You are given the following information about a 30-year bond:

1. The par value is 2000.
2. The redemption value is 2250.
3. Coupons are paid annually.
4. The annual coupon rate is twice the annual yield rate.
5. The purchase price is 3609.29.
6. Based on the yield rate, the Macaulay duration of the bond is 14.41 years

Calculate the modified duration of the bond, based on the yield rate.

• 12.40 years
• 13.07 years
• 13.71 years
• 14.41 years
• 15.15 years

Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Solution: C

Let [math]i[/math] be the yield rate. Then,

[[math]] \begin{aligned} & 3609.29=2000(2 i) a_{\overline{30}|i}+2250(1+i)^{-30} \\ & =4000\left[1-(1+i)^{-30}\right]+2250(1+i)^{-30} \\ & (1+i)^{-30}=(4000-3609.29) /(4000-2250)=0.22326 \\ & i=0.22326^{-1 / 30}-1=0.051251 . \end{aligned} [[/math]]

Modified duration is Macaulay duration divided by one plus the yield rate: [math]14.14 / 1.051251=[/math] 13.71.

Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.