A borrower takes out a 15-year loan for 400,000, with level end-of-month payments, at an annual nominal interest rate of 9% convertible monthly. Immediately after the 36th payment, the borrower decides to refinance the loan at an annual nominal interest rate of j, convertible monthly. The remaining term of the loan is kept at twelve years, and level payments continue to be made at the end of the month. However, each payment is now 409.88 lower than each payment from the original loan.

Calculate j.

• 4.72%
• 5.75%
• 6.35 %
• 6.90%
• 9.14%

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

Solution: D

The initial level monthly payment is

[[math]] R=\frac{400,000}{a_{\overline{15 \times 12}|0.09/12}}=\frac{400,000}{a_{\overline{180}|0.075}}=4,057.07. [[/math]]

The outstanding loan balance after the 36th payment is

[[math]] B_{3\mathrm{6}}=R a_{\overline{{{160-36}}}|0.0075}=4,057.07a_{\overline{{{144}}}|0.0075}=4,057.07(87.8711)=356,499.17. [[/math]]

The revised payment is 4,057.07 – 409.88 = 3,647.19. Thus,

[[math]] \begin{array}{l}{{356,{499}.17=3,647.19 a_{\overline{144}|j/12}}}\\ {{a_{\overline{144}|j/12}=356,499.17/3,647.19=97.7463.}}\end{array} [[/math]]

Using the financial calculator, j/12 = 0.575%, for j = 6.9%.

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