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 awhole life annuity. If there is to be a specified maximum term, it is called aterm annuityortemporary annuity.

5.4 Annual life annuities 143 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.

As with the insurance functions, we are primarily interested in the EPV of a cash flow, and we identify present value random variables for the annuity payments in terms of the future lifetime random variables from Chapters 2 and4,specifically,Tx,KxandK_x^(m).

5.4.1 Whole life annuity-due

Consider first a whole life annuity-due with annual payments of 1 per year, which depend on the survival of a life currently aged x. The first payment occurs immediately, the second payment is made 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 Figure 5.1 we show the payments and associated probabilities and discount functions in a time-line diagram.

If (x)dies between ages x+k andx+k+1, for some integer k ≥ 0, then annuity payments would be made at times 0, 1, 2,. . .,k, for a total of k+1 payments. From the definition of the curtate future lifetimeKx, we know that(x)dies between agesx+Kxandx+Kx+1, so the number of annuity payments isKx+1, including the initial payment. This means that the random variable representing the present value of the whole life annuity-due for(x)is Y = ¨a_K

x+1, and usingequation (5.1), we have Y = ¨a_K

x+1 = 1−v^Kx+1

d .

The expected value ofYis denoteda¨x.

Time 0 1 2 3

. . .

v

v v^{2 3}

Amount 1 1 1 1

Discount 1

Probability 1 p_x 2px ₃px

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

There are three useful ways to derive formulae for evaluatinga¨xfor a given survival model. We describe each of them below.

Using the insurance present value random variable,v^Kx+1 The expected value ofYis

¨ ax=E

!1−v^Kx+1 d

"

=1−E[v^Kx+1]

d .

Using the mean ofv^Kx+1which was derived inSection 4.4.2, we have

¨

ax=1−Ax

d . (5.3)

Similarly, we can immediately obtain the variance ofYfrom the variance of v^Kx+1as

V[Y]=V

!

1−v^Kx+1 d

"

= 1 d²V

v^Kx+1

= ²Ax−A²_x

d² . (5.4)

Summing EPVs of individual payments

InSection 4.4.2we stated that the EPV of any life contingent benefit can be found by considering each time point at which a benefit could be paid, and summing over all these time points the product of

(1) the amount of the benefit,

(2) the appropriate discount factor, and

(3) the probability that the benefit will be paid at that time.

For the annuity EPV, this approach is very helpful. Consider time t, where t = 0, 1, 2,. . .. There will be an annuity payment at timetif(x)survives to timet. The amount of the annuity payment, in this case, is 1. The discount factor for a payment made at timetisv^t, and the probability that the payment is made istpx, as it is contingent on the survival of(x)to agex+t. Note that v⁰=1 and0px=1. Then, summing over all possible payment times, we have

¨

ax=1+v px+v²2px+v³3px+ · · · =^∞

t=0

v^ttpx. (5.5) Equation (5.5)is the approach most commonly used in practice for evaluating

¨

ax. However, it does not lead to useful expressions for higher moments of the present value random variable. Each term in equation (5.5) is the EPV

5.4 Annual life annuities 145 of a single survival benefit of 1 at timet. That is, we can write the annuity present value random variableYas the sum of pure endowment present random variables,Zt, as

Y=Z0+Z1+Z2+ · · · , where

Zt=

v^t ifKx≥t, 0 ifKx<t, with

E[Zt]=v^ttpx and V[Zt]=v^2ttpx− v^ttpx

2

.

As the expected value of a sum of random variables is the sum of the expected values, we have

E[Y]=E[Z1]+E[Z2]+E[Z3]+ · · · ,

which gives us (5.5). However, we cannot obtain the variance of a sum of random variables by summing the variances unless the random variables are independent, and in our case they are not. We can easily see this if we consider, say,Z2 andZ3. If we are given the information that Z2 = 0, then we know that (x) died before time t = 2, which means we know that Z3 (and all subsequentZt’s) will also be zero, which means that the random variables are not independent.

Using the probability function forKx

We know that Pr[Kx=k]=k|qx, so that

¨ ax=E

a¨_K

x+1

=^∞

k=0

¨

a_k+1k|qx. (5.6) We can also use this approach to determine the variance of the annuity present value, although it does not reduce to a nice analytic form. We have

E (¨a_K

x+1)²

= ∞ k=0

(¨a_k+1)²k|qx

and hence

V a¨_K

x+1

=E (¨a_K

x+1)²

−(¨ax)².

Equation (5.6)is less often used in practice thanequations (5.3)and (5.5).

It is useful though to recognize the difference between the formulations fora¨x

inequations (5.5)and (5.6). Inequation (5.5)the summation is taken over all

the possible payment times; in (5.6) the summation is taken over the possible years of death.

Example 5.1 Show that equations (5.5) and (5.6) are equivalent – that is, show that

∞ k=0

¨

a_k+1 k|qx=^∞

k=0

v^kkpx. Solution 5.1 We 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.

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.

5.4 Annual life annuities 147 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,

¨

an 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, inSection 4.4.7. It is the present value of a unit benefit under ann-year endowment insurance, with death benefit payable at the end of the year of death. Its EPV isAx:n, so the EPV of the annuity is

¨

a_x:n =E[Y]= 1−E[v^min(Kx+1,n)]

d ,

that is,

¨

a_x:n = 1−Ax:n

d . (5.7)

The time-line for the term annuity-due cash flow is shown inFigure 5.2.

Notice that, because the payments are made in advance, there is no payment due at timen, the end of the annuity term.

UsingFigure 5.2,and summing the EPVs of the individual payments, we have

¨

ax:n =1+v px+v²2px+v³3px+ · · · +vⁿ⁻¹n−1px=

n−1

t=0

v^ttpx. (5.8)

Time 0 1 2 3

. . . .

n-1 n

Amount 1 1 1 1 1

Discount 1 v v² v³ vⁿ⁻¹

n−1p_x

Probability 1 px 2px 3px

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

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.

Using the third approach from the previous section, we can adapt equation (5.6) to write the EPV as

¨ a_x:n =

n−1

k=0

¨

a_k+1 k|qx+npxa¨_n.

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 an n-year annuity.

5.4.3 Immediate life annuities

Awhole life immediate annuityof 1 per year, under which the payments are at the end of each year rather than the beginning, is illustrated inFigure 5.3.

The actuarial notation for the EPV of this annuity isax.

We can see from the time-line that the difference in present value between the annuity-due and the immediate annuity is simply the first payment under the annuity-due, which is assumed to be paid at timet=0, with certainty.

So, ifYis 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. (5.9)

Also, fromequation (5.4)and the fact thatY^∗=Y−1, we have V[Y^∗]=V[Y]= ²Ax−A²_x

d² . (5.10)

The EPV of ann-year term immediate annuity of 1 per year is denoted a_x:n. Under this annuity payments of 1 are made at times k = 1, 2,. . .,n, conditional on the survival of the annuitant.