Assignment 2
1 of 2
1) You are given that
a) Deaths follow the constant force assumption over each year of age,
b) qx = 0.06,
c) qx+1 = 0.07,
d) 054.0
1
:
1=
x
A
Find
2:
2
1
x
A.
2) Assume that the mortality follows the DeMoivre’s Law where w = 100. Let Z be the
present value of the payment of a 10-year term insurance payable upon death for (20).
Find
a) the Z-density, fZ(z)
b) the distribution function of Z, FZ(z)
c) E[Zk], k = 1, 2, 3, …
d) the probability that the premium is not sufficient.
3) An annual decreasing 10-year term insurance pay (60) benefit of $(10 – k), k = 0, 1,
2, …, 9, if (60) dies in year k + 1. You are given that
a) ,96.0
1
:60 =A
b) The actuarial present value (premium) for this insurance is if q60 = 0.1.
Find the actuarial present value (premium) if q60 = 0.15 and if qx is unchanged for all
other ages.
HINT: Use
( ) ( )
1:
1
1
1
|
:
−+
+=
nx
DAvpnvqDA xx
nx
4) Assume there is constant force of mortality μ and constant force of interest δ, let Z
be the present value of one unit of whole life insurance that is payable at the end of
the year of death bought by a life aged x.
a) Find the mean of Z in terms of μ and δ.
b) Find the variance of Z in terms of μ and δ.
5) a) Express the expected present value of the benefit below (ie E[Z]) using standard
actuarial functions. T is the future lifetime RV for a life age x.
>
≤
=1520000
1510000
15 Tv
Tv
Z
T
b) You are given that, at an effective rate of interest of 6% per year, ,166117.0=
x
A
.314208.0
15=
+x
A You are also given that lx = 93132, lx+15 = 86409. Calculate the
expected value of Z.
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