Consider a 7-year loan to be repaid with equal payments made at the end of each year. The annual effective interest rate is 10%.

Calculate the Macaulay duration of the loan payments.

• 3.15
• 3.29
• 3.40
• 3.50
• 3.62

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

Solution: E

Let x be the annual payment amount. Macaulay duration is

[[math]] \frac{\frac{1 x}{1.1}+\frac{2 x}{1.1^2}+\frac{3 x}{1.1^3}+\cdots+\frac{7 x}{1.1^7}}{\frac{x}{1.1}+\frac{x}{1.1^2}+\frac{x}{1.1^3}+\cdots+\frac{x}{1.1^7}} =\frac{17.6315}{4.8684} = 3.62 [[/math]]

Alternatively, the duration can be calculated as

[[math]]\frac{(Ia)_{\overline{7}|0.01}}{a_{\overline{7}|0.01}}[[/math]]

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