The Foundation
5.10 Common Stock Valuation
Compared with valuing bonds and preferred stock, common stock valuation is more diffi- cult because of the uncertainty associated with the size and timing of future cash flows and the unknown nature of the required rate of return. Future cash flows may be in the form of cash dividend payments and/or changes in the stock’s price (gains or losses) over the holding period. Dividends are uncertain because corporations have no legal requirement to pay them unless declared by the board of directors. In addition, dividends may increase, remain stable, or decrease over time. Future stock prices are also uncertain. Because com- mon stock is generally riskier than bonds or preferred stock, investors require a higher rate of return (ks) to compensate for this risk.
Before discussing various valuation models, we want to stress that no one “best” model exists. The appropriate model to use in a specific situation depends on the characteristics of the asset being valued. These characteristics include such factors as the level of earnings, current growth rate in earnings, and the source of growth.
Discounted Cash Flow Models
Although numerous DCF valuation models are available for valuing common stock, these approaches generally differ based upon three factors:
• measure of cash flow – dividends and free cash flows to equity;
• expected holding period – finite (limited) and infinite; and
• pattern of expected dividends – zero (no) growth, stable (constant) growth, and supernormal growth.
We begin by narrowly defining the measure of cash flow as dividends and consider different expected holding periods and dividend patterns. Each of these models is a type of dividend discount model (DDM). Later, we use a broader measure of cash flow, namely, free cash flow to equity (FCFE) to estimate the value of common stock. FCFE valuation models are simply variants of the DDM, but use FCFE instead of dividends.
Dividend Discount Models
The most basic models for valuing equity are the dividend discount models. These models require two key inputs: expected dividends and the required rate of return on equity. The attractiveness of these models stems from their simplicity and intuitive logic.
Finite-period valuation model
A finite-period valuation model is a model that assumes an investor plans to buy a com- mon stock and hold it for a limited period. The holding period may be for one or more periods. Equation 5.15 shows that the intrinsic value of a common stock is the present value of the expected dividends during the holding period plus the present value of the terminal price, which is the expected price of a stock at the end of a specific holding period. If an investor sells the stock, the buyer is still acquiring the remaining dividend stream. Therefore, the basis for determining a stock’s terminal price at any point in time is the present value of the expected dividends after that point.
V D
k
P
s k
t s
t t
n
n s
= n
+ +
∑
( ) ( )+=1 1 1
(5.15)
where Pn is the terminal price of the common stock at the end of period n.
Determining the stock’s intrinsic value using a limited holding period model requires the following steps.
1 Forecast the dividends per share for each period. Companies pay dividends from earnings or cash flows. Thus, key inputs in estimating expected dividends per share are the firm’s expected earnings and dividend payout ratio. Various approaches are available for esti- mating earnings growth including historical growth, analyst projections, or the funda- mentals of the firm. The payout ratio should reflect changes in expected earnings growth.
For example, firms with high earnings growth typically have low payouts; while firms with stable earnings growth have high payouts.
The expected growth rate should be a function of the proportion of the earnings that a firm reinvests and the returns it earns on the projects undertaken using this money. Equation 5.16 gives the retention growth rate method, which we introduced
144 THEFOUNDATION
in Chapter 3. This formula shows that calculating the growth rate involves forecasting the retention rate and then multiplying the retention rate by the company’s expected future rate of return on equity.
g = RR(ROE) (5.16)
where g is the internal or sustainable growth rate; RR is the earnings retention rate, which is (1 – dividend payout); and ROE is the return on equity, which is net income/
equity. A common approach to forecasting the dividend is to use the current dividend and to increase it by the projected growth rate (g).
2 Forecast the expected price of the stock at the end of the holding period. The future estimated selling price of the stock (terminal value) reflects the present value of all future dividends after the selling date to infinity.
3 Estimate the required rate of return. Analysts often estimate the required rate of return that stockholders demand for holding a stock based on a risk and return model called the capital asset pricing model (CAPM). We discuss the CAPM further in Chapters 7 and 9.
Equation 5.17 presents the CAPM, which states that the required rate of return on equity is a function of the risk-free rate plus a premium based on the systematic risk of the security.
ks = Rf + βi(Rm − Rf) (5.17)
where ks is the required rate of return; Rf is the nominal risk-free rate of return (previously referred to as NRFR); Rm is the expected rate of return on the market; and βi is the beta coefficient (a measure of systematic risk) of the stock. Note that the risk premium for a specific stock i is βi(Rm− Rf). Other models are available for determining the required rate of return, such as arbitrage pricing models, but these are beyond the scope of this book.
