1 Suppose we are faced with a choice of four investment options with the net benefits shown in Table 22.1. Which, if any, do we choose if our budget is limited to 100?
If we had to choose only one from the four, we could be sure that it would never be A4, irrespective of the criterionused. For A3is as good as, or better than, A4
period for period. In the jargon, investment option A3‘dominates’ A4. Thus, if we subtract A4’s net benefit stream from that of A3, the difference is a series, 0, 20, 0, 0, 10, these figures showing the amounts by which A3’s net benefits exceed those of A4insuccessive periods. Inno period is A3’s net benefit less than that of A4.1
Let us now consider three rather crude investment criteria, which, however, are commonly employed in the business sector, especially where the venture contemplated is risky.
2 The cut-off period is perhaps the crudest possible criterionthat is used in business in order to decide whether or not to invest in a project. A suitable period is chosen over which the money invested must be fully recouped. The period could be tenyears, though usually a shorter period such as five years or evenless is chosen. Such a criterionmay be justified incases of innovationinproducts or methods that cannot be protected by a patent, and which innovations are likely to be copied by competing firms within two or three years. A cut-off period of
Table 22.1
0 1 2 3 4
A1 −100 115 0 0 0
A2 −100 20 30 50 170
A3 −100 100 110 −50 0 A4 −100 80 110 −50 −10
1 If, however, we had funds enabling us to choose two or more of these investment options, we might choose both A3and A4. But we should never include A4while rejecting A3.
QUAH: “CHAP22” — 2007/1/25 — 08:03 — PAGE 126 — #2 three years, for instance, may be chosen in the belief that, after three years, further profits are uncertain and increasingly unlikely. Glancing down the table, it is clear that a cut-off period of three years after the initial outlay would admit the A1
investment option. Indeed, more than the initial 100 is recouped in the first year after the outlay. The A2optiononly just scrapes home. A3would be able to recoup as much as 160 inthe three years, while A4would recoup 140.
The shortcomings of this criterion are easy to perceive. If the returns were not expected to accrue mainly in the first few years but mainly after the first few years, worthwhile projects would be rejected. A stream –100, 0, 0, 20, 40, 60, 80, 120,
… would be rejected. So also would a stream –100, 20, 20, 20, 20, 20, 20, . . . . 3 Pay-off period (or capital recovery method): instead of choosing an arbitrary cut-off period, we may rank the investment options according to the number of years necessary to recoup the initial outlay. Clearly, project A1would be ranked first, as its pay-off period is less thana year. For project A2it is exactly three years.
If we ignore subsequent outlays inthe last two projects, it is one year for A3and more thanone year for A4.
The pay-off period rate of return is but another way of expressing the above results. It is obtained simply by dividing 100 by the number of years in the pay-off period. Since the A1investment option pays off the initial outlay in less than a year; we divide 100 by something less than a year, which yields a pay-off period rate of returnof more than100 per cent. For project A2, the pay-off period rate of returnis equal to 100 divided by three, or 33.33 per cent. For the A3project, it is exactly 100 per cent, and for project A4, it is somewhat less than100 per cent.
The justification for either form of this ranking device is similar to that for the cut-off period. Whenimitationby competitors or rapid obsolescence is anticipated, or in circumstances of political uncertainty, one of the overriding considerations is safety. One looks for quick returns and prepares for a hasty exit. A project such as A1, which pays 115 within a year of 100 being invested, is likely, in such circumstances, to be looked onwith greater favour thanoptionA2, which would not show any profit until the fourth year. The method is easy to understand and gives a decisionquickly without much further analysis.
In the complete absence of uncertainty, however, it would be impossible to justify either of the above rules-of-thumb. If interest rates happened to be low, A2
would be far more profitable than A1and more profitable than A3for that matter.
And like the cut-off period, the method takes no account of the time value of money, and favours short-term investment over long-term ones.
4 The average rate of return is the simplest way of taking account of all the figures in the investment stream. Just because all the figures are taken at face value incalculating the average rate of return, there is animplied assumptionthat all the figures have been corrected for uncertainty.
