Although the principles behind discounted cash flow valuation are simple, applying the theory to equity valuation can be challenging. Four broad steps in applying DCF analysis to equity valuation are

In this chapter, we take the perspective that dividends—distributions to shareholders authorized by a company’s board of directors—are an appropriate definition of cash flows. The class of models based on this idea is called dividend discount models (DDMs). The basic objective of any DDM is to value a stock. The variety of implementations corresponds to different ways to model a company’s future stream of dividend payments. The steps of choosing a discount rate methodology and estimating the discount rate involve the same considerations for all DCF models, so they have been presented separately in Chapter 2, concerning return concepts.

This chapter is organized as follows: Section 2 provides an overview of present value models. A general statement of the dividend discount model follows in Section 3. Forecasting dividends, individually and in detail, into the indefinite future is not generally practicable, so the dividend-forecasting problem is usually simplified. One approach is to assign dividends to a stylized growth pattern. The simplest pattern—dividends growing at a constant rate forever—is the constant growth (or Gordon growth) model, discussed in Section 4. For some companies, it is more appropriate to view earnings and dividends as having multiple stages of growth; multistage dividend discount models are presented in Section 5 along with spreadsheet modeling. Section 6 lays out the determinants of dividend growth rates, and Section 7 summarizes the chapter.

For some assets, such as government debt, cash flows may be essentially known with certainty—that is, they are default risk free. The appropriate discount rate for such a risk-free cash flow is a risk-free rate of interest. For example, if an asset has a single, certain cash flow of $100 to be received in two years, and the risk-free interest rate is 5 percent a year, the value of the asset is the present value of $100 discounted at the risk-free rate: $100/(1.05)² = $90.70.

In contrast to risk-free debt, future cash flows for equity investments are not known with certainty—they are risky. Introducing risk makes applying the present value approach much more challenging. The most common approach to dealing with risky cash flows involves two adjustments relative to the risk-free case. First, discount the expected value of the cash flows, viewing the cash flows as random variables.⁶² Second, adjust the discount rate to reflect the risk of the cash flows.

The following equation expresses the concept that an asset’s value is the present value of its (expected) future cash flows: where

For simplicity, the discount rate in Equation 3-1 is represented as the same for all time periods (i.e., a flat term structure of discount rates is assumed). The analyst has the latitude in this model, however, to apply different discount rates to different cash flows.⁶³

Equation 3-1 gives an asset’s value from the perspective of today (t = 0). Likewise, an asset’s value at some point in the future equals the value of all subsequent cash flows discounted back to that point in time. Example 3-1 illustrates these points.

In the next section, we present an overview of three alternative definitions of cash flow. The selected cash flow concept defines the type of DCF model we can use: the dividend discount model, the free cash flow model, or the residual income model. We also broadly characterize the types of valuation problems for which analysts often choose a particular model. (Further details are supplied when each model is discussed individually.)

The dividend discount model defines cash flows as dividends. The basic argument for using this definition of cash flow is that an investor who buys and holds a share of stock generally receives cash returns only in the form of dividends.⁶⁴ In practice, analysts usually view investment value as driven by earnings. Does the definition of cash flow as dividends ignore earnings not distributed to shareholders as dividends? Reinvested earnings should provide the basis for increased future dividends. Therefore, the DDM accounts for reinvested earnings when it takes all future dividends into account. Because dividends are less volatile than earnings and other return concepts, the relative stability of dividends may make DDM values less sensitive to short-run fluctuations in underlying value than alternative DCF models. Analysts often view DDM values as reflecting long-run intrinsic value.

A stock either pays dividends or does not pay dividends. A company might not pay dividends on its stock because the company is not profitable and has no cash to distribute. Also, a company might not pay dividends for the opposite reason: because it is very profitable. For example, a company may reinvest all earnings—paying no dividends—to take advantage of profitable growth opportunities. As the company matures and faces fewer attractive investment opportunities, it may initiate dividends. Generally, mature, profitable companies tend to pay dividends and are reluctant to reduce the level of dividends (Lintner 1956; Grullon et al. 2007).

Dividend policy practices have international differences and change through time, even in one market. Typically, a lower percentage of companies in a given U.S. stock market index have paid dividends than have companies in a comparable European stock market index. Wanger (2007) noted a much higher propensity for European and Asian small-cap companies to pay dividends compared with U.S. companies. In addition, the following broad trends in dividend policy have been observed:

Analysts will frequently need to value non-dividend-paying shares. Can the DDM be applied to non-dividend-paying shares? In theory it can, as is illustrated later, but in practice it generally is not.

