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Risk aversion and the value of information.

Risk Aversion and the Value of Information

ABSTRACT

A "local" property is shown to govern the relationship between the expected value of

information and a risk aversion index. This property will be applied to study a simple

portfolio model and to discuss earlier results of Freixas and R. Kihlstrom.

Introduction

In a context of uncertainty, people usually gather information before making decision. Indeed, increasing knowledge before acting is one way to avoid the risks associated with decisions under uncertainty. As a matter of fact, information-gathering activities are close substitutes for insurance contracts as protection against certain types of risks.

One important concern is the relationship between information-seeking behavior and attitudes towards risk. Is there a clearcut relation between information demand and risk aversion? It seems, a priori, that more risk-averse decision makers should be willing to gather more information before acting, than less risk-averse agents, especially in situations where insurance opportunities fail to cover all types of risks. Thus, information demand should decrease with risk-aversion.

But this proposition is not always correct. Freixas and Kihlstrom (1984) showed that under specific assumptions the opposite relationship may hold. The demand for information may decrease with risk aversion. They gave the following intuitive argument for their result. Ex ante, when the amount of information to be demanded has to be planned for, the decision-maker ignores the message he or she is going to observe. Thus ex ante, information gathering is a risky activity that risk averse agents are less willing to bear, even if ex post it corresponds to a reduction in risk.

This intuitive argument can be explicitly modelled by adopting an alternative setting for the problem. The approach adopted differs from the seminal work of Kihlstrom (1974a, 1974b) about information demand. The present analysis is based on the expected value of information (EVI) or "asking price" earlier defined by La Valle (1968). The EVI measures the decision maker's willingness to pay for information. With this definition the difficulty of defining a controversial continuous variable representing the "amount of information" can be avoided. The author shows that for the class of constant absolute risk averse (CARA) utility functions considered by Freixas and Kihlstrom, the EVI increases (decreases) with risk aversion if and only if the expected risk associated with the decision maker's optimal course of action decreases (increases) with information.

This result may be relevant for explaining economic activities such as investment in search processes, buying insurance or speculating. For example, the result presented here tends to sustain Hirshleifer's (1975) analysis of speculation and hedging as information-seeking behaviors. Hedgers are those who invest more in information gathering only if they expect a decrease in the risk they are facing, while speculators are willing to buy more information if they anticipate higher variability of their payoffs. This is indeed reconcilable with being more or less risk-averse, respectively.

The study begins with some basic definitions in order to set out the problem, then shows that the above result is locally true, i.e. holds for small risks. Next a simple portfolio choice model is developed, expanding this result from more general types of risks, provided the further assumption of normally distributed random variables is made.

Definitions and Assumptions

Consider a decision maker with initial wealth w,(w [is greater than or equal to] o), whose preferences for random payoffs are representable by a strictly increasing and twice differentiable utility function u(.), defined on the real line R. Let D be the decision maker's choice set, and S a finite set of states of the world over which he has prior probability beliefs. Assume that these beliefs are represented by a probability measure on an algebra of subsets of S. An action D[Epsilon]D is a function from S into R, with d (s) denoting the payoff if state s obtains and d has been selected.

Restrict D to be compact and assume further than it contains the subset of all functions having at least two different values for R. This assumption is necessary to avoid the case where EVI is always null. If D contained all constant actions, the decision maker would always choose the one associated with the highest payoff regardless of the probability measure of S.

Risk-aversion is defined by the Arrow-Pratt index as: r(w) = [-u.sup.double prime] (w)/u[prime](w). If there is no opportunity to obtain further information before choosing d[Epsilon]D, the decision maker's optimal action is given by: Max [E.sub.s] u(w + d(s)) d[Epsilon]D where Es means expectation over S.

Now assume the decision maker is allowed to perform an experiment Y providing information before acting. An experiment is defined as a partition of the set of states of the world. Each element of an experiment is called an observation which tells the decision-maker to which subset the true state of the world belongs. Experiments are naturally ordered by fineness. Experiment Y is finer than experiment Y[prime] if any observation belonging to Y is contained in an observation belonging to Y[prime]. Thus experiment Y conveys at least as much information about the state of the world that will obtain as experiment Y[prime]. Experiment Y is at least as valuable as experiment Y[prime], if Y is finer thn Y[prime]. Notice that any partition of Y of S is valuable because it is at least as fine as S itself. The probability of a given observation belonging to an experiment Y, depends on the underlying probability measure defined on an algebra of subsets of S.

