Time Value Of Money

Several calculations are crucial to the study of finance.
The unknown future value, F, at some terminal time, T, that we can designate as $F_T$ given an initial value of $P_0$ can be found if we know the annual interest rate of r with the following equation.

\begin{equation} F_T=P_0(1+r)^T \end{equation}

This equation can be restructured to solve for an unknown present value when a future value is known. Both sides of the equation are divided by $(1+r)^T$ and the equations order is reversed to produce the following.

\begin{align} P_0=\frac{F_T}{(1+r)^T} \end{align}

Often a series of equal payments must be evaluated. When such a series of payments is characterized by its receipt at the end of each interval (year) the series is designated an ordinary annuity. If the payments occur at the beginning of each year it is known as an annuity due. An ordinary annuity lasting T years with equal payments of Pmt would be expressed by the following equation.

\begin{equation} F_T=Pmt(1+r)^T+Pmt(1+r)^{T-1}+Pmt(1+r)^{T-2}+...+Pmt(1+r)^0 \end{equation}