Introduction to statistical methods for financial models

Introduction to
Statistical Methods for
Financial Models

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Introduction to
Statistical Methods for
Financial Models
Thomas A. Severini
Northwestern University
Evanston, Illinois, USA

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Library of Congress Cataloging-in-Publication Data
Names: Severini, Thomas A. (Thomas Alan), 1959- author.
Title: Introduction to statistical methods for financial models / Thomas A.
Severini.
Description: Boca Raton, FL : CRC Press, [2018] | Includes bibliographical
references and index.
Identifiers: LCCN 2017003073| ISBN 9781138198371 (hardback) | ISBN
9781315270388 (e-book master) | ISBN 9781351981910 (adobe reader) | ISBN
9781351981903 (e-pub) | ISBN 9781351981897 (mobipocket)
Subjects: LCSH: Finance--Statistical methods. | Finance--Mathematical models.
Classification: LCC HG176.5 .S49 2017 | DDC 332.072/7--dc23
LC record available at />Visit the Taylor & Francis Web site at

and the CRC Press Web site at


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To Karla

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Contents

Preface

xv

1 Introduction
2 Returns
2.1 Introduction . . . . . . . . . . .
2.2 Basic Concepts . . . . . . . . . .
2.3 Adjusted Prices . . . . . . . . .
2.4 Statistical Properties of Returns
2.5 Analyzing Return Data . . . . .
2.6 Suggestions for Further Reading
2.7 Exercises . . . . . . . . . . . . .

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4 Portfolios
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . .
4.2 Basic Concepts . . . . . . . . . . . . . . . . . . . . . .
4.3 Negative Portfolio Weights: Short Sales . . . . . . . .
4.4 Optimal Portfolios of Two Assets . . . . . . . . . . .
4.5 Risk-Free Assets . . . . . . . . . . . . . . . . . . . . .
4.6 Portfolios of Two Risky Assets and a Risk-Free Asset
4.7 Suggestions for Further Reading . . . . . . . . . . . .
4.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . .

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5 Efficient Portfolio Theory
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Portfolios of N Assets . . . . . . . . . . . . . . . . . . . . . .
5.3 Minimum-Risk Frontier . . . . . . . . . . . . . . . . . . . . .

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3 Random Walk Hypothesis
3.1 Introduction . . . . . . . . . . . . . . . . .
3.2 Conditional Expectation . . . . . . . . . .
3.3 Efficient Markets and the Martingale Model
3.4 Random Walk Models for Asset Prices . .
3.5 Tests of the Random Walk Hypothesis . .
3.6 Do Stock Returns Follow the Random Walk
3.7 Suggestions for Further Reading . . . . . .
3.8 Exercises . . . . . . . . . . . . . . . . . . .

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xii

Contents

5.4
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5.6
5.7
5.8
5.9
5.10

The Minimum-Variance Portfolio
The Efficient Frontier . . . . . .
Risk-Aversion Criterion . . . . .
The Tangency Portfolio . . . . .
Portfolio Constraints . . . . . .
Suggestions for Further Reading
Exercises . . . . . . . . . . . . .

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6 Estimation
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . .
6.2 Basic Sample Statistics . . . . . . . . . . . . . . . . .
6.3 Estimation of the Mean Vector and Covariance Matrix
6.4 Weighted Estimators . . . . . . . . . . . . . . . . . .
6.5 Shrinkage Estimators . . . . . . . . . . . . . . . . . .
6.6 Estimation of Portfolio Weights . . . . . . . . . . . .
6.7 Using Monte Carlo Simulation to Study the Properties

of Estimators . . . . . . . . . . . . . . . . . . . . . . .
6.8 Suggestions for Further Reading . . . . . . . . . . . .
6.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . .

