Valuation in Practice Introduction The previous chapter investigated the theory of valuation. That theory has proved useful in valuing options, futures, and other derivative instruments, but has yet to be used for the valuation of equities. In this chapter, we will look at some of the quantitative methods that have been used for equity valuation. These will be ad hoc, although they will have some vague connection with theory. The reader should not be surprised that the chapter does not describe a ''right" way to value stocks. We described the theoretically correct approach in the previous chapter. This chapter is evidence of our inability to make the theoretically correct scheme operational. We have to turn to more ad hoc schemes. The reader should keep the humility principle in mind: The marketplace may be right and you may be wrong. The reader should also keep the fundamental law of active management in mind: You don't have to be right much more than 50 percent of the time to add value! With these modest goals in mind we shall commence. Insights included in this chapter are: • The basic theory of corporate finance provides ground rules for acceptable valuation models. • The standard valuation model is the dividend discount model, which focuses on dividends, earnings, and growth. Dividend discount models are only as good as their growth forecasts.
Page 226 • Comparative valuation models price attributes of a firm. • Returns-based analysis focuses directly on the ultimate goal of valuation models: forecasting exceptional returns. Returns-based analysis is related to APT models. The goal is to find assets that are unfairly valued by the market, hoping that the market will eventually correct itself. This requires some insight. Quantitative methods can help to focus that insight and use it efficiently, but they are not a substitute for insight. Corporate Finance The modern theory of corporate finance is based on the notion of market efficiency. Modigliani and Miller showed the power of market efficiency in their classic studies demonstrating that • Dividend policy influences only the scheduling of cash flows received by the shareholder. It is a "pay you now or pay you later" arrangement. Dividend policy doesn't affect the total value of the payments. • A firm's financing policy does not affect the total value of the firm. Financing policy will keep the total value of the firm's liabilities constant. These ideas were extremely controversial at first, but they have stood up for decades and are at least partially responsible for the authors' Nobel prizes. Active managers with market inefficiency in their veins can read these two precepts as: "Dividend and financing policy aren't very important." Any valuation method that hinges on some magic associated with dividends or debt financing may be dangerous. The economic value of a firm comes from its profitable activities. If the firm can transform inputs it buys for $1.00 into outputs it sells 3 months later for $1.45, then the firm can profit. If the firm can find additional projects that create value, then the firm can grow. This ability to generate profits and to make those profits grow is at the heart of attempts at valuing the firm. Much of the confusion about the Modigliani and Miller results arises from two issues: taxes and a failure to separate the operations of the firm from its financing activities.
Page 227 Let's ignore taxes for the moment. The failure to separate operations from financial decisions is quite natural. A firm borrows because it has capital expenditures that are needed to support future growth. A firm increases its dividend because its operations have been successful and future success is anticipated. We want to separate the operational considerations (the new plant, the successful product, etc.) from the financial. The new plant could have been financed from either retained earnings (reduced dividends), sale of new equity, or issuance of new debt. The benefits of the successful product launch could be used to retire debt, kept as retained earnings, or distributed as dividends. With the aid of the Modigliani and Miller principles, we can consider the firm's equity value as stemming from two sources: operational value and financial value: In a simple context, the financial value is the difference between the firm's capital surplus and its debt. The capital surplus is any money left after we have paid dividends and interest (if any) and paid for new investments that are needed in order to grow or sustain the operational side of the business. The operational value is derived from the revenue from operations (excluding interest), less the cost of those operations (labor, materials, and support, but not interest expense), and less the capital costs of maintaining and augmenting the capital stock (plant, machines, research, etc.). As an example, think of two firms operating side by side. The widget side of the business builds widgets, sells widgets, and invests in new and better widget-making equipment. The financing side of the business pays interest, pays dividends, retains earnings, issues (or repurchases) shares, and issues (or repurchases) bonds. If the financial side of the business is net long (retained earnings plus paid-in capital exceeds debt), then it invests the balance at the risk-free rate. If the financing side is net short, then it pays interest on the difference between the equity account and the bond account. As Modigliani and Miller point out, value is created on the widget side of the business. The financial side of the business moves money through time and allocates shares of the operational value between stock- and bondholders. A debt incurred today implies a
Page 228 sequence of interest payments in the future. The present value of those interest payments is equal to the present value of the debt. Figure 9.1 shows the flows into and out of our prototype firm. On the first level, we have the operating company. We call its output the cash flow. The cash flow is an input to the financial company. As we can see, the financial company pays dividends, pays interest if there is a debt position, collects interest on the capital surplus (if any), and either issues or repurchases dept and equity. Figure 9.1 might be called the world according to Modigliani and Miller. It is a conceptually useful way to look at the firm. Unfortunately, accountants don't share this view. Accounting information is based on the aggregate of the operational and financial sides of the firm. By adjusting dividend and debt policy, it is possible to manipulate not only dividends and debt/equity ratios, but earnings per share (EPS), the growth in EPS, earnings-to-price ratios, and book-to-price ratios. The reader should keep these Modigliani and Miller principles in mind in conjuring up and evaluating valuation schemes. Most of those schemes start with the notion of dividends. We'll start there too. Figure 9.1
