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# Bonds- Duration

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An insurance company is analyzing three bonds and is using duration as the measure of interest rate risk. All three bonds trade at a yield to maturity of 10% , have maturity of 5 years and have \$10,000 par values. The bonds differ only in the amount of annual coupon interest they pay: 8, 10, and 12%

a) What is the duration for each of the bonds?
b) What is the relationship between duration and the amount of coupon interest that is paid?

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#### Solution Preview

An insurance company is analyzing three bonds and is using duration as the measure of interest rate risk. All three bonds trade at a yield to maturity of 10% , have maturity of 5 years and have \$10,000 par values. The bonds differ only in the amount of annual coupon interest they pay: 8, 10, and 12%

a) What is the duration for each of the bonds?

Duration of bond
Duration is the average time of each payment
Duration=( 1 x PV(C1)+2 x PV(C2)+3 x PV(C3)+4 x PV(C4)+---)/V=
V=summation of PV ( C )
PV (C) is the present value of cash flow

Bond 1
Par value= ...

#### Solution Summary

Calculates duration of bonds and examines the relationship between duration and the amount of coupon interest that is paid.

\$2.19

## Given the current \$1,000 price of each bond, what is each bond's duration? Given each bond's duration, what is the forecasted change in the value of the bonds? If the rate on comparable bonds 6 %, what are the price and duration of each bond? Based on the duration of each bond, which portfolio is riskier?

1) You owe the following \$1,000 bonds:

Bond A 4% coupon due in three yrs
Bond B 5% coupon due in 5 yrs
Bond C 7% coupon due in 10 yrs

Currently the structure of yields is positive so that each bond sells for its par value. However, you expect that inflation will increase and cause interest rates to rise so that the structure of yields becomes inverted (i.e., a negatively sloped yield curve). You anticipate that interest rates will be 10,9, and 8 % for 3-,5- and 10- year bonds, respectively.

a) Given the current \$1,000 price of each bond, what is each bond's duration?
b) Given each bond's duration, what is the forecasted change in the value of the bonds?
c) If interest rates do change as you expected, what is the new price of each bond?
d) What is the forecasting error based on duration and the actual change in each bond's price?

2) Portfolio A consists entirely of \$1,000 zero coupons that mature in 8, 9, and 10 years. Portfolio B consists of \$1,000 8 % coupon bonds that mature in 10,15, and 20 years.
a) If the rate on comparable bonds 6 %, what are the price and duration of each bond?
b) Based on the duration of each bond, which portfolio is riskier?
c) Verify your answer to (b) by determining the percentage loss you would sustain if the interest rate rose to 10 %.

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