An investor buys a perpetuity-immediate providing annual payments of 1, with an annual effective interest rate of i and Macaulay duration of 17.6 years.

Calculate the Macaulay duration in years using an annual effective interest rate of 2i instead of i.

• 8.8
• 9.3
• 9.8
• 34.2
• 35.2

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

Solution: B

The Macaulay duration of the perpetuity is

[[math]] \frac{\sum_{n=1}^{\infty}n\nu^{n}}{\sum_{n=1}^{\infty}\nu^{n}}=\frac{(Ia)_{\overline{\infty}|}}{a_{\overline{\infty}|}}=\frac{\left(1+i\right)/i^{2}}{i}=\frac{1+i}{i}=1+1/i=17.6. [[/math]]

This implies that i = 1/16.6. With i = 2i = 2/16.6, the duration is 1 + 16.6/2 = 9.3.

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