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of return criterion

1 Consider the three alternative investment streams A, B and C, shownin Table 25.1. The undiscounted net benefit ratio (B − K)/K, inwhich B repre-sents the net benefit in the first and only year in which benefits accrue, with K representing the initial capital outlay in year zero, will here serve as the criterion, one that will rank C above B, an d B above A. Why the undiscounted net benefit ratio? For the simple reasonthat, whatever the rate of discount used, it will affect the net benefit at t1of A, B and C inexactly the same proportion. We may there-fore infer that, irrespective of the discount rate used, the resulting discounted net benefit ratio would give the same ranking as the undiscounted net benefit ratio.

This conclusion is valid, however, only for a two-period investment in which the outlay appears in the first period and the benefit in the second. Add but one more period, and the ranking will, in general, depend upon the discount rate. For instance, a stream −100, 10, 100 cannot be ranked in relation to the stream −100, 90, 10 without knowing the discount rate. If this were 1 per cent, the first would clearly yield a larger net benefit ratio than the second. If, however, the discount rate were 50 per cent, the second would yield a larger net benefit ratio than the first.

The original two-period investment stream has another property: the internal rates of returnof each of two-period streams A, B and C (as showninTable 25.1) are equal to their corresponding net benefit ratios (also shown in the table), and therefore produce the same ranking, C, B, A. There is no mystery about this: for the net or excess benefit (B − K) produced over the year, takenas a fractionof the capital cost K, is of course equivalent to one year’s growth of the initial capital K. Thus the capital of 100 invested in A will have beenperceived to grow by 5 per cent over the year, in B by 15 per cent, and in C by 25 per cent. A discount rate of 5 per cent for A, of 15 per cent for B and of 25 per cent for C will reduce

Table 25.1

t₀ t₁ (B − K)/K Internal rate of return

A −100 105 5/100 5%

B −100 115 15/100 15%

C −100 125 25/100 25%

QUAH: “CHAP25” — 2007/1/25 — 08:02 — PAGE 140 — #2 the magnitudes of their respective benefits to their original outlays, 100 in each case – such discount rates being, therefore, by definition, the respective internal rates of returnof A, B and C.

For such two-period investment streams, then, the ranking C, B, A is the same whether we use the IRR or the DPV method. Moreover, whatever the rate of discount that is employed, whether positive, zero or negative, the ranking remains unchanged.

2 This harmony between the net present value criterion and the IRR criterion will, however, as the reader probably suspects, break down if any of the investment streams being compared contains more than two periods. Indeed, this implication accords with the proposition exemplified above: that, for investment streams in excess of two periods, the ranking will vary with the rate of discount used. The IRR ranking does not, however, at all depend on the adopted rate of discount, but is independently determined. If it then so happens that at the ruling discount rate a number of investment streams show the same ranking by the two criteria, an alteration of the discount rate, which changes the net present value ranking of the investment projects, will also produce a discrepancy between this new net present value ranking and the ranking by IRR.

We illustrate this latter statement using two different investment streams, A and B, as in Table 25.2. Both of these are ranked equally by the IRR criterion, being 10 per cent in each case. Not surprisingly then, if the discount rate employed also happened to be 10 per cent, the discounted net benefit ratio (B − K)/K would be zero for both A and B, as (B − K) would equal zero using a 10 per cent discount rate. Were the discount rate equal to only 1 per cent, the three-period B stream would show a (B − K)K ratio of 19/100 and would rank above the (B − K)/K ratio for A, of 9/100, the reverse ranking being produced if the discount rate were doubled to become 20 per cent. In that case, B’s discounted net benefit ratio would be equal to −16/100, which is therefore ranked below that of −8/100 for the A stream.

A diagrammatic representation of the variation in the discounted net present value, PVr(B −K) with respect to the rate of discount for each of these investment streams, A and B, is displayed inFigure 25.1, where PVr(B−K) is measured along the vertical axis and the rate of discount r along the horizontal axis. It can be seen at a glance that discounted net benefit PVr(B − K) for each of the two streams of investment – indeed, for any investment stream – varies inversely with the rate of discount r.

QUAH: “CHAP25” — 2007/1/25 — 08:02 — PAGE 141 — #3 Alleged superiority of DPV vs IRR 141

B PV_r(B – K)20

10

0 0.01 0.10 0.20 r

–10

–20

A

Figure 25.1

It will be noticed that there is some rate of discount, here 10 per cent, at which the two investment streams will have exactly the same discounted net present value. In this particular case, moreover, it so happens that this same discounted net present value is equal to zero; hence, the IRR for each will be 10 per cent. For discount rates below 10 per cent, B’s PVr(B − K) exceeds that of A, the reverse being true for discount rates above 10 per cent.

