ACTUARIAL 2

8/13/2019 ACTUARIAL 2

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Dr Sanjeewa Perera

Department of Mathematics

University of Colombo

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CHAPTER 2

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Introduction "What is the value today of a random sum of

money which will be paid at a random time in the

future?

A=Random amount, T=Random Time

Present Value=AvT

Expected Present Value = E[AvT]

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A

T


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Meaning of E[AvT]

Average present value of the actual payment.

Averages are reasonable in the insurance context

since, from the company's point of view.

The average cost (and income) per policy is therefore

a reasonable starting point from which to determine

the premium. The expected present value is usually referred to as

the Actuarial present value in the insurance context.

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Insurance Contracts

Consist of two parts:

Benefit payments,

Premium payments

A life insurance policy paying benefit bt if death

occurs at time thas actuarial present value E[bTvT].

The actuarial present value is also called theNet single premium.

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Life Insurance

The net single premium is the average valueof the benefit payments, in today's money.

The net single premium is the amount of the

single premium payment which would berequired on the policy issue date by an

insurer with no expenses (and no profit

requirement). The actuarial present value ofthe premium payments must be at least equal

to the actuarial present value of the benefit

payments, or the company would go

bankrupt.

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What we want to do now?

Next two methods are developed for

computing the net single premium forcommonly issued life insurances and

develops methods for computing the

actuarial present value of the premium

payments. These two sets of methods are

then combined to enable the computation of

insurance premiums.

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Important Facts

In the case of life insurance, the amount that is paidat the time of death is usually fixed by the policy and

is non-random. The net single premium for an insurance which pays

1 at the time of death is then E[vT].

In insurance notation: bar: an insurance paid at thetime of death, subscripts: statuswhose death causesthe insurance to be paid.

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1

exp( )1v i


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-Year Pure Endowment Insurance This pays the full benefit amount at the end of the nth

year if the insured survives at least nyears.

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1, if ( ) , if ( ),

0, if ( ) 0, if ( )

n

t

T x n v T x nb Z

T x n T x n


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Show that

Since if the benefit is paid, the benefit paymentoccurs at time n,

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nn x n xE v p


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Use the life table to find the net single premium for a 5

year pure endowment policy for (30) assuming aninterest rate of 6%.

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5

5 30 5 30

The net single premium for the pure endowment polic

0.7409

y i

.

s

1E v p


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n-Year Term Insurance

This type insurance provides for a benefit paymentonly if the insured dies within n years of policy

inception.

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Rule of Moments FM 2002 ACTUARIALMATHEMATICS I13


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Whole Life Insurance The full benefit is paid no matter when the insured

dies in the future.

The whole life benefit can be obtained by taking

the limit as n (approaches to infinity) in the n-yearterm insurance setting.

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Suppose that T(x) has an exponential distribution withmean 50. If the force of interest is 5%, find the net singlepremium for a whole life policy for (x), if the benefit of $1000 is payable at the moment of death.

T(x) is exponentially distributed with 1/50.

Net single premium = E[1000 vT]

Net single premium = E[1000 exp(-T(x))]

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0

1[1000exp( ( ))] 1000exp( 0.05 ) exp50 50

=285.71

tE T x t dt

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For a whole life insurance of 1000 on (x) with benefitspayable at the moment of death, you are given:

Calculate the single benefit premium for this insurance.

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n-Year Endowment Insurance

Provides for the payment of the full benefit at thetime of death of the insured if this occurs before time

n and for the payment of the full benefit at time n

otherwise.

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: [ ]T n

x nA E v


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Another idea Let Z1 , Z2 and Z3be the PV of term, pure endowment

and endowment insurances.

Clearly Z3 =Z1 + Z2 . E[Z3]= E[Z1]+E[Z2].

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Deferred Insurance For an n-year deferred whole insurance, a benefit ispayable if the insured dies at least n years followingissue:

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m-Year Deferred n-Year Term Insurance

Provides the same benefits as nyear term insurance

between times m and m + n provided the insured

lives myears.

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| ( , ][ 1 ( ( ))]

T

m n x m m nA E v T x

|

m n

t

m n x t x x t

m

A v p dt

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Constant mortalityFM 2002 ACTUARIALMATHEMATICS I22

Assume mortality is based on a constant force andinterest is also based on a constant force of interest . Find expressions for the APV for the following types ofinsurances:

Whole life insurance -year term life insurance -year endowment life insurance -year deferred life insurance

Find the expression for the above insurance if weassume De oivr slaw.

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Non Fixed Constant Benefits Insurances All of the insurances discussed thus far have a fixedconstant benefit. Increasing whole life insuranceprovides a benefit which increase linearly in time.Similarly, increasing and decreasing n-year terminsurance provides for linearly increasing(decreasing) benefit over the term of the insurance.

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Insurances Payable the End of the Year of Death Direct computation of the net single premium for

an insurance payable at the time of death is

impossible using only the life table.

These net single premiums can be easily related to

the net single premium for an insurance that is

payable at the end of the year of death. For these insurance Z depends on Curtate life time

K.

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Consider n-year term insuranceFM 2002 ACTUARIALMATHEMATICS I

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Whole life FM 2002 ACTUARIALMATHEMATICS I26

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- year endowment insurance The APV of a (discrete) endowment life insurance is the sum ofthe APV of a (discrete) term and a pure endowment:

The policy pays a death benefit of 1 at the end of the year ofdeath, if death is prior to the end of n years, and a benefit of 1 ifthe insured survives at least n years.

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Summary

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Computation using Life Table Consider whole life insurance policy

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1

1

0

1

0

[ ]

= ( ( ) )

=

K

x

k

k

k x k

k x

A E v

v P K x k

dv

l

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Show that

How would you compute this using the life table?

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1 1: :x x

n n

A A

1 [ , ):

[ , ) 1:

[ 1 ( ( ))]

= [ 1 ( ( ))]

nn

xn

n

nx

n

A E v T x

E v K x A

1 1: :

=

x xn n

n n x nn x

x

A A

lv p v

l


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Use the life table to find the net single premium fora 5 year endowment policy for (30), with death

benefit paid at the moment of death, assuming an

interest rate of 6%.

The net single premium for a pure endowment

policy is

For the endowment policy, the net single premiumfor a 5 year term policy must be added to this

amount. From the relation given earlier,

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The relationship between insurances payable at

the time of death and insurances payable at the

end of the year of death is used to complete the

calculation. This gives

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