Guofu   Zhou

International Stock Return Predictability: What is the Role of the United States?

with David E. Rapach and Jack K. Strauss (current version: November, 2009).

We present significant evidence of out-of-sample equity premium predictability for a host of industrialized countries over the postwar period. There are important differences, however, in the nature of equity premium predictability between the United States and other developed countries. Taken collectively, U.S. economic variables are significant out-of-sample predictors of the U.S. equity premium, while lagged international stock returns have no predictive power. In contrast, lagged international stock returns-- especially lagged U.S. returns--substantially outperform economic variables as out-of-sample equity premium predictors for non-U.S. countries, pointing to a leading role for the United States with respect to international return predictability. The leading role of the United States is consistent with information frictions in international equity markets. In addition, the predictability patterns are enhanced during economic downturns, linking return predictability to business-cycle fluctuations and the diffusion of news on macroeconomic fundamentals across countries. The leading role of the United States stands out during the recent global financial crisis: lagged U.S. stock returns deliver especially sizable gains for forecasting the monthly equity premium in other countries, evidenced by out-of-sample R² statistics of 10% or greater, more than triple the postwar average.

A Long-run Risks Model with Long- and Short-run Volatilities: Explaining Predictability and Volatility Risk Premium

with Yingzi Zhu (current version: June, 2009).

In this paper, we extend the long-run risks model of Bansal and Yaron (2004) as well as the improved version of Bansal, Kiku and Yaron (2007a) by decomposing the volatility of consumption growth, long-run risk, and dividends into two components: a long-run and a short-run volatilities. In doing so, the new model can retain much of the good properties of the original models, and yet is capable of justifying simultaneously the large negative market variance risk premium, differing predictability in excess returns, consumption and dividends, as well as their volatilities, all of which are difficult to explain previously.

How Predictable are Components of the Aggregate Market Portfolio?

With Aiguo Kong, David Rapach, Jack Strauss and Jun Tu (current Version: August, 2009).

We analyze return predictability for components of the aggregate market, including portfolios sorted on industry, size, and book-to-market. Considering a variety of economic variables and lagged industry returns as predictors, we find that returns for certain component portfolios are substantially more predictable using both in-sample and out-of-sample tests. Among industry portfolios, construction, textiles, apparel, furniture, printing, automobiles, and manufacturing exhibit the most predictability, while portfolios of small-cap and high book-to-market firms typically display greater predictability. We provide economic explanations for differences in component predictability and further show that predictability can be exploited to improve portfolio performance for component-rotation investment strategies.

Being Naive about Naive Diversification: Can Investment Theory Be Consistently Useful? (2008 EFA version) Latest version available upon request

with Jun Tu; June, 2009.

The modern portfolio theory pioneered by Markowitz (1952) is widely used in practice and extensively taught to MBAs. But the estimated Markowitz's portfolio rule and most of its extensions not only underperform the naive 1/N rule proposed by Talmud (that invests equally across N assets) in simulations, but also lose money on a risk-adjusted basis in many real data sets. In this paper, we propose an optimal combination of the naive 1/N rule with each of the four sophisticated strategies: the Markowitz rule, the Jorion (1986) rule, the McKinlay and Pastor (2000) rule, and the Kan and Zhou (2007) rule, as a way to improve the performances. We find that the combination rules improve over both their uncombined counterparts (the Markowitz rule and its extensions) and the 1/N rule substantially in most scenarios, and they perform well consistently across real data sets and simulations. Since the newly proposed rules employ theory-based components in the combinations, our study may be interpreted as reaffirming the usefulness of the Markowitz theory in practice.

New Title: Markowitz Meets Talmud: A Combination of Sophisticated and Naive Diversification Strategies

How Much Return Predictability Do We Expect from An Asset Pricing Model?

First draft, September, 2008; current version, September, 2009.

