Philip J. McDonnell (Sammamish, WA) is a trader and software and trading methodologies developer who has created proprietary data collection and analysis tools for real time analysis of market direction and stock selection with an emphasis on options analysis. Prior, he handled network operations for a venture capital incubator, The Inception Group, and developed and sold an options analysis software package. He has also developed option risk management software for Charles Schwab & Co. McDonnell served as research assistant at University of California, Berkeley, School of Business, under Victor Niederhoffer. He holds degrees in mathematics and computer science from University of California, Berkeley.

Optimal Portfolio Modeling

Models to Maximize Returns and Control Risk in Excel and R, CD-ROM includes Models Using Excel and RBy Philip McDonnell

John Wiley & Sons

Copyright © 2007 Philip McDonnell
All right reserved.
ISBN: 978-0-470-11766-8

Chapter One

Modeling Market Microstructure-Randomness in Markets

Traditionally, portfolio modeling has been the domain of highly quantitative people with advanced degrees in math and science. On Wall Street, such people are commonly called rocket scientists. Optimal Portfolio Modeling was written to provide an easily accessible introduction to portfolio modeling for readers who prefer an intuitive approach. This book can be read by the average intelligent person who has only a modest high school math background. It is designed for people who wish to understand rocket science with a minimum of math.

The focus of this book is on money management. It is not a book about market timing, nor is it designed to help you pick stocks. There are numerous other books that address those subjects. Rather, this work will show the reader how to define models to help manage money and control risk. Stock selection is really just the details. The big picture is actually about achieving your overall portfolio goals.

Included with this book is a CD-ROM that includes numerous examples in both Excel and R, the statistical modeling language. The book assumes the user has a beginner’s level knowledge of Excel and focuses mainly on those specific areas that apply to portfolio modeling and optimization. There are many books that offer an introduction to Excel, and the interested reader is encouraged to investigate those.

R is an open-source language that offers powerful graphics and statistics capabilities. Two appendices in this book offer introductory support for users who wish to download R at no cost and learn how to program. Because R is powerful, many functions and graphs can be done with very few command lines. Often, only a single line will create a graph or perform a statistical analysis.

The overriding philosophy of all of the examples is simplicity and ease of understanding. Consequently, each example typically focuses on a single simple problem or calculation. It is the job of the computer to know how to perform the calculations. The user only needs to know how to invoke the right computer function and to understand the results. Understanding and intuition are the primary goals of this book.

This chapter introduces the important background of market microstructure and randomness. This is a foundation for the ideas developed later in this book. The discussion starts with a thorough introduction to the idea of randomness and what a random walk is. The topic of randomness is presented as an essential element in understanding how and why a portfolio works. After all, the primary rationale for a portfolio is intelligent diversification.

From there, the book moves to a discussion of market microstructure and how it affects the operation of markets. Later, the reader is introduced to the efficient market hypothesis, along with its history and development, starting with early pioneers in the field. Augmenting this is the discussion on arbitrage pricing theory and its modern applications. This latter topic shows how the market identifies and eliminates any risk, less arbitrage opportunities.

Trading speculative markets has always been difficult. Over the years, several studies have shown that some 70 to 80 percent of all mutual funds underperform the averages. A study by Professor Terrance Odean of the University of California at Berkeley demonstrated that most individual investors actually lose money. This study analyzed thousands of real-life individual investor brokerage accounts. Thus, it provides a comprehensive look at how real individual traders operate. The inescapable conclusion is that both professional and individual investors find that trading the markets is challenging.

Successful trading is predicated on one thing. Traders must predict the direction of price changes in the future. At a minimum, a successful trader must predict prices so that each trade has an expectation of yielding a profit. This does not mean that each trade must be successful, but, rather, that a succession of trades would usually be expected to result in a profit. This should not be taken to mean that having a positive expectation for each trade is the only thing a successful trader needs. The astute reader will note that the use of words such as usually, average, and expectation naturally implies that the art of forecasting is far from perfect. In fact, it is best studied from a statistical perspective with a view to identifying what is random and what is predictable.