Page 229 The Dividend Discount Model John Burr Williams, in his classic Theory of Investment Value, anticipated much of modern financial theory. In particular, Williams stressed the role of dividends as a determinant of value in an early version of the dividend discount model. An investor is paid for an investment either through dividends or through the sale of the asset. The sale price is based on the market's assessment of the firm's ability to pay future dividends. Other variables such as earnings may be important in valuing a firm, but their importance is derived from their ability to predict the future flow of dividends to the investor. So high current earnings may signal the firm's ability to increase dividends, and low current earnings signal a possible future decrease in dividends, or at least delays in future increases. This emphasis on dividends appears to clash with the Modigliani and Miller principle on dividend policy. This is not really the case. You can read the Modigliani and Miller principle as "pay me now or pay me later." What Modigliani and Miller say is that the firm is free to schedule the payment of the dividends to the investor in any way it likes. The scheduling will not (or should not) affect the market's perception of the value of the firm. The emphasis on dividends may also appear jarring to today's U.S. equity investors, conditioned to focus almost exclusively on price appreciation, almost to the point of disdaining current dividends. This wasn't always the case. In the 1950s and earlier, dividend yields exceeded bond yields. Equities are riskier than bonds, so investors required added incentives to purchase equities, according to the logic then. Even now, high-yield bonds follow this trend. Certainty If p(0) is the price of the firm at time 0, iF is the per-period interest rate, and d(t) is the certain dividend to be paid at time t, then both Williams and our theoretical valuation formula would say that
Page 231 The Constant-Growth Dividend Discount Model The constant-growth, or Gordon-Shapiro, dividend discount model assumes that dividends grow at a constant rate g. In other words, Substituting Eq. (9.4) into Eq. (9.3) and applying some algebra, we find the simplified formula This is the fundamental result of the constant-growth dividend discount model. Given a dividend d (1) and a growth rate g, the stock price increases as the dividend discount rate decreases, and vice versa. Low prices imply high dividend discount rates, and high prices imply low dividend discount rates or high growth rates. We will now consider another approach that leads to the same destination and provides further insight into Eq. (9.5). We can split the return on a stock into two parts, the dividend yield and the capital appreciation: where p = the price of the stock at the beginning of the period = the price of the stock at the end of the period d = the dividend paid in the period (assumed to be paid at the end) iF = the risk-free rate of interest r = the excess return ξ = the uncertain amount of capital appreciation Let g = E{ξ} be the expected rate of capital appreciation, f = E{r} be the expected excess return, and y = iF + f be the expected total rate of return. Taking expected values, Eq. (9.6) becomes We are assuming that the dividend is known or that d represents the expected dividend. We are also assuming that the expected total rate of return over the period, iF + f, equals the internal rate of
Page 232 return y, which is the (constant) average return calculated over the entire future stream of dividends. Solving Eq. (9.7) for the price leads back to the constant-growth dividend discount model result: Equations (9.5) and (9.8) imply that the expected rate of capital appreciation is identical to the expected rate of dividend growth. So far, this is something of a tautology, since we have, in fact, defined g to make this work. But we will next introduce a model for growth g, to show that we can equate it to the expected rate of capital appreciation. Modeling Growth We can use a simple model to show that the expected rate of capital appreciation equals the growth in the company's earnings per share and the growth in the company's dividends. Let e(t) = earnings in period t d(t) = dividends paid out at the end of period t κ = company's payout ratio I(t) = amount reinvested ρ = return on reinvested earnings and assume that the payout ratio κ and reinvestment rate of return ρ remain constant. In particular, assume that $1 of investment produces an expected perpetual stream of ρ dollars per period. Then, earnings either flow into dividends or are reinvested: The dividends constitute a fraction κ of the earnings: The reinvested earnings constitute the remaining fraction (1 – κ) of the earnings:
Page 233 And, since reinvestment produces returns ρ, we can determine e(t + 1) based on e(t) and the fraction reinvested: Equation (9.12) simply states that next year's earnings equal this year's earnings plus an increase due to the return on the portion of this year's earnings that was reinvested (the increase in equity). Of course, e(t) and e(t + 1) lead to the growth rate: The payout ratio is constant, and hence dividends are proportional to earnings. Therefore, the dividend growth rate [in Eq. (9.15)] is also the earnings growth rate [in Eq. (9.13)]. Moreover, this growth rate is determined by both the reinvestment rate 1 – κ (a measure of opportunity) and the average return on reinvested capital ρ [from Eq. (9.14)]. This average return on invested capital is, in turn, linked to the return on equity. Suppose b(t) is the book value at time t, and we start with e(1) = ρ · b(0); then book value will grow at rate g as well, and ρ will be the constant return on equity: e (t) = ρ · b(t – 1). Multiple Stocks The more general form of the dividend discount model for multiple stocks indexed by n = 1, 2, . . . N is The multiple-stock version of Eq. (9.7) becomes
Page 234 We can give this analysis some teeth by combining it with the consensus expected returns. The expected return fn includes both consensus expected returns2 and alphas: fn = βn · fB + αn. Substituting into Eq. (9.17), we see that We illustrate this relationship in Fig. 9.2, for a 4 percent risk-free rate, a 6 percent expected excess return on the benchmark, and an asset beta of 1.2. If the asset's yield is 2.5 percent, then it will be fairly priced, αn = 0, if the expected rate of capital appreciation is 8.7 percent. Figure 9.2 2Here we will assume that the expected benchmark return fB matches the long-run consensus expected benchmark return µB; i.e., we are assuming no benchmark timing.
