## 812 Discounting of Future Cash Flows

As mentioned above, even if there were no inflation, a future cash amount F is not equal to its present value P; it must be discounted. The relationship between P and its future value Fn n years from now is given by the discount rate rd, defined such that

The greater the discount rate, the smaller the present value of future transactions. To determine the appropriate value of the discount rate, one has to ask at what value of rd one is indifferent between an amount P today and an amount Fn = P/(1 + rd)n a year from now. That depends on the circumstances and on individual preferences. Consider a consumer who would put her money in a savings account with 5% interest. Her discount rate is 5%, because by putting $1000 into this account, she, in fact, accepts the alternative of (1 + 5%) x $1000 a year from now. If, instead, she would use that money to pay off a car loan at 10%, then her discount rate would be 10%; paying off the loan is like putting the money into a savings account which pays at the loan interest rate. If the money would allow her to avoid an emergency loan at 20%, then her discount rate would be 20%.

The situation becomes more complex when there are several different investment possibilities offering different returns at different risks, such as savings accounts, stocks, real estate, or a new business venture. By and large, if one wants the prospect of a higher rate of return, one has to accept a higher risk. Thus, a more general rule would state that the appropriate discount rate for the analysis of an investment is the rate of return on alternative investments of comparable risk. In practice, that is sometimes quite difficult to determine, and it may be desirable to have an evaluation criterion that bypasses the need to choose a discount rate. Such a criterion is obtained by calculating the profitability of an investment in terms of an unspecified discount rate and then solving for the value of the rate at which the profitability goes to zero. That method, called internal rate of return method, will be explained later.

Just as with other growth rates, one can specify the discount rate with or without inflation. If Fn is given in terms of constant currency, designated as Fn0, then it must be discounted with the real discount rate rd0. The latter is, of course, related to the market discount rate rd by r _ r d - rinf do _ T-—

according to Eq. 8.4. Present values can be calculated with real rates and real currency or with market rates and inflating currency; the result is readily seen to be the same because multiplying the numerator and denominator of Eq. 8.6 by (1 + rinf)n yields

according to Eq. 8.3.

The ratio P/Fn of present and future value is called the present worth factor, which is designated here with the mnemonic notation

It is plotted in Figure 8.2. Its inverse

is called the compound amount factor. These factors are the basic tool for comparing cash flows at different times. Note that the so-called end-of-year convention has been chosen here by designating Fn as the value at the end of the nth year. Also, annual intervals have been assumed, generally an adequate time step for engineering economic analysis; accountants, by contrast, tend to work with monthly intervals, corresponding to the way most regular bills are paid. The basic formulas are the same, but the numerical results differ slightly because of differences in the compounding of interest; this point will be explained more fully later when we pass to the continuous limit by letting the time step approach zero.

0.8 -0.6 0.4 0.2 0

FIGURE 8.2

The present worth factor (P/F,r,N) as function of rate r and number of years N.

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