A five-year loan has an annual nominal interest rate of 30%, convertible monthly. The loan is scheduled to be repaid with level monthly payments of 500, beginning one month after the date of the loan.

The borrower misses the thirteenth through the eighteenth payments, but increases the next six payments to X so that the final 36 payments of 500 will repay the loan.

Calculate X.

• 1070
• 1075
• 1080
• 1150
• 1160

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

Solution: C

After one year the outstanding balance is [math]500 a_{\left.\overline{48}\right|_{0.025}}=13,886.58[/math]. This must match the present value of the revised payments:

[[math]] \begin{aligned} & 13,886.58=X v^6 a_{\overline{6}|0.025}+500 v^{12} a_{\overline{36}|0.025}=4.74964 X+8,757.69 \\ & X=(13,886.58-8,757.69) / 4.74964=1,079.85 \end{aligned} [[/math]]

Alternatively, each missing payment is being replaced with a larger payment six months later. The larger payment should be the payment due plus the missed payment with interest, or [math]500+500(1.025)^6=1,079.85[/math]

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