Introduction to Statistical Methods for Financial Models

Thomas A Severini

• Publisher: Chapman and Hall/CRC
• Year: 2017
• eBook ISBN: 9781351981903
• Edition: 1
• Topic: Business
• Subtopic: Finance
• Index: Business

This book provides an introduction to the use of statistical concepts and methods to model and analyze financial data. The ten chapters of the book fall naturally into three sections. Chapters 1 to 3 cover some basic concepts of finance, focusing on the properties of returns on an asset. Chapters 4 through 6 cover aspects of portfolio theory and the methods of estimation needed to implement that theory. The remainder of the book, Chapters 7 through 10, discusses several models for financial data, along with the implications of those models for portfolio theory and for understanding the properties of return data.

The audience for the book is students majoring in Statistics and Economics as well as in quantitative fields such as Mathematics and Engineering. Readers are assumed to have some background in statistical methods along with courses in multivariate calculus and linear algebra.

1

Introduction

The goal of this book is to present an introduction to the statistical methodology used in investment analysis and financial econometrics, which are concerned with analyzing the properties of financial markets and with evaluating potential investments. Here, an “investment” refers to the purchase of an asset, such as a stock, that is expected to generate income, appreciate in value, or ideally both. The evaluation of such an investment takes into account its potential financial benefits, along with the “risk” of the investment based on the fact that the asset may decrease in value or even become worthless.

A major advance in the science of investment analysis took place beginning in the 1950s when probability theory began to be used to model the uncertainty inherent in any investment. The “return” on an investment, that is, the proportional change in its value over a given period of time, is modeled as a random variable and the investment is evaluated by the properties of the probability distribution of its return. The methods used in this statistical approach to investment analysis form an important component of the field known as quantitative finance or, more recently, financial engineering. The methodology used in quantitative finance may be contrasted with that based on fundamental analysis, which attempts to measure the “true worth” of an asset; for example, in the case of a stock, fundamental analysis uses financial information regarding the company issuing the stock, along with more qualitative measures of the firm’s profitability.

For instance, in the statistical models used in analyzing investments, the expected value of the return on an asset gives a measure of the expected financial benefit from owning the asset and the standard deviation of the return is a measure of its variability, representing the risk of the investment. It follows that, based on this approach, an ideal investment has a return with a large expected value and a small standard deviation or, equivalently, a large expected value and a small variance. Thus, the analysis of investments using these ideas is often referred to as mean-variance analysis.

Concepts from probability and statistics have been used to develop a formal mathematical framework for investment analysis. In particular, the properties of the returns on a portfolio, a set of assets owned by a particular investor, may be derived using properties of sums of the random variables representing the returns on the individual assets. This approach leads to a methodology for selecting assets and constructing portfolios known as modern portfolio theory or Markowitz portfolio theory, after Harry Markowitz, one of the pioneers in this field.

A central concept in this theory is the risk aversion of investors, which assumes that, when choosing between two investments with the same expected return, investors will prefer the one with the smaller risk, that is, the one with the smaller standard deviation; thus, the optimal portfolios are the ones that maximize the expected return for a given level of risk or, conversely, minimize the risk for a given expected return. It follows that numerical optimization methods, which may be used to minimize measures of risk or to maximize an expected return, play a central role in this theory.

An important feature of these methods is that they do not rely on accurate predictions of the future asset returns, which are generally difficult to obtain. The idea that asset returns are difficult to predict accurately is a consequence of the statistical model for asset prices known as a random walk and the assumption that asset prices follow a random walk is known as the random walk hypothesis. The random walk model for prices asserts that changes in the price of an asset over time are unpredictable, in a certain sense. The random walk hypothesis is closely related to the efficient market hypothesis, which states that asset prices reflect all currently available information. Although there is some evidence that the random walk hypothesis is not literally true, empirical results support the general conclusion that accurate predictions of future returns are not easily obtained.

Instead, the methods of modern portfolio theory are based on the properties of the probability distribution of the returns on the set of assets under consideration. In particular, the mean return on a portfolio depends on the mean returns on the individual assets and the standard deviation of a portfolio return depends on the variability of the individual asset returns, as measured by their standard deviations, along with the relationship between the returns, as measured by their correlations. Thus, the extent to which the returns on different assets are related plays a crucial role in the properties of portfolio returns and in concepts such as diversification.

