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.
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.
Euler's number or Euler's constant e is expressed as a limit:
$$e=\lim_{n\rightarrow \infty }\left ( 1+\frac{1}{n} \right )^{n}=2,71828\; 18284\; 59045\; 23536... $$
Leonhard Euler (April 15, 1707, Basel – September 18, 1783, St. Petersburg) was a Swiss mathematician and physicist who spent much of his life in Russia, in St. Petersburg, and in Germany, in Berlin. Euler proved that e is an irrational number and computed the first 18 digits of the constant in 1748.