An experiment is sometimes alternatively defined as a set of random variables correlated with the state of nature. As shown by Green and Stockey (1978), both descriptions of an experiment are however equivalent, in the following sense: experiments defined with random variables which are ordered according to Blackwell's (1951) criterion, can be alternatively described as partitions of a fundamental set preserving the ordering in terms of fineness.

Assume that observation costs C(Y), associated with an experiment Y, are non-random and bounded. This means that the costs for acquiring an experiment have to be incurred in advance and do not depend on the outcome of the experiment. If the decision maker observes y[Epsilon]Y his or her expected utility, after taking an optimal action, will be:

U(S,w,y,C(Y)) = Max [E.sub.s/y] u(w + d(s) - C(Y)),

d[Epsilon]D where [E.sub.s/y] is the conditional expectation over S, knowing y[Epsilon]Y. Before y is observed, the decision maker is uncertain about the true value of Y. Therefore the ex ante expectation is:

[E.sub.y]U(S,w,y,C(Y)) = [E.sub.y] max [E.sub.s/y] u(w + d(s) - C(Y)).

d[Epsilon]D

Now define EVI as V. For an experiment Y, V is the maximum price the decision maker is willing to pay for observing the outcome of Y. Assume that the cost of no information is zero and let U(S,w,[Phi],O) be the expected utility without information. V is defined as the cost which equates [E.sub.y] U(S,w,y,.) with U(S,w,[Phi],O). Thus V is the solution of equation (1) below: (1) [E.sub.y] U(S,w,y,V) = U(S,w,[Phi],O). Since u(.) is strictly increasing and hence U(.) is strictly decreasing in C(Y), and since [E.sub.y] U(S,w,y,O) [is greater than or equal to] U(S,w,[Phi],O), it can be shown[1] that there is a unique and non-negative number V satisfying equation (1). Thus EVI is a useful and simple criterion for deciding whether or not to acquire experiment Y. Y is purchased if and only if V [is greater than or equal to] C(Y).

Notice that in the linear utility case, EVI is simply the difference between [E.sub.y] U(S,w,y,V) and U(S,w,[Phi],O). But in general, this latter definition is not correct because the difference is measured in utility units, whereas V should be measured in cost units as pointed out by Arrow (1970b). Therefore V is only implicitly defined by equation (1), and generally cannot be explicitly calculated(2). One way to obtain a more explicit formula for V, is to introduce cash-equivalents for random payoffs. For that purpose define CE([d.sup.*]) the cash-equivalent for the random payoff associated with [d.sup.*], the optimal action in the case of no information. Similarly, let CE[Mathematical Expression Omitted] be the cash-equivalent of the random payoff corresponding to the optimal strategy with experiment Y. By definition: [Mathematical Expression Omitted] where [E.sub.s][d.sup.*] is the expected payoff of action [d.sup.*], and [Pi](w;[d.sup.*]) the corresponding risk premium at wealth-level w.

Similarly: [Mathematical Expression Omitted] where [Mathematical Expression Omitted] is the joint random payoff for strategy [Mathematical Expression Omitted]. Equation (1) can now be rewritten as: (2) [Mathematical Expression Omitted] with: (3) [Mathematical Expression Omitted] V is still implicitly defined since it appears on the left-hand side in (3). However, risk premiums are independent on wealth when restricted to the class of CARA utility functions(3), and thus: (4) V = CE ([d|.sub.y.sup.*]) - CE ([d.sup.*])

= [Delta]E - [Delta][Pi], where [Delta]E is the change in expected payoff resulting from experiment Y, [Mathematical Expression Omitted] and [Delta][Pi] as the corresponding change in risk premium [Mathematical Expression Omitted]

This notation shows that the EVI differs from one decision maker to another, first because they have a different [Delta]E since they do not take the same optimal actions; and second because, even if they face the same distributions, they have different risk premiums. Thus in order to characterize the relation between EVI and risk aversion, it would be useful to know how different agents value a given change in risk, i.e. a change in the probability vector on S.

The Value of a Change in Risk

Suppose all agents face the same exogenous risk [y|]. How do they price a move from risk [y|] to another risk [z|]? Obviously, for a decision maker having utility function u(.) and wealth w, the answer is given by equation (5) below, in which v measures the value of exchanging risk [y|] against risk [z|]. (5) Eu(w + [z|] - v) = Eu(w + [y|]). The variable, v, may be written by means of a difference in expected payoffs and a difference in risk-premiums: (6) v = E[z|] - E[y|] - ([Pi](w - v;[z|]) - [Pi](w;[y|])), If utility functions are CARA, v may also be written in the following form:

v = [Delta]E - [Delta][Pi], knowing that [Delta]E is now the same, by hypothesis, for all utility functions.