Predicting the timing of dividend initiation and the magnitude of future dividends without any prior dividend data or specifics about dividend policy to guide the analysis is generally not practical. For a non-dividend-paying company, analysts usually prefer a model that defines returns at the company level (as free cash flow or residual income—these concepts are defined shortly) rather than at the stockholder level (as dividends). Another consideration in the choice of models relates to ownership perspective. An investor purchasing a small ownership share does not have the ability to meaningfully influence the timing or magnitude of the distribution of the company’s cash to shareholders. That perspective is the one taken in applying a dividend discount model. The only access to the company’s value is through the receipt of dividends, and dividend policy is taken as a given. If dividends do not bear an understandable relation to value creation in the company, applying the DDM to value the stock is prone to error.

Generally, the definition of returns as dividends, and the DDM, is most suitable when three conditions are met:

A second definition of returns is free cash flow. The term cash flow has been given many meanings in different contexts. Earlier the term was used informally, referring to returns to ownership (equity). We now want to give it a more technical meaning, related to accounting usage. Over a given period of time, a company can add to cash (or use up cash) by selling goods and services. This money is cash flow from operations (for that time period). Cash flow from operations is the critical cash flow concept addressing a business’s underlying economics. Companies can also generate (or use up) cash in two other ways. First, a company affects cash through buying and selling assets, including investment and disinvestment in plant and equipment. Second, a company can add to or reduce cash through its financing activities. Financing includes debt and equity. For example, issuing bonds increases cash, and buying back stock decreases cash (all else equal).⁶⁷

Assets supporting current sales may need replacement because of obsolescence or wear and tear, and the company may need new assets to take advantage of profitable growth opportunities. The concept of free cash flow responds to the reality that, for a going concern, some of the cash flow from operations is not “free” but rather needs to be committed to reinvestment and new investment in assets. Free cash flow to the firm(FCFF) is cash flow from operations minus capital expenditures. Capital expenditures—reinvestment in new assets, including working capital—are needed to maintain the company as a going concern, so only that part of cash flow from operations remaining after such reinvestment is “free.” (This definition is conceptual; Chapter 4 defines free cash flow concepts in detail.) FCFF is the part of the cash flow generated by the company’s operations that can be withdrawn by bondholders and stockholders without economically impairing the company. Conceptually, the value of common equity is the present value of expected future FCFF—the total value of the company—minus the market value of outstanding debt.

Another approach to valuing equity works with free cash flow to equity. Free cash flow to equity(FCFE) is cash flow from operations minus capital expenditures, or FCFF, from which we net all payments to debtholders (interest and principal repayments net of new debt issues). Debt has a claim on the cash of the company that must be satisfied before any money can be paid to stockholders, so money paid on debt is not available to common stockholders. Conceptually, common equity can be valued as the present value of expected FCFE. FCFF is a predebt free cash flow concept; FCFE is a post-debt free cash flow concept. The FCFE model is the baseline free cash flow valuation model for equity, but the FCFF model may be easier to apply in several cases, such as when the company’s leverage (debt in its capital structure) is expected to change significantly over time.

Valuation using a free cash flow concept is popular in current investment practice. Free cash flow (FCFF or FCFE) can be calculated for any company. The record of free cash flows can also be examined even for a non-dividend-paying company. FCFE can be viewed as measuring what a company can afford to pay out in dividends. Even for dividend-paying companies, a free cash flow model valuation may be preferred when dividends exceed or fall short of FCFE by significant amounts.⁶⁸ FCFE also represents cash flow that can be redeployed outside the company without affecting the company’s capital investments. A controlling equity interest can effect such redeployment. As a result, free cash flow valuation is appropriate for investors who want to take a control perspective. (Even a small shareholder may want to take such a perspective when potential exists for the company to be acquired, because stock price should reflect the price an acquirer would pay.)

Just as there are cases in which an analyst would find it impractical to apply the DDM, applying the free cash flow approach is a problem in some cases. Some companies have intense capital demands and, as a result, have negative expected free cash flows far into the future. As one example, a retailer may be constantly constructing new outlets and be far from saturating even its domestic market. Even if the retailer is currently very profitable, free cash flow may be negative indefinitely because of the level of capital expenditures. The present value of a series of negative free cash flows is a negative number: The use of a free cash flow model may entail a long forecast horizon to capture the point at which expected free cash flow turns positive. The uncertainty associated with distant forecasts may be considerable. In such cases, the analyst may have more confidence using another approach, such as residual income valuation.

Generally, defining returns as free cash flow and using the FCFE (and FCFF) models are most suitable when

The third and final definition of returns that we discuss in this overview is residual income. Conceptually, residual income for a given time period is the earnings for that period in excess of the investors’ required return on beginning-of-period investment (common stockholders’ equity). Suppose shareholders’ initial investment is $200 million, and the required rate of return on the stock is 8 percent. The required rate of return is investors’ opportunity cost for investing in the stock: the highest expected return available from other equally risky investments, which is the return that investors forgo when investing in the stock. The company earns $18 million in the course of a year. How much value has the company added for shareholders? A return of 0.08 x $200 million = $16 million just meets the amount investors could have earned in an equivalent-risk investment (by the definition of opportunity cost). Only the residual or excess amount of $18 million - $16 million = $2 million represents value added, or an economic gain, to shareholders. So $2 million is the company’s residual income for the period. The residual income approach attempts to match profits to the time period in which they are earned (but not necessarily realized as cash). In contrast to accounting net income (which has the same matching objective in principle), however, residual income attempts to measure the value added in excess of opportunity costs.

