Advances in Risk Management

288 NPV PROBABILITY DISTRIBUTION OF RISKY INVESTMENTS

15.5 THE NPV PROBABILITY DISTRIBUTION AND THE CLT:

SIMULATION MODELS AND STATISTICAL TESTS

Our fundamental assumption is to the effect that as long as the discount
rate does not exceed a threshold value (to be determined statistically), then
the CLT applies to the NPV, for example, the NPV probability distribution
does not differ significantly from a Normal distribution. To test such an
assumption we have resorted to a simulation experiment using three prob-
ability distributions: (a) a uniform distribution, (b) a double exponential
distribution, and finally (c) a normal distribution. The uniform probability
distribution was chosen because it represents the case where uncertainty is
at its maximum (for example, maximum entropy). This distribution is sym-
metrical and represents any extreme case for which decision-makers have
very limited information. Thus, this case would apply when probability
distributions are relatively symmetrical and are bell-shaped. The double
exponential has the feature of having a thick tail. This might reveal particu-
larly important and instructive considering that risk analysis, in the context
of financial distress, focuses on the lower-tail of a probability distribution.
However, we are quite aware that the double exponential distribution does
provide a realistic description for net cash flows. As for the normal probabil-
ity distribution, it was used to ensure, in accordance with our demonstration
in Appendix 2 and contrary to what many authors have stated, the validity
of the CLT even in situations with highly correlated cash flows.
The density of the uniform probability distribution is defined in the
following fashion:

fU(u ̃t)=







1 /2 foru ̃t∈[−1,+1]

0 otherwise

withEU(u ̃)=0 andσU(u ̃)=

√

3 / 3

The double exponential distribution has the following density:

fDE(u ̃t)=

1

2

e−|u| withEDE(u ̃)=0 andσDE(u ̃)=

√

2

The normal probability distribution is given the standardized form:

fN(u ̃t)=

1

√

2 π

e−

u^2

(^2) with EN(u ̃)=0 andσN(u ̃)= 1
To generate these random variables, we have used the University of Water-
loo’s Maple.8 simulation software. Simulations of 5,000 runs (NS: number of
simulations) were carried out respectively on first-order and second-order
autoregressive processes for increasing values of the NPV discount ratekc.
The first-order autoregressive process is defined by the following model:
ε ̃t=ρε ̃t− 1 +u ̃t for : 0≤ρ< 1