Stock market predictability is of considerable interest in both academic research and investment practice. Ross (2005) provides a simple and elegant upper bound on the predictive regression R-squared that R² <= (1 + R_f)² Var(m) for a given asset pricing model with kernel m, where R_f is the riskfree rate of return. In this paper, we tighten this bound by a squared factor of the correlation between the default pricing kernel and the state variables of the economy. Since the correlation can be substantially smaller than one, our bound can be much tighter than Ross's. An empirical application illustrates that while Ross's bound is not binding, our bound does.

Why Is the Recent Financial Crisis a "Once-in-a-century" Event?

with Yingzi Zhu; First version, June, 2009; current version, August, 2009.

In the recent financial crisis, the Dow Jones stock market index dropped about 54% from a high of 14164.53 on October 9, 2007 to a low of 6547.05 on March 9, 2009. Alan Greenspan calls this a ``once-in-a century" crisis. While we do not know how he drew his conclusion, we show that the probability of a stock market drop of 50% from its high within a century is about 90% based on the popular random walk model of the stock prices. With a broad market index of the S&P500 and a more sophisticated asset pricing model which captures more risks in the economy, the probability rises to above 99%. The message of this paper is that a market drop of 50% or more is very likely in long-run stock market investments, and the investors should be prepared for it.

Optimal Portfolio Choice Under Recursive Utility: A New Analytical Approach

with Yingzi Zhu; April, 2007.

In this paper, we study the optimal allocation problem and provide an approximate analytical solution when tradable volatility follows the Heston (1993) stochastic volatility model. Using daily data of S&P 500 and volatility index from 1990 to 2005, we calibrate the model and find that the approximate solution is very accurate for parameters of practical interest. Economically, we find that volatility trading generates substantial hedging demand and horizon effect. Unlike existing studies, the impact of elasticity of intertemporal substitution on hedging demand for derivatives can be of first-order importance.

Modeling Non-normality Using Multivariate t: Implications for Asset Pricing

with Raymond Kan; February, 2006.

Many important findings in empirical finance are based on the normality assumption, but this assumption is firmly rejected by the data due to fat tails of asset returns. In this paper, we propose the use of a multivariate t-distribution as a simple and powerful tool to examine the robustness of results that are based on the normality assumption. In particular, we find that, after replacing the normality assumption with a reasonable t-distribution, the most efficient estimator of the expected return of an asset is drastically different from the sample average return. For example, the annual difference in the estimated expected returns under normal and t is 2.964% for the Fama and French's (1993, 1996) smallest size and book-to-market portfolio. In addition, there are also substantial differences in estimating Jensen's alphas, choosing optimal portfolios, and testing asset pricing models when returns follow a multivariate t-distribution instead of a multivariate normal.

Hansen-Jagannathan Distance: Geometry and Exact Distribution

with Raymond Kan; November, 2002.

This paper provides an in-depth analysis of the Hansen-Jagannathan (HJ) distance, which is a measure that is widely used for diagnosis of asset pricing models, and also as a tool for model selection. In the mean and standard deviation space of portfolio returns, we provide a geometric interpretation of the HJ-distance. In relation to the traditional regression approach of testing asset pricing models, we show that the HJ-distance is a scaled version of the aggregate pricing errors, and it is closely related to Shanken's (1985) cross-sectional regression test (CSRT) statistic, with the only major difference in how the zero-beta rate is estimated. For the statistical properties, we provide the exact distribution of the sample HJ-distance and also a simple numerical procedure for computing its distribution function. In addition, we propose a new test of equality of HJ-distance for two nested models. Simulation evidence shows that the asymptotic distribution for sample HJ-distance is grossly inappropriate for typical number of test assets and time series observations, making the small sample analysis empirically relevant.

Tests of Mean-Variance Spanning

with Raymond Kan; March, 2008.

The paper presents a thorough study on the spanning: points out years old errors in the literature and provides geometrical/economic interpretations, small sample distributions and power analysis for likelihood ratio, Wald, and Lagrange multiplier tests, and a comparison among them and between the stochastic discount factor approach, in addition to a new sequential test that weighs explicitly economic significance into the size of the test.

Toward a Better Understanding of the Beta Method and the Stochastic Discount Factor Method

with Raymond Kan; May, 2002.

