## Singte Amounts

An understanding of the present value concept should enable us to answer a question that was posed at the very beginning of this chapter: which should you prefer - \$1,000 today or \$2,000 ten years from today?*' Assume that both sums are completely certain and your opportunity cost of funds is 8 percent per annum (i.e., you could borrow or lend at 8 percent). The present worth of \$1,000 Received today is easy - it is worth \$1,000. However, what is \$2,000 received at the end of 10 years worth to you today? We might begin by asking what amount (today) would grow to be \$2,000 at the end of 10 years at 8 percent compound interest. This amount is called the present value of \$2,000 payable in 10 years, discounted at 8 percent. In present value problems such as this, the interest rate is also known as the discount rate (or capitalization rate).

We can now make use of Eq. (3.7) and Table 3.4 to solve for the present value of 52,000 to be received at the end of 10 years, discounted at 8 percent. In Table 3.4, the intersection of the 8% column with the 10-period row pinpoints PVIFmM - 0.463. This tells us that \$1 received 10 years from now is worth roughly 46 cents to us today. Armed with this information, we get.

Finally, if we compare this present value amount (S926) with the promise of \$1,000 to be received today, we should prefer to take the \$1,000. In present value terms we would be better off by \$74 (\$1,000 - \$926).

Discounting future cash flows turns out to be very much like the process of handicapping. That is, we put future cash flows at a mathematically determined disadvantage relative to current dollars. For example, in the problem just addressed, every future dollar was handicapped to such an extent that each was worth only about 46 cents. The greater the disadvantage assigned to a future cash flow, the smaller the corresponding present value interest factor (PVIF). Figure 3.2 illustrates how both time and discount rate combine to affect present value; the present value of \$100 received from 1 to 10 years in the future is graphed for discount rates of 5, 10, and 15 percent. The graph shows that the present value of \$100 decreases by a decreasing rate the further in the future that it is to be received. The greater the interest rate, of course, the lower the present value but also the more pronounced the curve. At a 15 percent discount rate, \$100 to be received 10 years hence is worth only \$24.70 today- or roughly 25 cents on the (future) dollar.

Unknown Interest (or Discount) Rate. Sometimes we are faced with a time-value-ofmoney situation in which we know both the future and present values, as well as the number of time periods involved. What is unknown, however, is the compound interest rate (;') implicit in the situation.

Unknown Number of Compounding lor Discounting! Periods. At times we may need to know how long it will take for a dollar amount invested today to grow to a certain future value given a particular compound rate of interest. For example, how long would it take for an investment of \$1,000 to grow to \$1,900 if we invested it at a compound annual interest rate of 10 percent? Because we know both the investment's future and present value, the number of compounding (or discounting) periods (n) involved in this investment situation can be determined by rearranging either a basic future value or present value equation.