Krzys’ Ostaszewski, http://www.math.ilstu.edu/krzysio/.
Author of a study manual for exam FM available at:
http://smartURL.it/krzysioFM (paper) or http://smartURL.it/krzysioFMe (electronic)
Instructor for online seminar for exam FM: http://smartURL.it/onlineactuary
If you find these exercises valuable, please consider buying the manual or attending our
seminar, and if you can’t, please consider making a donation to the Actuarial Program at
Illinois State University: https://www.math.ilstu.edu/actuary/giving/. Donations will be
used for scholarships for actuarial students. Donations are tax-deductible to the extent
allowed by law.
Questions about these exercises or the FM Manual written by Dr. Ostaszewski? E-mail:
[email protected]
November 2005 Course FM/2 Examination, Problem No. 8, and Dr. Ostaszewski’s
online exercise 263 posted May 29, 2010
Matthew makes a series of payments at the beginning of each year for 20 years. The first
payment is 100. Each subsequent payment through the tenth year increases by 5% from
the previous payment. After the tenth payment, each payment decreases by 5% from the
previous payment. Calculate the present value of these payments at the time the first
payment is made using an annual effective rate of 7%.
A. 1375
B. 1385
C. 1395
D. 1405
E. 1415
Solution.
The present value of the first ten payments is
10
⎛ 1.05 ⎞
1− ⎜
9
⎝ 1.07 ⎟⎠
1.05
1.05
100 + 100 ⋅
+ … + 100 ⋅
=
100
⋅
.
1.05
1.07 9
1.07
1−
1.07
The present value of the final ten payments is
1.05 9 ⋅ 0.95
1.05 9 ⋅ 0.95 2
1.05 9 ⋅ 0.9510
100 ⋅
+
100
⋅
+
…
+
100
⋅
=
1.0710
1.0711
1.0719
10
⎛ 0.95 ⎞
1− ⎜
9
⎝ 1.07 ⎟⎠
1.05 ⋅ 0.95
⋅
.
= 100 ⋅
0.95
1.0710
1−
1.07
The sum of the two is
10
10
⎛ 1.05 ⎞
⎛ 0.95 ⎞
1− ⎜
1
−
⎜⎝
⎟
⎝ 1.07 ⎟⎠
1.05 9 ⋅ 0.95
1.07 ⎠
100 ⋅
+ 100 ⋅
⋅
≈ 1384.64.
10
1.05
0.95
1.07
1−
1−
1.07
1.07
Answer B.
© Copyright 2010 by Krzysztof Ostaszewski.
All rights reserved. Reproduction in whole or in part without express written
permission from the author is strictly prohibited.
Exercises from the past actuarial examinations are copyrighted by the Society of
Actuaries and/or Casualty Actuarial Society and are used here with permission.