In a standardized factor model, Kan and Zhou (1999) show the stochastic discount factor (SDF) method yields less efficient estimates than the beta method when both are based on the generalized method of moments (GMM). By modifying the common use of the SDF [via adding more moment conditions to the practice before the publication of Kan and Zhou (1999)], Jagannathan and Wang (2001) and Cochrane (2000a,b) find that the two methods have the same asymptotic variance in parameter estimation. But their analysis relies on a joint normality assumption of both the asset returns and factors. In this paper, we show that: 1) once the normality assumption is relaxed, the modified SDF method is highly sensitive to factor skewness and kurtosis whereas the beta method is not, implying that the SDF estimates can be less reliable in realistic situations where the factors are leptokurtic; 2) in conditional asset pricing models, the modified SDF is in general still strictly dominated by the beta method in terms of estimation accuracy; 3) while it is not well understood and almost never used in the SDF formulation of an asset pricing model, the maximum likelihood method is well defined and has both strictly more efficient estimates and more powerful tests than the SDF method; 4) the SDF tests can have much less power than the beta method in conditional asset pricing models. In short, while the SDF set-up is an elegant theoretical formation, empirical estimation and tests should pay as much attention to the beta method as to the SDF if not more (one more reason is that, as shown by Jagannathan and Wang (2001), estimated model pricing errors have smaller variance by using the beta method than the SDF one).

Limited Participation, Consumption, and Saving Puzzles: A Simple Explanation and the Role of Insurance

(with Todd Gormley and Hong Liu)

In this paper, we use a simple model to illustrate that the existence of a large, negative wealth shock and insufficient insurance against such a shock can potentially explain both the limited stock market participation puzzle and the low-consumption-high-savings puzzle that are widely documented in the literature. We then conduct an extensive empirical analysis on the relation between household portfolio choices and access to private insurance and various types of government safety nets, including social security and unemployment insurance. The empirical results demonstrate that a lack of insurance against large, negative wealth shocks is strongly correlated with lower participation rates and higher saving rates. Overall, the evidence suggests an important role of insurance in household investment and savings decisions.

Journal of Financial Economics (forthcoming)

Out-of-Sample Equity Premium Prediction: Combination Forecasts and Links to the Real Economy

(with David Rapach and Jack Strauss).

While a host of economic variables have been identified in the literature with the apparent in-sample ability to predict the equity premium, Goyal and Welch (2008) find that these variables fail to deliver consistent out-of-sample forecasting gains relative to the historical average. Arguing that substantial model uncertainty and instability seriously impair the forecasting ability of individual predictive regression models, we recommend combining individual model forecasts to improve out-of-sample equity premium prediction. Combining delivers statistically and economically significant out-of-sample gains relative to the historical average on a consistent basis over time. We provide two empirical explanations for the benefits of the forecast combination approach: (i) combining forecasts incorporates information from numerous economic variables while substantially reducing forecast volatility; (ii) combination forecasts of the equity premium are linked to the real economy.

Review of Financial Studies (forthcoming)

Incorporating Economic Objectives into Bayesian Priors: Portfolio Choice Under Parameter Uncertainty

with Jun Tu (First version, April 2004)

Economic objectives are often ignored when estimating parameters, though the loss of doing so can be substantial. This paper proposes a way to allow Bayesian priors to reflect the objectives. Using monthly returns of the Fama-French 25 size and book-to-market portfolios and their three factors from January 1965 to December 2004, we find that investment performance under the objective-based priors can be significantly different from that under alternative priors, with differences in terms of annual certainty-equivalent returns greater than 10% in many cases. In terms of out-of-sample performance, the Bayesian rules under the objective-based priors can outperform substantially some of the best rules developed in the classical framework.

Journal of Financial and Quantitative Analysis (forthcoming)

Beyond Black-Litterman: Letting the Data Speak

(First Version, January, 2008)

Current Version: August, 2008.