Page 235 If we solve Eq. (9.18) for alpha, then we have a simple model for expected exceptional return in terms of yield, risk (as measured by beta), and growth: This formula points out the most important insight we must keep in mind while using a dividend discount model: The Golden Rule of the Dividend Discount Model: g in, g out. Each additional 1 percent of growth adds 1 percent to the alpha. The alphas that come out of the dividend discount model are as good as (or as bad as) the growth estimates that go in. If it's garbage in, then it's garbage out. Implied Growth Rates We can use Eq. (9.19) in a novel way: starting with the presumption that the assets are fairly priced, and determining the growth rates necessary to fairly price the assets. We call these the implied growth rates: We know the risk-free rate iF, and we can estimate the beta and the yield with reasonable accuracy. We can also make a reasonable estimate of the expected excess return on the market fB. The implied growth rates are handy in several ways. First they provide a rational feel for what growth rates should be, and help to point out consistent biases in analysts' estimates, either within sectors or across all stocks. Second, the implied growth rates can identify companies whose prices reflect unrealistic growth prospects. For example, in Table 9.1 we have listed the Major Market Index stocks along with their yield, 60month historical beta, and implied growth rates as of December 1992. The table assumes an expected excess return on the Major Market Index, fB, of 6 percent and uses the risk-free rate (3month Treasury bills) of 3.1 percent, which was the rate at the end of December 1992.
Page 236 TABLE 9.1 Stock Yield Beta Implied Growth Rate American Express 4.00% 1.21 6.36% AT&T 2.60% 0.96 6.26% Chevron 4.70% 0.45 1.10% Coca-Cola 1.30% 1.00 7.80% Disney 0.50% 1.24 10.04% Dow Chemical 4.50% 1.11 5.26% DuPont 3.70% 1.09 5.94% Eastman Kodak 4.90% 0.60 1.80% Exxon 4.70% 0.47 1.22% General Electric 2.90% 1.31 8.06% General Motors 2.50% 0.90 6.00% IBM 9.60% 0.64 –2.66% International Paper 2.50% 1.16 7.56% Johnson & Johnson 1.80% 1.15 8.20% McDonalds 0.80% 1.07 8.72% Merck 2.30% 1.09 7.34% 3M 3.20% 0.74 4.34% Philip Morris 3.40% 0.97 5.52% Procter & Gamble 2.10% 1.01 7.06% Sears 4.40% 1.04 4.94% We can see that reasonable growth rates average about 5.5 percent and that the standard deviation of growth rates in the Major Market Index is 3.1 percent. One of the difficulties in using a dividend discount model is implicit in the Golden Rule: The growth inputs can be wildly unrealistic. They tend to be too large in general, since Wall Street research (which drives the consensus) is interested in selling stocks, and positive bullish outlooks help to sell stocks. Dealing with Unrealistic Growth Rates We can treat this problem of unrealistic growth estimates in three ways. The first is a direct approach: 1. Group stocks into sectors.
Page 237 2. Calculate the implied growth rate for each stock in the sector. 3. Modify the growth forecasts so that they have the same mean and standard deviation as the implied growth rates in each sector. This approach linearly transforms the growth estimate gn to where we choose a and b to match the mean and standard deviation of the implied growth rates The result is where we calculate the mean and standard deviation cross-sectionally over stocks in the sector. The revised growth estimates are the sector average implied growth rate plus a term proportional to the difference between the initial growth estimates and that sector average implied growth rate. Table 9.2 shows the results of this approach for the stocks in the Major Market Index, using the 5year historical earnings-per-share growth rates (as of December 1992) as inputs. In this table, we have treated all these stocks as one sector. Table 9.2 also shows the alphas from Eq. (9.19), based on the modified growth inputs. One possible difficulty with this approach is that it may delete valuable sector timing information. Remember that the implied growth rates assume zero alphas, so this scheme will move the sector alphas toward zero.3 If you have no sector timing ability, then this is not a problem, but if you are trying to time sectors, you must account for sector alphas when generating target mean growth rates for each sector. The second approach to dealing with unrealistic growth estimates is a variant of the first approach which can, in principle, 3Issues of coverage and capitalization weighting keep the sector alphas from exactly equaling zero after this procedure.
