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

Reflections on reswitching and roundaboutness 191setting the present value equal to zero would allow for the calculation of the internalrate of return.With little or no grounding in economics, the students could easily see that if apresent-value equation took the form of a polynomial of second-degree (or of somehigher degree), there was the distinct possibility of multiple internal rates of return.Further, it turns out that if a single project whose break-even point (zero presentvalue) corresponds to more than one interest rate, that project’s outlays and revenuescan be decomposed mathematically into two outlay-and-revenue sequences torepresent two projects that exhibit the supposedly troublesome phenomenon ofreswitching.The illustrative examples I offer below differ from Samuelson’s in three respects.(1) I deal first with a single project that entails multiple rates of return. (2) I workwith numbers that constitute plausible interest rates: r = 2 percent and r = 8 percent.(Samuelson worked with 50 percent and 100 percent.) And (3) I start with themultiple rates and work backward to see what temporal characteristics the projectmust have. Then, having identified a temporal sequence of revenues and outlays, Idecompose the sequence into two projects that will exhibit switching and reswitching,the switch points occurring at those same two rates of interest, i.e., r = 2 percentand r = 8 percent.Multiple rates of returnIf a present-value reckoning yields two solutions for the internal rate of return, say r= 2 percent and r = 8 percent, then that reckoning must ultimately resolve itself intothe equation(r – 0.02)(r – 0.08) = 0 (1)or r 2 – 0.10r + 0.0016 = 0 (2)Rewriting to express this equation in terms of the discount factor (1 + r), we get[(1 + r) 2 – 2r – 1] – 0.10r + 0.0016 = 0 (3)(1 + r) 2 – 2.10r – 0.9984 = 0 (4)(1 + r) 2 – [2.10(1 + r) – 2.10] – 0.9984 = 0 (5)(1 + r) 2 – 2.10(1 + r) + 1.1016 = 0 (6)Dividing by the highest power of the discount factor (1 + r) 2 puts the present-valueequation in standard form:1 – 2.10/(1 + r) + 1.1016/(1 + r) 2 = 0 (7)Finally, we can scale equation (7) by 100 so as to avoid fractions of pennies.