The residual income model states that a stock’s value is book value per share plus the present value of expected future residual earnings. (Book value per share is common stockholders’ equity divided by the number of common shares outstanding.) In contrast to the dividend and free cash flow models, the residual income model introduces a stock concept, book value per share, into the present value expression. Nevertheless, the residual income model can be viewed as a restatement of the dividend discount model, using a company-level return concept. Dividends are paid out of earnings and are related to earnings and book value through a simple expression.⁶⁹ The residual income model is a useful addition to an analyst’s toolbox. Because the record of residual income can always be calculated, a residual income model can be used for both dividend-paying and non-dividend-paying stocks. Analysts may choose a residual income approach for companies with negative expected free cash flows within their comfortable forecast horizon. In such cases, a residual income valuation often brings the recognition of value closer to the present as compared with a free cash flow valuation, producing higher value estimates.

The residual income model has an attractive focus on profitability in relation to opportunity costs.⁷⁰ Knowledgeable application of the residual income model requires a detailed knowledge of accrual accounting; consequently, in cases for which the dividend discount model is suitable, analysts may prefer the DDM as the simpler choice. Management sometimes exercises its discretion within allowable accounting practices to distort the accuracy of its financials as a reflection of economic performance. If the quality of accounting disclosure is good, the analyst may be able to calculate residual income by making appropriate adjustments (to reported net income and book value, in particular). In some cases, the degree of distortion and the quality of accounting disclosure can be such that the application of the residual income model is error-prone.

Generally, the definition of returns as residual income, and the residual income model, is most suitable in either of the following two situations:

In summary, the three most widely used definitions of returns to investors are dividends, free cash flow, and residual income. Although claims are often made that one cash flow definition is inherently superior to the rest—often following changing fashions in investment practice—a more flexible viewpoint is practical. The analyst may find that one model is more suitable to a particular valuation problem. The analyst may also develop more expertise in applying one type of model. In practice, skill in application—in particular, the quality of forecasts—is frequently decisive for the usefulness of the analyst’s work.

In the next section, we present the general form of the dividend discount model as a prelude to discussing the particular implementations of the model that are suitable for different sets of attributes of the company being valued.

The DDM is the simplest and oldest present value approach to valuing stock. In a survey of CFA Institute⁷¹ members by Block (1999), 42 percent of respondents viewed the DDM as “v ery important” or “moderately important” for determining the value of individual stocks. Beginning in 1989, the Merrill Lynch Institutional Factor Survey has assessed the popularity of 23 valuation factors and methods among a group of institutional investors. The highest recorded usage level of the DDM was in the first survey in 1989, when more than 50 percent of respondents reported using the DDM. Although DDMs have had five years of popularity increases since 1989 (with a notable rebound to 39 percent usage in 2002), the long-term trend has been one of decline, with usage slightly more than 20 percent in 2006—still a significant presence. Besides its continuing significant position in practice, the DDM has an important place in both academic and practitioner equity research. The DDM is, for all these reasons, a basic tool in equity valuation.

If an investor wishes to buy a share of stock and hold it for one year, the value of that share of stock today is the present value of the expected dividend to be received on the stock plus the present value of the expected selling price in one year: where

Equation 3-2 applies to a single holding period the principle that an asset’s value is the present value of its future cash flows. In this case, the expected cash flows are the dividend in one year (for simplicity, assumed to be received as one payment at the end of the year)⁷² and the price of the stock in one year.

With a finite holding period, whether one, two, five, or some other number of years, the dividend discount model finds the value of stock as the sum of (1) the present values of the expected dividends during the holding period, and (2) the present value of the expected stock price at the end of the holding period. As the holding period is increased by one year, we have an extra expected dividend term. In the limit (i.e., if the holding period extends into the indefinite future), the stock’s value is the present value of all expected future dividends.

Equation 3-7 is the general form of the dividend discount model, first presented by John Burr Williams (1938). Even from the perspective of an investor with a finite investment horizon, the value of stock depends on all future dividends. For that investor, stock value today depends directly on the dividends the investor expects to receive before the stock is sold and indirectly on the expected dividends after the stock is sold, because those future dividends determine the expected selling price.

Equation 3-7, by expressing the value of stock as the present value of expected dividends into the indefinite future, presents a daunting forecasting challenge. In practice, of course, analysts cannot make detailed, individual forecasts of an infinite number of dividends. To use the DDM, the forecasting problem must be simplified. Two broad approaches exist, each of which has several variations:

Spreadsheets are particularly convenient tools for implementing a DDM with individual dividend forecasts, but are useful in all cases. We address spreadsheet modeling in Section 5.