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7 Capital Asset Pricing Model
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . .
7.2 Security Market Line . . . . . . . . . . . . . . . . . . .
7.3 Implications of the CAPM . . . . . . . . . . . . . . . .
7.4 Applying the CAPM to a Portfolio . . . . . . . . . . . .
7.5 Mispriced Assets . . . . . . . . . . . . . . . . . . . . . .
7.6 The CAPM without a Risk-Free Asset . . . . . . . . . .
7.7 Using the CAPM to Describe the Expected Returns on
a Set of Assets . . . . . . . . . . . . . . . . . . . . . . .
7.8 Suggestions for Further Reading . . . . . . . . . . . . .
7.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . .
8 The
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8.12

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Market Model
Introduction . . . . . . . . . . . . . . . . . . . . . . . . .
Market Indices . . . . . . . . . . . . . . . . . . . . . . . .
The Model and Its Estimation . . . . . . . . . . . . . . .
Testing the Hypothesis that an Asset Is Priced Correctly
Decomposition of Risk . . . . . . . . . . . . . . . . . . .
Shrinkage Estimation and Adjusted Beta . . . . . . . . .
Applying the Market Model to Portfolios . . . . . . . . .
Diversification and the Market Model . . . . . . . . . . .
Measuring Portfolio Performance . . . . . . . . . . . . . .
Standard Errors of Estimated Performance Measures . .
Suggestions for Further Reading . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Contents
9 The
9.1
9.2
9.3
9.4
9.5
9.6
9.7
9.8

Single-Index Model
Introduction . . . . . . . . . . . . . .
The Model . . . . . . . . . . . . . . .
Covariance Structure of Returns under
Single-Index Model . . . . . . . . . .
Estimation . . . . . . . . . . . . . . .

Applications to Portfolio Analysis . .
Active Portfolio Management and the
Treynor–Black Method . . . . . . . .
Suggestions for Further Reading . . .
Exercises . . . . . . . . . . . . . . . .

10 Factor Models
10.1 Introduction . . . . . . . . . . . . . .
10.2 Limitations of the Single-Index Model
10.3 The Model and Its Estimation . . . .
10.4 Factors . . . . . . . . . . . . . . . . .
10.5 Arbitrage Pricing Theory . . . . . . .
10.6 Factor Premiums . . . . . . . . . . .
10.7 Applications of Factor Models . . . .
10.8 Suggestions for Further Reading . . .
10.9 Exercises . . . . . . . . . . . . . . . .

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References

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Index

363

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Preface

This book provides an introduction to the use of statistical concepts and
methods to model and analyze financial data; it is an expanded version of
notes used for an advanced undergraduate course at Northwestern University,
“Introduction to Financial Statistics.” A central theme of the book is that by

modeling the returns on assets as random variables, and using some basic concepts of probability and statistics, we may build a methodology for analyzing
and interpreting financial data.
The audience for the book is students majoring in statistics and economics
as well as in quantitative fields such as mathematics and engineering; the book
can also be used for a master’s level course on statistical methods for finance.
Readers are assumed to have taken at least two courses in statistical methods
covering basic concepts such as elementary probability theory, expected values, correlation, and conditional expectation as well as introductory statistical
methodology such as estimation of means and standard deviations and basic
linear regression. They are also assumed to have taken courses in multivariate calculus and linear algebra; however, no prior experience with finance or
financial concepts is required or expected.
The 10 chapters of the book fall naturally into three sections. After a
brief introduction to the book in Chapter 1, Chapters 2 and 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, with Chapter 4 containing the basic ideas and Chapter 5 presenting a more mathematical treatment
of efficient portfolios; the estimation of the parameters needed to implement
portfolio theory is the subject of Chapter 6. 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. These models begin with the capital asset pricing
model in Chapter 7; its more empirical version, the market model, is covered
in Chapter 8. Chapter 9 covers the single-index model, which extends the
market model to the returns on several assets; more general factor models are
the topic of Chapter 10.
In addition to building on the basic concepts covered in math and statistics courses, the book introduces some more advanced topics in an applied
setting. Such topics include covariance matrices and their properties, shrinkage estimation, the use of simulation to study the properties of estimators,

