiency of modern capital markets. With this in mind, the Encyclopedia of Financial Models has been created to help a broad spectrum of individuals—ranging from finance professionals to academics and students—understand financial modeling and make use of the various models currently available.

Incorporating timely research and in-depth analysis, Volume 2 of the Encyclopedia of Financial Models covers both established and cutting-edge models and discusses their real-world applications. Edited by Frank Fabozzi, this volume includes contributions from global financial experts as well as academics with extensive consulting experience in this field. Organized alphabetically by category, this reliable resource consists of forty-four informative entries and provides readers with a balanced understanding of today's dynamic world of financial modeling.

• Volume 2 explores Equity Models and Valuation, Factor Models for Portfolio Construction, Financial Econometrics, Financial Modeling Principles, Financial Statements Analysis, Finite Mathematics for Financial Modeling, and Model Risk and Selection
• Emphasizes both technical and implementation issues, providing researchers, educators, students, and practitioners with the necessary background to deal with issues related to financial modeling
• The 3-Volume Set contains coverage of the fundamentals and advances in financial modeling and provides the mathematical and statistical techniques needed to develop and test financial models

Financial models have become increasingly commonplace, as well as complex. They are essential in a wide range of financial endeavors, and the Encyclopedia of Financial Models will help put them in perspective.

• Language: English
• Publisher: Wiley
• Release date: Oct 1, 2012
• ISBN: 9781118539682

Book preview

Guide to the Encyclopedia of Financial Models

The Encyclopedia of Financial Models provides comprehensive coverage of the field of financial modeling. This reference work consists of three separate volumes and 127 entries. Each entry provides coverage of the selected topic intended to inform a broad spectrum of readers ranging from finance professionals to academicians to students to fiduciaries. To derive the greatest possible benefit from the Encyclopedia of Financial Models, we have provided this guide. It explains how the information within the encyclopedia can be located.

ORGANIZATION

The Encyclopedia of Financial Models is organized to provide maximum ease of use for its readers.

Table of Contents

A complete table of contents for the entire encyclopedia appears in the front of each volume. This list of titles represents topics that have been carefully selected by the editor, Frank J. Fabozzi. The Preface includes a more detailed description of the volumes and the topic categories that the entries are grouped under.

Index

A Subject Index for the entire encyclopedia is located at the end of each volume. The subjects in the index are listed alphabetically and indicate the volume and page number where information on this topic can be found.

Entries

Each entry in the Encyclopedia of Financial Models begins on a new page, so that the reader may quickly locate it. The author’s name and affiliation are displayed at the beginning of the entry. All entries in the encyclopedia are organized according to a standard format, as follows:

Title and author

Abstract

Introduction

Body

Key points

Notes

References

Abstract

The abstract for each entry gives an overview of the topic, but not necessarily the content of the entry. This is designed to put the topic in the context of the entire Encyclopedia, rather than give an overview of the specific entry content.

Introduction

The text of each entry begins with an introductory section that defines the topic under discussion and summarizes the content. By reading this section, the reader gets a general idea about the content of a specific entry.

Body

The body of each entry explains the purpose, theory, and math behind each model.

Key Points

The key points section provides in bullet point format a review of the materials discussed in each entry. It imparts to the reader the most important issues and concepts discussed.

Notes

The notes provide more detailed information and citations of further readings.

References

The references section lists the publications cited in the entry.

Equity Models and Valuation

Dividend Discount Models

PAMELA P. DRAKE, PhD, CFA

J. Gray Ferguson Professor of Finance, College of Business, James Madison University

FRANK J. FABOZZI, PhD, CFA, CPA

Professor of Finance, EDHEC Business School

Abstract: Dividends are cash payments made by a corporation to its owners. Though cash dividends are paid to both preferred and common shareholders, most of the focus of the attention is on the dividends paid to the residual owners of the corporation, the common shareholders. Dividends paid to common and preferred shareholders are not legal obligations of a corporation, and some corporations do not pay cash dividends. But for those companies that pay dividends, changes in dividends are noticed by investors—increases in dividends are viewed favorably and are associated with increases in the company's stock price, whereas decreases in dividends are viewed quite unfavorably and are associated with decreases in the company's stock price. Most models that use dividends in the estimation of stock value use current dividends, some measure of historical or projected dividend growth, and an estimate of the required rate of return. Popular models include the basic dividend discount model that assumes a constant dividend growth, and the multiple-phase models, which include the two-stage dividend growth model and the stochastic dividend discount models.

In this entry, we discuss dividend discount models and their limitations. We begin with a review of the various ways to measure dividends and then take a look at how dividends and stock prices are related.

DIVIDEND MEASURES

Dividends are measured using three different measures:

Dividends per share

Dividend yield

Dividend payout

The value of a share of stock today is the investors’ assessment of today’s worth of future cash flows for each share. Because future cash flows to shareholders are dividends, we need a measure of dividends for each share of stock to estimate future cash flows per share. The dividends per share is the dollar amount of dividends paid out during the period per share of common stock:

Unnumbered Display Equation

If a company has paid $600,000 in dividends during the period and there are 1.5 million shares of common stock outstanding, then

Unnumbered Display Equation

The company paid out 40 cents in dividends per common share during this period.

The dividend yield, the ratio of dividends to price, is

Unnumbered Display Equation

The dividend yield is also referred to as the dividend-price ratio. Historically, the dividend yield for U.S. stocks has been a little less than 5%, according to a study by Campbell and Shiller (1998). In an exhaustive study of the relation between dividend yield and stock prices, Campbell and Shiller find that:

There is a weak relation between the dividend yield and subsequent 10-year dividend growth.

The dividend yield does not forecast future dividend growth.

The dividend yield predicts future price changes.

The weak relation between the dividend yield and future dividends may be attributed to the effects of the business cycle on dividend growth. The tendency for the dividend yield to revert to its historical mean has been observed by researchers.

