How to Calculate the Present Value of a One-Year Project

Suppose that you will graduate exactly one year from today and you will need $1,000 to rent your first apartment. In order to have $1,000 one year from now, how much do you need today? It’s not $1,000, and the reason why has to do with the interest rate. The interest rate, which we will denote by r, is the price charged a borrower for borrowing money expressed as a percentage of the amount borrowed. And let’s use X to denote the amount you need today in order to have $1,000 one year from now. If you put X in the bank today and earn an interest rate r on it, then after one year the bank will pay you X × (1 + r). If the amount paid to you by the bank one year from now is $1,000, then the amount you need to deposit with the bank today is given by the following equation:

When someone borrows money for a year, the interest rate is the price, calculated as a percentage of the amount borrowed, charged by the lender.

You can apply some basic algebra to find that:

So the amount you need today to be assured of having $1,000 one year from now, X, is equal to $1,000 divided by (1 + r). Notice that the value of X depends on the interest rate, r, which is always greater than zero. This fact implies that X is always less than $1,000. For example, if r = 5% (that is, r = 0.05), then X = $1,000/1.05 = $952.38. In other words, $952.38 is the value today of receiving $1,000 one year from now given an interest rate of 5%.

The present value of X is the amount of money needed today in order to receive X at a future date given the interest rate.

Now we can define the present value of X: it is the amount of money needed today in order to receive X in the future given the interest rate. In this example, $952.38 if the present value of $1,000 today given an interest rate of 5%.

The concept of present value is very useful when making decisions that require paying upfront costs now for benefits that arrive in the future. Say you had two options, A and B: the choice of taking a one-year job that pays $10,000 immediately (option A) or taking a one-year course that costs $1,000 now but allows you to earn a one-time payment of $12,000 one year from now (option B). Which one should you take?

On the one hand, the present value of option A is simply $10,000 because you receive its payoff immediately. On the other hand, the present value of option B, with an interest rate of 5%, is:

Since the present value of option B ($10,429) is greater than the present value of option A ($10,000), you should choose option B.

This example illustrates a general principle: when evaluating choices where the costs and/or benefits arrive over time, make your choice by converting the payoffs into their present values and choose the one with the highest present value. Next we will see how to use present value when projects have a time span of more than one year.