Models for Real-World Investors Some of you may have come here today expecting me to present a rash of mathematical formulas, theorems, and proofs, to enable you to make money in the financial markets without ever again having to think. If so, I am only mildly sorry to say that you're going to be disappointed. My "models" are not mathematical in nature. Still, please don't rush to the exits yet, for I think you'll benefit from our discussion. When I use the word "model," I refer to the way in which "real-world investors" should "invest" their clients' funds. Some of the clients are people like you, living in Ohio, surviving in long and cold winters, having worked really hard to save that money, and hoping that the money manager will do his very best for them. Or the clients are people like me, living in and liking Pennsylvania, but aware that it's not as warm as Florida (but at least we tend to lose less sleep over hurricanes). When I talk about a "real-world" money manager, I refer to one who is handling, say, the pension, college-tuition, or grandchildren-savings funds of clients who are Ohio or Pennsylvania residents. Real-world money managers should always keep in mind that they're handling somebody's hard-earned pension, college-tuition, funds, etc. These clients may, and have EVERY right to, request their funds at ANY time. They hope that, at all times, the manager acted in a way as to minimize potential losses, and they should be allowed to have good expectation of a small increase in their funds after five years, say, has elapsed. In short, the managers are being expected to exercise fiduciary responsibility at all times. As a matter of fact, the phrase "real-world investor" surely is an oxymoron, for all "investors" are in the real world. If not, their real-world clients are going to regret it. Notice that I haven't yet defined the word "investor." But I'll get to that. The old days ("The more things change, the more they remain the same") Traders (speculators or investors) generated the full range of emotions in the public eye. "Investors" were admired when things went well, and "speculators" were cursed when things went badly. I noticed that Exxon was blamed recently for the high price of gasoline. The old days are back with us! Fair disclosure: My pension fund owns a few shares of Exxon. The Good Old Days: There were few recorded systematic rules on how to be systematically successful at investing or speculation. Whatever existed was passed from generation to generation, usually as word-of-mouth rules described broadly as "wisdom," "experience," etc. This gave rise to the old cliches, e.g., Buy low, sell high Buy cheap and sell dear (the Rothschilds) Buy when there's blood in the streets Buy when the cannons boom; sell when the trumpets sound See Dickson G. Watts, "Speculation as a Fine Art." Gauss knew something. One of his biographers reported that, on a relatively small salary, Gauss left a very large estate. Gauss traded in government bonds; he left no records. Let's move forward to the 20th century. More precisely, the 1930's: Benjamin Graham The father of financial analysis The father of value investing The Dean of Wall Street Graham was a math/history (?) student at Columbia. His father died when he was quite young, and his mother raised the family on her own. Graham was offered a place in the mathematics graduate program, but declined it since he needed to work to help support his family. Graham lost money in the late 1920's stock market decline (i.e., The Great Crash). He wrote of how he was greatly affected by the loss of his clients' money and how hard he had to work to make up for it. Graham later prepared a comprehensive set of rules to educate and guide others. Graham and Dodd (1934), "Security Analysis: Principles and Techniques," McGraw-Hill. The bible of value investing. Graham's central idea was the concept of "MARGIN OF SAFETY." You didn't cut it close when you bought a stock or bond; you bought only when the company's financial condition AND stock price were enormously in your favor. With such a purchase, you had chosen to participate in the economic future of a particular enterprise, and so you wanted to buy only if the worst-case scenario (bankruptcy? sleepless nights?) was highly improbable. Even then, it would be nice if you at least got your money back in real terms. One of my books contains a story told by Joe Granville, who used to "move" the markets in the early 80's, of the guy who bought RCA stock in 1929, just before the market crashed. When the stock returned to his purchase level, the man promptly sold his RCA stock. Unfortunately, that did not happen until 1962. Graham and Dodd developed rules on to how to estimate the intrinsic value of a company, and a decision to purchase the stock was based on that estimate. Graham and Dodd's model: Purchase a company's shares if the price is lower than the proceeds you would receive by shutting down the company's operations and liquidating its assets in a fire sale. Calculate the company's net current assets (total current assets minus all liabilities) per share; buy the shares if the price on the stock exchange drops below the net current asset value per share. For over 30 years, Graham used this formula with lots of success. His students (including Warren Buffett) and followers have done very nicely too with the formula. Today, few companies' shares ever meet Graham's criteria, for managers are very quick to buy up shares if they get even close to bargain levels. Today, Graham's formula seems to work only if stock markets are at exceptionally low levels (e.g., in Japan during the 1990's). See Martin Whitman's 2006 First Quarter letter to shareholders of the Third Avenue Value Fund for comments and variations on Graham's "net-net" stock valuation procedures. Buffett saw early on that few stocks reached Graham's bargain basement levels anymore, so Buffett developed variations on Graham's approach. Every stock market book has its own model for how real-world money managers should invest. Graham's "The Intelligent Investor: A Book of Practical Counsel" is superb. Let me repeat: Go read this book! Now! Ah, I used the word "invest." Here's Graham's definition, taken from "The Intelligent Investor," of what it means to make an investment. "An investment operation is one which, upon