3.2 HISTORY AND GENESIS

These assumptions re¬‚ect the fact that the managers on every level supervise a

constant number of people on the level below them (3.24), and that the salary of a

certain manager is a constant proportion of the aggregate salary of the people whom

he or she supervises directly (3.25). It is then natural to assume that

xiþ1 =xi ¼ np . 1. In the highest level there will be one person, in the next level

there will be n persons, then n2 , etc. Hence,

yi ¼ nkÀi :

The total number Ni of persons on levels li or above is therefore

nkÀiþ1 À 1

Ni ¼ 1 þ n þ n2 þ Á Á Á þ nkÀi ¼

nÀ1

and the proportion of all employees, Qi , in the ¬rm who are working on level li or

above is

Ni nkÀiþ1 À 1

% n1Ài :

Qi ¼ ¼ k À1

N1 n

From (3.25) we see that

xi ¼ (np)iÀ1 Á x1 :

Hence,

log n log n

log xi ¼: log c À a log xi

log Qi ¼ log x1 À

log np log np

and therefore,

Qi ¼ c Á xÀa ,

i

which is the Pareto distribution. Here the levels li are discrete, but it is easy to cover

the continuous case as well (Lydall, 1968, Appendix 4).

3.2.5 Further Approaches

Mandelbrot (1964) derived a Pareto distribution of the amount of ¬re damage from

the assumption that the probability of the ¬re increasing its intensity at any instant of

time is constant. He assumed that the intensity is described by an integer-valued

random variable N ”there is no ¬re when N ¼ 0, the ¬re starts when N becomes

equal to 1, and it ends when either N becomes equal to zero again or when all that

possibly can be destroyed has already been burned. Assume further that, at any

instant in time, there is a probability p ¼ 1=2 that the ¬re encounters new material,

70 PARETO DISTRIBUTIONS

increasing its intensity by 1, and a probability q ¼ 1=2 that the absence of new

materials or the actions of dedicated ¬re¬ghters decreases the ¬re™s intensity by 1. In

the absence of a maximum extent of damage and of a lower bound on recorded

damages, the duration of the ¬re will be an even number given by a well-known

result (familiar from the context of coin tossing),

1=2

( À 1)x=2À1 ,

P(D ¼ x) ¼ 2

x=2

which is proportional to xÀ3=2 for reasonably large x. Under the assumption that

small damages, below a threshold x0, say, are not even properly recorded, the extent

of damage is then given by

P(D . x) ¼ (x=x0 )À1=2 , x0 x,

a Pareto distribution with shape parameter a ¼ 1=2. [The value of the parameter

a ¼ 1=2 may seem somewhat extreme here; however, values in the vicinity of 0.5

were found to describe the distribution of ¬re damage in post-war Sweden (Benckert

and Sternberg, 1957).]

Shpilberg (1977) presented a further argument leading to a Pareto distribution as

the distribution of ¬re loss amount. Suppose that the “mortality rate” l(t) of the ¬re

is constant, equal to a, say, so that the duration of the ¬re T is exponentially

distributed. Under the assumption that the resulting damage X is exponentially

related to the duration of the ¬re,

X ¼ x0 exp (kT ),

for some x0 , k . 0, the c.d.f. of X is given by

x a=k

0

F(x) ¼ 1 À , x ! x0 ,

x

and so it follows a Pareto distribution. (Clearly, different speci¬cations of the hazard

(mortality) rate can be used to motivate other loss models, and indeed we shall

encounter the Weibull and Benini distributions that can be derived along similar

lines.)

3.3 MOMENTS AND OTHER BASIC PROPERTIES

The Pareto density has a polynomial right tail; speci¬cally, it is regularly varying at

in¬nity with index Àa À 1. Thus, the right tail is heavier as a is smaller, implying

that only low-order moments exist. In particular, the kth moment of the Pareto

distribution exists only if k , a, in that case, it equals

axk

k 0

:

E(X ) ¼ (3:26)

aÀk

71

3.3 MOMENTS AND OTHER BASIC PROPERTIES

Speci¬cally, the mean is

ax0

E(X ) ¼ (3:27)

aÀ1

and the variance equals

ax20

:

var(X ) ¼ (3:28)

2

a(a À 1) (a À 2)

Hence, the coef¬cient of variation is given by

1

CV ¼ p¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬ , (3:29)

a(a À 2)

and the shape factors are

r¬¬¬¬¬¬¬¬¬¬¬¬

p¬¬¬¬¬ aþ1 2

b1 ¼ 2 1 À , a . 3, (3:30)

aÀ3 a

3(a À 2)(3a2 þ a þ 2)

b2 ¼ a . 4:

, (3:31)

a(a À 3)(a À 4)

p¬¬¬¬¬

It follows that b1 ! 2 and b2 ! 9, as a ! 1.

For the extremely heavy-tailed members (with a , 1) of this class of

distributions, other measures of location than the mean must be used. Options

include the geometric mean xg ¼ exp {E( log X )}, here given by