296 NPV PROBABILITY DISTRIBUTION OF RISKY INVESTMENTSUsing the development of log(1+z), given by log(1+z)=z−z
2
2 +
z^3
3 −
z^4
4 +..., we obtain
the logarithm of the characteristic function in terms of its cumulants:
√∑αt ̃εt
∑α 2
t=∑n
t= 1logφ ̃εt
αth
√∑
α^2 t
=∑nt= 1
i√αt
∑
α^2 thK 1 −
1
2
√αt
∑
α^2 t
2
h^2 K 2 −
i
3!
√αt
∑
α^2 t
3
h^3 K 3+
1
4!
√αt
∑
α^2 t
4
h^4 K 4 +...
whereKjis the cumulant of orderj. By assumption, we have setK 1 =0etK 2 =1 whereas
it is obvious that
∑n
t= 1(
√∑αt
α^2 t) 2
=1. Hence: √∑αt ̃εt
∑α 2
t=∑n
t= 1logφε
αth
√∑
α^2 t
=−h^2
2
−
i
3!∑n
t= 1
αt
√∑
α^2 t
3
h^3 K 3+
1
4!∑nt= 1
√αt
∑
α^2 t
4
h^4 K 4 +...which may be written as:
√∑αt ̃εt
∑α 2
t=∑nt= 1logφε
αth
√∑
α^2 t
=−h^2
2
−
i
3!∑nt= 1(
α^2
∑
α^2 t) 3 / 2
h^3 K 3+
1
4!∑n
t= 1(
α^2 t
∑
α^2 t) 2
h^4 K 4 +...and quite obviously:
nlim→∞α^21
∑n
t= 1α^2 t=nlim→∞
(1+kc)−^2
∑n
t= 1(1+kc)−^2 t=
(1+kc)^2 − 1
(1+kc)^2
= 1 −
1
(1+kc)^2
= 0wheneverkc =0.