Consider two loans. Loan A has an initial principal of P₀ and an annual nominal interest rate of i, convertible monthly. Loan B also has an annual nominal interest rate of i, but the interest is convertible daily. At the end of the first month, a payment of m is made on Loan A, which includes one month of interest. The remaining balance on Loan A is then P₁ . Let j be the monthly effective interest rate of Loan B, assuming there are 12 equal months in a year and 365 days in a year.

Determine which of the following represents j.

• [[math]]\quad\left[1+\frac{1}{12}\left(\frac{P_1-P_0+m}{P_0}\right)\right]^{12}-1[[/math]]
• [[math]]\left[1+12\left(\frac{P_1-P_0+m}{P_0}\right)\right]^{1 / 12}-1[[/math]]
• [[math]]\left[1+\frac{12}{365}\left(\frac{P_1-P_0+m}{P_0}\right)\right]^{12}-1[[/math]]
• [[math]]\left[1+\frac{365}{12}\left(\frac{P_1-P_0+m}{P_0}\right)\right]^{12 / 365}-1[[/math]]
• [[math]]\quad\left[1+\frac{12}{365}\left(\frac{P_1-P_0+m}{P_0}\right)\right]^{365 / 12}-1[[/math]]

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

Solution: E

Let I be the amount of interest in the first month.

[[math]] P_0-m+I=P_1, I=P_1-\left(P_0-m\right) [[/math]]

In the first month, the interest [math]P_1-\left(P_0-m\right)[/math] was charged on a principal of [math]P_0[/math], so the effective monthly interest rate (expressed as a decimal) of the first loan is

[[math]]\frac{P_1-\left(P_0-m\right)}{P_0}=\frac{P_1-P_0+m}{P_0}[[/math]]

.

The nominal annual interest rate (expressed as a decimal) for both loans is therefore [math]12\left(\frac{P_1-P_0+m}{P_0}\right)[/math], so the effective daily rate (expressed as a decimal) for the second loan is

[[math]] \frac{12}{365}\left(\frac{P_1-P_0+m}{P_0}\right) [[/math]]

Finally, the effective monthly rate (expressed as a decimal) for the second loan is

[[math]] \left[1+\frac{12}{365}\left(\frac{P_1-P_0+m}{P_0}\right)\right]^{365 / 12}-1 [[/math]]

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