Advances in Risk Management

JEAN-PAUL PAQUIN, ANNICK LAMBERT AND ALAIN CHARBONNEAU 289

while the second-order autoregressive process is defined by:

̃εt=ρ 1 ̃εt− 1 +ρ 2 ε ̃t− 2 +u ̃t for:ρ 1 +ρ 2 < 1

ρ 1 −ρ 2 < 1

|ρ 2 |< 1

Thus, these two models comply with the conditions required for stochastic
processes to be stationary in mean and in variance (Kendall, 1976; Nelson,
1973). All initial conditions of the simulation experiment were set at: ̃ε 1 =u ̃ 1.
The chi-square distribution was used to test all the simulated NPV results
for any significant statistical difference between the simulated distribu-
tion and a standardized Normal distribution. For the null hypothesis to be
accepted, the differences between the theoretical and observed results must
be attributable to sampling variability at the designated level of significance.
The Normal probability distribution had been subdivided symmetrically
into 8 classes with nearly equal probability.
The null hypothesis H 0 is the following: The simulated NPV probability
distribution is Normal. The null hypothesis is to be rejected at the 1% level
of significance for a chi-square distribution with 7 degrees of liberty when
the calculated chi-square values are greater than 18.48. Otherwise, the null
hypothesis is not rejected. The calculated chi-square statistic is given by:

χ^2 c=

∑r=^8

i= 1

(ni−Nπi)^2

Nπi

whereN=total number of simulation runs;πi=theoretical probability
from the standardized Normal distribution; ni=number of simulation
observations in classi; andr=total number of classes (r=8).

15.6 THE NPV PROBABILITY DISTRIBUTION AND

THE CLT: SIMULATION RESULTS

Let us consider the case of normally distributed cash flows. Table 15.2 sum-
marizes the chi-square statistical test results for cash flows governed by a
first-order autoregressive process extending over a 10-year period.
These results make it quite clear that highly correlated cash flows do
not invalidate the normality of the NPV probability distribution and conse-
quently the effectiveness of the CLT. Naturally, using a high discount rate
does not invalidate the CLT. However, at the 5% level of significance, we
would have found six cases for which the null hypothesis would have been
rejected. This is quite normal, for this represents 6.66 percent of the total
number of simulation trials, a percentage in agreement with such a level of
significance.