Statistics and Finance

The book grew out of a course first called “Empirical Research Methods in Financial Engineering.” Empirical means derived from experience, observation, or experiment, so the book is about working with data and doing statistical analysis. Financial engineering is the construction of financial products such as stock options, interest rate derivatives, and credit derivatives. The course has been renamed “Operations Research Tools for Financial Engineering,” because it also covers applications of probability, simulation, and optimization to financial engineering.

It is assumed that the reader is already at least somewhat familiar with the basics of probability and statistics. The goals of this chapter are to

A time series is a sequence of observations taken over time, for example, a sequence of daily log returns on a stock. In this chapter, we study statistical models for times series. These models are widely used in econometrics as well as in other areas of business and operations research. For example, time series models are routinely used in operations research to model the output of simulations and are used in supply chain management for forecasting demand.

How should we invest our wealth? Portfolio theory is based upon two principles:¹

Regression is one of the most widely used of all statistical methods. The available data are one response variable and p predictor variables, all measured on each of n observations. We let Y _i be the value of the response variable for the ith observation and X_{i, 1},..., X_{i, p} be the values of predictor variables 1 through p for the ith observation. The goals of regression modeling include investigation of how Y is related to X₁,..., X _p , estimation of the conditional expectation of Y given X₁,..., X _p , and prediction of future Y values when the corresponding values of X₁,..., X _p are already available. These goals are closely connected.

The CAPM (capital asset pricing model) has a variety of uses. It provides a theoretical justification for the widespread practice of “passive” investing known as indexing. Indexing means holding a diversified portfolio in which securities are held in the same relative proportions as in a broad market index such as the S&P 500. Individual investors can do this easily by holding shares in an index fund.¹ 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.²

Corporations finance their operations by selling stock and bonds. Owning a share of stock means partial ownership of the company. You share in both the profits and losses of the company, so nothing is guaranteed.

Computer simulation is widely used in all areas of operations research and, in particular, applications of simulation to statistics have become very widespread. In this chapter we apply a simulation technique called the “bootstrap” or “re-sampling” to study the effects of estimation error on portfolio selection. The term “bootstrap” was coined by Bradley Efron and comes from the phrase “pulling oneself up by one’s bootstraps” that apparently originated in the eighteenth century story “Adventures of Baron Munchausen” by Rudolph Erich Raspe.¹ In this chapter, “bootstrap” and “resampling” are treated as synonymous.

The financial world has always been risky, but for a variety of reasons the risks have increased over the last few decades. One reason is an increase in volatility. Equity returns are more volatile, as can be seen in Figure 11.1 where the average absolute value of daily log returns of the S&P 500 has approximately doubled over the period from 1993 to 2003. Foreign exchange rates are more volatile now than before the breakdown in the 1970s of the Bretton Woods agreement of fixed exchange rates.¹ Interest rates rose to new levels in the late 1970s and early 1980s, have risen and fallen several times since then, and are now (in 2003) extremely low. Figure 4.7 shows that interest rate volatility has itself varied over time but has certainly been higher since 1975 than before.

Figure 12.1 illustrates how volatility can vary dramatically over time in financial markets. This figure is a semilog plot of the absolute values of weekly changes in AAA bond interest rates. Larger absolute changes occur in periods of higher volatility. In fact, the expected absolute change is proportional to the standard deviation. Because many changes were zero, 0.005% was added so that all data could plot on the log scale. A spline was added to show changes in volatility more clearly. The volatility varies by an order of magnitude over time; e.g., the spline (without the 0.005% added) varies between 0.017% and 0.20%. Accurate modeling of time-varying volatility is of utmost importance in financial engineering. The ARMA time series models studied in Chapter 4 are unsatisfactory for modeling volatility changes and other models are needed when volatility is not constant.

As we have seen in Chapter 6, regression is about modeling the conditional expectation of a response given predictor variables. The conditional expectation is called the regression function and is the best possible predictor of the response based upon the predictor variables. Linear regression assumes that the regression function is a linear function and estimates the intercept and slope, or slopes if there are multiple predictors. Nonlinear parametric regression¹ does not assume linearity but does assume that the regression function is of a known parameter form, for example, an exponential function. In this chapter, we study nonparametric regression where the form of the regression function is also nonlinear but, unlike nonlinear regression, not specified by a model but rather estimated from data. Nonparametric regression is used when we know or suspect that the regression function is curved, but we do not have a model for the curve.

Behavioral finance is the application of cognitive psychology to the study of the participants in financial markets. The question being investigated in this field is how humans actually perceive risk and make investment decisions. Economists have long assumed that people act so as to maximize the utility of their wealth, but we are learning that human behavior is more complex than this. People use rules-of-thumb, called heuristics, as short-cuts to reasoning. Moreover, how a person makes a financial decision depends on how the decision problem is stated, a phenomenon called frame-dependence. For example, people have a strong aversion to losses and a decision might depend on whether a decision problem is posed in a way that mentions the word “loss.” Shefrin (2000) mentions the example of a stock broker who realized that clients were extremely reluctant to sell stocks at a loss in order to buy other stocks, even if this were the best investment decision for them. However, this broker found that clients were willing to sell at a loss when told that they were “transferring assets” rather than selling losers. Whether selling losers is a good thing for an investor could be debated, but the point is that the decision made by clients depends on how the problem is framed by the broker.