Multiperiod Models

Equity Valuation 22 min readModule 3 of 3Save

What you'll learn

LOS 18.k
Explain the growth phase, transition phase, and maturity phase of a business.
LOS 18.l
Explain the assumptions and justify the selection of the two stage DDM, the H model, the three stage DDM, or spreadsheet modeling to value a company’s common shares.
LOS 18.m
Describe terminal value and explain alternative approaches to determining the terminal value in a DDM.
LOS 18.n
Calculate and interpret the value of common shares using the two stage DDM, the H model, and the three stage DDM.
LOS 18.o
Explain the use of spreadsheet modeling to forecast dividends and to value common shares.
LOS 18.i
Estimate a required return based on any DDM, including the Gordon growth model and the H model.
LOS 18.p
Calculate and interpret the sustainable growth rate of a company and demonstrate the use of DuPont analysis to estimate a company’s sustainable growth rate.
LOS 18.j
Evaluate whether a stock is overvalued, fairly valued, or undervalued by the market based on a DDM estimate of value.

While the basic GGM assumes constant growth, most firms go through a pattern of growth that includes several phases:

An initial growth phase, where the firm has rapidly increasing earnings, little or no dividends, and heavy reinvestment.
A transition phase, in which earnings and dividends are still increasing but at a slower rate as competitive forces reduce profit opportunities and the need for reinvestment.
A mature phase, in which earnings grow at a stable but slower rate, and payout ratios are stabilizing as reinvestment matches depreciation and asset maintenance requirements.

The level and pattern of specific fundamental variables during the three phases, as well as the appropriate valuation model to apply in each phase, are shown in Figure 18.1.

Figure 18.1: Growth, Transition, and Maturity Phase

Variable	Initial growth	Transition	Maturity
Earnings growth	Very high	Above average but falling	Stable at long run level
Capital investment	Significant requirements	Decreasing	Stable at long run level
Profit margin	High	Above average but falling	Stable at long run level
FCFE	Negative	May be positive, and growing	High
ROE vs. required return	ROE > r	ROE approaching r	ROE = r
Dividend payout	Low or zero	Increasing	Stable at long run level
Appropriate model	Three stage	Two stage	Gordon growth

This pattern is not predestined because many firms are successful in constantly adapting and entering into new growth opportunities. Mature firms may develop technology that forms the basis for a whole new product and market. The point is that a multistage model is required in order to value many firms.

Fortunately, the GGM is easily adaptable to multistage growth.

For most companies, the Gordon growth model assumption of constant dividend growth that continues into perpetuity is unrealistic. For example, many companies experience growth rates in excess of the required rate of return for short periods of time as a result of a competitive advantage they have developed. We need more realistic multistage growth models to estimate value for companies with several stages of future growth.

The appropriate model is the one that most closely matches the firm’s expected pattern of growth. However, whichever multistage model we use, there are two important points to keep in mind:

We’re still just forecasting dividends into the future and discounting them back to today to find intrinsic value.
Over the long term, growth rates tend to revert to a long run rate approximately equal to the long term growth rate in real gross domestic product (GDP) plus the long term inflation rate. Historically, that number has been between 2% and 5%. Anything higher than 5% as a long run perpetual growth rate is difficult to justify.

Two Stage DDM: The most basic multistage model is a two stage DDM in which we assume the company grows at a high rate for a relatively short period of time (the first stage) and then reverts to a long run perpetual growth rate (the second stage). The length of the high growth phase is a function of the visibility of the company’s operations; in other words, it tells how far into the future the analyst can predict growth rates with a certain degree of confidence. An example in which the two stage model would apply is a situation in which a company has a patent that will expire.

For example, suppose a firm is expected to grow at 15% until patents expire in four years, then immediately revert to a long run growth rate of 3% in perpetuity. This stock should be modeled by a two stage model, with dividends growing at 15% before the patent expires and 3% thereafter (see Figure 18.2).