The Black-Litterman model is a popular approach for asset allocation by blending an investor's proprietary views with the views of the market. However, their model ignores the data-generating process whose dynamics can have significant impact on future portfolio returns. This paper extends, in two ways, the Black-Litterman model to allow Bayesian learning to exploit all available information-- the market views, the investor's proprietary views as well as the data. Our framework allows practitioners to combine insights from the Black-Litterman model with the data to generate potentially more reliable trading strategies and more robust portfolios. Further, we show that many Bayesian learning tools can now be readily applied to practical portfolio selections in conjunction with the Black-Litterman model.

Journal of Portfolio Management 36 (1), 2009, 36--45.

What Will the Likely Range of My Wealth Be?

With Raymond Kan

The median is a better measure than the mean in evaluating the long-term value of a portfolio. However, the standard plug-in estimate of the median is too optimistic. It has a substantial upward bias that can easily exceed a factor of two. In this paper, we provide an unbiased forecast of the median of the long-term value of a portfolio. In addition, we also provide an unbiased forecast of an arbitrary percentile of the long-term portfolio value distribution. This allows us to construct the likely range of the long-term portfolio value for any given confidence level. Finally, we provide an unbiased forecast of the probability for the long-term portfolio value falling into a given interval. Our unbiased estimators provide a more accurate assessment of the long-term value of a portfolio than the traditional estimators, and are useful for long-term planning and investment.

Financial Analysts Journal 65 (4), 2009, 68--77.

Technical Analysis: An Asset Allocation Perspective on the Use of Moving Averages

with Yingzi Zhu

In this paper, we analyze the usefulness of technical analysis, specifically the widely used moving average trading rule from an asset allocation perspective. We show that when stock returns are predictable, technical analysis adds value to commonly used allocation rules that invest fixed proportions of wealth in stocks. When there is uncertainty about predictability which is likely in practice, the fixed allocation rules combined with technical analysis can outperform the prior-dependent optimal learning rule when the prior is not too informative. Moreover, the technical trading rules are robust to model specification, and they tend to substantially outperform the model-based optimal trading strategies when there is uncertainty about the model governing the stock price.

Journal of Financial Economics 92, 2009, 519--544.

On the Fundamental Law of Active Portfolio Management: How to Make Conditional Investments Unconditionally Optimal?

The fundamental law of active portfolio management tells an active manager how to transform his alpha forecasts into the valued-added of his active portfolio by using a linear strategy with active positions proportional to the forecasts. This linear strategy is conditionally optimal because it is optimal each period conditional on the forecasts at that time. However, the unconditional value-added (the valued-added over the long haul or over multiple periods) is what usually the manager strives earnestly for. Under this unconditional objective, the linear strategy can approach zero value-added if the forecasts or signals have a high kurtosis. To overcome this problem, we provide an investment strategy that maximizes the unconditional value-added with the optimal use of conditional information. Our strategy is nonlinear in the forecasts, but has a simple economic interpretation. When the alpha forecasts are high, we invest less aggressively than the linear strategy, and when the forecasts are low, we invest more aggressively. In this way, we tend to smooth our value-added over time, and hence, on a risk-adjusted basis, our long-term unconditional value-added will in most cases be substantially higher than that based on the linear strategy, particularly when the alpha forecasts experience high kurtosis.

On the Fundamental Law of Active Portfolio Management: What Happens if Our Estimates Are Wrong?

The fundamental law of active portfolio management pioneered by Grinold (1989) provides profound insights on the value creation process of managed funds. However, a key weakness of the law and its various extensions is that they ignore the estimation risk associated with the parameter inputs of the law. We show that the estimation errors have a substantial impact on the value-added of an actively managed portfolio, and they can easily destroy all the value promised by the law if they are not dealt with carefully. For bettering the chance of active managers to beat benchmark indices, we propose two methods, scaling and diversification, that can be used effectively to minimize the impact of the estimation errors significantly.

