1 The internal rate of return (IRR) is a more respectable form of the average rate of returnmentioned inChapter 22 inthat, like the DPV method, it takes account of time.
A simple example will illustrate how the IRR is calculated. If we have a stream of net benefits −100, 50, 86.4, we can discount each of these net benefits to the present, t = 0, using a discount rate of 20 per cent. The present value of the net benefit of 50 in year 1, when discounted at 20 per cent, is 50/(1 + 0.2), or 40, while the present value of 86.4 in year 2, when discounted at 20 per cent, is 86.4/(1 + 0.2)2, or 60. The present value of both 50 in year 1 and 86.4 in year 2 is, therefore, 40 + 60, or 100; which is exactly equal to the initial neg-ative net benefit, or net outlay, of 100. This 20 per cent discount, just because it equates the present value of the positive net benefits to the present value of the net outlay, is taken to be the internal rate of return of the above stream of net benefits.
The above example serves to illustrate a commondefinitionof the IRR as being equal to that rate of discount, say λ, which when applied to a stream of net benefits, would make them equal to the initial outlay K; hence, the formula
K =n
t=1
Bt
(1 + λ)t or, alternatively, K −
n t=1
Bt
(1 + λ)t = 0
inwhich we solve for λ.
2 There could, however, be additional outlays in future years. If, for example there were additional outlays, say K2and K5inyears 2 and 5, respectively, these outlays have somehow to be discounted to the present and added to the initial outlay K in year 0. And if discounted, the relevant rate must also be λ which has, it may seem, yet to be determined.
Yet, on reflection, it will be understood that these later outlays can be left in place and, regarding them as negative net benefits to be discounted to the present along with the positive net benefits, so retaining the above formula. For example, given an initial outlay of 100 at time zero, a net benefit of 220 at the
QUAH: “CHAP24” — 2007/1/25 — 19:07 — PAGE 136 — #2 end of the first year and an outlay of 121 at the end of the second year – thus an investment stream equal to −100, 220, −121 – a discount rate equal to 10 per cent, reduces the present value of that stream to zero. The IRR is, therefore, equal to 10 per cent.
Inasmuch, then, as in conforming to the formula, both outlays and benefits are to be discounted to the present at the IRR, it will sometimes be neater to obviate mention of the initial outlay K and any subsequent outlays, treating them instead as negative net benefits in the year they occur. The formula then becomes
n t=1
Bt
(1 + λ)t = 0, inwhich Bt canbe positive, negative or zero
It will be convenient, henceforth to continue using negative Bs for any subse-quent outlays. Nonetheless, it should be understood that, although there can also be negative net benefits in future years that are in fact not cash outlays, being instead possible losses arising from environmental damage or compensatory payment, it makes no difference in the calculation of the IRR – or, for that matter, of the DPV, giventhe discount rate.1
3 The sense in which the internal rate of return, so defined, is an average over time is conveyed by the example of a man investing, say, 100 for five years. If the internal rate of return of some given investment stream were 25 per cent per annum, the manwould have inmind an equivalent, though simpler, investment in which his 100 in the present grows by 25 per cent each year. He sees his 100 in the present becoming 125 by the end of the first year, 15614by the end of the second year, and so on, to reach 100(1 + 0.25)5by the end of the fifth year. More generally, if the investment stream in question were −100, B1, B2, B3, B4, B5, where the Bs are any pattern of benefits, and the internal rate of return were known to be 25 per cent, then an equivalent investment stream would be −100, 0, 0, 0, 0, 100[(1 + 0.25)5], for this given investment stream, when discounted to its present value at 25 per cent, is, by assumption, equal to zero, and so also is the equivalent stream. Consequently, if a man is told that the internal rate of an investment stream over n years is equal to λ, he is justified in thinking of the investment as equivalent to one in which his initial outlay is compounded forward at the rate of λ per annum for n years.
Thinking of the IRR in this way, the man will want to compare any such invest-ment with the opportunities for putting his money into other securities, either equities or government bonds. If the only alternative open to him, or the only
1 Later, when we come to introduce a normalization procedure in which net benefits, positive or negative, are to be compounded forward, the composition of each net benefit must be considered.
In that case, a negative net benefit that is a cash outlay will, in general, be treated differently from a negative net benefit that is, say, a collective bad (such as the ambient pollution) that is inflicted on the community.
QUAH: “CHAP24” — 2007/1/25 — 19:07 — PAGE 137 — #3 Internal rate of return 137
Log of revenue
P
C
O G0 M9 t M
E
N9 N
Q G
Q9
Figure 24.1
alternative he will consider, is long-term government bonds, perpetuities say, yield-ing 6 per cent per annum,2an investment yielding a rate of return of more than 6 per cent (always assuming certainty or, at least, equal certainty) will be preferred to the purchase of these 6 per cent government bonds.
4 Let us now return briefly to the growth curve of the preceding chapter, depicted here in Figure 24.1. The reader will recall that using the tangency condition between the growth curve G0G and the highest 5 per cent discount curve at Q determined the optimal gestation or investment, period. Here however, we have followed the convention of drawing the discount curve as straight lines by measuring the loga-rithms of the values along the vertical axis – thus successive x per cent differences appear along it as equal distances.
If the highest discounted present value OP, at the givensocial rate of discount (which we continue to suppose is 5 per cent) determines the optimal period OQ, do we obtain the same result using, instead, the IRR? We should hardly expect so, since this optimal OQperiod itself will vary with the particular magnitude of the rate of discount adopted, being longer the lower the rate of discount.
But first, how do we represent the IRR on this diagram? The answer is that it can be represented by the slope of a straight line from C to any point of the growth curve, say M. For this slope is determined by tan θ, which is equal to the excess
2 In the modern economy there is, of course, a wide diversity of government bonds even if we restrict ourselves to long-term issues. We simplify the treatment, for the time being, by assuming there is only one type of long-term government bond, say ‘perpetuities’, i.e. interest-bearing bonds with no redemptiondate, such as British Consols.
QUAH: “CHAP24” — 2007/1/25 — 19:07 — PAGE 138 — #4 benefit EM at time M(that is, total benefit MM less the cost OC at time zero) divided by time OMinyears. This ratio givenby tanθ must be the IRR simply because, as required by definition, it is the rate of interest which reduces the total future benefit MM to a present value that is equal to the initial cost OC.
Now the highest IRR possible is, on this construction, determined by the slope of the straight line from C that just touches the growth curve G0G at point N, there being no straight lines from C steeper than CN that canalso just touch this growth curve.
If the optimal investment period is now defined as that yielding the highest IRR, this will be a period equal to ON. This optimal IRR period is clearly shorter than OQ, the optimal period onthe net present value criterionwith a givenrate of discount. So which is it to be? Do we let the tree grow for a period OQ, or do we cut it downafter ON years? This is not the only sort of problem in which the results obtained using these two investment criteria differ. We shall defer the resolution of this apparent discrepancy,3however, until we have illustrated other discrepancies in the results obtained using IRR as compared with using DPV.
3 A hint may be allowed the impatient reader, however. If, after time ONthe proceeds NNcould be reinvested in an identical tree-growing project, there would be a loss by, instead, letting the tree continue to grow to QQ. What is at issue, then, are the reinvestment possibilities whenever the tree is cut down. This reinvestment aspect of the problem is treated in some detail in later chapters on the ‘normalization’ procedure.
QUAH: “CHAP25” — 2007/1/25 — 08:02 — PAGE 139 — #1