1 Since, for the purpose in hand, any one of the five criteria discussed in the preceding chapter will serve, we shall use criterion (28.2), PVρ(B) > K, for demonstration.
Given, therefore, the initial outlay K in year zero, and the subsequent stream of (net) benefits, B1, B2, . . . , BT, we canre-write criterion(28.2) as
T t=1
Bt
(1 + ρ)t > K (29.1)
By multiplying through by a scalar (1 + ρ)T, we obtain the equivalent inequality
T t=1
Bt(1 + ρ)T−t> K(1 + ρ)T (29.2)
which may be summarized as TVρ(B) > TVρ(K), where TVρ(B) stands for the terminal value of the stream of benefits when each is compounded forward to terminal date T at rate ρ, an d TVρ(K) stands for the terminal value of the outlay K when it also is compounded forward to terminal date T at rate ρ.
If and only if PVρ(B) > K does TVρ(B) > TVρ(K); one form of the criterion, that is, entails the other. But the latter form is far more revealing: it makes clear that, for the criterionto be met, the aggregate of the benefits, B1, B2, BT, when each benefit is wholly and continually reinvested to time T at this same weighted rate of return ρ, must exceed the sum which K amounts to when it also is wholly invested and reinvested to the terminal year T. Such a criterionwould, of course, be applicable in the rare case when, in fact, both the benefits and the initial outlay of the project were to be used inexactly this way. It could be justified only if all benefits were encashed and wholly invested and reinvested at ρ, the returnto private investment, until the terminal date, and similarly for the amount K.
Inasmuch as this implicit requirement is seldom complied with, the use of a criterionthat is valid only if such a requirement is, infact, assured canbe seriously misleading. Certainly, any of these four criteria is misleading when it is applied to a public project without informationinthe particular case about the disposal of
QUAH: “CHAP29” — 2007/1/25 — 18:37 — PAGE 159 — #2 Critique of DPV criterion (II) 159 the returns to the project and without information also about the sort of stream that would have beengenerated by the sum K had it not beentakenfrom the economy.1 To illustrate, suppose that the outlay K required by a particular public investment is to be drawn entirely from the private investment sector, where it would otherwise have been reinvested continually at ρ to reach a terminal value at T of TVρ(K).
Suppose also that the project’s benefits, in contrast, are expected to be wholly consumed as they occur over time. The value of such benefits therefore grows in value at the rate of time preference r to reach a terminal value at T of TVr(B).
Now if TVr(B), as is likely, happens to be smaller than TVρ(K), the terminal value of the sum K, the project has to be rejected ona Pareto criterion: society would be better off leaving the amount K in the private investment sector (there to be continually reinvested at ρ) rather than using it to finance a project whose benefits are consumed as they occur.
It should be evident that the use of criterion (28.1), PVr(B)K, would fail to reveal this possibility. Employing it could then sanction projects that would be rejected on a Pareto criterion. By reversing these suppositions in the above example, so that the outlay K is raised entirely through a reduction in present consumption so that it is to be compounded to the terminal year T at society’s rate of time preference while, in contrast, all the returns over time to the project are to be wholly invested and reinvested at ρ to the terminal year, the employment of the PVr(B) > K criterioncould reject projects that do, infact, meet the Pareto criterion.
2 Thus transforming the criterion PVr(B) > K into its compounded termi-nal form TVr(B) > TVr(K) enables us immediately to appreciate that, for its valid employment, all the returns from the project should be wholly consumed as they occur and that the sum K should be raised entirely from current consump-tion. Similarly, transforming the other limiting case PVρ(B) > K, into the form TVρ(B) > TVρ(K) enables us also to appreciate at once that its Pareto validity is assured if, infact, it is applied to a case inwhich the benefits, as they occur, are wholly invested and reinvested in the private investment sector at prevailing yield ρ, until the terminal date T, and if the sum K raised from the private sector is also wholly invested and reinvested at yield ρ until T.