Figure 18.2: Example of a Two Stage DDM

H Model: The problem with the basic two stage DDM is that it is usually unrealistic to assume that a stock will experience high growth for a short period, then immediately fall back to a long run level. The H model utilizes a more realistic assumption: the growth rate starts out high and then declines linearly over the high growth stage until it reaches the long run average growth rate. For example, consider a firm that generates high profit margins, faces little competition from within its industry, and is currently growing at 15%.

We might forecast that the firm’s growth rate will decline by 3% per year as competitors enter the market until it reaches 3% at the end of the fourth year, when the industry matures and growth rates stabilize (see Figure 18.3).

Figure 18.3: Example of an H Model

Three stage DDM: Three stage models are appropriate for firms that are expected to have three distinct stages of earnings growth. A three stage model is a slightly more complex refinement of a two stage model. For example, suppose we forecast that a biotech company will experience supernormal growth of 25% for three years, then 15% growth for five years, and finally slow down to a stable, long run rate of 3% (see Figure 18.4).

Alternatively, in Stage 2, the growth rate may also linearly decay to the Stage 3 stable, long run growth rate.

Figure 18.4: Example of a Three Stage DDM

Spreadsheet modeling: The two and three stage models we’ve discussed so far are really just models in which we’ve simplified the growth pattern to make the calculations doable. Obviously that’s an important consideration on the exam. However, in practice we can use spreadsheets to model any pattern of dividend growth we’d like with different growth rates for each year because the spreadsheet does all the calculations for us.

Spreadsheet modeling is applicable to firms about which you have a great deal of information and can project different growth rates for differing periods, such as construction firms and defense contractors with many long term contracts. Figure 18.5 is an example of three different spreadsheet models.

Figure 18.5: Examples of Spreadsheet Modeling

Growth rates	Year 1	Year 2	Year 3	Year 4	Year 5	Year 6	Year 7 and after
Scenario 1	20%	19%	13%	5%	5%	5%	5%
Scenario 2	20%	19%	13%	11%	5%	5%	5%
Scenario 3	20%	19%	13%	11%	8%	7%	5%

No matter which dividend discount model we use, we have to estimate a terminal value at some point in the future. There are two ways to do this: using the Gordon growth model and using the market multiple approach.

The most common method (on the exam) is to estimate the terminal value with the Gordon growth model. In other words, at some point in the future, we assume dividends will begin to grow at a constant, long term rate. Then the terminal value at that point is just the value derived from the Gordon growth model.

Many analysts also use market price multiples to estimate the terminal value rather than use the GGM method of discounting dividends. For example, we could forecast earnings and a P/E ratio at the forecast horizon and then estimate the terminal value as the P/E multiplied by the earnings estimate.

Worked example: Estimating terminal value

Level Partners is expected to have earnings in ten years of $12 per share, a dividend payout ratio of 50%, and a required return of 11%. At that time, the dividend growth rate is expected to fall to 4% in perpetuity, and the trailing P/E ratio is forecasted to be eight times earnings. Estimate the terminal value at the end of ten years using the Gordon growth model and the P/E multiple.

Solution

The dividend at the end of ten years is expected to be $6 ($12 multiplied by 50%). The dividend in Year 11 is then $6.00 × 1.04 = $6.24. The terminal value using the Gordon growth model is therefore:

terminal value in Year 10 = \frac{D _{10} ( 1 + g )}{r - g} = \frac{D _{11}}{r - g} = \frac{$6.24}{0.11 - 0.04} = $89.14

The terminal value given forecasted earnings of $12 and a P/E ratio of 8 is:

terminal value in Year 10 (trailing P/E multiple) = $12.00 × 8 = $96.00

Valuation Using the Two Stage Model

The two stage fixed growth rate model is based on the assumption that the firm will enjoy an initial period of high growth, followed by a mature or stable period in which growth will be lower but sustainable:

