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CHAPTER 1

FM 2002 ACTUARIAL MATHEMATICS I 2


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Introduction

Insurance policy: Two types of RISK

life insurance: the variability in the claim

is only the time at which the claim is

made, since the amount of the claim isspecified by the policy.

Other types of insurance (such as auto or

casualty): there is variability in both thetime and amount of the claim.



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Why a Survival Model is Necessary Annuity base: to find present value, use discount

factor.

Life insurance:


iscounting Process


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Actuarial Present Value

Present value

Actuarial Present Value

In life insurance: Random Event is


nPv

( )nPv P E


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Survival Distribution

0

( ) ( ) (newborn dies before age ) ( )

x

X XF x P X x P x f s ds



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Survival Function The survival function sX(x) is defined as theprobability that a newborn survives to age x.Since this is the event that X>xwe have


( ) ( ) 1 ( ) 1 ( )X X

s x P X x P X x F x


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Suppose for example that sX 75)=0.12,so that FX 75)=0.88. MEANS?



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2If ( ) 1 ,for 0 100, find (75), (75)

100 X X

xs x x F f


2

2

2

( )=1- ( ) 1 1100

75(75)=1- 1 1 0.25100

(75)=0.9375

X

X

X

xF x s x

F

F

2( )= ( ) 1

100 1002 75

(75)= 1100 100

(75)=0.005

X X

X

X

d xf x F x

dx

f

f


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DeMoivre Law Historical Model

is the limiting age by which all have died.

X: uniform distribution on the interval (0,

)


( ) 1 , for 0

x

s x x


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The Future Lifetime of Age x) Life insurance is usually issued on a person who hasalready attained a certain age x. Life age x). Future life time of age x): T x)


age

x(today)

X(death)

years of past lifex years of future lifeT


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The Future Lifetime of Age x)

( )

( )

We are given that , so

Since T is a function of , its density function ( )

and distribution function ( ) shoud be related to .

T x

T x

X xT X x

X f t

F t X



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The Future Lifetime of Age x)

This gives the conditional probability that a newborn will diebetween the ages x and x+t. OR

age x) dies before reaching to age x+t, OR age x) dies within next t years.


the probability of survival to age given survival to age .

= [ ( ) ]t xp x t x

P T x t

the probability of death before age given survival to age .

= [ ( ) ] 1

t x

t x

q x t x

P T x t p


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RemarksThe symbol, can be interpreted as the probability that ( ) will die within years;

that is is the distribution function of ( ).

t x

t x

q x t

q T x


The symbol, can be interpreted as the probability that ( ) will survive another years;that is is the survival function of ( ).

t x

t x

p x tp T x

When 1 the prefix is omitted and one just writes and respectively.= [ ( ) 1], the probability that ( ) survives another year.

[ ( ) 1], the probability that ( ) will die within next ye

x x

x

x

t p qp P T x x

q P T x x

ar.


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Probability Concept

1t x t xq p



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Special symbol (x)will survivetyears and die within thefollowing uyears:i.e. x) will die between agesx+tandx+t+u.


| [ ( ) ] [ ]t u xq P t T x t u P x t X x t u

|t u x t u x t x t x t u xq q q p p

|t u x t x u x t q p q


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Prove that|t u x t x u x t q p q



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Compute tpx for the DeMoivre law of mortality.Conclude that under the DeMoivre law T x) hasthe uniform distribution on the interval 0,-x)



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Force of Mortality Consider

Now take t=x


( ) ( )[ | ]

( )t x

s x s x tq P x X x t X x

s x

( ) ( )

[ | ] 1 ( )

( ) ( ) ( ) =

1 ( ) 1 ( )

X X

x xX

X X X

X X

F x x F x

q P x X x x X x F x

F x x F x f x xx

x F x F x


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Force of Mortality

Force of mortality

This means the probability that x dies in the nextinstant delta(x).


For each age x, it gives the value of the conditional p.d.f.

of X at exact age x, given survival

( )

1

to that a e.