Whether analysts are using dividends or some other definition of cash flow, they generally use one of the preceding forecasting approaches when valuing stock. The challenge in practice is to choose an appropriate model for a stock’s future dividends and to develop quality inputs to that model.

Both equations are equivalent because D₁ = D₀(1 + g ). In Equation 3-10, it must be specified that the required return on equity must be greater than the expected growth rate: r> g. If r is equal to or less than g, Equation 3-10 as a compact formula for value assuming constant growth is not valid. If r equals g, dividends grow at the same rate at which they are discounted, so the value of the stock (as the undiscounted sum of all expected future dividends) is infinite. If r is less than g, dividends grow faster than they are discounted, so the value of the stock is infinite. Of course, infinite values do not make economic sense; so constant growth with r equal to or less than g does not make sense.

To illustrate the calculation, suppose that an annual dividend of €5 has just been paid (D₀ = €5). The expected long-term growth rate is 5 percent and the required return on equity is 8 percent. The Gordon growth model value per share is D₀(1 + g)/(r -g ) = (€5 x 1.05)/(0.08 - 0.05) = €5.25/0.03 = €175. When calculating the model value, be careful to use D₁and not D₀in the numerator.

The Gordon growth model (Equation 3-10) is one of the most widely recognized equations in the field of security analysis. Because the model is based on indefinitely extending future dividends, the model’s required rate of return and growth rate should reflect long-term expectations. Further, model values are very sensitive to both the required rate of return, r, and the expected dividend growth rate, g. In this model and other valuation models, it is helpful to perform a sensitivity analysis on the inputs, particularly when an analyst is not confident about the proper values.

Earlier we stated that analysts typically apply DDMs to dividend-paying stocks when dividends bear an understandable and consistent relation to the company’s profitability. The same qualifications hold for the Gordon growth model. In addition, the Gordon growth model form of the DDM is most appropriate for companies with earnings expected to grow at a rate comparable to or lower than the economy’s nominal growth rate. Businesses growing at much higher rates than the economy often grow at lower rates in maturity, and the horizon in using the Gordon growth model is the entire future stream of dividends.

To determine whether the company’s growth rate qualifies it as a candidate for the Gordon growth model, an estimate of the economy’s nominal growth rate is needed. This growth rate is usually measured by the growth in gross domestic product(GDP). (GDP is a money measure of the goods and services produced within a country’s borders.) National government agencies as well as the World Bank (www.worldbank.org) publish GDP data, which are also available from several secondary sources. Exhibit 3-2 shows the recent real GDP growth record for a number of major developed markets.

Based on historical and/or forward-looking information, nominal GDP growth can be estimated as the sum of the estimated real growth rate in GDP plus the expected long-run inflation rate. For example, an estimate of the underlying real growth rate of the Canadian economy is 3 percent as of early 2007. By using the Bank of Canada’s inflation target of 2 percent as the expected inflation rate, an estimate of the Canadian economy’s nominal annual growth rate is 3 percent + 2 percent = 5 percent. Publicly traded companies constitute varying amounts of the total corporate sector, but always less than 100 percent. As a result, the overall growth rate of the public corporate sector can diverge from the nominal GDP growth rate during a long horizon; furthermore, within the public corporate sector, some subsectors may experience persistent growth rate differentials. Nevertheless, an earnings growth rate far above the nominal GDP growth rate is not sustainable in perpetuity.

When forecasting an earnings growth rate far above the economy’s nominal growth rate, analysts should use a multistage DDM in which the final-stage growth rate reflects a growth rate that is more plausible relative to the economy’s nominal growth rate, rather than using the Gordon growth model.

Example 3-6 illustrates a Gordon growth model valuation introducing some problems the analyst might face in practice. The example refers to adjusted beta; the most common calculation adjusts raw historical beta toward the overall mean value of one for beta.

As mentioned earlier, an analyst needs to be aware that Gordon growth model values can be very sensitive to small changes in the values of the required rate of return and expected dividend growth rate. Example 3-7 illustrates a format for a sensitivity analysis.

Examples 3-6 and 3-7 illustrate the application of the Gordon growth model to a utility, a traditional source for such illustrations because of the stability afforded by providing an essential service in a regulated environment. Before applying any valuation model, however, analysts need to know much more about a company than industry membership. For example, another water utility, Aqua America Inc. (NYSE: WTR), is expected to be on a greater than 10 percent per year growth path for an extended period as a result of an aggressive acquisition program. Furthermore, many utility holding companies in the United States have major, nonregulated business subsidiaries so the traditional picture of steady and slow growth often does not hold.

In addition to individual stocks, analysts have often used the Gordon growth model to value broad equity market indexes, especially in developed markets. Because the value of publicly traded issues typically represents a large fraction of the overall corporate sector in developed markets, such indexes reflect average economic growth rates. Furthermore, in such economies, a sustainable trend value of growth may be identifiable.