V_{0} = [t = 1 \sum n \frac{D _{0} ( 1 + g _{S} ) ^{t}}{( 1 + r ) ^{t}}] + \frac{D _{0} ( 1 + g _{S} ) ^{n} ( 1 + g _{L} )}{( 1 + r ) ^{n} ( r - g _{L} )}

Symbol	Meaning
gₛ	Short term growth rate
gₗ	Long term growth rate
r	Required return
n	Length of high growth period

Worked example: Calculating value with a two stage DDM

Sea Island Recreation currently pays a dividend of $1.00. An analyst forecasts growth of 10% for the next three years, followed by 4% growth in perpetuity thereafter. The required return is 12%. Calculate the current value per share.

Solution

We could solve the problem by plugging the appropriate numbers into the formula as follows:

If we were robots instead of humans, this would be fine. However, because we are human beings (and not mindless machines), it might be better to actually try to understand what we are doing, limit the need to remember yet another formula, and reduce the possibility of error. This can be accomplished by drawing a time line and placing the appropriate cash flows on the line, followed by the fairly straightforward computation on our financial calculators that we did earlier (in the multiperiod DDM).

The forecasted dividends are shown in the following figure.

Dividend Cash Flows

Constant growth at 4% begins after the third year, and we can employ the DDM to determine the value of the stock at time t = 3. Accordingly:

V_{3} = \frac{D _{3} ( 1 + g )}{r - g} = \frac{D _{4}}{r - g} = \frac{$1.3842}{0.12 - 0.04} = $17.30

Now the problem is exactly like the three period DDM we solved in an earlier LOS: we know the dividends in Years 1, 2, and 3, the terminal value in Year 3, and the discount rate. The cash flows that we need to solve the problem are shown in the following figure.

Dividend and Terminal Value Cash Flows

The financial calculator does the hard work for us: CF0 = 0; C01 = 1.10; C02 = 1.21; C03 = 18.63; I = 12; CPT → NPV = 15.21.

We arrived at an estimated value of $15.21 using the calculator, which is exactly the same answer we got with the ugly formula. After a bit of practice, you should find that the calculator method is easier than the complicated formula, and, just as importantly, it will be less prone to error.

The value of a firm that doesn’t currently pay a dividend is a simple version of the two- stage DDM, where the firm pays no dividends in the first stage. Therefore, the value of the firm is just the present value of the terminal value computed at the point in time at which dividends are projected to start.

Worked example: Valuing a non dividend paying stock

Arena Distributors is a new company and currently pays no dividends. The company recently reported earnings of $1.50 per share and is expected to grow at a 15% rate for the next four years. Beginning in Year 5, Arena is expected to distribute 20% of its earnings in the form of dividends and to have a constant growth rate of 5%.

The required rate of return is 12%. Calculate the value of Arena shares today.

Solution

First forecast the earnings in Year 5. Then calculate the dividends in Year 5 as 20% of Year 5 earnings. Applying the Gordon growth model to the Year 5 dividend gives us an estimate of the terminal value in Year 4. The terminal value discounted back four years is the current value of the stock.

Valuation Using the H Model

The earnings growth of most firms does not abruptly change from a high rate to a low rate as in the two stage model but tends to decline over time as competitive forces come into play. The H model approximates the value of a firm assuming that an initially high rate of growth declines linearly over a specified period. The formula for this approximation is:

V_{0} = \frac{D _{0} ( 1 + g _{L} )}{r - g _{L}} + \frac{D _{0} H ( g _{S} - g _{L} )}{r - g _{L}}

Symbol	Meaning
H = t/2	Half life, in years, of high growth period
t	Length of high growth period
gₛ	Short term growth rate
gₗ	Long term growth rate
r	Required return

Note that the first term is what the shares would be worth if there were no high growth period and the perpetual growth rate was $g_{L}$ . The second term is an approximation of the additional value that results from the high growth period.