(

g

)

X

X

f x

F x

( ) ( )1 ( )

X

X

f x xF x

(

()

1 ( )) ( )Xx x

X

f x xq

Fx

xx


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Force of Mortality

In Actuarial Science (x) is called the force ofmortality. In reliability theory, the study of thesurvival probabilities of manufactured partsystem, (x) is called the failure rate or hazardrateor, more fully, thehazard rate function.


( ) ( )( )1 ( ) ( )

X

X

f x s xxF x s x


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What is the meaning of qx= 0.008 versus the

mean ing o f (x) = 0.008?



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Prove that


0

( ) exp ( )x

s x t dt

0

0 0

( )

( )( )( ) ( )

( )( ) ( )

( )

ln( ( )) ( ) ( ) ( ) exp ( ) ( )

x

x x

ds x

s x dxxs x s x

ds xt d t

s x

s x t d t s x t d t

E l


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Example

( ) ( )

2If ( ) for 0 100, find ( ), ( ), ( )

100

Let 40, find ( ) and ( )

X X

T x T x

x x s x F x f xx

x F t f t



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Properties of x)0

( ) exp ( )

t

s x t dt


( ) 0x x

0

( )x dx


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Show that density function of T(x) can be

written in the following form


( )( ) ( )

T x t xf t p x t


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If the force of mortality is constant then the life randomvariable X has an exponential distribution. Further T x) isalso exponentially distributed.



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Find the force of mortality for DeMoivers law.



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Complete Expectation of Life The expected value of T x), E T x)) is known as

the complete expectation of life at age x. Prove


0

Complete expectation life= ( ( ))xe E T x0

0( ( ))x t xe E T x p dt


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Show that

Remark:


2

0

( ( ) ) 2 t xE T x t p dt

2 2( ( )) ( ( ) ) ( ( ( )))Var T x E T x E T x


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If X follows DeMoivres law, compute (T(x))


0

0

( ) , 0

( ) ( ) , 0 -( )

( ( )) 2

t x

x

xt x

xs x x

s x t x tp t xs x x

x

e E T x p dt


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Life Tables In practice the survival distribution is estimated bycompiling mortality data in the form of a life table.Here is the conceptual model behind the entries inthe table. Imagine that at time 0 there are l0newborns. Here l0is called the radix of the life tableand is usually taken to be some large number such as100,000. Denote by lx the number of these original

newborns who are still alive at age x. Similarly ndxdenotes the number of persons alive at age xwho diebefore reaching age x + n.FM 2002 ACTUARIAL MATHEMATICS I 38


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Show

Since ndxis the number alive at age x who dieby age x + n, this is simply the number alive atage x, which is lx, minus the number alive at agex + n, which is lx+n.


n x x x nd l l


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Consider a group of newborns l0. Each newborn's age-at-death has a distribution

specified by survival function s(x). L(x) - random number of survivors at age x. Each newborn is viewed as a Bernoulli trial: survive

success, death fail. Hence L(x) has a binomial distribution: n= l0 ,p=P(success) =s(x). lx means expected # of survivors:



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Basic Relationships1

( )

( )

xx

x nx

x

x

n

ls x

lp

np

x l

l

s


1

( ) ( )

( )

x x n n xn x

x x

x x xx

x x

l l d

l

s x s x nq

sl

x

q

l

l l

ld

1

|

|

x n x n m m x nn m x

x x

x n x n x nn x

x x

l l dq l

l d

l

lq

l l

( )

( )( )

x

x

ls xx

s x l


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Curtate Future Lifetime A discrete random variable associated with thefuture lifetime is the number of future yearscompleted by x) prior to death. It is called the

curtate future lifetime of x), denoted by K(x), isdefined by the relation:

Here [ ] denote the greatest integer function.


( ) ( )K x T x


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The curtate future lifetime of x), K x) is a discreterandom variable with density:

The curtate lifetime, K x), represents the number ofcomplete future years lived by x).



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Given the following portion of a life Table, findthe distribution of K for x=90.