Page 238 TABLE 9.2 Stock Growth Forecast Modified Forecast Alpha American Express –3.20% 4.90% –1.46% AT&T –19.21% 2.21% –4.05% Chevron 3.18% 4.91% 3.81% Coca-Cola 19.31% 8.68% 0.88% Disney 12.18% 7.49% –2.55% Dow Chemicals –23.22% 1.54% –3.72% DuPont –7.46% 4.19% –1.75% Eastman Kodak –37.51% –0.86% –2.66% Exxon 3.32% 6.00% 4.78% General Electric 16.14% 8.15% 0.09% General Motors 4.72% 6.23% 0.23% IBM –32.53% –0.03% 2.63% International Paper –10.86% 3.62% –3.94% Johnson & Johnson 15.27% 8.01% –0.19% McDonalds 12.09% 7.47% –1.25% Merck 22.95% 9.30% 1.96% 3M 4.48% 6.19% 1.85% Philip Morris 22.92% 9.29% 3.77% Procter & Gamble 23.30% 9.35% 2.29% Sears –7.96% 4.10% –0.84% Mean 0.58% 5.54% -0.01% Standard deviation 18.40% 3.09% 2.70% account for the investor's skill in predicting growth. As we will discover in Chap. 10, a basic linear forecasting result is Unlike in Eq. (9.22), we start with the stock's implied growth rate, not the sector average, and shift away from it based on comparing the initial growth forecast to the stock's implied growth rate. Furthermore, as we will learn in Chap. 10, the constant c depends in part on the investor's skill in predicting growth, where we measure skill using the correlation between forecast and realized
Page 239 growth rates. With no skill, we set c to zero. We leave the details to Chap. 10. A third, more conventional and more elaborate, approach to getting realistic growth inputs is the three-stage dividend discount model. The Three-Stage Dividend Discount Model The three-stage dividend discount model is built on the premise that we may have some insights about a company's prospects over the short run (1 to 4 years), but we have little long-range information. Therefore, we should use something like the implied growth rate for the company (or the sector) as the long-run value, and interpolate between the long and the short. The idea stems from the initial dividend discount formula [Eq. (9.3)], which would naturally lead one to distinguish between long-range and short-range growth rates. The three-stage dividend discount model is complicated. The reader who is uninterested in the details can skip to the next subsection, where we comment (negatively) on the net effect of the three-stage approach. There are several ways to build a three-stage dividend discount model. What follows is typical, and we don't believe that different model structures give materially different results. The required inputs include the following: T1, T2 = times which define the stages—stage 1 runs from 0 to T1, stage 2 from T1 to T2, and stage 3 from T2 onward gIN = the short-term (0 to T1) forecast growth in earnings per share gEQ = the long-range or equilibrium (T2 onward) forecast growth in earnings per share κ0 = the current (0 to T1) payout ratio κEQ = the long-run (T2 onwards) payout ratio EPS(1) = the forecast of next year's earnings per share yEQ = the equilibrium expected rate of return, iF + β · fB
Page 240 We input the short-term and long-term growth forecasts. For the growth during the intermediate period, we linearly interpolate: for T1 ≤ t ≤ T2. We similarly interpolate the payout ratios κ(t). Then the earnings per share and dividend paths will be We now proceed by first finding a time T2 value of the company, and then focusing on the current value of the company. At timeT2, we can value the company using the equilibrium growth and dividend discount rates: Then we can solve for the dividend discount rate y that discounts to the current price p(0): The Three-Stage Dividend Discount Model Evaluated The entire three-stage dividend discount procedure is motivated by the tension between a short-run growth rate gIN and a long-run rate gEQ. In spirit, it is just another approach to adjusting gIN toward gEQ and applying Eq. (9.3). A cynical view would even be that this is an elaborate (and confusing) way to smooth the growth inputs. It is important to remember that the three-stage dividend discount
Page 241 Figure 9.3 Three stage dividend discount model. Alpha versus short term growth forecast. model is not magic. It cannot transform bad growth forecasts into good growth forecasts. The golden rule still applies. To understand this better, consider the following example. We will set up a typical three-stage dividend discount model4 and use it to estimate internal rates of return and alphas (by subtracting iF + β · fB). Figure 9.3 illustrates the almost linear relationship between gIN and alpha. In this case, the implied growth rate is 12.62 percent, and each additional 1 percent of growth adds 0.54 percent to the alpha. The golden rule has been modified slightly, to say ''g in, 0.54 · (g – 12.62 percent) out." We can apply the same analysis to observe the sensitivity of alpha to the initial earnings per share forecast. Figure 9.4 illustrates the result. The relationship between the initial earnings per share forecast and the alpha is slightly nonlinear; however, 30 basis points of alpha for each $0.20 of earnings per share is a reasonable guide. 4The parameters are 5 and 15 years for the stage times, 0.0 for the short-run payout, and 0.3 for the long-run payout. The long-run expected market excess return was 5.5 percent, the stock's beta was 1.2, and the risk-free rate was 4 percent. This means that the stock's long-run fair expected return was 10.60 percent. The first year's EPS number was $4.80, and the market price was $50.00.
Page 242 Figure 9.4 Three stage dividend discount model. Alpha versus initial earnings per share forecast. Dividend Discount Models and Returns To use a dividend discount model in the context of active portfolio management, the information in the model must be converted into forecasts of exceptional return. There are two standard approaches, based on calculating either internal rates of return or net present values. These approaches differ in their assumption of the horizon over which any misvaluation will disappear. In the internal rate of return mode, we use the stream of dividends and the current market price as inputs, and solve for the internal rate of return y which matches the dividends to the market price. We then obtain asset alphas in two steps: first aggregate the rates of returns yn to estimate the predicted benchmark excess return fB, and then convert the internal rates of return to expected residual returns: The underlying assumption is that the misvaluation of the asset will persist; i.e., after 1 year, the proper discount rate will still be
Page 243 yn and not the "fair" rate iF + βn · fB. To see this, we can verify, using Eq. (9.3), that the expected total rate of return for the asset will be yn, if we assume that yn remains constant. Mathematically, The expected price in 1 year is simply the dividend stream from year 2 on, discounted at the same internal rate of return yn. The alphas from Eq. (9.27) presume that the mispricing will persist indefinitely, and that one will continue to reap the benefits. In the net present value mode, we take the stream of dividends and a fair discount rate yn = iF + βn · fB as inputs, and solve for the fair market price as we would for a net present value calculation. We then compare this price to the current market price, to determine the degree of overvaluation or under-valuation. To convert these "pricing errors" into alphas, we could use the following two-step process. First we adjust fB (and hence yn) so that the aggregate fair market value of all the assets equals the aggregate current market value of the assets. Second, assuming that the pricing error disappears after 1 year, we define alpha as or approximately as5 The alpha forecast in Eq. (9.33) presumes that all of the misvaluation 5For monthly alpha forecasts, Eqs. (9.33) and (9.34) differ by about a factor of 1.01.
