A loan of 5,000,000 is to be repaid by installments of X at the end of each quarter over a period of ten years. The annual nominal interest rate for the loan is 8% compounded quarterly. The actual quarterly payment for the first five years is X rounded up to the next higher 1000. After that, each quarterly payment is X rounded up to the next higher 100,000, until the loan is paid off with a drop payment.

Calculate the total number of payments, including the drop payment, needed to repay the loan.

• 36
• 37
• 38
• 39
• 40

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

Solution: C

[[math]] \begin{aligned} & 5,000,000=X a_{\overline{40} \mid 0.02 } \\ & X=182,778.74 \\ & O B_{20}=5,000,000(1.02)^{20}-183,000 s_{\overline{20}|0.02} \\ & O B_{20}=2,983,318.31 \\ & 2,983,318.31=200,000 a_{\overline{n}|0.02} \\ & n=17.89 \end{aligned} [[/math]]

20 original payments plus 18 with the drop payment equals 38 total payments.

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