Creación del Real Seminario de Nobles de Madrid

1 If we have an investment −100, 50, 150, and we are given a rate of interest or discount rate of 10 per cent per annum, the DPV of the net benefits alone that are generated by the initial outlay of 100−50 at the end of the first year, and 150 after the second year is given by the calculation

50

(1 + 0.1)+ 150

(1 + 0.1)² = 169.4

In this connection, an initial outlay of 100 has a present value also of 100, being incurred at the end of the zeroth year.

The DPV of the entire stream, −100, 50, 150, is equal to the DPV of the net benefits (positive or negative): here, 169.4, minus the DPV of the outlays, here 100, and is therefore equal to 169.4−100, or 69.4.

In more general terms, given a stream of benefits generated by only a single initial outlay K0, but which for the time being may be designated as B0, so enabling us to represent the entire stream as B0, B1, B2, . . . , Bn, where the Bs are positive, negative or zero, its net DPV is given by

B0+ B1

(1 + r) + B2

(1 + r)² + . . . Bn

(1 + r)ⁿ or more briefly by

t=n t=0

Bt

(1 + r)^t

where r is the rate of discount.

The necessary instrument in this criterion is the appropriate rate of interest or rate of discount by which the net benefit at any point of time is weighted. It is commonly assumed that the correct rate of interest is that which reflects society’s rate of time preference. (If, for example, society is taken to be indifferent between having $100 million today and $106 million next year, the social rate of time preference is 6 per cent per annum.) We shall, for the present, go along with this assumption, though later on it will be argued that it is correct only under special

QUAH: “CHAP23” — 2007/1/25 — 08:01 — PAGE 130 — #2 conditions. In a Crusoe economy, if 120 bushels of corn next year are deemed by Crusoe to be equivalent insatisfactionto 100 bushels of corntoday – by which is meant that he is perfectly indifferent, to having either 100 bushels of corn today or 120 bushels of cornina year’s time – Crusoe’s rate of discount is 20 per cent per annum. Until Man Friday arrives and has some say in the decision, Crusoe’s individual rate of discount can also be thought of as the social rate of discount.

2 To be more accurate, however, Crusoe’s reactionto the choice presented to him gives us no more than the social rate of discount for the one year and, for that matter, is strictly valid only for 100 bushels of corn this year, not for more or for less. If, indeed, the same rate of discount did hold for successive years, then Crusoe would be indifferent as between 100 today, 120 next year, 144 in the year following that, and so on. It is, however, quite possible that his discount rate rises with the passage of time. Instead of being indifferent as between 100 today and 144 intwo years’ time, he might specify 150 intwo years’ time. This would mean that for the first year his rate of discount is 20 per cent, but for the second year he uses a discount rate of roughly 25 per cent per annum.

Again, even if we confine ourselves to the one year, it is not true that the same rate of discount holds for any amount of corn. If Crusoe agrees, though only just, to postpone consumptionof 100 bushels of cornthis year inorder to have an additional 120 bushels next year, it does not follow that he will be prepared to forgo another 100 bushels of corn this year in exchange for another additional 120 bushels next year. It is more plausible to suppose that he should want more thananadditional 120 bushels next year to persuade him to forgo this year the consumption of yet another 100 bushels; say an additional 140 bushels next year.

We could say that Crusoe’s marginal willingness to sacrifice 100 today for 120 next year reflects a discount rate of 20 per cent per annum, while his marginal willingness to sacrifice 200 today for 260 tomorrow reflects a discount rate of 30 per cent overall. Put otherwise, we could say that for the first 100 bushels the marginal discount rate was 20 per cent, and that the marginal discount rate for another 100 bushels was 40 per cent.

These possibilities are to be noted before passing on. For it is also the case in society at large that, however the social rate of discount is determined, it is invariant neither with respect to the magnitudes of the inter-temporal exchange of goods nor to the length of time involved. If we have information about the variationinthe rate of discount with respect to either magnitude or time, however, there is no difficulty, in principle, in adapting our chosen investment criterion accordingly. In the meantime, our task will be simplified by assuming but a single social rate of discount. Moreover, since we are to examine this concept in Chapter 25, we shall also assume that this social rate of discount is known to us. If the reader prefers, he cansuppose, provisionally, that it has arisenfrom the interplay of market forces plus, perhaps, some form of government intervention that has the object of ensuring that the resulting rate of interest in the economy correctly reveals society’s preference as between present and future goods. If, for instance, the social rate of discount is 10 per cent per annum, we shall take it that society

QUAH: “CHAP23” — 2007/1/25 — 08:01 — PAGE 131 — #3 Discounted present value criterion 131 as a whole is indifferent as between100 today, 110 ina year’s time, 121 intwo years’ time, and so on. And that, therefore, the present value of 110 to be received inone year’s time, or 121 intwo years’ time, is exactly 100.

