## Residual Income Using Fcfe

Negative

May be positive, and growing

Stabilize at long-run level

ROE vs. Required Return

ROE > r

ROE approaching r

ROE = r

Dividend Payout

Low or zero

Increasing

Stabilize at long-run level

Model

Three-stage

Two-stage

This pattern is not predestined, since 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.

LOS 41.1: Explain terminal value and discuss alternative approaches to determining the terminal value in a discounted dividend model.

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 forecasr horizon and then estimate the terminal value as the P/E times the earnings estimate.

Example: Estimating terminal value

Level Partners is expected to have earnings in ten years of \$ 12.00 per share, a dividend payout ratio of 50%, and an expected 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.

The dividend at the end of 10 years is expected to be \$6 (\$12 times 50%). The dividend in year 11 is then \$6.00 x 1.04 = \$6.24. The terminal value using the Gordon growth model is therefore:

terminal value in year 10 (Gordon growth model) = —^.24— = \$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 x 8 = \$96.00

Professor's Note: Remember that terminal value in year 10 is equal to:

Multistage Growth Models: Valuation

LOS 4l.m: Calculate the value of common shares using the two-stage DDM, the H-model, and the three-stage DDM.

### 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:

where:

gs = short-term growth rate gL = long-term growth rate r = required return n = length of high growth period

Example: Calculating value with a two-stage DDM

Sea Island Recreation currently pays a dividend of \$ 1 .OC. 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.

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

\$1.00x(1.10)' Sl.OOx(l.lO)2 \$1.00x(l.l0)3 \$1.00x(1.10)3x(1.04)

° (1.12)1 ' (1.12)2 ' (1.12)3 ' (l.l2)3 x(0.12—0.04) V0 =\$15.21

If we were robots instead of humans, this would be fine. However, since 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 multi-period DDM). The forecasted dividends are shown in the following figure.

Dividend Cash Flows

SI * 1.10 \$1.10x1.10 \$1-21x1.10 = \$1.10 -\$1.21 =\$1,331

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:

Now the problem is exactly like the three-period DDM we solved in LOS 4l.h: 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 arc projected to start.

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.

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:

half-life (in years) of high-growth period length of high growth period short-term growth rate long-term growth rate 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 gL. The second term is an approximation of the additional value that results from the high-growth period.

where:

where:

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 10 years to a stable rate of 5% thereafter. The required return is 12%. Calculate the current value of Omega.

„„„„ , „ €2.00 x (—] x (0.20 - 0.05) = €2.00x1.05 +-W-- = =

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 firsr 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, gs-gL.

### 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 timeline or an equivalent cash flow table will help the intuition. Your speed and accuracy will develop with practice.

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 race for the next four years before settling down to a constant 4% growth rate thereafter. Calculace the current value of R&M.