Inother words, the correct terminal value of a project’s benefit stream and the correct terminal value of the opportunity cost of its outlay are both functions, in the simplest possible case, of three variables, r, ρ and c, where c is the fraction of any income or of any return on investment2that is consumed, the remainder (1 − c) being invested (unless otherwise determined) in the private sector at ρ.
1 This critique, incidentally, applies as well to the usual DPV methods employed by private corporations for evaluating alternative investment streams – though in so far as the returns over time are likely to be treated more uniformly, the error may be less important.
2 For there may be public projects for which all or part of any of the expected returns over the future are required to be invested in designated public enterprises. Again, the amount K may be raised wholly or in part using the sums that are available simply in consequence of the non-renewal of existing public investments.
QUAH: “CHAP29” — 2007/1/25 — 18:37 — PAGE 160 — #3 Incontrast, criterion(28.2), PVρ(B) > K, is valid only if both the initial outlay K and the stream of returns that it generates are all wholly invested and re-invested to the terminal year at ρ. And this condition is transparent once this criterion (28.2) is transformed into the more explicit form TVρ(B) > TVρ(K).
By subtracting TV (K) from TV (B), we obtainthe net terminal value of the project in question. Once this net terminal value is correctly calculated for a number of projects, all with aninitial outlay K, the resultant ranking can be maintained whatever rate of discount is then used to discount them to the present.
3 In order to complete the critique, we need to re-examine the standard IRR criterion. Although, onthe face of it, there should be anadvantage inbeing able to calculate the IRR without reference to the prevailing interest rates or investment yields in the economy, it has fallen into disfavour among economists since, as we have seen, we can derive more than one IRR for a given investment stream.3
The more important reason, however, is that, even in the more usual case in which all net benefits are positive, the standard IRR calculated for an investment stream does not accord with the true average rate of return of the net benefit stream.
It transpires that, as conventionally defined, the IRR suffers from the same defect as common DPV criteria; namely, that the implied reinvestment rate of the net benefits has no necessary relation to the actual rates.
Given the standard definition of the IRR as that λ for which
T t=1
Bt
(1 + λ)t = K
If we multiply through by the scalar (1 + λ)t we obtain
T t=1
Bt(1 + λ)T−t= K(1 + λ)T
So explicated, the standard IRR is shown to be defined as that rate λ which, when it is used to compound each of the benefits Bt to the terminal year T, produces a terminal outlay that itself is equal to K compounded forward to year T also at λ. But this resulting λ has no necessary relation to the average rate at which the benefits Btare being actually compounded forward to T.
Since in any given project it cannot be assumed that each of the benefits Bt is wholly invested at λ when it occurs, the standard definition is misleading. In fact, the disposal of each Bt as it occurs depends upon behavioural and institutional factors, ingeneral onthe values r, ρ and c. Inorder, then, to calculate the IRR as a uniquely determined average rate of growth of the initial outlay K over period T,
3 The reader is reminded that a necessary though not a sufficient condition for the standard IRR to have more than one value is that one or more of the net benefits be negative, a contingency not often encountered.
QUAH: “CHAP29” — 2007/1/25 — 18:37 — PAGE 161 — #4 Critique of DPV criterion (II) 161 we must first calculate the actual terminal value of each of the benefits that are generated by the outlay K.
Acorrect calculation of the IRR, consistent with the normalized procedure being proposed, must therefore be defined as that rate of discount λ which would reduce the actual terminal value of the sum of each of the benefits, so compounded, to equality with the initial outlay K. This canbe formulated as that λ for which
TV (B) (1 + λ)T = K
It follows that if, with a givenoutlay K, the terminal value of the benefit stream of project X exceeds that of project Y , which inturnexceeds that of project Z – which we canwrite as X > Y > Z – then, by our definition above, their respective IRRs λx, λyand λz are those for which
X
(1 + λx)T = Y
(1 + λy)T = Z
(1 + λz)T = K from which it follows that λx> λy> λz.
However, for ranking purposes, at least, it would be pointless to calculate these normalized IRRs, as they will follow that of the terminal value of their respective benefit streams.
QUAH: “CHAP30” — 2007/1/25 — 18:59 — PAGE 162 — #1