Another way of describing dividends paid out during a period is to state the dividends as a portion of earnings for the period. This is referred to as the dividend payout ratio:

Unnumbered Display Equation

If a company pays $360,000 in dividends and has earnings available to common shareholders of $1.2 million, the payout ratio is 30%:

Unnumbered Display Equation

This means that the company paid out 30% of its earnings to shareholders.

The proportion of earnings paid out in dividends varies by company and industry. For example, the companies in the steel industry typically pay out 25% of their earnings in dividends, whereas the electric utility companies pay out approximately 75% of their earnings in dividends.

If companies focus on dividends per share in establishing their dividends (e.g., a constant dividends per share), the dividend payout will fluctuate along with earnings. We generally observe that companies set the dividend policy such that dividends per share grow at a relatively constant rate, resulting in dividend payouts that fluctuate.

DIVIDENDS AND STOCK PRICES

If an investor buys a common stock, he or she has bought shares that represent an ownership interest in the corporation. Shares of common perpetual security—there is no maturity. The investor who owns shares of common stock has the right to receive a certain portion of any dividends—but dividends are not a sure thing. Whether or not a corporation pays dividends is up to its board of directors—the representatives of the common shareholders. Typically, we see some pattern in the dividends companies pay: Dividends are either constant or grow at a constant rate. But there is no guarantee that dividends will be paid in the future.

Preferred shareholders are in a similar situation as the common shareholders. They expect to receive cash dividends in the future, but the payment of these dividends is up to the board of directors. But there are three major differences between the dividends of preferred and common shares. First, the dividends on preferred stock usually are specified at a fixed rate or dollar amount, whereas the amount of dividends is not specified for common shares. Second, preferred shareholders are given preference: their dividends must be paid before any dividends are paid on common stock. Third, if the preferred stock has a cumulative feature, dividends not paid in one period accumulate and are carried over to the next period. Therefore, the dividends on preferred stock are more certain than those on common shares.

It is reasonable to figure that what an investor pays for a share of stock should reflect what he or she expects to receive from it—return on the investor’s investment. What an investor receives are cash dividends in the future. How can we relate that return to what a share of common stock is worth? Well, the value of a share of stock should be equal to the present value of all the future cash flows an investor expects to receive from that share. To value stock, therefore, an investor must project future cash flows, which, in turn, means projecting future dividends. This approach to the valuation of common stock is referred to as the discounted cash flow approach and the models used are referred to as dividend discount models.

Dividend discount models are not the only approach to valuing common stock. There are fundamental factor models, also referred to as multifactor equity models.

BASIC DIVIDEND DISCOUNT MODELS

As discussed above, the basis for the dividend discount model (DDM) is simply the application of present value analysis, which asserts that the fair price of an asset is the present value of the expected cash flows. This model was first suggested by Williams (1938). In the case of common stock, the cash flows are the expected dividend payouts. The basic DDM model can be expressed mathematically as:

(1) Numbered Display Equation

where

P = the fair value or theoretical value of the common stock

Dt = the expected dividend for period t

rt = the appropriate discount or capitalization rate for period t

The dividends are expected to be received forever.

Practitioners rarely use the dividend discount model given by equation (1). Instead, one of the DDMs discussed below is typically used.

THE FINITE LIFE GENERAL DIVIDEND DISCOUNT MODEL

The DDM given by equation (1) can be modified by assuming a finite life for the expected cash flows. In this case, the expected cash flows are the expected dividend payouts and the expected sale price of the stock at some future date. The expected sale price is also called the terminal price and is intended to capture the future value of all subsequent dividend payouts. This model is called the finite life general DDM and is expressed mathematically as:

(2) Numbered Display Equation

where

PN = the expected sale price (or terminal price) at the horizon period N

N = the number of periods in the horizon

and P, Dt, and rt are the same as defined above.

Assuming a Constant Discount Rate

A special case of the finite life general DDM that is more commonly used in practice is one in which it is assumed that the discount rate is constant. That is, it is assumed each rt is the same for all t. Denoting this constant discount rate by r, equation (2) becomes:

(3) Numbered Display Equation

Equation (3) is called the constant discount rate version of the finite life general DDM. When practitioners use any of the DDM models presented in this entry, typically the constant discount rate version form is used.

Let’s illustrate the finite life general DDM assuming a constant discount rate assuming each period is a year. Suppose that the following data are determined for stock XYZ by a financial analyst:

Unnumbered Display Equation

Based on these data, the fair price of stock XYZ is

Unnumbered Display Equation

Required Inputs

The finite life general DDM requires three forecasts as inputs to calculate the fair value of a stock:

1. The expected terminal price (PN)

2. The dividends up to the assumed horizon (D1 to DN)

3. The discount rates (r1 to rN) or r (in the case of the constant discount rate version)

Thus the relevant question is, How accurately can these inputs be forecasted?

The terminal price is the most difficult of the three forecasts. According to theory, PN is the present value of all future dividends after N; that is, DN+1, DN+2, … , Dinfinity. Also, the future discount rate (rt) must be forecasted. In practice, forecasts are made of either dividends (DN) or earnings (EN) first, and then the price PN is estimated by assigning an appropriate requirement for yield, price-earnings ratio, or capitalization rate. Note that the present value of the expected terminal price PN/(1 + r)N becomes very small if N is very large.

The forecasting of dividends is somewhat easier. Usually, past history is available, management can be queried, and cash flows can be projected for a given scenario. The discount rate r is the required rate of return. Forecasting r is more complex than forecasting dividends, although not nearly as difficult as forecasting the terminal price (which requires a forecast of future discount rates as well). As noted above, in practice for a given company r is assumed to be constant for all periods and typically generated from the capital asset pricing model (CAPM). The CAPM provides the expected return for a company based on its systematic risk (beta).

Assessing Fair Value

Given the fair price derived from a dividend discount model, the assessment of the stock proceeds along the following lines. If the market price is below the fair price derived from the model, the stock is undervalued or cheap. The opposite holds for a stock whose market price is greater than the model-derived price. In this case, the stock is said to be overvalued or expensive. A stock trading equal to or close to its fair price is said to be fairly valued.

