# Introductory Finance

9.1

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

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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 doesnt 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)
doesnt 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 wont pay dividends for the next
few years. Kirks 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 the firm = \$900,000 \$100,000 = \$800,000
Value
per share = \$800,000/100,000 = \$8

A.

The
firm pays out dividends of \$1 per share.
When dividend is paid out, the value of for the shareholder does not change.
The value of the share is marked down by the amount of the dividend by most
exchanges.

B.

The
firm buys back 10,000 shares for \$10 cash each, and you choose to sell your
share back to the company.
If you choose to sell your share for \$10 when the
value if \$8, then your wealth increases by \$2 per share (\$10-\$8)

C.

The
firm buys back 10,000 shares for \$10 cash each, and you choose not to sell
your share back to the company.
If you do not choose to sell, then your wealth remains unchanged.

D.

The
firm declares a 2-for-1 stock split.
The overall wealth does not change. The number of stocks will now double up
to 200,000 and the number of stocks owned by you will now be 2. Since total
value of the firm remains the same (\$800,000), your wealth remains the same.

E.

The
firm declares a 10% stock dividend.
When dividend is paid out, the value of for the shareholder does not change.

F.

The
firm buys new equipment for \$100,000, which will be used to earn a return
equal to the firm’s discount rate.
If
\$100,000 cash is used to buy equipment which is then used to earn a return
equal to the firms discount rate, then the value of the firm will increase
from \$800,000 to \$900,00. Hence the value per share will increase to \$9
(\$900,000/100,000) and the wealth would increase by \$1.
(12 marks)

Do not submit these questions for
grading until you have completed all parts of Assignment 3, which is due after
Lesson 11.

Lesson 10: Assignment Problems

10.1

A.

Calculate the mean and standard
deviation of the following securities returns:

Year

Computroids
Inc.

Blazers
Inc.

1

10%

5%

2

5%

6%

3

3%

7%

4

12%

8%

5

10%

9%

B.

Assuming these observations are drawn from a normally distributed probability
space, we know that about 68% of values drawn from a normal distribution are
within one standard deviation away from the mean or expected return; about
95% of the values are within two standard deviations; and about 99.7% lie
within three standard deviations.

part A, calculate the 68%, 95%, and 99% confidence intervals for the two
stocks. To calculate the 68%, you would calculate the top of the confidence
interval range by adding one standard deviation to the expected return, and
calculate the bottom of the confidence interval by subtracting one standard
deviation from the expected return. For 95%, use two standard deviations, and
for 99%, use three.

ranges from the bottom of the confidence interval to the top of the
confidence interval.

C.

For
each security, would a return of 14% fall into the 68% confidence interval
range? If not, what confidence interval range would it fall into, or would it
be outside all three confidence intervals?

[This is the same as asking
whether a return of 14% has less than a 68% probability of occuring by chance
for that security. If its not inside the 68% confidence interval, its unlikely
to occur, since it will only occur by chance 32% of the time. Of course, the
99% confidence interval is much more likely to include the observed return,
simply by chance. Only 1% of the time will it fall outside the 99% CI. Pretty
rare.]
(14 marks)

10.2

Some
Internet research may be required to answer this question, although its not
absolutely necessary.

What could you do to protect your bond portfolio against the following kinds
of risk?

A.

Risk of
an increasing interest rate

B.

Risk of
inflation increasing

C.

Risk of
volatility in the markets
(6
marks)

10.3

You are
starting a new business, and you want to open an office in a local mall. You
have been offered two alternative rental arrangements. You can pay the
landlord 10% of your sales revenue, or you can pay a fixed fee of \$1,000 per
month. Describe the circumstances in which each of these arrangements would
(10 marks)

Do not submit these questions for
grading until you have completed all parts of Assignment 3, which is due after
Lesson 11.

Lesson 11: Assignment Problems

11.1

In the
northeast United States and in eastern Canada, many people heat their houses
with heating oil. Imagine you are one of these people, and you are expecting
a cold winter, so you are planning your heating oil requirements for the
season. The current price is \$2.25 per US gallon, but you think that in six
months, when you’ll need the oil, the price could be \$3.00, or it could be
\$1.50.

A.

If you
need 350 gallons to survive the winter, how much difference does the
potential price variance make to your heating bills?

B.

If your
friend Tom is running a heating oil business, and selling 100,000 gallons
over the winter season, how does the price variance affect Tom?

C.

Which
one of you benefits from the price increase? Which of you benefits from price
decrease?

D.

What
are two strategies you can use to reduce the risk you face? Could you make an
agreement with Tom to mitigate your risk?

E.

Assuming
you are both risk-averse, does such an agreement make you both better off?
(10 marks)

11.2

You
have just received good news. You have a rich uncle in France who has decided
to give you a monthly annuity of 2,000 per month. You are concerned that you
will become accustomed to having these funds, but if the currency exchange
rate moves against you, you may have to make do with less.

A.

If you
are living in Canada, what does it mean for the currency exchange rate to
move against you?

B.

Would
moving to France mitigate some of the risk? If so, how? If not, why not?

C.

If you
a Canadian pension of C\$1100 each, what could you do to reduce the risk for
all of you?
(9 marks)

11.3

You have learned about a number
of ways of reducing risk, specifically hedging, insuring, and diversifying.
In the table below, place an X in the cell for the technique being used to
reduce risk.

Hedging

Insuring

Diversifying

1

Placing
an advance order with Amazon.ca, which agrees to charge you the lower of
the advance price, and the price at the time your order is filled.

X

2

a call option on a stock you think may go up in price.

X

3

Selling
200 shares of IBM and buying a mutual fund that holds the same stocks as
the S&P index.

X

4

Selling
a debt owed to you for \$.50 per dollar owed.

X

5

Agreeing
to a long-term contract with a supplier at a fixed price.

X

6

Agreeing
to a no-trade clause with the sports team that employs you.

X

7

a Mac and a PC.

X

8

Paying
a clown to perform for your child’s birthday party six months before the
birthday.

X

(16 marks)

11.4

Suppose
you own 100 shares of Dell Inc. stock. Today it is trading at \$15 per share,
but you’re worried Michael Dell might retire again, causing the price to go
down. How would you protect yourself against his retirement, assuming you
don’t want to sell the shares today?
(5 marks)

One option is to hedge which can be done by
creating a futures contract that states you will sell your stock at a set price
There by taking out the risk of market fluctuation. Investors commonly do this
to reduce the risk when they are unsure what the market will do.