Introduction

We begin with what may be viewed as a small miracle. Two investors, Ann and David, expect random incomes amounting to random variables (r.v.) X1 and X2, respectively. We do not exclude the case where the X’s may take on negative values, which corresponds to losses. For simplicity, suppose X1 and X2 are independent with the same probability distribution. Then X1 and X2 have the same expected value m = EfXig and variance s2 = VarfXig. Assume that Ann and David evaluate the riskiness of their investments by the variance

of income, and being risk averse, they want to reduce the riskiness of their future incomes. To this end, Ann and David decide to divide the total income into equal shares, so each will have the random income

Y = 1 2 (X1+X2):

Then for both, Ann and David, the expected value of the new income will be

EfYg= 1 2 (m+m) = m;

that is, the same as before sharing the risk. On the other hand (see Chapter 0 for more details), the variance

VarfYg= 1 4 (s2+s2) =

2 ;

is half as large. Although this result is easy to prove, this is a key fundamental fact. And it is indeed

quite astonishing. The riskiness of the system as a whole did not change, the r.v.’s X1 and X2 remained as they were, but the level of risk faced by each participant has decreased. Now, consider n participants of a mutual risk exchange, and denote their random incomes

by X1; :::;Xn. Assume again that the X’s are independent and identically distributed, and set m = EfXig and s2 = VarfXig. If the participants divide their total income into equal shares, then the income for each is

Y = X1+ :::+Xn

n :

In this case (see again details in Chapter 0),

EfYg= m, while VarfYg= s 2

n ;

and for large n, the variance is close to zero. Thus, for a large number of participants, the risk of each separate participant may be reduced nearly to zero. The phenomenon we observed in its simplest form is called redistribution of risk. It

is at the heart of most stabilization financial mechanisms, certainly including insurance.

People use insurance because they can redistribute the risk, making it small for each if the number of participants is large. Insurance companies play the role of organizers of such a redistribution. Of course, they do it for profit, although there are non-profit organizations of mutual insurance. With some exaggeration, one may say that the theory we study in this book deals with

various generalizations of the scheme above. To have a general picture, let us consider a brief outline of the book.