Advances in Risk Management

298 NPV PROBABILITY DISTRIBUTION OF RISKY INVESTMENTS

We require the series to be stationary in mean and in variance which implies that they have
a constant mean and that they fluctuate about the mean with a constant variance. Such
constraints imply that the generating mechanism of the series remains constant through
time. We therefore consider series from which the trend has been removed, or in which it
was never present (for example,μX=0). The cash flows are therefore expressed in terms
of their deviation to such a trend, and:

X ̃t= ̃εt fort=1, 2, 3,...,n (A.2)

We furthermore assume that the random termsε ̃tobey a first-order autoregressive process
defined by:

ε ̃t=ρ ̃εt− 1 +u ̃t fort=2, 3, 4,...,n (A.3)

Stationarity in variance imposes the following additional condition:

0 ≤ρt,t− 1 < 1

Since the process begins at a specific date, we impose the following initial condition:

ε ̃ 1 =u ̃ 1

Finally, we introduce the following probabilistic assumptions:

E(u ̃t)= 0

V(u ̃t)=σ^2 u=1 constant for allt

Cov(u ̃θ,u ̃τ)=0 for:θ=τ

Cov(u ̃t, ̃εt− 1 )=0 for:t=2, 3, 4,...,n

Under such conditions, we demonstrate that the density probability function ofX ̃ ̄con-
verges towards a Normal distribution. In other words, first-order autocorrelation between
undiscountedcashflowsdoesnotinvalidatetheCentralLimitTheorem. Todemonstration
such an assertion we may write from equation (A.3):

ε ̃t=

∑t−^1

τ= 0

ρτu ̃t−τ fort=1, 2, 3,...,n (A.4)

and equation (1) may be rewritten as:

̃X=

∑n

t= 1

̃εt

n

=

1

n

∑n

t= 1

t∑− 1

τ= 0

ρτu ̃t−τ

Thus yielding:

̃X=^1

n

[

u ̃ 1

n∑− 1

τ= 0

ρτ+u ̃ 2

n∑− 2

τ= 0

ρτ+···+u ̃n− 2

∑^2

τ= 0

ρτ+u ̃n− 1

∑^1

τ= 0

ρτ+u ̃n

]

for example:

̃X=^1

n

[

u ̃ 1

(

1 −ρn

1 −ρ

)

+u ̃ 2

(

1 −ρn−^1

1 −ρ

)

+u ̃ 3

(

1 −ρn−^2

1 −ρ

)

+···

+u ̃n− 2

(

1 −ρ^3

1 −ρ

)

+u ̃n− 1

(

1 −ρ^2

1 −ρ

)

+u ̃n

(

1 −ρ

1 −ρ

)]