## Example Saving Up Revisited

-1—' You think you will be able to deposit \$4,000 at the end of each of the next three years in a bank account paying 8 percent interest. You currently have \$7,000 in the account. How much will you have in three years? In four years?

Ross et al.: Fundamentals I III. Valuation of Future I 6. Discounted Cash Flow of Corporate Finance, Sixth Cash Flows Valuation

Edition, Alternate Edition

CHAPTER 6 Discounted Cash Flow Valuation 159

At the end of the first year, you will have:

At the end of the second year, you will have:

Repeating this for the third year gives:

Therefore, you will have \$21,803.58 in three years. If you leave this on deposit for one more year (and don't add to it), at the end of the fourth year, you'll have:

When we calculated the future value of the two \$100 deposits, we simply calculated the balance as of the beginning of each year and then rolled that amount forward to the next year. We could have done it another, quicker way. The first \$100 is on deposit for two years at 8 percent, so its future value is:

\$100 X 1.082 = \$100 X 1.1664 = \$116.64 The second \$100 is on deposit for one year at 8 percent, and its future value is thus:

The total future value, as we previously calculated, is equal to the sum of these two future values:

Based on this example, there are two ways to calculate future values for multiple cash flows: (1) compound the accumulated balance forward one year at a time or (2) calculate the future value of each cash flow first and then add them up. Both give the same answer, so you can do it either way.

To illustrate the two different ways of calculating future values, consider the future value of \$2,000 invested at the end of each of the next five years. The current balance is zero, and the rate is 10 percent. We first draw a time line, as shown in Figure 6.2.

On the time line, notice that nothing happens until the end of the first year, when we make the first \$2,000 investment. This first \$2,000 earns interest for the next four (not five) years. Also notice that the last \$2,000 is invested at the end of the fifth year, so it earns no interest at all.

Figure 6.3 illustrates the calculations involved if we compound the investment one period at a time. As illustrated, the future value is \$12,210.20.

Figure 6.4 goes through the same calculations, but the second technique is used. Naturally, the answer is the same.

Time Line for \$2,000 per Year for Five Years

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