3 Having made these provisional simplifications, let us go on to consider the following propositions, all of them commonplace in the literature on the subject.

The net present value¹(or excess of present value over cost) of a particular investment stream depends upon the rate of discount used. If, for instance, the stream of net benefits is −100, 0, 150, the net present value of the stream would be a little less than 48 if the discount rate were 1 per cent, but would be −33¹₃ if the discount rate were 50 per cent.

It follows that, which one of a number of alternative investment streams yields the largest net DPV must depend, in general, on the rate of discount that is employed. Only if there is one investment stream that is “dominant” over all the others being contemplated, will it have a higher DPV irrespective of the dis-count rate. Thus, if there are but two investment streams, A having the stream −50, 20, 80, and B the stream −50, 20, 70, thenstream A, being dominant, will have a higher present value irrespective of the discount rate. But if, instead, there is an A stream of −100, 0, 180, and a B stream of −100, 165, 0, a discount rate of 0.01, or 1 per cent, ranks A, with a net present value of about 76, above B, which has a net present value of about 63. If, however, the appropriate rate of discount is 0.5, or 50 per cent, the net present value of the A stream is −20 and is therefore ranked below the B stream, which has a net present value of 10. From these two examples, it should be manifest that there is a particular social rate of discount between 1 per cent and 50 per cent for which the two streams have exactly the same present value. Let us call this social rate of discount r^∗. Then r^∗is easily determined by equating the net present value formulae for the two streams, i.e. we set

−100 + 180

(1 + r^∗)² = −100 + 165 (1 + r^∗)

and solve for r^∗, which turns out to be about 9 per cent.

In general, we can determine a net present value of a particular investment stream, say A, for each conceivable rate of discount. The resulting relationship canbe plotted inFigure 23.1, where the vertical axis measures PVr, or net present value of the investment stream in question, and the horizontal axis measures r, the social rate of discount. The net present value of the A stream becomes smaller, the larger the rate of discount r; hence the negative slope of the A curve. It will be noted that the negative slope crosses the horizontal axis and continues below it into the south-east quadrant. This indicates that, at discount rates above some critical rate of discount, the net present value of the stream becomes negative (for example,

1 The term ‘net present value’ will be used occasionally as an abbreviation of ‘net present discounted value’ or ‘net discounted present value’.

QUAH: “CHAP23” — 2007/1/25 — 08:01 — PAGE 132 — #4 A

PV_r

0 0.01 r* r_a r_b r

B

0.5

Figure 23.1

at a 50 per cent discount rate, the stream −100, 0, 180, has a net present value of

−20). A similar relationship can be plotted for the B investment stream.

If one of these two investment streams were dominant, it would lie above the other at all rates of discount. In the absence of dominance, the A and B curves will intersect, either inthe positive quadrant, as inthe figure, or else inthe negative quadrant (not shown).

For all conceivable (positive) discount rates – save one, r^∗– the present values of the two streams differ. At discount rates below r^∗, the A stream has a higher net present value than the B stream, the reverse being true for discount rates above r^∗. Only at r^∗do both streams have the same net present value. It is obvious that if the rate of discount, from being a little above r^∗, fell to a figure below r^∗, the net present value of the A stream would change from being less than that of the B stream to being greater than it.

It may be observed finally, that there is a discount rate corresponding to each investment stream, raand rbrespectively, for which the net present values of the A and B streams are both zero. By definition, therefore, ra and rbare, respectively, the internal rates of return of the A and B investment streams.

