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Practice Problem for October 13, 2007
May 2007 SOA Course MLC Examination, Problem No. 8
Kevin and Kira excel at the newest video game at the local arcade, “Reversion”. The
arcade has only one station for it. Kevin is playing. Kira is next in line. You are given:
(i) Kevin will play until his parents call him to come home.
(ii) Kira will leave when her parents call her. She will start playing as soon as Kevin
leaves if he is called first.
(iii) Each child is subject to a constant force of being called: 0.7 per hour for Kevin; 0.6
per hour for Kira.
(iv) Calls are independent.
(v) If Kira gets to play, she will score points at a rate of 100,000 per hour.
Calculate the expected number of points Kira will score before she leaves.
A. 77,000
B. 80,000
C. 84,000
D. 87,000
E. 90,000
Solution.
Let us rephrase the terms of the problem:
• Kevin is life-age (x), which “dies” when Kevin is called home. Kevin is subject to
constant force of mortality of 0.7, and the length of Kevin’s life is counted in years.
• Kira is life-age (y), which “dies” when Kira is called home. Kira is subject to constant
force of mortality of 0.6, and the length of Kira’s life is counted in years.
After (x) dies, if (y) is still alive (i.e., after Kevin is called home, if Kira has not been
called home yet), (y) receives a life annuity at the rate of 100,000 per hour (which is the
unit of time for measurement of (x)’s and (y)’s length of life). There is no interest, i.e., the
interest rate is zero, as all points are worth the same no matter when they are earned.
The game is called “Reversion” because (y) reseives a reversionary life annuity, i.e., a life
annuity paid after first death, if (x) dies first, until the death of (y), if (y) dies second. The
actuarial present value of that annuity is:
+"
APV =
# 100000 ! v
0
+"
t
! t qx ! t py dt =
# 100000 ! (1 $ e ) ! e
$0.7t
$0.6t
dt =
0
1 (
% 1
= 100000 '
$
+ 89743.59.
& 0.6 1.3 *)
Answer E.
© Copyright 2007 by Krzysztof Ostaszewski. All rights reserved. Reproduction in
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Exercises from the past actuarial examinations are copyrighted by the Society of
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