Page 244 will disappear in 1 year. In fact, we derive Eq. (9.33) by defining alpha using and assuming i.e., the end-of-year price is the net present value of the future stream of dividends, discounted at the fair rate iF + βn · fB. Beneficial Side Effects of Dividend Discount Models Beyond their possible value in forecasting exceptional returns, dividend discount models also have some beneficial institutional side effects. The dividend discount model is a process. Used properly, it entails keeping records and viewing all assets on an equal footing. Hence, it can identify the judgmental input to a portfolio. We can compare the portfolio that the dividend discount model would have purchased with the actual portfolio, and attribute the difference to judgmental overrides and implementation costs (trading, etc.). In the longer run, this process can lead to evaluation and improvement of the inputs. On the downside, total devotion to the dividend discount model as the only appoach to valuation is a form of tunnel vision. In the remainder of this chapter we will consider some other practical approaches to valuation. Comparative Valuation The theoretical valuation formula of Chap. 8 and the dividend discount model look to future dividends as the source of value. There is an alternative, which involves looking at the current characteristics of a company and asking if that company is fairly valued relative to other companies with similar characteristics. The most obvious and simple example compares companies solely on the basis of current
Page 245 price-earnings ratios. The Wall Street Journal provides these numbers every day. A second example is the rules of thumb quoted by investment bankers: "Consulting firms sell for two times revenues," "Asset management firms sell for 5 percent of assets," etc. These examples are familiar, unidimensional (and possibly static) approaches to valuation based on current attributes. Comparative valuation, as we will describe it, is a more systematic and sophisticated (multidimensional) application of this idea: pricing by comparison with analogous companies, or as a function of sales and/or earnings multiples. We will motivate6 this approach using the dividend discount model as a starting point. The "clean surplus" equation of accounting7 connects a company's dividends, earnings, and book value according to where we measure the book values b(t – 1) and b(t) at the beginning and end of period t (running from t – 1 to time t), the earning se(t) accrue during period t, and the company pays the dividend d(t) at the end. We can use the dividend discount rate y to split earnings into two parts: required and exceptional: Anticipating that these exceptional earnings e*(t) will die out gradually leads to where δ < 1. Now combine Eqs. (9.37), (9.38), and (9.39) to estimate the expected dividend, which is what the dividend discount model requires: 6In this case, "motivate" will involve a set of assumptions and logic leading to a formula, rather than any rigorous derivation. We will then use the formula in an empirical way. 7See Ohlson (1989) for an in-depth discussion.
Page 246 Now substitute this expected dividend stream into the dividend discount model [Eq. (9.3)] to find (after some manipulation) Equation (9.41) expresses the current price as a linear combination of anticipated earnings e(1) and current book value b(0). The coefficients are company-specific and depend on the dividend discount rate and the persistence of exceptional earnings. This particular derivation is motivation for a more general application. In the general case, we would like to explain today's price p(0) in terms of several characteristics: earnings, book, anticipated earnings, cash flow, dividends, sales, debt, etc. The idea is to use a cross-sectional comparison of similar companies and thus isolate those that are overpriced or underpriced. For example, suppose we wished to apply the ideas contained in Eq. (9.41) to a group of similar medium-sized electrical utilities. We are looking for coefficients c1 and c2 to fit Starting from Eq. (9.41), the idea is that for these similar companies, the coefficients should be identical, and so the pricing error ∈n will identify misvaluations. The exact procedure for implementing Eq. (9.42) could vary, from an ordinary least-squares (OLS) regression, to a generalized least squares (GLS) regression that takes into account company prices, to a pooled cross-sectional and longitudinal regression that looks at the relationship across time and across assets. In general, the goal of comparative valuation is to estimate a relationship
Page 247 where the fitted price depends upon a variety of company characteristics. Equation (9.43) leads to an exceptional return forecast of presuming that the fitted price is more accurate than the market price, and that the asset will move to the fitted price over the horizon of the alpha. We can provide an arbitrage interpretation of this comparative analysis. Suppose we split the companies into two groups: the overvalued companies (with market price exceeding fitted price) and the undervalued companies (with market price less than fitted price). We can then construct a portfolio of the overvalued companies and a portfolio of the undervalued companies with identical attributes: e.g., the same sales, the same debt, and the same earnings. This creates two megacompanies—call them OV and UV, respectively—that are alike in all important attributes. However, we can purchase megacompany UV more cheaply than megacompany OV. Two "identical" companies selling for different prices is an arbitrage opportunity. This comparative valuation procedure requires only current and forecast information, e.g., current book value and anticipated earnings, and thus is independent of the past. It is not an extrapolation of historical data. It also is a high-tech way of mimicking the investment banking valuation procedures, which are typically grounded on multiples of sales and earnings. The hope is that the set of characteristics in the model completely determines company valuations. If the model omits some important attribute, the valuation may be misleading: The pricing errors may measure just the missing component, not an actual misvaluation. One example of a missing factor might be brand value. If we ignored brand name value and applied comparative valuation to a consumer goods sector, we might find some companies that were overvalued on average over