4 Discount the expected dividends and terminal price at the required rate of return.
Example 5.17 Two-year Holding Period
An investor plans to buy common stock of Potomac Company and to sell it at the end of 2 years. Potomac Company just paid a $1 dividend (D0). The firm’s return on equity (ROE) is 10 percent and the investor expects it to stay at that rate in the future.
Potomac has a stable dividend payout policy of 40 percent. The investor forecasts that the stock price will be $50 at the end of the year. The current nominal risk-free rate is 5 percent, the expected market return is 11 percent, and Potomac’s beta is 1.2. What is the intrinsic value of Potomac’s common stock?
Solution: We can follow the four-step process to estimate the value of Potomac’s common stock.
1 Forecast the dividends per share for each period. This step requires forecasting the dividends for the next 2 years. To determine the growth rate (g) requires first
finding the retention rate = (1 − dividend payout) = 1 − 0.4 = 0.6 and then substituting RR = 0.60 and ROE = 0.10 percent in Equation 5.16 to get g = (RR)(ROE) = (0.60)(0.10) = 0.06 or 6 percent. Now calculate the dividends for the next 2 years: D1 = D0(1 + g) = $1.00(1.06) = $1.06 and D2 = D1(1 + g) = $1.06(1.06)
= $1.1236 or $1.12.
2 Forecast the expected price of the stock at the end of the holding period. In this situation, the terminal price at the end of the first year estimated to be $50.
3 Estimate the required rate of return. Using Equation 5.17 for the CAPM results in a required rate of return of ks = 0.05 + 1.2(0.11 − 0.05) = 0.122 or 12.2 percent.
4 Discount the expected dividends and terminal price at the required rate of return.
Substituting D1 = $1.06, D2 = $1.12, P2 = $50.00, and ks = 0.122 in Equation 5.15 results in a current value of $41.55.
Vs= $ . + + ( . ) $ . $ .
( . ) 1 06
1 122
1 12 50 00
1 1222 = $0.94 + $40.61 = $41.55
Infinite-period valuation model
For investors who do not contemplate selling their stock in the near future, a finite-period valuation model would be inappropriate. Therefore, we need to accommodate long-term holders. An infinite-period valuation model is a model that assumes an investor plans to buy a common stock and hold it indefinitely. As Equation 5.18 shows, the basic dividend discount model states that the intrinsic value of a share of common stock is equal to the discounted valued of all future dividends.
V D
s k
t s t
= +
∑
∞ ( )=1 1
(5.18) where Vs is the intrinsic value of a share of common stock; Dt is the expected dividends per share on common stock in period t; and ks is the investor’s required rate of return on common stock (cost of equity).
The basic DDM shows that the intrinsic value of a common stock depends on the ex- pected future dividends and the required rate of return (discount rate). Holding all other variables constant, an increase in expected dividends would cause Vs to increase, and a decrease in expected dividends would cause Vs to decrease. Likewise, an increase in ks would cause Vs to decrease, and a decrease in ks would cause Vs to increase (again holding all other variables constant).
This model applies both to firms that pay current dividends and to those that do not. This is because the expectations of all future dividends, not just the near ones, determine the intrinsic value of the stock. This model suggests that investors must forecast dividends to infinity and then discount them back to present value at the required rate of return to estimate the value of common stock. In practice, investors cannot accurately project dividends through infinity.
This does not present an insurmountable problem if investors can efficiently model the expected dividend stream by making appropriate assumptions about future growth. Here are three growth rate cases – zero growth, constant growth rate, and variable growth rates.
146 THEFOUNDATION
Zero growth
Some companies have highly stable expected dividend streams. The zero growth valuation model is a valuation approach that assumes dividends remain a fixed amount over time.
That is, the growth rate (g) of dividends is zero. Equation 5.19 for the zero growth DDM is a simplified version of the infinite-period DDM given. This formula is similar to Equation 5.13 for preferred stock because both treat dividends as a perpetuity.
V D
s k
s s
= (5.19)
where Ds is the expected dividends per share on common stock.
A limitation of this model comes from the fact that companies tend to pay a different amount of dividends over the course of their life cycle. Obviously, the model is best suited for companies that pay a constant dividend over time.
Example 5.18 Zero Growth Dividend Discount Model
United Industries expects to pay a $3.60 cash dividend at the end of each year, indefinitely into the future. If investors require a 12 percent return, what is the intrinsic value of the common stock of United Industries?
Solution: Substituting Ds = $3.60 and ks = 0.12 in Equation 5.19 results in a stock value of $30.00.
Vs= $ . =$ . .
3 60
0 12 30 00
Constant growth
Many companies have expected dividend streams that tend to grow at a constant rate for long periods. The constant growth valuation model is a valuation approach that assumes dividends per share grow at a constant rate each period that is never expected to change.
