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Economic and Financial Decisions under Risk

Princeton University Press

Chapter One

This chapter looks at a basic concept behind modeling individual preferences in the face of risk. As with any social science, we of course are fallible and susceptible to second-guessing in our theories. It is nearly impossible to model many natural human tendencies such as "playing a hunch" or "being superstitious." However, we can develop a systematic way to view choices made under uncertainty. Hopefully, our models can capture the basic human tendencies enough to be useful in understanding market behavior towards risk. In other words, even if we are not correct in predicting behavior under risk for every individual in every circumstance, we can still make general claims about such behavior and can still make market predictions, which after all are based on the "marginal consumer."

To use (vaguely) mathematical language, the understanding of this chapter is a necessary but not sufficient condition to go further into the analysis. Because of the importance of risk aversion in decision making under uncertainty, it is worthwhile to first take an "historical" perspective about its development and toindicate how economists and decision scientists progressively have elaborated upon the tools and concepts we now use to analyze risky choices. In addition, this "history" has some surprising aspects that are interesting in themselves. To this end, our first section in this chapter broadly covers these retrospective topics. Subsequent sections are more "modern" and they represent an intuitive introduction to the central contribution to our field, that of Pratt (1964).

1.1 An Historical Perspective on Risk Aversion

As it is now widely acknowledged, an important breakthrough in the analysis of decisions under risk was achieved when Daniel Bernoulli, a distinguished Swiss mathematician, wrote in St Petersburg in 1738 a paper in Latin entitled: "Specimen theoriae novae de mensura sortis," or "Exposition of a new theory on the measurement of risk." Bernoullis paper, translated into English in Bernoulli (1954), is essentially nontechnical. Its main purpose is to show that two people facing the same lottery may value it differently because of a difference in their psychology. This idea was quite novel at the time, since famous scientists before Bernoulli (among them Pascal and Fermat) had argued that the value of a lottery should be equal to its mathematical expectation and hence identical for all people, independent of their risk attitude.

In order to justify his ideas, Bernoulli uses three examples. One of them, the "St Petersburg paradox" is quite famous and it is still debated today in scientific circles. It is described in most recent texts of finance and microeconomics and for this reason we do not discuss it in detail here. Peter tosses a fair coin repetitively until the coin lands head for the first time. Peter agrees to give to Paul 1 ducat if head appears on the first toss, 2 ducats if head appears only on the second toss, 4 ducats if head appears for the first time on the third toss, and so on, in order to double the reward to Paul for each additional toss necessary to see the head for the first time. The question raised by Bernoulli is how much Paul would be ready to pay to Peter to accept to play this game.

Unfortunately, the celebrity of the paradox has overshadowed the other two examples given by Bernoulli that show that, most of the time, the value of a lottery is not equal to its mathematical expectation. One of these two examples, which presents the case of an individual named "Sempronius," wonderfully anticipates the central contributions that would be made to risk theory about 230 years later by Arrow, Pratt and others.

Let us quote Bernoulli:

Sempronius owns goods at home worth a total of 4000 ducats and in addition possesses 8000 ducats worth of commodities in foreign countries from where they can only be transported by sea. However, our daily experience teaches us that of [two] ships one perishes.

In modern-day language, we would say that Sempronius faces a risk on his wealth. This wealth may represented by a lottery [??], which takes on a value of 4000 ducats with probability 1/2 (if his ship is sunk), or 12 000 ducats with probability 1/2. We will denote such a lottery [??] as being distributed as (4000, 1/2; 12 000, 1/2). Its mathematical expectation is given by:

E[??] [equivalent to] 1/2 4000 + 1/2 12000 = 8000 ducats.

Now Sempronius has an ingenious idea. Instead of "trusting all his 8000 ducats of goods to one ship," he now "trusts equal portions of these commodities to two ships." Assuming that the ships follow independent but equally dangerous routes, Sempronius now faces a more diversified lottery [??] distributed as

(4000, 1/4; 8000, 1/2; 12000, 1/4).

Indeed, if both ships perish, he would end up with his sure wealth of 4000 ducats. Because the two risks are independent, the probability of these joint events equals the product of the individual events, i.e. [(1/2).sup.2] = 1/4. Similarly, both ships will succeed with probability 1/4 , in which case his final wealth amounts to 12000 ducats. Finally, there is the possibility that only one ship succeeds in downloading the commodities safely, in which case only half of the profit is obtained. The final wealth of Sempronius would then just amount to 8000 ducats. The probability of this event is 1/2 because it is the complement of the other two events which have each a probability of 1/4.