Page 248 time. This might not be an opportunity to sell, but rather the appearance of brand value. One way to account for this would be to compare current overvaluation or undervaluation to historical average valuations. Comparative valuation is quite flexible in practice. We can build separate valuation models for banks, mining and extraction companies, industrials, electric utilities, etc. This can help us focus on comparability within groups of similar companies. Returns-based Analysis The linear valuation model described above can serve as a useful introduction to returns-based analysis. The valuation model made a link between a purported percentage pricing error (the difference between the fitted and the current price divided by the current price) and the future residual return. Why not attack the problem directly and try to fit the price in a way that best forecasts residual returns? Suppose company n has attributes (earnings, dividends, book, sales, etc.) An,k(t) at time t. We might try the following model to directly explain residual returns using these attributes: Equation (9.45) is one example of returns-based analysis. A more typical returns-based analysis works with excess returns rn(t) rather than the residual returns θn(t), and uses risk control factors to separate residual and benchmark returns. Its basic form is similar to Eq. (9.45): but here the company exposures Xn,k(t) include both the attributes An,k(t) and the risk control factors. Equation (9.46) has the form of an APT model, as discussed in Chap. 7. At the same time, Eq. (9.46) may be somewhat more
Page 249 flexible. According to Chap. 7, we can fairly easily find qualified APT models—any factor model that can explain the returns to a diversified portfolio should probably do. But Eq. (9.46) could include all the APT model factors as risk factors, plus a new signal to forecast specific returns from that APT model. The APT theory doesn't allow forecasting of specific returns. But we are now in the ad hoc, empirical, nonefficient world of active management. We have the flexibility to forecast specific returns [although in Eq. (9.46) we express that forecast as another factor]. There are three important points to note concerning the use of the linear model [Eq. (9.46)] to forecast returns. First, in a true GLS regression, the factor return bk(t) is the return on a factor portfolio with unit exposure to factor k, zero exposure to the other factors, and minimum risk. If we know the exposures Xn,k(t) at the start of the period, even without knowing the returns rn(t), we can determine the holdings in the factor portfolio at the start of the period necessary to achieve the factor return bk(t). Second, the equations aggregate. In particular, looking at the benchmark return, We will utilize this linear aggregation property to help separate residual from benchmark returns. Third, when estimated using least-squares regression, the linear model [Eq. (9.46)] can be unduly influenced by outliers. One rule of thumb is to pull in all outliers in the Xn,k(t) to ±3 standard deviations about the mean. This will help avoid having all the explanatory power arise out of only one or two (possibly suspect) observations. The simplest approach to separating residual from benchmark returns is to model the residual returns directly, as in Eq. (9.45).
Page 250 Even then, we should be careful. The aggregate equation for the benchmark is Since θB(t) ≡ 0, we should adjust8 the variables An,k(t) so that aB,k(t) = 0. In the more typical situation where we are modeling excess returns, to separate benchmark from residual returns, we must include one or more variables that will pick up the benchmark return. The simplest approach is to include predicted beta as a risk control factor, and make sure that all other characteristics have a benchmark-weighted average of zero. The factor return associated with predicted beta will tend to pick up the benchmark return, leaving the other factors to describe residual returns. A more complicated procedure is to include a group of industry or sector variables as risk control factors. The factor returns associated with these industry or sector variables will pick up the benchmark effect.9 These industry or sector assignments add up to an intercept term in the regression. To separate bench8If Zn,k(t) is the raw description of the exposure of asset n to attribute k at time t, and is the benchmark exposure to attribute k at time t, then replace Zn,k(t) with If the beta of asset n is βn(t), then a slight variant is to replace Zn,k(t) with 9The remaining factor portfolio returns may still be correlated with the benchmark, even if the benchmark's exposure to the factor is zero. For example, factors associated with volatility (even after adjusting for market neutrality) tend to exhibit strong positive correlation with the market.
Page 251 mark from residual returns, the choice10 is between a beta and an intercept term. Using Returns-based Analysis Once set up, returns-based analysis examines past returns and attributes those returns to firm characteristics. The active manager starts the process by searching for appropriate characteristics. Some characteristics might directly explain residual returns—for example, alpha forecasts from dividend discount models or comparative valuation models. Other characteristics include reversal effects (e.g., last month's realized residual returns), momentum effects (e.g., the past 12 months' realized returns), earnings surprise effects, or particular accounting variables possibly connected with exceptional returns.11 In this case, the hope is that the factor returns bk(t) associated with these characteristics exhibit consistent positive returns. Alternatively, the manager could choose characteristics associated with investment themes, sometimes in favor and sometimes not—for example, growth forecasts or company size. Then the associated factor returns bk(t) would not consistently exhibit positive returns, but the manager would hope to forecast the sign and magnitude of these returns. For example, how will growth stocks or small-capitalization stocks perform relative to the market as a 10A word of caution if you try to include both betas and an intercept (or sectors or industries). There will be no problem with the factor portfolios that have benchmark-neutral attributes. Those factor portfolios will have both a zero beta and zero net investment. The difficulty is that the benchmark return will be spread between the beta factor and the intercept. We can avoid this confusion by introducing an equivalent regression. Suppose Zn,k (t) is the allocation of company n to sector k and: is the benchmark exposure to sector k at time t. If we substitute Xn,k(t) = Zn,k(t) – βn(t) · zB,k(t) for Zn,k(t), we can interpret the factor portfolio returns as the residual (the factor portfolio's beta will be zero) return on sector k when we control for the other variables. The returns to the benchmark-neutral factors will not change. 11For examples, see Ou and Penman (1989) and Lev and Thiagarajan (1993).
