An annuity provides the following payments:

1. X at the beginning of each year for 20 years, starting today
2. 4X at the beginning of each year for 30 years, starting 20 years from today

Calculate the Macaulay duration of this annuity using an annual effective interest rate of 2%

• 27.32
• 27.87
• 28.30
• 33.53
• 35.41

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

Solution: B

The denominator of the duration is the present value of the annuity

[[math]] X\ddot{a}_{\overline{20}|0.02}+4Xv^{20}\ddot{a}_{\overline{30}|2}=78.1729X. [[/math]]

The numerator is the time-weighted present value of the annuity. In units of X we need the present value of 0, 1, ..., 19, 80, 84, ..., 196. One way to view this is as four times a 49-year increasing immediate annuity (so payments of 4, 8, ..., 76, 80, 84, ..., 196) less three times a 19- year increasing immediate annuity (so payments of 3, 6, ..., 57). The present value is:

[[math]] 4X(I a)_{\overline{49}|0.02}-3X(I a)_{\overline{{{19}}}|0.02}^{}=[4(655.2078) -3(147.4923)]X=2,178.3542X [[/math]]

The duration is the ratio, 2,178.3542/78.1729 = 27.87.

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