Andrew W. Lo

The Adaptive Markets Hypothesis: Market Efficiency from an Evolutionary Perspective

One of the most influential ideas in the past 30 years is the efficient markets hypothesis, the idea that market prices incorporate all information rationally and instantaneously. The emerging discipline of behavioral economics and finance has challenged the EMH, arguing that markets are not rational, but rather driven by fear and greed. Research in the cognitive neurosciences suggests these two perspectives are opposite sides of the same coin. An adaptive markets hypothesis that reconciles market efficiency with behavioral alternatives applies the principles of evolution?competition, adaptation, and natural selection?to financial interactions. Extending Simons notion of ?satisficing? with evolutionary dynamics, the author argues that much of what behaviorists cite as counter-examples to economic rationality?loss aversion, overconfidence, overreaction, mental accounting, and other behavioral biases?is in fact consistent with an evolutionary model of individual adaptation to a changing environment via simple heuristics. The adaptive markets hypothesis offers a number of surprisingly concrete implications for portfolio management.

Optimal control of execution costs

We derive dynamic optimal trading strategies that minimize the expected cost of trading a large block of equity over a fixed time horizon. Specifically, given a fixed block SM of shares to be executed within a fixed finite number of periods „, and given a price-impact function that yields the execution price of an individual trade as a function of the shares traded and market conditions, we obtain the optimal sequence of trades as a function of market conditions — closed-form expressions in some cases — that minimizes the expected cost of executing SM within „ periods. Our analysis is extended to the portfolio case in which price impact across stocks can have an important e⁄ect on the total cost of trading a portfolio. ( 1998 Elsevier Science B.V. All rights reserved.

Nonparametric risk management and implied risk aversion

Typical value-at-risk (VaR) calculations involve the probabilities of extreme dollar losses, based on the statistical distributions of market prices. Such quantities do not account for the fact that the same dollar loss can have two very different economic valuations, depending on business conditions. We propose a nonparametric VaR measure that incorporates economic valuation according to the state-price density associated with the underlying price processes. The state-price density yields VaR values that are adjusted for risk aversion, time preferences, and other variations in economic valuation. In the context of a representative agent equilibrium model, we construct an estimator of the risk-aversion coefficient that is implied by the joint observations on the cross-section of option prices and time-series of underlying asset values.

The Size and Power of the Variance Ratio Test in Finite Samples: a Monte Carlo Investigation

We examine the finite sample properties of the variance ratio test of the random walk hypothesis via Monte Carlo simulations under two null and three alternative hypotheses. These results are compared to the performance of the Dickey-Fuller t and the Box-Pierce Q statistics. Under the null hypothesis of a random walk with independent and identically distributed Gaussian increments, the empirical size of all three tests are comparable. Under a heteroscedastic random walk null, the variance ratio test is more reliable than either the Dickey-Fuller or Box-Pierce tests. We compute the power of these three tests against three alternatives of recent empirical interest: a stationary AR(1), the sum of this AR(1) and a random walk, and an integrated AR( 1). By choosing the sampling frequency appropriately, the variance ratio test is shown to be as powerful as the Dickey-Fuller and Box-Pierce tests against the stationary alternative, and is more powerful than either of the two tests against the two unit-root alternatives.