The DDM tells us the fair price but does not tell us when the price of the stock should be expected to move to this fair price. That is, the model says that based on the inputs generated by the analyst, the stock may be cheap, expensive, or priced appropriately. However, it does not tell us that if it is mispriced how long it will take before the market recognizes the mispricing and corrects it. As a result, an investor may hold on to a stock perceived to be cheap for an extended period of time and may underperform a benchmark during that period.

While a stock may be mispriced, an investor must also consider how mispriced it is in order to take the appropriate action (buy a cheap stock and sell or sell short an expensive stock). This will depend on the degree of mispricing and transaction costs.

CONSTANT GROWTH DIVIDEND DISCOUNT MODEL

If future dividends are assumed to grow at a constant rate (g) and a single discount rate (r) is used, then the finite life general DDM assuming a constant growth rate given by equation (3) becomes

(4)

Numbered Display Equation

and it can be shown that if N is assumed to approach infinity, equation (4) is equal to:

(5) Numbered Display Equation

Equation (5) is the constant growth dividend discount model (Gordon and Shapiro, 1956). An equivalent formulation for the constant growth DDM is

(6) Numbered Display Equation

where D1 is equal to D0(1 + g).

Consider a company that currently pays dividends of $3.00 per share. If the dividend is expected to grow at a rate of 3% per year and the discount rate is 12%, what is the value of a share of stock of this company? Using equation (5),

Unnumbered Display Equation

If the growth rate for this company’s dividends is 5%, instead of 3%, the current value is $45.00:

Unnumbered Display Equation

Therefore, the greater the expected growth rate of dividends, the greater the value of a share of stock.

In this last example, if the discount rate is 14% instead of 12% and the growth rate of dividends is 3%, the value of a share of stock is:

Unnumbered Display Equation

Therefore, the greater the discount rate, the lower the current value of a share of stock.

Let’s apply the model as given by equation (5) to estimate the price of three companies: Eli Lilly, Schering-Plough, and Wyeth Laboratories. The discount rate for each company was estimated using the capital asset pricing model assuming (1) a market risk premium of 5% and (2) a risk-free rate of 4.63%. The market risk premium is based on the historical spread between the return on the market (often proxied with the return on the S&P 500 Index) and the risk-free rate. Historically, this spread has been approximately 5%. The risk-free rate is often estimated by the yield on U.S. Treasury securities. At the end of 2006, 10-year Treasury securities were yielding approximately 4.625%. We use 4.63% as an estimate for the purposes of this illustration. The beta estimate for each company was obtained from the Value Line Investment Survey: 0.9 for Eli Lilly, 1.0 for Schering-Plough and Wyeth. The discount rate, r, for each company based on the CAPM is:

The dividend growth rate can be estimated by using the compounded rate of growth of historical dividends.

The compound growth rate, g, is found using the following formula:

Unnumbered Display Equation

This formula is equivalent to calculating the geometric mean of 1 plus the percentage change over the number of years. Using time value of money math, the 2006 dividend is the future value, the starting dividend is the present value, the number of years is the number of periods; solving for the interest rate produces the growth rate.

Substituting the values for the starting and ending dividend amounts and the number of periods into the formula, we get:

Unnumbered Table

The value of D0, the estimate for g, and the discount rate r for each company are summarized below:

Unnumbered Table

Substituting these values into equation (5), we obtain:

Unnumbered Display Equation

Schering-Plough estimated price

Unnumbered Display EquationUnnumbered Display Equation

Comparing the estimated price with the actual price, we see that this model does not do a good job of pricing these stocks:

Notice that the constant growth DDM is considerably off the mark for all three companies. The reasons include: (1) the dividend growth pattern for none of the three companies appears to suggest a constant growth rate, and (2) the growth rate of dividends in recent years has been much slower than earlier years (and, in fact, negative for Schering-Plough after 2003), causing growth rates estimated from the long time periods to overstate future growth. And this pattern is not unique to these companies.

Another problem that arises in using the constant growth rate model is that the growth rate of dividends may exceed the discount rate, r. Consider the following three companies and their dividend growth over the 16-year period from 1991 through 2006, with the estimated required rates of return:

Unnumbered Table

For these three companies, the growth rate of dividends over the prior 16 years is greater than the discount rate. If we substitute the D0 (the 2006 dividends), the g, and the r into equation (5), the estimated price at the end of 2006 is negative, which doesn’t make sense. Therefore, there are some cases in which it is inappropriate to use the constant rate DDM.

The potential for misvaluation using the constant rate DDM is highlighted by Fogler (1988) in his illustration using ABC prior to its being taken over by Capital Cities in 1985. He estimated the value of ABC stock to be $53.88, which was less than its market price at the time (of $64) and less than the $121 paid per share by Capital Cities.

MULTIPHASE DIVIDEND DISCOUNT MODELS

The assumption of constant growth is unrealistic and can even be misleading. Instead, most practitioners modify the constant growth DDM by assuming that companies will go through different growth phases. Within a given phase, dividends are assumed to grow at a constant rate. Molodovsky, May, and Chattiner (1965) were some of the pioneers in modifying the DDM to accommodate different growth rates.

Two-Stage Growth Model

The simplest form of multi-phase DDM is the two-stage growth model. A simple extension of equation (4) uses two different values of g. Referring to the first growth rate as g1 and the second growth rate as g2 and assuming that the first growth rate pertains to the next four years and the second growth rate refers to all years following, equation (4) can be modified as:

Unnumbered Display Equation

which simplifies to:

Unnumbered Display Equation

Because dividends following the fourth year are presumed to grow at a constant rate g2 forever, the value of a share at the end of the fourth year (that is, P4) is determined by using equation (5), substituting D0(1 + g1)⁴ for D0 (because period 4 is the base period for the value at end of the fourth year) and g2 for the constant rate g:

(7)

Numbered Display Equation

Suppose a company’s dividends are expected to grow at 4% rate for the next four years and then 8% thereafter. If the current dividend is $2.00 and the discount rate is 12%,

Unnumbered Display Equation

If this company’s dividends are expected to grow at the rate of 4% forever, the value of a share is $26.00; if this company’s dividends are expected to grow at the rate of 8% forever, the value of a share is $52.00. But because the growth rate of dividends is expected to increase from 4% to 8% in four years, the value of a share is between those two values, or $46.87.