Asymmetries in Stock Returns: Statistical Tests and Economic Evaluation

with Yongmiao Hong and Jun Tu

In this paper, we provide a model-free test for asymmetric correlations in which stocks move more often with the market when the market goes down than when it goes up. We also provide such tests for asymmetric betas and covariances. In addition, we evaluate the economic significance of incorporating asymmetries into investment decisions. When stocks are sorted by size, book-to-market and momentum, we find strong evidence of asymmetry for both the size and momentum portfolios, but no evidence for the book-to-market portfolios. Moreover, the asymmetries can be of substantial economic importance for an investor with a disappointment aversion preference of Ang, Bekaert and Liu (2005). If the investors's felicity function is of the power utility form and if his coefficient of disappointment aversion is between 0.55 and 0.25, he can achieve over 2% annual certainty-equivalent gains when he switches from a belief in symmetric stock returns into a belief in asymmetric ones.

Optimal Portfolio Choice with Parameter Uncertainty

with Raymond Kan

In this paper, we analytically derive the expected loss function associated with using sample means and covariance matrix of returns to estimate the optimal portfolio. Our analytical results show that the standard plug-in approach that replaces the population parameters by their sample estimates can lead to very poor out-of-sample performance. We further show that with parameter uncertainty, holding the sample tangency portfolio and the riskless asset is never optimal. An investor can benefit by holding some other risky portfolios that help reduce the estimation risk. In particular, we show that a portfolio that optimally combines the riskless asset, the sample tangency portfolio, and the sample global minimum-variance portfolio dominates a portfolio with just the riskless asset and the sample tangency portfolio, suggesting that the presence of estimation risk completely alters the theoretical recommendation of a two-fund portfolio.

Journal of Financial and Quantitative Analysis 42, 2007, 621--656.

Estimating and Testing Beta Pricing Models: Alternative Methods and Their Performance in Simulations

with Jay Shanken

In this paper, we provide a comprehensive theoretical and small sample study of the Fama and MacBeth (1973) two-pass procedure that is fundamental in understanding to what extent cross-sectional expected returns/values are explained by certain factor attributes. While existing studies use almost exclusively this procedure, we show that alternative two-pass methods can have better small sample performance. In addition, we provide tractable GMM approaches that accommodate conditional heteroscedasticity of the data. Moreover, the risk premium estimates and t-ratios of the Fama and MacBeth procedure provide no information on whether the model is misspecified or not, and they can be misleadingly interpreted if the model is indeed misspecified. We not only provide formal model misppecification tests, but also how that various estimation methods are useful in detecting model misppecification.

Journal of Financial Economics 84, 2007, 40--86.

Using Bootstrap to Test Portfolio Efficiency

with Pin-Huang Chou

To facilitate wide use of the bootstrap method in finance, this paper shows by intuitive arguments and by simulations how it can improve upon existing tests to allow less restrictive distributional assumptions on the data and to yield more reliable (higher-order accurate) asymptotic inference. In particular, we apply the method to examine the efficiency of CRSP value-weighted stock index, and to test the well-known Fama and French (1993) three-factor model. We find that existing tests tend to over-reject.

Annals of Economics and Finance 7, 2006, 217--249.

Portfolio Optimization under Asset Pricing Anomalies

with Pin-Huang Chou and Wen-Shen Li

Fama and French (1993) find that the SMB and the HML factors explain much of the cross-section stock returns that are unexplained by the CAPM, whereas Daniel and Titman (1997) show that it is the characteristics of the stocks that are responsible rather than the factors. But both arguments are largely based only on expected return comparisons, and little is known about how important each of the two explanations matters to an investor's investment decisions in general and portfolio optimization in particular. In this paper, we show that a mean-variance maximizing investor who exploits the asset pricing anomaly of the CAPM can achieve substantial economic gains than simply holding the market index. Indeed, using Japanese data over the period 1980-1997, we find that the optimized portfolio constructed from characteristics-based model and based on the first 200 largest stocks is the best performing one and has monthly returns more than 0.81% (10.16% annualized) over the Nikkei 225 index with no greater risk.

Japan & The World Economy 18, 2006, 121--142.

