Assurances and annuities

Introduction to Assurances and Annuities

Life insurance contracts (policies) are made between life offices and one or more policyholders.Policyholder pays premium(s) to insurer

In return the insurer pays benefit(s), to the policyholder(s) on the occurrence of a specified event.

Main Benefit Types: AssurancesThe sum assured (benefit) is payable on/after the death of the policyholder.

AnnuitiesThe benefit is payable (at regular intervals) while the policyholder is still alive.

Other benefit types include: near-life contingencies : benefit is paid on occurrence of specified serious illness(e.g. stroke, heart attack) income protection (PHI): benefit is paid when policyholder is unemployed due to illness or redundancy

non-life contingencies: eg: motor insurance, house contents insurance etcPricing Contracts

The simplest approach to pricing contracts is to use the Equation of Value:

Money inMoney out

PremiumsBenefits

Investment incomeExpenses

Allowance also needs to be made for profit in pricing a contract.In calculating the equation of value we must consider:(a)

the time value of money, and

(b)the uncertainty attached to payments to be made in the future, depending on the death or survival of a given life

We rely on the topics covered in the Financial Mathematics Subject (CT1), in particular compound interest, and the topics covered in the Models Subject (CT4), in particular the unknown future lifetime and its associated probabilities.

Allowing for the time value of money:There are 3 main approaches to determine the interest rate to use to allow for the time value of money.i) Assume constant, known interest rate

ii) Non-constant deterministic rate

iii) Stochastic model

We will use the first method in this course.

Where interest rate is i, we will use v = (1+ i)-1 for discounting (or perhaps e-(, where (=loge(1+ i)).

To allow for the uncertainty attached to payments we will also rely on a number of basic probabilities introduced in Survival Models.

Tx:complete future lifetime random variable and it is continuous

tpx:

Prob. [Tx > t]

ie: the probability that a life aged x survives for at least another t years.

tqx:

Prob. [Tx ( t]

ie: the probability that a life aged x dies within the next t years.

= force of mortality at age x+t

p.d.f. of Tx:

tpx(x+t

n|mqx= Prob. [n< Tx < n+m]

=(npx)(mqx+n)

Kx = [Tx]curtate future lifetime random variable and it is discrete

P[Kx = k] =(kpx)(qx+k)

Initially we will concentrate on valuing benefits.

Life Assurance Contracts

Whole Life Assurance

Term Assurance

Pure Endowment

Endowment

Critical Illness Assurance

In each case we will derive formula for: Expected Present Value (EPV) of benefits payable under the contract

the variance of the benefits payable

in terms of the random variables Tx and Kx.

Whole Life AssuranceA contract which pays the sum assured on the policyholder's death.

Usually the present value at time 0 of a payment of 1 to be made at time t is vt.

Here the sum assured (assume S = 1) is payable at Kx+1, and the present value of the benefit is .This is a random variable as we do not know when x will die.

The Expected Present Value of the Benefit is:

E[] =

By definition of t px and t qx it follows that

P[Kx = k] = (kpx)(qx+k) = k(qx (k = 0, 1, 2, ...)

Therefore

This is the expected present value (EPV) of a sum assured of 1, payable at the end of the year of death. These EPVs play a central role in life insurance mathematics and are included in the standard actuarial notation.

We define:

[Note that, for brevity, we have written the sum as instead of . These are equivalent since k px = 0 for k x.]If the sum assured is S, then the EPV of the benefit is (S)Ax.

Values of Ax at various rates of interest are tabulated in (for example) the AM92 tables, which can be found in Formulae and Tables for Examinations.

4%6%

A300.160230.07328

A600.456400.32692

A900.841960.77843

Variance

In general:Var[x] = E[x2]-(E[x])2

But since (vk+1)2 = (v2)k+1, the first term is just Ax calculated at a rate of interest (1 + i)2 ( 1.

We denote this as 2Ax

=>

So provided we can calculate EPVs at any rates of interest, it is easy to find the variance of a whole life benefit.

Values of 2Ax are tabulated at various rates of interest in (for example) AM92 Formulae and Tables for Examinations.

Note that

.

Term AssuranceA contract that pays the sum assured on the policyholder's death, provided death occurs during a specified period, the term. Payment of the benefit is NOT certain.

Let F denote the present value of this payment. F is a random variable.

The term is n years and is usually integral.

F=

Kx < n

0

Kx ( n

Expected Present Value of benefit:

E[F] =

E[F]=

=

These policies are sold widely, often used in connection with loans or generally making provision for dependents.

Variance of benefit:

Var[F] =

Where the 2 prefix means that the EPV is calculated at a rate of interest (1+i)2-1.

