A loan is amortized over five years with monthly payments at an annual nominal interest rate of 9% compounded monthly. The first payment is 1000 and is to be paid one month from the date of the loan. Each succeeding monthly payment will be 2% lower than the prior payment.

Calculate the outstanding loan balance immediately after the 40^th payment is made.

• 6750
• 6890
• 6940
• 7030
• 7340

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

Solution: B

Monthly payment at time t is [math]1000(0.98)^{t-1}[/math]

Because the loan amount is unknown, the outstanding balance must be calculated prospectively. The value at time 40 months is the present value of payments from time 41 to time 60:

[[math]] \begin{array}{c}{{O B_{40}=1000[0.98^{40} v^{1}+\cdots+0.98^{59} v^{20}]}}\\ {{=1000\frac{0.98^{40} v^{1}-0.98^{60} v^{21}}{1-0.98 v}, v=1/(1.0075)}}\\ {{=10000\frac{0.44238-0.25434}{1-0.97270}=688.}}\end{array} [[/math]]

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