This model, also called the Gordon growth model, represents a single-stage growth pat- tern. Substituting D0(1 +g)t for Dt in Equation 5.18 produces the following constant growth DDM.
V D g
s k
t
st t
= +
+
∑
∞ 0( ) 11 1 ( )
=
(5.20) where D0 is the dividends per share in the current period; and g is the constant dividend growth rate.
If ks is greater than g, Equation 5.20 can be simplified as follows:
V D g
k g
D
k g
s
s s
= +
− =
− ( )
01 1
(5.21)
Although the constant growth model is a simple and convenient way to value common stock, it has several limiting assumptions. First, in order for the dividend to increase at a constant growth rate, the firm’s earnings must increase by at least that rate, because a firm pays dividends from earnings. Second, D1 cannot be equal to zero, because the model would not apply. Third, the discount rate (ks) must be greater than the growth rate (g). Otherwise, the model breaks down and the results are nonsense. Finally, value goes to infinity as g approaches ks. Thus, underestimating or overestimating the growth rate can lead to large valuation errors.
Because the constant growth DDM requires using some highly restrictive assumptions, it is best suited to firms that are growing at a steady rate that is comparable to or lower than the nominal growth rate in the economy. These firms should also have well-established dividend policies. As discussed in Chapter 12, some companies have an explicit goal of steady growth in dividends. Other firms are unlikely to meet these assumptions. For example, the constant growth DDM does not work with growth stocks.
Example 5.19 Constant Growth Dividend Discount Model
Kogod Enterprises just paid a $2.00 dividend last year. Analysts expect the firm’s dividends to grow at a constant rate of 6 percent a year. If investors require a 14 percent return, what is the intrinsic value of Kogod’s common stock?
Solution: Substituting D1 = $2.00(1.06) = $2.12, ks = 0.14, and g = 0.06 in Equa- tion 5.21 results in a stock value of $26.50.
Vs=
− =
. $ .
$ . .
2 12
0 14 0 06 26 50
Practical Financial Tip 5.9
An important relationship exists between the required rate of return (ks) and the growth rate (g).
• As the difference between ks and g widens, the stock value falls.
• As the difference between ks and g narrows, the stock value rises.
• Small changes in the difference between ks and g can lead to large changes in the stock’s value.
Variable growth
A company may experience different levels of growth in earnings and dividends over time.
A variable growth valuation model is a valuation approach that allows for a change in the dividend growth rate. That is, different growth rates occur during specific segments of the overall holding period. Thus, a characteristic of such a model is a multi-stage growth pattern. For example, a two-stage valuation model might involve a high growth period
148 THEFOUNDATION
followed by a stable or “steady state” growth period. Because a company cannot maintain high growth forever, it is likely to return to a more sustainable rate of growth at some time in the future. The model is suitable for companies that cannot sustain high dividend pay- ments. Limitations of such a model include the practical difficulty of defining the length of the high growth period and the abrupt drop to stable growth. To reduce these limitations, other models depend on different assumptions. For example, a three-stage valuation model can allow for an initial period of high growth, a transitional period where dividend growth has declined to more moderate growth, and a final stable-growth stage. Such a model could be appropriate for valuing firms with very high current growth rates.
One type of two-stage model is the temporary supernormal growth valuation model.
As Equation 5.22 shows, the temporary supernormal growth DDM states that the value of a firm’s common stock equals the present value of the expected dividends during the above- normal growth period plus the present value of the terminal price, which is the value of all remaining dividends to infinity starting at the beginning of the constant growth.
V D g
k
P
s k
t
st t
n
n sn
= +
+ +
∑
0( )1 +1
1
1 1
( ) ( )
=
(5.22)
where D0 is the dividends per share in period 0; g1 is the supernormal growth rate; n is the length of the supernormal growth period; Pn is the terminal price of the stock = Dn+1/(ks − g2) where Dn+1 is the first dividend at the resumption of normal growth; and g2 is the normal growth rate.
Equation 5.22 also applies to a two-stage DDM with a delayed dividend stream. For example, a company may pay no dividends during the first stage but then pay out dividends at a constant rate thereafter.
Practical Financial Tip 5.10
Linkages exist between growth rates in earnings or cash flows and dividend payout ratios. In practice, high growth firms typically have low dividend payouts. They prefer to reinvest their earnings in profitable projects instead of paying cash to stockholders.
After growth rates stabilize, firms tend to adopt more liberal dividend payout policies.
Example 5.20 Temporary Supernormal Growth Dividend Discount Model
Analysts expect dividends at LAB Corporation to grow at a rate of 20 percent for the next 4 years and 5 percent a year thereafter. The company just paid a $2 per share dividend. If investors require a 12 percent return, what is the value of the stock today?
