Math 325
Additional Problems – Ch 10 & 11
1. The current prices for a one-year, two-year, and three-year $1000 zero coupon bonds are $943.40, X,
and $805.08, respectively. If the one-year forward rate for year 2 (the one year effective rate during
year 2) is 8%, find X.
2. The one-year spot rate is 5.5%. A two-year bond maturing at par, with 7% annual coupons, is
currently selling for its par value. Determine the 2-yr spot rate.
3. A $1000 par value bond with 10% annual coupons matures at par in two years. The one-year spot
rate is 9% and the one-year forward rate for year 2 is 10.5%. Determine the price of the bond.
4. A $1000 bond with 8% annual coupons matures in 3 years at $1100. It is purchased at a price to yield
9% effective. Determine the modified duration of the bond.
5. A perpetuity-immediate with level payments has a duration of 13.5 years at an effective interest rate
of i. Determine i.
6. A perpetuity-due with level payments has a duration of 25 years at an effective interest rate of i.
Determine i.
7. Company ABC must pay liabilities of $1000 due one year from now and $2000 due two years from
now. There are two investments available : one-year zero-coupon bonds and 2-year bonds with 10%
annual coupons maturing at par. The one-year spot rate is 8% and the 1-year forward rate is 9%. What
is the company's total cost of the bonds to exactly match the liabilities.
8. William plans to attend college for 3 years. His first day of college will be one year from today. He
expects tuition to cost $15,000 in the first year. He also estimates that the college will increase his
tuition by 4% each year for the next two years. William would like to exactly match these liabilities
using the following assets:
I.a one-year coupon bond with annual coupon of 3% and a yield to maturity of 4%
II.a two-year zero coupon bond with a yield to maturity of 4.5%
III.a three-year coupon bond with annual coupons of 7% and a yield to maturity of 5%
What is the total cost of the asset portfolio that will exactly match the liabilities?
9. A company must pay a benefit of $1000 to a customer in two years. To provide for this benefit, the
company will buy one-year and 3-year zero-coupon bonds. The 1-year and 3-year spot rates are 8% and
10%, respectively. The company wants to immunize itself from small changes in interest rates on either
side of 10%. What amount should it invest in the 1-year bonds?