Variable insurance benefits

For all the insurance benefits studied in this chapter, the EPV of the benefit can be expressed as the sum over all the possible payment dates of the product of three terms:

• the amount of benefit paid,

• the appropriate discount factor for the payment date, and

• the probability that the benefit will be paid at that payment date.

This approach works for the EPV of any traditional benefit – that is, where the future lifetime is the sole source of uncertainty. It will not generate higher moments or probability distributions.

The approach can be justified technically using indicator random vari- ables. Consider a life contingent eventE– for example,Ecould be the event that a life agedxdies in the interval(k,k+1]. The indicator random variable is

I(E)=

1 ifEis true, 0 ifEis false.

In this example, Pr[Eis True] =k|qx, so the expected value of the indicator random variable is

E[I(E)]=1(k|qx)+0(1−k|qx)=k|qx,

4.6 Variable insurance benefits 127 and, in general, the expected value of an indicator random variable is the probability of the indicator event.

Consider, for example, an insurance that pays $1 000 after 10 years if(x)has died by that time, and $2 000 after 20 years if(x)dies in the second 10-year period, with no benefit otherwise.

We can write the present value random variable as

1 000I(Tx≤10)v¹⁰+2 000I(10<Tx≤20)v²⁰ and the EPV is then

1 00010qxv¹⁰+2 00010|10qxv²⁰.

Indicator random variables can also be used for continuous benefits. Here we consider indicators of the form

I(t<Tx≤t+dt) for infinitesimaldt, with associated probability

E[I(t<Tx≤t+dt)]=Pr[t<Tx≤t+dt]

=Pr[Tx>t] Pr[Tx<t+dt|Tx>t]

≈tpxμx+tdt.

Consider, for example, an increasing insurance policy with a death benefit of Txpayable at the moment of death. That is, the benefit is exactly equal to the number of years lived by an insured life from agexto his or her death. This is a continuous whole life insurance under which the benefit is a linearly increasing function.

To find the EPV of this benefit, we note that the payment may be made at any time, so we consider all the infinitesimal intervals(t,t+dt), and we sum over all these intervals by integrating fromt=0 tot= ∞.

First, we identify the amount, discount factor and probability for a benefit payable in the interval(t,t+dt). The amount ist, the discount factor ise^−δt; the probability that the benefit is paid in the interval(t,t+dt)is the probability that the life survives fromxtox+t, and then dies in the infinitesimal interval (t,t+dt), which for the purpose of constructing the integral istpxμx+tdt.

So, we can write the EPV of this benefit as ∞

0

t e^−δttpxμx+tdt. (4.32)

In actuarial notation we write this as(¯IA)¯ x. TheIhere stands for ‘increasing’

and the bar over theIdenotes that the increases are continuous.

An alternative approach to derivingequation (4.32)is to identify the present value random variable for the benefit, denoted byZ, say, in terms of the future lifetime random variable,

Z=Txe^−δTx. Then any moment ofZcan be found from

E[Z^k]= ∞ 0

t e^−δtk

tpxμx+tdt.

The advantage of the first approach is that it is very flexible and generally quick, even for very complex benefits.

If the policy term ceases after a fixed term ofnyears, the EPV of the death benefit is

(¯IA)¯ _x:n¹ = n

0

t e^−δttpxμx+tdt.

There are a number of other increasing or decreasing benefit patterns that are fairly common. We present several in the following examples.

Example 4.6 Consider an n-year term insurance policy issued to(x)under which the death benefit isk+1, payable at the end of the year of death, if death occurs between agesx+kandx+k+1, fork=0, 1, 2,. . .,n−1.

(a) Derive a formula for the EPV of the benefit using the first approach described, that is multiplying together the amount, the discount factor and the probability of payment, and summing for each possible payment date.

(b) Derive a formula for the variance of the present value of the benefit.

Solution 4.6 (a) If the benefit is paid at timek+1, then the benefit amount is

$(k+1), the discount factor isv^k+1. The probability that the benefit is paid atk+1 is the probability that the policyholder died in the year(k,k+1], which isk|qx. Summing over all the possible years of death, to the end of the policy term, we have the EPV of the death benefit is

n−1

k=0

v^k+1(k+1)k|qx. In actuarial notation the above EPV is denoted(IA)_x:n¹ .

