Expectations (Finance)

Expectations arise when the economic agents make decisions in a world involving uncertainty. If we were living in a world of perfect information with unbounded rationality, the notion of expectation would be irrelevant. Unfortunately, the reality of the world is much more complex than captured by theoretical models. The concept of expectations, like love, has many splendorous dimensions. In finance and economics, its application include the theories of intertemporal consumption, labour supply decisions, theory of firm’s pricing, sale, investment, or inventory decisions, theories of insurance, financial markets, and search behavior, signaling, agency theory, and corporate takeovers, etc. In fact, expectations are implicit in the study of all behavioral models.

The use of expectations appears to be similar in all academic disciplines. However, there are some important distinctions. For example, the term “expectations” used in economics does not necessarily conform to the term “expectations” used in the statistical theory of probability. Linear utility functions exhibit risk-neutral behavior and may give rise to models in which agents care only about the mathematical expectations of variables. In the models in finance, mathematical expectations of returns on various assets are equalized. Quadratic expected utility functions may also produce a model in which mathematical expectations become important.

The concept of expectations has a wide range of applications in economics, business, and finance. Simple expectations proxies are proposed by Fisher (1930), and a variation on Fisher’s expectation mechanism adaptive expectations is given by Cagan (1956). Alternatives to adaptive expectations are regressive expectations, expectations of experts, rational expectations versus mechanical expectation proxies, mathematical expectations, etc. In fact, expectations are the reality of modeling of stochastic processes. One of the most useful applications of expectations in economic and finance is expected utility theory. According to the expected utility hypothesis, the individual decision makers possess a “Von Neumann-Morgenstern utility function” defined over every conceivable outcome. Individuals faced with alternative risky “lotteries” over these outcomes will choose that prospect which maximizes the expected value of utility function {Ui}.

The expected utility hypothesis can be app lied to a variety of situations, because the outcomes of “lotteries” could be alternative wealth levels, multidimensional commodity bundles, time streams of consumption, or even qualitative consequences (e.g. a trip to Pink City Jaipur), etc. Most research in the economics of uncertainty and all applied research in the field of optimal trade, investment, or search under uncertainty, is carried out in the framework of expected utility. In Arrow-Debreu general equilibrium theory, the expected utility model proceeds by specifying a set of objects of choice. It is assumed that the individual possesses a preference ordering over these objects (in the sense that one object is preferred to another if, and only if, it is assigned a higher value by this preference function) which can be represented by a real valued function V(.).

The expected utility model (under uncertainty) differs from the theory of choice over non-stochastic commodity bundles in two important ways. First, in the expected utility model choice is made under uncertainty; the objects of choice are not deterministic outcomes, but rather probability distributions over the outcomes. Second, unlike the non-stochastic case, the expected utility model imposes a very specific restriction on the functional form of the preference function V(.). Mathematically speaking, the hypothesis that the preference function V(.) takes the form of a statistical expectation is equivalent to the condition that it be “linear in the probabilities.” In the Von Neumann-Morgenstern utility function, this assumption is a primary feature of the expected utility model and provides the basis for many of its observable implications and predictions . At this stage, it is important to distinguish between the preference function V(.) and the Von Neumann-Morgenstern utility function U(.) of an expected utility maximizer, particularly with respect to the mistaken belief that expected utility preferences are somehow “cardinal” in a sense that is not represented by preferences over non-stochastic commodity bundles. An expected utility function V(.) is “ordinal” because it may be subject to any in creasing transformation without affecting the validity of the representation. One important p roperty stems from the above characterization of utility function that U(x) be an increasing function of x. Rothschild and Stiglitz (1970, 1971) have generalized the notion of a mean preserving increase in risk to density functions or cumulative distribution functions. The algebraic condition for risk aversion generalizes to the condition that U”(x) < 0 for all x which implies that the Von Neumann-Morgenstern utility function U(.) is concave.

Arrow (1971) and Pratt (1964) have contributed a great deal to analytical capabilities of the expected utility model for the study of behavior under uncertainty. They showed that the “degree” of concavity of the Von Neumann-Morgenstern utility function can be used to provide a measure of an expected utility maximizer’s degree of risk aversion. The curvature measure of R(x) – U”(x)/U’(x) is known as the Arrow-Pratt index of absolute risk aversion. The certainty equivalence and asset demand c onditions makes the Arrow-Pratt measure an important result in expected utility theory. Ross (1981) gave an alternative and stronger formulation of comparative risk aversion. According to Hey (1979), the application of the expected utility model extends to virtually all branches of economic theory, but much of the flavor of these can be sensed from Arrow’s (1974) analysis of the portfolio problem. If R(x) is a decreasing (increasing) function of the individual’s wealth level x, then it would mean that an increase in initial wealth will always increase (decrease) the demand for the risky asset if, and only if, U(.) exhibits “decreasing (increasing) absolute risk aversion in wealth.”

Finally, we focus on axiomatic developments in expected utility theory. There exist quite a few formal axiomatizations of the expected utility model in different contexts in Von Neumann-Morgenstern (1944), Marschak (1950), Herstein and Milnor (1953), and Savage (1954). Most of these axiomatizations proceed by specifying an outcome space and postulating that the individual’s preferences over probability distributions on the outcome space satisfy the following four axioms: completeness, transitivity, continuity, and the independence axiom. It is beyond the scope of this short piece to provide a derivation and explanation of these axioms and sketch a proof. Researchers have begun to develop alternative formulations of expected utility models by taking into account first order stochastic dominance preference and risk aversion.

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