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.

Introduction

In this chapter we discuss the counterparts in the multidate setting of the results of Chapter 18 deriving beta pricing and of Chapter 19 deriving the capital asset pricing model, each in the two-date setting.

We identified the CAPM in the two-date setting as having the property that the return on the market portfolio lies on the mean-variance frontier. We showed that the CAPM holds in equilibrium in the two-date model if agents have quadratic utilities or more general mean-variance preferences. This implies that the beta pricing relation holds with the market portfolio as the reference portfolio, that is, the security market line.

We show that a suitably defined one-period return on the aggregate endowment lies on the mean-variance frontier in equilibrium in the multidate model if agents have quadratic utilities. Consequently, a beta pricing relation with the return on the aggregate endowment as the reference return holds. However, the reference portfolio differs from the market portfolio. We show in Section 28.6 that even with quadratic utility the return on the market portfolio generally does not lie on the mean-variance frontier in the multidate setting. In this case the beta pricing relation with the market portfolio as the reference portfolio cannot be invoked to derive the security market line. Therefore CAPM fails.

The beta pricing relation we derive is the conditional beta pricing relation.

Desire to improve drives many human activities. Optimization can be seen as a means for identifying better solutions by utilizing a scientific and mathematical approach. In addition to its widespread applications, optimization is an amazing subject with very strong connections to many other subjects and deep interactions with many aspects of computation and theory. The main goal of this textbook is to provide an attractive, modern, and accessible route to learning the fundamental ideas in optimization for a large group of students with varying backgrounds and abilities. The only background required for the textbook is a first-year linear algebra course (some readers may even be ready immediately after finishing high school). However, a course based on this book can serve as a header course for all optimization courses. As a result, an important goal is to ensure that the students who successfully complete the course are able to proceed to more advanced optimization courses.

Another goal of ours was to create a textbook that could be used by a large group of instructors, possibly under many different circumstances. To a degree, we tested this over a four-year period. Including the three of us, 12 instructors used the drafts of the book for two different courses. Students in various programs (majors), including accounting, business, software engineering, statistics, actuarial science, operations research, applied mathematics, pure mathematics, computational mathematics, computer science, combinatorics and optimization, have taken these courses. We believe that the book will be suitable for a wide range of students (mathematics, mathematical sciences including computer science, engineering including software engineering, and economics).

In Chapter 2, we have seen how to solve LPs using the simplex algorithm, an algorithm that is still widely used in practice. In Chapter 3, we discussed efficient algorithms to solve the special class of IPs describing the shortest path problem and the minimum cost perfect matching problem in bipartite graphs. In both these examples, it is sufficient to solve the LP relaxation of the problem.

Integer programming is widely believed to be a difficult problem (see Appendix A). Nonetheless, we will present algorithms that are guaranteed to solve IPs in finite time. The drawback of these algorithms is that the running time may be exponential in the worst case. However, they can be quite fast for many instances, and are capable of solving many large-scale, real-life problems.

These algorithms follow two general strategies. The first attempts to reduce IPs to LPs – this is known as the cutting plane approach and will be described in Section 6.2. The other strategy is a divide and conquer approach and is known as branch and bound and will be discussed in Section 6.3. In practice, both strategies are combined under the heading of branch and cut. This remains the preferred approach for all general purpose commercial codes.

In this chapter, in the interest of simplicity we will restrict our attention to pure IPs where all the variables are required to be integer. The theory developed here extends to mixed IPs where only some of the variables are required to be integer, but the material is beyond the scope of this book.

Possible outcomes

Consider an LP (P) with variables x1, …, xn. Recall that an assignment of values to each of x1, …, xn is a feasible solution if the constraints of (P) are satisfied. We can view a feasible solution to (P) as a vector x = (x1, …, xn)T. Given a vector x, by the value of x we mean the value of the objective function of (P) for x. Suppose (P) is a maximization problem. Then recall that we call a vector x an optimal solution if it is a feasible solution and no feasible solution has larger value. The value of the optimal solution is the optimal value. By definition, an LP has only one optimal value; however, it may have many optimal solutions. When solving an LP, we will be satisfied with finding any optimal solution. Suppose (P) is a minimization problem. Then a vector x is an optimal solution if it is a feasible solution and no feasible solution has smaller value.

If an LP (P) has a feasible solution, then it is said to be feasible, otherwise it is infeasible. Suppose (P) is a maximization problem and for every real number α there is a feasible solution to (P) which has value greater than α, then we say that (P) is unbounded. In other words, (P) is unbounded if we can find feasible solutions of arbitrarily high value. Suppose (P) is a minimization problem and for every real number α there is a feasible solution to (P) which has value smaller than α, then we say that (P) is unbounded.