Annual life annuities

The annual life annuity is paid once each year, conditional on the survival of a life (theannuitant) to the payment date. If the annuity is to be paid throughout the annuitant’s life, it is called a whole life annuity. If there is to be a specified maximum term, it is called a term or temporary annuity.

Annual annuities are quite rare. We would more commonly see annuities payable monthly or even weekly. However, the annual annuity is still important in the situation where we do not have full information about mortality between integer ages, for example because we are working with an integer age life table.

Also, the development of the valuation functions for the annual annuity is a good starting point before considering more complex payment patterns.

5.4 Annual life annuities 109 As with the insurance functions, we are primarily interested in the EPV of a cash flow, and we also identify the present value random variables in terms of the future lifetime random variables from Chapters2and4, specifically,Tx,Kx

andKx^(m).

5.4.1 Whole life annuity-due

Consider first an annuity of 1 per year payable annuallyin advancethrough- out the lifetime of an individual now agedx. The life annuity with payments in advance is known as awhole life annuity-due. The first payment occurs immediately, the second in one year from now, provided that(x)is alive then, and payments follow at annual intervals with each payment conditional on the survival of(x)to the payment date. In Figure5.1we show the payments and associated probabilities and discount functions in a time-line diagram.

We note that if(x)were to die between agesx+kandx+k+1, for some positive integerk, then annuity payments would be made at times 0, 1, 2,. . .,k, for a total ofk+1 payments. We definedKxsuch that the death of(x)occurs betweenx+Kxandx+Kx+1, so, the number of payments isKx+1, including the initial payment. This means that, fork =0, 1, 2,. . ., the present value of the annuity isa¨_k+1 ifKx = k. Thus, using equation (5.1), the present value random variable for the annuity payment series,Y, say, can be written as

Y = ¨a_K

x+1 =1−v^Kx+1

d .

There are three useful ways to derive formulae for calculating the expected value of this present value random variable.

First, the mean and variance can be found from the mean and variance of v^Kx+1, which were derived in Section4.4.2. For the expected value ofY, which

Time 0 1 2 3

. . .

Amount 1 1 1 1

Discount 1 v v² v³

Probability 1 px 2px 3px

Figure 5.1 Time-line diagram for whole life annuity-due.

is denoteda¨x, we have

¨ ax =E

1−v^Kx+1 d

= 1−E[v^Kx+1]

d .

That is,

¨

ax= 1−Ax

d . (5.4)

This is a useful approach, as it also immediately gives us the variance ofY as V[Y] =V

1−v^Kx+1 d

= 1

d²V[v^Kx+1]

= ²Ax−A²_x

d² . (5.5)

Secondly, we may use the indicator random variable approach from Section 4.6. The condition for the payment atk, say, is that(x)is alive at agex+k, that is, thatTx>k. The present value random variable can be expressed as

Y =I(Tx>0)+vI(Tx>1)+v²I(Tx>2)+v³I(Tx>3)+ · · · (5.6) and the EPV of the annuity is the sum of the expected values of the individual terms. Recall that E[I(Tx>t)] =Pr[Tx>t] =tpx, so that

¨

ax=1+vpx+v²2px+v³3px+ · · ·, that is

¨ ax=^∞

k=0

v^kkpx. (5.7)

This is a very useful equation fora¨x. However, this approach does not lead to useful expressions for the variance and higher moments ofY. This is because the individual terms in expression (5.6) are dependent random variables.

Finally, we can work from the probability function for Kx, that is using Pr[Kx=k] =k|qx, so that

¨ ax =^∞

k=0

¨

a_k+1k|qx. (5.8)

5.4 Annual life annuities 111 This is less used in practice than equations (5.4) and (5.7). The difference between the formulations fora¨xin equations (5.7) and (5.8) is that in equation (5.7) the summation is taken over the possible payment dates, and in equation (5.8) the summation is taken over the possible years of death.

Example 5.1Show that equations (5.7) and (5.8) are equivalent – that is, show that

∞ k=0

¨

a_k+1 k|qx=^∞

k=0

v^kkpx.

Solution 5.1We can show this by using

¨ a_k+1 =

k t=0

v^t

and

∞ k=t

k|qx=^∞

k=t

(kpx−k+1px)=tpx.

Then

∞ k=0

¨

a_k+1k|qx = ∞ k=0

k t=0

v^tk|qx

=qx+(1+v)1|1qx+(1+v+v²)2|1qx

+(1+v+v²+v³)3|1qx+ · · ·.