As you can see from this example, the basic valuation model can be modified to accommodate different patterns of expected dividend growth.

Three-Stage Growth Model

The most popular multiphase model employed by practitioners appears to be the three-stage DDM. (The formula for this model is derived in Sorensen and Williamson [1985].) This model assumes that all companies go through three phases, analogous to the concept of the product life cycle. In the growth phase, a company experiences rapid earnings growth as it produces new products and expands market share. In the transition phase the company’s earnings begin to mature and decelerate to the rate of growth of the economy as a whole. At this point, the company is in the maturity phase in which earnings continue to grow at the rate of the general economy.

Different companies are assumed to be at different phases in the three-phase model. An emerging growth company would have a longer growth phase than a more mature company. Some companies are considered to have higher initial growth rates and hence longer growth and transition phases. Other companies may be considered to have lower current growth rates and hence shorter growth and transition phases.

In the typical investment management organization, analysts supply the projected earnings, dividends, growth rates for earnings, and dividend and payout ratios using fundamental security analysis. The growth rate at maturity for the entire economy is applied to all companies. As a generalization, approximately 25% of the expected return from a company (projected by the DDM) comes from the growth phase, 25% from the transition phase, and 50% from the maturity phase. However, a company with high growth and low dividend payouts shifts the relative contribution toward the maturity phase, while a company with low growth and a high payout shifts the relative contribution toward the growth and transition phases.

STOCHASTIC DIVIDEND DISCOUNT MODELS

As we noted in our discussion and illustration of the constant growth DDM, an erratic dividend pattern such as that of Wyeth can lead to quite a difference between the estimated price and the actual price. In the case of the pharmaceutical companies, the estimated price overstated the actual price for Eli Lilly, but understated the price of Schering-Plough and Wyeth.

Hurley and Johnson (1998a, 1998b) have suggested a new family of valuation model. Their model allows for a more realistic pattern of dividend payments. The basic model generates dividend payments based on a model that assumes that either the firm will increase dividends for the period by a constant amount or keep dividends the same. The model is referred to as a stochastic DDM because the dividend can increase or be constant based on some estimated probability of each possibility occurring. The dividend stream used in the stochastic DDM is called the stochastic dividend stream.

There are two versions of the stochastic DDM. One assumes that dividends either increase or decrease at a constant growth rate. This version is referred to as a binomial stochastic DDM because there are two possibilities for dividends. The second version is called a trinomial stochastic DDM because it allows for an increase in dividends, no change in dividends, and a cut in dividends. We discuss each version below.

Binomial Stochastic Model

For both the binomial and trinomial stochastic DDM, there are two versions of the model—the additive growth model and the geometric growth model. The former model assumes that dividend growth is additive rather than geometric. This means that dividends are assumed to grow by a constant dollar amount. So, for example, if dividends are $2.00 today and the additive growth rate is assumed to be $0.25 per year, then next year dividends will grow to $2.25, in two years dividends will grow to $2.50, and so on. The second model assumes a geometric rate of dividend growth. This is the same growth rate assumption used in the earlier DDMs presented in this entry.

Binomial Additive Stochastic Model

This formulation of the model is expressed as follows:

Unnumbered Display Equation

where

Dt = dividend in period t

Dt+ 1 = dividend in period t+1

C = dollar amount of the dividend increase

p = probability that the dividend will increase

Hurley and Johnson (1998a) have shown that the theoretical value of the stock based on the additive stochastic DDM assuming a constant discount rate is equal to:

(8) Numbered Display Equation

For example, consider once again Wyeth. In the illustration of the constant growth model, we used D0 of $1.01 and a g of 3.533%. We estimate C by calculating the dollar increase in dividends for each year that had a dividend increase and then taking the average dollar dividend increase. The average of the increases is $0.0373.

In the 15-year span 1991 through 2006, dividends increased 11 of the 14 year-to-year differences. Therefore, p = 11/15 = 73.3333%. Substituting these values into equation (8), we find the estimated price to be:

Unnumbered Display Equation

Applying this model to the other two pharmaceutical companies, we see that the model produces an estimated price that is closer to the actual price than the fair value based on the constant growth model:

Unnumbered Table

Binomial Geometric Stochastic Model

Letting g be the growth rate of dividends, then the geometric dividend stream is

Unnumbered Display Equation

Hurley and Johnson (1998b) show that the price of the stock in this case is:

(9) Numbered Display Equation

Equation (9) is the binomial stochastic DDM assuming a geometric growth rate and a constant discount rate.

Trinomial Stochastic Models

The trinomial stochastic DDM allows for dividend cuts. Within the Hurley-Johnson stochastic DDM framework, Yao (1997) derived this model that allows for a cut in dividends. He notes that is not uncommon for a firm to cut dividends temporarily. In fact, an examination of the dividend record of the electric utilities industry as published in Value Line Industry Review found that in the aggregate firms cut dividends three times over a 15-year period.

Table 1 Fit of the Different Dividend Models Applied to Five Electric Utilities

Table 1-7

Trinomial Additive Stochastic Model

The additive stochastic DDM can be extended to allow for dividend cuts as follow:

Unnumbered Display Equation

where

pU = probability that the dividend will increase

pD = probability that the dividend will decrease

pC = probability that the dividend will be unchanged

The theoretical value of the stock based on the trinomial additive stochastic DDM then becomes:

(10) Numbered Display Equation

Notice that when pD is zero (that is, there is no possibility for a cut in dividends), equation (10) reduces to equation (8).

Trinomial Geometric Stochastic Model

For the trinomial geometric stochastic DDM allowing for a possibility of cuts, we have:

Unnumbered Display Equation

and the theoretical price is:

(11) Numbered Display Equation

Once again, substituting zero for pD, equation (11) reduces to equation (9)—the binomial geometric stochastic DDM.