Worked example: Calculating value with the H model

Omega Foods currently pays a dividend of €2.00. The growth rate, which is currently 20%, is expected to decline linearly over the next ten years to a stable rate of 5% thereafter. The required return is 12%. Calculate the current value of Omega.

Solution

V_{0} = \frac{€ 2.00 \times 1.05}{0.12 - 0.05} + \frac{€ 2.00 \times ( 10/2 ) \times ( 0.20 - 0.05 )}{0.12 - 0.05} = € 30.00 + € 21.43 = € 51.43

Remember that the H model provides only an approximation of the value of Omega shares. To find the exact answer, we’d have to forecast each of the first ten dividends, applying a different growth rate to each, and then discount them back to the present at 12%. In general, the H value approximation is more accurate the shorter the high- growth period, t, and/or the smaller the spread between the short term and long- term growth rates, $g_{S}$ − $g_{L}$ .

Valuation Using the Three Stage DDM

A three stage model can be used to estimate the value of a firm that is projected to have three stages of growth with a fixed rate of growth for each stage. The approach is the same as the two stage model, with the projected dividends and the terminal value of the shares discounted to their present value at the required rate of return. Again, a time line or an equivalent cash flow table will help the intuition.

Your speed and accuracy will develop with practice.

Worked example: Calculating value with the three stage DDM

R&M has a current dividend of $1.00 and a required rate of return of 12%. A dividend growth rate of 15% is projected for the next two years, followed by a 10% growth rate for the next four years before settling down to a constant 4% growth rate thereafter. Calculate the current value of R&M.

Solution

Relevant Cash Flows for R&M Example

Time	Value	Calculation	Dₜ or Vₜ
1	D₁	$1.00(1.15)	$1.150
2	D₂	$1.150(1.15)	$1.323
3	D₃	$1.323(1.10)	$1.455
4	D₄	$1.455(1.10)	$1.600
5	D₅	$1.600(1.10)	$1.760
6	D₆	$1.760(1.10)	$1.936
6	V₆	[$1.936(1.04)] / (0.12 − 0.04)	$25.168

Now we enter the cash flows into our calculator, noting that the total cash flow at Time 6 is $1.936 + $25.168 = $27.104: CF0 = 0; C01 = 1.150; C02 = 1.323; C03 = 1.455; C04 = 1.600; C05 = 1.760; C06 = 27.104; I = 12; CPT → NPV = 18.864.

According to the three stage model, R&M is worth $18.864 today. This question is tedious, but it is not a question to be feared, as long as your calculator batteries hold up.

Worked example: Three stage growth model with linear growth decline in Stage 2

As an analyst, you have gathered the following information on a company you are tracking. The current annual dividend is $0.75. Dividends are expected to grow at a rate of 12% over the next three years, decline linearly to 4% over the next six years, and then remain at a long term equilibrium growth rate of 4% in perpetuity. The required return is 9%.

Calculate the value of the company.

Solution

Let’s start by valuing the last two stages using the H model. We know that:

It follows that:

Now we have a series of three cash flows to discount in order to find the current value of the stock, and our financial calculator does the rest of the work.

CF0 = 0; C01 = $D_{1}$ = $0.75(1.12) = $0.84; C02 = $D_{2}$ = $0.75(1.12) $^{2}$ = $0.9408; and C03 = $D_{3}$ + $V_{3}$ = $1.0537 + $26.9747 = $28.0284; I = 9; CPT → NPV = 23.2056.

The price of the stock is $23.2056.

Valuation Models Using Spreadsheets

If you have been calculating along with the examples, you already recognize that the use of these models can be computationally intensive, though the formulas are straightforward. These characteristics make DDM models ideally suited to being solved with spreadsheet software. A spreadsheet allows the analyst to easily calculate values based on models with many stages, growth rates, and required rates of return.

We have previously noted that, in practice, financial analysts are much more likely to use a spreadsheet than any of the stylized models presented here when valuing equity securities. The reason for this is the inherent flexibility and computational accuracy of spreadsheet modeling.

