Rhonda purchases a perpetuity providing a payment of 1 at the beginning of each year. The perpetuity’s Macaulay duration is 30 years.

Calculate the modified duration of this perpetuity

• 28.97
• 29.00
• 29.03
• 29.07
• 29.10

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

Solution: C

[[math]] \begin{align*} 30 = \textrm{MACD} = \frac{\sum_{n=0}^{\infty}nv^n}{\sum_{n=0}^{\infty} v^n} = \frac{Ia_{\overline{\infty}|}}{\ddot{a}_{\overline{\infty}|}} = \frac{1/\left(d i\right)}{1/d}=\frac{\left(1+i\right)/\,i^{2}}{\left(1+i\right)/\,i}=\frac1i\ , \implies i = 1/30 \\ \textrm{ModD} = \frac{\textrm{MacD}}{1+i} = \frac{30}{1 + \frac{1}{30}} = 29.032. \end{align*} [[/math]]

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