The decision maker's risk sensitivity, a property indicating the magnitude of his or her reaction to an exogenous change in risk is measured by [Delta][Pi]. There is unfortunately no clearcut relation between risk sensitivity as measured by [Delta][Pi] and r, the Arrow-Pratt index of risk aversion. All that can be said is that risk-neutral agents are not risk sensitive ([Delta][Pi] = O); it cannot be said that more risk-averse agents are more risk sensitive. This is because risk sensitivity depends both on an agent's risk aversion and on the type of risk change that he or she faces. However, for the class of CARA utility functions and small risks (i.e., random variables having small variance), one can obtain a local property relating [Delta][Pi] to r. Indeed, Pratt (1964) showed that for such small risks, [Pi](w;z) could be approximated by: (7) [Mathematical Expression Omitted] where [Mathematical Expression Omitted] is the variance of [z|], and r the Arrow-Pratt index which amounts to being a constant with respect to wealth under the CARA assumption.

Now for two agents facing the same risks, with risk-aversion index [r.sub.1] and [r.sub.2] respectively, and such that [r.sub.1] [is greater than or equal to] [r.sub.2]: (8) [Mathematical Expression Omitted] The following proposition may now be established: in a local sense (i.e., for small risks), if u has constant absolute risk aversion, v increases with risk aversion if and only if the risk (measured by variance) decreases with information.

Indeed, with constant risk-aversion r, the following relation develops: [Mathematical Expression Omitted] Since: [r.sub.1] - [r.sub.2] [is greater than or equal to] O, [Mathematical Expression Omitted] This proposition also holds for global risks if the second order approximation for the risk premium is sufficient. It is well-known that second-order approximations are sufficient in cases for which the mean-variance approach of decision making under uncertainty is reconcilable with the expected utility hypothesis(4). Next a case is presented for which the proposition holds for global risks even if agents differ in their choice of optimal actions, i.e. holds not only for v (the value of a risk change) but also for V (the EVI).

The EVI in a Simple Portfolio Choice Model

In this section a simple portfolio choice model is presented in which the riskiness of a portfolio is unambiguously measured by the variance of the payoffs. As will be demonstrated, more risk-averse agents should be willing to pay more for expected information if the expected variance of their portfolio decreases with that information, and less in the opposite case.

Suppose there is only one risky asset, having rate of return p(s) when state s occurs, and one safe asset (e.g. money) with zero rate of return in any state. The decision maker wishes to allocate his or her initial wealth between these assets in order to maximize expected utility. He or she chooses to invest an amount [partial derivative.sub.s] [is less than or equal to] w of total wealth to invest in the risky asset to solve the following program: (9) Max [E.sub.s]u(w + [a.sub.s]p(s))

[a.sub.s] [is less than or equal to] w. Let [Mathematical Expression Omitted] be the solution of (9). Assume that p is normally distributed with mean [p|.sub.s] and variance [Theta.sub.s], and that u is negatively exponential. Then

u(w) = - exp( - r.w). Notice that u(.) is CARA and has risk-aversion coefficient r [is greater than] O.

With these assumptions it can be shown that: (10) [Mathematical Expression Omitted] Notice that [a.sub.s][p|.sub.s] is simply the expected payoff of the portfolio, and [Pi] = [Mathematical Expression Omitted] the decision maker's risk premium with [Mathematical Expression Omitted] the variance of the portfolio if he or she chooses to invest amount [a.sub.s] in the risky asset.

The first order condition(5) for a maximum is: (11) [P|.sub.s] - [ra.sub.s][Theta.sub.s] = O, and the optimal amount to invest in the risky asset is: (12) [Mathematical Expression Omitted] Substituting this expression in (10) one obtains: (13) [Mathematical Expression Omitted]

Assume now that conditionally on each observation y[Epsilon]Y the returns are normally distributed with mean [P[bar].sub.y] and variance [Theta.sub.y]. Applying for each observation y[Epsilon]Y, a similar reasoning as for (13), the following expectation is obtained: (14) [Mathematical Expression Omitted] Recall that EVI is the maximum acceptable cost for experiment Y, i.e. the cost V which equates (13) and (14) above. This gives the following equation for EVI: (15) [Mathematical Expression Omitted] with: (16) [Mathematical Expression Omitted] The expression between the heavy brackets in (16) is positive since u(w) is an increasing function of w and expected utility is non decreasing with

information. Thus, expression (14) is greater than or equal to expression (13). Consequently: (17) [Mathematical Expression Omitted] Thus, in the present model, EVI decreases with risk aversion. More risk averse investors are less willing to pay for expected information because the expected variance of the portfolio is always greater with the information. Information is valuable because it increases expected returns of the portfolio, but ex ante, information is itself risky because the decision maker does not know in advance which observation he or she is going to make. Thus the optimal ex post portfolio is also unknown and may be more or less risky according to the type of observation made. To see that result (17) is due in fact to a systematic increase in expected variance, notice that, since V is non-negative, (16) yields: [Mathematical Expression Omitted] By concavity of the Log function: [Mathematical Expression Omitted] [Mathematical Expression Omitted] The expression on the left side is the variance of the portfolio without information (see expression 10), and that on the right side is the expected variance with experiment Y.

