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	$\$4,000$	$\$6,000$

${}\end{center}Whichoptionwouldyouselectinordertohavethemaximumgain?Ofcourse,% ifthereisnointerest,bothoptionsareequal.Ifanynon-zerointerestratesareinvolved,% oneoptionmaybepreferablethantheother.\inner@par Bycalculatingthe\emph{present % values}oftheseoptions,onemaybeabletocomparethe``present^{\prime\prime}% valuesofthesepaymentsandfigureoutwhichisthepreferableoption.Sowhatisa\emph{% present value}?\inner@par\textbf{Definition}.Let$ P $b e t h e a m o u n t o f a p a y m e n t a t s o m e t i m e$ t¿0 $inthefuture.thenthe\emph{present value}$ PV(P) $o f$ P $i s s i m p l y t h e v a l u e o f t h i s p a y m e n t a t t i m e$ t=0 $. S p e c i f i c a l l y, i f t h e i n t e r e s t r a t e f r o m$ 0 $t o$ t $i s$ r $, t h e n$ $\operatorname{PV}(P)=\frac{P}{1+r}.$ $I n o t h e r w o r d s, i f w e i n v e s t$ PV(P) $t o d a y, e a r n i n g a n i n t e r e s t a t a r a t e o f$ r $b e t w e e n t i m e s$ 0 $a n d$ t $, t h e n a t t i m e$ t $, w e w o u l d h a v e m a d e$ P $.\inner@par Now,supposeintheexampleabove,% bothoptionshaveaneffectiveannualinterestrate(\texttt{http://planetmath.org/% InterestRate})of$ 5% $compoundedannually(\texttt{http://planetmath.org/CompoundInterest}),% thenthepresentvalueofoption1is$ $\frac{\$6,000}{1.05}+\frac{\$4,000}{(1.05)^{2}}\approx\$9,342.40$ $w h e r e a s t h e s e c o n d o p t i o n h a s p r e s e n t v a l u e$ $\frac{\$4,000}{1.05}+\frac{\$6,000}{(1.05)^{2}}\approx\$9,251.70$ $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 $asa``negative^{\prime\prime}return,$ A=-P_0=-PV(P_0) $, t h e n t h e n e t p r e s e n t v a l u e o f t h e i n v e s t m e n t c a n b e w r i t t e n$ $\operatorname{NPV}(I)=\operatorname{PV}(P_{0})+\operatorname{PV}(P_{1})+\cdots% +\operatorname{PV}(P_{n})=\sum_{i=0}^{n}\operatorname{PV}(P_{i}).$ ${{{Onewouldusuallywanttoinvestinsomethingwithapositivenetpresentvalue.% Netpresentvaluesarecommonlyusedwhenoneisinterestedincomparingcarloansorhomemortgages% .\begin{flushright}\begin{tabular}[]{|ll|}\hline Title&present value\\ Canonical name&PresentValue\\ Date of creation&2013-03-22 16:40:59\\ Last modified on&2013-03-22 16:40:59\\ Owner&CWoo (3771)\\ Last modified by&CWoo (3771)\\ Numerical id&9\\ Author&CWoo (3771)\\ Entry type&Definition\\ Classification&msc 91B28\\ Defines&net present value\\ Defines&future value\\ \hline}\end{tabular}}$}\end{flushright}\end{document}$