A firm’s dividends (or cash flows) often do not grow at a smooth rate for an extended period. When changes in dividends can be predicted, there can obviously be more than two or three stages of change involved. Moreover, there are often idiosyncratic events that, even if they can be predicted, do not fit neatly into any of the patterns required by these models.

Using a spreadsheet is relatively straightforward and can accommodate nearly any pattern that the analyst can imagine.

Step 1: Establish the base level of cash flows or dividends. In the case of dividends, this would ordinarily be either the amount paid over the preceding year or some normalized level based upon projected firm earnings.

Step 2: Estimate changes in the firm’s dividends for the foreseeable future (also known as the supernormal growth period) and project future cash dividends on the basis of these estimates. Because the spreadsheet can be programmed in a virtually infinite series of combinations, any dividend pattern desired can be achieved.

Step 3: Because an equity security has an infinite life, the analyst needs to estimate what normalized level of growth will occur at the end of the supernormal growth period. This allows for an estimate of a terminal value, representing the cash flow (i.e., the firm’s value if sold at this time) to be received at the end of the supernormal growth period.

Step 4: Discount all projected dividends and the terminal value back to today to obtain an estimate of the firm’s current value.

The last step is where the use of the spreadsheet really pays off. The analyst is in position to conduct detailed scenario analyses wherein the model inputs can be altered to see how changes in the pattern of future dividends, interest rates, and firm risk affect firm valuation estimates.

The bottom line is that performing the analysis just listed for a period of 10 or 20 years is relatively easy with a spreadsheet but would be all but impossible with any of the stylized models presented.

We have been using DDMs to determine the value of a stock, assuming that we know the dividends and required rates of return. The models are just as useful in determining the required rate of return, given the current value and dividends of a stock. No matter which model you’re using (whether it’s a two stage DDM, an H model, a three stage DDM, or a spreadsheet model), in theory this is easy: Given all the other inputs to the model, we can back into the expected return that makes the present value of the forecasted dividend stream equal to the current market price.

For example, if the dividend growth rate is constant forever, we can use the Gordon growth model to calculate the implied expected return given the expected dividend, the current market price, and the expected growth rate:

r = \frac{D _{1}}{P _{0}} + g

Worked example: Calculating expected return with the Gordon growth model

Smyth & Weston Explosives’ stock is expected to pay a dividend of $1.60, has a current price of $40.00, and has a projected growth rate of 9%. Calculate S&W’s implied required return.

Solution

r = \frac{$1.60}{$40.00} + 0.09 = 0.13 = 13%

The H model can be rewritten in terms of r and used to solve for r given the other model inputs:

r = {\frac{D _{0}}{P _{0}} [(1 + g_{L}) + H (g_{S} - g_{L})]} + g_{L}

Worked example: Solving for expected return with the H model

Beluga Fisheries, Inc., just paid a current dividend of $0.75, which has been growing at a rate of 10%. This growth rate is expected to decline to 5% over the next five years and then remain at 5% indefinitely. Calculate the implied required return for Beluga based on the current price of $30.00.

Solution

r = {\frac{$0.75}{$30.00} [(1 + 0.05) + \frac{5}{2} (0.10 - 0.05)]} + 0.05 = 0.0794 = 7.94%

Using the general two stage model is more difficult because we have to solve for r with an iterative process because there is no closed form solution. Here is an example of how to approach the problem using the two stage DDM.

Worked example: Solving for expected return with the two stage DDM

Ozone Laboratories, Inc., recently paid a dividend of $1.00. Dividends are expected to grow at a rate of 11% for the next two years and 8% thereafter. The implied required return for Ozone based on the current price of $35.60 is closest to:

A. 9.5%.

B. 10.0%.

C. 11.2%.

Solution

We have to solve for r in the following equation:

This actually requires an iterative process. The easiest way to do this is to start with the middle value from the three answer choices (10%, in this case):

Use your calculator to value the stock:

CF0 = 0, C01 = 1.11, C02 = 1.23 + 66.55 = 67.78

NPV (I = 10%) = $57.03 (too high relative to the given market price of $35.60)

Hence, the discount rate has to be higher. The only higher value (from the multiple choices) is 11.2%.

