You are here
Homepresent value
Primary tabs
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  $\$4,000$  $\$6,000$ 
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 nonzero interest rates are involved, one option may be preferable than the other.
By calculating the 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 present value?
Definition. Let $P$ be the amount of a payment at sometime $t>0$ in the future. then the 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 effective annual interest rate of $5%$ compounded annually, 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.
Remarks.

Of course, the result will be the same if one instead computes the 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 future value is
$\operatorname{FV}(P)=P(1+r),$ if $r$ is the interest rate from $0$ to $t$.

An accompanying concept is that of the 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
$\operatorname{NPV}(I)=\Big(\operatorname{PV}(P_{1})+\operatorname{PV}(P_{2})+% \cdots+\operatorname{PV}(P_{n})\Big)A.$
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.
Mathematics Subject Classification
91B28 no label found Forums
 Planetary Bugs
 HS/Secondary
 University/Tertiary
 Graduate/Advanced
 Industry/Practice
 Research Topics
 LaTeX help
 Math Comptetitions
 Math History
 Math Humor
 PlanetMath Comments
 PlanetMath System Updates and News
 PlanetMath help
 PlanetMath.ORG
 Strategic Communications Development
 The Math Pub
 Testing messages (ignore)
 Other useful stuff
 Corrections