## Present Values for Multiple Periods

Suppose you need to have \$1,000 in two years. If you can earn 7 percent, how much do you have to invest to make sure that you have the \$1,000 when you need it? In other words, what is the present value of \$1,000 in two years if the relevant rate is 7 percent?

Based on your knowledge of future values, you know the amount invested must grow to \$1,000 over the two years. In other words, it must be the case that:

\$1,000 = PV X 1.07 X 1.07 = PV X 1.072 = PV X 1.1449

Given this, we can solve for the present value:

Therefore, \$873.44 is the amount you must invest in order to achieve your goal.

### Saving Up

You would like to buy a new automobile. You have \$50,000 or so, but the car costs \$68,500. If you can earn 9 percent, how much do you have to invest today to buy the car in two years? Do you have enough? Assume the price will stay the same.

What we need to know is the present value of \$68,500 to be paid in two years, assuming a 9 percent rate. Based on our discussion, this is:

You're still about \$7,655 short, even if you're willing to wait two years.

As you have probably recognized by now, calculating present values is quite similar to calculating future values, and the general result looks much the same. The present value of \$1 to be received t periods into the future at a discount rate of r is:

The quantity in brackets, 1/(1 + r)', goes by several different names. Because it's used to discount a future cash flow, it is often called a discount factor. With this name, it is not surprising that the rate used in the calculation is often called the discount rate. We will tend to call it this in talking about present values. The quantity in brackets is also called the present value interest factor (or just present value factor) for \$1 at r percent for t periods and is sometimes abbreviated as PVIF(r, t). Finally, calculating the present value of a future cash flow to determine its worth today is commonly called discounted cash flow (DCF) valuation.

To illustrate, suppose you need \$1,000 in three years. You can earn 15 percent on your money. How much do you have to invest today? To find out, we have to determine discount rate

The rate used to calculate the present value of future cash flows.

discounted cash flow (DCF) valuation

Calculating the present value of a future cash flow to determine its value today.

Ross et al.: Fundamentals of Corporate Finance, Sixth Edition, Alternate Edition

III. Valuation of Future Cash Flows

5. Introduction to Valuation: The Time Value of Money