What Discounted Cash Flow Rate of Return Never Did Require (includes associated papers 15324 and 15327 and 15331 and 15332 and 15681 and 15910)
- E.L. Dougherty (U. of Southern California)
- Document ID
- Society of Petroleum Engineers
- Journal of Petroleum Technology
- Publication Date
- January 1986
- Document Type
- Journal Paper
- 85 - 87
- 1986. Society of Petroleum Engineers
- 7.4 Energy Economics
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- 156 since 2007
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Summary. A misunderstanding exists on the reinvestment of cash flows from a project to meet a calculated rate of return for an investment. Several textbooks state that cash flow from project must be reinvested at the calculated rate of return in order for that cash flow rate of return for an investment to be realized. We demonstrate mathematically and logically that such reinvestment is not required.
For an investment to yield the calculated discounted cash flow rate of return for an investment, ir, must the cash flows be reinvested from receipt to the project's end at ir?. The correct answer is .. no." But a recent article cites frequently used financial textbooks that teach the incorrect interpretation. Confusion occurs, because if ir, is to be earned on the initial investment. the correct answer is "yes." This requirement is sensible only when a single cash flow occurs at the project's end, such as with zero coupon bonds, so that reinvestment is impossible. For any project with cash flow before its end, the sensible requirement is for i, to be earned on the unamortized investment, in which case the disposition of the cash flows is not a consideration.
This confusion pervades SPE literature. At a recent symposium, while I was proving the correct answer mathematically, two other speakers were stating the contrary. Boyle and Schenck state, "IRR (ir) ..... assumes that all cash thrown off by a project will be reinvested at the high IRR of that Project. Because such reinvestment is unlikely, . . projects with a high IRR . . . will return significantly less than is forecast.....". My mathematical demonstration is repeated here, and the correct intepretation illustrated with an example.
Continuous Discounting Procedures
The use of continuous discounting, simplifies the mathematical demonstration. Discounting formulas measure economic equivalence between cash flows at different times while taking into account interest earned during the intervening time. With interest compounded continuously at rate i, the rate of increase in wealth, (future worth) is given by
The solution to Eq. 1, giving PFt at any time, t, is
Multiplying Eq. 2 by exp( -it) gives the present worth of PFt at t=0: (3)
Thus, if PF, is a lump sum cash flow at t, the discount factor that gives its present worth is exp(-it). With a uniform and continuous payment of P. the rate of increase of future worth is given by
From Eq. 4, when P is received from t=0 to t=t, PFt is given by
From Eq. 3, it follows that the present worth of this payment is
so that the discount factor that gives present worth of a continuous payment at its start is [1-exp(-it)]/i.
Rate of Return and Present Worth Equation
With a single investment (Ci, at t=0), the all-important net present value (PNPV) is given by
The discount rate for which PNPV = 0 is ir. The numerical value of ir, depends upon the discounting method. but this variation need not concern us if the same discounting procedure is used with all projects.
Meaning of Rate of Return
We use the cumulative cash balance, CBt, to demonstrate that i, is the earnings rate on the unrecovered portion of the investment regardless of what is done with the project's cash flows. CBt the unamortized investment at any time, equals the future worth at t of the project's discounted cumulative cash flow at t. The following example with ir=0 illustrates CBt- SUP-pose an investment (Ci = $1,000) yields P= $500/yr for 2 years, so that at t=2, Ci is returned with no interest.
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