xv

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Preface

multiple testing, estimation of standard errors using resampling, and optimization methods. The discussion of such methods focuses on their use and
the interpretation of the results, rather than on the underlying theory.
Data analysis and computation play a central role in the book. There are
detailed examples illustrating how the methods presented may be implemented
in the statistical software R; the methods described are applied to genuine
financial data, which may be conveniently downloaded directly into R. These
examples include both the use of R packages when available and the writing
of small R programs when necessary. I have tried to provide sufficient details
so that readers with even minimal experience in R can successfully implement
the methodology; however, those with no R experience will likely benefit from
one of the many introductory books or online tutorials available.
Each chapter ends with exercises and suggestions for further reading. The
exercises include both questions requiring analytic solutions and those requiring data analysis or other numerical work; in nearly all cases, any R functions
needed have been discussed in the examples in the text. Finance and financial statistics are well-studied fields about which much has been written. The
books and papers given as suggestions for further reading were chosen based
on the expected background of the reader, rather than to reference the most
definitive treatments of a topic.
I would like to thank Karla Engel who was instrumental in preparing the
manuscript and who provided many useful comments and corrections; it is
safe to say that this book would not have been completed without her help.
I would like to thank Matt Davison (University of Western Ontario) for a
number of valuable comments and suggestions. Several anonymous reviewers
made helpful comments at various stages of the project and their contributions
are gratefully acknowledged. I would also like to thank Rob Calver and the
staff at CRC Press/Taylor & Francis for suggestions and other help throughout

the publishing process.

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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

1

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Introduction to Statistical Methods for Financial Models

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

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Introduction

3

as the market portfolio, and measured by a suitable market index, such as the
Standard & Poors (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.

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Introduction to Statistical Methods for Financial Models

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 tmethods.
net/index.html, contains much useful information for both the beginner and
experienced user.

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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. The statistical foundations for the analysis

of such data are presented, along with statistical methods that are useful for
investigating the properties of return data.

2.2

Basic Concepts

Consider an asset, such as one share of a particular stock, and let Pt denote
the price of the asset at time t, t = 0, 1, 2, . . . so that P0 is the initial price,
P1 is the price at time 1, P2 is the price at time 2, and so on. Some assets
pay dividends, a specified amount at a given time. For example, one share of
IBM stock may pay a dividend of $1.20 each quarter. These dividends make
the asset worth more than simply the price. For now, assume that there are
no dividends.
The net return or, simply, the return, on the asset over the period from
time t − 1 to time t is defined as
Rt =

Pt − Pt−1
Pt
=
− 1,
Pt−1
Pt−1

t = 1, 2, . . . .

That is, the return on the asset is simply the proportional change in its price
over a given time period; the return is positive if the price increased and is
negative if the price decreased.

Example 2.1 Suppose that, for a given asset, P0 = 60, P1 = 62.40, P2 = 63.96,
P3 = 61.40, and P4 = 66; assume that all prices are in dollars but, for
5

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simplicity, the dollar sign is omitted. Then the returns are
R1 =

62.40 − 60
= 0.040,
60

R2 =

63.96 − 62.40
= 0.025,
62.40

61.40 − 63.96
66 − 61.40
= −0.040, R4 =
= 0.075.
63.96
61.40

The revenue from holding the asset is given by
R3 =

revenue = (investment) × (return).
Therefore, in Example 2.1, if the initial investment is $100, the revenue over
the period from t = 0 to t = 1 is
100(0.04) = $4.
Normally, we focus on the return rather than on the revenue, which
depends on the amount invested.
The gross return on the asset over the period from time t − 1 to time t is
Pt
= 1 + Rt
Pt−1
so that, for example, the gross return corresponding to R1 = 0.04 is simply
1.04.
We may be interested in returns over a length of time longer than one
period. The return over the time period from time t − k to time t, known as
the k-period return at time t, is defined as the proportional change in price
over that time period. Let Rt (k) denote the k-period return at time t. Then
Rt (k) =

Pt − Pt−k
Pt
=
− 1,
Pt−k
Pt−k

t = k, k + 1, . . . .

Multiperiod returns are related to one-period returns by
Pt
Pt Pt−1
Pt−k+1
=
···
Pt−k
Pt−1 Pt−2
Pt−k
= (1 + Rt )(1 + Rt−1 ) · · · (1 + Rt−k+1 ).

1 + Rt (k) =

Note that 1 + Rt (k) is the gross return from t − k to t and 1 + Rt , 1 + Rt−1 , . . .,
are the single-period gross returns.
Example 2.2 Using the sequence of prices given in Example 2.1, the twoperiod return at time 4 is
R3 (2) =

P4 − P2
66 − 63.96
= 0.032.
=
P2
63.96

Recall that R3 = −0.040, and R4 = 0.075. Then
R4 (2) = (1 + 0.075)(1 − 0.040) − 1 = 0.032.

