Suppose you are going to receive $\$ 10,000$, to be paid in two payments at the end of the next two years. You have the following two options \begin{center} \begin{tabular}{|c||c|c|} \hline options & year 1 & year 2 \\ \hline\hline option 1 & $ $6,000$ & $ $4,000$ \\ \hline option 2 & $ $4,000$ & $ $6,000$ \\ \hline \end{tabular} \end{center} Which option would you select in \htmladdnormallink{order}{http://planetmath.org/encyclopedia/UniformizingElement.html} to have the maximum gain? Of course, if there is no \htmladdnormallink{interest}{http://planetmath.org/encyclopedia/Interest.html}, both options are equal. If any non-zero \htmladdnormallink{interest rates}{http://planetmath.org/encyclopedia/InstantaneousInterestRate.html} are involved, one option may be preferable than the other. By calculating the \emph{present values} of these options, one may be able to compare the ``present'' values of these payments and figure out which is the preferable option. So what is a \emph{present value}? {Definition}. Let $ P$ be the amount of a payment at sometime $ t>0$ in the future. then the \emph{present value} $ PV(P)$ of $ P$ is simply the value of this payment at time $ t=0$. Specifically, if the interest rate from $ 0$ to $ t$ is $ r$, then $ $\operatorname{PV}(P)=\frac{P}{1+r}.$ $ In other words, if we invest $ PV(P)$ today, earning an interest at a rate of $ r$ between times $ 0$ and $ t$, then at time $ t$, we would have made $ P$. Now, suppose in the example above, both options have an \htmladdnormallink{effective annual interest rate}{http://planetmath.org/encyclopedia/InterestRate.html} of $ 5%$ \htmladdnormallink{compounded annually}{http://planetmath.org/encyclopedia/CompoundInterest.html}, then the present value of option 1 is $ $\frac{\$ 6,0001.05+$4,000(1.05)^2&ap#approx;$9,342.40$$ whereas the second option has present value $$ $4,0001.05+$6,000(1.05)^2&ap#approx;$9,251.70$$ Clearly, the first option is superior than the second one. \textbf{Remarks}. \begin{itemize} \item Of course, the result will be the same if one instead computes the \emph{future values} of these options, which are the values of the payments at a specific future time $t>0$: if payment is valued at $P$ at time $0$, its value at some future time $t>0$, or its \emph{future value} is $$ FV(P)=P(1+r),$$ if $r$ is the interest rate from $0$ to $t$. \item An accompanying concept is that of the \emph{net present value} $\operatorname{NPV}$. It is the present value of all the future payments minus the initial investment: suppose an investment $I$ is made where an initial amount of $A$ is made at time $0$, and payments $P_1,\ldots,P_n$ are returns as a result of this investment. Then $$ NPV(I)=(PV(P_1)+PV(P_2)+&cdots#cdots;+PV(P_n))-A.$$ \end{itemize} If we treat the initial invsetment $A$ as a ``negative'' return, $A=-P_0=-\operatorname{PV}(P_0)$, then the net present value of the investment can be written $$ NPV(I)=PV(P_0)+PV(P_1)+&cdots#cdots;+PV(P_n)=&sum#sum;_i=0^n PV(P_i).$$ One would usually want to invest in something with a \htmladdnormallink{positive}{http://planetmath.org/encyclopedia/Negative.html} net present value. Net present values are commonly used when one is interested in comparing car loans or home mortgages. \end{document} $$