The World According to Andrew Lo
Andrew Lo is the Harris & Harris Group Professor of Finance at MIT's Sloan School of Management and the director of MIT's Laboratory for Financial Engineering. He has spent most of his academic career developing models of financial markets as well as developing computational algorithms for implementing those models. He is a coauthor of The Econometrics of Financial Markets, a popular graduatelevel textbook on the statistical methods most useful for financial engineering. Most recently, he published A NonRandom Walk Down Wall Street with A. Craig MacKinlay (Princeton University Press, 1999), which examines some of the major market anomalies of the last three decades. He spoke with editor Joe Kolman in January.
Derivatives Strategy: You've done a lot of work on detecting predictable components in the stock market and the implications for derivatives pricing. What have you discovered?
Andrew Lo: In 1988, Craig MacKinlay and I started thinking about the random walk hypothesis. We wanted to construct a more robust test of the random walk, a test that would be able to detect departures—even small departures—from randomness. When we constructed this test and applied it to U.S. stock returns, we found some dramatic violations of the random walk. At first, we thought we must have made a programming error.
DS: You found inefficiencies in the market that were dramatic.
AL: I wouldn't call them inefficiencies. An inefficiency implies that somehow people are being irrational or stupid, or that there is something wrong with the market. Quite to the contrary, what we discovered was a “market opportunity” rather than a market inefficiency. It's not as if individuals were being lazy or irrational, but rather that there was predictability in the marketplace that might have been created by changes in supply and demand, or by some kind of demand for liquidity. By trading in certain ways, and therefore by providing liquidity to the marketplace, one could earn a pretty handsome return.
This was in the late 1980s, and D.E. Shaw had just been created. Although Craig and I didn't know it at the time, Shaw's main focus was to take advantage of these patterns.
DS: How would you describe these particular patterns?
AL: They were rather stark. On a weekly basis, there seemed to be a significant amount of positive correlation from one week to the next. That is, if the market return was high one week, it tended to be high the next week as well. And if it was low one week, it tended to be low the next week. So there was a certain amount of persistence in weekly stock index returns.
The curious thing was that when you looked at individual stock returns such as IBM or Microsoft, there was little in the way of predictability. This puzzled us for a long time. In fact, it was one of the reasons we felt we had made a programming error. When you looked at portfolio returns, you got sharp rejections of the random walk, but when you looked at individual securities, it seemed like the random walk was a pretty good fit. It wasn't until we started teasing apart some of our statistics that we realized what was going on.
“There was a certain amount of persistence in weekly stock index returns, but when you looked at individual stock returns, there was little in the way
of predictability. This puzzled us for a
long time.” 
The predictability came largely from crossstocks/crossweek correlations. In other words, if you look at stock A this week and next, there's not a lot of correlation between those two returns. If you look at stock B this week and next, there's not a lot of correlation between those two returns. But it turns out that there actually is a significant correlation between stock A this week and stock B next week. It was these cross effects that were giving us the rejections of the random walk hypothesis.
Right around that time, again unbeknownst to us, a trading strategy called “pairs trading” was becoming popular among equity traders. They would trade pairs of securities, one long and the other short, to take advantage of certain pricing anomalies. Many traders engaged in these kinds of activities, including D.E. Shaw.
DS: And LTCM...
AL: I believe they were doing similar types of trades in the fixedincome markets. We had inadvertently stumbled upon a set of trades that people had also come upon through entirely different means. Interestingly enough, over the past 10 years, since we published our paper, the kind of predictabilities that we first discovered have become considerably less significant.
DS: They've been arbitraged away.
AL: Yes, quite a lot has been arbitraged away, but not all of it. There are still some aspects that are present. So this is not to say that you can't continue doing it. Obviously, D.E. Shaw is quite successful with its equity portfolio. The firm has lost money on its fixedincome trades, of course.
DS: What specific trade would a trader do to take advantage of this opportunity?
AL: Well, if a trader discovered that stock A and stock B were positively correlated across weeks, then he or she would use stock A this week to forecast stock B next week. An example of a trading strategy that benefits from such crossweek predictabilities is a contrarian strategy where you buy the losers and sell the winners.
