What Is an Annuity Factor?

Figure the annuity factor

An annuity is a financial instrument that pays income in a series of regular payments in return for an initial capital investment. The payments accumulate interest up until the time they are paid out. An annuity factor is the present value of an income stream that generates one dollar of income each period for a specified number of periods. The annuity factor can therefore be multiplied by the periodic annuity payment to determine the present value of the remaining annuity payments.

Present Value

The present value of an annuity is the value of all future payments at the current time, or before they have earned interest for the investor receiving the payments. Therefore, to determine the present value of future payments, the interest they have earned by the time of their scheduled payout has to be backed out.

Payments earn interest up until the time they are paid out.

For example, the first future payment's present value is P/(1+i), if P is the payout amount and i is the interest accumulated in one time period. If the interest is compounded, then the present value of the second future payment is P/(1+i)^2.

Present Value of an Annuity

If there are n future payments left, the total present value, PV, of an annuity is therefore the accumulation of all payments reduced by the amount of interest they have earned by the time of payment:

PV = P/(1+i) + P/(1+i)^2 + P/(1+i)^3 + P/(1+i)^4 + ... + P/(1+i)^n

A Useful Formula

Because n tends to be a high number, a useful formula for accumulating these terms would be helpful.

Note that

(1 - x) * (1 + x + x^2 + x^3 + ... + x^n) = 1 - x ^ (n+1).

Why?

Because all terms in (1 + x + x^2 + x^3 + ... + x^(n-1)) are multiplied by -x

and (x + x^2 + x^3 + ... + x^n) is multiplied by 1.

Adding these two gives zero.

So all that remains is 1 - x ^ (n+1).

So now we have a simple formula for 1 + x + x^2 + x^3 + ... + x^n, which can save much time if n is large.

Present Value Formula

PV = P/(1+i) + P/(1+i)^2 + P/(1+i)^3 + P/(1+i)^4 + ... + P/(1+i)^n

can now be rewritten as

P/(1+i) * [ 1 + P/(1+i) + P/(1+i)^2 + P/(1+i)^3 + P/(1+i)^4 + ... + P/(1+i)^(n-1) ] (to make the first term 1; notice the change to n-1)

P/(1+i) * [1 - 1/(1+i)^n] / [1- 1/(1+i)]

Multiplying through the denominator of P gives

P * [1 - 1/(1+i)^n] / i

Note that this formula is not applicable if i=0, but then if i=0 in the first place, solving for PV was trivial.

Annuity Factor Formula

Therefore, the formula for the annuity factor is [1 - 1/(1+i)^n] / i, because by definition, the annuity factor is what is multiplied by P to get PV.

Notation

The annuity factor is represented by a small a. It has two subscripts, the number of payments, boxed apart from the interest. It is pronounced a-angle-n-at-i.

References

• Stephen G. Kellison; The Theory of Interest; 2000