PresentValue 2007-02-09 17:02:44 2007-12-18 10:26:13 Definition net present value future value \usepackage{amssymb,amscd} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{tabls} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics \usepackage{xypic} \usepackage{pst-plot} \usepackage{psfrag} % define commands here 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 order to have the maximum gain? Of course, if there is no interest, both options are equal. If any non-zero interest rates 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}? \textbf{Definition}. Let $P$ be the amount of a payment at sometime $t>0$ in the future. then the \emph{present value} $\operatorname{PV}(P)$ 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 $\operatorname{PV}(P)$ 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 \PMlinkname{effective annual interest rate}{InterestRate} of $5\%$ \PMlinkname{compounded annually}{CompoundInterest}, then the present value of option 1 is $$\frac{\$6,000}{1.05}+\frac{\$4,000}{(1.05)^2}\approx \$9,342.40$$ whereas the second option has present value $$\frac{\$4,000}{1.05}+\frac{\$6,000}{(1.05)^2}\approx \$9,251.70$$ 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 $$\operatorname{FV}(P)=P(1+r),$$ 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 $$\operatorname{NPV}(I)=\Big(\operatorname{PV}(P_1)+\operatorname{PV}(P_2)+\cdots +\operatorname{PV}(P_n)\Big)-A.$$ \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 $$\operatorname{NPV}(I)=\operatorname{PV}(P_0)+\operatorname{PV}(P_1)+\cdots +\operatorname{PV}(P_n)=\sum_{i=0}^n \operatorname{PV}(P_i).$$ One would usually want to invest in something with a positive net present value. Net present values are commonly used when one is interested in comparing car loans or home mortgages.