Contrarians think that prices overreact—that what goes up must come down and vice versa—so buying the losers and selling the winners can be profitable. But the existence of crossweek predictabilities provides another source of contrarian profits. To see this, consider a stock market with only two stocks, A and B, and let A and B have positive crossweek correlations. Suppose A is up this week and B is down; a contrarian would then sell A and buy B. But when there's positive crossweek correlation, an upweek for A implies an upweek for B next week, and a downweek for B implies a downweek for A next week, hence a contrarian profits from both positions—but it's because of these crossstock predictabilities, not overreaction. If you are trading 150 stocks simultaneously, it's much more complicated to take into account all of these possible cross effects—there are some real combinatorial challenges involved. That's what D.E. Shaw does so successfully: it uses various computational algorithms to optimize these sorts of trades. Nowadays, there are many more sophisticated varieties.
DS: What are the implications of this phenomenon for derivatives pricing?
AL: My colleague Jiang Wang and I wrote a paper four years ago that made what we thought was a rather simple and uncontroversial point. In derivatives pricing models, the expected return of the underlying security does not enter into the formula of the derivatives price.
DS: The important thing in the BlackScholesMerton model is volatility.
AL: That's right. But while it's true that volatility may be the only thing that matters in the formula, from an empirical point of view the expected return of the underlying asset certainly affects the volatility.
DS: It's an indirect influence.
AL: Exactly. I'll give you a simple example. Suppose we can perfectly predict what next week's stock price is going to be. In that case, what's the volatility? It's zero—there is no volatility. That's an example of an indirect relationship between knowing the expected return and having a handle on the volatility.
DS: Of course, we couldn't possibly know the expected return.
AL: That's right. But what if we could know a little bit about it? The BlackScholesMerton formula and most derivatives pricing models assume you can't predict returns at all. They assume that returns are what we academics call “independently and identically distributed,” or IID. If you assume that returns are IID, then the only thing that matters for derivatives pricing is the volatility.
But what if they're not IID? What if you know something about next week's return? For example, what if you've got five factors that tell you a lot about the behavior of Microsoft. It turns out that the existence of such predictability is going to influence the volatility. The more predictability there is, the less volatility there's going to be.
I gave you an example of an extreme case, in which there's perfect predictability and therefore no volatility. But you don't have to look at such an extreme. Even if there's a little bit of predictability, it will tend to reduce the volatility.
DS: I suppose you're taking this discovery you and others made about correlation and setting it up as a kind of subcomponent of the BlackScholesMerton model.
AL: Yes, but we also did it more generally for other kinds of derivatives pricing formulas. Let's say you can explain next week's returns to a small degree using factors from this week. And the measure of predictability is R2. That's a common statistical measure. If the R2 of a simple regression equation is 100 percent, then you've explained it perfectly. If it's 0, you haven't explained it at all. And if it's anywhere in between, you've got some predictability. What we found was that even if you had, say, 10 percent predictability in next week's returns, that could have fairly dramatic implications for derivatives prices.
DS: How dramatic?
AL: It will affect prices by 20 percent, and in some cases 30 percent, for even modest levels of predictability. That seems to suggest that this is an important feature. What we don't know is whether people have fully incorporated this into their models. We suspect that some have. Maybe D.E. Shaw, maybe Salomon Smith Barney, maybe JP Morgan or some of the more advanced derivatives houses. But since we don't do much consulting in that area, we don't really have a handle on what people in the industry are doing with it.
“While volatility may be the only thing that matters in the [BlackScholesMerton] formula, the expected return of the underlying asset certainly affects the volatility.” 
DS: Have you ever thought of altering the BlackScholesMerton model in some way or coming up with a new version on your own?
AL: We have. In the paper Wang and I published, we have a derivative pricing formula that adjusts the BlackScholesMerton formula for predictability.
DS: But how likely is perfect predictability?
AL: We argue in our paper that you'll never get perfect predictability. That's one of the most beautiful aspects of financial markets. In some sense, arbitrage provides you with an upper boundary on how much predictability there can ever be in securities markets. Having said that, there is still some predictability—it's not zero. That's what Craig MacKinlay and I found in our research, and we've spent 10 years and written 12 research papers documenting this fact in a number of different ways.
DS: So the arbitrage opportunities you talked about have diminished. What does that say about the random walk hypothesis in general?
AL: It says that the random walk hypothesis is an approximation of reality and that the approximation errors increase and decrease through time. In the 1980s, the approximation error was rather large. The random walk was actually a pretty poor approximation of the way stock indexes behaved. Over the last 10 years, the approximation has improved because there have been arbitrageurs out there taking advantage of these errors.
DS: Exceedingly sophisticated arbitrageurs who are fulfilling their function in the market.