4 If now, the annual net benefits that are generated by an initial outlay (even-tually) become smaller over time and, also, instead of dividing the relevant time span into years or other unit periods, it is treated as a continuum, the resulting net benefit profile over time can be envisaged as a growth path, it being understood that no benefit is reaped at any date earlier than some given point of time, at which point the cumulative benefit may be discounted. Thus, rather than plot a profile of marginal net benefits over time, we can plot a profile over time of the total benefit, ineffect a growth path as represented inFigure 23.2, where time is measured along the horizontal axis and the total net benefit, or total value, reached at any point of

QUAH: “CHAP23” — 2007/1/25 — 08:01 — PAGE 133 — #5 Discounted present value criterion 133 V

Q M G

C

O 1 2 3 4 5 6 7

G₀

V₃

V₂

V₃ V2

V₁

V1

t (years)

Figure 23.2

time, is measured vertically. The vertical axis OV, which cuts the horizontal axis at time zero, can be used to measure the DPV reached at any point of time, given the rate of discount.

Common examples of such growth curves are those of trees or wine. Once a tree is planted, its value increases after a point in time, roughly in proportion to the increase in the volume of its timber (assuming a constant price of timber).

As for a barrel of wine, its value increases over time (up to a point) in con-sequence of the improvement in its flavour. Let us consider the timber example.

A continuous growth path is represented by G0G inFigure 23.2. At time zero, total costs OC (measured above O along the vertical axis OV) are incurred in purchasing the sapling and in employing the labour to plant the tree. Although the sapling may begin to grow immediately after planting, its wood will be worth nothing until, say, the end of the second year, from which point of time it grows invalue – at first more rapidly – to a maximum, after which it declines. Thus, the net value of the timber at any point in time is given by the vertical distance from that point of time.

Inorder to appreciate better the connectionbetweenthis growth curve and the preceding Figure 23.1, along with its examples, we could split the time axis into discrete units, say years, and measure vertically the total value, of the timber at the end of each successive year. Instead of a continuous growth path, we should then have a succession of vertical lines increasing in height up to point M. The heights of each of these vertical lines could then be regarded as the value of alternative investment streams. For instance, the vertical line above 4 on the

QUAH: “CHAP23” — 2007/1/25 — 08:01 — PAGE 134 — #6 horizontal axis could measure exactly 100. If OC measured aninitial cost of 50, the investment option corresponding to t = 4 would be −50, 0, 0, 0, 100.

The investment option corresponding to t = 5 could be −50, 0, 0, 0, 0, 112. The investment option corresponding to t = 6 could be −50, 0, 0, 0, 0, 0, 120, and so onfor t = 7, 8, 9, . . . , n.²Which of all the investment options would we choose?

Bearing in mind the preceding proposition, we need have no hesitation in affirming that, in general, it will depend upon the social rate of discount.

For any given discount rate, we may construct a number of V curves over time, each corresponding to a different present value. For a social discount rate that is equal to 5 per cent, one such discount curve V1V1would measure, say, 80 along the vertical axis at time zero. At a point directly above t = 1, the height of the curve would be 80 (1 + 0.5), or 84; at t = 2 the height of the curve would be 84 (1 + 0.5), or 88.2, and so on, the height at the end of each successive year being 5 per cent greater thanthat of the preceding year.

At a social discount rate that is supposed to equal to the social rate of time preference of, say, 5 per cent, society is deemed indifferent to receiving timber worth, say, 100 as measured at some point in time by the height of the V1V1curve, and receiving timber worth 105 a year later. Other 5 per cent discount curves such as V2V2, V3V3 and so on may be constructed on the same principle. The family of such 5 per cent VV curves is conceived as being ‘infinitely dense’, and optimizationrequires we select among them the highest VV curve that just touches the G0G growth curve. InFigure 23.2 this is V2V2, which just touches G0G at Q. The optimal growth period, or gestationperiod, is thenexactly six years. And the net present value of the timber that is cut down at the end of the sixth year is measured by the height OQ less the initial cost OC. It will be correctly surmised that point Q is one of mutual tangency between a VV discount rate curve and the growth curve.

It will be understood that, if the tree were not cut down at the end of the sixth year, it would continue to grow for a number of years. After point Q is reached at the end of the sixth year, however, the increase in its value falls below 5 per cent per annum. There is, therefore, more to be gained by cutting down the tree at the end of the sixth year, selling the timber and investing the proceeds at 5 per cent than the alternative of cutting the trees at a later date.

2 For expository convenience, we are ignoring the costs of tending the tree while it is growing. If these were constant at, say, 10 each year, the investment option correction to t = 4 would be −50, −10,

−10, −10, 100.

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