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C H A P T E R !" Equity Valuation Models #$#

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This equation can be simplified to2

V0!=! D0(1!+!g) _________

k!"!g !=! D1 _____

k!"!g (18.4)

Notice in Equation 18.4 that we calculate intrinsic value by dividing!D1 (not D0) by k " g. If the market capitalization rate for Steady State is 12%, this equation implies that the intrinsic value of a share of Steady State stock is

$3.81(1!+!.05 ) _____________ .12!"!.05

!=! $4.00 ________ .12!"!.05

!=!$57.14

Equation 18.4 is called the constant-growth DDM, or the Gordon model, after Myron J. Gordon, who popularized the model. It should remind you of the formula for the present value of a perpetuity. If dividends were expected not to grow, then the dividend stream would be a simple perpetuity, and the valuation formula would be3 V0 = D1/k. Equation 18.4 generalizes the perpetuity formula for the case of a growing perpetuity. As g increases (for a given value of D1), the stock price also rises.

2Here is a proof. By definition,

V0!=! D1 _____

1!+!k !+! D1(1!+!g) _________

(1!+!k)2 !+! D1(1!+!g)

2 _________

(1!+!k)3 !+!#!#!# (a)

Multiplying through by (1 + k)/(1 + g), we obtain

(1!+!k) ______ (1!+!g)

V0!=! D1 ______

(1!+!g) !+! D1 ______

(1!+!k) !+! D1(1!+!g) _________

(1!+!k)2 !+!#!#!# (b)

Subtracting equation (a) from equation (b), we find that

1!+!k _____ 1!+!g

V0!"!V0!=! D1 ______

(1!+!g)

which implies

(k!"!g)V0 ________ (1!+!g)

!=! D1 ______ (1!+!g)

V0!=! D1 _____

k!"!g

3Recall from introductory finance that the present value of a $1 per year perpetuity is 1/k. For example, if k = 10%, the value of the perpetuity is $1/.10 = $10. Notice that if g = 0 in Equation 18.4, the constant-growth DDM formula is the same as the perpetuity formula.

Preferred stock that pays a fixed dividend can be valued using the constant-growth dividend discount model. The constant-growth rate of dividends is simply zero. For example, to value a preferred stock paying a fixed dividend of $! per share when the discount rate is "%, we compute

V#!=! $! ______

.#"!"!# !=!$!$

Example 18.1 Preferred Stock and the DDM

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The constant-growth DDM is valid only when g is less than k. If dividends were expected to grow forever at a rate faster than k, the value of the stock would be infinite. If an analyst derives an estimate of g greater than k, that growth rate must be unsustainable in the long run. The appropriate valuation model to use in this case is a multistage DDM such as those discussed below.

The constant-growth DDM is so widely used by stock market analysts that it is worth exploring some of its implications and limitations. The constant-growth rate DDM implies that a stock’s value will be greater:

1. The larger its expected dividend per share. 2. The lower the market capitalization rate, k. 3. The higher the expected growth rate of dividends. Another implication of the constant-growth model is that the stock price is expected to

grow at the same rate as dividends. To see this, suppose Steady State stock is selling at its intrinsic value of $57.14, so that V0 = P0. Then

P0!=! D1 _____

k!"!g

Notice that price is proportional to dividends. Therefore, next year, when the dividends paid to Steady State stockholders are expected to be higher by g = 5%, price also should increase by 5%. To confirm this, we can write

D2!=!$4(1.05)!=!$4.20

P1!=! D2 _____

k!"!g !=! $4.20 ________

.12!"!.05 !=!$60.00

which is 5% higher than the current price of $57.14. To generalize,

P1!=! D2 _____

k!"!g !=! D1(1!+!g) _________

k!"!g !=! D1 _____

k!"!g (1!+!g)!=!P0(1!+!g)

High Flyer Industries has just paid its annual dividend of $! per share. The dividend is expected to grow at a constant rate of "% indefinitely. The beta of High Flyer stock is #.$, the risk-free rate is %%, and the market risk premium is "%. What is the intrinsic value of the stock? What would be your estimate of intrinsic value if you believed that the stock was riskier, with a beta of #.&'?

Because a $! dividend has just been paid and the growth rate of dividends is "%, the forecast for the year-end dividend is $! # #.$" = $!.&(. The market capitalization rate (using the CAPM) is %% + #.$ # "% = #(%. Therefore, the value of the stock is

V$!=! D# _____

k!"!g !=! $!.&(

_______ .#(!"!.$" !=!$'(

If the stock is perceived to be riskier, its value must be lower. At the higher beta, the market capitalization rate is %% + #.&' # "% = #%%, and the stock is worth only

$!.&(

_______ .#%!"!.$" !=!$($.'$

Example 18.2 The Constant-Growth DDM

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Convergence of Price to Intrinsic Value Now suppose that the current market price of ABC stock is only $48 per share and, therefore, that the stock is undervalued by $2 per share. In this case the expected rate of price appreciation depends on an additional assumption about whether the discrepancy between the intrinsic value and the market price will disappear, and if so, when.

One fairly common assumption is that the discrepancy will never disappear and that the market price will trend upward at rate g forever. This implies that the discrepancy between intrinsic value and market price also will grow at the same rate. In our example:

Now Next Year

V!"="$#! V$"="$#!"!"$.!%"="$#& P!"="$%' P$"="$%'"!"$.!%"="$%(.(& V!")"P!"="$& V$"#"P$"="$&"!"$.!%"="$&.!'

Under this assumption the expected HPR will exceed the required rate, because the dividend yield is higher than it would be if P0 were equal to V0. In our example the divi- dend yield would be 8.33% instead of 8%, so that the expected HPR would be 12.33% rather than 12%:

E(r)"=" D1 ___ P0

"+"g"=" $4 ____ $48

"+".04"=".0833"+".04"=".1233

An investor who identifies this undervalued stock can get an expected dividend that exceeds the required yield by 33 basis points. This excess return is earned each year, and the market price never catches up to intrinsic value.

An alternative assumption is that the gap between market price and intrinsic value will disappear by the end of the year. In that case we would have P1 = V1 = $52, and

E(r )"=" D1 ___ P0

"+" P1"#"P0 _______ P0

"=" 4 ___ 48

"+" 52"#"48 _______ 48

"=".0833"+".0833"=".1667

The assumption of complete catch-up to intrinsic value produces a much larger 1-year HPR. In future years, however, the stock is expected to generate only fair rates of return.

Many stock analysts assume that a stock’s price will approach its intrinsic value gradu- ally over time—for example, over a 5-year period. This puts their expected 1-year HPR somewhere between the bounds of 12.33% and 16.67%.

Stock Prices and Investment Opportunities Consider two companies, Cash Cow, Inc., and Growth Prospects, each with expected earnings in the coming year of $5 per share. Both companies could in principle pay out all of these earnings as dividends, maintaining a perpetual dividend flow of $5 per share. If the market capitalization rate were k = 12.5%, both companies would then be valued at D1/k = $5/.125 = $40 per share. Neither firm would grow in value, because with all earn- ings paid out as dividends, and no earnings reinvested in the firm, both companies’ capital stock and earnings capacity would remain unchanged over time; earnings4 and dividends would not grow. 4Actually, we are referring here to earnings net of the funds necessary to maintain the productivity of the firm’s capital, that is, earnings net of “economic depreciation.” In other words, the earnings figure should be interpreted as the maximum amount of money the firm could pay out each year in perpetuity without depleting its productive capacity. For this reason, the net earnings number may be quite different from the accounting earnings figure that the firm reports in its financial statements. We explore this further in Chapter 19.

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When Growth Prospects pursued a no-growth policy and paid out all earnings as divi- dends, the stock price was only $40. Therefore, you can think of $40 as the value per share of the assets the company already has in place.

When Growth Prospects decided to reduce current dividends and reinvest some of its earnings in new investments, its stock price increased. The increase in the stock price reflects the fact that the planned investments provide an expected rate of return greater than the required rate. In other words, the investment opportunities have positive net pres- ent value. The value of the firm rises by the NPV of these investment opportunities. This net present value is also called the present value of growth opportunities, or PVGO.

Therefore, we can think of the value of the firm as the sum of the value of assets already in place, or the no-growth value of the firm, plus the net present value of the future investments the firm will make, which is the PVGO. For Growth Prospects, PVGO = $17.14 per share:

Price!=!No-growth!value!per!share!+!PVGO P0!=!