Applications of the Stochastic DDM

Yao (1997) applied the stochastic DDMs to five electric utility stocks that had regular dividends from 1979 to 1994 and found that the models fit the various utility stocks differently.

We see similar results in an updated example using five electric utilities, as shown in Table 1. For three of the five utilities, the binomial model provides an estimate closest to the actual stock price, whereas for the other two utilities, the trinomial model offers the closest estimate. In none of the cases, however, did the constant dividend growth model offer the closest approximation to the actual stock price.

Advantages of the Stochastic DDM

The stochatic DDM developed by Hurley and Johnson is a powerful tool for the analyst because it allows the analyst to generate a probability distribution for a stock’s value. The probability distribution can be used by an analyst to assess whether a stock is sufficiently mispriced to justify a buy or sell recommendation. For example, suppose that a three-phase DDM indicates that the value of a stock trading at $35 is $42. According to the model, the stock is underpriced and the analyst would recommend the purchase of this stock. However, the analyst cannot express his or her confidence as to the degree to which the stock is undervalued.

Hurley and Johnson show how the stochastic DDM can be used to overcome this limitation of traditional DDMs. An analyst can use the derived probability distribution from the stochastic DDM to assess the probability that the stock is undervalued. For example, an analyst may find from a probability distribution that the probability that the stock is greater than $35 (the market price) is 90%.

To employ a stochastic DDM an analyst must be prepared to make subjective assumptions about the uncertain nature of future dividends. Monte Carlo simulation available on a spread sheet (@RISK in Excel, for example) can then be used to generate the probability distribution.

EXPECTED RETURNS AND DIVIDEND DISCOUNT MODELS

Thus far, we have seen how to calculate the fair price of a stock given the estimates of dividends, discount rates, terminal prices, and growth rates. The model-derived price is then compared to the actual price and the appropriate action is taken.

The analysis can be recast in terms of expected return. This is found by calculating the return that will make the present value of the expected cash flows equal to the actual price. Mathematically, this is expressed as follows:

(12)

Numbered Display Equation

where

PA = actual price of the stock

ER = expected return

The expected return (ER) in equation (12). For example, consider the following inputs used at the outset of this entry to illustrate the finite life general DDM as given by equation (3). For stock XYZ, the inputs assumed are:

Unnumbered Display Equation

We calculated a fair price based on equation (3) to be $24.90. Suppose that the actual price is $25.89. Then the expected return is found by solving the following equation for ER:

Figure 1 The Relation between the Fair Value of a Stock and the Stock’s Expected Return

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The expected return is 9%.

The expected return is the discount rate that equates the present value of the expected future cash flows with the present value of the stock. The higher the expected return—for a given set of future cash flows—the lower the current value. The relation between the fair value of a stock and the expected return of a stock is shown in Figure 1.

Given the expected return and the required return (that is, the value for r), any mispricing can be identified. If the expected return exceeds the required return, then the stock is undervalued; if it is less than the required return then the stock is overvalued. A stock is fairly valued if the expected return is equal to the required return. In our illustration, the expected return (9%) is less than the required return (10%); therefore, stock XYZ is overvalued.

With the same set of inputs, the identification of a stock being mispriced or fairly valued will be the same regardless of whether the fair value is determined and compared to the market price or the expected return is calculated and compared to the required return. In the case of XYZ stock, the fair value is $24.90. If the stock is trading at $25.89, it is overvalued. The expected return if the stock is trading at $25.89 is 9%, which is less than the required return of 10%. If, instead, the stock price is $24.90, it is fairly valued. The expected return can be shown to be 10%, which is the same as the required return. At a price of $23.95, it can be shown that the expected return is 11%. Since the required return is 10%, stock XYZ would be undervalued.

While the illustration above uses the basic DDM, the expected return can be computed for any of the models. In some cases, the calculation of the expected return is simple since a formula can be derived that specifies the expected return in terms of the other variables. For example, for the constant growth DDM given by equation (5), the expected return (r) can be easily solved to give:

Unnumbered Display Equation

Rearranging the constant growth model to solve for the expected return, we see that the required rate of return can be specified as the sum of the dividend yield and the expected growth rate of dividends.

KEY POINTS

Dividends are measured in a number of ways, including dividends per share, dividend yield, and dividend payout.

The discounted cash flow approach to valuing common stock requires projecting future dividends. Hence, the model used to value common stock is called a dividend discount model.

The simplest dividend discount model is the constant growth model. More complex models include the multiphase model and stochastic models.

Stock valuation using a dividend discount model is highly dependent on the inputs used.

A dividend discount model does not indicate when the current market price will reach its fair value.

The output of a dividend discount model is the fair price. However, the model can be used to generate the expected return.

The expected return is the interest rate that will make the present value of the expected dividends plus terminal price equal to the stock’s market price. The expected return is then compared to the required return to assess whether a stock is fairly priced in the market.

REFERENCES

Campbell, J. Y., and Shiller, R. J. (1998). Valuation ratios and the long-run stock market outlook. Journal of Portfolio Management 24 (Winter): 11–26.

Fogler, R. H. (1988). Security analysis, DDMs, and probability. In Equity Markets and Valuation Methods (pp. 51–52). Charlottesville, VA: The Institute of Chartered Financial Analysts.

Gordon, M., and Shapiro, E. (1956). Capital equipment analysis: The required rate of profit. Management Science 3: 102–110.

Hurley, W. J., and Johnson, L. (1994). A realistic dividend valuation model. Financial Analysts Journal 50 (July–August): 50–54.

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Discounted Cash Flow Methods for Equity Valuation

GLEN A. LARSEN Jr., PhD, CFA

Professor of Finance, Indiana University Kelley School of Business–Indianapolis

Abstract: Most applied methods of valuing a firm’s equity are based on discounted cash flow and relative valuation models. Although stock and firm valuation is very strongly tilted toward the use of discounted cash flow methods, it is impossible to ignore the fact that many analysts use other methods to value equity and entire firms. The primary alternative valuation method is relative valuation. Both discounted cash flow and relative valuation methods require strong assumptions and expectations about the future. No one single valuation model or method is perfect. All valuation estimates are subject to model error and estimation error.