The Gordon growth model can also be used to value the noncallable form of a traditional type of preferred stock, fixed-rate perpetual preferred stock(stock with a specified dividend rate that has a claim on earnings senior to the claim of common stock, and no maturity date). Perpetual preferred stock has been used particularly by financial institutions such as banks to obtain permanent equity capital while diluting the interests of common equity. Generally, such issues have been callable by the issuer after a certain period, so valuation must take account of the issuer’s call option. Valuation of the noncallable form, however, is straightforward.

If the dividend on such preferred stock is D, because payments extend into the indefinite future a perpetuity(a stream of level payments extending to infinity) exists in the constant amount of D. With g = 0, which is true because dividends are fixed for such preferred stock, the Gordon growth model becomes

The discount rate r capitalizes the amount D, and for that reason is often called a capitalization rate in this expression and any other expression for the value of a perpetuity.

A perpetual preferred stock has a level dividend, thus a dividend growth rate of zero. Another case is a declining dividend—a negative growth rate. The Gordon growth model also accommodates this possibility, as illustrated in Example 3-9.

To summarize, g in the Gordon growth model is the rate of value or capital appreciation (sometimes also called the capital gains yield ). Some textbooks state that g is the rate of price appreciation. If prices are efficient (price equals value), price is indeed expected to grow at a rate of g. If there is mispricing (price is different from value), however, the actual rate of capital appreciation depends on the nature of the mispricing and how fast it is corrected, if at all. This topic is discussed in Chapter 2.

Another characteristic of the constant growth model is that the components of total return (dividend yield and capital gains yield) will also stay constant through time, given that price tracks value exactly. The dividend yield, which is D₁/P₀at t = 0, will stay unchanged because both the dividend and the price are expected to grow at the same rate, leaving the dividend yield unchanged through time. For example, consider a stock selling for €50.00 with a forward dividend yield(a dividend yield based on the anticipated dividend during the next 12 months) of 2 percent based on an expected dividend of €1. The estimate of g is 5.50 percent per year. The dividend yield of 2 percent, the capital gains yield of 5.50 percent, and the total return of 7.50 percent are expected to be the same at t = 0 and at any future point in time.

Earnings growth may increase, leave unchanged, or reduce shareholder wealth depending on whether the growth results from earning returns in excess of, equal to, or less than the opportunity cost of funds. Consider a company with a required return on equity of 10 percent that has earned €1 per share. The company is deciding whether to pay out current earnings as a dividend or to reinvest them at 10 percent and distribute the ending value as a dividend in one year. If it reinvests, the present value of investment is €1.10/1.10 = €1.00, equaling its cost, so the decision to reinvest has a net present value (NPV) of zero. If the company were able to earn more than 10 percent by exploiting a profitable growth opportunity, reinvesting would have a positive NPV, increasing shareholder wealth. Suppose the company could reinvest earnings at 25 percent for one year: The per-share NPV of the growth opportunity would be €1.25/1.10 - €1 ≈ €0.14. Note that any reinvestment at a positive rate below 10 percent, although increasing EPS, is not in shareholders’ interests. Increases in shareholder wealth occur only when reinvested earnings earn more than the opportunity cost of funds (i.e., investments are in positive net present value projects).⁷⁸ Thus, investors actively assess whether and to what degree companies will have opportunities to invest in profitable projects. In principle, companies without prospects for investing in positive NPV projects should distribute most or all earnings to shareholders as dividends so the shareholders can redirect capital to more attractive areas.

A company without positive expected NPV projects is defined as a no-growth company (a term for a company without opportunities for profitable growth). Such companies should distribute all their earnings in dividends because earnings cannot be reinvested profitably and earnings will be flat in perpetuity, assuming a constant return on equity (ROE). This flatness occurs because earnings equal ROE x Equity, and equity is constant because retained earnings are not added to it. E₁is t= 1 earnings, which is the constant level of earnings or the average earnings of a no-growth company if return on equity is viewed as varying about its average level. The no-growth value per share is defined as E₁/r, which is the present value of a perpetuity in the amount of E₁where the capitalization rate, r, is the required rate of return on the company’s equity. E₁/r can also be interpreted as the per-share value of assets in place because of the assumption that the company is making no new investments because none are profitable. For any company, the actual value per share is the sum of the no-growth value per share and the present value of growth opportunities (PVGO):

Exhibit 3-4, based on data from early October 2008, illustrates that the value of growth represented about 65 percent of the market value of technology company Google; a much smaller percentage of McDonald’s value; and for Macy’s, the value of growth appeared to be negative. The negative value for Macy’s PVGO could be explained in several ways. The value could reflect an expectation that management’s investment policy would destroy value (e.g., analysts were generally negative on the company’s business prospects and reinvestment in it might not cover all costs including the cost of funds); it could indicate that this stock’s price in a severe market break had lost contact with fundamentals (one month earlier the company’s share price was approximately double); or it might indicate that the estimated no-growth value per share was too high because the earnings estimate was too high and/or the required return on equity estimate was too low.

