MATH 471: Actuarial Theory I
Homework #6: Fall 2010
Assigned September 29, due October 13
1. Assume
μ
x
(
t
) =
μ
and
δ
t
=
δ
for
t
≥
0. Show:
¯
A
x
=
μ
μ
+
δ
.
2.
Z
is the present value random variable for a whole life insurance of
b
payable at the moment of death of (x):
(i)
δ
= 0.04
(ii)
μ
x
(
t
) = 0.02 for
t
≥
0
(iii) The single benefit premium for this insurance is equal to
var
(
Z
).
Calculate:
b
.
(3.75)
3. Let
Z
be the present value random variable for a special continuous whole
life insurance on (x), where for
t
≥
0:
(i)
b
t
= 1000 exp[0
.
05
t
]
(ii)
μ
x
(
t
) = 0.01
(iii)
δ
t
= 0.06
(a) Calculate the actuarial present value of this insurance.
(500)
(b) Calculate the variance of
Z
.
(83,333.33)
4. Let
Z
be the present value random variable for a whole life insurance on
(x) with a benefit of 10,000 payable at the moment of death.
Assume
μ
x
(
t
) = 0.03 and
δ
t
= 0.06 for
t
≥
0.
(a) Determine the cumulative distribution function of
Z
.
(b) Calculate the 65th percentile of the distribution of
Z
.
(4225)
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Let
Z
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 Fall '08
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 Math, Normal Distribution, value random variable

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