Sound investing requires that an investor does not pay more for an asset than its worth. There are those who argue that value is in the eyes of the beholder, which is simply not true when it comes to financial assets. Perceptions may be all that matter when the asset is an art object or antique automobile, but investors should not buy financial assets for aesthetic or emotional reasons; financial assets are acquired for the cash flows expected from them in future periods. Consequently, perceptions of value have to be backed up by reality, which implies that the price paid for any financial asset should reflect the cash flows that it is expected to generate.

Realize that at the end of the most careful and detailed valuation, there will be uncertainty about the final numbers, biased as they are by the assumptions that we make about the future of the company and the economy. It is unrealistic to expect or demand absolute certainty in valuation, since cash flows and discount rates are estimated with error. This also means that you have to give yourself a reasonable margin for error in making recommendations on the basis of valuations. Most importantly, realize that the degree of precision in valuations is likely to vary widely across investments. For example, the valuation of a large and mature company, with a long financial history, will usually be much more precise than the valuation of a young company or of a company that is in a sector that is in turmoil.

Implicit often in the act of valuation is the assumption that markets make mistakes and that we can find these mistakes, often using information that tens of thousands of other investors can access. Thus, the argument that those who believe that markets are inefficient should spend their time and resources on valuation whereas those who believe that markets are efficient should take the market price as the best estimate of value, seems to be reasonable. This statement, though, does not reflect the internal contradictions in both positions. Those who believe that markets are efficient may still feel that valuation has something to contribute, especially when they are called upon to value the effect of a change in the way a firm is run or to understand why market prices change over time.

Furthermore, it is not clear how markets would become efficient in the first place, if investors did not attempt to find under- and overvalued stocks and trade on these valuations. In other words, a precondition for market efficiency seems to be the existence of millions of investors who believe that markets are not.

Stock-pricing models are not physical or chemical laws of nature. There is, however, a strong principle of investing that must eventually hold true for all firms over time if they are to have a positive value. This principle is that you should always be able, in your mind, to construct some sort of logical connection between a positive stock price today and a stream of future cash flows to the investor. The logical chain might be long. You might assume that years of start-up losses (earnings are zero or negative) will be followed by more years of all profits being reinvested. But you should be able to envision some connection between today’s positive stock price and a stream of cash flows that will commence someday in the future.

In this entry, we discuss practical methods of valuing a firm’s equity based on discounted cash flow (DCF) models. Although stock and firm valuation is very strongly tilted toward the use of DCF methods, it is impossible to ignore the fact that many analysts use other methods to value equity and entire firms. The DCF model is the subject of this entry. The primary alternative valuation method is relative valuation (RV). Both DCF and RV valuation methods require strong assumptions and expectations about the future. No one single valuation model or method is perfect. All valuation estimates are subject to model error and estimation error. Nevertheless, investors use these models to help form their expectations about a fair market price. Markets then generate an observable market clearing price based on investor expectations, and this market clearing price constantly changes along with investor expectations.

DIVIDEND DISCOUNT MODEL

The dividend discount model (DDM) is the most basic DCF stock approach to equity valuation, originally formulated by Williams (1938). It states that the stock price should equal the present value of all expected future dividends into perpetuity under the assumption that a firm has an infinite life. But you may also have ignored the DDM once you recognized how difficult it is to apply in the real world. The next several paragraphs simply review the basic concepts in order to highlight the complexities that surround implementing the DDM in practice.

Consider an investor who buys a share of stock, planning to hold it for one year. As you know from previous studies, the intrinsic value of the share is the present value, P(0), of the expected dividend to be received at the end of the first year, ED(1), and the expected sales price, EP(1).

(1) Numbered Display Equation

Keep in mind that since we live in a world of uncertainty and no human can perfectly forecast the future, future prices and dividends are unknown. Specifically, we are dealing with expected values, not certain values. Under the assumption that dividends can be predictable, given a company’s dividend history, the expected future dividend in the next period, ED(1), can be estimated based on historical trends. You might ask how we can estimate EP(1), the expected year-end price.

According to equation (1), the year-end intrinsic value, P(1), will be

(2) Numbered Display Equation

If we assume the stock will be selling for its intrinsic value next year, then P(1) = EP(1), and we can substitute equation (2) into equation (1), which gives

(3) Numbered Display Equation

Equation (3) may be interpreted as the present value of dividends plus the expected sales price at the end of a two-year holding period. Of course, now we need to come up with a forecast of EP(2). Continuing in the same way, we can replace the expected price at the end of two years by the intrinsic value at the end of two years. That is, replace EP(2) by [ED(3)+EP(3)]/(1+R), which relates P(0) to the value of dividends over three years plus the expected sales price at the end of a three-year holding period.

More generally, for a holding period of T years, we can write the stock value as the present value of dividends over the T years discounted at an appropriate discount rate, R, that is assumed to remain constant, plus the present value of the ultimate sales price, EP(T):

(4) Numbered Display Equation

In short, the intrinsic price of a share of stock is the present value of a stream of payments (dividends in the case of stocks) and a final payment (the sales price of the stock at time T).

The key problems with implementing this model are the uncertainty of future dividends, the lack of a fixed maturity date, and the unknown sales price at the horizon date and the appropriate discount rate. Indeed, one can continue to substitute for a terminal price on out to infinity (INF):

(5) Numbered Display Equation

Equation (5) states that the stock price should equal the present value of all expected future dividends in perpetuity. This formula is the DDM in mathematical form. It is tempting, but incorrect, to conclude from the equation that the DDM focuses exclusively on dividends and ignores capital gains as a motive for investing in stock. Indeed, we assume explicitly in equation (4), the finite version of the DDM, that capital gains (as reflected in the expected sales price, EP(T)) are part of the stock’s value. EP(T) is the present value at time T of all dividends expected to be paid after the horizon date. That value is then discounted back to today, time T = 0. The DDM asserts that stock prices are determined ultimately by the cash flows accruing to stockholders, and those are dividends.