3 Inspite of this discrepancy betweenthe two criteria, the IRR has beenrec-ommended in some circumstances, particularly as a method of allocating a given capital budget among a number of potential investment projects. Thus, one might select a number of public investment streams, subject to a budget, provided that the IRR on each investment stream that is chosen exceeds the adopted rate of discount. The scheme is illustrated in Table 25.3, which shows five investment streams in declining order of IRR. The DPV of their net benefit ratios is also given for a discount rate equal to 3 per cent.

QUAH: “CHAP25” — 2007/1/25 — 08:02 — PAGE 142 — #4

If the available budget were 1,000, on the IRR criterion, only 350 of it would be spent. We should admit A, B, C and D, but not E, since the latter has an IRR of only 2 per cent, whereas the discount rate is taken to be 3 per cent. The reader will doubtless observe that the IRR ranking A, B, C, D and E differs from that resulting from the DPV criterion, which is B, A, C, D and E. Yet, this latter ranking holds only for rates of discount close to 3 per cent. As we move further from a 3 per cent discount rate, so the ranking may change, in general, for any set of investment streams.

Were the available budget only 100, the IRR criterion would choose investment option A, incontrast to the selectionof the B optionif, instead, the DPV criterion were used. But which is the better option?

4 Although the ranking of a number of alternative investment streams will, in general, differ according to whether we use one criterion or the other, we have no reason, so far, for preferring one to the other. We may now introduce at least one consideration that tells against the use of the IRR criterion as usually defined:¹ more than a single IRR may be yielded by a given investment stream. A necessary, though not sufficient condition, for this to occur is that not all outlays (or negative net benefits) take place in the initial period. There have to be negative net benefits inlater periods.

A simple example of such an investment stream, call it the H stream, could be

−100, 420, −400. This stream is one that yields two IRRs, λ1of 46 per cent, and λ2of 174 per cent, since using either of these rates as a discount rate would reduce the discounted net present value of the H stream to zero.²

1 A more accommodating definition is proposed in Chapter 28, following the critique of the DPV criterion.

2 From this definition of the IRR, say λ, we require a λ for which

−100 + 420

(1 + λ)− 400 (1 + λ)²= 0

The reader will recognize the expressionas a quadratic equationwith two solutions, λ = 0.46 and λ = 1.74. A negative net benefit in the second period is not a sufficient condition for two IRRs, however. If it were small enough, there would still be a single IRR. An example would be the stream

−100, 121, −1, which yields anIRR of a little over 20 per cent.

QUAH: “CHAP25” — 2007/1/25 — 08:02 — PAGE 143 — #5 Alleged superiority of DPV vs IRR 143 Figure 25.2 depicts the curve relating the net present value of the H stream to the rate of discount r. The curve will be seen to cut the horizontal axis, not once (as does each of the investment streams in Figure 25.3), but twice: once at the point where r is 0.46 and once where r is 1.74. Since either of these two discount rates reduces the present value of the H stream to zero, they are identified as the two IRRs λ1and λ2.

Of course, the reader might think that, of these two IRRs, λ1(46 per cent) is the more reasonable. If he were obliged to adopt an IRR for such a stream, he would probably choose 46 per cent. But he would find it difficult to justify such a choice, if he were not allowed to draw on intuition. Moreover, even if the reader did feel confident about the 46 per cent IRR for the H stream, this example of two IRRs is only a special case. For one can devise investment streams to yield three, four or indeed any number of IRRs.³However open-minded the reader may wish to remain, he cannot deny that the case for preferring the present value criterion above the IRR criterionlooks very strong.

1.0 2.0 r

0

– 150 – 100 – 50 +50 +100 +150

1= 0.46 2=1.74 PV_r(B – K )

Figure 25.2

3 Determining the values of the IRRs corresponding to any investment stream implies solving for the roots of a polynomial. Any investment stream with n periods can be transformed into a polynomial with a maximum of n−1 different roots, each being a possible IRRs. Only those that are positive will matter for investment criteria. Negative IRRs make sense, but are not usually of much importance.

Complex roots do not appear to make sense in this context.

QUAH: “CHAP25” — 2007/1/25 — 08:02 — PAGE 144 — #6 5 If, provisionally, we accept the DPV criterion, there remains the question of whether we are to rank (i) by excess benefit over cost (B^− K^), (ii) by the ratio of benefit to cost B^/K^, or (iii) by the ratio of excess benefit to cost (B^− K^)/K^. These alternative ranking methods are worked out in Table 25.4 for investment options A and B, where K^ is taken to be the DPV of all outlays, initial and subsequent ones, if any, and B^is understood to be the DPV of all net benefits, the discount rate being given.