For all investment options with only an initial outlay of 100, such as A1and A2
in Table 22.1, there is no ambiguity in the method. One simply adds together all the subsequent positive net benefits, divides this sum by the number of years and
QUAH: “CHAP22” — 2007/1/25 — 08:03 — PAGE 127 — #3 Crude investment criteria 127 expresses the resulting figure as a percentage of the initial investment outlay. For option A1, the sum of positive benefits is 115. This sum divided by the number of years, in this case only one year, gives an average net benefit of 115 per annum.
Expressed as a percentage of the outlay of 100, it is 115 per cent. For A2, the sum of positive net benefits over four years is 270 and, dividing by four, gives an annual average returnof 67.5. Expressed againas a percentage of the outlay of 100, it is 67.5 per cent.
For investment options A3and A4, which happento have outlays also inlater years, we add together all the figures, both positive and negative, after the initial outlay of 100 and proceed as before. For A3, the algebraic sum of net benefits over three years (years 1, 2 and 3) comes to 160. This sum divided by three yields an average of 53.33 per cent per annum. Similarly for option A4which yields an average return of 32.5 per cent per annum.
The weakness of this method is apparent at once, for it is by no means evident to anyone thinking of investing 100 that A1with anaverage of returnof 115 per cent is superior to A2. It might be added in passing, however, that the weakness is not particular to this method, but arises also in the more familiar internal-rate-of returnmethod, which will be treated later.
5 The average rate of return including the initial outlay is anobvious modifi-cationof the preceding criterion. The average rate of returnis calculated inthe same way except that the initial outlay of 100 has to be included as a negative net benefit before dividing by the number of years.
Thus, inoptionA1, we subtract the outlay of 100 from 115 to give 15, this being a 15 per cent net benefit on the outlay of 100. In A2, subtracting the 100 outlay from the 270 of net benefits leaves us with 170 net benefits, and dividing by four yields a net average return of 42.5 per cent per annum on the outlay of 100. Similar calculations yield a net average return of 20 per cent per annum for three years on the 100 outlay for the A3option, and an average of 7.5 per cent per annum for the four years for option A4.
Under conditions of certainty, at least, this criterion, although clearly superior to the others, is unsatisfactory for two reasons. First, the results depend critically upon the number of years counted on this criterion. But determining the length of the investment stream by reference to the number of years for which there is a net benefit, positive or negative, is arbitrary. If, for example, option A1were altered so as to yield a slight positive net benefit, say 0.1, in year 2, the annual net benefit, spread over two years, would be little more than7.5 per cent onthe outlay of 100, which makes this slightly altered A1optionfar less attractive.
Second, a less apparent but no less serious defect is that this criterion takes no cognizance of the profile of the net benefits over time. Given the algebraic aggregate of all the net benefits following the initial outlay of 100, it makes no difference to the calculation whether the net benefits are bunched together over the first years, spread evenly over the time span, or are bunched together toward the end of the time span. Thus, an investment stream of –100, 5, 20, 25, 250 is valued as highly onthis criterionas one of –100, 250, 25, 20, 5; or, for that matter,
QUAH: “CHAP22” — 2007/1/25 — 08:03 — PAGE 128 — #4 an investment stream of –100, 1, 1, 1, 297 is valued as highly as an investment stream of –100, 297, 1, 1, 1.
Yet, who would not prefer the latter stream to the former! People do indeed take account of the timing of benefits: they are not indifferent as between receiving 100 now and 100 in ten years’ time. Once we take account of the time dimension of the net-benefit stream, we are impelled to move away from these rather primitive investment criteria and move on to those that are more familiar to economists – those inwhich 100 inany year is valued more than100 inany subsequent year.
6 Although all investment criteria resort to the common procedure of reducing a stream of net benefits to a single figure at one point in time, usually the present time,2the economist invariably uses some rate of interest (or some combination of rates) as a means of weighting the net benefits in successive years.
The more familiar criteria fall into two categories: (i) those that draw on a given rate (or rates) of interest in reducing the investment stream to a present discounted value, and (ii) those that, in contrast, are used to discover just what is the average rate of return on the initial outlay. This latter criterion, known as the internal rate of return, may be defined as that rate of discount which reduces the entire investment stream (including the initial outlay) to zero.
These two popular criteria, the present discounted value and the internal rate of return, will be compared in the following two chapters.
2 However, as we shall propose later, the future compounded value, taken at a terminal year, offers certainadvantages.
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