E1 ___ k !+!PVGO (18.6)

57.14!=!40!+!17.14 We know that in reality, dividend cuts almost always are accompanied by steep drops

in stock prices. Does this contradict our analysis? Not necessarily: Dividend cuts are usu- ally taken as bad news about the future prospects of the firm, and it is the new information about the firm—not the reduced dividend yield per se—that is responsible for the stock price decline.

For example, when J.P. Morgan cut its quarterly dividend from 38 cents to 5 cents a share in 2009, its stock price actually increased by about 5%. The company was able to convince investors that the cut would conserve cash and prepare the firm to weather a severe recession. When investors were convinced that the dividend cut made sense, the stock price actually increased. Similarly, when BP announced in the wake of the massive 2010 Gulf oil spill that it would suspend dividends for the rest of the year, its stock price did not budge. The cut already had been widely anticipated, so it was not new information. These examples show that stock price declines in response to dividend cuts are really a response to the information conveyed by the cut.

It is important to recognize that growth per se is not what investors desire. Growth enhances company value only if it is achieved by investment in projects with attractive profit opportunities (i.e., with ROE > k). To see why, let’s now consider Growth Pros- pects’s unfortunate sister company, Cash Cow, Inc. Cash Cow’s ROE is only 12.5%, just equal to the required rate of return, k. Therefore, the net present value of its investment opportunities is zero. We’ve seen that following a zero-growth strategy with b = 0 and g = 0, the value of Cash Cow will be E1/k = $5/.125 = $40 per share. Now suppose Cash Cow chooses a plowback ratio of b = .60, the same as Growth Prospects’s plowback. Then g would increase to

g!=!ROE!"!b!=!.125!"!.60!=!.075 but the stock price is still $40:

P0!=! D1 _____

k!#!g !=! $2 __________

.125!#!.075 !=!$40

which is no different from the no-growth strategy. When Cash Cow reduced dividends to free funds for reinvestment in the firm, it gener-

ated only enough growth to maintain the stock price at its current level. This makes sense: If the firm’s projects yield only what investors can earn on their own, shareholders cannot be

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To value companies with temporarily high growth, analysts use a multistage version of the dividend discount model. For example, a two-stage dividend discount model allows for an initial high-growth period before the firm settles down to a sustainable growth trajectory. The combined present value of dividends in the initial high-growth period is calculated first. Then, once the firm is projected to settle down to a steady- growth phase, the constant-growth DDM is applied to value the remaining stream of dividends.

Let’s try this approach with an actual firm.!Figure 18.2 is a Value Line Investment Sur- vey report on General Electric. Some of the relevant information in 2016 is highlighted.

GE’s beta appears at the circled A, its recent stock price at the B, the per-share divi- dend payments at the C, the ROE (referred to as return on shareholder equity) at the D, and the dividend payout ratio (referred to as all dividends to net profits) at the E.!The rows ending at C, D, and E are historical time series. The boldfaced, italicized entries under 2017 are estimates for that year. Similarly, the entries in the far right column (labeled 15–17) are forecasts for some time between 2019–2021, which we will take to be 2020.

Value Line projects fairly rapid growth in the near term, with dividends rising from $1.04 in 2017 to $1.60 in 2020. This growth rate cannot be sustained indefinitely. We can obtain dividend inputs for this initial period by using the explicit forecasts for 2017 and 2020 and linear interpolation for the years between:

2017 $1.04 2019 $1.41 2018 $1.22 2020 $1.60

Now let us assume the dividend growth rate levels off in 2020. What is a good guess for that steady-state growth rate? Value Line forecasts a dividend payout ratio of .53 and an ROE of 19.5%, implying long-term growth will be

g!=!ROE!"!b!=!19.5%!"!(1!#!.53)!=!9.17% Value Line rounds this estimate off to 9% (see the entry for “Retained [Earnings] to Com- mon Equity” near the circled D), so we too will set g = 9%.

Our estimate of GE’s intrinsic value using an investment horizon of 2020 is therefore obtained from Equation 18.2, which we restate here:

V2016!=! D2017 _____ 1!+!k

!+! D2018 _______ (1!+!k)2

!+! D2019 _______ (1!+!k)3

!+! D2020!+!P2020 ___________ (1!+!k)4

=! 1.04 _____ 1!+!k

!+! 1.22 _______ (1!+!k)2

!+! 1.41 _______ (1!+!k)3

!+! 1.60!+!P2020 ___________ (1!+!k)4

Here, P2020!represents the forecast price at which we can sell our shares at the end of 2020, when dividends are assumed to enter their constant-growth phase. That price, according to the constant-growth DDM, should be

P2020!=! D2021 _____ k!#!g

!=! D2020(1!+!g) ___________ k!#!g

!=! 1.60!"!1.09 __________ k!#!.09

The only variable remaining to be determined to calculate intrinsic value is the market capitalization rate, k.

One way to obtain k is from the CAPM. Observe from the Value Line report that GE’s beta is 1.10. The risk-free rate on long-term Treasury bonds in 2016 was about 2.5%.6

6When valuing long-term assets such as stocks, it is common to treat the long-term Treasury bond, rather than short-term T-bills, as the risk-free asset.

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The Price–Earnings Ratio and Growth Opportunities Much of the real-world discussion of stock market valuation concentrates on the firm’s price–earnings multiple, the ratio of price per share to earnings per share, commonly called the P/E ratio. Our discussion of growth opportunities shows why stock market analysts focus on the P/E ratio. Both companies considered, Cash Cow and Growth Prospects, had earnings per share (EPS) of $5, but Growth Prospects reinvested 60% of earnings in prospects with an ROE of 15%, whereas Cash Cow paid out all earn- ings as dividends. Cash Cow had a price of $40, giving it a P/E multiple of 40/5 = 8.0, whereas Growth Prospects sold for $57.14, giving it a multiple of 57.14/5 = 11.4. This observation suggests the P/E ratio might serve as a useful indicator of expectations of growth opportunities.

We can see how growth opportunities are reflected in P/E ratios by rearranging Equation 18.6 to

P0 ___ E1

!=! 1 __ k ( 1!+!PVGO _________ E / k ) (18.7)

When PVGO = 0, Equation 18.7 shows that P0 = E1/k. The stock is valued like a non- growing perpetuity of E1, and the P/E ratio is just 1/k. However, as PVGO becomes an increasingly dominant contributor to price, the P/E ratio can rise dramatically.

The ratio of PVGO to E/k has a straightforward interpretation. It is the ratio of the com- ponent of firm value due to growth opportunities to the component of value due to assets already in place (i.e., the no-growth value of the firm, E/k). When future growth oppor- tunities dominate the estimate of total value, the firm will command a high price relative to current earnings. Thus a high P/E multiple indicates that a firm enjoys ample growth opportunities.

P/E multiples do vary with growth prospects. Between 1996 and 2015, for example, FedEx’s P/E ratio averaged about 17.4 while Consolidated Edison’s (an electric utility) average P/E was only 14.0. These numbers do not necessarily imply that FedEx was over- priced compared to Con Ed. If investors believed FedEx would grow faster than Con Ed, the higher price per dollar of earnings would be justified. That is, an investor might well pay a higher price per dollar of current earnings if he or she expects that earnings stream to grow more rapidly. In fact, FedEx’s growth rate has been consistent with its higher P/E multiple. Over this period, its earnings per share grew at 10.5% per year while Con Ed’s earnings growth rate was only 1.7%. (Later in the chapter, where we turn to practical issues in interpreting P/E ratios,!Figure 18.4!presents the EPS history of the two companies.)

We conclude that the P/E ratio reflects the market’s optimism concerning a firm’s growth prospects. Analysts must decide whether they are more or less optimistic than the belief implied by the market multiple. If they are more optimistic, they will recommend buying the stock.

There is a way to make these insights more precise. Look again at the constant-growth DDM formula, P0 = D1/(k " g). Now recall that dividends equal the earnings that are not reinvested in the firm: D1 = E1(1 " b). Recall also that g = ROE # b. Hence, substituting for D1 and g, we find that

P0!=! E1(1!"!b) ___________

k!"!ROE!#!b

18.4 The Price–Earnings Ratio

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Notwithstanding these fine points, P/E ratios frequently are taken as proxies for the expected growth in dividends or earnings. In fact, a Wall Street rule of thumb is that the growth rate ought to be roughly equal to the P/E ratio. In other words, the ratio of P/E to g, often called the PEG ratio, should be about 1.0. Peter Lynch, the famous portfolio manager, puts it this way in his book One Up on Wall Street:

The P/E ratio of any company that’s fairly priced will equal its growth rate. I’m talking here about growth rate of earnings here.!.!.!. If the P/E ratio of Coca Cola is 15, you’d expect the company to be growing at about 15% per year, etc. But if the P/E ratio is less than the growth rate, you may have found yourself a bargain.

