A mortgage for 125,000 has level payments at the end of each month and an annual nominal interest rate compounded monthly. The balances owed immediately after the first and second payments were 124,750 and 124,498, respectively.

Calculate the number of payments needed to pay off the mortgage.

• 198
• 199
• 200
• 201
• 202

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

Solution: E

Let m be the monthly payment and i be the monthly interest rate. The interest in the first payment is 125,000i and the principal repaid is 125,000 – 124,750 = 250. Thus m = 125,000i + 250. Similarly, for the second payment, m = 124,750i +252. Thus, 250i = 2 for i = 2/250 = 0.008 and then m = 1250. To obtain the number of payments, the equation to solve is

[[math]] \begin{array}{c}{{125,000=1250a_{\overline{n}|0.008} }}\\ {{100 = \frac{1-1.008^{-n}}{0.008} }} \\ {{0.2=1.008^{-n}}}\\ {{n=-\ln(0.2)/\ln(1.008)=202}}\end{array} [[/math]]

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