A 16-year loan of L is repaid with a payment at the end of each year. During the first eight years, the payment is 100. During the final eight years, the payment is 300. Interest is charged on the loan at an annual effective rate of i, such that [math]1/(1+i)^8 \gt 0.3.[/math]

After the first payment of 100 is made, the outstanding principal is L + 25.

Calculate the outstanding balance on the loan immediately after the eighth annual payment of 100 has been made.

• 1660
• 1760
• 1870
• 1970
• 2080

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

Solution: A

The principal repaid in the first payment is 100 – iL. The outstanding principal is

[[math]] L – 100 + iL = L + 25 [[/math]]

Hence, iL = 125. Also

[[math]] \begin{array}{l l}{{L=300(a_{\overline{16}|}-200a_{\overline{8}|})-\frac{300(1- v^{16})-200(1- v^{8})}{i}}}\\ {{125=i L=100+200 v^{8} - 300v^{16}}}\\ {{300 v^{16}-200 v^{8} + 25 = 0}}& \\ {{ v^{8}=\frac{200\pm\sqrt{200^{2}-4(300)(25)}}{600}=\frac{200\pm100}{600}=0.5. }} \end{array} [[/math]]

The larger of the two values is used due to the value being known to exceed 0.3. The outstanding valance at time eight is the present value of the remaining payments:

[[math]] 300a_{\overline{{{8}}}|}=300\frac{1-0.5}{2^{1/8}-1}=1657. [[/math]]

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