What Discounted Cash Flow Rate of Return Never Did Require (includes associated papers 15324 and 15327 and 15331 and 15332 and 15681 and 15910)
Authors: Dougherty E.L. 
Published: December 2007

Summary
A misunderstanding exists on the reinvestment of cash flows froma project to meet a calculated rate of return for an investment. Severaltextbooks state that cash flow from project must be reinvested at thecalculated rate of return in order for that cash flow rate of return for aninvestment to be realized. We demonstrate mathematically and logicallythat such reinvestment is not required.

Introduction
For an investment to yield the calculated discounted cashflow rate of return for an investment, ir, must the cashflows be reinvested from receipt to the project's end atir?. The correct answer is .. no." But a recent articlecites frequently used financial textbooks that teach the incorrect interpretation. Confusion occurs, because if ir, is to be earned on the initial investment. the correctanswer is "yes." This requirement is sensible only whena single cash flow occurs at the project's end, such aswith zero coupon bonds, so that reinvestment isimpossible. For any project with cash flow before its end, thesensible requirement is for i, to be earned on theunamortized investment, in which case the disposition ofthe cash flows is not a consideration.
This confusion pervades SPE literature. At a recentsymposium, while I was proving the correct answer mathematically, two other speakers were stating thecontrary. Boyle and Schenck state, "IRR (ir)..... assumes that all cash thrown off by a project will bereinvested at the high IRR of that Project. Because suchreinvestment is unlikely, . . projects with a highIRR . . . will return significantly less than isforecast.....". My mathematical demonstration isrepeated here, and the correct intepretation illustratedwith an example.

Continuous Discounting Procedures
The use of continuous discounting, simplifies themathematical demonstration. Discounting formulasmeasure economic equivalence between cash flows at different times while taking into account interest earned during theintervening time. With interest compoundedcontinuously at rate i, the rate of increase in wealth,(future worth) is given by
(1)
The solution to Eq. 1, giving PFt at any time, t, is
(2)
Multiplying Eq. 2 by exp( -it) gives the present worth ofPFt at t=0:(3)
Thus, if PF, is a lump sum cash flow at t, the discountfactor that gives its present worth is exp(-it). With auniform and continuous payment of P. the rate ofincrease of future worth is given by
(4)
From Eq. 4, when P is received from t=0 to t=t, PFt is given by
(5)
From Eq. 3, it follows that the present worth of this payment is
(6)
so that the discount factor that gives present worth of acontinuous payment at its start is [1-exp(-it)]/i.

Rate of Return andPresent Worth Equation
With a single investment (Ci, at t=0), the all-importantnet present value (PNPV) is given by
(7)
The discount rate for which PNPV = 0 is ir. Thenumerical value of ir, depends upon the discountingmethod. but this variation need not concern us if thesame discounting procedure is used with all projects.

Meaning of Rate of Return
We use the cumulative cash balance, CBt, todemonstrate that i, is the earnings rate on theunrecovered portion of the investment regardless of whatis done with the project's cash flows. CBt theunamortized investment at any time, equals the future worth at tof the project's discounted cumulative cash flow at t. The following example with ir=0 illustratesCBt- SUP-pose an investment (Ci = $1,000) yields P= $500/yr for2 years, so that at t=2, Ci is returned with no interest.
JPT
P. 85^