Page 252 whole? These forecasts might rely on extrapolations of past factor returns, e.g., using moving averages. More sophisticated forecasting might use economic indicators such as long and short interest rates, changes in interest rates, corporate bond spreads, or average dividend yields versus interest rates. Sometimes the manager may wish to forecast factor returns even for characteristics that generally exhibit consistent positive factor returns. For example, the BARRA Success factor in the United States, a momentum factor, generally exhibits positive returns. However, these returns are often negative in the month of January.12 Managers who wish to bet on momentum should take into account (forecast) this January effect. Managers can of course combine these different types of signals, forecasting factor returns for investment themes as well as using company-level valuation models. The relative emphasis will depend on the information ratios associated with these various efforts. Chapter 12, ''Information Analysis," and Chap. 17, "Performance Analysis," describe this research and evaluation effort in more detail. Chapters 10 and 11, which cover forecasting, will greatly expand on the technical issues surrounding forecasting of returns. The notes section of this chapter includes a list of popular investment signals. Summary This chapter began with a discussion of corporate finance, to motivate the development of valuation models based on appropriate corporate attributes. It then discussed dividend discount models in considerable detail, especially the critical issues of dealing with growth and expected future dividend payments, and converting information from the dividend discount model into exceptional return forecasts. The chapter went on to empirically expand the dividend discount concept with comparative valuation models, less rigorous in principle but more flexible in practice. The chapter ended with a discussion of returns-based analysis—which focuses 12We attribute this to investors' tendency to "window dress" portfolios by keeping winners for the year-end and selling losers in December, and to accelerate capital losses (take them in December) and postpone capital gains (take them in January).
Page 253 directly on forecasts of exceptional return—and alternative empirical approaches that are available. Notes Given the nature of active management, we can't provide recipes for finding superior information. Hence, this chapter has focused on structure and methodology. But we can provide some indication of what types of information U.S. equity investors have found useful in the past. According to an annual survey of institutional investors conducted by Merrill Lynch, these factors, attributes, and models include • EPS variability • Return on equity • Foreign exposure • Beta • Price/earnings ratios • Price/sales ratios • Size • Earnings estimate revisions • Neglect • Rating revision • Dividend yield • EPS momentum • Projected growth • Low price • Duration • Relative strength • EPS torpedo • Earnings estimation dispersion • Debt/equity ratio • EPS surprise • DDM • Price/cash flow ratios
• Price/book ratios
Page 254 Most investors responding to the survey relied on five to seven of these factors. See Bernstein (1998) for details. Problems 1. According to Modigliani and Miller (and ignoring tax effects), how would the value of a firm change if it borrowed money to repurchase outstanding common stock, greatly increasing its leverage? What if it changed its payout ratio? 2. Discuss the problem of growth forecasts in the context of the constant-growth dividend discount model [Eq. (9.5)]. How would you reconcile the growth forecasts with the implied growth forecasts for AT&T in Tables 9.1 and 9.2? 3. Stock X has a beta of 1.20 and pays no dividend. If the risk-free rate is 6 percent and the expected excess market return is 6 percent, what is stock X's implied growth rate? 4. You are a manager who believes that book-to-price (B/P), earnings-to-price (E/P), and beta are the three variables that determine stock value. Given monthly B/P, E/P, and beta values for 500 stocks, how could you implement your strategy (a) using comparative valuation? (b) using returnsbased analysis? 5. A stock trading with a P/E ratio of 15 has a payout ratio of 0.5 and an expected return of 12 percent. What is its growth rate, according to the constant-growth DDM? References Bernstein, Peter L. "Off the Average." Journal of Portfolio Management, vol. 24, no. 1, 1997, p. 3. Bernstein, Richard. "Quantitative Viewpoint: 1998 Institutional Factor Survey."Merrill Lynch Quantitative Publication, Nov. 25, 1998. Brennan, Michael J. "Stripping the S&P 500 Index." Financial Analysts Journal, vol. 54, no. 1, 1998, pp. 12–22. Chugh, Lal C., and Joseph W. Meador. "The Stock Valuation Process: The Analyst's View." Financial Analysts Journal, vol. 40, no. 6, 1984, pp. 41–48. Durand, David. "Afterthoughts on a Controversy with Modigliani and Miller, plus Thoughts on Growth and the Cost of Capital." Financial Management, vol. 18, no. 2, 1989, pp. 12–18.
