# Among stocks A, B, C, X, Y, Z, and a portfolio consisting of all six stocks, which one do you expect to have a beta closest to one? Explain why

1. Among stocks A, B, C, X, Y, Z, and a portfolio consisting of all six stocks, which one do you expect to have a beta closest to one? Explain why. (3 marks)

2. Stocks J, K, and L all have the same expected rate of return and standard deviation. The correlation coefficients between each pair of these stocks are as follows:

Stocks J K L

J 0.8 0.2

K -0.5

L

Given these correlation coefficients, which pairs of stocks should be combined together to form a minimum variance portfolio? Explain your choice.

3. The universe of available securities includes only two risky stock funds, X and Y, as well as T-bills. The data for the investment universe are shown in the table below. The correlation coefficient between the two funds is -0.3.

Expected return Standard deviation

X 0.15 0.25

Y 0.35 0.55

T-bills 0.06 0

a. Find the optimal risky portfolio and its expected return and standard deviation.

b. Find the slope of the Capital Allocation Line supported by T-bills and the optimal risky portfolio.

c. If an investor’s utility can be represented by , how much will the investor invest in funds X and Y and in T-bills?

4. Use the information in the following tables and the method as shown in section 7.5, “A Spreadsheet Model” (textbook, pp. 224–229), to find portfolio variance, portfolio standard deviation, and the weights for stocks in the portfolio. Attach a copy of the spreadsheet(s) you use to compute your answers. (See Hints.)

5. Use the information in the following table to determine the betas of the two stocks.

Market condition Market return Aggressive stock Defensive stock

Bear 0.06 0.01 0.07

Bull 0.18 0.37 0.14

Assume that portfolios A and B are well diversified, and that their expected rates of return are at 0.13 and 0.09 respectively. If the economy has only one factor, and the betas for the two portfolios are 1.2 and 0.8, what must the risk free rate be? (5 marks)

7. If financial markets are all efficient, do investors need portfolio managers? Why or why not? (5 marks)

8. Derive the relationships among coupon rate, current yield, and yield to maturity for bonds selling at discount from par, at par, and at premium over par. (Hint: You may assume that the bonds have one year to maturity.) (6 marks)

9. A bond pays a coupon rate of 8% per year semiannually when the prevailing market interest rate is at 6%. The bond is four years away from its maturity. Find the bond’s price now, and six months from now after the next coupon is paid. What will the holding period rate of return for the next six months be? (5 marks)

10. Define short rate, spot rate, forward rate, and yield to maturity. If the expectations hypothesis holds, how should these rates be related? (6 marks)

11. Suppose the prices of zero-coupon bonds are as given in the table below, and each bond has a face value of $1,000.

Bond Price Maturity (years)

A $955.94 1

B $870.22 2

C $790.50 3

D $715.28 4

E $644.14 5

a. Calculate the yields to maturity for the five bonds.

b. Compute the forward rate for each year.

c. How would you construct a one-year forward loan beginning in year 2?

12. A Treasury bond has three years to maturity and a coupon rate of 8%. The coupon payments are made on semiannual basis. The current market interest rate is 10%.

a. If the market interest rate goes up to 10.05%, what would the change in bond price be if the bond valuation formula (equation 13.2) on page 421 of the textbook is used?

b. If the market interest rate goes up to 10.25%, what would the change in bond price be if the bond valuation formula (equation 13.2) on page 421 of the textbook is used?

c. Using formula 15.2 on page 481 of the textbook, redo (a) and (b).

d. Does formula 15.2 perform better under (a) than (b)? Why?

(10 marks)

13. Prove Rule 2 for Duration: Holding maturity constant, a bond’s duration is higher when the coupon rate is lower. You may complete the proof using either mathematical notation or numerical examples.

14. Given the two bonds in the table below, calculate duration and convexity for each of them, and decide which bond an investor should purchase if the investor is allowed to hold only one of them. Explain why.

Face Value Coupon Rate Maturity Yield to Maturity

Bond A $1,000 9.0% 10 years 9.0%

Bond B $1,000 3.1% 8 years 9.0%

15. Given the information in the table below, does the Law of One Price hold? If not, what action should an investor take?

Bond Price Cash Flow in Year 1 Cash Flow in Year 2

A $970 $100 $1100

B $936 $80 $1080

C $980 $90 $1090