Conclusion

For a restricted class of utility functions (CARA) and for normally distributed random variables, there is a clearcut relation between risk aversion and the expected value of information. Information demand increases with risk aversion if the expected variance of the payoffs resulting from the optimal course of action decreases with information. This result may explain Freixas and Kihlstrom's earlier result. Indeed their assumptions imply a systematically increased riskiness of the optimal course of action with the information.[6]

These results may explain hedging and speculative behaviors as well as insurance strategies. In the general case, however, there is no evident link between information demand and risk aversion. This suggests that the relation is highly model specific. Further research should identify other classes of utility functions and/or random variables for which the present result may hold.

(1)For a rigorous proof see La Valle (1968) or Marschak and Radner (1972) pp. 86. (2)Ziemba and Butterworth (1975) and Hausch and Ziemba (1983) studied upper bounds for V, but they are not very helpful for this problem. (3)CARA utility functions are either defined by u(x) = ax + b, u(x) = [e.sup.rx] or by u(x) = [-e.sup.-rx], where r is the (constant) risk aversion coefficient. (4)It should be noticed however that the quadratic case does not show constant risk aversion. Thus our proposition does not apply for this case. (5)It is easily verified that the second order condition holds. (6)One of the assumptions of their model implies null variance if the decision maker acts solely with his or her prior knowledge that s[Epsilon]S.

References [1]Arrow K. J., (1970a), "The Theory of Risk Aversion," in Essays in the Theory of Risk-Bearing, K. J. Arrow ed., (North-Holland), pp. 90-109. [2]Arrow K. J., (1970b), "The Value of and Demand for Information," in Essays in the Theory of Risk-Bearing, K. J. Arrow ed., (North-Holland), pp. 267-78. [3]Blackwell D., (1951), "Comparisons of Experiments," Proceedings of the 2nd Berkeley Symposium on Mathematical Statistics and Probability, (Berkeley: University of California Press) pp. 93-102. [4]Freixas X. and Kihlstrom R., (1984), "Risk Aversion and Information Demand," in Boyer M. and Kihlstrom R.E. eds, Bayesian Models in Economic Theory, (North-Holland), pp. 93-104. [5]Green J. and Stockey N., (1978), "Two Representations of Information Structures and their Comparisons," Technical Report, No. 271, Stanford University. [6]Hausch D. B. and Ziemba W. T., 1983, "Bounds on the Value of Information in Uncertain Decision Problems II," Stochastics, Vol. 10, pp. 181-217. [7]Hirshleifer J., 1975, "Speculation and Equilibrium: Information, Risk and Markets," The Quarterly Journal of Economics, Vol. 89, No. 4, pp. 519-42. [8]Kihlstrom R., (1974a), "A Bayesian Model of Demand for Information about Product Quality," International Economic Review, Vol. 15, pp. 99-118. [9]Kihlstrom R., (1974b), "A General Theory of Demand for Information about Product Quality," Journal of Economic Theory, Vol. 8, pp. 413-39. [10]La Valle I. H., (1968), "On Cash Equivalents and Information Evaluation in Decisions under Uncertainty: Part I: Basic Theory," American Statistical Association Journal, Vol. 63, No. 321, pp. 252-76. [11]Marschak J. and Radner R., (1972), Economic Theory of Teams, Cowles Foundation monograph No. 22, (New Haven: Yale University Press). [12]Pratt J., (1964), "Risk Aversion in the Small and in the Large," Econometrica, Vol. 32, pp. 122-36. [13]Ziemba W. T. and Butterworth J.E., (1975), "Bounds on the Value of Information in Uncertain Decisions Problems," Stochastics, Vol. 1, pp. 361-78.

Marc Willinger is Charge de Recherche at Universite Louis Pasteur in Strasbourg, France. He thanks R. Dos Santos Ferreira. W. Hildebrand and two anonymous referees fo their valuable comments on an earlier version of this note.
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Title Annotation:information gathering prior to decision making
Author:Willinger, Marc
Publication:Journal of Risk and Insurance
Date:Mar 1, 1989
Words:3426
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