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 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.

Using your calculations from

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.

Your answer should show 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

be your preferred choice.

(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

want to stay in Canada, and your grandparents, who have retired to Provence, receive

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

Purchasing

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

Buying

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.