Since common wisdom suggests that diversification is a good idea, we would expect that the value attached to [??] exceeds that attributed to [??]. However, if we compute the expected profit, we obtain that

E [??] = 1/4 4000 + 1/2 8000 + 1/4 12000 = 8000 ducats,

the same value as for E[??]! If Sempronius would measure his well-being ex ante by his expected future wealth, he should be indifferent about whether to diversify or not. In Bernoullis example, we obtain the same expected future wealth for both lotteries, even though most people would find [??] more attractive than [??]. Hence, according to Bernoulli and to modern risk theory, the mathematical expectation of a lottery is not an adequate measure of its value. Bernoulli suggests a way to express the fact that most people prefer [??] to [??]: a lottery should be valued according to the "expected utility" that it provides. Instead of computing the expectation of the monetary outcomes, we should use the expectation of the utility of the wealth. Notice that most human beings do not extract utility from wealth. Rather, they extract utility from consuming goods that can be purchased with this wealth. The main insight of Bernoulli is to suggest that there is a nonlinear relationship between wealth and the utility of consuming this wealth.

What ultimately matters for the decision maker ex post is how much satisfaction he or she can achieve with the monetary outcome, rather than the monetary outcome itself. Of course, there must be a relationship between the monetary outcome and the degree of satisfaction. This relationship is characterized by a utility function u, which for every wealth level x tells us the level of "satisfaction" or "utility" u(x) attained by the agent with this wealth. Of course, this level of satisfaction derives from the goods and services that the decision maker can purchase with a wealth level x. While the outcomes themselves are "objective," their utility is "subjective" and specific to each decision maker, depending upon his or her tastes and preferences. Although the function u transforms the objective result x into a perception u(x) by the individual, this transformation is assumed to exhibit some basic properties of rational behavior. For example, a higher level of x (more wealth) should induce a higher level of utility: the function should be increasing in x. Even for someone who is very altruistic, a higher x will allow them to be more philanthropic. Readers familiar with indirect utility functions from microeconomics (essentially utility over budget sets, rather than over bundles of goods and services) can think of u(x) as essentially an indirect utility of wealth, where we assume that prices for goods and services are fixed. In other words, we may think of u(x) as the highest achievable level of utility from bundles of goods that are affordable when our income is x.

Bernoulli argues that if the utility u is not only increasing but also concave in the outcome x, then the lottery [??] will have a higher value than the lottery [??], in accordance with intuition. A twice-differentiable function u is concave if and only if its second derivative is negative, i.e. if the marginal utility u'(x) is decreasing in x. In order to illustrate this point, let us consider a specific example of a utility function, such as u(x) = [square root of x], which is an increasing and concave function of x. Using these preferences in Semproniuss problem, we can determine the expectation of u(x):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Because lottery [??] generates a larger expected utility than lottery ??, the former is preferred by Sempronius. The reader can try using concave utility functions other than the square-root function to obtain the same type of result. In the next section, we formalize this result.

Notice that the concavity of the relationship between wealth x and satisfaction/ utility u is quite a natural assumption. It simply implies that the marginal utility of wealth is decreasing with wealth: one values a one-ducat increase in wealth more when one is poorer than when one is richer. Observe that, in Bernoullis example, diversification generates a mean-preserving transfer of wealth from the extreme events to the mean. Transferring some probability weight from x = 4000 to x = 8000 increases expected utility. Each probability unit transferred yields an increase in expected utility equaling u(8000) - u(4000). On the contrary, transferring some probability weight from x = 12 000 to x = 8000 reduces expected utility. Each probability unit transferred yields a reduction in expected utility equaling u(12 000) - u(8000). But the concavity of u implies that

u(8000) - u(4000) > u(12 000) - u(8000), (1.1)

i.e. that the positive effect of these combined transfers must dominate the negative effect. This is why all investors with a concave utility would support Semproniuss strategy to diversify risks.

