homework for macroeconomics

1. Homework 7
Theory of Interest
Duration/Convexity/Immunization/Term Structure of Interest Rates
1. The current price of a bond is $114.72 and the current yield is 6.00%. The
modified duration of the bond is 7.02. Use the modified duration to estimate the
price of the bond if the yield increases to 6.10%.
A.114 B.120 C.210 D.220 E.300
2. A two-year bond has 8% annual coupons payable semiannually. The bond’s yield
is 10% compounded semiannually. Calculate the modified duration of the bond.
A.0.8 B.1.0 C.1.5 D.1.8 E.2.0
3. A 22-year bond pays 7% annual coupons and has a current price of $81.12. The
annual effective yield on the bond is 9%. The Macaulay duration of the bond is
10.774. Estimate the new price if the yield falls to 8.95%.
A.76 B.78 C.80 D.82 E.84
4. A 15-year mortgage is repaid with level monthly payments. The yield is 12%
compounded monthly. Calculate the Macaulay duration of the mortgage.
A.3.2 B.5.4 C.7.1 D.9.7 E.10.3
5. A 20-year bond yielding 9% has a price of $127.79. If the bond’s yield falls to
8.75%, then the price of the bond will increase to $130.65. If bond’s yield
increases to 9.25%, then the price of the bond will fall to $125.02. Calculate the
effective duration of the bond.
A.8.6 B.8.7 C.8.8 D.8.9 E.9.0
6. A five-year bond with a coupon of 6.7% pays coupons semiannually. It is
currently yielding 6.4%. Its current price is $101.2666. If the bond’s yield
increases by 10 basis points, then its price falls to $100.8422. If the bond’s yield
falls by 10 basis points, then its price rises to $101.6931. Calculate the effective
convexity of the bond.
A.15.67 B.17.75 C.18.52 D.19.32 E.20.74
7. The modified duration of an 8-year bond is 5.35 and its convexity is 39.19.
Estimate the percentage change in the price of the bond if its yield increases by 63
basis points.
A. -3.29% B.-1.09% C.1.09% D.3.29% E.5.14%

2. 8. An insurance company has committed to make a payment of $100,000 in 5 years.
The insurance company can fund this liability only through the purchase of 4-year
zero-coupon bonds and 10-year zero-coupon bonds. The annual effective yield for
all assets and liabilities is 12%. Determine how much the bank should invest in
the 4-year zero-coupon bond in order to immunize its position.
A.26401 B.26933 C.47286 D.47813 E.48005
9. You are given the following information with respect to a callable bond:
Time Expected Cash Flows at a 7% Annual Yield
1 8.00
2 7.90
3 107.80
Annual Yield Bond Price
6% 104.33
7% 102.37
8% 99.76
The current yield is 7%.
Calculate the ratio of the modified duration to the effective duration of this bond.
A.1.01 B.1.17 C.1.32 D.1.38 E.1.50
10. The current price of a bond is 100. The derivative of the price with respect to the
yield to maturity is -700. The yield to maturity is 8%.
Calculate the Macaulay duration.
A.8 B.9 C.10 D.11 E.12
11. Given the following annual effective spot rates, find the present value of a 3-year
bond paying 15% annual coupons and having a par value of $100.
Time Annual effective spot
1 7.000%
2 6.000%
3 8.000%
A.100 B.110 C.120 D.130 E.140
12. Given the following yields and coupons for bonds with $100 of par value,
determine the 2-year annual effective spot rate.
Maturity Annual coupon Annual effective yield
1 10.000% 8.000%
2 4.000% 8.979%
3 20.000% 9.782%
A.7% B.8% C.9% D.10% E.11%

3. 13. Given the following table of forward rates, find the present value of a 3-year bond
paying 15% annual coupons and having a par value of $100.
t ft
0 7.000%
1 5.009%
2 12.114%
A.118.66 B.120.24 C.132.86 D.144.12 E.151.97
14. The table below provides the prices of 5 zero-coupon bonds:
Maturity Price per $100 of par
1 97.0874
2 90.7029
3 81.6298
4 73.5030
5 64.2529
Determine the value of f 3 , the annual effective forward rate applicable from time
3 to time 4.
A.7% B.8% C.9% D.10% E.11%
15. The following spot rates are expressed as rates compounded twice per year:
s0.5  12.00%
(2)
s1.0  13.50%
(2)
s1.5  14.66%
(2)
s2.0  16.00%
(2)
Calculate the 1-year implied forward rate for the second year, expressed as a rate
that is effective over six months.
A.7% B.8% C.9% D.10% E.11%