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How to find fractional part???Three approaches: Uniform Distribution of Deaths in the Year ofthe Death UDD) Constant Force of Mortality Balducci Assumption



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Uniform Distribution of Deaths in the Year of the DeathUDD) The number alive at age x + t, where x is an integer and 0

< t < 1, is given by:

The UDD assumption means that the age at death ofthose who will die at curtate age x is uniformlydistributed between the ages x and x + 1. In terms of thesurvival function the UDD assumption means:

Where x is an integer and 0


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tShow that

1

( )1

x x

t x x

x

x

q tq

p tq

qx t

tq



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Consider Previous Example The ideas here will be introduced in the contextof previous Exercise , a 3-year, discrete survivalmodel for 90-year-old. The UDD linearly

interpolates among these 4 points to obtain thecomplete graph of lxfor all x between 90 and 93.



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Now let T be the complete future lifetime of a90-year-old from previous problem where wehave extended the life table to a continuousmodel via the UDD assumption.


90 90 90( ) 90

90 90 90

( ) (90 ) t t tT x tt

l l lf t p tl l l


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Curtate Life Expectancy E K) is known as the complete life expectancy and isdenoted by ex


1 1

( )

0 0

[ ] ( )x x

x K x k x x k

k k

e E K kf k k p q

00

0 0

x

x tx x

x tx t x

x x

l dtl

e p dt dt l l

1 1

1 1 2 11

0 0

...x xx k x x

x k x

k k x x

l l l l e p

l l


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With UDD Assumption Let T=K+S, then Sis uniformlydistributed

over [0, 1).

Find the complete and curtate life expectancies at age 90for the survival model



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Constant Force of Mortality The assumption of a constant force of mortality ineach year of age means that x+t)= x),for each integeragexand 0


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Balducci Assumption1 1

( ) ( ) ( 1)

t t

s x t s x s x


Find expressions for tqxand (x+t), under this

assumption


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The Expected Number of Years Lived by x)

:

0

:

The expected number of complete years lived by ( )

before age

The expected number of years lived by ( )

before age

x

x

n

n

e x

x n

e x

x n


if

if

T T nT

n T n

if 1

if

K K nK

n K n


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0

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2If ( ) , for 0 100, compute

100x x e

x


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Select Mortality and the Underwriting Process x) may pass the medical test to buy insurancepolicy. Survival function is actually dependent on two

variables. The age at the selection application for insurance) The amount of time passed after the time ofselection A life table which takes this effect into account iscalled a select table.



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Notations q[x]+idenotes the probability that a person dies between years x +i and x + i + 1 given that selection occurred at age x.

q25- Probability that an insured 25-old will die in thenext year. q25values for individuals underwritten at ages 0, 1, 2,...,24, 25 are respectively denoted by q[0]+25, q[1]+24, ,q[25].



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A select mortality table is based on this idea. As onemight expect, after a certain period of time the effect ofselection on mortality is negligible. The length of timeuntil the selection effect becomes negligible is called theselect period. The Society of Actuaries uses a 15 yearselect period in its mortality tables. The Institute ofActuaries in UK uses a 2 year select period. Theimplication of the select period of 15 years incomputations is that for each 0



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Aggregate Table A life table in which the survival functions aretabulated for attained ages only is called an aggregate

table. Generally, a select life table contains a finalcolumn which constitutes an aggregate table. Thewhole table is then referred to as a select andultimate table and the last column is usually called anultimate table.



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Consider:

3-year select period

85%, 90%, 95% and 100% of general mortality in policyyear 1,2,3 and 4, respectively.

With a 3-year select period an individual underwritten at

age 21 would be subject to mortality rates

at age 21, 22 and 23.



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You are given the following extract from a 3

year select and ultimate mortality table.

Assume that the ultimate table follows

DeMoivreslaw and that d[x]=d[x]+1=d[x]+2for

all x. Find 1000( 2|2q[71])



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2|2 [71] Probability of age 71 survies two years and

will die the following 2 years.

q



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