Krzys’ Ostaszewski, http://www.math.ilstu.edu/krzysio/, Exercise 223, 8/22/2009
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November 2005 Course FM/2 Examination, Problem No. 1, and Dr. Ostaszewski’s online exercise No. 223 posted August 22, 2009
An insurance company earned a simple rate of interest of 8% over the last calendar year based on the following information:
Assets, beginning of year
Sales revenue
Net investment income
Salaries paid
Other expenses paid
25,000,000
X
2,000,000
2,200,000
750,000
All cash flows occur at the middle of the year. Calculate the effective yield rate.
A. 7.7% B. 7.8% C. 7.9% D. 8.0% E. 8.1%
Solution.
You are supposed to guess that net investment income is not a cash flow, while only these accounting entries: sales revenue, salaries paid, and other expenses paid, contribute to the cash flows. On the other hand, you are supposed to include net investment income as income obtained at the end of the year (not before, as it apparently did not result in a cash flow). The net cash flow occurring in the middle of the year is
X
−
2, 200, 000
−
750, 000
=
X
−
2, 950, 000.
Therefore, the rate of return based on simple interest, i.e., the dollar-weighted rate of return, is
Assets BOY, exposed for the whole year to earning interest
⋅
  
X
−
2, 950, 000
1
+ (
X
−
Year-end investment
  
+
2, 000, 000

Middle-year cash flow,
 exposed to be earning interest for half a year
) ⋅
1
2
+
 
Year-end investment income, exposed to earning interest for
0 years, i.e., not exposed
⋅
0
=
0.08.
As 8% of 25 million is 2 million, this results in
2, 000, 000
+
0.04
⋅ (
X
−
2, 950, 000
) =
X
−
950, 000, or
0.04
⋅ (
X
−
2, 950, 000
) =
1
⋅ (
X
−
2, 950, 000
)
.
But we know that 0.04 is not equal to one, so unless X = 2,950,000, the above equation results in a contradiction. Hence X = 2,950,000 and the middle-year cash flow is zero.
This means that the initial 25,000,000 brings 2,000,000 of income during the year,

resulting in the effective yield rate of 8%.
Answer D.
© Copyright 2009 by Krzysztof Ostaszewski. All rights reserved. Reproduction in whole or in part without express written permission from the author is strictly prohibited.
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