Let’s try Lynch’s rule of thumb. Assume that

rf!=!"% (roughly the value when Peter Lynch was writing) rM!"!rf!=!"% (about the historical average market risk premium)

b!=!.# (a typical value for the plowback ratio in the United States)

Therefore, rM = rf + market risk premium = "% + "% = $%%, and k = $%% for an average (# = $) company. If we also accept as reasonable that ROE = $%% (the same value as the expected return on the stock), we conclude that

g!=!ROE!$!b!=!$%%!$!.#!=!%.#%

and

P

__ E

!=! $!"!.#

________ .$%!"!.&%# !=!%.'%

Thus, the P/E ratio and g are about equal using these assumptions, consistent with the rule of thumb.

However, note that this rule of thumb, like almost all others, will not work in all circum- stances. For example, the yield on long-term Treasury bonds today is more like '.(%, so a comparable forecast of rM today would be

rf!+!Market!risk!premium!=!'.(%!+!"%!=!$&.(%

If we continue to focus on a firm with # = $, and if ROE still is about the same as k, then

g!=!$&.(%!$!.#!=!#.'%

while

P

__ E

!=! $!"!.# _________

.$&(!"!.&#' !=!).(

The P/E ratio and g now diverge and the PEG ratio is now '.*. Nevertheless, lower-than- average PEG ratios are still widely seen as signaling potential underpricing.

Example 18.5 P/E Ratio versus Growth Rate

The importance of growth opportunities is most evident in the valuation of start-up firms. For example, in the dot-com boom of the late 1990s, many companies that had yet to turn a profit were valued by the market at billions of dollars. The perceived value of these companies was exclusively as growth opportunities. For example, the online auction firm eBay had 1998 profits of $2.4 million, far less than the $45 million profit earned by the traditional auctioneer Sotheby’s; yet eBay’s market value was more than 10 times greater: $22 billion versus $1.9 billion. As it turns out, the market was quite right to value eBay so

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Pro forma earnings are calculated ignoring certain expenses, for example, restructur- ing charges, stock-option expenses, or write-downs of assets from continuing operations. Firms argue that ignoring these expenses gives a clearer picture of the underlying profit- ability of the firm. Comparisons with earlier periods probably would make more sense if those costs were excluded.

But when there is too much leeway for choosing what to exclude, it becomes hard for investors or analysts to interpret the numbers or to compare them across firms. The lack of standards gives firms considerable leeway to manage earnings.

Even generally accepted accounting principles (GAAP) allow firms considerable discre- tion to manage earnings. For example, in the late 1990s, Kellogg took restructuring charges, which are supposed to be one-time events, nine quarters in a row. Were these really one-time events, or were they more appropriately treated as ordinary expenses? Given the available leeway in managing earnings, the justified P/E multiple becomes difficult to gauge.

Another confounding factor in the use of P/E ratios is related to the business cycle. We were careful in deriving the DDM to define earnings as being net of economic deprecia- tion, that is, the maximum flow of income that the firm could pay out without depleting its productive capacity. But reported earnings are computed in accordance with GAAP and need not correspond to economic earnings. Beyond this, however, notions of a normal or justified P/E ratio, as in Equation 18.7 or 18.8, assume implicitly that earnings rise at a constant rate, or, put another way, on a smooth trend line. In contrast, reported earnings can fluctuate dramatically around a trend line over the course of the business cycle.

Another way to make this point is to note that the “normal” P/E ratio predicted by Equation 18.8 is the ratio of today’s price to the trend value of future earnings, E1. The P/E ratio reported in the financial pages of the newspaper, by contrast, is the ratio of price to the most recent past accounting earnings. Current accounting earnings can differ considerably from future economic earnings. Because ownership of stock conveys the right to future as well as current earnings, the ratio of price to most recent earnings can vary substantially over the business cycle, as accounting earnings and the trend value of economic earnings diverge by greater and lesser amounts.

As an example, Figure 18.4 graphs the earnings per share of FedEx and Con Ed since 1996. Note that FedEx’s EPS is far more variable. Because the market values the entire stream of future dividends generated by the company, when earnings are temporarily depressed, the P/E ratio should tend to be high—that is, the denominator of the ratio responds more sensitively to the business cycle than the numerator. This pattern is borne out well.

Figure 18.5 graphs the P/E ratios of the two firms. FedEx has greater earnings vola- tility and more variability in its P/E ratio. Its clearly higher average growth rate shows up in its generally higher P/E ratio. The only year in which Con Ed’s ratio exceeded FedEx’s was 2012, during which FedEx’s earnings rose at a far faster rate than its underlying trend. The market seems to have decided that this earnings performance was not likely to be sustainable, and FedEx’s price rose less dramatically than its annual earnings. Consequently, its P/E ratio declined.

This example shows why analysts must be careful in using P/E ratios. There is no way to say the P/E ratio is overly high or low without referring to the company’s long-run growth prospects, as well as to current earnings per share relative to the long-run trend line.

Nevertheless, Figures 18.4 and 18.5 demonstrate a clear relation between P/E ratios and growth. Despite considerable short-run fluctuations, FedEx’s EPS clearly trended upward over the period. Con Ed’s earnings were essentially flat. FedEx’s growth prospects are reflected in its consistently higher P/E multiple.

This analysis suggests that P/E ratios should vary across industries, and in fact they do. Figure 18.6 shows P/E ratios in 2016 for a sample of industries. Notice that the

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The free cash flow to equity approach yields a similar intrinsic value for the stock.10 Free cash flow to equity (row 18) is obtained from FCFF by subtracting after-tax interest expense and net debt repurchases. FCFE in the early years is negative because the firm is devoting $5 billion a year to reducing debt, which more than absorbs the profits the firm is expected to earn. However, profits are expected to increase rapidly, and FCFE turns positive after 2019. FCFE is discounted at the equity rate. Like WACC, the cost of equity changes each period as leverage changes. The present value factor for equity cash flows is presented in row 34. Equity value is reported in cell J38, which is put on a per share basis in cell L38.

Spreadsheet 18.2 is available in Connect!or through your course instructor.

Comparing the Valuation Models In principle, the free cash flow approach is fully consistent with the dividend discount model and should provide the same estimate of intrinsic value if one can extrapolate to a period in which the firm begins to pay dividends growing at a constant rate. This was demonstrated in two famous papers by Modigliani and Miller.11 However, in practice, you will find that values from these models may differ, sometimes substantially. This is due to the fact that in practice, analysts are always forced to make simplifying assumptions. For example, how long will it take the firm to enter a constant-growth stage? How should depreciation best be treated? What is the best estimate of ROE? Answers to questions like these can have a big impact on value, and it is not always easy to maintain consistent assumptions across the models.

We have now valued GE using several approaches, with estimates of intrinsic value as follows:

Model Intrinsic Value

Two-stage dividend discount model $!".#$ DDM with earnings multiple terminal value ""."% Three-stage DDM "!.&$ Free cash flow to the firm '#.(' Free cash flow to equity '&.!' Market price (from Value Line) "$.)(

What should we make of these differences? Aside from the two-stage DDM, which clearly assumes GE will enter a sustainable growth period earlier and with a higher growth rate than is realistic, the estimates of intrinsic value cluster pretty much symmetrically around the actual market price. The dividend discount models tend to give estimates that exceed price, while the estimates from the free cash flow models are less than price. But by and large, there is no clear signal that GE is substantially mispriced.

On balance, therefore, this valuation exercise suggests that finding bargains is not gen- erally going to be easy. Although applying these models is straightforward, establishing 10Over the 2016–2020 period, Value Line predicts that GE will retire a considerable fraction of its outstanding debt. The implied debt repurchases are a use of cash and reduce the cash flow available to equity. Such repur- chases cannot be sustained indefinitely, however, for debt outstanding would soon be run down to zero. Therefore, in our estimate of the terminal value of equity, we compute the final cash flow assuming that, starting in 2020, GE will begin issuing enough debt to maintain its debt-to-value ratio. This approach is consistent with the assumption of constant growth and constant discount rates after 2020. 11Franco Modigliani and M. Miller, “The Cost of Capital, Corporation Finance, and the Theory of Investment,” American Economic Review, June 1958; and “Dividend Policy, Growth, and the Valuation of Shares,” Journal of Business, October 1961.

