9.1 The Constant-Growth-Rate Discounted Dividend Model, as described equation 9.5 on page 247, says that: P 0 = D 1 / (k – g) A. rearrange the terms to solve for: i. G ii. D 1. As an example, to solve...

9.1 The Constant-Growth-Rate Discounted Dividend Model, as described
equation 9.5 on page 247, says that:

P0 = D1 / (k – g)
A. rearrange the terms to solve for:

i. G

ii. D1.

As an example, to solve for k, we would do the following:

1. Multiply both sides by (k – g) to get: P0 (k – g) = D1

2. Divide both sides by P0 by to get: (k – g) = D1 / P0

3. Add g to both sides: k = D1 / P0 + g
(8 marks)
Given,
P0 = D1 / (k – g)
 (k-g) * P0 = (k-g)* D1 / (k – g)
 k- g = D1/P0
 g = k - D1/P0
Given,
P0 = D1 / (k – g)
 (k-g) * P0 = (k-g)* D1 / (k – g)
 K*P0- g*P0 = D1


9.2 Notation: Let

Pn = Price at time n
Dn = Dividend at time n
Yn = Earnings in period n

r = retention ratio = (Yn– Dn) / Yn = 1 – Dn/ Yn = 1 - dividend payout
ratio

En = Equity at the end of year n

k = discount rate
g = dividend growth rate = r x ROE
ROE = Yn / En-1 for all n>0.

We will further assume that k and ROE are constant, and that r and g
are constant after the first dividend is paid.


A. Using the Discounted Dividend Model, calculate the price P0 if

D1 = 20, k = .15, g = r x ROE = .8 x .15 = .12, and Y1 = 100 per
share

P0 = D1/(k-g) = 20/(0.15-0.12) = 666.67
B. What, then, will P5 be if:
D6 = 20, k = .15, and g = r x ROE = .8 x .15 = .12?
P5 = D6/(k-g) = 20/(0.15-0.12) = 666.67
C. If P5 = your result from part B, and assuming no dividends are paid until D6, what
would be P0? P1? P2?
P0 = P5/(1+k)^5 = 666.67/(1.15)^5 = 331.45
P1 = P5/(1+k)^4 = 666.67/(1.15)^4 = 381.17
P2 = P5/(1+k)^3 = 666.67/(1.15)^3 = 438.34


D. Again, assuming the facts from part B, what is the relationship
between P2 and P1 (i.e., P2/P1)? Explain why this is the result.

If we look at the formulae in part c, we can see the below relationship:
P2/P1 = (1+k)
This is because if no dividends are paid between the two periods, the only factor that
remains is the discount factor.
E. If k = ROE, we can show that the price P0 doesn’t depend on r. To
see this, let

g = r x ROE, and ROE = Yn / En-1, and

since r = (Yn – Dn) / Yn , then D1 = (1 – r) x Y1 and
P0 = D1 / (k – g)

P0 = [(1 – r) x Y1] / (k – g)

P0 = [(1 – r) x Y1] / (k – g), but, since k = ROE = Y1 / E0

P0 = [(1 – r) x Y1] / (ROE– r x ROE)

P0 = [(1 – r) x Y1] / (Y1 / E0 – r x Y1 / E0)

P0 = [(1 – r) x Y1] / (1 – r) x Y1 / E0), and cancelling (1 – r)

P0 = Y1 / (Y1/E0) = Y1 x (E0 / Y1) = E0

So, you see that r is not in the final expression for P0, indicating that r
(i.e., retention ration or, equivalently, dividend policy) doesn’t matter if
k = ROE.
Check that changing r from .8 to .6 does not change your answer in part
A of this question by re-calculating your result using r = .6.

We can see above that when r=0.8, then P0 = 666.67

If we change r to 0.7, we have g = 0.7*0.15 = 0.105 and D1 = 30
Hence, P0 = D1/(k-g) = 30/(0.15-0.105) = 666.67
If we change r to 0.6, we have g = 0.6*0.15 = 0.09 and D1 = 40
Hence, P0 = D1/(k-g) = 40/(0.15-0.09) = 666.67
Hence, we note that changing the retention ratio does not have an impact on the price as
the ROE = k = 0.15.
(10 marks)
9.3 You are considering an investment in the shares of Kirk's Information
Inc. The company is still in its growth phase, so it won’t pay dividends
for the next few years. Kirk’s accountant has determined that their first
year's earnings per share (EPS) is expected to be $20. The company
expects a return on equity (ROE) of 25% in each of the next 5 years but
in the sixth year they expect to earn 20%. In the seventh year and
forever into the future, they expect to earn 15%. Also, at the end of the
sixth year and every year after that, they expect to pay dividends at a
rate of 70% of earnings, retaining the other 30% in the company. Kirk's
uses a discount rate of 15%.



A. Fill in the missing items in the following table:
Year EPS ROE Expected
Dividend
Present
Value Of
Dividend
(end of
year)
(at time 0)
0 n/a n/a n/a n/a
1 20 25% 0 0
2 25 = 1.25 x
20
25% 0 0
3 31.25 25% 0 0
4 39.06 25% 0 0
5 48.83 25% 0 0
6 58.59 20% 41.02 17.73
7 67.38 15% 47.17 17.73
8 77.49 15% 54.24 17.73



B.

What would the dividend be in year 8?
The dividend at the end of year 8 would be 70% of earnings in year 8. Hence, as we see
from the table above, we note that dividend would be 54.24.

C. Calculate the value of all future dividends at the beginning of year
8. (Hint: P7 depends on D8.)
P7 = D8/(0.15-0.15*0.7) = 54.24/0.045 = 1205.33

D. What is the present value of P7 at the beginning of year 1?
Present value of P7 = P7/1.15^7 = 1205.33/1.15^7 = 453.13


E. What is the value of the company now, at time 0?

We note that the dividend at the end of year 6 is 41.02. This dividend
grows at a rate of 10.5% (ROE*retention ratio=7%*15%). The
discount rate id 15%. Hence,
P5 = 41.02/(0.15-0.105) = 911.56

Now we discount this to time 0. Hence, P0 = 911.56/(1.15)^5 =
453.20

(10 marks)
9.4 You own one share in a company called Invest Co. Inc. Examining the
balance sheet, you have determined that the firm has $100,000 cash,
equipment worth $900,000, and 100,000 shares outstanding.

Calculate the price/value of each share in the firm, and explain how your
wealth is affected if:

Value of...