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# Summary of Part II (formulas) All the formulas you need for part two of the course organized in a pdf file

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University of Waterloo

Actuarial Science

ACTSC 232

Jun Cai

Fall

Description

Summary of Lecture Notes - ACTSC 232, Winter 2010
Part 2 - Life Benets
2.1 Life insurances on (x)
A life insurance on (x) is a contract or policy issued by an insurer to a life currently aged
x. The insurer will pay benets to the beneciaries of (x) in the future. The payment
times of the benets are contingent on the death time of (x). Such benets are called
death benets or life benets.
Generally speaking, a life insurance on (x) is called a continuous life insurance if benets
are payable at the moment of the death of (x). A life insurance is called a discrete life
insurance if benets are payable at the end of the death year of (x).
Review of the expectations of the functions of T and K : For a function g,
x x
Z 1 Z 1
E[g(T )] = g(t)f (t)dt = g(t) p dt
x x t x x+t
0 0
and
1 1
X X
E[g(K )x = g(k) PrfK =xkg = g(k) k x x+k:
k=0 k=0
Review of present values: Let v denote present value (PV) at time 0 of 1 (dollar or unit)
t
to be paid at time t and v is called discount function. If the force of interest = is a
t t
constant, then v = v = e t= ( 1) , where v = 1 = e and 1 + i = e .
t 1+i 1+i
t t
Unless stated otherwise, we assume that t is a constant or v t v = e .
(a) A general continuous life insurance on (x) pays death benets at the death time of
(x). Denote the benet by b tf T x t or (x) dies at time t; t > 0.
Let Z denote the present value at time 0 or at age x of the benets to be paid by
Tx Tx
the insurance. Then Z = b Txv = b Txe and Z is a random variable, where T x
is the death time of (x).
The expectation or mean of the present value Z is
Z 1 Z 1
Tx t t
E[Z] = E bTxe = bte fx(t)dt = bte tpx x+tdt
0 0
1 which is called the actuarial present value (APV) of the insurance, or the expected
present value (EPV) of the insurance, or the pure premium of the insurance, or the
net premium of the insurance, or the single benet premium of the insurance.
The second moment of the present value Z is
Z 1 Z 1
2 2 2x 2 2t 2 2t
E[Z ] = E b Txe = bte fx(t)dt = bte tpx x+tt
0 0
and V ar[Z] = E[Z ] (E[Z]) :
The distribution function of Z is denoted bZ F (z) = PrfZ zg. The distribution
function may be continuous, or discrete, or mixed.
(b) A general discrete life insurances on (x) pays death benets at the end of the death
year of (x). Denote the death benet by k+1 if Kx= k or (x) dies in year k + 1,
k = 0;1;2;::::
Let Z denote the present value at time 0 or at age x of the benets. Then,
Z = bK +1 vKx+1 = bK +1e(Kx+1)
x x
The APV or EPV of the insurance is
X1
E[Z] = E[b K +1vKx+1] = bk+1vk+1 PrfK x kg
x
k=0
X1 X1
k+1 (k+1)
= bk+1v kpxqx+k = bk+1e kpxq x+k:
k=0 k=0
This expectation is also called the pure premium of the insurance, or the net pre-
mium of the insurance, or the single benet premium of the insurance.
The second moment of the present value is given by
1 1
2 2 2(Kx+1) X 2 2(k+1) X 2 2(k+1)
E[Z ] = E[bK x1 v ] = bk+1v kpxq x+k= bk+1e kpxq x+k:
k=0 k=0
2.2 Level Benet Life Insurances
A life insurance on (x) is called a level benet life insurance if benets are constant and
independent of the payment times of the benets.
2(a) A continuous whole life insurance of 1 on (x) pays 1 at the moment of death of (x).
Tx
The PV of the benet is Z = v and the APV of the insurance is denoted by
Z 1 Z 1
Tx t t
Ax= E[v ] = v fx(t)dt = e tpxx+tdt:
0 0
The second moment of Z is denoted by
Z Z
1 1
A x E[v 2Tx] = v fx(t)dt = e2ttp x+tdt
0 0 x
2 2
and V ar[Z] = A x (A ) x
If the mortality force of (x) follows the constant force lawxor = for all x > 0,
t
then fx(t) = t x x+t= e for 0 < t < 1 and
2
A x + and A x + 2:
If the mortality force of (x) follows De Moivres law with the limiting age !, then
1
fx(t) = t x x+t= !x; 0 < t < ! x and
Z !x
t 1
A x e ! xdt
0
and

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