Kylie bought a 7-year, 5000 par value bond with an annual coupon rate of 7.6% paid semiannually. She bought the bond with no premium or discount.

Calculate the Macaulay duration of this bond with respect to the yield rate on the bond.

• 5.16
• 5.35
• 5.56
• 5.77
• 5.99

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

Solution: C

If the bond has no premium or discount, it was bought at par so the yield rate equals the coupon rate, 0.038.

[[math]] \begin{align*} d &= \frac{\frac{1}{2}\biggl(1(190)v+2(190)v^{2}+\cdots+14(190)v^{14}+14(5000)v^{14}\biggr)}{190v+190v^{2}+\cdots+190v^{14}+5000v^{14}} \\ d &= \frac{95\bigl(Ia\bigr)_{\overline{{{14}}}|}+7\bigl(5000\bigr)v^{14}}{190a_{\overline{14}|}+5000v^{14}}\\ d = 5.5554 \end{align*} [[/math]]

Or, taking advantage of a shortcut:

[[math]]d=\ddot{a}_{\overline{14}|0.038}=11.1107[[/math]]

This is in half years, so dividing by two [math]d = 11.1107/2 = 5.5554[/math].

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