Of course, in practice, parameters such as means, standard deviations, and correlations are unknown and must be estimated from historical data. Thus, statistical methodology plays a central role in the mean-variance approach to investment analysis. Although, in principle, the estimation of these parameters is straightforward, the scale of the problem leads to important challenges. For instance, if a portfolio is based on 100 assets, we must estimate 100 return means, 100 return standard deviations, and 4950 return correlations.

The properties of the returns on different assets are often affected by various economic conditions relevant to the assets under consideration. Hence, statistical models relating asset returns to available economic variables are important for understanding the properties of potential investments. For instance, the theoretical capital asset pricing model (CAPM) and its empirical version, known as the market model, describe the returns on an asset in terms of their relationship with the returns on the equity market as a whole, known as the market portfolio, and measured by a suitable market index, such as the Standard & Poor’s (S&P) 500 index. Such models are useful for understanding the nature of the risk associated with an asset, as well as the relationship between the expected return on an asset and its risk. The single-index model extends this idea to a model for the correlation structure of the returns on a set of assets; in this model, the correlation between the returns on two assets is described in terms of each asset’s correlation with the return on the market portfolio.

The CAPM, the market model, and the single-index model are all based on the relationship between asset returns and the return on some form of a market portfolio. Although the behavior of the market as a whole may be the most important factor affecting asset returns, in general, asset returns are related to other economic variables as well. A factor model is a type of generalization of these models; it describes the returns on a set of assets in terms of a few underlying “factors” affecting these assets. Such a model is useful for describing the correlation structure of a set of asset returns as well as for describing the behavior of the mean returns of the assets. The factors used are chosen by the analyst; hence, there is considerable flexibility in the exact form of the model. The parameters of a factor model are estimated using statistical techniques such as regression analysis and the results provide useful information for understanding the factors affecting the asset returns; the results from an analysis based on a factor model are important in analyzing potential investments and constructing portfolios.

Data Analysis and Computing

Data analysis is an important component of the methodology covered in this book and all of the methods presented are illustrated on genuine financial data. Fortunately, financial data are readily available from a number of Internet sources such as finance.yahoo.com and the Federal Reserve Economic Data (FRED) website, fred.stlouisfed.org. Experience with such data is invaluable for gaining a better understanding of the features and challenges of financial modeling.

The analyses in the book use the statistical software R which can be downloaded, free of charge, at www.r-project.org. Analysts often find it convenient to use a more user-friendly interface to R such as RStudio, which is available at www.rstudio.com; however, the examples presented here use only the standard R software. R includes many functions that are useful for statistical data analysis; in addition, it is a programming language and users may define their own functions when convenient. Such user-defined functions will be described in detail and implemented as needed; no previous programming experience is necessary.

There are two features of R that make it particularly useful for analyzing financial data. One is that stock price data may be downloaded directly into R. The other is that there are many R packages available that extend its functionality; several of these provide functions that are useful for analyzing financial data.

Suggestions for Further Reading

A detailed nontechnical introduction to financial analysis based on statistical concepts is given in Bernstein (2001). Chapter 1 of Fabozzi et al. (2006) gives a concise account of the history of financial modeling. Malkiel (1973) contains a nontechnical discussion of the random walk hypothesis and its implications, as well as many of the criticisms of the random walk hypothesis that have been raised.

For readers with limited experience using R, the document “Introduction to R,” available on the R Project website at https://cran.r-project.org/doc/manuals/r-release/R-intro.pdf, is a good starting point. Dalgaard (2008) provides a book-length treatment of basic statistical methods using R with many examples. The “Quick-R” website, at http://www.statmethods.net/index.html, contains much useful information for both the beginner and experienced user.

2

Returns

2.1 Introduction

As discussed in Chapter 1, the goal of this book is to provide an introduction to the statistical methodology used in modeling and analyzing financial data. This chapter introduces some basic concepts of finance and the types of financial data used in this context. The analyses focus on the returns on an asset, which are the proportional changes in the price of the asset over a given time interval, typically a day or month. Th...

Table of contents

• Title Page
• Copyright Page
• Dedication
• Table of Contents
• Preface
• 1 Introduction
• 2 Returns
• 3 Random Walk Hypothesis
• 4 Portfolios
• 5 Efficient Portfolio Theory
• 6 Estimation
• 7 Capital Asset Pricing Model
• 8 The Market Model
• 9 The Single-Index Model
• 10 Factor Models
• References
• Index