Krzys’ Ostaszewski, http://www.math.ilstu.edu/krzysio/, Exercise 39, 2/11/6
Author of the Course FM manual available at:
http://smartURL.it/krzysioFM (paper) or http://smartURL.it/krzysioFMe (electronic)
Instructor for online seminar for exam FM: http://smartURL.it/onlineactuary
Society of Actuaries Sample Examination 140-83-94, Problem No. 7
Perpetuity X has annual payments of 1, 2, 3, … at the end of each year. Perpetuity Y has
annual payments of q, q, 2q, 2q, 3q, 3q, … at the end of each year. The present value of X
is equal to the present value of Y at an annual effective interest rate of 10%. Calculate q.
A. 1.1
B. 1.3
C. 1.5
D. 1.7
E. 1.9
Solution.
1
, we have
10
i
0.10 1
d=
=
= ,
1 + i 1.10 11
Note that for i = 0.10 =
and
1
= 121.
d2
The first perpetuity is worth
1 1 121
!
=
= 110.
1 + i d 2 1.1
Let us observe also that j = 1.12 ! 1 = 0.21 is the effective interest rate over a two-year
0.21 21
period, and its corresponding discount rate is d j =
=
. The second perpetuity can
1.21 121
be viewed as a sum of two identical perpetuities: q, 2q, 3q, …, the first one starting at
time 1 (year) and the second one starting at time 2 (years). Therefore the value of the
second perpetuity is
1 q
1 q
q 1212
q 1212
q
1210q
! 2+ 2! 2 =
! 2 + 2 ! 2 = 2 ! (13310 + 12100 ) =
.
1.1 d j 1.1 d j 1.1 21
1.1 21
21
21
This gives
1210q
= 110,
21
and
2310
q=
! 1.9090909.
1210
Answer E.
© Copyright 2006 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.