Advances in Risk Management

294 NPV PROBABILITY DISTRIBUTION OF RISKY INVESTMENTS

However and as soon as a positive discount rate is introduced into the
NPV equation, then the CLT does not apply in a strict mathematical sense.
In fact, the higher the investment project discount rate and the less the
CLT would be applicable to the NPV probability distribution. The authors
explore through simulation runs and statistical testing, using the normal,
uniform and double exponential probability distributions, the boundaries
limiting the applicability of the CLT in ensuring convergence towards a
Normal distribution.
In summary, the results are the following: managers and analysts are justi-
fied in invoking the CLT when assigning the normal probability distribution
to an investment project probability distribution. As long as the cash flows
are bell-shaped or uniformly distributed, the CLT may be invoked however
highly serially correlated are the project cash flows and the discount rate.
However, those projects whose cash flows have extremely high or low val-
ues with high probability may invalidate the CLT whenever the discount
rate exceeds 15 to 20 percent; for low discount rates, the CLT would still be
effective and reliable.

APPENDIX 1: THE CLT AND THE NPV PROBABILITY

DISTRIBUTION

LetPbe the present value of net cash flowsXtover a period ofnyears. Now, let us assume
that these net cash flows are random variablesX ̃twith the additional features of being
independent in probability and stationary in mean and in variance. The present valueP
must therefore be considered as a random variableP ̃equal to the weighted sum of then
net random cash flowsX ̃t:

P ̃=

∑n

t= 1

X ̃t(1+kc)−t

wherekc, the cost of capital, is the appropriate discount or hurdle rate. We posit that:

X ̃t=μX+ ̃εt (μXis a constant or a trend)

We further require the cash flow series to be stationary in mean and in variance. The cash
flows are therefore expressed in terms of their deviation to such a trend, and without loss
in generality, we may write:

X ̃t= ̃εt fort=1, 2, 3,···,n

These random error terms are assumed independent in probability and obey the following
probabilistic assumptions:

E(ε ̃t)= 0

V(ε ̃t)=σu^2 =1 constant for allt

with Cov(ε ̃τ,ε ̃θ)=0, forτ=θ.