Introductory Finance

9.1

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

P0 = D1 / (k g)

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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)

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.

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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

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What, then, will P5 be if:

D6 = 20, k = .15, and g = r x ROE = .8 x .15 = .12?

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If P5 = your result from part B, and assuming no dividends are paid until D6, what would be P0? P1? P2?

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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.

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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

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D1 / (k g)

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[(1 r) x Y1] / (k g)

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

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[(1 r) x Y1] / (ROE r x ROE)

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[(1 r) x Y1] / (Y1 / E0 r x Y1 / E0)

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

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

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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.

(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%.

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Fill in the missing items in the following table:

Year

EPS

ROE

Expected Dividend
(end of year)

Present Value Of Dividend
(at time 0)

n/a

25%

25 = 1.25 x 20

25%

20%

15%

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What would the dividend be in year 8?

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Calculate the value of all future dividends at the beginning of year 8. (Hint: P7 depends on D8.)

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What is the present value of P7 at the beginning of year 1?

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

(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:

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The firm pays out dividends of $1 per share.

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The firm buys back 10,000 shares for $10 cash each, and you choose to sell your share back to the company.

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The firm buys back 10,000 shares for $10 cash each, and you choose not to sell your share back to the company.

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The firm declares a 2-for-1 stock split.

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The firm declares a 10% stock dividend.

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The firm buys new equipment for $100,000, which will be used to earn a return equal to the firm’s discount rate.

(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

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

Year

Computroids Inc.

Blazers Inc.

10%

12%

10%

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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.

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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?

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Risk of an increasing interest rate

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Risk of inflation increasing

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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.

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

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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?

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

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

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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.

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

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

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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.

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Hedging

Insuring

Diversifying

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.

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

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

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

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

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

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Buying a Mac and a PC.

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

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(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)

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