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proper inputs is not. This should not be surprising. In even a moderately efficient market, finding profit opportunities will be more involved than analyzing Value Line data for a few hours. These models are extremely useful to analysts, however, because they provide ballpark estimates of intrinsic value. More than that, they force rigorous thought about underlying assumptions and highlight the variables with the greatest impact on value and the greatest payoff to further analysis.

The Problem with DCF Models Our estimates of GE’s intrinsic value are all based on discounted cash flow (DCF) models, in which we calculate the present value of forecasted cash flows and a terminal sales price at some future date. It is clear from our calculations that most of the action in these models is in the terminal value and that this value can be highly sensitive to even small changes in some input values (see, e.g., Concept Check 18.4). Therefore, you must recognize that DCF valuation estimates are almost always going to be imprecise. Growth opportunities and future growth rates are especially hard to pin down.

For this reason, many value investors employ a hierarchy of valuation. They view the most reliable components of value as the items on the balance sheet that, in principle, can be sold and for which estimates of market value are readily available. Real estate, plant, and equipment would fall in this category.

A somewhat less reliable component of value is the economic profit on assets already in place. For example, a company like Intel earns a far higher ROE on its investments in chip-making facilities than its cost of capital. The present value of these “economic profits,” or economic value added,12 is a major component of Intel’s market value. This component of value is less certain than its balance sheet assets, however, because there is always a concern that new competitors will enter the market, force down prices and profit margins, and reduce the return on Intel’s investments. Thus, one needs to carefully assess the barriers to entry that protect Intel’s pricing and profit margins. We noted some of these barriers in the last chapter, where we discussed the role of industry analysis, market structure, and competitive position (see Section 17.6).

Finally, the least reliable components of value are growth opportunities, the purported ability of firms like Intel to invest in positive-NPV ventures that contribute to high market valuations today. Value investors don’t deny that such opportunities exist, but they are skeptical that precise values can be attached to them and, therefore, tend to be less willing to make investment decisions that depend on the value of those opportunities.

12We discuss economic value added in greater detail in Chapter 19.

The most popular approach to valuing the overall stock market is the earnings multiplier approach applied at the aggregate level. The first step is to forecast corporate profits for the coming period. Then we derive an estimate of the earnings multiplier, the aggregate P/E ratio, based on a forecast of long-term interest rates. The product of the two forecasts is the estimate of the end-of-period level of the market.

The forecast of the P/E ratio of the market is sometimes derived from a graph simi- lar to that in Figure 18.8, which plots the earnings yield (earnings per share divided by price per share, the reciprocal of the P/E ratio) of the S&P 500 and the yield to maturity on 10-year Treasury bonds. The two series clearly move together over time and suggest that this

18.6 The Aggregate Stock Market

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book value liquidation value replacement cost Tobin’s q intrinsic value (of a share) market capitalization rate

KEY TERMS dividend discount model (DDM)

constant-growth DDM dividend payout ratio plowback ratio earnings retention ratio

present value of growth opportunities (PVGO)

two-stage dividend discount model

price–earnings multiple earnings management

Intrinsic value: V0!=! D1 _____

1!+!k !+! D2 _______

(1!+!k)2 !+!"!"!"!+! DH!+!PH ________

(1!+!k)H

Constant-growth DDM: V0!=! D1 _____

k!#!g

Growth opportunities: Price!=! E1 ___ k !+!PVGO

Determinant of P/E ratio: P0 ___ E1

!=! 1 __ k (1!+! PVGO ______ E1 / k )

Free cash flow to the firm:

FCFF!=!EBIT(1!#!tc )!+!Depreciation!#!Capital!expenditures!#!Increases!in!NWC

Free cash flow to equity: FCFE = FCFF # Interest expense $ (1 # tc) + Increases!in!net!debt!

KEY EQUATIONS

a middle, or transition, period in which the dividend growth rate declines from its initial high rate to the lower sustainable rate.

6. Stock market analysts devote considerable attention to a company’s price-to-earnings ratio. The P/E ratio is a useful measure of the market’s assessment of the firm’s growth opportunities. Firms with no growth opportunities should have a P/E ratio that is just the reciprocal of the capitalization rate, k. As growth opportunities become a progressively more important compo- nent of the total value of the firm, the P/E ratio will increase.

7. The expected growth rate of earnings is related both to the firm’s expected profitability and to its dividend policy. The relationship can be expressed as

g!=!(ROE!on!new!investment )!$!(1!#!Dividend!payout!ratio )

8. You can relate any DDM to a simple capitalized earnings model by comparing the expected ROE on future investments to the market capitalization rate, k. If the two rates are equal, then the stock’s intrinsic value reduces to expected earnings per share (EPS) divided by k.

9. Many analysts form their estimates of a stock’s value by multiplying their forecast of next year’s EPS by a predicted P/E multiple. Some analysts mix the P/E approach with the dividend dis- count model. They use an earnings multiplier to forecast the terminal value of shares at a future date, and add the present value of that terminal value with the present value of all interim divi- dend payments.

10. The free cash flow approach is the one used most often in corporate finance. The analyst first estimates the value of the firm as the present value of expected future free cash flows to the entire firm and then subtracts the value of all claims other than equity. Alternatively, free cash flows to equity can be discounted at a discount rate appropriate to the risk of the stock.

11. The models presented in this chapter can be used to explain and forecast the behavior of the aggregate stock market. The key macroeconomic variables that determine the level of stock prices in the aggregate are interest rates and corporate profits.

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1. In what circumstances would you choose to use a dividend discount model rather than a free cash flow model to value a firm?

2. In what circumstances is it most important to use multistage dividend discount models rather than constant-growth models?

3. If a security is underpriced (i.e., intrinsic value > price), then what is the relationship between its market capitalization rate and its expected rate of return?

4. Deployment Specialists pays a current (annual) dividend of $1.00 and is expected to grow at 20% for 2 years and then at 4% thereafter. If the required return for Deployment Specialists is 8.5%, what is the intrinsic value of its stock?

5. Jand, Inc., currently pays a dividend of $1.22, which is expected to grow indefinitely at 5%. If the current value of Jand’s shares based on the constant-growth dividend discount model is $32.03, what is the required rate of return?

6. A firm pays a current dividend of $1.00 which is expected to grow at a rate of 5% indefinitely. If current value of the firm’s shares is $35.00, what is the required return based on the constant- growth dividend discount model (DDM)?

7. Tri-coat Paints has a current market value of $41 per share with earnings of $3.64. What is the present value of its growth opportunities (PVGO) if the required return is 9%?

8. a. Computer stocks currently provide an expected rate of return of 16%. MBI, a large computer company, will pay a year-end dividend of $2 per share. If the stock is selling at $50 per share, what must be the market’s expectation of the dividend growth rate?

b. If dividend growth forecasts for MBI are revised downward to 5% per year, what will happen to the price of MBI stock?

c. What (qualitatively) will happen to the company’s price–earnings ratio? 9. a. MF Corp. has an ROE of 16% and a plowback ratio of 50%. If the coming year’s earnings

are expected to be $2 per share, at what price will the stock sell? The market capitalization rate is 12%.

b. What price do you expect MF shares to sell for in three years? 10. The market consensus is that Analog Electronic Corporation has an ROE = 9%, a beta of 1.25,

and plans to maintain indefinitely its traditional plowback ratio of 2/3. This year’s earnings were $3 per share. The annual dividend was just paid. The consensus estimate of the coming year’s market return is 14%, and T-bills currently offer a 6% return. a. Find the price at which Analog stock should sell. b. Calculate the P/E ratio. c. Calculate the present value of growth opportunities. d. Suppose your research convinces you Analog will announce momentarily that it will

immediately reduce its plowback ratio to 1/3. Find the intrinsic value of the stock. e. The market is still unaware of this decision. Explain why V0 no longer equals P0 and why V0

is greater or less than P0. 11. The FI Corporation’s dividends per share are expected to grow indefinitely by 5% per year.

a. If this year’s year-end dividend is $8 and the market capitalization rate is 10% per year, what must the current stock price be according to the DDM?

b. If the expected earnings per share are $12, what is the implied value of the ROE on future investment opportunities?

c. How much is the market paying per share for growth opportunities (i.e., for an ROE on future investments that exceeds the market capitalization rate)?

