PENSIONES

The previous discussion provides us with the necessary framework for the rigorous examination of the various rules of thumb proposed in the literature.

Take for example the case where the only significant budget constraint is in the first year. This implies that all the duals 0. are zero apart' from 0 and the solution to the dual L.P is

* 1 NTV

given by merely accepting projects in the ranked order of i.

TVj(1) which is the familiar ratio of terminal value to initial outlay, the Lorie-Savage (55) solution*. In figure one this rank is generated by descending the axis.

On the other hand the ratio of discounted benefits to discounted benefits to discounted costs might be considered more appropriate for cash flows spread over several years**.

This is equivalent to setting 3 ^ = 0 and 32 > 0 in the example and can be achieved by the rank ^X^-2 ^ or equivalently by using the rank defined by descending the 02 axis.

A third familiar rule of thumb is ranking projects by internal rate of return. This is equivalent to making another approximation to the dual, namely by putting 0fc = 0t+^(1+i) t=l,2,...,T-2 3.3.1 and

Bw . - (i-r) 3.:

T-l

and using i as a parameter. In terms of figure 3.2.1. this is equivalent to ranking along the parameterised curve

* Bernhard (71) correctly analysed this ratio using a method of analysis similar to the one developed here, though he failed to extend his analysis to the case where the binding constraint was otlier than in the first year.

** See Quirin (67)

t The section is based on analysis carried by Atkins in the paper by Ashton and Atkins (74).

8t - (i-r)(l+i)T_1" t t- 1 .... T-l 3.3.3

more simply for the two dimensional case under discussion

B1 " 62 (l+r+B2) 3.3.4

Another frequently suggested rule is to rank by some measure NTV as discounted benefits/discounted costs, that is b y _.. and

TV^ (T-l) to calculate the IRR of the marginally accepted project. The suggestion is now to rerank projects again by NTV/TV(T-1) but using the internal rate of return of the marginally rejected project as the new discount rate, in this case r = 20% as the project is F.

The idea behind this is that this rate is a better aDproximation to the 'true' opportunity cost of funds. The assumptions behind this idea were discussed in section 1.2. This is equivalent to a second approximation to the dual by making

BT-1 3 + (i-r) 3.3.6

BT-2 (i-r)(B+l+i) 3.3.7

Bt “ (l+i)Bt+1 for t=l,2.... T-3 3.3.8

K - (i-r) (B+l+i) (l+i)T-2_t for t=l, 2,...,T-3 3.3.9 where i is now a constant, the internal rate of return of the

marginally rejected project, in this case 20% and 3 is the parameter.*

In the example i=0.2, r=0.1 and the reranking is equivalent to ranking along the line defined by B2 » 3 + 0.1 3.3.10

*The proof of this was first derived by Atkins in the paper by Ashton and Atkins (74). It is reproduced in appendix XIV.

88

■= 0.1 (6+1 .2) 3.3.11

or equivalently the line

ex - 0.1B2 + 0.11 3.3.12

which is shown dotted in the diagram. The new ranks, which could be calculated from the original data as being in the order A B F G D C E, corresponds to the ranks along this line. The implication behind this approach is of course to continue to rerank until no further changes occur.

It should now be plain that not only can many of the traditional rules of thumb be investigated by means of the approximations that they imply to the dual, but also conversely that almost any continuous monotonic non-decreasing function of the 6^,'s has an implication as some form of ranking procedure.

Now such an observation would be of practical significance only if rankings obtained from the various rules of thumb were roughly similar.

In this type of model, this is likely to be true since the rankings in each period are computed from the relative values of NTVj/TVj(t) where

the least weight is given to the most recent. This smoothes the

Further simplification occurs because we need only to consider the

a net absorber of funds. Typically this is for only the first few TV

j

+ (1+r)TV. (t-1) with TV. (t) - -c

3 33

3.3.13

ranking of a project whilst TV^(t) >0, i.e. whilst the project is

All these factors help to reduce the number of intersections of the lines and hence to reduce the number of alternative possible rankings. In this context it can be noted that the axis-ranks play a very special role in that they really define extreme project ranks and hence span all possible rankings. Thus if the axial ranks are quite similar so also will be any other rank, including such 'average' ranks as internal rate of return. This result alone can often simplify problems.

Take the example above, and accept projects in the ranked order along the axes B^ and B^ •

TABLE 3.2.2

NTV T V (1)

NTV T V (2)

Totally accepted Partially accepted Rejected

A,B,G,F A,B,D

C F

D,E,(H) C,E,G,(H)

Thus immediately A and B can be accepted, E and H rejected, leaving just C,D,F and G as possible marginal projects. In fact

more than this can be claimed as can be seen by inspection of the actual NTV/TV(t) ratios as below.

Year 1 Year 2

C 0.20 0.09

D 0.16 0.12

F 00 0.10

G oo 0.05

Project F clearly dominates both C and G in the sense of having a higher rank in each year and will always be chosen in preference,

90

which leaves the principal choice to be between D' and F or even both. In this way mere inspection of the axis ranks can often reduce the number of likely combinations down to very few. In

this case only two real options remain, either to accept D completely and F partially at 0.26 or F completely with D at 0.43, the latter being also the IRR solution incidently. This simple case also illustrates a point worthy of further consideration. Once the marginal projects have been identified, a task which it is argued is not laborious for most financial models, then the final choice is most likely to be made on the grounds of criteria other than the purely financial. Thus the two remaining options above differ by about *j% in the final plan value, which is likely to be of much less practical significance than many other features of projects D and F that have not been considered in this simple model.

A further observation supports the claim that in practice the number of plausible rankings might be quite small. In the large number of experiments carried out on these types of models in the development of this thesis seldom were there solutions in which the 8 t are non-zero in more than two or three years. In fact, Heingartner’ s own result, in which a twenty-six year horizon model ultimately had only one non-zero is by no means untypical.

It is, of course, simple enough to artificially generate a project set in which every 8t is positive, it need only contain as many projects as years. The point is that this seldom seems to occur on real project sets. This will be returned to below, but its practical importance will be emphasized here.

Firstly, knowing which 6t are likely positive means that the dominance analysis above need only be done in those years. Secondly, and somewhat conversely, the dominance analysis usually helps to highlight the years in which 6fc > 0 anyway. Thus in the example above, the two options of D or F partially accepted both imply year two as the bottleneck. In which case only the NTV/TV(2) ranking is relevant, leading to the optimal solution below

A B D F

V 1 W

l V 2 W

2 D

Year 1 -100 -100 -50 0 0 70 0 0 180

Year 2 - 50 - 50 -10 -14 0 -77 0 100 100

Return 246 256 89 28 -110 - 407.6

where project A, B, D are fully accepted, with project F partially accepted at 28%. Any deficit or surplus funds result in borrowing and lending decisions.

3.4 A rule of thumb solution to Weingartner's Horizon Model