Stocks That Currently Pay No Dividend

If investors never expected a dividend to be paid, then this model implies that the stock would have no value. To reconcile the fact that stocks not paying a current dividend do have a positive market value with this model, one must assume that investors expect that someday, at some time T, the firm must pay out some cash, even if only a liquidating dividend.

CONSTANT-GROWTH DDM

The general form of the DDM, as it stands, is still not very useful in valuing a stock because it requires dividend forecasts for every year into the indefinite future. To make the DDM practical, we need to introduce some simplifying assumptions. One useful and common first pass at the problem is to assume that dividends are trending upward at a stable or constant growth rate, g.

For example, if g = 0.05 and the most recently paid dividend was D(0) = 3.81, expected future dividends are

unNumbered Display Equation

and so on. Using these dividend forecasts, we can solve for intrinsic value as

unNumbered Display Equation

Since the basic form of this equation stretches to infinity, basic algebra allows this equation to be written as

(6) Numbered Display Equation

Equation (6) is called the constant-growth DDM, or the Gordon-Shapiro model, after Myron Gordon and Eli Shapiro, who popularized the model [see Gordon (1962) and Gordon and Shapiro (1956)].

Equation (6) should remind you of the formula for the present value of perpetuity. If dividends were expected not to grow, g = 0, then the dividend stream would be a simple perpetuity, and the valuation formula would be

Unnumbered Display Equation

P(0) = ED(1)/(R−g) is a generalization of the perpetuity formula to cover the case of a perpetuity growing at a constant rate, g. As g increases, for a given value of ED(1), the stock price rises. The constant-growth DDM is valid only when g is less than R. If dividends were expected to grow forever (to infinity) at a rate faster than R, the value of the stock would be infinite. Further, in all of the DDM equations presented, R is also assumed to be constant forever.

NONCONSTANT-GROWTH DDM

If you feel that you know the future growth rates in each period for a firm, then you can certainly use unique growth rates, g(T) and required rates of return, R(T), in the present value equation and discount all unique dividends and future selling price back to the present. The problem becomes one of time, effort, and estimation risk. At some future point in time, what you believe to be a better unique estimate of a future dividend or a future discount rate will in reality be no better than an assumption of constant growth and constant discount rate.

INTUITION BEHIND THE DDM

In a market economy, common sense dictates that you should go into business only if you expect to make money. In a sole proprietorship, everything left over from the revenue you earned, minus expenses, is yours. In other forms of a business organization, you need to be a bit more formal because there are other owners. In a partnership, partners draw money out of the business. And shareholders get money out of a corporation by receiving dividends. Using the corporate form as an example, the value per share is determined by the value of the dividends distributed to each shareholder. That is, the value per share is determined by the present value of each shareholder’s expected share of the profits.

Here is a simple example that illustrates several of the uncertainties involved with the basic DCF valuation process for a share of common stock. Let’s say you consider buying shares of a corporation. How much will you pay if the expected annual dividend forever is $10 per share? That depends on how much of an annual return you want. If you want a 10% return, you’ll offer $100 (that is, a $10 dividend divided by a $100 investment equals a return of 10%). But just because you offer to pay $100 doesn’t mean someone will sell to you at that price.

Financial capital is subject to principles of market supply and demand, just like commodities. Suppose market conditions are such that prevailing rates of return for corporate shares in this particular risk class are in the 5% range. If I’m selling stock that commands a $10 per share dividend I can demand a price of $200, and someone will give it to me. Suppose this corporation is a bit riskier than most others. A buyer may say, "If I’m willing to accept the prevailing 5% return, there are hundreds upon hundreds of better-quality corporations I can invest in. So if you want me to buy your shares, you need to give me incentive to bypass all the others. The buyer and seller may settle on a 7% return, which is equivalent to a price of about $143. The appropriate required rate of return, R, is therefore critical, and R can vary with market conditions.

In all cases, assuming that the life of the corporation is infinite, the current price, P(0), is computed as the constant dividend in perpetuity, D, divided by the required rate of return, R, that is, the present value of all future constant dividends. Often, though, investors use return, R, as the basis for comparing and pricing investments. R is often estimated from observable information as D (dividend) divided by current price P(0). Mathematically, it looks like this:

Unnumbered Display Equation

You’ve seen this before. It is the dividend yield.

COMPLICATIONS IN IMPLEMENTING THE DDM IN THE REAL WORLD

As you can see by now, there are essentially four major issues that complicate finding the present value of all future dividends and, therefore, in implementing the DDM.

Expected Growth of Dividends

As profits grow over time (as we hope they will), dividends can be expected to grow and not remain constant forever. If profits and dividends are growing by 10% every year, the dividend this year may be $10, but by next year, it will be $11. If we divide $11 by today’s $200 purchase price, next year’s yield will be 5.5% (11/200). The year after, assuming further 10% growth, the dividend will be $12.10. Dividing that by the $200 purchase price produces a yield of 6.05%. The buyer might smile, but the seller won’t accept it. The seller wants a price that truly is consistent with the prevailing 5% yield. At $200, the buyer gets too much of a good deal. If the latter holds the stock over time, he’ll wind up with an annual return well in excess of 5%.

Appropriate Expected Required Rate of Return

Simply stated, present value is a tool for computing today’s equivalent of a cash payment to be made tomorrow. As stated earlier, this is often referred to as DCF valuation. If I offer you $10 today or $10 a year from now, you’ll probably choose $10 today. But if the choice is $10 today or $11.50 a year from now, you have to pause. If you can invest today’s $10 payment for one year at 5%, at the end of the year you’ll have $10.50. But if you bypass the $10 for now and wait, you can get $11.50 a year hence. That’s a better deal. The way to decide if you should wait is to do some mathematics that helps you decide how much you must receive today to allow you to invest and wind up with $11.50 a year hence. In this example, the present value of $11.50 one year from now, assuming a 5% return, is $10.95. If I take $10.95 and invest it for one year at 5%, I’ll wind up with $11.50 at the end of the year. If interest rates rise, to say 8%, it’ll take less money today to generate $11.50 a year hence ($10.65 will be sufficient). So as interest rates rise, present values fall, and vice versa.

