ANALYSIS OF THE PERFORMANCE OF SEVERAL SETS OF PROJECTS IN A STOCHASTIC WORLD.

Uncertainty Is Introduced through the use of model (3.37) presented In chapter III. The model Is applied to generate the cash flow P 1t of project 1 In time period t In the way described in section IV.3.1. The sets in each group are going to be compared in terms of their expected NPV and in terms of a percentile taken as a measure of risk. To calculate these statistics the control variable technique of variance reduction was used in the way detailed in chapter V.

Table 6.3 presents the results for the expected values of the NPV obtained from 250 evaluations of this summary measure with the stochastic cash flows. The interest rate used was 6% , as in the deterministic situation. The three tax situations previously

considered, the old tax, the new tax and no tax systems, were also examined. Again, as one would expect, the Introduction of taxation greatly reduces the NPV value, and much more so with the old tax system than with the new tax system. Comparing the results of tables 6.Z and 6.3 shows that there is a clear decrease In the NPV values under the tax regimes, but no change in the no tax situation. The values of the groups of projects are system atically lower under uncertainty and taxation so that any values obtained deterministically are biased. This may occur because the system of tax allowances ensures that the high revenues Incur a proportionaly greater tax liability than low ones and also because the benefits of the tax system cannot be so accurately exploited, when stochastic values are present. It is also an important result given that in practice deterministic evaluation is common. The reduction due to uncertainty is 183 for group 1 under the old tax system, and 6.7» for group 2 under the new tax system.

Uncertainty also changes the rankings of the groups of projects. Group 1 is now only ranked third under the old tax system despite that system being used in its generation. Under the new tax system group 2 (generated under that system) Is

edged Into second place by Group 3 set 1. Group 3 set 1 Is now

Table 6.3. Summary of results for the stochastic values of the NPV calculated with an interest rate of 6%.

Old tax system New tax system No tax Group 1 Group 2 Group 3: Set 1 Set 2 Group 4 Group 5 630.852 (3) 652.327 (2) 655.509 (1) 619.510 (4) 612.045 (5) 61 1.417 (6) 928.034 (5) 967.706 (2) 968.514 (1) 948.493 (6) 915.462 (6) 943.065 (4) 1893.34 (4) 1912.58 (3) 1892.05 (5) 1928.36 (1) 1880.15 (6) 1917.95 (2)

Inside th e p arenthesis Is th e ranking of the s e t In term s of the NPV. The groups w ere obtained a s follows:

Group 1 _ M ILP model fo r th e old tax system ; Group 2 _ M ILP model fo r th e new tax system ; Group 3 _ P ro je c t s w ith th e highest NPV values:

se t 1 _ the b e st 15 projects; s e t 2 _ the 21 p ro jects w ith p o s itive NPV;

Group 4 _ M ILP model fo r th e new tax sy s te m w it h an upper bound on the unused c apital allowance;

Group 5 _ M axim isin g th e p ro b a b ility th a t the to ta l N P V e x c e e d a ta rg e t value Po-910.

ranked 1 under both tax systems despite the fact that the selection criterion (best is projects) Is somewhat arbitrary.

Group 3 set 2 (all projects with a positive NPV) does not perform well under tax and uncertainty. The findings on rankings are thus inconclusive but the preeminence of selection by deterministic mathematical programming incorporating taxation is lost when the projects are placed in an uncertain world.

To assess the riskiness of a set of projects it can be Important to know the probability of falling below a certain fixed value, K, which represents a threshold of financial trouble for the firm, in real situations. In the situation of this research, a value Ktt for K was fixed so that there was an a % probability of the best set of projects In each of the three tax situations being less than Ka . The values of a were taken to be 1,5, 10, 15 and 20 and the Ka values, given in table 6.^, were estimated from normal distributions with parameters estimated during the expected values calculations.

The percentiles obtained are given in table 6.5, The best performers within both tax systems, in terms of the means, continue to be the best performers in terms of the percentiles: group 3 - set 1 and group 2. Group 5, which was constructed

controlling the variability in some way, behaved well with the new tax system but less well with the old tax system. The other group for which it was hoped to control the variability, group 4, behaved rather badly. In the no tax situation there is a change in the ranking which is similar when considering the percentiles compared to the deterministic and stochastic mean results.

Table 6.4. Ka values used in the percentile calculations. a Old tax system New tax system No tax

1 % -1247.973 -1362.765 -1499.631 5% -707.139 -701.322 -533.140 10% -418.784 -348 662 -17 839 15% -224.167 -1 10.644 329.951

T ab le 6.5. Sum m ary of r e s u lt s fo r the p e rc e n tile va lu e s .

Old tax system New tax s ystem No tax

Group 1 .0414 (5) .0336 (5) .0157 (3) Group 2 .0302 (2) .0303 (2) .0088 (1) Group 3: set 1 .0087 (1) .0088 (1) .0088 (1) set 2 .0316 (4) .0313 (4) .0161 (5) Group 4 .0503 (6) .0384 (6) .0231 (6) Group 5 .0309 (3) .0304 (3) .0157 (3) Group 1 .1035 (6) .1046 (5) .0605 (4) Group 2 .0698 (2) .0643 (2) .0483 (2) Group 3: set 1 .0551 (1) .0514 (1) .0308 (1) set 2 .0958 (4) .0922 (4) .0604 (3) Group 4 .1051 (5) .1056 (6) .0724 (6) Group 5 .0926 (3) .0829 (3) .0618 (5) Group 1 .1407 (5) .1437 (5) .1152 (4) Group 2 .1141 (2) .1063 (2) .0854 (2) Group 3: set 1 .0995 (1) .0961 (1) .0743 (1) set 2 .1348 (3) .1291 (4) .1210 (6) Group 4 .1489 (6) .1659 (6) .1142 (3) Group 5 1378 (4) .1238 (3) .1161 (5) Group 1 .1717 (3) .1772 (3) .1718 (4) Group 2 .1540 (2) .1603 (2) .1213 (2) Group 3: set 1 .1230 (1) .1233 (1) .1148 (1) set 2 .1866 (5) .1898 (6) .1663 (3) Group 4 .1849 (4) .1846 (4) .1872 (6) Group 5 .1900 (6) .1889 (5) .1758 (5) Group 1 .21 15 (3) .2418 (5) .2061 (3) Group 2 .1835 (2) .1809 (2) .1837 (2) Group 3: set 1 .1587 (1) .1671 (1) .1710 (1) set 2 .2234 (5) .2273 (4) .2205 (6) Group 4 .2309 (6) .2509 (6) .2167 (4) Group 5 .2198 (4) .2267 (3) .2174 (5)