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CHAPTER 9
STOCK VALUATION
Answers to Concept Questions
1. The value of any investment depends on the present value of its cash flows; i.e., what investors will
3. In general, companies that need the cash will often forgo dividends since dividends are a cash
expense. Young, growing companies with profitable investment opportunities are one example;
4. The general method for valuing a share of stock is to find the present value of all expected future
dividends. The dividend growth model presented in the text is only valid (i) if dividends are expected
to occur forever; that is, the stock provides dividends in perpetuity, and (ii) if a constant growth rate
of dividends occurs forever. A violation of the first assumption might be a company that is expected
5. The common stock probably has a higher price because the dividend can grow, whereas it is fixed on
6. The two components are the dividend yield and the capital gains yield. For most companies, the
7. Yes. If the dividend grows at a steady rate, so does the stock price. In other words, the dividend
8. The three factors are: 1) The company’s future growth opportunities. 2) The company’s level of risk,
10. Presumably, the current stock value reflects the risk, timing and magnitude of all future cash flows,
Solutions to Questions and Problems
NOTE: All end-of-chapter problems were solved using a spreadsheet. Many problems require multiple
steps. Due to space and readability constraints, when these intermediate steps are included in this
solutions manual, rounding may appear to have occurred. However, the final answer for each problem is
found without rounding during any step in the problem.
Basic
1. The constant dividend growth model is:
So, the price of the stock today is:
The dividend at Year 4 is the dividend today times the FVIF for the growth rate in dividends and four
years, so:
We can do the same thing to find the dividend in Year 16, which gives us the price in Year 15, so:
P15 = D15(1 + g) / (R – g) = D0(1 + g)16 / (R – g)
There is another feature of the constant dividend growth model: The stock price grows at the
dividend growth rate. So, if we know the stock price today, we can find the future value for any time
P3 = P0(1 + g)3
And the stock price in 15 years will be:
2. We need to find the required return of the stock. Using the constant growth model, we can solve the
equation for R. Doing so, we find:
3. The dividend yield is the dividend next year divided by the current price, so the dividend yield is:
Dividend yield = D1 / P0
The capital gains yield, or percentage increase in the stock price, is the same as the dividend growth
rate, so:
4. Using the constant growth model, we find the price of the stock today is:
P0 = D1 / (R – g)
5. The required return of a stock is made up of two parts: The dividend yield and the capital gains yield.
So, the required return of this stock is:
6. We know the stock has a required return of 10.8 percent, and the dividend and capital gains yield are
equal, so:
Now we know both the dividend yield and capital gains yield. The dividend is the stock price times
the dividend yield, so:
This is the dividend next year. The question asks for the dividend this year. Using the relationship
between the dividend this year and the dividend next year:
7. The price of any financial instrument is the PV of the future cash flows. The future dividends of this
stock are an annuity for 13 years, so the price of the stock is the PVA, which will be:
8. The price of a share of preferred stock is the dividend divided by the required return. This is the
same equation as the constant growth model, with a dividend growth rate of zero percent. Remember
that most preferred stock pays a fixed dividend, so the growth rate is zero. Using this equation, we
find the price per share of the preferred stock is:
9. The growth rate of earnings is the return on equity times the retention ratio, so:
g = ROE × b
To find next year’s earnings, we multiply the current earnings times one plus the growth rate, so:
Next year’s earnings = Current earnings(1 + g)
10. Using the equation to calculate the price of a share of stock with the PE ratio:
So, with a PE ratio of 18, we find:
And with a PE ratio of 21, we find:
Intermediate
11. This stock has a constant growth rate of dividends, but the required return changes twice. To find the
value of the stock today, we will begin by finding the price of the stock at Year 6, when both the
dividend growth rate and the required return are stable forever. The price of the stock in Year 6 will
be the dividend in Year 7, divided by the required return minus the growth rate in dividends. So:
Now we can find the price of the stock in Year 3. We need to find the price here since the required
return changes at that time. The price of the stock in Year 3 is the PV of the dividends in Years 4, 5,
and 6, plus the PV of the stock price in Year 6. The price of the stock in Year 3 is:
12. Here we have a stock that pays no dividends for 9 years. Once the stock begins paying dividends, it
will have a constant growth rate of dividends. We can use the constant growth model at that point. It
is important to remember that the general form of the constant dividend growth formula is:
This means that since we will use the dividend in Year 10, we will be finding the stock price in Year
