PlanetMath: present value

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present value

(Definition)

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 non-zero 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 be the amount of a payment at sometime in the future. then the present value $\operatorname{PV}(P)$ of is simply the value of this payment at time . Specifically, if the interest rate from 0 to is , then

$\displaystyle \operatorname{PV}(P)=\frac{P}{1+r}.$

In other words, if we invest $\operatorname{PV}(P)$ today, earning an interest at a rate of

between times 0 and

, then at time

, we would have made

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

$\displaystyle \frac{\$6,000}{1.05}+\frac{\$4,000}{(1.05)^2}\approx \$9,342.40$

whereas the second option has present value

$\displaystyle \frac{\$4,000}{1.05}+\frac{\$6,000}{(1.05)^2}\approx \$9,251.70$

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 : if payment is valued at at time 0, its value at some future time , or its future value is
$\displaystyle \operatorname{FV}(P)=P(1+r),$
if is the interest rate from 0 to .
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 is made where an initial amount of is made at time 0, and payments $P_1,\ldots,P_n$ are returns as a result of this investment. Then
$\displaystyle \operatorname{NPV}(I)=\Big(\operatorname{PV}(P_1)+\operatorname{PV}(P_2)+\cdots +\operatorname{PV}(P_n)\Big)-A.$

If we treat the initial invsetment

as a “negative” return, $A=-P_0=-\operatorname{PV}(P_0)$ , then the net present value of the investment can be written

$\displaystyle \operatorname{NPV}(I)=\operatorname{PV}(P_0)+\operatorname{PV}(P_1)+\cdots +\operatorname{PV}(P_n)=\sum_{i=0}^n \operatorname{PV}(P_i).$

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.

"present value" is owned by CWoo.

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Also defines:

net present value, future value

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Cross-references: positive, interest rates, interest, order
There are 4 references to this entry.

This is version 6 of present value, born on 2007-02-09, modified 2007-12-18.
Object id is 8893, canonical name is PresentValue.
Accessed 1866 times total.

Classification:

AMS MSC:

91B28 (Game theory, economics, social and behavioral sciences :: Mathematical economics :: Finance, portfolios, investment)

Pending Errata and Addenda

None.

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