## Present Value with Multiple Cash Flows

It will turn out that we will very often need to determine the present value of a series of future cash flows. As with future values, there are two ways we can do it. We can either discount back one period at a time, or we can just calculate the present values individually and add them up.

Suppose you need \$1,000 in one year and \$2,000 more in two years. If you can earn 9 percent on your money, how much do you have to put up today to exactly cover these amounts in the future? In other words, what is the present value of the two cash flows at 9 percent?

The present value of \$2,000 in two years at 9 percent is: \$2,000/1.092 = \$1,683.36 The present value of \$1,000 in one year is:

\$1,000/1.09 = \$917.43 Therefore, the total present value is: \$1,683.36 + 917.43 = \$2,600.79

To see why \$2,600.79 is the right answer, we can check to see that after the \$2,000 is paid out in two years, there is no money left. If we invest \$2,600.79 for one year at 9 percent, we will have:

We take out \$1,000, leaving \$1,834.86. This amount earns 9 percent for another year, leaving us with:

This is just as we planned. As this example illustrates, the present value of a series of future cash flows is simply the amount that you would need today in order to exactly duplicate those future cash flows (for a given discount rate).

An alternative way of calculating present values for multiple future cash flows is to discount back to the present, one period at a time. To illustrate, suppose we had an investment that was going to pay \$1,000 at the end of every year for the next five years. To find the present value, we could discount each \$1,000 back to the present separately and then add them up. Figure 6.5 illustrates this approach for a 6 percent discount rate; as shown, the answer is \$4,212.37 (ignoring a small rounding error).

Alternatively, we could discount the last cash flow back one period and add it to the next-to-the-last cash flow:

(\$1,000/1.06) + 1,000 = \$943.40 + 1,000 = \$1,943.40

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

III. Valuation of Future Cash Flows

6. Discounted Cash Flow Valuation

© The McGraw-Hill Companies, 2002

PART THREE Valuation of Future Cash Flows

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