12. The stock of Nogro Corporation is currently selling for $10 per share. Earnings per share in the coming year are expected to be $2. The company has a policy of paying out 50% of its earnings each year in dividends. The rest is retained and invested in projects that earn a 20% rate of return per year. This situation is expected to continue indefinitely.

PROBLEM SETS

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a. Assuming the current market price of the stock reflects its intrinsic value as computed using the constant-growth DDM, what rate of return do Nogro’s investors require?

b. By how much does its value exceed what it would be if all earnings were paid as dividends and nothing were reinvested?

c. If Nogro were to cut its dividend payout ratio to 25%, what would happen to its stock price? d. What if Nogro eliminated the dividend?

13. The risk-free rate of return is 8%, the expected rate of return on the market portfolio is 15%, and the stock of Xyrong Corporation has a beta coefficient of 1.2. Xyrong pays out 40% of its earnings in dividends, and the latest earnings announced were $10 per share. Dividends were just paid and are expected to be paid annually. You expect that Xyrong will earn an ROE of 20% per year on all reinvested earnings forever. a. What is the intrinsic value of a share of Xyrong stock? b. If the market price of a share is currently $100, and you expect the market price to be equal

to the intrinsic value one year from now, what is your expected 1-year holding-period return on Xyrong stock?

14. The Digital Electronic Quotation System (DEQS) Corporation pays no cash dividends currently and is not expected to for the next five years. Its latest EPS was $10, all of which was reinvested in the company. The firm’s expected ROE for the next five years is 20% per year, and during this time it is expected to continue to reinvest all of its earnings. Starting in year 6, the firm’s ROE on new investments is expected to fall to 15%, and the company is expected to start paying out 40% of its earnings in cash dividends, which it will continue to do forever after. DEQS’s market capitalization rate is 15% per year. a. What is your estimate of DEQS’s intrinsic value per share? b. Assuming its current market price is equal to its intrinsic value, what do you expect to

happen to its price over the next year? c. What do you expect to happen to price in the following year? d. What effect would it have on your estimate of DEQS’s intrinsic value if you expected

DEQS to pay out only 20% of earnings starting in year 6? 15. Recalculate the intrinsic value of GE in each of the following scenarios by using the three-stage

growth model of Spreadsheet 18.1 (available in Connect; link to Chapter 18 material). Treat each scenario independently. a. The terminal growth rate will be 7%. b. GE’s actual beta is 1.0. c. The market risk premium is 7.5%.

16. Recalculate the intrinsic value of GE shares using the free cash flow model of Spreadsheet 18.2 (available in Connect; link to Chapter 18 material) under each of the following assumptions. Treat each scenario independently. a. GE’s P/E ratio starting in 2020 (cell G3) will be 16. b. GE’s unlevered beta (cell B22) is 0.8. c. The market risk premium (cell B27) is 7.5%.

17. The Duo Growth Company just paid a dividend of $1 per share. The dividend is expected to grow at a rate of 25% per year for the next three years and then to level off to 5% per year forever. You think the appropriate market capitalization rate is 20% per year. a. What is your estimate of the intrinsic value of a share of the stock? b. If the market price of a share is equal to this intrinsic value, what is the expected dividend

yield? c. What do you expect its price to be one year from now? d. Is the implied capital gain consistent with your estimate of the dividend yield and the market

capitalization rate? 18. The Generic Genetic (GG) Corporation pays no cash dividends currently and is not expected

to for the next four years. Its latest EPS was $5, all of which was reinvested in the company. The firm’s expected ROE for the next four years is 20% per year, during which time it is

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expected to continue to reinvest all of its earnings. Starting in year 5, the firm’s ROE on new investments is expected to fall to 15% per year. GG’s market capitalization rate is 15% per year. a. What is your estimate of GG’s intrinsic value per share? b. Assuming its current market price is equal to its intrinsic value, what do you expect to

happen to its price over the next year? 19. The MoMi Corporation’s cash flow from operations before interest and taxes was $2 million

in the year just ended, and it expects that this will grow by 5% per year forever. To make this happen, the firm will have to invest an amount equal to 20% of pretax cash flow each year. The tax rate is 35%. Depreciation was $200,000 in the year just ended and is expected to grow at the same rate as the operating cash flow. The appropriate market capitalization rate for the unleveraged cash flow is 12% per year, and the firm currently has debt of $4 million outstanding. Use the free cash flow approach to value the firm’s equity.

20. Chiptech, Inc., is an established computer chip firm with several profitable existing products as well as some promising new products in development. The company earned $1 a share last year, and just paid out a dividend of $.50 per share. Investors believe the company plans to maintain its dividend payout ratio at 50%. ROE equals 20%. Everyone in the market expects this situation to persist indefinitely. a. What is the market price of Chiptech stock? The required return for the computer chip

industry is 15%, and the company has just gone ex-dividend (i.e., the next dividend will be paid a year from now, at t = 1).

b. Suppose you discover that Chiptech’s competitor has developed a new chip that will elimi- nate Chiptech’s current technological advantage in this market. This new product, which will be ready to come to the market in two years, will force Chiptech to reduce the prices of its chips to remain competitive. This will decrease ROE to 15%, and, because of falling demand for its product, Chiptech will decrease the plowback ratio to .40. The plowback ratio will be decreased at the end of the second year, at t = 2: The annual year-end dividend for the second year (paid at t = 2) will be 60% of that year’s earnings. What is your estimate of Chiptech’s intrinsic value per share? (Hint: Carefully prepare a table of Chiptech’s earnings and dividends for each of the next three years. Pay close attention to the change in the payout ratio in t = 2.)

c. No one else in the market perceives the threat to Chiptech’s market. In fact, you are confi- dent that no one else will become aware of the change in Chiptech’s competitive status until the competitor firm publicly announces its discovery near the end of year 2. What will be the rate of return on Chiptech stock in the coming year (i.e., between t = 0 and t = 1)?!(Hint for parts c through e: Pay attention to when the market catches on to the new situation. A table of dividends and market prices over time might help.)

d. What will be the rate of return on Chiptech stock in the second year (between t = 1 and t = 2)? e. What will be the rate of return on Chiptech stock in the third year (between t = 2 and t = 3)?

1. At Litchfield Chemical Corp. (LCC), a director of the company said that the use of divi- dend discount models by investors is “proof ” that the higher the dividend, the higher the stock price. a. Using a constant-growth dividend discount model as a basis of reference, evaluate the direc-

tor’s statement. b. Explain how an increase in dividend payout would affect each of the following (holding all

other factors constant): i. Sustainable growth rate. ii. Growth in book value. 2. Helen Morgan, CFA, has been asked to use the DDM to determine the value of Sundanci, Inc.

Morgan anticipates that Sundanci’s earnings and dividends will grow at 32% for two years and 13% thereafter. Calculate the current value of a share of Sundanci stock by using a two-stage dividend discount model and the data from Tables 18A and 18B.

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3. Abbey Naylor, CFA, has been directed to determine the value of Sundanci’s stock using the Free Cash Flow to Equity (FCFE) model. Naylor believes that Sundanci’s FCFE will grow at 27% for two years and 13% thereafter. Capital expenditures, depreciation, and working capital are all expected to increase proportionately with FCFE. a. Calculate the amount of FCFE per share for the year 2016, using the data from Table 18A. b. Calculate the current value of a share of Sundanci stock based on the two-stage FCFE

model. c. i. Describe one limitation of the two-stage DDM model that is addressed by using the two-

stage FCFE model. ii. Describe one limitation of the two-stage DDM model that is not addressed by using the

two-stage FCFE model. 4. Christie Johnson, CFA, has been assigned to analyze Sundanci using the constant dividend

growth price/earnings (P/E) ratio model. Johnson assumes that Sundanci’s earnings and dividends will grow at a constant rate of 13%. a. Calculate the P/E ratio based on information in Tables 18A and 18B and on Johnson’s assump-

tions for Sundanci. b. Identify, within the context of the constant dividend growth model, how each of the following

factors would affect the P/E ratio: ! Risk (beta) of Sundanci. ! Estimated growth rate of earnings and dividends. ! Market risk premium.