Expected Future Selling Price

Thus far, we have thought about a stream of dividends stretching into the infinite future. Even long-term investors prefer a holding period that’s something short of infinity. So we need to account for the fact that someday you’ll want to sell your shares. As such, the proceeds you expect to get when you sell are included, along with dividends, in the stream of cash you expect to get, and that goes into the present value calculation.

Let’s think about a projection of the future sale price. If you think you may sell in two years, imagine how a prospective buyer, two years into the future, will value the dividend stream that he’ll get. Continuing with the preceding example, he’ll be looking at an initial payout of $12.10 and a 5% return. So a price of $244 seems a reasonable starting point. Of course, you’ll need to make adjustments for probable growth beyond year two. And perhaps 5% won’t be appropriate as a rate of return. Market rates may rise or fall, and/or the quality of the corporation may improve or deteriorate relative to alternative investments. And two years hence, the growth forecast may change. But in any case, we do have a $244 starting point. The changes may bring it up, perhaps to $275, or down, possibly to $175. But if an exuberant analyst publishes a target price of $1,000, you ought to raise an eyebrow and insist that the analyst get serious about justifying his presumably bold assumptions about market rates, growth, or company quality.

Reinvestment of Profits/Internal Financing that Support Growth

It is standard for corporations to refrain from paying out all annual profit as dividend. Some money is held in the business for a rainy day. And some money is simply reinvested for future growth. Either way, profits not paid out as dividends are known as retained earnings. Reinvestment is more desirable than dividend payments if the corporation can earn a higher return on the money than the shareholder could get (by reinvesting the dividends). If all goes well, the reinvestment will enable the corporation to pay a higher dividend in the future than would otherwise have been the case. Going back to the preceding example, if reinvestment gives the corporation the ability to set a year-five payout at $18 rather than $12.10, that raises the starting-point target price to $360. A shareholder who accepts a forecast like that would likely forgo all or some immediate dividend payments in order to get that bigger future reward. As you can see, even if a corporation currently pays little or no dividend, we still have to acknowledge dividends as a major factor in our thoughts about share pricing.

For better or worse, many corporations now see themselves as growth companies. And many shareholders have accepted a situation where these publicly traded growth companies pay out very little of their profits, if anything, as dividends, and reinvest most or all profits back into the business. Many companies do not deliver nearly as well on the growth dream as everybody hopes. But the growth culture remains alive and well, and the dividend payout ratio has declined.

ADAPTING TO THE COMPLICATIONS: THE EARNINGS PER SHARE APPROACH

As a result of the four complications listed, modern stock prices have become uncoupled from dividends. So, in the real world, it is difficult to compute a fair price through the basic dividend formulas presented.

Here is one solution. It involves substituting earnings per share (EPS) for dividends. This doesn’t really work in a theoretical DDM sense, but it does work within the context of a growth culture. Shareholders have so thoroughly accepted and adopted growth that they act as if all corporate EPS (whether paid as dividends or reinvested back into the business) is in their hands. So, instead of working with a dividend yield as presented earlier, we can substitute an earnings (E) yield, which is computed as follows:

Unnumbered Display Equation

Does the E/P ratio look familiar? It should. Turn it upside down and we get something you see all the time: the P/E (price/earnings) ratio.

It is important to emphasize that P/E ratios are not just one of those things we use for the heck of it. They have a serious and solid intellectual underpinning. They are equivalent to earnings yields, which are the modern-day substitute for dividend yields—the true basis for valuing ownership of corporate stock. So when somebody states that P/E ratios are no longer relevant, you’d best turn away. Buying any stock without addressing the P/E ratio is not sensible.

When we flip P/E back over and think of earnings yield, we can understand, from the prior discussion of dividend yield, that a bad company’s stock will have to offer a higher yield to attract buyers. Similarly, the yield for a great company will be low (otherwise, there would be too many would-be buyers). Let’s see how this works when we flip the earnings yields back to P/Es.

If EPS equals $3.00 and the earnings yield is 5%, the price will be $60. If it’s a bad company and the yield is higher, at 8%, the stock price will be $37.50. If it’s a good company and the yield is lower, say 3%, the stock price will be $100. The starting number translates to a P/E as follows: a $60 price divided by $3.00 EPS gives us a P/E of 20. A bad-company stock price of $37.50 divided by EPS of $3.00 produces a P/E of 12.5. A good-company stock price of $100 divided by EPS of $3.00 produces a P/E of 33.3.

That’s the basis for the generally recognized phenomenon of good stocks having higher P/Es and bad stocks generally having lower P/Es. So, once again, this isn’t just one of those things. It’s an inevitable result of the basic principles of finance and math. When evaluating companies, good or bad is usually determined based on growth prospects and risk.

We handled the complicating factors by treating EPS as if it were the same as a dividend. But notwithstanding, we still have a reasonably rational basis for stock prices. We can argue over what the growth prospects are and what the market return ought to be (based on differing assessments of market conditions and company-quality issues). So there will always be disagreement on what, exactly, a fair stock price ought to be. But all rational investors should be somewhere in the same ballpark. We may have a big ballpark and debate if a stock that commands $25 today is worth $15 or $35. But we are unlikely to seriously consider a price of, say, $350.

FREE CASH FLOW DCF MODEL—TOTAL FIRM VALUATION

While estimating future cash flows to an individual share of stock can seem daunting, some investors prefer to estimate the free cash flow to the entire firm. Doing this allows investors to estimate the value of the entire firm and then back out an estimated value of a share of stock. This is called the free cash flow (FCF) model. While legitimate accounting rules do enable managers and auditors some range of choices, at the end of the day, good companies wind up looking good and bad companies wind up looking bad. In short, there’s no one number in an income report that truly gives you the necessary information to value a firm from a discounted expected