Page 255 Fama, Eugene F., and Kenneth R. French. "Taxes, Financing Decisions, and Firm Value." Journal of Finance, vol. 53, no. 3, 1998, pp. 819–843. Fouse, William. "Risk and Liquidity: The Keys to Stock Price Behavior." Financial Analysts Journal, vol. 32, no. 3, 1976, pp. 35–45. ———. "Risk and Liquidity Revisited." Financial Analysts Journal, vol. 33, no. 1, 1977, pp. 40– 45. Gordon, Myron J. The Investment, Financing, and Valuation of the Corporation (Homewood, Ill.: Richard D. Irwin, 1962). Gordon, Myron J., and E. Shapiro. "Capital Equipment Analysis: The Required Rate of Profit." Management Science, vol. 3, October 1956, pp. 102–110. Grant, James L. "Foundations of EVA for Investment Managers." Journal of Portfolio Management, vol. 23, no. 1, 1996, pp. 41–48. Jacobs, Bruce I., and Kenneth N. Levy. "Disentangling Equity Return Opportunities: New Insights and Investment Opportunities." Financial Analysts Journal, vol. 44, no. 3, 1988, pp. 18–44. ———. "Forecasting the Size Effect." Financial Analysts Journal, vol 45, no. 3, 1989, pp. 38–54. Lev, Baruch, and S. Ramu Thiagarajan. "Fundamental Information Analysis."Journal of Accounting Research, vol. 31, no. 2, 1993, pp. 190–215. Modigliani, Franco, and Merton H. Miller. "Dividend Policy, Growth, and the Valuation of Shares." Journal of Business, vol 34, no. 4, 1961, pp. 411–433. Ohlson, James A. "Accounting Earnings, Book Value, and Dividends: The Theory of the Clean Surplus Equation (Part I)." Columbia University working paper, January 1989. Ou, Jane, and Stephen Penman. "Financial Statement Analysis and the Prediction of Stock Returns." Journal of Accounting and Economics, vol. 11, no. 4, November 1989, pp. 295–329. Rosenberg, Barr, Kenneth Reid, and Ronald Lanstein. "Persuasive Evidence of Market Inefficiency." Journal of Portfolio Management, vol. 11, no. 3, 1985, pp. 9–17. Rozeff, Michael S. "The Three-Phase Dividend Discount Model and the ROPE Model." Journal of Portfolio Management, vol. 16, no. 2, 1989, pp. 36–42. Sharpe, William F., and Gordon J. Alexander. Investments (Englewood Cliffs, N.J.: Prentice-Hall, 1990). Wilcox, Jarrod W. "The P/B-ROE Valuation Model." Financial Analysts Journal, vol. 40, no. 1, 1984, pp. 58–66. Williams, John Burr. The Theory of Investment Value (Amsterdam: North-Holland Publishing Company, 1964). Technical Appendix This technical appendix will discuss how to handle nonlinear models and fractiles in the (linear) returns-based framework. The idea will be to capture the nonlinearities in the exposures to linearly
Page 256 estimated factors. We can even make these exposures benchmark-neutral. Let's start with an example. Suppose you think that the relationship between earnings yield and return is not linear. Here is one approach to testing that hypothesis. Divide the assets into three categories, high earnings yield, low earnings yield, and average earnings yield,13 and consider two earnings yield variables, high yield and average yield. So the model is The high-earnings-yield assets have Xn,high = 1 and Xn,ave = 0. The average-earnings-yield assets have Xn,high = 0 and Xn,ave = 1. We can determine the exposures of the low-yield assets by imposing benchmark neutrality. Assuming that each group has equal capitalization, the condition θB = 0 leads to So for the low-yield assets, we must have Xn,high = – 1 and Xn,ave = – 1. Once we estimate the factor returns, we can determine whether there exist any nonlinearities. For example, one sign of a nonlinear effect would be the return to the high-yield assets relative to the average assets differing significantly from the return of the low-yield assets relative to the average assets. We can clearly extend this approach to quartiles, quintiles, or deciles. In fact, this returns-based fractile analysis has the advantage of allowing for controls. We can make sure that each fractile portfolio is sector- and/or benchmark-neutral, and perhaps neutral on some other dimension that we believe may be confounding the effect. Other methods for analyzing nonlinear effects also exist. Let's start with an attribute Xn,1 that is benchmark-neutral: We wish to analyze nonlinearities while still ensuring 13The choice of three categories is arbitrary, and mainly done for ease of explanation. If we considered four categories, this would be quartile analysis. As in quartile analysis, we also have a choice in how to form the group: equal number in each group or equal capitalization in each group.
Page 257 benchmark neutrality. A simple approach is to include a new variable: The squared exposure has the disadvantage of placing undue emphasis on outliers. Other alternatives would include using the absolute value of Xn,1, the square root of the absolute value of Xn,1, or the Max{0, Xn,1}, in each case benchmark-neutralized as in Eq. (9A.3). Each of these would reduce the nonlinear effect for the more extreme positive and negative values of Xn,1. Exercise 1. You forecast an alpha of 2 percent for stocks that have E/P above the benchmark average and IBES growth above the benchmark average. On average, what must your alpha forecasts be for stocks that do not satisfy these two criteria? If you assume an alpha of zero for stocks which have either above-average E/P or above-average IBES growth, but not both, what is your average alpha for stocks with E/P and IBES growth both below average? Applications Exercise 1. Use appropriate software (e.g. BARRA's Aegis and Alphabuilder products) to determine the current dividend-to-price ratio, dividend growth, and beta (with respect to the CAPMMI) for GE and Coke. Using these data, a risk-free rate of 6 percent, and expected benchmark excess return of 6 percent, what are the prices implied by the constant-growth DDM? What is the dividend discount rate? Estimate alphas for GE and Coke using both methods described in the section "Dividend Discount Models and Returns."
Page 259 PART THREE— INFORMATION PROCESSING
Page 261