290 NPV PROBABILITY DISTRIBUTION OF RISKY INVESTMENTSTable 15.2 Calculated Chi-square table normal distributionε ̃=ρ ̃εt− 1 +u ̃t
kc|rho 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.00 5.0 2.7 7.9 5.5 3.6 1.4 3.6 10.9 5.7 6.4
0.05 7.3 14.7 9.0 7.3 5.7 0.6 8.6 3.7 3.9 3.9
0.10 4.8 3.9 6.9 2.3 3.3 12.2 17.2 15.9 11.7 14.2
0.15 4.8 11.1 3.0 10.2 5.1 11.8 4.1 3.3 7.1 1.7
0.20 2.9 4.2 7.8 2.5 2.0 3.1 3.2 4.8 3.7 7.3
0.25 7.9 4.3 3.9 4.6 5.1 1.5 8.5 4.8 1.7 9.9
0.30 3.2 2.0 15.7 4.6 5.2 3.5 2.2 0.9 1.9 11.0
0.35 4.1 4.5 4.0 3.6 8.9 7.6 14.0 5.9 4.1 8.3
0.40 3.0 13.5 15.1 2.6 8.1 2.5 1.8 11.3 9.2 4.4
n= 10 NS=5,000 χ^27 (α= 0. 01 )= 18. 48 χ^27 (α= 0. 05 )= 14. 07
Table 15.3Calculated Chi-square table uniform distributionε ̃t=ρ ̃εt− 1 +u ̃t
kc|rho 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.00 1.4 12.1 8.7 12.6 4.2 3.5 7.8 10.3 8.1 5.4
0.05 5.1 8.6 4.1 3.1 5.8 7.8 5.6 8.3 7.6 9.9
0.10 7.8 15.6 1.1 5.8 6.5 4.2 5.4 4.1 3.9 13.5
0.15 9.9 6.2 7.3 9.2 8.4 5.4 0.8 6.8 9.6 4.9
0.20 7.2 6.4 7.4 7.2 18.6(∗) 14.8 12.9 7.0 11.2 11.4
0.25 6.7 11.8 16.9 16.1 8.7 5.8 7.3 10.2 10.0 11.7
0.30 6.1 10.0 15.6 12.9 9.8 14.3 15.9 16.0 14.6 17.5
0.3 11.9 13.0 13.8 13.0 16.1 20.7(∗) 24.1(∗) 7.7 20.6(∗) 13.9
0.40 24.1(∗) 14.4 18.0 32.8(∗) 15.4 17.4 14.8 14.5 24.0(∗) 20.0(∗)
n= 10 NS=5,000 χ^27 (α= 0. 01 )= 18. 48 χ^27 (α= 0. 05 )= 14. 07
The following three Tables (15.3, 15.4 and 15.5) illustrate the results for
uniformly distributed cash flows over three different time periods: 10, 20 and
40 years for a first-order autoregressive process. The bolded single asterisked
(*) values represent cases where the null hypothesis has been rejected at the
1% level of significance.
These results make it quite clear that highly correlated cash flows do
not invalidate the effectiveness of the CLT at ensuring the converging of
the NPV probability distribution towards a normal distribution. We also
observe that the CLT is not invalidated for discount rates lower than 30%,
which is still quite high. Only at extremely high discount rates, such as 40%,
can we say that the CLT is invalidates and the more so when a 5% level