As of 2002 tremendous upheavals in corporate institutions that involve great

¬rms are taking place throughout the world especially in the United States and

Germany. This will no doubt result in drastic changes in the near future in the size

distribution of ¬rms, and the recent frequent mergers and occasional breakdowns of

¬rms may even require a new methodology. We will not address these aspects, but it

is safe to predict new theoretical and empirical research along these lines.

Distributions of Actuarial Losses

Coincidentally, the unprecedented forest ¬res that recently occurred in the western

United States (especially in Colorado and Arizona) may challenge the conventional

wisdom that “¬re liabilities are rare.” The model of the total amount of losses in a

6 INTRODUCTION

given period presented below may undergo substantial changes: In particular, the

existing probability distributions of an individual loss amount F(x) will no doubt be

reexamined and reevaluated.

In actuarial sciences, the total amount of losses in a given period is usually

modeled as a risk process characterized by two (independent) random variables: the

number of losses and the amount of individual losses. If

pn (t) is the probability of exactly n losses in the observed period [0, t],

.

F(x) is the probability that, given a loss, its amount is x,

.

F Ãn (x) is the nth convolution of the c.d.f. of loss amount F(x),

.

then the probability that the total loss in a period of length t is x can be expressed

as the compound distribution

X

1

pn (t)F Ãn (x):

G(x, t) ¼

n¼0

Although the total loss distribution G(x, t) is of great importance for insurers in

their task of determining appropriate premiums or reinsurance policies, it is the

probability distribution of an individual loss amount, F(x), that is relevant when a

property owner has to decide whether to purchase insurance or when an insurer

designs deductible schedules. Here we are solely concerned with the distributions of

individual losses.

1.3 BRIEF HISTORY OF THE MODELS FOR STUDYING

ECONOMIC SIZE DISTRIBUTIONS

A statistical study of personal income distributions originated with Pareto™s

´

formulation of “laws” of income distribution in his famous Cours d™economie

politique (1897) that is discussed in detail in this book and in Arnold™s (1983) book

Pareto Distributions.

Pareto was well aware of the imperfections of statistical data, insuf¬cient

reliability of the sources, and lack of veracity of income tax statements. Nonetheless,

he boldly analyzed the data using his extensive engineering and mathematical

training and succeeded in showing that there is a relation between Nx ”the number

of taxpayers with personal income greater or equal to x”and the value of the income

x given by a downward sloping line

log Nx ¼ log A À a log x (1:1)

or equivalently,

A

A . 0, a . 0, x . x0 ,

Nx ¼ , (1:2)

xa

7

1.3 HISTORY OF ECONOMIC SIZE DISTRIBUTIONS

x0 being the minimum income (Pareto, 1895). Economists and economic statisticians

(e.g., Brambilla, 1960; Dagum, 1977) often refer to a (or rather Àa) as the elasticity

of the survival function with respect to income x

d log {1 À F(x)}

¼ Àa:

d log x

Thus, a is the elasticity of a reduction in the number of income-receiving units when

moving to a higher income class. The graph with coordinates (log x, log Nx ) is often

referred to as the Pareto diagram. An exact straight line in this display de¬nes the

Pareto distribution.

Pareto (1896, 1897a) also suggested the second and third approximation equations

A

A . 0, a . 0, x0 þ x . 0,

Nx ¼ , (1:3)

(x þ x0 )a

and

A Àbx

, A . 0, a . 0, x . x0 , b . 0:

Nx ¼ ae (1:4)

(x þ x0 )

Interestingly enough, equation (1.2) provided the most adequate ¬t for the income

distribution in the African nation of Botswana, a republic in South Central Africa, in

1974 (Arnold, 1985).

The fact that empirically the values of parameter a remain “stable” if not

constant (see Table 1.1”based on the ¬tting of his equations for widely diverse

economies such as semifeudal Prussia, Victorian England, capitalist but highly

diversi¬ed Italian cities circa 1887, and the Communist-like regime of the Jesuits

in Peru during Spanish rule (1556 “ 1821)”caused Pareto to conclude that human

nature, that is, humankind™s varying capabilities, is the main cause of income

inequality, rather than the organization of the economy and society. If we were to

examine a community of thieves, Pareto wrote (1897a, p. 371), we might well ¬nd

an income distribution similar to that which experience has shown is generally

obtained. In this case, the determinant of the distribution of income “earners”

would be their aptitude for theft. What presumably determines the distribution in a

community in which the production of wealth is the only way to gain an income is

the aptitude for work and saving, steadiness and good conduct. This prevents

necessity or desirability of legislative redistribution of income. Pareto asserted

(1897a, p. 360),

This curve gives an equilibrium position and if one diverts society from this position

automatic forces develop which lead it back there.

In the subsequent version of his Cours, Pareto slightly modi¬ed his position by

asserting that “we cannot state that the shape of the income curve would not change

8 INTRODUCTION

Table 1.1 Pareto™s Estimates of a

a

Country Date

England 1843 1.50

1879“ 1880 1.35

Prussia 1852 1.89

1876 1.72

1881 1.73

1886 1.68

1890 1.60

1894 1.60

Saxony 1880 1.58

1886 1.51

Florence 1887 1.41

Perugia (city) 1889 1.69

Perugia (countryside) 1889 1.37

Ancona, Arezzo, Parma, Pisa (total) 1889 1.32

Italian cities (total) 1889 1.45

Basle 1887 1.24

Paris (rents) 1887 1.57

Augsburg 1471 1.43

1498 1.47

1512 1.26