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7

Log-Returns
It is sometimes convenient to work with log-returns, defined by rt =
log (1 + Rt ), t = 1, 2, . . .; note that throughout the book, “log” will denote
natural logarithms.
Let pt = log Pt , t = 0, 1, . . . denote the log prices. Then the log-returns are
defined as
Pt
rt = log (1 + Rt ) = log
= pt − pt−1 .
Pt−1
That is, log-returns are simply the change in the log-prices.
One advantage of working with log-returns is that it simplifies the analysis
of multi-period returns. Let rt (k) denote the k-period log-return at time t.
Then, by analogy with the single-period case, rt (k) = log(1 + Rt (k)) and
rt (k) = log (1 + Rt (k))
= log ((1 + Rt )(1 + Rt−1 ) · · · (1 + Rt−k+1 ))
= log (1 + Rt ) + log (1 + Rt−1 ) + · · · + log (1 + Rt−k+1 )
= rt + rt−1 + · · · + rt−k+1 ;
that is, the k-period log-return at time t is simply the sum of the k
single-period log-returns, rt−k+1 , rt−k+2 , . . . , rt . Alternatively, because rt =
pt − pt−1 , the k-period log-return is the change in the log-price from period
t − k to period t,
rt (k) = pt − pt−k .
Example 2.3 Using the sequence of prices given in Example 2.1, P0 = 60,
P1 = 62.40, P2 = 63.96, P3 = 61.40, and P4 = 66, the log-prices are given by

p0 = log(60) = 4.0943,
p1 = log(62.40) = 4.1336,
p2 = log(63.96) = 4.1583,
p3 = 4.1174,
and

p4 = 4.1897.

It follows that the log-returns are
r1 = p1 − p0 = 4.1336 − 4.0943 = 0.0393,
r2 = p2 − p1 = 4.1583 − 4.1366 = 0.0217,
r3 = p3 − p2 = 4.1174 − 4.1583 = −0.0409,
and

r4 = p4 − p3 = 4.1897 − 4.1174 = 0.0723.

Alternatively, the log-returns may be calculated from the returns; for example
R1 = 0.04 so that
r1 = log(1 + R1 ) = log(1 + 0.04) = log(1.04) = 0.0392,

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with the difference between this and our previous result due to round-off
error. The three-period log-return at time 4 is
r4 (2) = r3 + r4 = −0.0409 + 0.0723 = 0.0314;

alternatively, using the result from Example 2.2,
r4 (2) = log(1 + R4 (3)) = log(1 + 0.032) = 0.0315.

Dividends
Now suppose that there are dividends. Let Dt represent the dividend paid
immediately prior to time t, that is, after time t − 1 but before time t; for
convenience, we will refer to such a dividend as being paid “at time t.” Then
the gross return from time t − 1 to time t takes into account the payment of
the dividend, along with the change in price; it is defined as
1 + Rt =

Pt + Dt
.
Pt−1

The net return is given by
Rt =

Pt
Dt
−1 +
Pt−1
Pt−1

Pt + Dt
−1 =
Pt−1

= (proportional change in price)
+ (dividend as a proportion of price at time t − 1).

Thus, it is possible to make money from an investment in an asset even if the
asset’s price declines over time.
The multiperiod return from period t − k to period t is defined by an
analogy with the no-dividend case:
1 + Rt (k) = (1 + Rt )(1 + Rt−1 ) · · · (1 + Rt−k+1 )
=

Pt + Dt
Pt−1

Pt−1 + Dt−1
Pt−2

···

Pt−k+1 + Dt−k+1
Pt−k

.

Example 2.4 Suppose that, as in Example 2.1, P0 = 60, P1 = 62.40, P2 =
63.96, and suppose that there are dividends D1 = 2 and D2 = 1. Then
R1 =

P1 + D1
62.40 + 2
− 1 = 0.073
−1 =
P0
60

and

P2 + D2
63.96 + 1
− 1 = 0.041.
−1 =
P1
62.40
The two-period return at time 2 is
R2 =

R2 (2) =

P2 + D2
P1

P1 + D1
P0

− 1 = (1.073)(1.041) − 1 = 0.117.

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