Present Value Is An Important Financial Concept

It forms the basis for understanding and evaluating all financial products characterized by constant streams of payments. But first you ask, what is a "stream of payment"? The term refers to payments that come in periodically over some time.

For example, a check of a fixed amount from your CD earnings paid out every month for a year period is a stream of payment. Other examples include salaries, membership dues, mortgages, lottery winnings, bank account and CD interest payments, and of course annuities. The difficulty with income streams is the comparison part. If someone offers you money periodically paid out at certain intervals, how do you compare it to another payment schedule and amount, or how do you equate to a sum of money at present? Someone evaluating two jobs with varying payouts over time is advised to calculate the present value of each job for level comparisons.

A Real Simple Example To Illustrate The Concept Of Present Value

Let us begin with a very simple example: suppose you are to receive two future payments of $1000 each. The first payment of $1000 will come in 12 months, and the second payment of $1000 in 24 months. The naive assumption is that you are going to be receiving $2000 total, but we will now learn that this naive assumption is wrong if you consider the concept of present value.

Money Promised In The Future Is Worth Less Than Money In Your Hands Now

The basic reason why you are not going to receive $2000 is because money promised in the future is not worth the same if received now. Suppose you receive $1000 right now. You can put the $1000 immediately into a risk-free Treasury bill or bank CD account bearing 2% annual interest. After 12 months, it will have accumulated about 2% interest which is $20. In other words, your $1000 now will become $1020 in 12 months. Before 2007 when government bond rates were pushing 4-5%, the present value shoots up to $1050 in 12 months for $1000 in hand now.

A Backwards Way Of Thinking About It

Let's go back to 2% and think about present value in another way: $1020 in 12 months is $1000. An easy way to calculate the present value of $1020 in 12 months is to divide it by 1.02 (because 2% is the interest after 12 months), which equals $1000. Consider now what happens after 24 months, there will be two interest periods, resulting in $1040.40. In this case, one would say that the present value of $1040.40 in 2 years is $1000. Therefore, the present value depends on both the length of time to the payment and prevailing risk-free interest rates.

Recalculating Our Simple Example

Let us return to the simple example of two payments, $1000 in 12 months plus another $1000 in 24 months. We will again assume the annual risk-free interest rate is 2%. The present value of $1000 in 12 months is $1000/(1.02) = $980. The present value of $1000 in 24 months is $1000/1.02 and then divided by 1.02 again = $961. The conclusion is that the two promised payments total $980 + $961 = $1941, rather than $2000.

Mathematical Statement

In general, the present value of a payment promised 'n' years into the future when the annual interest rate is 'r' is given by the expression

Where PV is the present value and payment is the amount rendered. The caret symbol '^' means to make the following 'n' an exponent. Try this formula with r = 2%, n = 2 years, and Payment = $1000. You should get a PV close to $961. Let us make our calculations more precise. If the payments are instead given in a number of months 'm' into the future, the formula is modified in this way.

A Good Approximation

This is actually an approximation but is very close to the correct answer. It works because one month is just 1/12th of a year, the interest-bearing period. This should be obvious because if 'm' were 12 months (1 year), the second formula would become identical to the first in which the number of years 'n' were 1. Of course if interest rates were computed monthly, then the expression would be

where 'r' is the monthly interest rate and 'm' is the number of months.

We have created a little web app that allows you to compute the present value of annuities. Check out our annuity formula calculator.