Seth has two retirement benefit options. His first option is to receive a lump sum of 374,500 at retirement. His second option is to receive monthly payments for 25 years starting one month after retirement. For the first year, the amount of each monthly payment is 2000. For each subsequent year, the monthly payments are 2% more than the monthly payments from the previous year. Using an annual nominal interest rate of 6%, compounded monthly, the present value of the second option is P.

Determine which of the following is true.

• P is 323,440 more than the lump sum option amount.
• P is 107,170 more than the lump sum option amount.
• The lump sum option amount is equal to P.
• The lump sum option amount is 60 more than P.
• The lump sum option amount is 64,090 more than P.

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

Solution: D

The accumulated value of the first year of payments is

[[math]]2000 s_{\overline{12}|0.005} = 24, 671.12[[/math]]

. This amount increases at 2% per year. The effective annual interest rate is 1.005¹² -1 = 0.061678. The present value is then

[[math]] \begin{align*} P=24,671.12\sum_{k=1}^{2s}1.02^{k-1}(1.061678)^{-k}=24,671.12\frac{1}{1.02}\sum_{k=1}^{2s}\Biggl(\frac{1.02}{1.061678} \Biggr)^{k} \\ =24,187.37{\frac{0.960743-0.960743^{26}}{1-0.960743}}=374,444. \end{align*} [[/math]]

This is 56 less than the lump sum amount.

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