Krzys’ Ostaszewski, http://www.math.ilstu.edu/krzysio/, Exercise 138, 1/5/8
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November 2001 Course 2 Examination, Problem No. 9, also Study Note FM-09-05,
Problem No. 16
A loan is amortized over five years with monthly payments at a 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.
A. 6751
B. 6889
C. 6941
D. 7030
E. 7344
Solution.
The key issue is to determine the outstanding balance of a loan when the payments are
not level. The monthly payment at time t (in months) is 1000 ! 0.98 t "1. As the actual
amount of the loan is not given, the outstanding balance must be calculated prospectively.
The balance at time 40 is the present value of the future payments made at times 41 to 60.
This equals (using a monthly rate 0.75%):
0.98 40
0.98 41
0.98 59
1000 !
+ 1000 !
+ ... + 1000 !
=
1.00751
1.0075 2
1.0075 20
2
19
0.98 " 0.98 %
0.98 40 "
" 0.98 % %
=
! 1+
+$
= 1, 000 !
' + ... + $#
'
1.00751 $# 1.0075 # 1.0075 &
1.0075 & '&
" 0.98 %
1( $
40
# 1.0075 '&
0.98
= 1, 000 !
!
0.98
1.00751
1(
1.0075
20
) 6889.11.
Answer B.
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