Introduction

Expected utility provides a framework for the analysis of agents' attitudes toward risk. In this chapter we expand on the formal definition of risk aversion presented in Chapter 8 and introduce measures of the intensity of risk aversion such as the Arrow–Pratt measures and risk compensation. The main result of this chapter, the Pratt Theorem, establishes the equivalence of these different measures of risk aversion.

Agents' preferences for risky consumption plans are assumed – except in Section 9.10 – to have a state-independent expected utility representation with continuous von Neumann–Morgenstern utility functions. The consumption plans in the domain of an expected utility function may be defined either narrowly or broadly. The axioms of expected utility imply that any consumption plan can be viewed narrowly as a random variable on the set S of states equipped with an agent's subjective probability measure. Thus, if the objects of choice are specified as the consumption plans that emerge from the axioms of expected utility, they are appropriately defined narrowly as random variables that can take S values with given probabilities. However, the analysis of this chapter applies equally well if consumption plans are broadly interpreted as arbitrary random variables (that is, as random variables with an arbitrary number of realizations and arbitrary probabilities). The choice between these interpretations is a matter of taste. In Section 9.10 we discuss risk aversion for a multiple-prior expected utility.

Introduction

We have thus far limited ourselves to models of two-date security markets in which securities are traded only once before their payoffs are realized. These models are suitable for the introductory study of the risk-return relation for securities and the role of securities in the equilibrium allocation of risk. However, two-date models require the assumption that all uncertainty is resolved at once. It is more realistic to assume that uncertainty is resolved only gradually. As the uncertainty is resolved, agents trade securities again and again. The multidate model of this and the following chapters assumes that there are a finite number of future dates. This specification allows for the gradual resolution of uncertainty and the retrading of securities as new information about security prices and payoffs becomes available.

Uncertainty and Information

In the multidate model, just as in the two-date model, uncertainty is specified by a set of states S. Each of the states is a description of the economic environment for all dates t = 0, 1, …, T. At date 0 agents do not know which state will be realized. But as time passes, they obtain more and more information about the state. At date T they learn the actual state.

Formally, the information of agents at date t is described by a partition Ft of the set of states S (a partition Ft of S is a collection of subsets of S such that each state s belongs to exactly one element of Ft).

Introduction

Beta pricing (see Section 18.5) implies that the risk premium on any security or portfolio is proportional to the covariance of its return with a frontier return. However, beta pricing by itself gives no guidance as to which returns are frontier returns. We use the term “capital asset pricing model” (CAPM) if the market return is a frontier return. Note that the CAPM is here identified with a property of equilibrium security prices, not with a class of models of security markets (that is, the restriction involves endogenous variables, not exogenous variables). Therefore it is necessary to determine what restrictions on preferences or payoffs give rise to equilibria that conform to the CAPM definition.

Under the CAPM the market return, being a frontier return, can be taken as the reference portfolio in the beta pricing equation. Doing so leads directly to the security market line, which relates the risk premium on any security to the covariance between the return on that security and the market return.

In Chapter 14 we derived the equation of the security market line by applying consumption-based security pricing under the assumption that agents have quadratic utilities. The derivation was generalized in Chapter 16. In this chapter we derive the CAPM in an equilibrium under the assumption that agents take variance as a measure of consumption risk (mean-variance preferences).

Introduction

In Chapter 5 we showed that the payoff pricing functional – and also its extension, the valuation functional – can be represented either by state prices or by risk-neutral probabilities. In this chapter we derive another representation of the payoff pricing functional, the pricing kernel. The existence of the pricing kernel is a consequence of the Riesz representation theorem, which, in the present context, says that any linear functional on a vector space can be represented by a vector in that space.

We begin by introducing the concepts of inner product, orthogonality, and orthogonal projection. These concepts are associated with an important class of vector spaces, the Hilbert spaces, to which the Riesz representation theorem applies. In the finance context, the Riesz representation theorem implies that any linear functional on the asset span can be represented by a payoff. Two linear functionals are of particular interest: the payoff pricing functional, which maps every payoff into its date-0 value, and the expectations functional, which maps every payoff into its expectation. Their representations are the pricing kernel and the expectations kernel, respectively.

Hilbert space methods are important for the study of the capital asset pricing model and factor pricing in the following chapters. Our treatment of these methods is mathematically superficial, because our interest lies in arriving quickly at results that are applicable in finance. In particular, the finite-dimensional contingent claims space ℛS is for us the primary example of a Hilbert space.

Introduction

In this chapter we relax the assumption made in Chapter 21 that the number of dates is finite and consider a model of security markets with an infinite time horizon.

The existence of a (finite) terminal date when all securities are liquidated has an effect on agents' trading strategies at all dates. The optimal portfolio generally depends on how distant the terminal date is. This dependence on a terminal date can be avoided by assuming that the time horizon is infinite. Many securities – stocks being one example – do not have a specific maturity date and are appropriately analyzed in an infinite-time model.

In infinite-time security markets a new problem arises that has no counterpart in the multidate model: in the absence of trading restrictions, agents can borrow and roll over the debt indefinitely from one date to the next. If such Ponzi schemes are permitted, there do not exist optimal portfolios under the usual specifications of preferences, implying that there can be no equilibrium. Equilibrium in security markets exists only if trading restrictions are invoked that render Ponzi schemes impossible at equilibrium security prices under the usual specifications of preferences – in particular, under strict monotonicity of preferences. One type of trading restrictions is a debt constraint that puts a limit on agents' debt at every date. Equilibrium under debt constraints is the subject of this chapter.