Statistics and Data Analysis for Financial Engineering

Frontmatter

1. Introduction

Abstract

This book is about the analysis of financial markets data. After this brief introductory chapter, we turn immediately in Chapters 2 and 3 to the sources of the data, returns on equities and prices and yields on bonds. Chapter 4 develops methods for informal, often graphical, analysis of data. More formal methods based on statistical inference, that is, estimation and testing, are introduced in Chapter 5.

David Ruppert

2. Returns

Abstract

The goal of investing is, of course, to make a profit. The revenue from investing, or the loss in the case of a negative revenue, depends upon both the change in prices and the amounts of the assets being held. Investors are interested in revenues that are high relative to the size of the initial investments. Returns measure this, because returns on an asset, e.g., a stock, a bond, a portfolio of stocks and bonds, are changes in price expressed as a fraction of the initial price.

David Ruppert

3. Fixed Income Securities

Abstract

Corporations finance their operations by selling stock and bonds. Owning a share of stock means partial ownership of the company. Stockholders share in both the profits and losses of the company. Owning a bond is different. When you buy a bond you are loaning money to the corporation, though bonds, unlike loans, are tradeable. The corporation is obligated to pay back the principal and to pay interest as stipulated by the bond.

David Ruppert

4. Exploratory Data Analysis

Abstract

This book is about the statistical analysis of financial markets data such as equity prices, foreign exchange rates, and interest rates. These quantities vary random thereby causing financial risk as well as the opportunity for profit.

David Ruppert

5. Modeling Univariate Distributions

Abstract

As seen in Chapter 4, usually the marginal distributions of financial time series are not well fit by normal distributions. Fortunately, there are a number of suitable alternative models, such as t-distributions, generalized error distributions, and skewed versions of t- and generalized error distributions. All of these will be introduced in this chapter.

David Ruppert

6. Resampling

Abstract

Finding a single set of estimates for the parameters in a statistical model is not enough. An assessment of the uncertainty in these estimates is also needed. Standard errors and confidence intervals are common methods for expressing uncertainty. In the past, it was sometimes difficult, if not impossible, to assess uncertainty, especially for complex models. Fortunately, the speed of modern computers, and the innovations in statistical methodology inspired by this speed, have largely overcome this problem.

David Ruppert

7. Multivariate Statistical Models

Abstract

Often we are not interested merely in a single random variable but rather in the joint behavior of several random variables, for example, returns on several assets and a market index. Multivariate distributions describe such joint behavior. This chapter is an introduction to the use of multivariate distributions for modeling financial markets data.

David Ruppert

8. Copulas

Abstract

Copulas are a popular method for modeling multivariate distributions. A copula models the dependence|and only the dependence|between the variates in a multivariate distribution and can be combined with any set of univariate distributions for the marginal distributions. Consequently, the use of copulas allows us to take advantage of the wide variety of univariate models that are available.

David Ruppert

9. Time Series Models: Basics

Abstract

A time series is a sequence of observations in chronological order, for example, daily log returns on a stock or monthly values of the Consumer Price Index (CPI). In this chapter, we study statistical models for time series. These models are widely used in econometrics, business forecasting, and many scientific applications.

David Ruppert

10. Time Series Models: Further Topics

Abstract

Economic time series often exhibit strong seasonal variation. For example, an investor in mortgage-backed securities might be interested in predicting future housing starts, and these are usually much lower in the winter months compared to the rest of the year. Figure 10.1(a) is a time series plot of the logarithms of quarterly urban housing starts in Canada from the first quarter of 1960 to final quarter of 2001. The data are in the data set Hstarts in R's Ecdat package.

David Ruppert

11. Portfolio Theory

Abstract

How should we invest our wealth? Portfolio theory provides an answer to this question based upon two principles:

• we want to maximize the expected return; and
• we want to minimize the risk, which we define in this chapter to be the standard deviation of the return, though we may ultimately be concerned with the probabilities of large losses.

David Ruppert

12. Regression: Basics

Abstract

Regression is one of the most widely used of all statistical methods. For uni- variate regression, the available data are one response variable and p predictor variables, all measured on each of n observations.

David Ruppert

13. Regression: Troubleshooting

Abstract

Many things can, and often do, go wrong when data are analyzed. There may be data that were entered incorrectly, one might not be analyzing the data set one thinks, the variables may have been mislabeled, and so forth. In Example 13.5, presented shortly, one of the weekly time series of interest rates began with 371 weeks of zeros, indicating missing data. However, I was unaware of this problem when I first analyzed the data.

David Ruppert

14. Regression: Advanced Topics

Abstract

When residual analysis shows that the residuals are correlated, then one of the key assumptions of the linear model does not hold, and tests and confidence intervals based on this assumption are invalid and cannot be trusted. Fortunately, there is a solution to this problem: Replace the assumption of independent noise by the weaker assumption that the noise process is station- ary but possibly correlated. One could, for example, assume that the noise is an ARMA process. This is the strategy we will discuss in this section.

David Ruppert

15. Cointegration

Abstract

Cointegration analysis is a technique that is frequently applied in econometrics. In finance it can be used to find trading strategies based on meanreversion.

David Ruppert

16. The Capital Asset Pricing Model

Abstract

The CAPM (capital asset pricing model) has a variety of uses. It provides a theoretical justification for the widespread practice of passive investing by holding index funds. The CAPM can provide estimates of expected rates of return on individual investments and can establish \fair" rates of return on invested capital in regulated firms or in firms working on a cost-plus basis.

David Ruppert

17. Factor Models and Principal Components

Abstract

High-dimensional data can be challenging to analyze. They are difficult to visualize, need extensive computer resources, and often require special statistical methodology. Fortunately, in many practical applications, high-dimensional data have most of their variation in a lower-dimensional space that can be found using dimension reduction techniques.

David Ruppert

18. GARCH Models

Abstract

As seen in earlier chapters, financial markets data often exhibit volatility clustering, where time series show periods of high volatility and periods of low volatility; see, for example, Figure 18.1. In fact, with economic and financial data, time-varying volatility is more common than constant volatility, and accurate modeling of time-varying volatility is of great importance in financial engineering.

David Ruppert

19. Risk Management

Abstract

The financial world has always been risky, and financial innovations such as the development of derivatives markets and the packaging of mortgages have now made risk management more important than ever but also more difficult.

David Ruppert

20. Bayesian Data Analysis and MCMC

Abstract

Bayesian statistics is based up a philosophy different from that of other methods of statistical inference. In Bayesian statistics all unknowns, and in particular unknown parameters, are considered to be random variables and their probability distributions specify our beliefs about their likely values. Estimation, model selection, and uncertainty analysis are implemented by using Bayes's theorem to update our beliefs as new data are observed.

David Ruppert

21. Nonparametric Regression and Splines

Abstract

As discussed in Chapter 12, regression analysis estimates the conditional expectation of a response given predictor variables. The conditional expectation is called the regression function and is the best predictor of the response based upon the predictor variables, because it minimizes the expected squared prediction error.

David Ruppert

Backmatter