What determines PVGO? One determinant is the value of a company’s options to invest, captured by the word opportunities. In addition, the flexibility to adapt investments to new circumstances and information is valuable. Thus, a second determinant of PVGO is the value of the company’s options to time the start, adjust the scale, or even abandon future projects. This element is the value of the company’s real options(options to modify projects, in this context). Companies that have good business opportunities and/or a high level of managerial flexibility in responding to changes in the marketplace should tend to have higher values of PVGO than companies that do not have such advantages. This perspective on what contributes to PVGO can provide additional understanding of the results in Exhibit 3-4.

As an additional aid to an analyst, Equation 3-12 can be restated in terms of the familiar P/E ratio based on forecasted earnings:

As analysts, the distinction between no-growth and growth values is of interest because the value of growth and the value of assets in place generally have different risk characteristics (as the interpretation of PVGO as incorporating the real options suggests).

Leading and trailing justified P/E expressions can be developed from the Gordon growth model. Assuming that the model can be applied for a particular stock’s valuation, the dividend payout ratio is considered fixed. Define b as the retention rate, the fraction of earnings reinvested in the company rather than paid out in dividends. The dividend payout ratio is then, by definition, (1 - b) = Dividend per share/Earnings per share = D_t/E_t. If P₀ = D₁/(r - g) is divided by next year’s earnings per share, E₁, we have

Later in this chapter we present multistage DDMs. Expressions for the P/E can be developed in terms of the variables of multistage DDMs, but the usefulness of these expressions is not commensurate with their complexity. For multistage models, the simple way to calculate a justified leading P/E is to divide the model value directly by the first year’s expected earnings. In all cases, the P/E is explained in terms of the required return on equity, expected dividend growth rate(s), and the dividend payout ratio(s). All else equal, higher prices are associated with higher anticipated dividend growth rates.

The Gordon growth model is a single-stage DDM because all future periods are grouped into one stage characterized by a single growth rate. For many or even the majority of companies, however, future growth can be expected to consist of multiple stages. Multistage DDMs are the subject of the next section.

A company may attempt and succeed in restarting the growth phase by changing its strategic focuses and business mix. Technological advances may alter a company’s growth prospects for better or worse with surprising rapidity. Nevertheless, this growth-phase picture of a company is a useful approximation. The growth-phase concept provides the intuition for multistage discounted cash flow (DCF) models of all types, including multistage dividend discount models. Multistage models are a staple valuation discipline of investment management firms using DCF valuation models.

In the following sections, we present three popular multistage DDMs: the two-stage DDM, the H -model (a type of two-stage model), and the three-stage DDM. Keep in mind that all these models represent stylized patterns of growth; they are attempting to identify the pattern that most accurately approximates an analyst’s view of the company’s future growth.

In the second version, called the H-model, the dividend growth rate is assumed to decline from an abnormal rate to the mature growth rate during the course of stage 1. For example, the growth rate could begin at 15 percent and decline continuously in stage 1 until it reaches 7 percent. The second model will be presented after the general two-stage model.

The first two-stage DDM provides for a high growth rate for the initial period, followed by a sustainable and usually lower growth rate thereafter. The two-stage DDM is based on the multiple-period model where V_nis used as an estimate of P_n. The two -stage model assumes that the first n dividends grow at an extraordinary short-term rate, g_S:

The two-stage DDM is useful because many scenarios exist in which a company can achieve a supernormal growth rate for a few years, after which time the growth rate falls to a more sustainable level. For example, a company may achieve supernormal growth through possession of a patent, first-mover advantage, or another factor that provides a temporary lead in a specific marketplace. Subsequently, earnings will most likely descend to a level that is more consistent with competition and growth in the overall economy. Accordingly, that is why in the two-stage model, extraordinary growth is often forecast for a few years and normal growth is forecast thereafter. A possible limitation of the two-stage model is that the transition between the initial abnormal growth period and the final steady-state growth period is abrupt.

The accurate estimation of V_n, the terminal value of the stock (also known as its continuing value), is an important part of the correct use of DDMs. In practice, analysts estimate the terminal value either by applying a multiple to a projected terminal value of a fundamental, such as earnings per share or book value per share, or they estimate V_n using the Gordon growth model. In Chapter 6 we discuss using price-earnings multiples in this context.

In the examples, a single discount rate, r, is used for all phases, reflecting both a desire for simplicity and the lack of a clear objective basis for adjusting the discount rate for different phases. Some analysts, however, use different discount rates for different growth phases.

Example 3-14 values E. I. DuPont de Nemours and Company by combining the dividend discount model and a P/E valuation model.

