Sam buys an eight-year, 5000 par bond with an annual coupon rate of 5%, paid annually. The bond sells for 5000. Let [math]d_1[/math] be the Macaulay duration just before the first coupon is paid. Let [math]d_2[/math] be the Macaulay duration just after the first coupon is paid.

Calculate [math]d_1/d_2[/math].

• 0.91
• 0.93
• 0.95
• 0.97
• 1.00

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

Solution: C

Let [math]d_0[/math] be the Macaulay duration at time 0.

[[math]] \begin{array}{l}{{d_{0}=\ddot{a}_{\overline{8}|0.05}=6.7864}}\\ {{d_{1}=d_{0}-1=5.7864}}\\ {{d_{2}=\ddot{a}_{\overline{7}|0.05}}}=6.0757 \\ {{\frac{d_1}{d_2}=\frac{5.7864}{6.0757}=0.9524}}\end{array} [[/math]]

This solution employs the fact that when a coupon bond sells at par the duration equals the present value of an annuity-due. For the duration just before the first coupon the cash flows are the same as for the original bond, but all occur one year sooner. Hence the duration is one year less.

Alternatively, note that the numerators for [math]d_1[/math] and [math]d_2[/math] are identical. That is because they differ only with respect to the coupon at time 1 (which is time 0 for this calculation) and so the payment does not add anything. The denominator for [math]d_2[/math] is the present value of the same bond, but with 7 years, which is 5000. The denominator for [math]d_1[/math] has the extra coupon of 250 and so is 5250. The desired ratio is then 5000/5250 = 0.9524.

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