1.2 Definition and Characterization of Risk Aversion

We assume that the decision maker lives for only one period, which implies that he immediately uses all his final wealth to purchase and to consume goods and services. Later in this book, we will disentangle wealth and consumption by allowing the agent to live for more than one period. Final wealth comes from initial wealth w plus the outcome of any risk borne during the period.

Definition 1.1. An agent is risk-averse if, at any wealth level w, he or she dislikes every lottery with an expected payoff of zero: [for all]w, [for all][??] with E[??] = 0, Eu(w + [??]) [less than or equal to] u(w).

Observe that any lottery [??] with a non-zero expected payoff can be decomposed into its expected pay off E[??] and a zero-mean lottery [??] - E[??]. Thus, from our definition, a risk-averse agent always prefers receiving the expected outcome of a lottery with certainty, rather than the lottery itself. For an expected-utility maximizer with a utility function u, this implies that, for any lottery [??] and for any initial wealth w, Eu(w + [??]) [less than or equal to] u(w + E[??]). (1.2)

If we consider the simple example from Semproniuss problem, with only one ship the initial wealth w equals 4000, and the profit [??] takes the value 8000 or 0 with equal probabilities. Because our intuition is that Sempronius must be risk averse, it must follow that

1/2 u (12000) + 1/2 u(4000) [less than or equal to] u(8000). (1.3)

If Sempronius could find an insurance company that would offer full insurance at an actuarially fair price of E[??] = 4000 ducats, Sempronius would be better off by purchasing the insurance policy. We can check whether inequality (1.3) is verified in Figure 1.1. The right-hand side of the inequality is represented by point 'f on the utility curve u. The left-hand side of the inequality is represented by the middle point on the arc 'ae', i.e. by point 'c'. This can immediately be checked by observing that the two triangles 'abc' and 'cde' are equivalent, since they have the same base and the same angles. We observe that 'f' is above 'c': ex ante, the welfare derived from lottery [??] is smaller than the welfare obtained if one were to receive its expected payoff E[??] with certainty. In short, Sempronius is risk-averse. From this figure, we see that this is true whenever the utility function is concave. The intuition of the result is very simple: if marginal utility is decreasing, then the potential loss of 4000 reduces utility more than the increase in utility generated by the potential gain of 4000. Seen ex ante, the expected utility is reduced by these equally weighted potential outcomes.

It is noteworthy that Equations (1.1) and (1.3) are exactly the same. The preference for diversification is intrinsically equivalent to risk aversion, at least under the Bernoullian expected-utility model.

Using exactly the opposite argument, it can easily be shown that, if u is convex, the inequality in (1.2) will be reversed. Therefore, the decision maker prefers the lottery to its mathematical expectation and he reveals in this way his inclination for taking risk. Such individual behavior will be referred to as risk loving. Finally, if u is linear, then the welfare Eu is linear in the expected payoff of lotteries. Indeed, if u(x) = a + bx for all x, then we have

Eu(w + [??]) = E]a + b(w + [??])] = a + b(w + E]??]) = u(w + E]??]),

which implies that the decision maker ranks lotteries according to their expected outcome. The behavior of this individual is called risk-neutral.

In the next proposition, we formally prove that inequality (1.2) holds for any lottery [??] and any initial wealth w if and only if u is concave.

Proposition 1.2. A decision maker with utility function u is risk-averse, i.e. inequality (1.2) holds for all w and [??], if and only if u is concave.

Proof. The proof of sufficiency is based on a second-order Taylor expansion of u(w + z) around w + E[??]. For any z, this yields

u(w + z) = u(w + E[??]) + (z - E[??])u'(w + E[??]) + 1/2 [(z - E[??]).sup.2] u"([xi](z))

for some [xi] (z) in between z and E[??]. Because this must be true for all z, it follows that the expectation of u(w + [??]) is equal to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Observe now that the second term of the right-hand side above is zero, since E([??] - E[??]) = E[??] - E[??] = 0. In addition, if u" is uniformly negative, then the third term takes the expectation of a random variable ([??] - E[??]).sup.2] u" ([xi]([??])) that is always negative, as it is the product of a squared scalar and negative u". Hence, the sum of these three terms is less than u(w + E[??]). This proves sufficiency.

(Continues...)

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Excerpted from Economic and Financial Decisions under Risk by Louis Eeckhoudt Christian Gollier Harris Schlesinger Copyright © 2005 by Princeton University Press . Excerpted by permission.
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