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Appendix F Present Value Calculation I n simple terms, a dollar received in the future is worth less than a dollar received today. One reason for this is inflation—a general increase in the prices of all goods and services. Suppose we assume, however, that there is no inflation or, equivalently that amounts measured in nominal (sometimes called current) dollars are converted into amounts measured in real (sometimes called constant) dollars. Individuals would still prefer a real (inflation-adjusted) dollar today to a real dollar in the future. There are two main reasons. First, today’s dollar could be invested and would yield a positive real return, thereby providing the opportunity to buy more goods in the future. Second, all things being equal, individuals would rather consume now than in the future. This means that the value of a dol- lar received in the future is discounted relative to a dollar received now. Mathematically, the present value, PV, of $1 received in one year is 1 PV = 1+ i where i is the appropriate real discount rate; it might, for example, reflect a company’s real return on investment or an individual’s real saving rate. The present value of $1 received in n years’ time is 1 PV = (1 + i )n 194

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APPENDIX F 195 This term is called the present value factor or the discount factor. It equals the present value of $1 received in n years when the discount rate is i, compounded annually. For example, if a company receives $1 in 30 years time, and it uses a discount rate of 7 percent, then the present value factor is 1/(1 + .07)30 = 0.13. In other words, $1 in 30 years’ time is equivalent to 13 cents today. As amounts are received further in the future, n increases and the present value of that amount decreases. Table 10.1 supposes that firms receive an incremental increase in rev- enues each year over a fixed number of years, 55 or 30. Such payment streams are called an annuity. The present value of an annuity of $1 ­ eceived each year for 30 years, denoted ain , equals r 1 1 1 ain = + + + (1 + i ) (1 + i ) 1 2 (1 + i )n This can be shown to equal 1 − (1 + i ) −n ain = i Thus, for example, the present value of an annuity of $1 per year received for 30 years at a discount rate of 7 percent would equal $12.41. Consequently, the present value of $7.02 million per year for 30 years at a discount rate of 7 percent would equal $7.02 × 12.41 million = $87.1 million. This amount is rounded down to $85 million in Table 10.1.   $225 × 312,000 × 0.10 = $7.02 million.