• Table of Contents • Introduction • 1-Time Value • 2-Risk/Return • 3-Accounting • 4-Securities • 5-Business • 6-Regulatory • Case Studies • Student Papers

## Chapter 2 - Problems

 Expected returns 1 Calculate returns 2 Required returns Return and risk 3 Return variation 4 Risk/return comparison Portfolio volatility 5 Calculate beta 6 Risk/return comparison Using beta 7 Calculate E(r) 8 With CAPM Combining returns and risk 9 Problem Solutions Attached spreadsheet

1 - Calculate returns

Your client is interested in buying a business. You help structure the transaction and draw up the appropriate dcouments. The client has been offered to ways to pay the \$400,000 purchase price. One is for cash, which your client could finance by borrowing from the bank with a 9% loan payable over 10 years -- annuial payments due on December 31 of each year. THe other is a strcutrued purchase as follows:  \$40,000.00/yr. for 10 years, payable by December 31 of each year, beginning next year plus each year accrued interest on the outstanding balance at an interest rate of 6%/year for the first five years, then 10% for years 6 through 10.

Which is the better deal for your client?

Assuming that past returns predict future returns, what is the expected return for the following investments?

 Investment A Investment B Market price (year ago) \$20,000 \$55,000 Market price (present) 21,000 55,000 Cash flow (current year) 1,500 6,800

2 - Required returns

You are considering buying a car for \$8,500. You live in the city and taking the bus costs you \$1,700 in fares and aggravation. You expect the car will cost about \$1,950 in annual maintenance and insurance costs. After five years you expect to sell the car for \$5,000. To buy the car, you will dip into your savings account, which earns 4.5% after taxes. You ask yourself whether you should buy the car or keep on taking the bus.

• What are the cash flows and expected return applicable to buying the car?
• Which is a better choice - buying the car (with attendant costs and benefits) or keep on riding the bus and investing the money?
• What if you plan to use money from your tax-free municipal bond mutual fund, which has had steady after-tax returns of 7.8%. How much should you spend on the car given your alternative use of the purchase price?

3 - Return variation

You are considering three stock investments. You have studied the stocks' historical returns over the last five years.

 Year Stock 1 Stock 2 Stock 3 1996 -12% 0% -10% 1997 0% 5% 4% 1998 10% 12% 12% 1999 25% 20% 20% 2000 50% 25% 35%

Assuming that historical returns are a measure of future returns --

• Calculate the expected return, the standard deviation (s) and coefficient of variation for each stock.
• Which stock has the highest expected return? the highest risk?
• Which is the best combination of risk and return?

4 - Risk/return comparison

Portfolio A has an expected return of 8.4% and a standard deviation of 12.1%. Portfolio B has an expected return of 10.7% and a standard deviation of 15.8%. The risk free rate of return is 5.1%.

• Assuming CAPM, what is the slope of the market line if Portfolio A lies on the line? What about Portfolio B?
• What do these slopes tell you? Which is the better portfolio -- that is, the better combination of risk and return?

5 - Calculate beta

You want to calculate the betas for two companies. You have the following return data:

 Expected return (%) Year Market Stock X Stock Y 1991 6 11 16 1992 2 8 11 1993 -13 -4 -10 1994 -4 3 3 1995 -8 0 -3 1996 16 19 30 1997 10 14 22 1998 15 18 29 1999 8 12 19 2000 13 17 26
• How might you compute beta?
• Draw the characteristic line for each stock and estimate slope.
• Use regression analysis to compute the line's slope.
• Are the stocks riskier, less risky than the market?
• What is the beta of a portfolio containting with equal amounts of Stock X and Stock Y? Is this portfolio riskier than the market?
• You're not sure if the market will be up or down. What will be the return on Stock X and Stock Y, assuming either an up market next year with returns of +20.0% or a down market with returns of -6.0%?

6 - Risk/return comparison

Assume the risk-free rate of return is 10%, and the expected return on the market is 16%. An investment manager can choose between two portfolios, A and B, having the following properties:

 A B Expected return .17 .24 Beta 1.50 2.00

Under the CAPM, which portfolio should the investment manager choose?

7 - Calculate E(r)

You compute the historical betas for various investments and assume they are good predictors of expected volatility -- thus risk. You also assume various risk-free rates and expected market returns. In each case calculate the expected return for the investment.

 Investment Beta Risk-free rate Market return A 1.30 5.2 8.7 B .90 8.5 13.4 C -.20 9.1 12.1 D 1.00 10.9 15.5 E .60 6.2 10.8

If the expected return is greater than the promised return - what should you do? Buy or sell?

8 - With CAPM

Use the CAPM equation to compute the unknown variable --

 Investment Risk-free return Market return Beta Required return A 4.2 12.3 .85 ? B ? 15.8 1.35 17.4 C 3.6 ? 1.10 16.1 D 5.2 12.6 ? 15.2

9 - Problem

You compare two stocks, both traded on the stock exchange. You are thinking of selling one. Which investment is a better combination of risk and return? You look at returns (cash flow and market appreciation) over the last ten years. You predict that future returns will be comparable.

 Stock X Stock Y Year Cash flow Starting value Ending value Cash flow Starting value Ending value 1 1.0 20.3 22.5 1.5 20.8 20.9 2 1.5 22.5 21.0 4.2 20.9 22.4 3 1.8 21.0 24.8 3.6 22.4 19.8 4 2.2 24.8 22.0 5.1 19.8 22.1 5 2.4 22.0 28.5 4.8 22.1 22.5 6 2.5 28.5 30.2 3.2 22.5 19.3 7 2.5 30.2 26.4 5.9 19.3 24.1 8 2.5 26.4 28.1 4.6 24.1 22.7 9 1.8 28.1 27.7 3.9 22.7 24.1 10 1.5 27.7 31.2 5.7 24.1 25.6
• Calculate the annual rate of return for each asset for each year, and then compute the average return.
• Use these results to find standard deviation and coefficient of variation for each investment.
• Based on these results, which investment offers a better mix of risk and return?
• Assuming betas (X = 1.60, Y = .85) and a risk-free rate of 4.8% and an expected market return of 14.2%, use the CAPM to find the required return for each investment.
• Using these CAPM results and comparing them to past returns, which investment seems to offer a better mix of risk and returns?