286 NPV PROBABILITY DISTRIBUTION OF RISKY INVESTMENTSvariables, but rather a sum of discounted cash flows, then the shape of the
NPV distribution would be dominated by the early cash flows, the more so
the higher the discount rate. Wagle (1967) would correctly conclude, without
providing any mathematical proof:
Thus even if we had independently distributed cash flows continuing forever,
the variance of the present value of the first n cash flows would remain finite as
n→∞, and in this case it is known that the distribution of the present value will
not tend to normality unless each of the net cash flows is normally distributed.
(Wagle, 1967: 18)It is true that most versions of the CLT apply to a direct sum of indepen-
dent random variables. However, as Wagle correctly argues, the fact that
the NPV is the sum of discounted random cash flows does invalidate the
CLT asymptotic convergence of the NPV probability distribution towards
a Normal distribution. As shown in Appendix 1, the CLT does not apply
strictly to the NPV probability distribution whenever the discount rate dif-
fers from zero; unless, of course, the probability distribution of cash flows is
Normal.
As for the assumption of probability independence between net cash
flows, a certain number of authors (Hoeffding and Robbins, 1948) have
extended the CLT to the case of dependent random variables. However,
the conditions under which these theorems are stated require conditional
distributions, are subjected to very restrictive conditions, or involve spe-
cial conditions which are difficult to comply with or to assess, most of all
in the context of cash flow analysis. We prove in Appendix 2 that proba-
bility independence is not a necessary condition for obtaining asymptotic
convergence towards a Normal distribution. More specifically, we show
that the sum of equally-weighted first-order autoregressive cash flows con-
verges toward a Normal probability distribution. Still, in the case where
one would be dealing with discounted and first-order autoregressive cash
flows, the NPV probability distribution would not strictly comply with the
CLT. Considering that in a strict sense the discount rate, however small, ulti-
mately invalidates the CLT, must we conclude, for practical purposes, that
the CLT should never be used? Before exploring such a matter by simulation,
let us consider (Appendix 1) the logarithm of the characteristic function in
terms of its cumulants:
√∑αtεt
∑α 2
t=∑nt= 1logφε
αth
√∑
α^2 t
=−h^2
2−i
3!∑nt= 1
αt
√∑
α^2 t
3h^3 K 3+1
4!∑nt= 1
αt
√∑
α^2 t
4h^4 K 4 +...