In a recent 500-day period, the stock market as measured by the Standard and Poor’s 500 index was generally a modestly up market. A statistical analysis of the daily compounded returns for the period shows:

Average daily return: .038 percent Standard deviation: .640 percent Probability of rise: 56 percent

The standard deviation is simply a measure of the variability of returns around the average. From this simple analysis, we can make some interesting observations:

1. The average daily return is small with respect to the standard deviation.

2. The daily variability is relatively large, at 16 times the return.

3. The market went up 56 percent of the time, or slightly more than half. It also went down the other 44 percent of the days. So even during up markets, the number of up days is only slightly better than 50-50.

4. The variability completely swamps the average return.

Observations such as these have led many early researchers in finance to propose a model for the markets that explicitly embraces randomness at its very core. A cornerstone of this idea is that markets represent all of the knowledge, information, and intelligent analysis that the many participants bring to bear. Thus, the market has already priced itself to correspond with the sum of all human knowledge. In order to outperform the market, a trader must have better information or analysis than the rest of the participants collectively. It would seem the successful trader must be smarter than everyone else in the world put together.

THE RANDOM WALK MODEL

To the typical layman, the random walk model is the best-known name for the idea that markets are very good at pricing themselves so as to remove excess profit opportunities. The academic community generally prefers the description the efficient market hypothesis (EMH). Either way, the idea is the same-it is very difficult to outperform the market. If someone does outperform, then it is likely only attributable to mere luck and not skill.

The history of the EMH is a rather long one. The first known work was by Louis Bachelier in 1900, in which he posited a normal distribution of price changes and developed the first-known option model based on the idea of a normal random walk (see Figure 1.1). His seminal paper in the field was quickly forgotten for some 60 years. As an interesting side note, the mathematics that Bachelier developed was essentially the same analysis that Albert Einstein reinvented in 1906 in his study of Brownian motion of microscopic particles. Einstein’s famous paper was published some six years after Bachelier’s work. However Bachelier’s paper languished in relative obscurity until its rediscovery in the 1960s.

Prof. Paul Samuelson of the Massachusetts Institute of Technology offered a Proof that Properly Anticipated Prices Fluctuate Randomly in the 1960s. This provided a theoretical basis for the EMH idea. However, it fell to M. F. M. Osborne to provide the modern theoretical basis for the efficient market hypothesis. Osborne was the first to posit the idea of a lognormal distribution and provide evidence that the price changes in the market were log normally distributed. Furthermore, he was the first modern researcher to draw the link between the fluctuations of the market and the mathematics of random walks developed by Bachelier and Einstein decades earlier.

Osborne was a physicist by training employed at the U.S. Naval Observatory. As such, he was not an academic, nor did he come from a traditional finance background. Thus, it is not surprising that he is rarely recognized as the father of the efficient market hypothesis in the lognormal form. However, it is very clear that his empirical and theoretical work that described the distribution of stock price changes as log normal and the underlying process of the market as being akin to the process described by Einstein called Brownian motion was the first to elucidate both concepts. Osborne deserves the honor of being the father of the EMH.

As so often happens in academia, others who published later and were fully aware of Osborne’s work have received much of the credit. Statistician and student of mathematical and statistical history, Stephen M. Stigler has whimsically called the phenomenon his law of eponymy. The wrong person is invariably credited with any given discovery.

One aspect of this phenomenon is that when a person is erroneously credited with a discovery for whatever reason, his or her name is attached to that discovery. After much widespread usage, the name tends to stick. So even when it is later discovered by historians that someone else actually discovered the idea first, it is usually just treated as a footnote and rarely adopted into common usage among practitioners in the field.

Such is the case for Osborne’s contribution to the efficient market hypothesis. It was partly because he was a physicist working in the field of astronomy. At the time of his publication, he was not really an accepted name in the field of finance.

One form of the EMH defines the relationship between today’s price [X.sub.t] and tomorrow’s price [X.sub.t+1] as follows:

[X.sub.t+1] = [X.sub.t] + e (1.1)

where e is a random error term. We note that this model is inherently an additive model. The usual academic assumption corresponding to this type of model is the normal distribution. The key concept is that the normal distribution is strongly associated with sums of random variables. In fact, there is a weak convergence theorem in probability theory that states that for any sums of independent identically distributed variables with finite variance, their distribution will converge to the normal distribution. This result virtually assures us that the normal distribution will remain ubiquitous in nature.

