Annuities, Perpetuities, and Compounding

An annuity consists of equal payments in equal time, as opposed to a mixed and varied stream of cash flows. This unique character of annuities makes it possible to compute infinite numbers of such payments in one calculation, whereas mixed streams require a separate calculation for each payment to determine the present or future value of the entire stream.

The mathematics of determining the single factor for the future values of a series of equal payments is the sum of the individual future value factors. For example, the single factor to determine five annual equal investments of $500 each year compounding at 8% annually would be computed as follows:

1+(1.08)+(1.08)^2+(1.08)^3+(1.08)^4= 5.8666, multiplied by $500, equals $2933.30, or the total value

This calculation assumes that the first payment is received at the end of the first year and the last payment is received on the last day of the last year. This type of annuity is called an ordinary annuity.If the first payment was received on the first day of the first year there would be one more compounding period:

(1.08)+(1.08)^2+(1.08)^3+(1.08)^4+(1.08)^5 =6.3359, multiplied by $500, equals $3167.95

This type of an annuity is called an annuity due. As you can see, the difference in the final sum is significant between the two types of annuities. While most investments do not produce cash until the end of a given period, there are several investment, such as rents on an apartment building, where the cash flow comes at the beginning of the period, and which can significantly impact total future value.

On the present value side, all true interest lending, requiring equal payments, is computed by using the present value of an ordinary annuity. Solving for the equal payment on a mortage or a loan requires the same basic calculation, except we would now use present value of an annuity computations. Lets assume a 12% annual rate on a 6 month, equal payment loan for $1000. The present value factor is the sum of the individual present value factors as follows:

(1/(1.02))+(1/(1.02)^2)+(1/(1.02)^3)+(1/(1.02)^4)+(1/(1.02)^5) +(1/(1.02)^6)= 5.60143.

We then divide that factor into the amount of the loan to get the equal payments which will pay all the interest and all the principal in 6 equal payments:

$1000/5.60143= $178.53

Thus we see the importance of the annuity concept in all equal payment lending transactions.

The concept of perpetuity is a related idea having to do with the amount of money necessary to produce an equal flow of cash to infinity. For example, suppose one planned to provide a scholarship to a student of $10,000 forever. How much money would have to be under investment at 5%, for example, to provide that scholarship, without ever touching the principal?

The computation is straightforward: the annuity divided by the rate of return equals the principal required. For our example above, the calculation would be:

$10,000/.05= $200,000.

Therefore if $200,000 were invested in government bonds at 5%, an annual income of $10,000 would be provided into perpetuity and the principal would remain unchanged.

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