To value a non-dividend-paying company using a DDM, generally an analyst can use a multistage DDM model in which the first-stage dividend equals zero. Example 3-15 llustrates the approach.

If a company is not paying a dividend but is very profitable, an analyst might be willing to forecast its future dividends. Of course, for non-dividend-paying, unprofitable companies, such a forecast would be very difficult. Furthermore, as discussed in Section 2.2 (“Streams of Expected Cash Flows”), it is usually difficult for the analyst to estimate the timing of the initiation of dividends and the dividend policy that will then be established by the company. Thus, the analyst may prefer a free cash flow or residual income model for valuing such companies.

The first term on the right-hand side of Equation 3-20 is the present value of the company’s dividend stream if it were to grow at g_Lforever. The second term is an approximation of the extra value (assuming g_S> g_L) accruing to the stock because of its supernormal growth for years 1 through 2H (see Fuller and Hsia 1984 for technical details).⁷⁹ Logically, the longer the supernormal growth period (i.e., the larger the value of H, which is one-half the length of the supernormal growth period) and the larger the extra growth rate in the supernormal growth period (measured by g_S minus g_L ), the higher the share value, all else equal. To illustrate the expression, if the analyst in Example 3-13 had forecast a linear decline of the growth rate from 15 percent to 8 percent over the next 10 years, his estimate of value using the H -model would have been €11.78 (rather than €16.09 as in Example 3-13):

The H-model is an approximation model that estimates the valuation that would result from discounting all of the future dividends individually. In many circumstances, this approximation is very close. For a long extraordinary growth period (a high H) or for a large difference in growth rates (the difference between g_S and g_L), however, the analyst might abandon the approximation model for the more exact model. Fortunately, the many tedious calculations of the exact model are made fairly easy using a spreadsheet program.

Example 3-17 shows how the first type of the three-stage model can be used to value a stock, in this case IBM.

A second version of the three-stage DDM has a middle stage similar to the first stage in the H-model. In the first stage, dividends grow at a high, constant (supernormal) rate for the whole period. In the second stage, dividends decline linearly as they do in the H-model. Finally, in stage 3, dividends grow at a sustainable, constant growth rate. The process of using this model involves four steps:

In the first step, analysts often investigate the company more deeply, making explicit, individual earnings and dividend forecasts for the near future (often 3, 5, or 10 years), rather than applying a growth rate to the current level of dividends.

In problem 5 of Example 3-18, the analyst examined the consequences of 12 percent growth in year 1 and no growth in year 2, with 12 percent growth resuming in years 3, 4, and 5. In the first stage, analysts may forecast earnings and dividends individually for a certain number of years.

The three-stage DDM with declining growth in stage 2 has been widely used among companies using a DDM approach to valuation. An example is the DDM adopted by Bloomberg L.P., a financial services company that provides Bloomberg terminals to professional investors and analysts. The Bloomberg DDM is a model that provides an estimated value for any stock that the user selects. The DDM is a three-stage model with declining growth in stage 2. The model uses fundamentals about the company for assumed stage 1 and stage 3 growth rates, and then assumes that the stage 2 rate is a linearly declining rate between the stage 1 and stage 3 rates. The model also makes estimates of the required rate of return and the lengths of the three stages, assigning higher growth companies shorter growth periods (i.e., first stages) and longer transition periods, and slower growth companies longer growth periods and shorter transition periods. Fixing the total length of the growth and transition phases together at 17 years, the growth stage/transition stage durations for Bloomberg’s four growth classifications are 3 years/14 years for “explosive growth” equities, 5 years/12 years for “high growth” equities, 7 years/10 years for “average growth” equities, and 9 years/8 years for “slow/mature growth” equities. Analysts, by tailoring stage specifications to their understanding of the specific company being valued, should be able to improve on the accuracy of valuations compared to a fixed specification.

Spreadsheets allow the analyst to build complicated models that would be very cumbersome to describe using algebra. Furthermore, built-in spreadsheet functions (such as those for finding rates of return) use algorithms to get a numerical answer when a mathematical solution would be impossible or extremely challenging. Because of the widespread use of spreadsheets, several analysts can work together or exchange information by sharing their spreadsheet models. Example 3-19 presents the results of using a spreadsheet to value a stock with dividends that change substantially through time.

As the table in Example 3-19 shows, the total present value of Yang Co.’s dividends is $399.48. In this example, the terminal value of the company (V_n) at the end of the first stage is found using the Gordon growth model and a mature growth rate of 5 percent. Several alternative approaches to estimating g are available in this context:

In some cases, finding the IRR is very easy. In the Gordon growth model, r = D₁/P₀+ g. The required return estimate is the dividend yield plus the expected dividend growth rate. For a security with a current price of $10, an expected dividend of $0.50, and expected growth of 8 percent, the required return estimate is 13 percent.

