Acturial maths

DR SANJEEWA PERERADEPARTMENT OF MATHEMATICS UNIVERSITY OF COLOMBO

FM 2002 ACTUARIAL MATHEMATICS I

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Survival Distribution ModelsCHAPTER 1

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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 is specified by the policy.

Other types of insurance (such as auto or casualty): there is variability in both the time and amount of the claim.

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Life insurance and related topics will be discussed.

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

Life insurance:

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P

Discounting ProcessPV=PVN

DISCOUNTING PROCESS: DEPENDS ON RANDOM EVENT

P

PV=Pvn IF EVENT Occurs

Actuarial Present Value Present value

Actuarial Present Value

In life insurance: Random Event is

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nPv

( )nPv P E

AMOUNT DISCOUNT PROBABILITY

random future lifetime

A PROBABILITY MODEL FOR THIS RANDOM LIFETIME IS KNOWN AS A SURVIVAL MODEL.

Survival Function

The basic of a survival model is a random lifetime variable and its corresponding distribution.

X: Life time of a newborn. X: Cts, Probability density function Probability distribution

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[0, ]X ( )Xf x

( )XF x

Survival Distribution0

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

X XF x P X x P x f s ds

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FX(x) is continuous, increasing function with

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

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( ) ( ) 1 ( ) 1 ( )X Xs x P X x P X x F x

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

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Then 88% of newborns are expected to die prior to age 75, whereas 12% of newborns are expected to survive to age 75.

2

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

xs x x F f

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2

22

( )=1- ( ) 1 1100

75(75)=1- 1 1 0.25

100

(75)=0.9375

X

X

X

xF x s x

F

F

2( )= ( ) 1

100 100

2 75(75)= 1

100 100

(75)=0.005

X X

X

X

d xf x F x

dx

f

f

DeMoivre Law Historical Model

is the limiting age by which all have died. X : uniform distribution on the interval (0,)

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( ) 1 , for 0x

s x x

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

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age

0(birth)

x(today)

X(death)

years of past lifex years of future lifeT

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 x

T 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 die between the ages x and x+t. OR age (x) dies before reaching to age x+t, OR age (x) dies within next t years.

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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 .

= [ ( ) ] 1t x

t x

q x t x

P T x t p

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

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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 t

p 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 q

p P T x x

q P T x x

ar.

Probability Concept

1t x t xq p

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Probability Concept

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Special symbol (x) will survive t years and die within the following u years: i.e. (x) will die between ages x+t and x+t+u.

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| [ ( ) ] [ ]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 tq p q

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

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

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Force of Mortality Consider Now take t=Δx

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( ) ( )[ | ]

( )t x

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

s x

( ) ( )[ | ]

1 ( )

( ) ( ) ( ) =

1 ( ) 1 ( )

X Xx x

X

X X X

X X

F x x F xq P x X x x X x

F x

F x x F x f x xx

x F x F x

Force of Mortality

Force of mortality

This means the probability that x dies in the "next instant" delta(x). FM 2002 ACTUARIAL MATHEMATICS I 24

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 xx

F x

(

()

1 ( )) ( )X

x xX

f x xq

Fx

xx

Force of Mortality In Actuarial Science µ(x) is called the force of mortality. In reliability theory, the study of the survival probabilities of manufactured part system, µ(x) is called the failure rate or hazard rate or, more fully, the hazard rate function.

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( ) ( )( )

1 ( ) ( )X

X

f x s xx

F x s x

What is the meaning of qx = 0.008 versus the meaning of µ(x) = 0.008?

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If qx=0.008 then for every 1000 survivors to age x, 8 are expected to die in the next year. With µ(x)=0.008 we still have an annual rate, but we should only use it for short-term projections.

For example, viewing a week as (x)=(1/52)nd of a year, there would be 8 (1/52) expected deaths in the next week for every 1000 lives reaching age x.

Prove that

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0

( ) exp ( )x

s x t dt

0

0 0

( )( )

( )( ) ( )

( )( ) ( )

( )

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

x

x x

ds xs x dxxs x s x

ds xt d t

s x

s x t d t s x t d t

Example

( ) ( )

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

100Let 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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0<X<100, and x=40 and T=X-40, 0<T<100-40, 0<t<60.

