Über die Autorin bzw. den Autor

Svetlozar T. Rachev, PhD, Doctor of Science, is Chair-Professor at the University of Karlsruhe in the School of Economics and Business Engineering; Professor Emeritus at the University of California, Santa Barbara; and Chief-Scientist of FinAnalytica Inc.

John S. J. Hsu, PhD, is Professor of Statistics and Applied Probability at the University of California, Santa Barbara.

Biliana S. Bagasheva, PhD, has research interests in the areas of risk management, portfolio construction, Bayesian methods, and financial econometrics. Currently, she is a consultant in London.

Frank J. Fabozzi, PhD, CFA, is Professor in the Practice of Finance and Becton Fellow at Yale University's School of Management and the Editor of the Journal of Portfolio Management.

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An accessible overview of the theory and practice of Bayesian Methods in Finance

This first-of-its-kind book explains and illustrates the fundamentals of the Bayesian methodology and their applications to finance in clear and accessible terms.

Bayesian Methods in Finance provides a unified examination of the use of Bayesian theory and practice in portfolio and risk management explaining the concepts and techniques that can be applied to real-world financial problems.

This book is a guide to using Bayesian methods and, notably, the Markov Chain Monte Carlo toolbox to: incorporate prior views of an analyst or a fund manager into the asset allocation process; estimate and predict volatility; improve risk forecasts; and combine the conclusions of different models. Each application presentation begins with the basics, works through the traditional "frequentist" perspective, and then follows with the Bayesian treatment.

This invaluable resource presents a theoretically sound framework for combining various sources of information and a robust estimation setting that explicitly incorporates estimation risk, and brings within reach the flexibility to handle complex and realistic models.

Aus dem Klappentext

Recent years have seen an impressive growth in the variety and complexity of quantitative models and modeling techniques used in finance, particularly in portfolio and risk management. While criticisms of the excessive reliance on quantitative models resurface with each turmoil in the financial markets, the focus should be on employing techniques such that the likelihood of extreme events as well as the uncertainty of the decision-making environment are properly accounted for. Bayesian methods, coupled with heavy-tailed distributional assumptions, provide one theoretically sound avenue to achieve this goal.

Together with the ability to incorporate inform-ation from different sources and tackle complex estimation problems, dealing with estimation uncertainty has been a driving factor behind the increased popularity of Bayesian methods among academics and practitioners alike.

The aim of Bayesian Methods in Finance is to provide an overview of the theory of Bayesian methods and explain their real-world applications to financial modeling. While the principles and concepts explained in the book can be used in financial modeling and decision making in general, the authors focus on portfolio management and market risk management, since these are the areas in finance where Bayesian methods have had the greatest penetration to date.

Bayesian Methods in Finance offers both students of finance and practitioners an invaluable resource in the form of a previously unavailable, highly accessible, unified look at the use of the Bayesian methodology―as well as numerical computational methods―in financial models and asset management.

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Bayesian Methods in Finance

John Wiley & Sons

Chapter One

Quantitative financial models describe in mathematical terms the relationships between financial random variables through time and/or across assets. The fundamental assumption is that the model relationship is valid independent of the time period or the asset class under consideration. Financial data contain both meaningful information and random noise. An adequate financial model not only extracts optimally the relevant information from the historical data but also performs well when tested with new data. The uncertainty brought about by the presence of data noise makes imperative the use of statistical analysis as part of the process of financial model building, model evaluation, and model testing.

Statistical analysis is employed from the vantage point of either of the two main statistical philosophical traditions-"frequentist" and "Bayesian." An important difference between the two lies with the interpretation of the concept of probability. As the name suggests, advocates of frequentist statistics adopt a frequentist interpretation: The probability of an event is the limit of its long-run relative frequency (i.e., the frequency with which it occurs as the amount of data increases without bound). Strict adherence to this interpretation is not always possible in practice. When studying rare events, for instance, large samples of data may not be available and in such cases proponents of frequentist statistics resort to theoretical results. The Bayesian view of the world is based on the subjectivist interpretation of probability: Probability is subjective, a degree of belief that is updated as information or data are acquired.

