Research Dialogues Issue Number 52 September 1997 A publication of External Affairs — Corporate Research A Nonrandom Walk Down Wall Street: Recent Advances in Financial Technology (those who select managers of large December 1996, the $1 investment In this issue: portfolios) listened. But as time went on, would have grown to $14. If, on the Introduction it became apparent that they should have. other hand, the investor had put $1 into Because of fees and turnover, the the stock market, e.g., the S&P 500, and Stock Market Prices and managers they picked typically under- continued reinvesting the proceeds in the the Random Walk performed the market. And the worse an stock market month by month over this active manager did relative to a market same seventy-year period, the $1 The Martingale Model index, the more attractive seemed the low- investment would have grown to $1,370, The Random Walk Hypothesis cost alternative of buying and holding the a considerably larger sum. Now suppose index itself. that each month, an investor could tell in Rejecting the Random Walk advance which of these two investments But as luck would have it, just as Implications for Investment would yield a higher return for that indexing was gaining ground, a new month, and took advantage of this Management wave of academic research was being information by switching the running published that weakened some of the The Efficient Markets Hypothesis total of his initial $1 investment into the results of the earlier research and thereby higher-yielding asset, month by month. A Modern View of Efficient Markets undercut part of the justification for What would a $1 investment in such a indexing. It didn’t obviate all the “perfect foresight” investment strategy Practical Considerations reasons for indexing (indexing was still yield by December 1996? a low-cost way to create diversification for an entire fund or as part of an The startling answer—$2,303,981,824 In this issue of Research Dialogues, (yes, over $2 billion; this is no typo- Andrew W. Lo, Harris & Harris active/passive strategy), but it did tend to silence the index-because-you-can’t-do- graphical error)—often comes as a shock Group Professor, Sloan School of to even the most seasoned professional in- Management, Massachusetts Institute better school. vestment manager. Of course, few in- of Technology, takes a look at an old Andrew Lo was, and continues to be, at vestors have perfect foresight. But this issue in investment risk management in the forefront of this new research. In a new way. In the ’50s and ’60s, just extreme example suggests that even a 1988, for example, he and MacKinlay modest ability to forecast financial asset re- as the era of the professional portfolio (see references, page 7) published evidence manager was dawning, financial that resulted in the rejection of the turns may be handsomely rewarded: it economists were telling anyone who Random Walk Hypothesis. That does not take a large fraction of would listen that active management evidence is summarized below and $2,303,981,824 to beat $1,370! For this was probably a big mistake—a waste fittingly constitutes the first step in reason, quantitative models of the risks of time and money. Their research Professor Lo’s interesting nonrandom and rewards of financial investments— demonstrated that historical prices were walk. now known collectively as financial technol- of little use in helping to predict where ogy or financial engineering—have become future prices would go. Prices simply Introduction virtually indispensable to institutional in- took a “random walk.” The better part vestors throughout the world. If, in January 1926, an investor put of wisdom, they advised, was to be a Of course, financial engineering is still passive investor. $1 into one-month U.S. Treasury bills— one of the “safest” assets in the world— in its infancy when compared with the At first, not too many of the people who and continued reinvesting the proceeds mathematical and natural sciences. influence the way money is managed in Treasury bills month by month until However, it has enjoyed a spectacular © 1997 Teachers Insurance and Annuity Association I College Retirement Equities Fund period of growth over the past three model, whose origins lie in the history of information contained in past prices is in- decades, thanks in part to the break- games of chance and the birth of probabili- stantly, fully, and perpetually reflected in throughs pioneered by academics such as ty theory.1 The prominent Italian mathe- the asset’s current price. However, one of Fischer Black, John Cox, Harry Markowitz, matician Girolamo Cardano proposed an the central ideas of modern financial eco- Robert Merton, Stephen Ross, Paul elementary theory of gambling in his 1565 nomics is the necessity of some trade-off Samuelson, Myron Scholes, William