Changing the order of summation on the right-hand side (that is, collecting together terms in powers ofv) gives

∞ k=0

k t=0

v^tk|qx=^∞

t=0

∞ k=t

v^tk|qx

=^∞

t=0

v^t∞

k=t k|qx

= ∞ t=0

v^ttpx

as required. 2

5.4.2 Term annuity-due

Now suppose we wish to value a term annuity-due of 1 per year. We assume the annuity is payable annually to a life now agedxfor a maximum ofnyears.

Thus, payments are made at timesk=0, 1, 2,. . .,n−1, provided that(x)has survived to agex+k. The present value of this annuity isY, say, where

Y = a¨_K

x+1 ifKx=0, 1,. . .,n−1,

¨

a_n ifKx≥n.

that is

Y = ¨a_min(K

x+1,n) = 1−v^min(Kx+1,n)

d .

The EPV of this annuity is denoteda¨_x:n.

We have seen the random variablev^min(Kx+1,n)before, in Section4.4.7, where the EPVA_x:n is derived. Thus, the EPV of the annuity can be determined as

¨

a_x:n =E[Y] = 1−E[v^min(Kx+1,n)] d

that is,

¨

ax:n = 1−A_x:n

d . (5.9)

The time-line for the term annuity-due cash flow is shown in Figure5.2. Notice that, because the payments are made in advance, there is no payment due at timen, the end of the annuity term.

Using Figure5.2, and summing the EPVs of the individual payments, we have

¨

a_x:n =1+vpx+v²2px+v³3px+ · · · +vⁿ⁻¹n−1px

Time 0 1 2 3

. . . .

n-1 n

Amount 1 1 1 1 1

Discount 1 v v² v³ vⁿ⁻¹

Probability 1 px 2px 3px n−1px

Figure 5.2 Time-line diagram for term life annuity-due.

5.4 Annual life annuities 113 that is

¨ ax:n =

n−1

t=0

v^ttpx. (5.10)

Also, we can write the EPV as

¨ ax:n =

n−1

k=0

¨

a_k+1k|qx+npxa¨n

adapting equation (5.8) above. The second term here arises from the second term in the definition ofY – that is, if the annuitant survives for the full term, then the payments constitute ann-year annuity.

5.4.3 Whole life immediate annuity

Now consider a whole life annuity of 1 per year payable in arrear, conditional on the survival of(x)to the payment dates. We use the term immediate annuity to refer to an annuity under which payments are made at the end of the time periods, rather than at the beginning. The actuarial notation for the EPV of this annuity isax, and the time-line for the annuity cash flow is shown in Figure5.3.

LetY^∗denote the present value random variable for the whole life immediate annuity. Using the indicator random variable approach we have

Y^∗=vI(Tx>1)+v²I(Tx>2)+v³I(Tx>3)+v⁴I(Tx>4)+ · · ·. We can see from this expression and from the time-line, that the difference in present value between the annuity-due and the immediate annuity payable in arrear is simply the first payment under the annuity-due, which, under the annuity-due, is assumed to be paid at timet=0 with certainty.

Time 0 1 2 3

. . .

Amount 1 1 1

Discount v v² v³

Probability px 2px 3px

Figure 5.3 Time-line diagram for whole life immediate annuity.

So, ifY is the random variable for the present value of the whole life annuity payable in advance, andY^∗is the random variable for the present value of the whole life annuity payable in arrear, we haveY^∗ = Y −1, so that E[Y^∗] = E[Y] −1, and hence

ax= ¨ax−1. (5.11)

Also, from equation (5.5) and the fact thatY^∗=Y −1, we have V[Y^∗] =V[Y] = ²Ax−A²_x

d² .

5.4.4 Term immediate annuity

The EPV of a term immediate annuity of 1 per year is denoteda_x:n. Under this annuity payments of 1 are made at timesk = 1, 2,. . .,n, conditional on the survival of the annuitant.

The random variable for the present value is Y =a_min(K

x,n) ,

and the time-line for the annuity cash flow is given in Figure5.4.

Summing the EPVs of the individual payments, we have

a_x:n =vpx+v²2px+v³3px+ · · · +vⁿnpx= n

t=1

v^ttpx. (5.12)

The difference between the annuity-due EPV,a¨_x:n, and the immediate annuity EPV,a_x:n, is found by differencing equations (5.10) and (5.12), to give

¨

a_x:n −a_x:n =1−vⁿnpx

Time 0 1 2 3

. . . .

n-1 n

Amount 1 1 1 1 1

Discount v v² v³ vⁿ⁻¹ vⁿ

Probability px 2px 3px n−1px npx

Figure 5.4 Time-line diagram for term life immediate annuity.

5.5 Annuities payable continuously 115 so that

a_x:n = ¨a_x:n −1+vⁿnpx. (5.13) The difference comes from the timing of the first payment under the annuity due and the last payment under the immediate annuity.

5.5 Annuities payable continuously