A New Variance Bound on the Stochastic Discount Factor

with Raymond Kan

In this paper, we construct a new variance bound on any stochastic discount factor (SDF) of the form m=m(x) where x is a vector of random state variables. In contrast to the well known Hansen-Jagannathan bound that places a lower bound on the variance of m(x), our bound tightens it by a ratio of (1/ρ_x,m0)², where ρ_x,m0 is the multiple correlation coefficient between x and the standard minimum variance SDF, m₀. In many applications, the correlation is small, and hence our bound can be substantially tighter than Hansen-Jagannathan's. For example, when x is the gross growth rate of consumption, based on Cochrane's (2001) estimates of market volatility and ρ_x,m0, the new bound is 25 times greater than the Hansen-Jagannathan bound, making it much more difficult to explain the equity-premium puzzle based on existing asset pricing models. Another example is applying the new bound, with the growth rate of consumption as a state variable, to the 25 size and book-to-market sorted portfolios used by Fama and French (1993), then it is more than 100 times greater than the Hansen-Jagannathan bound.

Journal of Business 79, 2006, 941--961.

Data-generating Process Uncertainty: What Difference Does It Make in Portfolio Decisions?

with Jun Tu

As the usual normality assumption is firmly rejected by the data, investors encounter a data-generating process (DGP) uncertainty in making investment decisions. In this paper, we propose a novel way to incorporate uncertainty about the DGP into portfolio analysis. We find that accounting for fat tails leads to nontrivial changes in both parameter estimates and optimal portfolio weights, but the certainty–equivalent losses associated with ignoring fat tails are small. This suggests that the normality assumption works well in evaluating portfolio performance for a mean-variance investor.

Journal of Financial Economics 72, 2004, 385--421.

What Determines Expected International Asset Returns?

with Campbell Harvey and Bruno Solnik

This paper characterizes the forces that determine time-variation in expected international asset returns. We offer a number of innovations. By using the latent factor technique, we do not have to prespecify the sources of risk. We solve for the latent premiums and characterize their time-variation. We find evidence that the first factor premium resembles the expected return on the world market portfolio. However, the inclusion of this premium alone is not sufficient to explain the conditional variation in the returns. We find evidence of a second factor premium which is related to foreign exchange risk. Our sample includes new data on both international industry portfolios and international fixed income portfolios. We find that the two latent factor model performs better in explaining the conditional variation in asset returns than a prespecified two factor model. Finally, we show that differences in the risk loadings are important in accounting for the cross-sectional variation in the international returns.

Annals of Economics and Finance 3, 2002, 83--127.

On Rate of Convergence of Discrete-time Contingent Claims

with Steve Heston

This paper characterizes the rate of convergence of discrete-time multinomial option prices. We show that it depends on the smoothness of option payoff function, and is much lower than commonly believed because the payoff functions are often all-or-nothing type and not continuously differentiable. We propose two methods, one of which is to smooth the payoff function, that help to yield the same rate of convergence as smooth payoff functions.

Mathematical Finance 10, 2000, 53--75.

Investment Horizon and the Cross Section of Expected Returns: Evidence from the Tokyo Stock Exchange

With Pin-Huang Chou and Yuan-Lin Hsu

Using data from the Tokyo Stock Exchange, we study how beta, size, and ratio of book to market equity (BE/ME) account for the cross-section of expected stock returns over different lengths of investment horizons. We find that beta, adjusted for infrequent trading or not, fails to explain the cross-section of monthly expected returns, but does a much better job for horizons over half- and one-year. However, either the size or the BE/ME alone is still a significant factor in explaining the cross-section expected returns, but the size significantly diminishes for longer horizons when beta is included as an additional independent variable.

Annals of Economics and Finance 1, 2000, 79--100.

Security Factors as Linear Combinations of Economic Variables

A new framework is proposed to find the best linear combinations of economic variables that optimally forecast security factors. In particular, we obtain such combinations from Chen et al. (Journal of Business 59, 383--403, 1986) five economic variables, and obtain a new GMM test for the APT which is more robust than existing tests. In addition, by using Fama and French's (1993) five factors, we test whether fewer factors are sufficient to explain the average returns on 25 stock portfolios formed on size and book-to-market. While inconclusive in-sample, a three-factor model appears to perform better out-of-sample than both four- and five-factor models.

