# Present Value of Annuity

The present value of an annuity. An annuity is a series of periodic constant cash flows, either received or paid out, lasting for a fixed period. Ordinary annuity payments occur at the end of each payment period. The present value of an ordinary annuity is found using the following formula:

$$PV_{Ordinary\; Annuity}=C \times \left[\frac{1-(1+r)^{-t}}{r}\right]$$

C— periodic payment amount;
r— periodic discount rate;
t— number of periods.

Annuity due payments occur at the beginning of each payment period. Examples include rent or lease payments. The present value of an annuity due is calculated using the following formula:

$$PV_{Annuity\; Due}=C \times \left[\frac{1-(1+r)^{-t}}{r}\right]\times(1+r)$$

C— periodic payment amount;
r— periodic discount rate;
t— number of periods.

### Present Value of a Growing Annuity

Growing annuity is a series of periodic cash flows that lasts for a fixed period and increases at a constant rate each year. The present value of a growing annuity can be calculated using the formula:

$$PV=\frac{C_{0}}{r-g} \times \left[1-\left(\frac{1+g}{1+r}\right)^{t}\right]$$

C0— initial payment;
g— growth rate;
r— discount rate;
t— number of periods.

$$e=\lim_{n\rightarrow \infty }\left ( 1+\frac{1}{n} \right )^{n}=2,71828\; 18284\; 59045\; 23536...$$