Table 18A Sundanci actual #"$% and #"$! financial statements for fiscal years ending May &$ ($ million, except per-share data)

Income Statement #"$% #"$!

Revenue $ !"! $ #$% Depreciation &' &( Other operating costs ()% !)' Income before taxes %) **# Taxes &) (# Net income )' %' Dividends *% &! Earnings per share $'."*! $'.$#& Dividend per share $'.&*! $'.&%) Common shares outstanding (millions) %!.' %!.'

Balance Sheet #"$% #"$!

Current assets $ &'* $ (&) Net property, plant, and equipment !"! !%$ Total assets $ )"# $ %*# Current liabilities #" *!* Long-term debt ' ' Total liabilities $ +#" $ *!* Shareholders’ equity + )*% )"! Total liabilities and equity )"# %*# Capital expenditures (! (%

Required rate of return on equity *!% Growth rate of industry *(% Industry P/E ratio &)

Table 18B Selected financial information

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! $ Million

& '$!" '$!%

Cash and equivalents $ !!"#$ $ !%& Accounts receivable '$( ')( Inventory !'*( !'%( Total current assets $ !*"$ $ )&& Gross fixed assets $,&(+ %,$++ Accumulated depreciation (),")+) !((,)**) Net fixed assets $&,"$+ $&,''& Total assets $&,$+$ $&,%() Accounts payable $ !''% $ ''+ Notes payable + + Accrued taxes and expenses + + Total current liabilities $ ''% $ ''+ Long-term debt $",)(+ $",%++ Common stock (+ (+ Additional paid-in capital + + Retained earnings ",$%" !",*%) Total shareholders’ equity $',+&" $",%&) Total liabilities and shareholders’ equity $&,$+$ $&,%()

Table 18C Dynamic Communica- tion balance sheets

5. Dynamic Communication is a U.S. industrial company with several electronics divisions. The company has just released its 2018 annual report. Tables 18C and 18D present a summary of Dynamic’s financial statements for the years 2017 and 2018. Selected data from the financial statements for the years 2014 to 2016 are presented in Table 18E. a. A group of Dynamic shareholders has expressed concern about the zero growth rate of

dividends in the last four years and has asked for information about the growth of the company. Calculate Dynamic’s sustainable growth rates in 2015 and 2018. Your calculations should use beginning-of-year balance sheet data.

b. Determine how the change in Dynamic’s sustainable growth rate (2018 compared to 2015) was affected by changes in its retention ratio and its financial leverage. (Note: Your calcula- tions should use beginning-of-year balance sheet data.)

Table 18D Dynamic Communica- tion statements of income (U.S. $ millions except for share data)

'$!" '$!%

Total revenues $&,#'( $&,&++ Operating costs and expenses ',&*$ ',&"$ Earnings before interest, taxes, depreciation, and amortization (EBITDA)

$",+#) $ $%"

Depreciation and amortization #%& #(# Operating income (EBIT) $ ()& $ ('* Interest expense "+# "+* Income before taxes $ #($ $ #'+ Taxes (#+%) "%# ")% Net income $ '*( $ '(' Dividends $ %+ $ %+ Change in retained earnings $ "$( $ "*' Earnings per share $ '.*( $ '.(' Dividends per share $ +.%+ $ +.%+ Number of shares outstanding (millions) "++ "++

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6. Mike Brandreth, an analyst who specializes in the electronics industry, is preparing a research report on Dynamic Communication. A colleague suggests to Brandreth that he may be able to determine Dynamic’s implied dividend growth rate from Dynamic’s current common stock price, using the Gordon growth model. Brandreth believes that the appropriate required rate of return for Dynamic’s equity is 8%. a. Assume that the firm’s current stock price of $58.49 equals intrinsic value. What sustained

rate of dividend growth as of December 2018 is implied by this value? Use the constant- growth dividend discount model (i.e., the Gordon growth model).

b. The management of Dynamic has indicated to Brandreth and other analysts that the com- pany’s current dividend policy will be continued. Is the use of the Gordon growth model to value Dynamic’s common stock appropriate or inappropriate? Justify your response based on the assumptions of the Gordon growth model.

7. Peninsular Research is initiating coverage of a mature manufacturing industry. John Jones, CFA, head of the research department, gathered the following fundamental industry and market data to help in his analysis:

Forecast industry earnings retention rate !"% Forecast industry return on equity #$% Industry beta %.# Government bond yield &% Equity risk premium $%

a. Compute the price-to-earnings (P0 /E1) ratio for the industry based on this fundamental data.

b. Jones wants to analyze how fundamental P/E ratios might differ among countries. He gathered the following economic and market data:

Fundamental Factors Country A Country B

Forecast growth in real GDP $% #% Government bond yield %"% &% Equity risk premium $% !%

Determine whether each of these fundamental factors would cause P/E ratios to be generally higher for Country A or Country B.

8. Janet Ludlow’s firm requires all its analysts to use a two-stage dividend discount model (DDM) and the capital asset pricing model (CAPM) to value stocks. Using the CAPM and DDM, Ludlow has valued QuickBrush Company at $63 per share. She now must value SmileWhite Corporation.

Table 18E Dynamic Communica- tion selected data from financial statements (U.S. $ millions except for share data)

$"%! $"%& $"%'

Total revenues $',%($ $',"($ $',""" Operating income (EBIT) !)$ !!* !'' Interest expense %"! %"% )) Net income $ #'$ $ #"* $ #"" Dividends per share $ ".*" $ ".*" $ ".*" Total assets $',&#$ $',!%! $',#'" Long-term debt $%,($" $%,("" $%,&$" Total shareholders’ equity $%,&&! $%,$") $%,'*" Number of shares outstanding (millions) %"" %"" %""

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a. Calculate the required rate of return for SmileWhite by using the information in the following table:

& QuickBrush SmileWhite

Beta !."# !.!# Market price $$#.%% $"%.%% Intrinsic value $&".%% ? Notes: Risk-free rate $.#%% Expected market return !$.#%%

b. Ludlow estimates the following EPS and dividend growth rates for SmileWhite: First " years !'% per year Years thereafter (% per year

Estimate the intrinsic value of SmileWhite by using the table above and the two-stage DDM. Dividends per share in the most recent year were $1.72. c. Recommend QuickBrush or SmileWhite stock for purchase by comparing each company’s

intrinsic value with its current market price. d. Describe one strength of the two-stage DDM in comparison with the constant-growth DDM.

Describe one weakness inherent in all DDMs. 9. Rio National Corp. is a U.S.-based company and the largest competitor in its industry. Tables

18F through 18I present financial statements and related information for the company. Table 18J presents relevant industry and market data.

The portfolio manager of a large mutual fund comments to one of the fund’s analysts, Katrina Shaar: “We have been considering the purchase of Rio National Corp. equity shares, so I would like you to analyze the value of the company. To begin, based on Rio National’s past perfor- mance, you can assume that the company will grow at the same rate as the industry.” a. Calculate the intrinsic value of a share of Rio National equity on December 31, 2017, using

the Gordon constant-growth model and the capital asset pricing model. b. Calculate the sustainable growth rate of Rio National on December 31, 2017. Use 2017

beginning-of-year balance sheet values.

Table 18F Rio National Corp. sum- mary year-end balance sheets (U.S. $ millions)

'$!( '$!#

Cash $ !".%% $ #.)* Accounts receivable "%.%% '*.%% Inventory '%(.%& !)(.%& Current assets $'#'.%& $''!.(" Gross fixed assets $*$.$* $%(.$* Accumulated depreciation (!#$.!*) ((%.%%) Net fixed assets $"'%."% $"!(.$* Total assets $#*'."& $#$!.$% Accounts payable $ '#.%# $ '&.%# Notes payable %.%% %.%% Current portion of long-term debt %.%% %.%% Current liabilities $ '#.%# $ '&.%# Long-term debt '$%.%% '$#.%% Total liabilities $'&#.%# $'*!.%# Common stock !&%.%% !#%.%% Retained earnings !$*."! !'%."# Total shareholders’ equity $"%*."! $'*%."# Total liabilities and shareholders’ equity $#*'."& $#$!.$%

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Table 18G Rio National Corp. sum- mary income statement for the year ended December $", %#"& (U.S. $ millions)

Revenue $!"".#" Total operating expenses ($%!.%&) Operating profit $$'%."( Gain on sale &."" Earnings before interest, taxes, depreciation, & amortization (EBITDA) $$!$."( Depreciation and amortization (%$.$%) Earnings before interest & taxes (EBIT) $ )*.#* Interest ($(.#") Income tax expense ($'.*!) Net income $ !".$(

Table 18H Rio National Corp. supplemental notes for %#"&

Note $: Rio National had $%) million in capital expenditures during the year. Note ': A piece of equipment that was originally purchased for $$" million was sold for $% million at year-

end, when it had a net book value of $! million. Equipment sales are unusual for Rio National. Note !: The decrease in long-term debt represents an unscheduled principal repayment;

there was no new borrowing during the year. Note &: On January $, '"$%, the company received cash from issuing &"",""" shares of

common equity at a price of $')."" per share. Note ): A new appraisal during the year increased the estimated market value of land held for

investment by $' million, which was not recognized in '"$% income.