Journal of Financial Markets 2, 1999, 403--432.

Testing Multi-beta Pricing Models

with Raja Velu

This paper presents a complete solution to the estimation and testing of multi-beta models by providing a small sample likelihood ratio test when the usual normality assumption is imposed and an almost analytical GMM test when the normality assumption is relaxed. Using 10 size portfolios from January 1926 to December 1994, we reject the joint efficiency of the CRSP value-weighted and equal-weighted indices. We also apply the tests to analyze a new version of Fama and French’s [Fama, E.F., French, K.R. 1993. Common risk factors in the returns on stocks and bonds. Journal of Financial Economics 33, 3–-56] three-factor model in addition to two standard ones, and find that the new version performs the best.

Journal of Empirical Finance 6, 1999, 219--241.

A Critique of the Stochastic Discount Factor Methodology

with Raymond Kan

In this paper, we point out that the widely used stochastic discount factor (SDF) methodology ignores a fully specified model for asset returns. As a result, it suffers from two potential problems when asset returns follow a linear factor model. The first problem is that the risk premium estimate from the SDF methodology is unreliable. The second problem is that the specification test under the SDF methodology has very low power in detecting misspecified models. Traditional methodologies typically incorporate a fully specified model for asset returns, and they can perform substantially better than the SDF methodology.

Journal of Finance 54, 1999, 1021--1048.

Going to Extremes: Correcting Simulation Bias in Exotic Option Valuation

with Phil Dybvig and David Beaglehole

Monte Carlo simulation is widely used in practice to value exotic options for which analytical formulas are not available. When valuing those options that depend on extreme values of the underlying asset, convergence of the standard simulation is slow as the time grid is refined, and even a daily simulation interval produces unacceptable errors. This article suggests approximating the extreme value on a subinterval by a random draw from the known theoretical distribution for an extreme of a Brownian bridge on the same interval. This approach provides reliable option values and retains the flexibility of simulations, in that it allows great freedom in choosing a price process for the underlying asset or a joint process for the asset price, its volatility, and other asset prices.

Financial Analysts Journal 53, 1997, 62--68.

Temporary Components of Stock Returns: What Do the Data Tell Us?

with Chris Lamoureux

Within the past few years several articles have suggested that returns on large equity portfolios may contain a significant predictable component at horizons 3 to 6 years. Subsequently, the tests used in these analyses have been criticized (appropriately) for having widely misunderstood size and power, rendering the conclusions inappropriate. This criticism however has not focused on the data, it addressed the properties of the tests. In this article we adopt a subjectivist analysis - treating the data as fixed - to ascertain whether the data have anything to say about the permanent/temporary decomposition. The data speak clearly and they tell us that for all intents and purposes, stock prices follow a random walk.

Review of Financial Studies 9, 1996, 1033--1059.

Measuring the Pricing Error of the Arbitrage Pricing Theory

with John Geweke

This article provides an exact Bayesian framework for analyzing the arbitrage pricing theory (APT). Based on the Gibbs sampler, we show how to obtain the exact posterior distributions for functions of interest in the factor model. In particular, we propose a measure of the APT pricing deviations and obtain its exact posterior distribution. Using monthly portfolio returns grouped by industry and market capitalization, we find that there is little improvement in reducing the pricing errors by including more factors beyond the first one.

Review of Financial Studies 9, 1996, 553--583.

Time-to-Build Effects and the Term Structure

with Jack Strauss

This paper shows that real macroeconomic variables have power in predicting movements in the term structure of interest rates, complementing recent studies on the links of structure to expected stock returns. We find that up to 86 percent of the variation in the term premia are due to changes in the macroeconomy. The predictive power can be attributed to time-to-build effect of investments.

Journal of Financial Research 18, 1995, 115--127.

