Mathematics of Life Contingencies. Math 3281 3.00 W
Instructor: Edward Furman
Homework 1
Unless otherwise indicated, all lives in the following questions are
subject to the same law of mortality and their times until death are
independent random variables.
1. Assume that a decision maker’s current wealth is 10,000. Assign u(0)=-1
and u(10,000)=0.
a) When facing a loss of X with probability 0.5 and remaining at current
wealth with probability 0.5, the decision maker would be willing to pay up
to G for complete insurance. The values for X and G in three situations are
given below.
X
10 000
G
6000
6 000
3 300
3 300
1 700
Determine three values on the decision maker’s utility of wealth function u.
b) Calculate the slopes of the four line segments joining the five points
determined on the graph u(w). Determine the rates of change of the slopes
from segment to segment.
c) Put yourself in the role of a decision maker with wealth 10,000. In additional to the given values of u(0) and u(10,000), elicit three additional values
on your utility of wealth function u.
d) On the basis of the five values of your utility function, calculate the slopes
and the rates of change of the slopes as done in part (b).
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2. Consider a game of chance that consists of tossing a coin until a head appears.
The probability of a head is 0.5 and the repeated trails are independent. Let
the random variable N be the number of the trial on which the first head
occurs.
a) Show that the probability function of N is given by
1
f (n) = ( )n
2
n = 1, 2, 3, · · ·
b) Find E[N] and Var(N).
c) If a reward of X = 2N is paid, prove that the expectation of the reward
does not exist.
d) If this reward has utility u(w)=log w, find E[u(X)].
3. A utility function is given by
u(w) =


e−(w−100)2 /200
w < 100

2 − e−(w−100)2 /200
w ≥ 100
a) Is u0 (w) ≥ 0?
b) For what range of w is u00 (w) < 0?
4. If a utility function is such that u0 (w) > 0 and u00 (w) > 0, show that π[X] ≤
E[X]. A decision maker with preferences consistent with u00 (w) > 0 is a risk
lover.
5. You are given:
1) The benefit of 1 on a ten-year endowment insurance is payable at the
moment of death, or at the end of 10 years if (x) survives 10 years.
2) µx (t) = 0.01 for t > 0.
3) δ = 0.06.
2
Determine the single benefit premium.
solution
Note that under constant force of mortality, the expectation of random variable for 1 unit of whole life insurance is
Ax:10 =
µ
.
µ+δ
µ
(1 − e−n(µ+δ) ) + e−n(µ+δ) = 0.5685
µ+δ
6. A whole life insurance pays a benefit of 10e0.05t at the moment of death if
death occurs at time t. You are given:
1) µ = 0.02
2) δ = 0.04
Calculate the actuarial present value of the benefit.
solution
Netting the 0.05 rate of benefit increase against δ = 0.04 yields a net interest
rate of -0.01. Letting A be the actuarial present value of the benefit,
0.02
A = 10
= 20
0.01
7. An n-year term insurance payable at the moment of death has actuarial
present value of 0.0572. You are given:
1) µx (t) = 0.007, t > 0.
2) δ = 0.05.
Determine n.
solution
We have
0.05720 =
µ
0.007
(1 − e−n(µ+δ) ) =
(1 − e−0.057n )
µ+δ
0.057
n = 11
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8. For a whole life insurance of 1000 on (x) with benefits payable at the moment
of death:


0.04 0 < t ≤ 10
1) δt =

0.05 10 < t


0.06 0 < t ≤ 10
2) µx (t) =

0.07 10 < t
Calculate the single benefit premium for this insurance.
solution
The single benefit premium for 1 is
Ax = Ax:10
+10 Ex Ax+10 = Ax (1 −10 Ex ) +10 Ex Ax+10
1
= 0.593869
Therefore, the single benefit premium for a benefit of 1000 is 593.868.
9. Dave wants to purchase a 5 year pure endowment with a single benefit premium. The amount of the endowment is $1,000. His insurance agent convinces him instead to use the same money to purchase a year endowment
insurance policy which pays at the moment of death or at the end of five
years, whichever comes first. You are given that µ = 0.04 and δ = 0.06.
Calculate the benefit amount for this 5 year endowment.
solution
A
1
x:5
= e−5(0.04+0.06) = 0.606531, whereas Ax:5 =
0.04
(1
0.1
− e−0.5 ) + e−0.5 =
0.763918, so the benefit amount for endowment insurance is
100(0.606531/0.763918) = 793.97
.
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10. Bryon, a non-actuary, estimates the single benefit premium for a continuous
whole life policy with a benefit of $100,000 on (30) by calculating the present
value of $100,000 paid at the expected time of death.
(30) is subject to a constant force of mortality µx = 0.05, and the force of
interest is δ = 0.08.
Determine the absolute value of the error of Bryon’s estimate.
solution
The expected value of an exponential distribution is the reciprocal of the
force, or 1/0.05=20. The value of $100,000 paid in 20 years is 100, 000e−20(0.08) =100, 000e−1.6 =
0.05
20, 190. The true value of 100, 000A30 is 100, 000( 0.05+0.08
) = 38, 462. The
error is $18,272.
11. Given:
1) i=5%.
2) The force of mortality is constant.
◦
3) ex = 16
Calculate
20| Ax .
solution
The mean survival time is 16, so the force of mortality is 1/16. Under the
constant force of mortality, Ax+20 = Ax . Thus we have:
1/16
−(ln1.05+1/16)(20)
20| Ax =20 Ex Ax = e
1/16 + ln1.05
= 0.06065
12. For a continuous whole life insurance (Z = v T , T ≤ 0), E[Z]=0.25.
Assume the forces of mortality and interest are each constant.
Calculate Var(Z).
solution
5
Since E[Z]=µ/µ + δ = 0.25, it follows that δ = 3µ. Then,
µ
= 1/7
µ + 2δ
1
1
V ar(Z) = − ( )2 = 0.0804
7
4
E[Z 2 ] =
13. Z is the present-value random variable for a whole life insurance of b payable
at the moment of death of (x).
You are given:
1) µx+t = 0.01, t ≤ 0
2) δ = 0.05
3) The single benefit premium for this insurance is equal to Var(Z).
Calculate b.
solution
µ
For one unit, E[Z]= µ+δ
=
1
6
and
V ar(Z) =
25
µ
− (E[Z])2 =
µ + 2δ
396
So we need b such that
b
25b2
=
6
396
396
b=
= 2.64
6 ∗ 25
GOOD LUCK!
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