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# Measuring risk and rates of return

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6-10. (Measuring risk and rates of return):
a. Given the holding-period returns shown here, compute the average returns and the standard deviations for the Zemin Corporation and for the market.
MONTH ZEMIN CORP. MARKET
1 6% 4%
2 3 2
3 -1 1
4 -3 -2
5 5 2
6 0 2

b. If Zemin's beta is 1.54 and the risk-free rate is 8 percent, what would be an appropriate required return for an investor owning Zemin? (Note: Because the returns of Zemin Corporation are based on monthly data, you will need to annualize the returns to make them compatible with the risk-free rate. For simplicity, you can convert from monthly to yearly returns by multiplying the average monthly returns by 12.)

c. How does Zemin's historical average return compare with the return you believe to be a fair return, given the firm's systematic risk?

#### Solution Preview

6-10. (Measuring risk and rates of return)

a. Given the holding-period returns shown here, compute the average returns and the standard deviations for the Zemin Corporation and for the market.

MONTH ZEMIN CORP. MARKET
1 6% 4%
2 3% 2%
3 -1% 1%
4 -3% -2%
5 5% 2%
6 0% 2%

Zemin Corp
Month X= X 2 =
1 6% 0.0036
2 3% 0.0009
3 -1% 0.0001
4 -3% 0.0009
5 5% 0.0025
6 0% 0
Total= 10.00% 0.0080
n=no of observations= 6
Average return= 1.67% =10.%/6

variance={&#931;X 2 - n(Mean) 2 }/(n-1)= 0.001265 =(0.008-6*0.0167^2)/(6-1)
standard deviation of return (SD)=&#8730;Variance= 3.56% =&#8730;0.001265

These are monthly returns and standard ...

#### Solution Summary

Compute the average returns and the standard deviations for the Zemin Corporation and for the market. Calculates appropriate required return for an investor owning Zemin using Capital Asset Pricing Model (CAPM).

\$2.19

## Portfolio Return and Standard Deviation, SML and Required Rate of Return, Futures Contracts, Future Margin, S&P 500 index futures contracts, Futures Speculation,Futures Hedging, Call Options, Put Options, Portfolio Performance Measures (Sharpe Measure, Jensen Measure, Treynor Measure)

Portfolio Return and Standard Deviation:

1. You are trading in a market that has only two securities available. Security A has an expected return of 8 percent and a standard deviation of 40 percent. Security B has an expected return of 20 percent and a standard deviation of 120 percent.

a) If you place half of your money in each stock, what is your expected return?

b) If you place 40 percent of your money in A and the remaining 60 percent in B, what is your expected return?

c) If the correlation between the returns of Securities a. and b. above is 0.8
what are the variance and the standard deviation of the returns of each of the two portfolios
you found in parts a. and 1 b. above?

2. You are trading in a market that has only two securities. Security C has an expected return of 6% and a standard deviation of 2.5% while security D has an expected return of 15% and a standard deviation of 8%. The correlation of returns between the two securities is -1.

a) If you place half of the money in each stock, then what is the expected return of
the portfolio?

b) If you place 40% of the money in stock C and the remaining in stock D, then what is the expected return of the portfolio?

c) What is the portfolio standard deviation in b above?

SML and Required Rate of Return:

3. a) A security has a beta of 1.5 when the risk-free rate is 4 percent and the expected return on the market is 12 percent. Calculate the expected return on the security.

b) If the beta on the security in a) increases to 2, what is the new expected return? Why does the expected return increase as the beta increases?

SML and Security Selection:

4. Use the following information to answer the questions below:

STOCK
BETA
CURRENT PRICE
EXPECTED PRICE
EXPECTED DIVIDEND

A -1.50
\$ 40
\$ 32
\$ 0.00

B
0.75
\$ 40
\$ 46
\$ 1.50

C
1.75
\$ 80
\$92
\$ 1.45

Assume that the Return on the Market is 12% and the Risk-free rate is 2%.

a) What are the required rates of return for the three stocks?
b) What are the estimated rates of return for the three stocks?
c) What is your investment strategy concerning the three stocks?

Futures Margin:

5. Zack Wheat has just bought four September 5,000-bushel corn futures contracts at \$1.75 per bushel. The initial margin requirement is 3%. The maintenance margin requirement is 80% of the initial margin requirement.

a) How many dollars in initial margin must Zack put up?

b) If the September price of corn rises to \$1.85, how much equity is in Zack's commodity account? Compute the profit/loss

c) If the September price of corn falls to \$1.70, how much equity is in Zack's commodity account? Compute percentage profit/loss

Futures Speculation:

6. Sleeper Sullivan bought 10 December S&P 500 index futures contracts at 310. If the index rises to 318, what is Sleeper's dollar profit? The multiplier for the S&P 500 index futures contract is 250.

7. The margin requirements on the S&P 500 futures contract are 10%, and the stock index is currently at 1,200. Each contract has a multiplier of 250. How much margin must be put up for each contract sold? If the futures price falls by 1% to 1,188 what will happen to margin account of the investor who holds one contract? What will be the investor's percentage return based on the amount put up as margin?

Futures Hedging:

8. The S&P index is currently is at 1,400. You manage a \$7 million indexed equity portfolio. The S&P 500 futures contract has a multiplier of 250.

a) If you are bearish on the stock market, how many contracts should you sell to fully eliminate your risk over the next 6 months?

b) How would your hedging strategy change if instead of holding an indexed portfolio, you hold a portfolio of socks with a beta of 0.60? How many contracts would now choose to sell? Would your hedged position be riskless?

Stock Options:

Call Options:

9. When the underlying stock trades at \$40, a call with an exercise price of \$40 is priced at \$5. Assume that you buy this call. Calculate your profits/losses for stock prices at expiration of \$20, \$30, \$40, \$50, and \$60.

Put Options:

10. Joe Six-pack bought a put option on Cisco for the month of January 2001 with a strike price of \$50. He paid a premium of \$2.25 per option. If Cisco closes at \$46.50 on the exercise date, then compute the percent of profit/loss.

Portfolio Performance Measures:

11. Given the following:

Portfolio Return Standard Deviation Beta

1 9% 5% 0.80

2 13% 7% 1.10

3 22% 18% 1.25

Market 14% 12% 1.00

Risk-Free Security 10%

Compute:

1. Sharpe Measure
2. Jensen Measure
3. Treynor Measure

And Rank the portfolios by each measure. Are the rankings consistent? Why? Or Why not?

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