For multistage models and spreadsheet models, finding a single equation for the rate of return can be more difficult. The process generally used is similar to that of finding the IRR for a series of varying cash flows. Using a computer or trial and error, the analyst must find the rate of return such that the present value of future expected dividends equals the current stock price.

In general, multistage DDMs make stylized assumptions about growth based on a life-cycle view of business. The first stage of a multistage DDM frequently incorporates analysts’ individual earnings and dividend forecasts for the next two to five years (sometimes longer). The final stage is often modeled using the Gordon growth model based on an assumption of the company’s long-run sustainable growth rate. In the case of the H-model, the transition to the mature growth phase happens smoothly during the first stage. In the case of the standard two-stage model, the growth rate typically transitions immediately to mature growth rate in the second period. In three-stage models, the middle stage is a stage of transition. Using a spreadsheet, an analyst can model an almost limitless variety of cash flow patterns.

Multistage DDMs have several limitations. Often, the present value of the terminal stage represents more than three-quarters of the total value of shares. Terminal value can be very sensitive to the growth and required return assumptions. Furthermore, technological innovation can make the life-cycle model a crude representation.

The expression to calculate the sustainable growth rate is where

Example 3-22 is an illustration of the fact that growth in shareholders’ equity is driven by reinvested earnings alone (no new issues of equity and debt growing at the rate g).⁸¹

Equation 3-22 implies that the higher the return on equity, the higher the dividend growth rate, all else constant. That relation appears to be reliable. Another implication of the expression is that the lower (higher) the earnings retention ratio, the lower (higher) the growth rate in dividends, holding all else constant; this relationship has been called thedividend displacement of earnings.⁸²Of course, all else may not be equal—the return on reinvested earnings may not be constant at different levels of investment, or companies with changing future growth prospects may change their dividend policy. Arnott and Asness (2003) and Zhou and Ruland (2006), in providing U.S.-based evidence that dividend-paying companies had higher future growth rates during the period studied, indicate that caution is appropriate in assuming that dividends displace earnings.

A practical logic for defining sustainable in terms of growth through internally generated funds (retained earnings) is that external equity (secondary issues of stock) is considerably more costly than internal equity (reinvested earnings), for several reasons including the investment banker fees associated with secondary equity issues. In general, continuous issuance of new stock is not a practical funding alternative for companies.⁸³ Growth of capital through issuance of new debt, however, can sometimes be sustained for considerable periods. Further, if a company manages its capital structure to a target percentage of debt to total capital (debt and common stock), it will need to issue debt to maintain that percentage as equity grows through reinvested earnings. (This approach is one of a variety of observed capital structure policies.) In addition, the earnings retention ratio nearly always shows year-to-year variation in actual companies. For example, earnings may have transitory components that management does not want to reflect in dividends. The analyst may thus observe actual dividend growth rates straying from the growth rates predicted by Equation 3-22 because of these effects, even when his input estimates are unbiased. Nevertheless, the equation can be useful as a simple expression for approximating the average rate at which dividends can grow over a long horizon.

This model can be expanded further by breaking ROA into two components, profit margin and turnover (efficiency):

Theoretically, the sustainable growth rate expression and this expansion of it based on the DuPont decomposition of ROE hold exactly only when ROE is calculated using beginning-of-period shareholders’ equity, as illustrated in Example 3-22. Such calculation assumes that retained earnings are not available for reinvestment until the end of the period. Analysts and financial databases more frequently prefer to use average total assets in calculating ROE and, practically, DuPont analysis is frequently performed using that definition (see Robinson et al. 2009a). Example 3-23 illustrates the logic behind this equation.

If growth is being forecast for the next five years, an analyst should use the expectations of the four factors driving growth during this five-year period. If growth is being forecast into perpetuity, an analyst should use very long-term forecasts for these variables.

To illustrate the calculation and implications of the sustainable growth rate using the expression for ROE given by the DuPont formula, assume the growth rate is g = b x ROE = 0.60 x 15% = 9 percent. The ROE of 15 percent was based on a profit margin of 5 percent, an asset turnover of 2.0, and an equity multiplier of 1.5. Given fixed ratios of sales-to-assets and assets-to-equity, sales, assets, and debt will also be growing at 9 percent. Because dividends are fixed at 40 percent of income, dividends will grow at the same rate as income, or 9 percent. If the company increases dividends faster than 9 percent, this growth rate would not be sustainable using internally generated funds. Earning retentions would be reduced, and the company would not be able to finance the assets required for sales growth without external financing.

An analyst should be careful in projecting historical financial ratios into the future when using this analysis. Although a company may have grown at 25 percent per year for the past five years, this rate of growth is probably not sustainable indefinitely. Abnormally high ROEs, which may have driven that growth, are unlikely to persist indefinitely because of competitive forces and possibly other reasons, such as adverse changes in technology or demand. In Example 3-24, an above-average terminal growth rate is plausibly forecasted because the company has positioned itself in businesses that may have relatively high margins on an ongoing basis.