If the termn is infinite this is a whole life version of the increasing annual policy, with benefitKx+1 paid at the end of the year of death. The EPV of the death benefit is denoted(IA)xwhere

4.6 Variable insurance benefits 129 (IA)x =^∞

k=0

v^k+1(k+1)k|qx.

(b) We must go back to first principles. First, we identify the random variable as

Z=

(Kx+1)v^Kx+1 ifKx<n,

0 ifKx≥n.

So

E[Z²]=

n−1

k=0

v^2(k+1)(k+1)²k|qx, and the variance is

V[Z]=

n−1

k=0

v^2(k+1)(k+1)²k|qx−

(IA)_x:n¹ 2

.

Example 4.7 A whole life insurance policy offers an increasing death benefit payable at the end of the quarter year of death. If(x)dies in the first year of the contract, then the benefit is 1, in the second year it is 2, and so on. Derive an expression for the EPV of the death benefit.

Solution 4.7 First, we note that the possible payment dates are ¹₄, ²₄, ³₄, . . . . Next, if(x)dies in the first year, then the benefit payable is 1, if death occurs in the second year the benefit payable is 2, and so on. Third, corresponding to the possible payment dates, the discount factors arev^1/4,v^2/4,. . ..

The probabilities associated with the payment dates are 1 4qx, 1

4|¹

4qx, 2 4|¹

4qx,

3 4|¹

4qx,. . ..

Hence, the EPV, which is denoted(IA⁽⁴⁾)x, can be calculated as

1

4qxv¹⁴ +¹

4|¹

4qxv²⁴ +²

4|¹

4qxv³⁴ +³

4|¹

4qxv¹ +2

1|¹

4qxv¹¹⁴ +₁¹

4|¹

4qxv¹²⁴ +₁²

4|¹

4qxv¹³⁴ +₁³

4|¹

4qxv² +3

2|¹

4qxv²¹⁴ +₂¹

4|¹

4qxv²²⁴ +₂²

4|¹

4qxv²³⁴ +₂³

4|¹

4qxv³ + · · ·

=A⁽⁴⁾¹

x:1 +21|A⁽⁴⁾¹

x:1 +32|A⁽⁴⁾¹

x:1 + · · · .

We now consider the case when the amount of the death benefit increases in geometric progression. This is important in practice because compound reversionary bonuses will increase the sum insured as a geometric progression.

Example 4.8 Consider ann-year term insurance issued to(x)under which the death benefit is paid at the end of the year of death. The benefit is 1 if death occurs between agesxandx+1, 1+jif death occurs between agesx+1 and x+2,(1+j)²if death occurs between agesx+2 andx+3, and so on. Thus, if death occurs between agesx+kandx+k+1, the death benefit is(1+j)^k fork=0, 1, 2,. . .,n−1. Derive a formula for the EPV of this death benefit.

Solution 4.8 Using the technique of multiplying together the amount, the discount factor and the probability of payment, and summing for each possible payment date, the EPV can be constructed as

v qx+(1+j)v²1|qx+(1+j)²v³2|qx+ · · · +(1+j)ⁿ⁻¹vⁿn−1|qx

=

n−1

k=0

v^k+1(1+j)^kk|qx

= 1 1+j

n−1

k=0

v^k+1(1+j)^k+1k|qx

= 1

1+jA_{x:n i}¹ _∗ where i^∗=1+i

1+j−1= i−j

1+j. (4.33)

The notation A_{x:n i}¹ _∗ indicates that the EPV is calculated using the rate of interesti^∗, rather thani. In most practical situations,i>jso thati^∗>0.

Example 4.9 Consider an insurance policy issued to(x)under which the death benefit is(1+j)^twheretis the time of death (from the inception of the policy).

The death benefit is payable immediately on death.

(a) Derive an expression for the EPV of the death benefit if the policy is an n-year term insurance.

(b) Derive an expression for the EPV of the death benefit if the policy is a whole life insurance.

Solution 4.9 (a) The present value of the death benefit is (1 +j)^Txv^Tx if Tx<n, and is zero otherwise, so that the EPV of the death benefit is

n

0

(1+j)^tv^ttpxμx+tdt= ¯A_{x:n i}¹ _∗ wherei^∗= 1+i 1+j−1.

4.8 Notes and further reading 131 (b) Similarly, if the policy is a whole life insurance rather than a term

insurance, then the EPV of the death benefit would be ∞

0

(1+j)^tv^ttpxμx+tdt=(A¯x)i^∗ wherei^∗= 1+i 1+j −1.