Krzys’ Ostaszewski, http://math.illinoisstate.edu/krzysio/krzys.html
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Spring 2000 Course 2 Examination, Problem No. 16, also Dr. Ostaszewski’s online
exercise No. 190 posted January 3, 2009
On January 1, 1997, an investment account is worth 100,000. On April 1, 1997, the value
has increased to 103,000 and 8,000 is withdrawn. On January 1, 1999, the account is
worth 103,992. Assuming a dollar weighted method for 1997 and a time weighted
method for 1998, the annual effective interest rate was equal to x for both 1997 and 1998.
Calculate x.
A. 6.00%
B. 6.25%
C. 6.50%
D. 6.75%
E. 7.00%
Solution.
Remember that on this examination, the dollar-weighted method always assumes simple
interest. Let us write B for the balance in the account at the end of 1997. The dollar
weighted return for 1997 was
B + 8000 − 100000 B − 92000
=
= x.
3
94000
100, 000 − ⋅ 8000
4
The time weighted return for 1998 was
103, 992
− 1 = x.
B
103, 992
From this, we obtain B =
. Substituting this into the first equation, we get:
1+ x
103, 992
− 92000
1+ x
= x.
94000
Solving this, we obtain
103,992 − 92000 ⋅ (1 + x ) = 94000x ⋅ (1 + x ) ,
103,992 − 92000 − 92000x = 94000x + 94000x 2 ,
11,992 − 186000x − 94000x 2 = 0,
11750x 2 + 23250x − 1499 = 0,
with two solutions: x ≈ 0.0625, or x = −2.0412. Only the first solution is feasible.
Answer B.
© Copyright 2009 by Krzysztof Ostaszewski.
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