manuscript Liber de ludo aleae (The Book of between risk and expected return, and al- Sharpe, and many others. Moreover, paral- Games of Chance), in which he writes: though the martingale hypothesis places a lel breakthroughs in mathematics, statis- The most fundamental principle of all restriction on expected returns, it does not tics, and computational power have in gambling is simply equal conditions, account for risk in any way. enabled the financial community to imple- e.g., of opponents, of bystanders, of In particular, if an asset’s expected price ment such financial technology almost im- money, of situation, of the dice box, and change is positive, it may be the reward mediately, giving it an empirical relevance of the die itself. To the extent to which necessary to attract investors to hold the and practical urgency shared by few other you depart from that equality, if it is in asset and bear its associated risks. Indeed, if disciplines in the social sciences. your opponent’s favour, you are a fool, an investor is risk averse, he would gladly This brief article describes one simple and if in your own, you are unjust.2 pay to avoid holding an asset with the mar- example of modern financial technology: This clearly contains the notion of a “fair tingale property. Therefore, despite the in- testing the Random Walk Hypothesis— game,” a game which is neither in your tuitive appeal that the “fair game” the hypothesis that past prices cannot be favor nor your opponent’s, and this is the interpretation might have, it has been used to forecast future prices—for aggre- essence of a martingale, a precursor to the shown that the martingale property is nei- gate U.S. stock market indexes. Although ther a necessary nor a sufficient condition Random Walk Hypothesis (RWH). If Pt the random walk is a very old idea, dating represents one’s cumulative winnings or for rationally determined asset prices (see, back to the sixteenth century, recent studies wealth at date t from playing some game of for example, LeRoy [1973], and Lucas have shed new light on this important chance each period, then a fair game is one [1978]). model of financial prices. The findings in which the expected wealth next period is Nevertheless, the martingale has be- suggest that stock market prices do contain simply equal to this period’s wealth. come a powerful tool in probability and predictable components and that there may If P is taken to be an asset’s price at date statistics, and also has important applica- be significant returns to active investment t tions in modern theories of asset prices. For management. t, then the martingale hypothesis states that tomorrow’s price is expected to be example, once asset returns are properly ad- justed for risk, the martingale property Stock Market Prices and equal to today’s price, given the asset’s en- the Random Walk tire price history. Alternatively, the asset’s does hold (see Lucas [1978], Cox and Ross expected price change is zero when condi- [1976], and Harrison and Kreps [1979]), One of the most enduring questions of and the combination of this risk adjust- financial economics is whether or not finan- tioned on the asset’s price history; hence its price is just as likely to rise as it is to fall. ment and the martingale property has led cial asset prices are forecastable. Perhaps to a veritable revolution in the pricing of because of the obvious analogy between fi- From a forecasting perspective, the martin- gale hypothesis implies that the “best” fore- complex financial instruments such as op- nancial investments and games of chance, tions, swaps, and other derivative securities. mathematical models of asset prices have an cast of tomorrow’s price is simply today’s price, where the “best” forecast is defined to Moreover, the martingale led to the devel- unusually rich history that predates virtual- opment of a closely related model that has be the one that minimizes the average ly every other aspect of economic analysis. now become an integral part of virtually squared error of the forecast. The vast number of prominent mathemati- every scientific discipline concerned with cians and scientists who have plied their Another implication of the martingale dynamic uncertainty: the Random Walk considerable skills in forecasting stock and hypothesis is that nonoverlapping price Hypothesis. commodity prices is a testament to the fas- changes are uncorrelated at all leads and cination and the challenges that this prob- lags, which further implies the ineffective- The Random Walk Hypothesis lem poses. Indeed, the intellectual roots of ness of all linear forecasting rules for future The simplest version of the RWH states modern financial economics are firmly price changes that are based on the price that future returns cannot be forecast from planted in early attempts to “beat the mar- history.