The expected values are not normally tabulated, but can easily be obtained by using other functions.

=

=

=

=

=

Note that ( as n ( (Sample values

4%6%

0.047090.03251

0.028560.02002

0.041340.02683

Pure Endowment AssuranceA contract that pays the sum assured at the end of a specified period, the term, provided a life aged x is still alive. Payment of the benefit is NOT certain.

Let G denote the random variable of the present value of this payment.

G=

0

Kx < n

Kx ( n

E[G]=

=

Var[G] = where the 2 prefix denotes and EPV calculated at rate of interest (1+i)2-1.These policies are rare in practice but are very useful theoretically and also in computing other values.

The expected values are not normally tabulated directly but can easily be calculated, remembering that:

Sample values using AM924%6%

0.274880.15523

0.288500.16292

0.225230.11563

For the same life and common n, F and G are NOT independent random variables.

F > 0 ( G = 0 or F = 0 ( G > 0

Endowment Assurance

This contract combines a term assurance and a pure endowment. It pays a sum assured of 1 to a life now aged x at the end of the year of death, if death occurs during the next n years, or after n years if the life is then alive.

The benefit is certain to be paid.

Let H denote the random variable of the present value of this payment. H = F + G (previously defined).

H =

or H =

E[H] =

=E[F] + E[G]

=

=

Var[H]

= where the 2 prefix denotes an EPV calculated at rate of interest (1+i)2-1.

Note:F and G are not independent random variables therefore

Var[H] var[F] + var[G]Many expected values for endowments are tabulated for common values of x+n (60,65).

Sample values

4%6%

0.321970.18774

0.317060.18294

0.266570.14246

Also since

= + , we can rewrite as

= -

and this is a common way to obtain expected values for term assurances.Variance of a Portfolio of Policies:

Remember that F, G and H are all random variables depending directly on Kx, the curtate future lifetime of an individual aged x.

, etc. are all expected values and therefore apply to a portfolio of identical policies.

With a portfolio of policies, we also usually assume that the lives are independent with respect to mortality.

( Var.[portfolio] = ( var.(individual policies)

Critical Illness Assurance

This contract pays a sum assured at the end of the year of diagnosis, from a specified list of diseases, of a healthy life aged x, provided this occurs during the next n years.

Again payment of the benefit is NOT certain.

We proceed very much like a term assurance.

For a term assurance: E[F]

For a critical illness contract the Expected Present Value of the benefit is:

EPV

where

k(ap)x is the probability that (x) is still alive at time k and has not yet been diagnosed with a critical illness,and

is the probability that a life aged x+k is diagnosed with a critical illness in the coming year.The probabilities used in this evaluation are examples of dependent decrements, which we will study in more detail later in the course.Deferred Assurance Benefits

The benefit payment is deferred for a specified period of time.

Deferred Whole Life Assurance

A WL Assurance with sum assured of 1 payable to a life aged x but deferred for n years will pay a death benefit of 1 at the end of the year of death provided death occurs after age x+n.

Let J denote the random variable of the present value of this payment.

J =

The EPV[H] =

Variance

Note:

Var(X-Y) = Var[X] +Var[Y] -2Cov[X,Y]

Cov(XY)=E[XY]-E[X]E[Y]

Benefits Payable Immediately on Death

In practice, assurance death benefits are paid a short time after death, as soon as the validity of the claim can be verified.

It is imprudent to assume a delay and we should really assume no delay i.e. sum assured payable immediately on death.

Notation

The notation for EPVs is similar to before:

Whole life

Term Assurance

Endowment

= +

(clearly the pure endowment is unchanged)

Whole Life AssuranceConsider a whole life assurance with sum assured 1 payable immediately on the death of a life aged x.

The present value of the benefit is

=

In general is not tabulated but is calculated approximately from

The variance is:

Similarly EPVs and Variances can be derived for Term Assurances and Endowment Assurances where the benefit is payable immediately on death.

Useful Relationships

Claim Acceleration Approximation

Of deaths occurring between ages x + k and x + k + 1, say, (k = 0, 1, 2, ...) roughly speaking the average age at death will be x + k + . Under this assumption claims are paid on average 6 months before the end of the year of death. Therefore we obtain the approximate EPVs:-

( for small i)

Similarly

(only the death claim is accelerated)Further ApproximationsA second approximation is obtained by considering a WL ir term assurance to a sum of deferred term assurances each for a term of 1 years.Consider a WL assurance:

but

Under UDD, tpx+k(x+k+t = qx+k (0 t < 1) i.e. is constant

Hence

Similarly:

Expected present value of money in

Expected present value of money out

=

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