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FIN 3220A
Actuarial Models I
Chapter 2
Life Insurance
Page 2
FIN 3220A (20052006)
Chapter 2 Life Insurance
Introduction
±
The models for life insurances designed to reduce the financial impact of
the random event of untimely death are developed
±
Life insurances are characterized by their longterm nature
±
The amount of investment earnings, up to the time of payment, provides
a significant element of uncertainty
±
The uncertainty has two causes:
²
The unknown rate of earnings over the investment period
²
The unknown length of the investment period
±
In this course:
²
A deterministic model is used for the unknown investment earnings
²
A probability distribution is used to model the uncertainty in regards to the
investment period
±
Our model will be built in terms of functions of
T
, the insured’s future
lifetime random variable
Page 3
FIN 3220A (20052006)
Chapter 2 Life Insurance
Insurance Payable at the Moment of Death
±
We look at a life insurance with the benefit amount and the time of
payment depend only on the length of the interval from the issue of the
insurance to the death of the insured
±
There are two functions in the model:
²
A
benefit function
,
b
t
²
A
discount function
,
v
t
±
t
is the length of the interval from issue to death
±
v
t
is the interest discount factor from the time of payment back to the
time of policy issue
²
Here, the underlying force of interest is deterministic
²
Most of the time, we only look at the case of constant force of interest
Page 4
FIN 3220A (20052006)
Chapter 2 Life Insurance
Insurance Payable at the Moment of Death
±
The
presentvalue function
,
z
t
, is the present value, at policy issue, of the
benefit payment
±
The elapsed time from policy issue to the death of the insured is the
insured’s futurelifetime random variable,
T
=
T
(
x
)
²
So
z
T
is also a random variable
±
The model for the insurance is based on the equation
±
To develop a probability model of
Z
²
The first step will be to define
b
t
and
v
t
²
The next step is to determine some characteristics of the probability
distribution of
Z
that are consequences of an assumed distribution of
T
tt
t
z
bv
=
TT
Zb
v
=
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FIN 3220A (20052006)
Chapter 2 Life Insurance
Level Benefit Insurance
±
An
nyear term life insurance
provides a payment only if the insured dies
within the
n
year term of an insurance commencing at issue
±
If a unit is payable at the moment of death of (
x
), we have
±
Three conventions:
²
We define
b
t
,
v
t
and
Z
only on nonnegative values
²
For a
t
value where
b
t
is 0, the value of
v
t
is irrelevant
²
The forecast of interest is assumed to be constant, unless stated otherwise
±
The expectation of the presentvalue random variable,
Z
, is called the
actuarial present value
,
APV
, of the insurance
²
A more exact term would be
expectation of the present value of the payments
,
1
,
0,
0,
0.
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 Spring '05
 CSWong

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