WFU Law School
Law & Valuation
Chapter 2 - Risk and Return

Chapter 2 - Problems

Expected returns
Calculate returns
Required returns
Return and risk
Return variation
Risk/return comparison
Portfolio volatility
Calculate beta
Risk/return comparison
Using beta
Calculate E(r)
With CAPM
Combining returns and risk
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

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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?

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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?

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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?

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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%?

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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?

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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?

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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

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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?

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Chapter 2 - Risk and Return

©2003 Professor Alan R. Palmiter

This page was last updated on: November 7, 2007