# Present Value

Suppose that someone will give you a gift of \$100 either now or in four years. Which is better, the money now or the money four years from now, or are they the same? The rule that gifts with restrictions are of less value than gifts without restrictions suggests that money now is worth more than money in the future. Anything that one can do with the gift of \$100 four years from now one can do with \$100 now simply by saving it for four years. But there are many things that one can do with money now that one cannot do with money four years from now. Therefore, \$100 promised four years from now is not worth as much as \$100 right now.

One of the things that can be done with money now is to invest it so that it will earn interest. Because this cannot be done with money four years from now, this option of foregone interest is a cost of waiting for the money. When this cost is measured, one sees the amount by which money in the future must be discounted to obtain its present value.

If the interest rate is 10%, \$100 now can be turned into \$110 one year from now. Thus, \$100 now and \$110 a year from now have the same value. (You may have to think about this for a while.) This simple idea is vital in business and governmental decisions because a great many decisions have costs and benefits spread over time, and it is often necessary to compare sums in different time periods.

Computing the present value of future sums is nothing more than working compound interest problems backward. The formula for finding the future value of a present sum after one period is

(1) P + Pr = F

or

(2) P(1 + r) = F

where P is the present sum, r is the interest rate in decimal form, and F is the future sum. (Try the formula for P = \$100 and r = .10. You should get F = \$110.)

After two years the amount of money will be

(3) F1(1 + r) = F2

where F1 is the amount of money one year from now, and F2 is the amount of money two years from now. This may be rewritten as

(4) (P(1 + r))(1 + r) = P(1 + r)2 = F2.

(Try this formula for P = \$100 and r = .10. F2 should be \$121.) Using the same logic, the future value three years from now will be

(5) P(1 + r)3 = F3

and for any arbitrary period n, it will be

(6) P(1 + r)n = Fn.

Simple algebra allows us to solve this equation if we have the time period and two of the three remaining variables. (Logarithms help a lot if we are solving for r.) In particular, if we have a future sum of money and want to find its present value, the last equation can be rewritten as

(7) P = F / (1 + r)n

Using this formula in our case of \$100 four years from now and an interest rate of 10%, the present value is

P = 100/(1.10)4 = 100/1.464 = 68.30

This means that \$68.30 invested at 10% will grow to \$100 four years from now. Therefore, \$68.30 now and \$100 four years in the future have equivalent value if the interest rate is 10%.

Present value analysis explains bond prices.