Let’s confirm:

$P_{2}$ = 1.331 / (0.112 − 0.08) = 41.59

CF0 = 0, C01 = 1.11, C02 = 1.23 + 41.59 = 42.82

PV (I = 11.20%) = $35.63 (close enough)

The sustainable growth rate (SGR) is the rate at which earnings (and dividends) can continue to grow indefinitely, assuming that the firm’s debt to equity ratio is unchanged and it doesn’t issue new equity. SGR is a simple function of the earnings retention ratio and the return on equity (ROE):

SGR = b \times ROE

Symbol	Meaning
b	Earnings retention rate = 1 − dividend payout rate
ROE	Return on equity

The SGR is important because it tells us how quickly a firm can grow with internally generated funds.

Worked example: Calculating SGR

Biotechnica, Inc., is growing earnings at an annual rate of 9%. It currently pays out dividends equal to 20% of earnings. Biotechnica’s ROE is 15%.

Calculate its SGR.

Solution

g = (1 - 0.20) \times 15% = 12%

A firm’s rate of growth is a function of both its earnings retention and its return on equity. ROE can be estimated with the DuPont formula, which presents the relationship between margin, sales, and leverage as determinants of ROE:

ROE = \frac{net income}{stockholders’ equity} = \frac{net income}{sales} \times \frac{sales}{total assets} \times \frac{total assets}{stockholders’ equity}

If the other factors remain constant, we can see that the growth of a firm’s earnings (and dividends) is a function of its ROE and its retention rate:

g = \frac{net income - dividends}{net income} \times \frac{net income}{sales} \times \frac{sales}{total assets} \times \frac{total assets}{stockholders’ equity}

This has also been called the PRAT model, where SGR is a function of the profit margin (P), the retention rate (R), the asset turnover (A), and financial leverage (T). Two of these factors are functions of the firm’s financing decisions (leverage and earnings retention), and two are functions of performance (return on assets equals profit margin multiplied by asset turnover). These factors can be used as building blocks in developing an estimate of a firm’s growth.

If the actual growth rate is forecasted to be greater than SGR, the firm will have to issue equity unless the firm increases its retention ratio, profit margin, total asset turnover, or leverage.

Worked example: Calculating ROE and SGR

Halo Construction has been successful in a mature industry. Over the last three years, Halo has averaged a profit margin of 10%, a total asset turnover of 1.8, and a leverage ratio of 1.25. Assuming Halo continues to distribute 40% of its earnings as dividends, calculate its long term SGR.

Solution

g = P \times R \times A \times T

g = 0.10 (1 - 0.4) (1.8) (1.25) = 0.135 = 13.5%

Worked example: Calculating ROE and SGR

Given the following partial balance sheets and income statement for Far Horizons Company, calculate three components of the ROE (using the DuPont model) and the sustainable growth rate for 20X3 based on beginning balance sheet values. Use a dividend payout ratio of 30%. All values are in millions of USD.

Far Horizons Income Statement

Income statement for fiscal year ended 20X3	Amount
Sales	$40.0
Net income	$1.8

Far Horizons Balance Sheet

Balance sheet for fiscal year end	20X2	20X3
Assets	$30.0	$50.0
Total assets	$30.0	$50.0
Liabilities	$10.0	$20.0
Equity	$20.0	$30.0
Total liabilities and equity	$30.0	$50.0

Solution

If a stock is trading at a price (market price) higher than the price implied by a dividend discount model (model price), the stock is considered to be overvalued. Similarly, if the market price is lower than the model price, the stock is considered to be undervalued, and if the model price is equal to the market price, the stock is considered to be fairly valued.

In other words: market price > calculated value.

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