Mathematics and Economics

Archive for January, 2011|Monthly archive page

CCAPM I: The Basic Pricing Equation

In Asset Pricing Reading Group, Bubbles and Crashes, Economics, Finance on January 20, 2011 at 1:41 am

1. Introduction

Over a series of two posts, we will discuss the consumption capital asset pricing model (CCAPM) and some of its results concerning expected asset price returns. In particular, we derive a range within which asset returns lie. The model we will be discussing describes the simplified behavior of a trader who chooses to consume a portion of his income and invest the rest in market assets over two time periods of any kind (i.e. days, months, etc.)

In this post, we describe a simplified setting of the model and derive a formula for prices arising from the marginal cost and marginal benefit of the trader’s investment decisions. This equation will then show us how prices are affected by preferences. Furthermore, the resultant equation turns out to be robust to the initial simple setting of the model. Our exposition of the model borrows from John Cochrane’s book, Asset Pricing.

2. Investor Consumption Model

We begin our discussion of the Consumption Capital Asset Pricing Model (CCAPM) by first modeling our trader with a utility function defined over current and future values of consumption:

\displaystyle U(c_t,c_{t+1}) = u(c_t) + \beta E_t[u(c_{t+1})]. \ \ \ \ \ (1)

To simplify our discussion, we assume that there exists only one consumption good, denoted by {c}, and that all other asset values are quoted in terms of it. The utility function represents the “fundamental pleasure” that the investor derives from consuming the good in question. The period utility function {u(\cdot)} is assumed to be increasing, concave, and continuously differentiable (i.e. smooth). That the function is increasing describes, somewhat expectedly, the desire for greater consumption rather than lesser consumption, while concavity is a result of the decling marginal value of greater consumption. In other words, our investors will be happy to consume more rather than less, but enjoy each additional unit at a decreasing rate.
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246A: Complex Analysis, Notes 1 – Analytic functions, Cauchy’s formula and singularities.

In Mathematics on January 4, 2011 at 7:36 pm

The following series of posts comprises our introduction to complex analysis as taught by Professor Rowan Killip at the University of California, Los Angeles, during the Fall quarter of 2009. Where necessary, course notes have been supplemented with details written by the authors of this website using assistance from Complex Analysis by Elias Stein and Rami Shakarchi. The basic properties of complex numbers will be assumed allowing us to begin with the definition of a holomorphic (or complex-differentiable) function, the central notion in our study of complex analysis.

The basic properties of complex numbers will be assumed, allowing us to begin with the definition of a holomorphic (or complex-differentiable) function, the central notion in our study of complex analysis.

Definition 1.1 Suppose {\Omega \in \mathbb{C}} is an open set and {f:\Omega \rightarrow \mathbb{C}}. We say {f} is holomorphic (or complex-differentiable) at {z_0 \in \Omega} if there exists {f^{\prime}(z_0) \in \mathbb{C}: f(z) = f(z_0) + f^{\prime}(z_0)(z - z_{0}) + o(\left |z-z_0\right |).} We say {f} is holomorphic on {\Omega} if it has this property for all {z \in \Omega}.

We can rewrite this formula in terms of the real and imaginary parts of {f} to surmise the relationship between complex differentiability and real analytic differentiability. Let {z, z_{0} \in \mathbb{C}} with {z = x+iy} and {z_{0} = x_0 + iy_{0}}, {x, y, x_0, y_0 \in \mathbb{R}} and write {f(z) = u(x,y) + iv(x,y)} where {u,v: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}}. Then,

\displaystyle  \left[ \begin{array}{cc} u(x,y) \\ v(x,y) \end{array} \right] = \left[ \begin{array}{cc} u(x_0, y_0) \\ v(x_0, y_0) \end{array} \right] + \left[ \begin{array}{cc} {\rm Re}f^{\prime}(z_{0}) & -{\rm Im}f^{\prime}(z_{0}) \\ {\rm Im}f^{\prime}(z_{0}) & {\rm Re}f^{\prime}(z_{0}) \end{array} \right] \left[ \begin{array}{cc} x-x_{0} \\ y-y_{0} \end{array} \right] + o(\left |x-x_{0}\right | + \left |y-y_{0}\right |).

We first notice that this is stronger than the differentiability of the real map {(x, y) \mapsto (u(x,y), v(x,y))} in {\mathbb{R}^2 \rightarrow \mathbb{R}^2}. In the real, multivariable case, the derivative of this map is a linear operator, namely, the Jacobian, {J_{f}(x,y)}; in our equation above, the {2 \times 2} matrix on the right hand side is {J_{f}(x,y)}. Clearly, it is endowed with a distinct structure summarized in the following proposition.

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