7. Short-term interest rates are more volatile than long-term rates. Despite this, the rates of return of long-term bonds are more volatile than returns on short-term securities. How can these two empirical observations be reconciled?

8. Find the duration of a 6% coupon bond making annual coupon payments if it has three years until maturity and a yield to maturity of 6%. What is the duration if the yield to maturity is 10%?

9. A nine-year bond has a yield of 10% and a duration of 7.194 years. If the bond's yield changes by 50 basis points, what is the percentage change in the bond's price?

11. If the plan in the previous problem wants to fully fund and immunize its position, how much of its portfolio should it allocate to one-year zero-coupon bonds and perpetuities, respectively, if these are the only two assets funding the plan?

15. You will be paying $10,000 a year in tuition expenses at the end of the next two years. Bonds currently yield 8%.

a. What is the present value and duration of your obligation?

b. What maturity zero-coupon bond would immunize your obligation?

c. Suppose you buy a zero-coupon bond with value and duration equal to your obligation. Now suppose that rates immediately increase to 9%. What happens to your net position, that is, to the difference between the value of the bond and that of your tuition obligation?

d. What if rates fall to 7%?

16. Pension funds pay lifetime annuities to recipients. If a firm remains in business indefinitely, the pension obligation will resemble a perpetuity. Suppose, therefore, that you are managing a pension fund with obligations to make perpetual payments of $2 million per year to beneficiaries. The yield to maturity on all bonds is 16%.

a. If the duration of 5-year maturity bonds with coupon rates of 12% (paid annually) is 4 years and the duration of 20-year maturity bonds with coupon rates of 6% (paid annually) is 8 years, how much of each of these coupon bonds (in market value) will you want to hold to both fully fund and immunize your obligation?

b. What will be the par value of your holdings in the 20-year coupon bond?

19. You are managing a portfolio of $1 million. Your target duration is 10 years, and you can choose from two bonds: a zero-coupon bond with maturity five years, and a perpetuity, each currently yielding 5%.

a. How much of each bond will you hold in your portfolio?

b. How will these fractions change next year if target duration is now nine years?