However, the empirical work of Osborne showed us that the distribution of price changes was log normal. This type of distribution is consistent with a multiplicative model of price changes. In this model, the expression for price changes becomes

[X.sub.t+1] = [X.sub.t](1+e) (1.2)

WHAT YOU CANNOT PREDICT IS RANDOM TO YOU

Some would argue that the market is not random. Certainly, almost every single participant in the market believes he or she will achieve superior results. Most of these participants are smarter, richer, and better educated than average. Can they all achieve superior returns? Of course, it would be mathematically impossible for everyone to be above average. Can they all be deluded?

To answer this question it is helpful to look at the long-term history of the market. When we fit a regression line through the monthly Standard & Poor’s closing prices [P.sub.t] on the first trading day of each month since 1950 until November 2006, we find the following:

ln [P.sub.t] = .0059t + 3.06

In this case the t values are simply month numbers starting at 1, then 2, and so on for each of the 683 months in the study. The fitted coefficient .0059 can be interpreted as a simple monthly rate of increase in the series. So if we annualize, we get an annual rate of return of about 7.1 percent for the long-term growth rate of the Standard & Poor’s 500 average (see Figure 1.2). This is a very respectable long-term upward trend in the market. The [R.sup.2] for this regression was 97 percent. Given that 100 percent is a perfect fit, this indicates that the model is a very good one.

The underlying message here is that the market goes up over time. The fact that the natural log model fits well tells us that the growth in the market is compounded and presumably derived from a multiplicative model. But beyond that, it tends to make people think they are financial geniuses who might not be.

Bull markets make us all geniuses. -Wall Street maxim

From the perspective of the long-term time frame, the market has been in a bull phase for at least the entire last century. Human beings have a natural propensity to attribute good luck to their own innate skill. Psychologists call this the self-attribution fallacy. The long-term bull market has created a large group of investors who believe they have some superior gift for investing. Few investors ever stop to critically analyze their own results to verify that they are indeed performing better than the market.

Given that the market exhibits long-term compounded returns over time, it is clear the best model is a multiplicative one. This long-term return is often called the drift-the tendency of the market to move inexorably upward over time. However, to understand the shorter-term movements of the market, we must look to a different kind of model in which the short-term fluctuations appear to be more random. The reason for that is simply because the marketplace in general will anticipate all known information, and thus, the current market price is the best price available. Thus, by definition, any news that is material to the market and was not anticipated will appear as random shocks in either direction.

The key idea to understand is that the market will not respond to news that it already knows. Or if it does, that response will be contrary to what a rational analyst might have expected. These contrary movements are caused when a large group of investors was expecting a certain piece of news and thus, holding positions that were previously taken. When the news is announced, the entire group may try to unwind their positions, resulting in a market movement in exactly the opposite direction one might expect. Simply put, the market has already discounted the expected news and adjusted the price well in advance. Because this phenomenon is so prevalent, Wall Street has evolved the maxim, “Buy on the rumor, sell on the news.” Although one would never recommend relying on rumors for investment success, certainly buying on the correct anticipation of news is the better strategy.

This leaves us with the realization that, absent informed knowledge of upcoming news, the outcome of such events will be random and unpredictable to us. Some would argue that for most news someone knew the event in advance. Certainly for earnings announcements and government reports, someone did know the information to a certainty. For them, the news was not random but completely predictable. Assuming the information was not widely disclosed, then for the rest of investors, the information remains random and unpredictable.

There is a general principle at work here. If we cannot predict the news, then it is random to us. So even if others know the information, then insofar as we do not, and cannot predict it, it remains random for us.

MARKET MICROSTRUCTURE

Generally speaking, the market consists of the interactions between four broad classes of orders. These can be grouped into two categories each. There are market orders and there are limit orders. There are orders to buy and sell. Although there are variations and nuances on each, these characterize the main categories of trading orders.

Market order-A market order is an order to buy or sell that is to be executed immediately at the best available price

Limit order-This is an order to buy or sell that is only to be executed at the specified limit price or better. Limit orders may have an expiration, such as the end of the day or 60 days.

(Continues…)

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