9. The dividend growth model is similar to the PVA and the PV of a perpetuity: The equation gives
you the PV one period before the first payment. So, the price of the stock in Year 9 will be:
The price of the stock today is the PV of the stock price in the future. We discount the future stock
price at the required return. The price of the stock today will be:
13. The price of a stock is the PV of the future dividends. This stock is paying five dividends, so the
price of the stock is the PV of these dividends using the required return. The price of the stock is:
14. With differential dividends, we find the price of the stock when the dividends level off at a constant
growth rate, and then find the PV of the future stock price, plus the PV of all dividends during the
differential growth period. The stock begins constant growth in Year 5, so we can find the price of
the stock in Year 4, one year before the constant dividend growth begins, as:
The price of the stock today is the PV of the first four dividends, plus the PV of the Year 4 stock
price. So, the price of the stock today will be:
15. With differential dividends, we find the price of the stock when the dividends level off at a constant
growth rate, and then find the PV of the future stock price, plus the PV of all dividends during the
differential growth period. The stock begins constant growth in Year 4, so we can find the price of
the stock in Year 3, one year before the constant dividend growth begins as:
The price of the stock today is the PV of the first three dividends, plus the PV of the Year 3 stock
price. The price of the stock today will be:
16. Here we need to find the dividend next year for a stock experiencing differential growth. We know
the stock price, the dividend growth rates, and the required return, but not the dividend. First, we
need to realize that the dividend in Year 3 is the current dividend times the FVIF. The dividend in
Year 3 will be:
And the dividend in Year 4 will be the dividend in Year 3 times one plus the growth rate, or:
The stock begins constant growth after the 4th dividend is paid, so we can find the price of the stock
in Year 4 as the dividend in Year 5, divided by the required return minus the growth rate. The
equation for the price of the stock in Year 4 is:
Now we can substitute the previous dividend in Year 4 into this equation as follows:
When we solve this equation, we find that the stock price in Year 4 is 93.33 times as large as the
dividend today. Now we need to find the equation for the stock price today. The stock price today is
the PV of the dividends in Years 1, 2, 3, and 4, plus the PV of the Year 4 price. So:
Reducing the equation even further by solving all of the terms in the braces, we get:
This is the dividend today, so the projected dividend for the next year will be:
17. The constant growth model can be applied even if the dividends are declining by a constant
percentage, just make sure to recognize the negative growth. So, the price of the stock today will be:
18. We are given the stock price, the dividend growth rate, and the required return, and are asked to find
the dividend. Using the constant dividend growth model, we get:
Solving this equation for the dividend gives us:
19. The price of a share of preferred stock is the dividend payment divided by the required return. We
know the dividend payment in Year 5, so we can find the price of the stock in Year 4, one year before
the first dividend payment. Doing so, we get:
The price of the stock today is the PV of the stock price in the future, so the price today will be:
20. The dividend yield is the annual dividend divided by the stock price, so:
Dividend yield = Dividend / Stock price
The “Net Chg” of the stock shows the stock decreased by $.13 on this day, so the closing stock price
yesterday was:
To find the net income, we need to find the EPS. The stock quote tells us the PE ratio for the stock is
23. Since we know the stock price as well, we can use the PE ratio to solve for EPS as follows:
PE = Stock price / EPS
We know that EPS is just the total net income divided by the number of shares outstanding, so:
EPS = Net income / Shares
21. Here we have a stock paying a constant dividend for a fixed period, and an increasing dividend
thereafter. We need to find the present value of the two different cash flows using the appropriate
quarterly interest rate. The constant dividend is an annuity, so the present value of these dividends is:
PVA = C(PVIFAR,t)
Now we can find the present value of the dividends beyond the constant dividend phase. Using the
present value of a growing annuity equation, we find:
P12 = D13 / (R – g)
This is the price of the stock immediately after it has paid the last constant dividend. So, the present
value of the future price is:
The price today is the sum of the present value of the two cash flows, so:
22. Here we need to find the dividend next year for a stock with irregular growth in dividends. We know
the stock price, the dividend growth rate, and the required return, but not the dividend. First, we need
to realize that the dividend in Year 3 is the constant dividend times the FVIF. The dividend in Year 3
will be:
The equation for the stock price will be the present value of the constant dividends, plus the present
value of the future stock price, or:
$53 = D{1 / 1.11 + 1 / 1.112 + [(1.04) / (.11 – .04)] / 1.112}
Reducing the equation even further by solving all of the terms in the braces, we get:
