1 The normalized CTV procedure is designed to transform the stream of net benefits B1, B2, . . . , BT, arising from an initial outlay K into an equivalent stream, 0, 0, . . . , TV (B), this being shorthand for the terminal value of the stream of net benefits generated by the initial outlay K. From this TV (B) we are to subtract TV (K), or the terminal value of the outlay K, conceived as the terminal value of the opportunity cost of investing the sum K.
What we call normalization is the requirement that, in ranking two or more projects, not only must they have the same terminal date T, but also a common initial outlay K. This requirement is in no way restrictive.
If, for example, of two mutually exclusive projects, the initial outlay required for project Y is, say 80, this 80 being less than the outlay of the 100 that is required for project X , the outlay that is to be commonto both projects must be the larger, being 100 in this example. If the project Y is to be undertakenthenanadditional 20 of outlay must be spent. It may be possible to spend this 20 on an additional project that is a quarter the size of Y and yields benefits that are also a quarter the size of project Y . If, however, the project is not one that is divisible, the 20 to be spent can, at least, be returned to the private investment sector where, if returns are continually reinvested, will produce a terminal value of 20(1 + ρ)T, which must be added to the terminal value of project Y that requires anoutlay of only 80. If, however, this 20 is returned to the government, it will generate a terminal value that will depend upon how the government disposes of it.
Similarly, if the life-spanof project X is 20 years, and that of the alternative investment is only 16 years, the common terminal period is 20 years. The orig-inal termorig-inal value of the benefits of the project that is reached in the sixteenth year must then be compounded forward to the twentieth year. This compound-ing for the additional four years must, however, follow the relevant pattern of behaviour.
To illustrate, consider first the simple case in which all the benefits, B1, B2, . . . , B16, are wholly consumed. If their terminal value in the sixteenth year amounts to, say, 250, this being the equivalent worth in the sixteenth year whenall the preceding Bs are compounded to this sixteenth year at society’s rate of time deference r, thenthe terminal value of the Bs inthe twentieth year will be 250(1 + r)4.
QUAH: “CHAP30” — 2007/1/25 — 18:59 — PAGE 163 — #2 Normalized CTV criterion (I) 163 The other simple case is that in which all the annual returns are directed by the political decision maker to be invested and reinvested in the private sector at rate ρ until the sixteenth year. If the resulting terminal value in the sixteenth year comes to, say, 350 – this 350 being conceived as an increase in the capital stock – the terminal value in the twentieth year is equal to 350(1 + ρ)4.
Ina more general case inwhich, of the full returns annually paid to subscribers, fraction c is consumed, the remaining (1 − c) invested at ρ, the terminal value of the benefits in the sixteenth year may be divided into two parts:1(a) the equivalent terminal worth in the sixteenth year of all the amounts consumed, say 220, and (b) the increase in the capital stock in the sixteenth year that arises from the amounts each year that are invested in the private sector at the average yield ρ. We suppose this increase in the capital stock to be equal to 60.
Part (a) of the terminal value in the sixteenth year, equal to 220 will become equal to 220(1 + r)4 inthe twentieth year. As for part (b), equal to 60, of the returnto this additional capital ineach of the additional years, the fractionc is consumed and the remaining fraction (1−c) invested. Therefore, at the end of four more years, we must add, first, the value of consumption (equal to the amounts consumed over the four years compounded forward at rate r), say this is equal to 20. Then, we must also add the further increase in the capital stock for four more years that results from the amounts invested in each of those years. We suppose this to equal 8.
Thus, using the figures we have adopted, the terminal value in the sixteenth year, equal to 220 + 60, or 280, is extended to a terminal value in the twentieth year that is equal to 220(1 + r)4+ 60 + 20 + 8.
Needless to remark, similar principles must be used in extending for an additional four years the terminal value in the sixteenth year of the initial outlay K.
Finally, it should be noted, that this method of extending projects to a common terminal year where necessary is applicable also where the alternative to one or more projects is that of two (or several) successive projects that, usually, have relatively shorter lifespans: applicable also where one or more of the projects can be undertakenat a later date thanthe others.
2 The principle to be followed in compounding forward requires attention to the disposal of each of the benefits right through to the terminal year T. Inthe general case, the fraction c of the benefit in each year is consumed, the remaining fraction (1 − c) being invested at ρ (unless otherwise directed). The terminal value of the amount consumed in year t being cBt, it is compounded forward to become cBt(1 + r)T−tinthe terminal year T. The remaining part (1 − c)Btthat is invested
1 There can also be a general case with yet another behaviour pattern: namely, that in which each year only a part of the returns that year, say two-thirds, is paid out to subscribers, the remaining one-third being invested in the private sector at yield ρ (or else invested in some other designated public project). Of the amounts, received each year by the subscribers, fraction c is, again, consumed, the remainder being invested at yield ρ. Although the calculation required to extend the terminal value in the sixteenth year to that in the twentieth year is a bit more elaborate, no new principle is involved.
QUAH: “CHAP30” — 2007/1/25 — 18:59 — PAGE 164 — #3 inyear t is to be conceived as anadditionto the capital stock. As such, it yields an annual return that is equal to (1 − c)Bt(1 + ρ) in the following year and in each subsequent year to year T, of which annual returns, the fraction c is consumed and the remainder invested, and so on.
We may as well consider, first, the two simple cases mentioned earlier, (28.1) PVr(B) > K and (28.2) PVρ(B) > K, wheneach of these is transformed correctly into the CTV criterion. The first case, that in which c is equal to unity, requires that each of the benefits, being wholly consumed, is compounded to the terminal year at society’s rate of time preference r. Their terminal value is therefore equal to B1(1 + r)T−1 + B2(1 + r)T−2 + . . . BT, while the terminal value of the initial outlay is equal to K(1 + r)T.
The second simple case is that in which c is equal to zero, as a result of which each of the benefits is invested and reinvested at rate ρ. The stream of benefits givenby the project will thenhave a terminal value equal to B1(1 + ρ)T−1+ B2(1 + ρ)T−2+ . . . BT, this terminal value being conceived as the increase in the capital stock that is contributed by the project in question. As for the initial outlay K, its terminal value is equal to K(1 + ρ)T.
In the more general case in, which c is a positive fractiongreater thanzero but less than unity, we must treat the part of the benefit that is consumed differently from the part that is invested. The amount of the benefit Btthat is consumed has a terminal value of Bt(1 + r)T−t. The terminal value of the remaining part (1 − c)Bt
that is invested is not so easy to calculate. Although this much is to be added to the terminal capital stock, this additionto the capital stock inyear t also produces, in each of the following years until year T, an annual return equal to (1−c)Bt(1+ρ) of which, again, fraction c is consumed each year and the remainder invested at rate, ρ. An d so on .
3 The exact method of calculation will be easier to understand if we suppose that the project to be considered is one that generates a stream of benefits, B1, B2, . . . , BT, each annual benefit being equal to 10 million. In addition, we shall let c = 0.8, r = 0.05 and ρ = 0.1 and take T to equal 10.
(i) Of the B1of 10 million, therefore, 8 million will be consumed, the remaining 2 million being invested in the private sector at interest rate r = 0.05. The 8 million consumed has a terminal value, when compounded at the rate of time preference r, equal to 8 million (1.05)T−t. As for the 2 millionthat is invested that year, it is far more prolific, as we shall see, for it adds that much to the private capital stock which is thena part of the terminal value.
(ii) This 2 million of additional capital yields an annual return of 200,000 begin-ning in the second year and ending in the terminal year, or tenth year. Of each of these nine annual returns of 200,000, 160,000 is consumed, its ter-minal value therefore requiring that it be compounded to the tenth year at a rate equal to 0.05. Altogether, they contribute to the terminal value a total of 160,000(1.05)8+ 160, 000(1.05)7+ . . . + 160,000.
QUAH: “CHAP30” — 2007/1/25 — 18:59 — PAGE 165 — #4 Normalized CTV criterion (I) 165 The remaining 40,000 that is invested each year, from the second to the tenth year, must also be added to the capital stock.