Properties of µ(x)0

( ) exp ( )t

s x t dt

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( ) 0 x x

0

( )x dx

Show that density function of T(x) can be written in the following form

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( ) ( ) ( )T x t xf t p x t

If the force of mortality is constant then the life random variable X has an exponential distribution. Further T(x) is also 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

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0

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

0

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

Show that

Remark: FM 2002 ACTUARIAL MATHEMATICS I 36

2

0

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

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

If X follows DeMoivres’ law, compute E(T(x))

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0

0

( ) , 0

( ) ( ), 0 -

( )

( ( ))2

t x

x

x t x

xs x x

s x t x tp t x

s x x

xe E T x p dt

Life Tables In practice the survival distribution is estimated by compiling mortality data in the form of a life table. Here is the conceptual model behind the entries in the table. Imagine that at time 0 there are l0 newborns. Here l0 is called the radix of the life table and is usually taken to be some large number such as 100,000. Denote by lx the number of these original newborns who are still alive at age x. Similarly ndx denotes the number of persons alive at age x who die before reaching age x + n.

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Show Since ndx is the number alive at age x who die by age x + n, this is simply the number alive at age x, which is lx, minus the number alive at age x + n, which is lx+n.

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n x x x nd l l

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

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1

( ) ( )

( )x x n n x

n xx x

x x xx

x x

l l d

l

s x s x nq

s

l

x

q

l

l l

l

d

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

Curtate Future Lifetime A discrete random variable associated with the future lifetime is the number of future years completed by (x) prior to death. It is called the curtate future lifetime of (x), denoted by K(x), is defined by the relation: Here [ ] denote the greatest integer function.

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( ) ( )K x T x

The curtate future lifetime of (x), K(x) is a discrete random variable with density:

The curtate lifetime, K(x), represents the number of complete future years lived by (x).

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

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

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Uniform Distribution of Deaths in the Year of the Death (UDD) 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 of those who will die at curtate age x is uniformly distributed between the ages x and x + 1. In terms of the survival function the UDD assumption means:Where x is an integer and 0<t<1.

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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 context of previous Exercise , a 3-year, discrete survival model for 90-year-old. The UDD linearly interpolates among these 4 points to obtain the complete graph of lx for all x between 90 and 93.

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Now let T be the complete future lifetime of a 90-year-old from previous problem where we have extended the life table to a continuous model via the UDD assumption.

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90 90 90( ) 90

90 90 90

( ) (90 ) t t tT x t

t

l l lf t p t

l l l

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

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1 1

( )0 0

[ ] ( )x x

x K x k x x kk k

e E K kf k k p q

00

0 0

x

x tx xx t

x t xx x

l dtl

e p dt dtl l

1 1

1 1 2 11

0 0

...x xx k x x

x k xk k x x

l l l le p

l l

With UDD Assumption Let T=K+S, then S is uniformly distributed over [0, 1). Find the complete and curtate life expectancies at age 90 for the survival model

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Constant Force of Mortality The assumption of a constant force of mortality in each year of age means that μ(x+t)=μ(x), for each integer age x and 0<t<1

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( )

( ) ( ) exp( ), where =- ln

tt x x

x

p p

s x t s x t p

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Balducci Assumption1 1

( ) ( ) ( 1)

t t

s x t s x s x

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Find expressions for tqx and μ(x+t), under this assumption

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

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if

if

T T nT

n T n

if 1

if

K K nK

n K n

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0

50:252

If ( ) , for 0 100, compute 100

x x ex

That is, on average 50 year old lives 14.583 years between age 50 and 75

Select Mortality and the Underwriting Process (x) may pass the medical test to buy insurance policy. Survival function is actually dependent on two variables. The age at the selection (application for insurance) The amount of time passed after the time of selection A life table which takes this effect into account is called a select table.

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Notations q[x]+i denotes 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 the next year. q25 values 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 one might expect, after a certain period of time the effect of selection on mortality is negligible. The length of time until the selection effect becomes negligible is called the select period. The Society of Actuaries uses a 15 year select period in its mortality tables. The Institute of Actuaries in UK uses a 2 year select period. The implication of the select period of 15 years in computations is that for each j≥0FM 2002 ACTUARIAL MATHEMATICS I 60

Aggregate Table A life table in which the survival functions are tabulated for attained ages only is called an aggregate table. Generally, a select life table contains a final column which constitutes an aggregate table. The whole table is then referred to as a select and ultimate table and the last column is usually called an ultimate table.

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Consider: 3-year select period 85%, 90%, 95% and 100% of general mortality in policy

year 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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The mortality pattern for anindividual issued life insurance at age 21

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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 DeMoivre’s law and that d[x]=d[x]+1=d[x]+2 for 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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