The Statistics of Sharpe Ratios

The building blocks of the Sharpe ratio—expected returns and volatilities—are unknown quantities that must be estimated statistically and are, therefore, subject to estimation error. This raises the natural question: How accurately are Sharpe ratios measured? To address this question, I derive explicit expressions for the statistical distribution of the Sharpe ratio using standard asymptotic theory under several sets of assumptions for the return-generating process—independently and identically distributed returns, stationary returns, and with time aggregation. I show that monthly Sharpe ratios cannot be annualized by multiplying by √12 except under very special circumstances, and I derive the correct method of conversion in the general case of stationary returns. In an illustrative empirical example of mutual funds and hedge funds, I find that the annual Sharpe ratio for a hedge fund can be overstated by as much as 65 percent because of the presence of serial correlation in monthly returns, and once this serial correlation is properly taken into account, the rankings of hedge funds based on Sharpe ratios can change dramatically. The building blocks of the Sharpe ratio—expected returns and volatilities—are unknown quantities that must be estimated statistically and are, therefore, subject to estimation error. This raises the natural question: How accurately are Sharpe ratios measured? In this article, I provide an answer by deriving the statistical distributions of the usual Sharpe ratio estimator—sample mean excess return over sample standard deviation—using standard econometric methods under several different sets of assumptions for the statistical properties of the return series on which the ratio is based. Armed with these statistical distributions, I show that confidence intervals, standard errors, and hypothesis tests for the estimated Sharpe ratio can be computed in much the same way that they are computed for regression coefficients, such as portfolio alphas and betas. The accuracy of Sharpe ratio estimators hinges on the statistical properties of returns (e.g., time-series properties, such as mean reversion, momentum, or time-varying volatilities). Although this may seem like a theoretical exercise best left for statisticians, there is often a direct connection between the investment management process of a portfolio and its statistical properties. For example, a change in the portfolio managers style from a small-cap value orientation to a large-cap growth orientation will typically have an impact on the portfolios volatility, degree of mean reversion, and market beta. Even for a fixed investment style, a portfolios characteristics can change over time because of fund inflows and outflows, capacity constraints (e.g., a microcap fund that is close to its market-capitalization limit), liquidity constraints (e.g., an emerging market or private equity fund), and changes in market conditions (e.g., sudden increases or decreases in volatility, shifts in central banking policy, and extraordinary events, such as the default of Russian government bonds in August 1998). At a superficial level, it is obvious that the properties of the Sharpe ratio should depend on the investment style of the portfolio being evaluated; the performance of more volatile investment strategies is more difficult to gauge than that of less volatile strategies. Therefore, it should come as no surprise that the results derived in this article imply that Sharpe ratios are likely to be more accurately estimated for mutual funds than for hedge funds. A less intuitive implication is that the time-series properties of investment strategies (e.g., mean reversion, momentum, and other forms of serial correlation) can have a nontrivial impact on the Sharpe ratio estimator itself, especially when computing an annualized Sharpe ratio from monthly data. For example, the results derived in this article show that the common practice of annualizing Sharpe ratios by multiplying monthly estimates by √12 is correct only under very special circumstances and that the correct multiplier—which depends on the serial correlation of the portfolios returns—can yield Sharpe ratios that are considerably smaller (in the case of positive serial correlation) or larger (in the case of negative serial correlation). Therefore, Sharpe ratio estimators must be computed and interpreted in the context of the particular investment style with which a portfolios returns have been generated. Although the Sharpe ratio has become part of the canon of modern financial analysis, the results presented in this article suggest that a more sophisticated approach to interpreting Sharpe ratios is called for, one that incorporates information about the investment style that generated the returns and the market environment in which those returns were generated. For example, hedge funds have very different return characteristics from the characteristics of mutual funds; hence, the comparison of Sharpe ratios between these two investment vehicles cannot be performed naively. In light of the recent interest in alternative investments by institutional investors—investors that are accustomed to standardized performance attribution measures such as the annualized Sharpe ratio—there is an even greater need to develop statistics that are consistent with a portfolios investment style. The empirical example in this article underscores the practical relevance of proper statistical inference for Sharpe ratio estimators: Ignoring the impact of serial correlation of hedge fund returns can yield annualized Sharpe ratios that are overstated by more than 65 percent, understated Sharpe ratios in the case of negatively serially correlated returns, and inconsistent rankings across hedge funds of different styles and objectives. By using the appropriate statistical distribution for quantifying the performance of each return history, the Sharpe ratio can provide a more complete understanding of the risks and rewards of a broad array of investment opportunities.