Inthe (B^− K^) column, A, which has excess benefit over cost of 50, is ranked above C, which has an excess benefit over cost of only 30. In the next, B^/K^ column, however, C, with a ratio of benefit to cost of 2.5, is ranked above A, which has a ratio of benefit to cost of only 1.5. In the final column showing (B^−K^)/K^, the ratio of excess benefit to cost continues to show C ranked above A. A glance at the last two columns will assure the reader that B^/K^and (B^−K)/K^will give the same ranking, as the latter ratio is derived from the former simply by subtracting unity from it. We can, then, ignore the B^/K^ratio and compare the (B^− K^) with the (B^− K^)/K^ranking.

Now, if there is a capital budget of exactly 100, it may seem reasonable to be guided by the (B^− K^)/K^ratio ranking, and therefore to choose C rather than A. This is rational enough if it is established that one can have either A alone or, instead, five of the C streams. The outlay for five of the C streams uses up exactly the budget of 100, and produces a DPV of five times 50, or 250 – which is 100 more thancanbe obtained by choosing to put the whole of the 100 inoptionA.

But if it is not possible to have more than one C investment option, we could be misled by using the ranking method (B^− K^)/K^, for although this ratio is higher for C thanit is for A, the excess net benefit for A, 50, is greater thanthat for C, which is only 30.

Let us, therefore consider ranking by the (B^− K^) method and, to make things more awkward, let us assume also that option A is indivisible. In that case, by choosing option C, we are left with anoutlay of 80 from the original budget of 100. The relevant question now is: are there any opportunities for investing this remaining outlay of 80?

Allowing that there are no other public investment options available, we must recognize that there is always the private investment sector. If the average rate of returnover time inthis private sector happens to be equal to perpetuity of 8 per cent per annum, then a sum equal to 80 invested in the private sector could be said to yield a returnof 6.4 inperpetuity.

At a given rate of discount of 5 per cent, this perpetuity of 6.4 has a DPV of 128. Adding this B^of 128 to the B^of 50 (from putting 20 in the C option) gives a

Table 25.4

K^ B^ (B^− K^) B^/K^ (B^− K^)/K^

A 100 150 50 1.5 0.5

C 20 50 30 2.5 1.5

QUAH: “CHAP25” — 2007/1/25 — 08:02 — PAGE 145 — #7 Alleged superiority of DPV vs IRR 145 total B^of 178, which is more than can be obtained by investing the 100, instead, wholly inoptionA, which yields a B^ of only 150, at least if we ignore further reinvesting possibilities.

Thus, if we do adopt the ranking method (B^− K^)/K^, it is tacitly assumed that each of the investment options being considered may be multiplied in such a way as to ensure that the magnitude of the K^ is the same for each of them.

Where this cannot be assumed, we should be advised to use the (B^ − K^) method of ranking and use the stratagem above in order that each option use a K^ of the same magnitude. This can always be done, in the last resort if nec-essary, by investing the ‘spare’ funds of any option in the private investment sector.

In sum, for a valid ranking of alternative investment options, we must first make sure we have created for each optiona K^ of the same magnitude. Once this is done, we shall, in fact, obtain the same ranking, whichever of the three methods we use – (B^− K^), B^/K^or (B^− K^)/K^.

6 Finally, a brief word about the treatment of a set of investment streams that have different life spans, some beginning earlier than others, some later.

A correct ranking of such streams requires, ideally, only one particular adjust-ment: that each of these investment streams be compounded forward to a common terminal year, this common terminal year clearly being that year in which the final net benefit (positive or negative) occurs in that investment stream stretching furthest into the future.

To illustrate, if we measure time in years along a horizontal axis, four invest-ment streams to be compared can be represented as successive horizontal lines of different lengths, as in Figure 25.3.

It will be seen that investment stream A begins in year 0 and ends in year 7.

Two successive and complementary streams are generated by project B, the first beginning in year 3 and ending in year 7, the second being from year 8 to year 10.

0 1 2 3 4 5 6 7 8 9 10 11 12

A B

C D

Figure 25.3

QUAH: “CHAP25” — 2007/1/25 — 08:02 — PAGE 146 — #8 The C investment stream is from year 4 to year 8, while the D stream is from year 5 to year 12.

Year 12 is thento be accepted as the commonterminal date for the four investment streams. In Chapter 28, where the normalization procedure is elab-orated, we shall find that there is no difficulty in compounding the first three investment streams, A, B and C, to the terminal year 12.

QUAH: “CHAP26” — 2007/1/25 — 08:00 — PAGE 147 — #1

26 Investment criteria in an ideal