You are given the following information about a loan of L that is to be repaid with a series of 16 annual payments:

• The first payment of 2000 is due one year from now.
• The next seven payments are each 3% larger than the preceding payment
• From the 9th to the 16th payment, each payment will be 3% less than the preceding payment.
• The loan has an annual effective interest rate of 7%.

Calculate L.

• 20,689
• 20,716
• 20,775
• 21,147
• 22,137

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

Solution: A

The present value of the first eight payments is:

[[math]] 2000v+2000(1.03)v^{2}+...+2000(1.03)^{7}v^{8}={\frac{2000v-20000(1.03)^{8}v^{7}}{1-1.03v}} = 13,136.41. [[/math]]

The present value of the last eight payments is

[[math]] \begin{split} 2000(1.03)^{7}0.97v^{9}+2000(1.03)^{7}(0.97)^{2}v^{10}+\cdots+2000(1.03)^{7}(0.97^{9})v^{96} \\ =\frac{2000(1.03)^{7}0.97v^{9}-2000(1.03)^{7}(0.97)^{9}v^{17}}{1-0.97v}=7,552.22. \end{split} [[/math]]

Therefore, the total loan amount is L = 20,688.63.

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