John took out a 20-year loan of 85,000 on July 1, 2005 at an annual nominal interest rate of 6% compounded monthly. The loan was to be paid by level monthly payments at the end of each month with the first payment on July 31, 2005.

Right after the regular monthly payment on June 30, 2009, John refinanced the loan at a new annual nominal rate of 5.40% compounded monthly, and the remaining balance will be paid with monthly payments beginning July 31, 2009. The amount of each payment is 500 except for a final drop payment.

Calculate the date of John’s last payment.

• July 31, 2022
• April 30, 2030
• May 31, 2030
• April 30, 2031
• May 31, 2031

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

Solution: C

The original monthly payment is

[[math]]85,000 /a_{\overline{240}|0.005}[[/math]]

. On July 1, 2009 there has been 4 years of payments, hence 16x12 = 192 remaining payments. The outstanding balance is

The number of remaining payments after refinancing is determined as

[[math]] 608.97a_{\overline{240}|0.005} = 608.97(123.2380) = 75, 048.24. [[/math]]

The number of remaining payments after refinancing is determined as

[[math]] \begin{array}{l}{{75,048.24=500a_{\overline{{{n}}}|0.0045}=500\frac{1-1.0045^{-n}}{0.67543=1-1.0045^{-n}}}}\\ {{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}}\\ {{n=-\ln(0.32457)/\ln(1.0045)=250.62.}}\end{array} [[/math]]

Thus the final payment will be 251 months from June 30, 2009. This is 20 years and 11 months and so the final payment is May 31, 2030.

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