thorough analysis, promises safety of principal and an adequate return. Operations not meeting these requirements are speculative." Key words and phrases: "thorough analysis," "safety of principal," "adequate return." If any of these are missing from your operation then you're a speculator. Graham's Corollary: If it is a good investment then it is a good speculation. In other words, if it's a good investment then you might make much more than an adequate return. If you continually make good investments then you'll probably do better than the majority of speculators. Indeed, Thompson, et al (2003), "Models for Investors in Real World Markets," Wiley; they report on p. 220: "As some readers may know, [Warren Buffett] turned a $10,000 investment in 1955 into $250 million today ..." Buffett clearly lives by Graham's "Margin of Safety" philosophy. Buffett once commented: "You don't try to buy businesses worth $83 million for $80 million." Graham greatly deplored the way in which the term "investor" was used to describe anyone who is trading stocks. Me too. Graham also deplored the way the stock market was treated like a gambling casino, right down to the "blue chips" terminology. As I said before, every book on stock market investing represents the author's model on how investors should go about making investments. Before the week is out, I will show you a collection of books on investing and speculation, each presenting its own model. Let me now turn to mathematical models for real-world investors. I don't have enough time to give a comprehensive review of the literature developed by mathematicians, economists, probabilists, etc. I hope you will be content with some highlights. In 1900, Bachelier wrote his dissertation on the use of Brownian motion for the modelling the prices of financial securities. And then things took off from there. Later came: Markowitz's optimal portfolio selection theory The Efficient Market Hypothesis (EMH) The Capital Assets Pricing Model The Black-Scholes formula Modern portfolio theory Post-modern portfolio theory The enormous literature on the mathematics of arbitraging, options, and derivatives. There is now a large literature on financial mathematics, financial engineering, ... And there's also The behavioral finance crowd, looking at the matter of cognitive biases on the part of "investors." At least at this stage, I don't want to get into the controversy about whether or not the EMH is valid. I will say that I've seen times when the times when the market seems to be sensible/efficient and there are times when it's not. More importantly, I believe that whether or not the EMH is valid is completely irrelevant to the hardscrabble clients of Ohio and Pennsylvania, i.e., the little people like you and me. I'd hate to receive a pension fund report, e.g., "Dear Shareholder, we have experienced a 100% loss of your funds, that being caused by our discovery that the EMH is (or isn't) valid. Yours sincerely, etc." From what I've said so far, you might have sensed that I'm not a fan of "financial engineering." Nevertheless, I shall suggest four models which I think financial mathematicians could address, leading to results which would be of serious interest to the real-world girls and boys. I will pose the problems in a very broad way. I'm not after journal publications here; I'm looking for mathematical thinking and its applications which will make real-world, highly successful, value fund managers sit up straight. 1. Hidden Markov models: This approach uses none of the stochastic differential equations, measure theory, martingales, submartingales, Ito's lemma, you-name-it, which arises in financial mathematics. Let's start with a simple Markov chain model for the weather. Suppose we're tracking the daily temperature. We've noticed over the past 5 years that if the weather one day is good then there is a 65% relative frequency (which I will call "chance") that it will be good on the next day and a 35% chance that it will be bad. And if the weather one day is bad then there is a 55% chance that it will be bad the next day and a 45% relative frequency that it will be good. This gives us a "transition matrix": Tomorrow Good Bad Good .65 .35 Today Bad .45 .55 We could now ask: Given that the weather on Jan. 1 is good, then what is the chance that the weather will be good on Jan. 20? The calculation can be done by multiplying the transition matrix by itself 20 times, etc. We could also ask: What is the long-term stationary transition matrix, i.e., what are the "transition probabilities" which will take us from Jan. 1 this year to Jan. 1, 2016? We could replace the weather by the stock market, good/bad by up/down, and try to answer similar questions. Unfortunately, this kind of simple analysis doesn't work well for the stock market. There are too many "exogenous" events which cause the stock market to fluctuate. Also, the stock market seems to have pretty long-term memory in the sense that the "correlation" between its behavior on trading days 3 months apart can be very strong. Hidden Markov models are a generalization of Markov models. They were invented to deal with cases where we have long-range memory and there are many "hidden" decisions which affect the variability of the markets (e.g., a big corporation might decide to acquire a smaller competitor, and the news could even have leaked out and caused market fluctuations before the deal was announced officially). Technically, a hidden Markov model is a model in which we have a transition matrix, a finite set of "states," and a set of observed outcomes. We can see the actual outcomes of the process, but the states are hidden from us. Some years ago, I attended a talk given by a hedge fund manager. He told the audience of his experiences in trying to use SDE's to make a play on mortgage prepayments during a time when mortgage interest rates were going down. (The basic problem was to estimate the proportion of mortgagees who will prepay their mortgages as interest rates decrease, and then you arbitrage the spread between the mortgage rates and the prevailing interest rate.) When he told this to the audience, I laughed out loud and said something like, "If you'd asked me, I would've told you not to do that, for I was prepaying my mortgage as fast as I could, and I was calling all my friends