AL: Let me suggest something I find even more interesting that is happening today. I haven't done an exhaustive study, but I have done some casual empirical analysis. My sense is that over the past couple of years, the approximation is breaking down yet again. It looks like there are new kinds of predictabilities coming into play. I think that's because there are a lot of unsophisticated investors entering the market, thanks to this raging bull market.
DS: Every experienced professional investor has been infuriated by the ability of completely ignorant investors to make a lot of money on Internet stocks.
“Over the past couple years, there are new kinds of predictabilities coming into play because there
are a lot of unsophisticated investors coming into the market.” 
AL: Nobody understands how these Internet stocks are behaving, but they're going up, and there are certain predictable patterns. And individuals who are not as technologically sophisticated are getting involved. If you go to the Amazon.com web site and click on the 10 bestselling books for 1998, you'll find the usual suspects: Tom Clancy, Stephen King and Tom Wolfe. But you'll also find that one of the top–10 bestselling books of 1998 is “The Electronic Day Trader.”
What this tells me is that there's a new element in the market now. In the past, there were two different groups of investors: passive investors and active investors. Now, there is a new group: hyperactive investors.
DS: Maybe you'd call them the tulip bulb investors.
AL: I must say I'm tempted to do that because I think many individuals are trading on very little information. They're trading on emotion, and that's the easiest way to do a lot of damage to a portfolio. But with that kind of activity going on, is it any wonder there are new forms of predictability being created in the marketplace? And, as a result, new opportunities that are presented to sophisticated arbitrageurs like D.E. Shaw?
DS: You've also done quite a bit of research on risk preferences.
AL: I just published a paper in the Financial Analyst's Journal titled “The Three P's of Total Risk Management.” The point I make in that paper is that we've got a lot of sophisticated tools for calculating valueatrisk and doing scenario analysis and so on. But those are not the most important aspects of risk management.
Any complete risk management system really contains three components: prices, probabilities and preferences. You have to know about prices—How much does it cost to hedge a certain risk? You have to know about probabilities—that's VAR. But even if you know those two quantities perfectly, without any estimation error, that still leaves open the question of how much risk you should hedge.
How is a CFO going to figure out how much foreign currency risk to hedge? From an enduser's perspective, VAR doesn't give you the answer.
DS: Quite often people decide to hedge 60 percent or 50 percent or some other number based on their own criteria.
AL: But what is the right number? How do we know it's 50 percent and not 43 percent? Or 85 percent? I argue that a single answer doesn't exist. The answer depends on individual and corporate preferences for risk.
DS: How much pain they're willing to suffer.
AL: Exactly.
DS: Which gets us back to valueatrisk, doesn't it?
AL: No, because VAR does not tell you how much pain you're willing to suffer. It will tell you the likelihood of suffering a certain amount of pain. It will tell you with 95 percent probability that you will suffer at most a $100 million loss over the next month.
DS: But you can suffer more.
AL: You can suffer more—or less. More specifically, you can change the dial on your risk meter. You can adjust the VAR so it's not $100 million, but $50 million. Or maybe $150 million.
For example, the reason that LTCM increased the risk of its portfolio after returning money to investors was because it was actually bearing much less risk than it wanted. People there show an extremely compelling series of statistics in which the portfolio's standard deviation was quite a bit lower than that of the S&P500, which is about what it wanted the standard deviation to be.
The bottomline question for risk management is, “How much risk do you want to bear?” We really have not focused on that question yet, and you can't do proper risk management until you answer it. In my research, I'm beginning to explore the psychological aspects of risk bearing to understand how people determine their own risk tolerances.
DS: Are you trying to quantify preferences in some way?
AL: Yes, exactly. It involves conducting psychological surveys and having people fill out riskattitude questionnaires. It involves giving people choices between several lotteries, or several trading strategies, and getting them to pick which trading strategy is more valuable or less valuable. By asking a sequence of these lotterytype questions, you can construct a mathematical representation of their risk tolerances.
DS: It seems to me that the answers are going to be somewhat irrational.
AL: I think in many cases it's not a matter of irrationality but simply risk preferences. Interestingly enough, if you explain to people what their preferences are, in some cases they might actually change them. I'm not really sure what that means. Does that mean they were irrational before? Maybe it means that people change their preferences over time.
Some of my colleagues have expressed a bit of surprise and dismay that I am going down this track. My response is, “What else am I supposed to do? I've been led here by my research. Where else am I supposed to look for the answers to the questions I want answered?”
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