Krzys’ Ostaszewski, http://www.math.ilstu.edu/krzysio/, Exercise 137, 12/29/7
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November 2001 SOA/CAS Course 2 Examination, Problem No. 6, Problem No. 5,
also Study Note FM-09-05, Problem No. 15
A 10-year loan of 2000 is to be repaid with payments at the end of each year. It can be
repaid under the following two options:
(i) Equal annual payments at an annual effective rate of 8.07%.
(ii) Installments of 200 each year plus interest on the unpaid balance at an annual
effective rate of i.
The sum of the payments under option (i) equals the sum of the payments under option
(ii). Determine i.
A. 8.75%
B. 9.00%
C. 9.25%
D. 9.50%
E. 9.75%
Solution.
If P is the payment under option (i) then 2000 = Pa10 8.07% and therefore
P=
2000
! 299.
a10 8.07%
This implies that the total of the payments under this option is 2990. Under option (ii) the
total of all payments is also 2990, and since principal payments total 2000, interest paid
will be 990. The interest paid is equal to the sum of interest charged on all outstanding
principal amounts throughout loan’s existence, giving
990 = i ! ( 2000 + 1800 + 1600 + ... + 200 ) = i !11, 000.
!#####"#####$
11,000
Therefore i = 0.09.
Answer B.
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