ABSTRACT
This chapter concerns various rules of comparison and subsequent selection among risky alternatives.
What do we usually do when we choose an investment strategy in the presence of uncertainty? Consciously or not, we compare random variables (r.v.’s) of the future income, corresponding to different possible strategies, and we try to figure out which of these r.v.’s is the “best”. Suppose you are one of 2 million of people who buy a lottery ticket to win a single one
million dollar prize. Your gain is a random variable (r.v.)
x=
8<: 1;000;000 with probability 1=2;000;000;
0 with probability 11=2;000;000:
If the ticket’s price is $1, then your random profit is x1. If you have decided to buy the ticket, it means that, when comparing the r.v.’s X = x1 and Y = 0 (the profit if you do not buy a ticket), you have decided, perhaps at an intuitive level, that X is better for you than Y . The fact that the mean value EfXg = Efxg 1 = 12 1 = 12 is negative does not say
that the decision is unreasonable. You pay for hope or for fun. Suppose you buy auto insurance against a possible future loss x. Assume that with prob-
ability 0:9 the r.v. x = 0 (nothing happened), and with probability 0:1, the loss x takes on values between $0 and $2000, and all these values are equally likely. In this case, Efxg= 0:1 1000= 100. If the premium c you pay is equal, say, to $110, it means that the loss of x is worse for you than the loss of the certain amount c= 110. The fact that you pay $10 more than the mean loss, again, does not necessarily mean that you made a mistake. The additional $10 may be viewed as a payment for stability. For the insurance company the decision in this case is, in a sense, the opposite. The
company gets your premium c, and it will pay you a random payment x. The company
compares the r.v. eX = c x with the r.v. Y = 0, and if the company signs the insurance contract, it means that it has decided that the random income eX is better than zero income. In the reasoning above, we assumed that decision did not depend on the total wealth of
the client or the company but just on the r.v.’s under comparison. Suppose that in the case of insurance you also take into account your total wealth or a part of it, which we denote by w. Then the r.v.’s under comparison are wx (your wealth if you do not insure the loss) and w c (your wealth if you do insure the loss for the premium c). In the examples we considered, one of the variables under comparison was non-random.
Certainly, this is not always the case. For example, if you decide to insure only half of the future loss x for a lower premium c0, then the r.v.’s we should consider are X = wx (you do not buy an insurance) and Y = w x2 c0 (you insure half of the loss for c0). This chapter addresses various criteria for the comparison of risky alternatives. As a
rule, we will talk about possible values of future income. In this case, while the criteria may vary, they usually have one feature in common. When choosing a possible investment strategy, we have competing interests: we want the income to be large, but we also want the risk to be low. As a rule, we can reach a certain level of stability only by sacrificing a part of the income: we should pay for stability. So, our decision becomes a trade-off between the possible growth and stability. Let us consider the general framework where we deal with a fixed class X = fXg of
r.v.’s X . We assume that r.v.’s from X are all defined on some sample space W= fwg (see Section 0.1.3.1). That is, X = X(w). Defining a rule of comparison on the class X means that for pairs (X ;Y ) of r.v.’s from
X , we should determine whether X is better than Y , or X is worse than Y , or these two random variables are equivalent for us. Formally, this means that among possible pairs (X ;Y ) (the order of the r.v.’s in the pair
(X ;Y ) is essential), we specify a collection of those pairs (X ;Y ) for which X is preferable or equivalent to Y . In other words, “X is not worse than Y”, and as a rule we will use the latter terminology. We will write it as X % Y . If (X ;Y ) does not belong to the collection mentioned, we say “X is worse than Y” or
“Y is better than X”, writing X Y or Y X , respectively. If simultaneously X % Y and Y % X , we say that “X is equivalent to Y”, writing X ' Y . Not stating it each time explicitly, we will always assume that the relation % satisfies the
following two properties. (i) Completeness: For any X and Y fromX , either X %Y , or Y % X . (As was mentioned,
these relations may hold simultaneously); (ii) Transitivity: For any X , Y , and Z from X , if X % Y and Y % Z, then X % Z. The rule of comparison so defined is called a preference order on the class X . Before discussing examples, we state one general requirement on preference orders. This
requirement is quite natural when we view X’s as the r.v.’s of future income.
