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Nov 20'23
Exercise
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.
Nov 20'23
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.