## Annuity

An annuity is a series of equal payments made at equal intervals of time. Financial activities like installment payments, monthly rentals, life-insurance premium, monthly retirement benefits, are familiar examples of annuity.

Annuity can be certain or uncertain. In annuity certain, the specific amount of payments are set to begin and end at a specific length of time. A good example of annuity certain is the monthly payments of a car loan where the amount and number of payments are known. In annuity uncertain, the annuitant may be paid according to certain event. Example of annuity uncertain is life and accident insurance. In this example, the start of payment is not known and the amount of payment is dependent to which event.

Annuity certain can be classified into two, simple annuity and general annuity. In simple annuity, the payment period is the same as the interest period, which means that if the payment is made monthly the conversion of money also occurs monthly. In general annuity, the payment period is not the same as the interest period. There are many situations where the payment for example is made quarterly but the money compounds in another period, say monthly. To deal with general annuity, we can convert it to simple annuity by making the payment period the same as the compounding period by the concept of effective rates.

### Elements of Annuity

A = amount of periodic payment
P = present amount of all periodic payments
F = future worth of all periodic payments after the last payment is made
i = interest rate per compounding period
n = total number of payments
m = nominal rate (see compounded interest)
t = number of years

## Types of Simple Annuities

In engineering economy, annuities are classified into four categories. These are: (1) ordinary annuity, (2) annuity due, (3) deferred annuity, and (4) perpetuity. These four are actually simple annuities described above.

### Ordinary Annuity

In ordinary annuity, the equal payments are made at the end of each compounding period starting from the first compounding period. From the cash flow diagram shown above, the future amount F is the sum of payments starting from the end of the first period to the end of the nth period. Observe that the total number of payments is n and the total number of compounding periods is also n. Thus, in ordinary annuity, the number of payments and the number of compounding periods are equal.

Future amount of ordinary annuity, F

$F = \dfrac{A[ \, (1 + i)^n - 1 \, ]}{i}$

The factor   $\dfrac{(1 + i)^n - 1}{i}$   is called equal-payment-series compound-amount factor and is denoted by   $(F/A, \, i, \, n)$.

Present amount of ordinary annuity, P

$P = \dfrac{F}{(1 + i)^n} = \dfrac{A[ \, (1 + i)^n -1 \, ]}{(1 + i)^ni}$

The factor   $\dfrac{(1 + i)^n - 1}{(1 + i)^ni}$   is called equal-payment-series present-worth factor and is denoted by   $(P/A, \, i, \, n)$.

Periodic payment of annuity, A
Value of A if F is known:

$A = \dfrac{Fi}{(1 + i)^n - 1}$

The factor   $\dfrac{i}{(1 + i)^n - 1}$   is called equal-payment-series sinking-fund factor and is denoted by   $(A/F, \, i, \, n)$.

Value of A if P is known:

$A = \dfrac{P(1 + i)^ni}{(1 + i)^n - 1}$

The factor   $\dfrac{(1 + i)^ni}{(1 + i)^n - 1}$   is called equal-payment-series capital-recovery factor and is denoted by   $(A/P, \, i, \, n)$.

### Annuity Due

In annuity due, the equal payments are made at the beginning of each compounding period starting from the first period. The diagram below shows the cash flow in annuity due. As indicated in the figure above, F1 is the sum of ordinary annuity of n payments. The future amount F of annuity due at the end of nth period is one compounding period away from F1. In symbol, F = F1(1 + i).

Future amount of annuity due, F

$F = F_1(1 + i) = \dfrac{A[ \, (1 + i)^n -1 \, ]}{i}(1 + i)$

Present amount of annuity due, P

$P = \dfrac{F}{(1 + i)^n} = \dfrac{A[ \, (1 + i)^n -1 \, ]}{(1 + i)^ni}(1 + i)$

### Deferred Annuity

In deferred annuity the first payment is deferred a certain number of compounding periods after the first. In the diagram below, the first payment was made at the end of the kth period and n number of payments was made. The n payments form an ordinary annuity as indicated in the figure. Future amount of deferred annuity, F

$F = \dfrac{A[ \, (1 + i)^n -1 \, ]}{i}$

Present amount of deferred annuity, P

$P = \dfrac{F}{(1 + i)^{k + n}} = \dfrac{A[ \, (1 + i)^n -1 \, ]}{(1 + i)^{k + n}i}$

### Perpetuity

Perpetuity is an annuity where the payment period extends forever, which means that the periodic payments continue indefinitely. There is no definite future in perpetuity, thus, there is no formula for the future amount.

Present amount of perpetuity, P
From the present amount of ordinary annuity:
$P = \dfrac{A[ \, (1 + i)^n -1 \, ]}{(1 + i)^ni}$

$P = \dfrac{A(1 + i)^n}{(1 + i)^ni} - \dfrac{A}{(1 + i)^ni}$

$P = \dfrac{A}{i} - \dfrac{A}{(1 + i)^ni}$

When   $n \to \infty$,   $\dfrac{A}{(1 + i)^ni} \to 0$.   Thus,

$P = \dfrac{A}{i}$

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