# present value

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

options year 1 year 2
option 1 $\6,000$ $\4,000$
option 2 $\4,000$ $\6,000$

${}\end{center}Whichoptionwouldyouselectinordertohavethemaximumgain?Ofcourse,% ifthereisnointerest,bothoptionsareequal.Ifanynon-zerointerestratesareinvolved,% oneoptionmaybepreferablethantheother.\inner@par Bycalculatingthe\emph{present % values}oftheseoptions,onemaybeabletocomparethepresent^{\prime\prime}% valuesofthesepaymentsandfigureoutwhichisthepreferableoption.Sowhatisa\emph{% present value}?\inner@par\textbf{Definition}.Let$P$betheamountofapaymentatsometime$t¿0$inthefuture.thenthe\emph{present value}$PV(P)$of$P$issimplythevalueofthispaymentattime$t=0$.Specifically,iftheinterestratefrom$0$to$t$is$r$,then$$\operatorname{PV}(P)=\frac{P}{1+r}.$$Inotherwords,ifweinvest$PV(P)$today,earninganinterestatarateof$r$betweentimes$0$and$t$,thenattime$t$,wewouldhavemade$P$.\inner@par Now,supposeintheexampleabove,% bothoptionshaveaneffectiveannualinterestrate(\texttt{http://planetmath.org/% InterestRate})of$5%$compoundedannually(\texttt{http://planetmath.org/CompoundInterest}),% thenthepresentvalueofoption1is$$\frac{\6,000}{1.05}+\frac{\4,000}{(1.05)^{2}}\approx\9,342.40$$whereasthesecondoptionhaspresentvalue$$\frac{\4,000}{1.05}+\frac{\6,000}{(1.05)^{2}}\approx\9,251.70$$Clearly,thefirstoptionissuperiorthanthesecondone.\inner@par\textbf{Remarks}.% \begin{itemize} \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. \itemize@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}Ifwetreattheinitialinvsetment$A$asanegative^{\prime\prime}return,$A=-P_0=-PV(P_0)$,thenthenetpresentvalueoftheinvestmentcanbewritten$$\operatorname{NPV}(I)=\operatorname{PV}(P_{0})+\operatorname{PV}(P_{1})+\cdots% +\operatorname{PV}(P_{n})=\sum_{i=0}^{n}\operatorname{PV}(P_{i}).$${{{Onewouldusuallywanttoinvestinsomethingwithapositivenetpresentvalue.% Netpresentvaluesarecommonlyusedwhenoneisinterestedincomparingcarloansorhomemortgages% .\begin{flushright}\begin{tabular}[]{|ll|}\hline Title&present value\\ Canonical name&PresentValue\\ Date of creation&2013-03-22 16:40:59\\ Last modified on&2013-03-22 16:40:59\\ Owner&CWoo (3771)\\ Last modified by&CWoo (3771)\\ Numerical id&9\\ Author&CWoo (3771)\\ Entry type&Definition\\ Classification&msc 91B28\\ Defines&net present value\\ Defines&future value\\ \hline}\end{tabular}}}\end{flushright}\end{document}$