Closely related to the concept of probability is that of uncertainty. Proponents of the frequentist approach consider the source of uncertainty to be the randomness inherent in realizations of a random variable. The probability distributions of variables are not subject to uncertainty. In contrast, Bayesian statistics treats probability distributions as uncertain and subject to modification as new information becomes available. Uncertainty is implicitly incorporated by probability updating. The probability beliefs based on the existing knowledge base take the form of the prior probability. The posterior probability represents the updated beliefs.

Since the beginning of last century, when quantitative methods and models became a mainstream tool to aid in understanding financial markets and formulating investment strategies, the framework applied in finance has been the frequentist approach. The term "frequentist" usually refers to the Fisherian philosophical approach named after Sir Ronald Fisher. Strictly speaking, "Fisherian" has a broader meaning as it includes not only frequentist statistical concepts such as unbiased estimators, hypothesis tests, and confidence intervals, but also the maximum likelihood estimation framework pioneered by Fisher. Only in the last two decades has Bayesian statistics started to gain greater acceptance in financial modeling, despite its introduction about 250 years ago by Thomas Bayes, a British minister and mathematician. It has been the advancements of computing power and the development of new computational methods that has fostered the growing use of Bayesian statistics in finance.

On the applicability of the Bayesian conceptual framework, consider an excerpt from the speech of former chairman of the Board of Governors of the Federal Reserve System, Alan Greenspan:

The Federal Reserve's experiences over the past two decades make it clear that uncertainty is not just a pervasive feature of the monetary policy landscape; it is the defining characteristic of that landscape. The term "uncertainty" is meant here to encompass both "Knightian uncertainty," in which the probability distribution of outcomes is unknown, and "risk," in which uncertainty of outcomes is delimited by a known probability distribution. [...] This conceptual framework emphasizes understanding as much as possible the many sources of risk and uncertainty that policymakers face, quantifying those risks when possible, and assessing the costs associated with each of the risks. In essence, the risk management approach to monetary policymaking is an application of Bayesian [decision-making].

The three steps of Bayesian decision making that Alan Greenspan outlines are:

1. Formulating the prior probabilities to reflect existing information.

2. Constructing the quantitative model, taking care to incorporate the uncertainty intrinsic in model assumptions.

3. Selecting and evaluating a utility function describing how uncertainty affects alternative model decisions.

While these steps constitute the rigorous approach to Bayesian decision-making, applications of Bayesian methods to financial modeling often only involve the first two steps or even only the second step. This tendency is a reflection of the pragmatic Bayesian approach that researchers of empirical finance often favor and it is the approach that we adopt in this book. The aim of the book is to provide an overview of the theory of Bayesian methods and explain their applications to financial modeling. While the principles and concepts explained in the book can be used in financial modeling and decision making in general, our focus will be on portfolio management and market risk management since these are the areas in finance where Bayesian methods have had the greatest penetration to date.

A FEW NOTES ON NOTATION

Throughout the book, we follow the convention of denoting vectors and matrices in boldface.

We make extensive use of the proportionality symbol, '[varies]', to denote the cases where terms constant with respect to the random variable of interest have been dropped from that variable's density function. To illustrate, suppose that the random variable, X, has a density function

p(x) = 2x. (1.1)

Then, we can write

p(x) [varies] x. (1.2)

Now suppose that we take the logarithm of both sides of (1.2). Since the logarithm of a product of two terms is equivalent to the sum of the logarithms of those terms, we obtain

log(p(x)) = const + log(x), (1.3)

where const = log(2) in this case. Notice that it would not be precise to write log(p(x)) [varies] log(x). We come across the transformation in (1.3) in Chapters 10 through 14, in particular.

OVERVIEW

The book is organized as follows. In Chapters 2 through 5, we provide an overview of the theory of Bayesian methods. The depth and scope of that overview are subordinated to the methodological requirements of the Bayesian applications discussed in later chapters and, therefore, in certain instances lacks the theoretical rigor that one would expect to find in a purely statistical discussion of the topic.

In Chapters 6 and 7, we discuss the Bayesian approach to mean-variance portfolio selection and its advantages over the frequentist approach. We introduce a general framework for reflecting degrees of belief in an asset pricing model when selecting the optimal portfolio. We close Chapter 7 with a description of Bayesian model...