Advances in Risk Management

JEAN-PAUL PAQUIN, ANNICK LAMBERT AND ALAIN CHARBONNEAU 287

Table 15.1Discount rates and the first term factor of

the cumulants of the NPV probability distribution

kc K 3 K 4 K 5

0.01 0.00276 0.000388 0.000034

0.05 0.02834 0.008643 0.002635

0.10 0.07230 0.03012 0.01254

0.15 0.12042 0.05946 0.02936

0.20 0.16890 0.15277 0.05160

0.25 0.21600 0.12960 0.07776

0.30 0.26088 0.16669 0.10651

0.35 0.30310 0.20367 0.13682

0.40 0.34278 0.23990 0.16789

0.45 0.37972 0.27496 0.19916

0.50 0.41408 0.30860 0.23004

We show in Appendix 1 that the limit of the first term in the expansion of
the factor multiplying each cumulant is given by:

lim

n→∞

α^21

∑n

t= 1

α^2 t

= 1 −

1

(1+kc)^2

= 0 whenever kc= 0

Whenkc=0, then all the cumulants of an order higher than 3 are multi-
plied by a weight of 0, thus ensuring the asymptotic convergence of the
NPV probability distribution towards the Normal probability distribution.
So what happens as to the effectiveness of the CLT when starting from 0 the
discount ratekcis increased progressively? Table 15.1 gives, for increasing
values of the discount ratekc, the limit value of the first and the largest term
serving as weight for cumulants of order 3, 4 and 5.
FromTable15.1, itisobviousthattheimportanceofthevariouscumulants
decreases as their order increases. Therefore, there is no need to consider
higher order cumulants. On the other hand, the importance of the weights
of the various cumulants increases as the discount rate is increased. Also, the
relative difference between the cumulant factors decreases as the discount
rate is increased. In other words, higher-order cumulants acquire relative
importance as the discount rate is increased. For low values ofkc,from1%
to 10%, we could be justified in assuming that the cumulants of order higher
than 2 might not hamper the effectiveness of the CLT’s asymptotic properties
concerning the NPV probability distribution. For higher values ofkc, that is
rates over 30%, it would seem quite plausible to assume that cumulants of
order higher than 2 might invalidate the asymptotic properties of the NPV
probability distribution.