Money and Markets: Essays in Honor of Leland B. Yeager

192 Roger W. Garrison100 – 210/(1 + r) + 110.16/(1 + r) 2 = 0 (8)In its simplest interpretation (and taking positive and negative terms to indicaterevenues and outlays, respectively), equation (8) represents a three-period projectthat entails some up-front revenue. In the initial period, t 0, a contractually requiredpre-payment in the amount of $100 is received; in the next period, t 1, outlays of$210 are made; and in the last period, t 2, the project’s output is delivered and a finalpayment of $110.16 is received. Figure 13.1 illustrates the project in terms of bothtime and money. By construction, this is a break-even project at interest rates of2 percent and 8 percent.REVENUESt$10001$110.16t 0 t 2$100$200OUTLAYS$210Figure 13.1 A three-period project.At the middling interest rate of 5 percent, the project is in the red by $0.08. Atinterest rates below 2 percent or above 8 percent, the project is in the black – by$0.06 at a 1 percent rate of interest and by $0.07 at a 9 percent rate. The dependenceof present value on the rate on interest over the range of 0 percent to 14 percent isshown in Figure 13.2. Note that at a zero rate of interest, the present value, which issimply the algebraic sum of the undiscounted revenues and outlays, is $0.16.Except in the vicinity of the switch and reswitch points, the relationship ofpresent value to the rate of interest is well behaved. At extremely high rates ofinterest, where the terms containing positive powers of the discount factor becomenegligible, the profitability of this project approaches the initial receipt of $100.Figure 13.3 shows the dependence of present value on the rate of interest forinterest rates up to 1000 percent. Note that the variations of present value in thelow single digits – including the negative present values over the 2–8 percent range– are too small to be discernible in Figure 13.3.A mirror-image interpretation of our multiple-rate equation is produced byreversing all the signs:–100 + 210/(1 + r) – 110.16/(1 + r) 2 = 0 (9)This three-period project requires an initial outlay of $100 and generatesrevenue in the second period of $210. The outlay of $110.16 that occurs in the third