Table 18J Industry and market data December $", %#"&

Risk-free rate of return &.""% Expected rate of return on market index *.""% Median industry price/earnings (P/E) ratio $*.*" Expected industry earnings growth rate $'.""%

Table 18I Rio National Corp. common equity data for %#"&

Dividends paid (U.S. $ millions) $!.'" Weighted-average shares outstanding $(,""",""" Dividend per share $".'" Earnings per share $$.#* Beta $.#"

Note: The dividend payout ratio is expected to be constant.

10. While valuing the equity of Rio National Corp. (see CFA Problem 9), Katrina Shaar is consider- ing the use of either cash flow from operations (CFO) or free cash flow to equity (FCFE) in her valuation process. a. State two adjustments that Shaar should make to cash flow from operations to obtain free

cash flow to equity. b. Shaar decides to calculate Rio National’s FCFE for the year 2017, starting with net income.

Determine for each of the five supplemental notes given in Table 18H whether an adjust- ment should be made to net income to calculate Rio National’s free cash flow to equity for the year 2017, and the dollar amount of any adjustment.

c. Calculate Rio National’s free cash flow to equity for the year 2017. 11. Shaar (see CFA Problem 10) has revised slightly her estimated earnings growth rate for Rio

National and, using normalized (underlying trend) EPS, which is adjusted for temporary impacts on earnings, now wants to compare the current value of Rio National’s equity to that of the industry, on a growth-adjusted basis. Selected information about Rio National and the industry is given in Table 18K.

Compared to the industry, is Rio National’s equity overvalued or undervalued on a P/E-to- growth (PEG) basis, using normalized (underlying trend) earnings per share? Assume that the risk of Rio National is similar to the risk of the industry.

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Table 18K Rio National Corp. vs. industry

Rio National

Estimated earnings growth rate !!.""% Current share price $#$."" Normalized (underlying trend) EPS for #"!% $ !.%! Weighted-average shares outstanding during #"!% !&,""","""

Industry

Estimated earnings growth rate !#.""% Median price/Earnings (P/E) ratio !'.'"

E$INVESTMENTS EXERCISES !. Choose !% firms that interest you and download their financial statements from any of these

Web sites: finance.yahoo.com, finance.google.com, or money.msn.com.

a. For each firm, find the return on equity (ROE), the number of shares outstanding, the dividends per share, and the net income. Record them in a spreadsheet.

b. Calculate the total amount of dividends paid (Dividends per share ! Number of shares outstanding), the dividend payout ratio (Total dividends paid/Net income), and the plowback ratio (! " Dividend payout ratio).

c. Compute the sustainable growth rate, g = b ! ROE, where b equals the plowback ratio. d. Compare the growth rates (g) with the P/E ratios of the firms by plotting the P/Es against

the growth rates in a scatter diagram. Is there a relationship between the two? e. Find the price-to-book, price-to-sales, and price-to-operating-cash-flow ratios for your

sample of firms. Use a line chart to plot these three ratios on the same set of axes. What relationships do you see among the three series?

f. For each firm, compare the &-year growth rate of earnings per share with the growth rate you calculated above. Is the actual rate of earnings growth correlated with the sustainable growth rate you calculated?

'. Now calculate the intrinsic value of three of the firms you selected in the previous question. Make reasonable judgments about the market risk premium and the risk-free rate, or find estimates from the Internet.

a. What is the required return on each firm based on the CAPM? b. Try using a two-stage growth model, making reasonable assumptions about how future

growth rates will differ from current growth rates. Compare the intrinsic values derived from the two-stage model to the intrinsic values you find assuming a constant-growth rate. Which estimate seems more reasonable for each firm?

&. Now choose one of your firms and look up the other firms in the same industry. Perform(a “Valuation by Comparables” analysis by looking at the price/earnings, price/book value, price/sales, and price/cash flow ratios of the firms relative to each other and to the industry average. Which of the firms seem to be overvalued? Which seem to be undervalued? Can you think of reasons for any apparent mispricing?

). The actually expected return on a stock based on estimates of future dividends and future price can be compared to the “required” or equilibrium return given its risk. If the expected return is greater than the required return, the stock may be an attractive investment.

First calculate the expected holding-period return (HPR) on Target Corporation’s stock based on its current price, its expected price, and its expected dividend.

a. Go to MSN’s money central at www.msn.com/en-us/money. Get information for Target (enter TGT under quote search). Find the range of forecasted year-ahead prices. Find the range for estimated target price for the next fiscal year.

b. Collect information about today’s price and the dividend rate. What is the company’s expected dividend in dollars for the next fiscal year?

c. Use these inputs to calculate the range of Target’s HPRs for the next year.

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!"# P A R T V Security Analysis

SOLUTIONS TO CONCEPT CHECKS 1. a. Dividend yield = $2.15/$50 = 4.3%.

Capital gains yield = (59.77 ! 50)/50 = 19.54%. Total return = 4.3% + 19.54% = 23.84%.

b. k = 6% + 1.15(14% ! 6%) = 15.2%. c. V0 = ($2.15 + $59.77)/1.152 = $53.75, which exceeds the market price. This would indicate a

“buy” opportunity. 2. a. D1/(k ! g) = $2.15/(.152 ! .112) = $53.75.

b. P1 = P0(1 + g) = $53.75(1.112) = $59.77. c. The expected capital gain equals $59.77 ! $53.75 = $6.02, for a percentage gain of 11.2%.

The dividend yield is D1/P0 = 2.15/53.75 = 4%, for a holding-period return of 4% + 11.2% = 15.2%.

3. a. g = ROE " b = 20% " .60 = 12%. D1 = .4 " E1 = .4 " $5 = $2. P0 = $2/(.125 ! .12) = $400.

b. When the firm invests in projects with ROE less than k, its stock price falls. If b = .60, then g = 10% " .60 = 6% and P0 = $2/(.125 ! .06) = $30.77. In contrast, if b = 0, then P0 = $5/.125 = $40.

4. V2016#=# 1.04 _______

(1.113) #+# 1.22 _______

(1.113)2 #+# 1.41 _______

(1.113)3 #+# 1.60#+#P2020 ___________

(1.113)4

To estimate P2020#in this equation, use the constant-growth dividend discount model, but assuming the sustainable growth rate will be#g = 7%.

P2020#=# 1.60#"#(1#+#g) ____________

k#!#g #=# $1.712 _________

.113#!#.07 #=#$39.81

Therefore, V2016 = $29.93. 5. a. ROE = 12%.

b = $.50/$2.00 = .25. g = ROE " b = 12% " .25 = 3%. P0 = D1/(k ! g) = $1.50/(.10 ! .03) = $21.43. P0 /E1 = $21.43/$2.00 = 10.71.

b. If b = .4, then .4 " $2 = $.80 would be reinvested and the remainder of earnings, or $1.20, would be paid as dividends. g = 12% " .4 = 4.8%. P0 = D1/(k ! g) = $1.20/(.10 ! .048) = $23.08. P0 /E1 = $23.08/$2.00 = 11.54.

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C H A P T E R !" Equity Valuation Models #$!

bod77178_ch18_569-612.indd 571 04/04/17 03:21 PM

liabilities recognized in financial statements are based on historical—not current—values. For example, the book value of an asset equals the original cost of acquisition less some adjustment for depreciation, even if the market price of that asset has changed over time. Moreover, depreciation allowances are used to allocate the original cost of the asset over several years, but do not reflect loss of actual value.