(iii) Now each one of the successive annual investments of 40,000, beginning in the second year, will itself generate an annual return of 4,000, starting the year after the investment took place, and continuing until the tenth year. (Thus, the 40,000 invested in the second year will generate a return of 4,000 in year 3, 4,000 in year 4, and so on until year 10. The 40,000 invested in the third year will also generate a returnof 4,000 inthe fourth year, 4,000 inthe fifth year, and so on until year 10, and similarly for each of the 40,000 in subsequent years).
(iv) Of each of these annual returns of 4,000, the amount 3,200 will be consumed, the remaining 800 invested, each 800 invested giving rise in the following year to an annual return until the tenth year of 80, and so we can continue.
Thus, we may reckon up the total number of additional returns so far that have been generated by the 2 million invested from the first benefit of 10 million, as follows: (ii) 9 of 200,000 plus (iii) 8 of 4,000 plus (iv) 7 of 80, and so on to the terminal year.
Having completed all these calculations, we now recognize that the amount we have added to the capital stock and the terminal value of all amounts consumed are those that flow only from B1– from the 10 milliongenerated by the project in year 1. Clearly the same calculations must be undertaken for each of the subsequent benefits. B2, B3, . . . , BT that we have conveniently assumed to be also equal to 10 million. Clearly, the calculations required for each of the successive year’s benefits will, as we approach closer to the terminal year, be smaller than the calculations required for the preceding year.
4 Turning to the calculation of the terminal value of the initial outlay K, the procedure is no different from that above. Thus, only in two simple cases mentioned inwhich the two criteria, PVr(B) > K and PVρ(B) > K are correctly transformed into their corresponding CTVcriteria, will the terminal values of K be, respectively, K(1 + r) and K(1 + ρ)T.
Inall cases, the terminal value of K, so calculated, is conceived as the terminal value of the opportunity cost of any one of the public projects under consideration;
that is the terminal value of K is calculated as what it would amount to if it were left in its current use or in some specifically designated alternative use.
5 The preliminary calculations above have been undertaken to show how the cor-rect terminal value of the given ten-year stream of net benefits is to be determined onthe simple but commonassumptionthat, inordinary circumstances, people generally save a proportion of their annual incomes. And since it follows that their incomes grow over time, so also does the amount being saved annually.
Wherever the actual behaviour patterndiffers from this commonassumption, the terminal value of any given stream of net benefits will, of course, also be different.
Inparticular, it may be necessary to modify the simple assumptionthat people
QUAH: “CHAP30” — 2007/1/25 — 18:59 — PAGE 166 — #5 save a given proportion of each annual net benefit in two ways: (i) where only a proportion of the annual net benefits, say w, is paid out as income to subscribers to the project; and (ii) where, in addition, such income is subject to income tax.
Ineither case – ofteninboth cases – the calculationof the terminal value will be yet more exacting. Incase (ii), where each year a proportionof income received by the subscribers to the project is taxed, it is necessary to follow to the terminal year the disposal by the government of the additional revenues it receives in that year – at least if the government’s annual disposal of the additional revenues takes a pattern that is different from that which would be taken if, instead, such revenues were left to be disposed of inthe usual way by the subscribers to the project (since, if the patterns were the same, the terminal value would remain the same whether the annual income received by the subscribers were taxed or not).
6 Finally, a brief word about the possibility that some or all of the sum needed for the project is to be borrowed from abroad.
Clearly, the eventual repayment of the sum borrowed, say M, takes place in some future year or years. If the whole of M is repaid inthe terminal year T, it will feature as a negative benefit in year T. In addition, each of the annual interest payments to the foreign country will appear as negative benefits. Consequently, there may be negative net benefits in some years.
There can, of course, be different arrangements for the payment of interest on the sum borrowed and also for the eventual repayment. But the above guidelines will suffice to determine their treatment.
QUAH: “CHAP31” — 2007/1/25 — 07:59 — PAGE 167 — #1