# Mathematics and Economics

## CCAPM II: Returns Formulation and the Mean-Variance Frontier

In Asset Pricing Reading Group, Economics, Finance on March 29, 2011 at 1:58 am

Due to technical issues on getting our figure and latex to display and compile on wordpress, we will temporarily include a pdf version of this post: CAPM II: Returns & MVF. We hope to resolve this issue and post in our usual format soon.

## Bubbles and Crashes: A Survey of the Efficient Market Hypothesis, Part II

In Bubbles and Crashes, Economics, Finance on February 12, 2011 at 3:48 pm

1. Empirical Tests of the Efficient Market Hypothesis and their Results

The models developed so far provide the theoretic basis for our study. We have clearly defined efficient markets and discussed a paradox in their construction, the joint hypothesis problem. We have used the language of probability theory to imbed rigor into our definition. But, we also find that much of the inspiration for these definitions arose from experience. Germinated by a broad intuition for the structure of asset returns and supported by practical experience with traders and trading, the theory is something of a hybrid: equal parts practice and principle. It was sculpted by finely-tuned economic insight, and followed statistical study or grew alongside it. Mathematical rigor was one fiber in the thread of its development.

Historically, the empirical study of the efficient market hypothesis focused on whether prices fully reflect particular subsets of information, while the unpredictability of stock market returns played a lesser role. Weaker forms of the EMH were tested first. The first subset of information considered was that of past prices, and financial economists sought to determine the validity of the weak form efficient market hypothesis. The next subset of information considered concerned the speed at which asset prices adjusted to news and other information as well as past prices; these studies tested the semi-strong form of the hypothesis. Later work considered past prices, fundamental data, news, and insider information, in an attempt to study the strong form efficient market hypothesis.
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## 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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## Bubbles and Crashes: A Survey of the Efficient Market Hypothesis, Part I

In Asset Pricing Reading Group, Bubbles and Crashes, Economics, Finance on November 29, 2009 at 2:58 am

“A study of economics usually reveals that the best time to buy anything is last year.”

-Marty Allen

1. Introduction

A thorough study of asset price bubbles and market crashes necessitates the study of the efficient market hypothesis, one of the most controversial theories in modern finance and economics. In the wake of the 2008-2009 financial crisis, the efficient market hypothesis (EMH) has faced both opprobrium and found defense. In various editorials and journal articles, economists have criticized the financial community for what they argue was unreasonable and nearsighted adherence to the pronouncements of the hypothesis (Thaler, 2008). Others, however, hold that government interference, specifically its attempts to manage inefficiencies, have created the conditions necessary to promote price bubbles (Thompson, 2006).

In an ideal capital market, prices incorporate all available information necessary for the proper allocation of resources. However, the existence of asset bubbles and crashes may suggest that the allocation of resources is often improper. Indeed much of the discussion about the current financial crisis implies that efficient markets preclude asset price bubbles. However, definitions of the EMH vary and are implicitly based on general equilibrium principles that may be consistent with asset price bubbles. Therefore one must determine the extent to which the forms of the efficient market hypothesis preclude various assertions about bubbles. Of related importance are the relationships between financial efficiency, loosely defined as the absence of arbitrage opportunities, and other standards of efficiency (e.g. informational and welfare efficiency). The latter are the frameworks of normative economics, and our discussion of how resources should be allocated.
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## Bubbles and Crashes: Motivation and Initial Reading List

In Asset Pricing Reading Group, Bubbles and Crashes, Economics, Finance on November 10, 2009 at 3:57 pm

“I can calculate the motions of the heavenly bodies, but not the madness of people.”

— Sir Isaac Newton after losing £20,000 in the South Sea Bubble

Bubbles and their subsequent crashes have confounded historians, economists, financiers and the general populous throughout history.  Examples, often categorized as bubbles, include Tulip Mania, the South Seas bubble, the Dot Com bubble, and the recent housing bubble.  The importance of bubbles and crashes cannot be overlooked, and the housing bubble is a prime example: as a consequence of the crash, global GDP (the cumulative GDP of every country) was severely affected. Much of the literature in macroeconomics ignored the consequences of bubbles by ignoring financial intermediation and associated frictions.  In light of the recent crises, the literature is now shifting toward an approach that brings together financial economics, monetary economics, and standard macroeconomic techniques.

An asset bubble is formed when an asset’s price is significantly different from its fundamental value, also known as its intrinsic value. In practice it is sometimes calculated as the discounted sum of expected future income; however, this may not be a good estimate to the actual fundamental value. In order to calculate the true fundamental value, we must first construct a general equilibrium model such as Milgrom & Stokey 1982 or Tirole 1982. Read the rest of this entry »