to tell them to do the same. In fact, my mortgage company quickly sold my mortgage to another company when they saw how fast I was prepaying!" The hedge fund manager continued by saying something to the effect that the use of SDEs, Ito's lemma, etc., was like trying to shoot a mosquito with a rifle. And then he proceeded to mention very vaguely his fund's approach. Keeping the details to a minimum, what I remember him saying is: Each day, we check the financial weather and, on the basis of yesterday's weather we make a prediction about today's weather. At the end of today, we update our prediction using the error in our last prediction, etc. [I read recently that his fund made nearly $1 Billion in profits from trading crude oil futures last year.] After the talk was over, the guy came over to me and said, "hey, who are you?" That led to the rare opportunity of a nearly one-hour long interview by him. I've always felt that the hidden Markov models approach ought to be a good way to predict the markets. There's now a large literature on that subject. If you search Google, you can find enormous amounts of papers, software, etc. on HMMs. Recently, I started to work with Jia Li, a terrific colleague of mine at Penn State. We are trying to construct hidden Markov models to predict the stock and currency markets. So far, it's been so-so, possibly because we deliberately opted to build extremely risk-averse models (no short-selling allowed, for example). In back-testing, we have models which beat the S&P 500 during bear markets but not during bull markets. Factoring in commissions, taxes, and stratospheric hedge-fund-type salaries for the two of us, Jia and I fall way behind the S&P 500. We've also applied our HMMs to individual stocks but, so far, it's still so-so. Still, we're going to keep trying. Problem 1: Develop a hidden Markov model approach which beats the S&P 500 over rolling 10-year periods after taxes, commissions, and stratospheric managerial salaries, with no short-selling when the model predicts a decline in the S&P 500, no leverage, and no purchases or sales of puts, calls, options, synthetics, derivatives, or exotics. Even better than beating the S&P 500, your model should beat Warren Buffett over rolling 10-year periods: Good luck. (If you can't beat Warren Buffett then join him.) With regard to our restrictions on derivatives and other exotics, read Buffet's story on the headaches involved in understanding the complexities fo derivative contracts (he compared it to Mark Twain's comment on the pain involved in trying to carry a cat home by the tail). 2. Edgar Peters, is chief investment office of PanAgora Asset Management, in Boston, a large quantitative money manager. Peters has authored "Chaos and Order in the Capital Markets," (Wiley, 2nd. ed., 1996), and "Patterns in the Dark: Free Markets, Complexity, and the Need for Uncertainty," (Wiley, 1999). Peters' approach in "Chaos and Order" involves estimation of the Hurst parameter, which arises in time series analysis, as in the flooding of the Nile River (a weak joke: Maybe the flooding Nile is what causes stock markets to crash, or is it the other way around?) Ultimately, everyone wants to know if another crash is about to happen, and if so, when; this is similar to the constant fear of flooding harbored by residents of the Nile Valley, but for them, it's a life-and-death matter, while for mutual fund clients, hey, it's only paper! Peters' book describes a money management approach which rests on estimating or calculating the Hurst parameter for the stock market, As I recall, his approach uses data going back to the 1950's to estimate the Hurst coefficient. In short, my impression is that Peters calculates the Hurst parameter using the entire population of data. On the other hand, I've always felt that it should be possible to estimate the parameter more efficiently and just as accurately using far less data obtained through random sampling. Problem 2: Investigate whether there are more efficient schemes for carrying out Peters' calculations. Is is there a way to estimate, rather than to calculate the Hurst coefficient? If you can convince Peters that you have a better way to do it then I bet you'll be considered seriously for a good job at his firm. And when you can get an interview with him, be sure to call me first because I have a small comment to tell you about his book. By the way, at , Peters comments that the "overall investment philosophy at PanAgora is that market inefficiencies occur because of behavioral reasons. In general, people overweight information which they believe they understand, and they underweight information which they don't understand or that doesn't fit in with their concept of how things are. A good example of this is the recent tech bubble. ..." Note carefully: Peters is saying that the EMH is not valid and that PanAgora's overall investment philosophy depends on its invalidity. By the way, Peters and his group did pretty well all through the turbulent times of the early 2000's. 3. James R. Thompson, et al. (2003), "Models for Investors in Real World Markets." This is an intriguing book, and I will say more about it in the last lecture. This book first subjects an individual stock to a fundamental (Graham-style) balance sheet analysis. Using novel methods for estimating the stock's future earnings, the authors then apply stochastic differential equations to predict it's future stock price, it's likelihood of dropping sharply during short-term market downturns, etc. Problem 3: Can you improve on Thompson, et al. with NO increase in the complexity of the mathematical techniques used therein? 4. Joel Greenblatt, "The Little Book that Beats the Market." Another intriguing book. He recommends that we rank all stocks by return on capital (x) and earnings yield (y). Then rank all stocks by x+y. Problem 4: Buy and read Greenblatts' book. Find a better way to rank stocks based on x and y only. (Somehow, "x+y" seems too simple.) Note that your ranking should still be based on a very simple function of x and y; nothing complicated, please! It should be simple enough for your hardscrabble, blue collar clients to understand. http://www.maths.strath.ac.uk/~aas96106/fquotes.html Some interesting quotations on finance engineering http://www.pupress.princeton.edu/video/frankfurt/