Small Sample Rank Tests with Applications to Asset Pricing

This paper proposes small sample tests for rank restrictions that arise in many asset pricing models, economic fields and others, complementing the usual asymptotic theory which can be unreliable. Using monthly portfolio returns grouped by industry and using two sets of instrumental variables, we cannot reject a one-factor model for the industry returns.

Journal of Empirical Finance 2, 1995, 71--93.

Analytical GMM Tests: Asset Pricing with Time-Varying Risk Premiums

We propose alternative generalized method of moments (GMM) tests that are analytically solvable in many econometric models, yielding in particular analytical GMM tests for asset pricing models with time-varying risk premiums. We also provide simulation evidence showing that the proposed tests have good finite sample properties and that their asymptotic distribution is reliable for the sample size commonly used. We apply our tests to study the number of latent factors in the predictable variations of the returns on portfolios grouped by industries. Using data from October 1941 to September 1986 and two sets of instrumental variables, we find that the tests reject a one factor model but not a two-factor one.

Review of Financial Studies 7, 1994, 687--709.

Asset Pricing Test Under Alternative Distributions

Given the normality assumption, we reject the mean-variance efficiency of the Center for Research in Security Prices value-weighted stock index for three of the six consecutive ten-year subperiods from 1926 to 1986. However, the normality assumption is strongly rejected by the data. Under plausible alternative distributional assumptions of the elliptical class, the efficiency can no longer be rejected. When the normality assumption is violated but the ellipticity assumption is maintained, many tests tend to be biased toward over-rejection and both the accuracy of estimated beta and R² are usually overstated.

Journal of Finance 48, 1993, 1927--1942.

International Asset Pricing with Alternative Distributional Specifications

with Campbell Harvey

The unconditional mean-variance efficiency of the Morgan Stanley Capital International world equity index is investigated. Using data from 16 OECD countries and Hong Kong and maintaining the assumption of multivariate normality, we cannot reject the efficiency of the benchmark. However, residual diagnostics reveal significant departures from normality. We test the sensitivity of the results by specifying error structures that are t-distributed and mixtures of normal distributions. Even after relaxing the i.i.d. assumption, we cannot reject the mean-variance efficiency of the world portfolio. Our results suggest that differences in country risk exposure, measured against the MSCI world portfolio, will lead to differences in expected returns.

Journal of Empirical Finance 1, 1993, 107--131.

Small Sample Tests of Portfolio Efficiency

This paper presents an eigenvalue test of the efficiency of a portfolio when there is no riskless asset, complementing the test of Gibbons, Ross, and Shanken (1989). Besides optimal upper and lower bounds, an easily-implented numerical method is provided for computing the exact P-value. Our approach makes it possible to draw statistical inferences on the efficiency of a given portfolio both in the context of the zero-beta CAPM and with respect to other linear pricing models. As an application, using monthly data for every consecutive five-year period from 1926 to 1986, we reject the efficiency of the CRSP value-weighted index for most periods.

Journal of Financial Economics 30, 1991, 165--191.

Algorithms for the Estimation of Possibly Nonstationary Time Series

This paper presents efficient algorithms for computing time series projections, the maximum likelihood function and its gradient in possibly nonstationary vector times series model (VARMA).

Journal of Time Series Analysis 13, 1991, 171--188.

Bayesian Inference in Asset Pricing Tests

(and Unpublished TechAppendix)

with Campbell Harvey

We test the mean-variance efficiency of a given portfolio using a Bayesian framework. Our test is more direct than Shanken's (1987b), because we impose a prior on all the parameters of the multivariate regression model. The approach is also easily adapted to other problems. We use Monte Carlo numerical integration to accurately evaluate 90-dimensional integrals. Posterior-odds ratios are calculated for 12 industry portfolios from 1926–1987. The sensitivity of the inferences to the prior is investigated by using three different distributions. The probability that the given portfolio is mean-variance efficient is small for a range of plausible priors.

Journal of Financial Economics 26, 1990, 221--254.

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