Solution: Using Equation 5.22, finding the value of LAB Corporation’s stock involves three major steps.
1 Find the present value of the dividends during the supernormal growth period.
D g
k
t
st t
n t
t
0 1
1 4
1
1 4
1
2 00 1 20 1 10 ( )
.
+
+ =
∑
( )∑
= $ .( ( .) )=
Year t Dividend Discount factor Present value
$1.00(1.20)t= Dt 1/(1.12)t
[1] [2] [1] ××××× [2]
1 $2.00(1.20)= $2.40 0.8929 $2.14
2 2.00(1.20)2 = 2.88 0.7972 2.30
3 2.00(1.20)3 = 3.46 0.7118 2.46
4 2.00(1.20)4 = 4.15 0.6355 2.64
Total $9.54
2 Find the present value of the terminal price at the end of the supernormal growth period (year 4).
This step involves calculating the terminal price at the end of year 4 (P4).
P D
k g
D g
n s 4
1 2
41 2
0 12 0 05
4 151 05 0 07
4 358
0 07 62 25
= − = +
− = = =
+ ( )
. . $ . ( . ) . $ .
. $ .
Next, discount the terminal price to the present using the 12 percent required rate of return.
Present value of P4 =
P ksn 4
1 4
62 25 1 12
62 25
1 5735 39 56 ( ) $ .
( . ) $ .
. $ .
+ = = =
3 Sum the present value of both the dividends during the 4-year supernormal growth period and the terminal price in year 4.
Vs = $9.54 + $39.56 = $49.10
Free Cash Flow to Equity Models
An alternative measure of cash flows is free cash flow to equity (FCFE). When a firm’s dividends and FCFE differ, estimates of value will also differ when using DDM versus FCFE valuation. For example, the actual amount of dividends paid to stockholders often contrasts sharply with the FCFE that firms can afford to pay. Few firms follow a policy of paying out their entire FCFE as dividends. For example, some firms desire stability or are reluctant to change dividends. If a firm pays out all FCFE as dividends, this could lead to a highly volatile payout rate. Other firms want to hold back some FCFE to provide a reserve for future needs.
150 THEFOUNDATION
Thus, FCFE valuation models are more suited to value common stock when dividends are substantially higher or lower than the FCFE. FCFE model valuation also provides a better estimate of value in valuing firms for takeovers or where changing corporate control exists.
Although several approaches are available for measuring FCFE, Equation 5.23 shows a basic measure.
FCFE =Net income + Depreciation − Capital spending − Change in working
capital spending − Principal repayments spending − New debt issues (5.23) The major difference between dividend discount models and free cash flow to equity models lies in the measure of cash flows. The assumptions about growth and other inputs are similar. For example, the constant growth FCFE is the same as the constant growth DDM, except for substituting FCFE1 in Equation 5.21 in lieu of D1. Similar substitutions apply to the other DDM models. Because of the similarity between these two models, we do not repeat equations for FCFE model valuation here.
Relative Valuation Models
Another approach to valuation is relative valuation, which estimates value by finding simi- lar assets that are cheap or expensive and examining how the market currently prices these assets. Relative valuation involves defining “comparable” assets and choosing a standardized measure of value to compare firms. Usually value is in the form of some multiple of earnings, cash flow, book value of equity, or sales. Thus, an asset may be cheap (expensive) based on intrinsic value but expensive (cheap) based on relative valuation.
Although relative valuation appears to require fewer assumptions than DCF valuation methods, it actually does not. A major difference between these two valuation methods is that the assumptions underlying DCF valuation are explicit whereas they are implicit with relative valuation. The same variables that affect the estimated value using DCF valuation, such as the dividend payout ratio (D1/E1), risk as measured by the required rate of return (ks), and the expected growth rate of dividends (g), also affect multiples in relative valuation.
At the core of the relative valuation process is choosing comparable firms.10 A compar- able firm is one in the same business or sector with similar characteristics as the firm being valued. In selecting comparables, the analyst attempts to control for differences across firms, such as size. The analyst computes the multiple for each comparable firm and then averages them. Finally, the analyst compares the multiple of an individual firm to the average and attempts to explain any differences based on the firm’s individual characteristics such as growth or risk. Suppose comparable firms have an average P/E of 20 but the firm’s P/E is 12.
If analysts cannot explain these differences by the fundamentals, they will consider the stock as cheap (undervalued) if the multiple is less than average or expensive (overvalued) if the multiple is more than the average. Below are several common relative valuation models.
10 In contrast to the comparable firm approach, analysts can use multiple regression analysis using a cross- section of firms to predict various multiples such as P/E ratios.