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# Duration of bond

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Calculate the duration of an 8 percent, \$1,000 par bond that matures in the three years if the bond's YTM is 10 percent and interest is paid semi-annually.
a) Calculate this bond's modified duration.
b) Assuming the bond's YTM goes from 10 percent to 9.5 percent, calculate an estimate of the price change.

#### Solution Preview

Calculate the duration of an 8 percent, \$1,000 par bond that matures in the three years if the bond's TYM is 10 percent and interest is paid semiannually.

?In general, for a bond paying constant periodic coupons, the formula for Macaulay's Duration is:

?In the formula, C is the annual coupon rate, M is the bond maturity (in years), and YTM is the yield to maturity, n is the compounding frequency (i.e. n=1 for annual 2 for semiannual , 4 for quarterly) ...

#### Solution Summary

Calculates duration, modified duration and price change when interest rate changes.

\$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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