## The Single Period Case

We've seen that the future value of \$1 invested for one year at 10 percent is \$1.10. We now ask a slightly different question: How much do we have to invest today at 10 percent to get \$1 in one year? In other words, we know the future value here is \$1, but what is the present value (PV)? The answer isn't too hard to figure out. Whatever we invest today will be 1.1 times bigger at the end of the year. Because we need \$1 at the end of the year:

Present value X 1.1 = \$1 Or, solving for the present value:

In this case, the present value is the answer to the following question: What amount, invested today, will grow to \$1 in one year if the interest rate is 10 percent? Present value is thus just the reverse of future value. Instead of compounding the money forward into the future, we discount it back to the present.

Single-Period PV

Suppose you need \$400 to buy textbooks next year. You can earn 7 percent on your money. How much do you have to put up today?

We need to know the PV of \$400 in one year at 7 percent. Proceeding as in the previous example:

Present value X 1.07 = \$400

We can now solve for the present value:

Thus, \$373.83 is the present value. Again, this just means that investing this amount for one year at 7 percent will result in your having a future value of \$400.

CHAPTER 5 Introduction to Valuation:The Time Value of Money 139

From our examples, the present value of \$1 to be received in one period is generally given as:

We next examine how to get the present value of an amount to be paid in two or more periods into the future.

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