Maximum Likelihood Estimation of Generalized Ito Processes with Discretely Sampled Data

In this paper, we consider the parametric estimation problem for continuous time stochastic processes described by general first-order nonlinear stochastic differential equations of the Ito type. We characterize the likelihood function of a discretely-sampled set of observations as the solution to a functional partial differential equation. The consistency and asymptotic normality of the maximum likelihood estimators are explored, and several illustrative examples are provided.

An Econometric Model of Serial Correlation and Illiquidity in Hedge Fund Returns

The returns to hedge funds and other alternative investments are often highly serially correlated in sharp contrast to the returns of more traditional investment vehicles such as long-only equity portfolios and mutual funds. In this paper, we explore several sources of such serial correlation and show that the most likely explanation is illiquidity exposure, i.e., investments in securities that are not actively traded and for which market prices are not always readily available. For portfolios of illiquid securities, reported returns will tend to be smoother than true economic returns, which will understate volatility and increase risk-adjusted performance measures such as the Sharpe ratio. We propose an econometric model of illiquidity exposure and develop estimators for the smoothing profile as well as a smoothing-adjusted Sharpe ratio. For a sample of 908 hedge funds drawn from the TASS database, we show that our estimated smoothing coefficients vary considerably across hedge-fund style categories and may be a useful proxy for quantifying illiquidity exposure.

Systemic Risk and Hedge Funds

Systemic risk is commonly used to describe the possibility of a series of correlated defaults among financial institutions---typically banks---that occur over a short period of time, often caused by a single major event. However, since the collapse of Long Term Capital Management in 1998, it has become clear that hedge funds are also involved in systemic risk exposures. The hedge-fund industry has a symbiotic relationship with the banking sector, and many banks now operate proprietary trading units that are organized much like hedge funds. As a result, the risk exposures of the hedge-fund industry may have a material impact on the banking sector, resulting in new sources of systemic risks. In this paper, we attempt to quantify the potential impact of hedge funds on systemic risk by developing a number of new risk measures for hedge funds and applying them to individual and aggregate hedge-fund returns data. These measures include: illiquidity risk exposure, nonlinear factor models for hedge-fund and banking-sector indexes, logistic regression analysis of hedge-fund liquidation probabilities, and aggregate measures of volatility and distress based on regime-switching models. Our preliminary findings suggest that the hedge-fund industry may be heading into a challenging period of lower expected returns, and that systemic risk is currently on the rise.

What Happened to the Quants in August 2007?: Evidence from Factors and Transactions Data

During the week of August 6, 2007, a number of quantitative long/short equity hedge funds experienced unprecedented losses. It has been hypothesized that a coordinated deleveraging of similarly constructed portfolios caused this temporary dislocation in the market. Using the simulated returns of long/short equity portfolios based on five specific valuation factors, we find evidence that the unwinding of these portfolios began in July 2007 and continued until the end of 2007. Using transactions data, we find that the simulated returns of a simple marketmaking strategy were significantly negative during the week of August 6, 2007, but positive before and after, suggesting that the Quant Meltdown of August 2007 was the combined effects of portfolio deleveraging throughout July and the first week of August, and a temporary withdrawal of marketmaking risk capital starting August 8th. Our simulations point to two unwinds---a mini-unwind on August 1st starting at 10:45am and ending at 11:30am, and a more sustained unwind starting at the open on August 6th and ending at 1:00pm---that began with stocks in the financial sector and long Book-to-Market and short Earnings Momentum. These conjectures have significant implications for the systemic risks posed by the hedge-fund industry.

Logit versus discriminant analysis: A specification test and application to corporate bankruptcies☆

Abstract Two of the most widely used statistical techniques for analyzing discrete economic phenomena are discriminant analysis (DA) and logit analysis. For purposes of parameter estimation, logit has been shown to be more robust than DA. However, under certain distributional assumptions both procedures yield consistent estimates and the DA estimator is asymptotically efficient. This suggests a natural Hausman specification test of these distributional assumptions by comparing the two estimators. In this paper, such a test is proposed and an empirical example involving corporate bankruptcies is provided. The finite-sample properties of the test statistic are also explored through some sampling experiments.