Whereas book values are based on original cost, market values measure the! current values of assets and liabilities. The market value of the shareholders’ equity investment equals the difference between the current values of all assets and liabilities. We’ve empha- sized that current values generally will not match historical ones. Equally or even more important, many assets, such as the value of a good brand name or specialized expertise developed over many years, may not even be included on the financial statements. Market prices therefore reflect the value of the firm as a going concern. It would be unusual if the market price of a stock were exactly equal to its book value.

Can book value represent a “floor” for the stock’s price, below which level the market price can never fall? Although Microsoft’s book value per share in 2016 was less than its market price, other evidence disproves this notion. While it is not common, there are always some firms selling at a market price below book value. In 2016, for example, such unfortunate firms included Honda, Bank of America, Mitsubishi, and Citigroup.

A better measure of a floor for the stock price is the firm’s liquidation value per share. This represents the amount of money that could be realized by breaking up the firm, selling its assets, repaying its debt, and distributing the remainder to the shareholders. If the market price of equity drops below liquidation value, the firm becomes attractive as a takeover target. A corporate raider would find it profitable to buy enough shares to gain control and then actually to liquidate.

Another measure of firm value is the replacement cost of assets less liabilities. Some analysts believe the market value of the firm cannot remain for long too far above its replacement cost (sometimes called reproduction cost) because if it did, competitors would enter the market. The resulting competitive pressure would drive down the market value of all firms until they fell to replacement cost.

This idea is popular among economists, and the ratio of market price to replacement cost is known as Tobin’s q, after the Nobel Prize–winning economist James Tobin. In the long run, according to this view, the ratio of market price to replacement cost will tend toward 1, but the evidence is that this ratio can differ significantly from 1 for very long periods.

Although focusing on the balance sheet can give some useful information about a firm’s liquidation value or its replacement cost, the analyst must usually turn to expected future cash flows for a better estimate of the firm’s value as a going concern. We therefore turn to the quantitative models that analysts use to value common stock based on forecasts of future earnings and dividends.

The most popular model for assessing the value of a firm as a going concern starts from the observation that an investor in stock expects a return consisting of cash dividends and capital gains or losses. We begin by assuming a 1-year holding period and supposing that ABC stock has an expected dividend per share, E(D1), of $4; the current price of a share, P0, is $48; and the expected price at the end of a year, E(P1), is $52. For now, don’t worry about how you derive your forecast of next year’s price. At this point we ask only whether the stock seems attractively priced today given your forecast of next year’s price.

18.2 Intrinsic Value versus Market Price

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!"# P A R T V Security Analysis

bod77178_ch18_569-612.indd 572 04/04/17 03:21 PM

The expected holding-period return is E(D1) plus the expected price appreciation, E(P1) ! P0, all divided by the current price, P0:

Expected"HPR"="E(r)"=" E(D1)"+ [E(P1)"!"P0] __________________ P0

=" 4"+"(52"!"48) ____________ 48

"=".167, or 16.7%

Thus, the stock’s expected holding-period return is the sum of the expected dividend yield, E(D1)/P0, and the expected rate of price appreciation, the capital gains yield, [E(P1) ! P0]/P0.

But what is the required rate of return for ABC stock? The CAPM states that when stock market prices are at equilibrium levels, the rate of return that investors can expect to earn on a security is rf + #[E(rM) ! rf]. Thus, the CAPM may be viewed as providing an estimate of the rate of return an investor can reasonably expect to earn on a security given its risk as measured by beta. This is the return that investors will require of any other investment with equivalent risk. We will denote this required rate of return as k. If a stock is priced “correctly,” it will offer investors a “fair” return, that is, its expected return will equal its required return. Of course, the goal of a security analyst is to find stocks that are mispriced. An underpriced stock will provide an expected return greater than the required return.

Suppose that rf = 6%, E(rM) ! rf = 5%, and the beta of ABC is 1.2. Then according to the CAPM, the value of k is

k"="6%"+"1.2"$"5%"="12%

The expected holding-period return, 16.7%, therefore exceeds the required rate of return based on ABC’s risk by a margin of 4.7%. Naturally, the investor will want to include more of ABC stock in the portfolio than a passive strategy would indicate.

Another way to see this is to compare the intrinsic value of a share of stock to its market price. The intrinsic value, denoted V0, is defined as the present value of all cash payments (on a per share basis) to the stockholder, including dividends as well as the proceeds from the ultimate sale of the stock, discounted at the appropriate risk-adjusted interest rate, k. If the intrinsic value, or the investor’s own estimate of what the stock is really worth, exceeds the market price, the stock is considered undervalued and a good investment. For ABC, using a 1-year investment horizon and a forecast that the stock can be sold at the end of the year at price P1 = $52, the intrinsic value is

V0"=" E(D1)"+"E(P1) ____________

1"+"k "=" $4"+"$52 ________

1.12 "="$50

Equivalently, at a price of $50, the investor would derive a 12% rate of return—just equal to the required rate of return—on an investment in the stock. However, at the current price of $48, the stock is underpriced compared to intrinsic value. At this price, it provides better than a fair rate of return relative to its risk. Using the terminology of the CAPM, it is a positive-alpha stock, and investors will want to buy more of it than they would following a passive strategy.

If the intrinsic value turns out to be lower than the current market price, investors should buy less of it than under the passive strategy. It might even pay to go short on ABC stock, as we discussed in Chapter 3.

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!"# P A R T V Security Analysis

bod77178_ch18_569-612.indd 574 04/04/17 03:21 PM

More generally, for a holding period of H years, we can write the stock value as the present value of dividends over the H years, plus the ultimate sale price, PH:

V0!=! D1 _____

1!+!k !+! D2 _______

(1!+!k)2 !+!"!"!"!+! DH!+!PH ________

(1!+!k )H (18.2)

Note the similarity between this formula and the bond valuation formula developed in Chapter 14. Each relates price to the present value of a stream of payments (coupons in the case of bonds, dividends in the case of stocks) and a final payment (the face value of the bond, or the sales price of the stock). The key differences in the case of stocks are the uncertainty of dividends, the lack of a fixed maturity date, and the unknown sales price at the horizon date. Indeed, one can continue to substitute for price indefinitely, to conclude

V0!=! D1 _____

1!+!k !+! D2 _______

(1!+!k)2 !+! D3 _______

(1!+!k)3 !+!"!"!" (18.3)

Equation 18.3 states that the stock price should equal the present value of all expected future dividends into perpetuity. This formula is called the dividend discount model (DDM) of stock prices.

It is tempting, but incorrect, to conclude from Equation 18.3 that the DDM focuses exclusively on dividends and ignores capital gains as a motive for investing in stock. Indeed, we assume explicitly in Equation 18.1 that capital gains (as reflected in the expected sales price, P1) are part of the stock’s value. Don’t forget that the price at which you can sell a stock in the future depends on dividend forecasts at that time.

The reason only dividends appear in Equation 18.3 is not that investors ignore capital gains. It is instead that those capital gains will reflect dividend forecasts at the time the stock is sold. That is why in Equation 18.2 we can write the stock price as the present value of dividends plus sales price for any horizon date. PH is the present value at time H of all dividends expected to be paid after the horizon date. That value is then discounted back to today, time 0. The DDM asserts that stock prices are determined ultimately by the cash flows accruing to stockholders, and those are dividends.1

The Constant-Growth DDM Equation 18.3 as it stands is still not very useful in valuing a stock because it requires dividend forecasts for every year into the indefinite future. To make the DDM practical, we need to introduce some simplifying assumptions. A useful and common first pass is to assume that dividends are trending upward at a stable growth rate that we will call g. For example, if g = .05, and the most recently paid dividend was D0 = 3.81, expected future dividends are

D1!=!D0(1!+!g)! =!3.81!#!1.05! =!4.00 D2!=!D0(1!+!g)2!=!3.81!#!(1.05)2!=!4.20 D3!=!D0(1!+!g)3!=!3.81!#!(1.05)3!=!4.41

and so on. Using these dividend forecasts in Equation 18.3, we solve for intrinsic value as

V0!=! D0(1!+!g) _________

1!+!k !+! D0(1!+!g)

2 _________

(1!+!k)2 !+! D0(1!+!g)

3 _________

(1!+!k)3 !+!"!"!"

1If investors never expected a dividend to be paid, then this model implies that the stock would have no value. To reconcile the DDM with the fact that non-dividend-paying stocks do have a market value, one must assume that investors expect that some day it may pay out some cash, even if only a liquidating dividend.

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