Two borrowers obtain loans at the same time. Each loan is for the same amount and is repaid with level end-of month payments. The first borrower is charged a monthly effective interest rate of i and makes payments of P for k months to pay off the loan, where k is a positive integer. The second borrower is charged a monthly effective interest rate of j and makes payments of 120 for 5k months to pay off the loan.

Determine which statement about P is true.

• 400 < P ≤ 450
• 450 < P ≤ 500
• 500 < P ≤ 550
• 550 < P ≤ 600
• 600 < P ≤ 650

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

Solution: E

Let L = the loan amount. Note that [math][/math]. The equation of value is

[[math]] P\cdot a_{\overline{{{k}}}|\;i}=L=120\cdot a_{\overline{5k}|j} [[/math]]

so that

[[math]] \begin{aligned} P=\frac{120a_{\overline{{{5k}}}\mid j}}{a_{\overline{{{k}}}\mid i}} \\ =120\frac{1-(1+j)^{-5k}}{j}\frac{i}{1-(1+i)^{-k}} \\ =120\frac{1-\left(1+i\right)^{-k}}{j}\frac{i}{1-\left(1+i\right)^{-k}} \\ =120 \frac{i}{j} \\ =120\frac{\left(1+j\right)^{5}-1}{j} \end{aligned} [[/math]]

Next, using the fact that 0 < j < 0.04, we get

[[math]]5\lt{\frac{\left(1+j\right)^{5}-1}{j}}\lt5.41\,633[[/math]]

by plugging in a small value like 0.000000001 and 0.04 resulting in P equaling more than 600 but less than 650.

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