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Abstract:
Conventional wisdom views stocks as less volatile over long horizons than over short horizons due to mean reversion induced by return predictability. In contrast, we find stocks are substantially more volatile over long horizons from an investor's perspective. This perspective recognizes that parameters are uncertain, even with two centuries of data, and that observable predictors imperfectly deliver the conditional expected return. We decompose return variance into five components, which include mean reversion and various uncertainties faced by the investor. Although mean reversion makes a strong negative contribution to long-horizon variance, it is more than offset by the other components. Using a predictive system, we estimate annualized 30-year variance to be nearly 1.5 times the 1-year variance.

long-run, risk, stock, variance

Abstract:
Conventional wisdom views stocks as less volatile over long horizons than over short horizons due to mean reversion induced by return predictability. In contrast, we find stocks are substantially more volatile over long horizons from an investor's perspective. This perspective recognizes that parameters are uncertain, even with two centuries of data, and that observable predictors imperfectly deliver the conditional expected return. We decompose return variance into five components, which include mean reversion and various uncertainties faced by the investor. Although mean reversion makes a strong negative contribution to long-horizon variance, it is more than offset by the other components. Using a predictive system, we estimate annualized 30-year variance to be nearly 1.5 times the 1-year variance.

Abstract:
Conventional wisdom views stocks as less volatile over long horizons than over short horizons due to mean reversion induced by return predictability. In contrast, we find stocks are substantially more volatile over long horizons from an investor's perspective. This perspective recognizes that parameters are uncertain, even with two centuries of data, and that observable predictors imperfectly deliver the conditional expected return. Mean reversion contributes strongly to reducing long-horizon variance, but it is more than offset by various uncertainties faced by the investor, so that annualized 30-year variance is nearly 1.5 times the 1-year variance. The same uncertainties also make target-date funds undesirable to a class of investors who would otherwise find them appealing.

stock, volatility, target-date funds, Bayesian, predictive system, predictive variance

Abstract:
We construct optimal portfolios of mutual funds whose objectives include socially responsible investment (SRI). Comparing portfolios of these funds to those constructed from the broader fund universe reveals the cost of imposing the SRI constraint on investors seeking the highest Sharpe ratio. This SRI cost depends crucially on the investor's views about asset pricing models and stock-picking skill by fund managers. To an investor who believes strongly in the CAPM and rules out managerial skill, i.e. a market-index investor, the cost of the SRI constraint is typically just a few basis points per month, measured in certainly-equivalent loss. To an investor who still disallows skill but instead believes to some degree in pricing models that associate higher returns with exposures to size, value, and momentum factors, the SRI constraint is much costlier, typically by at least 30 basis points per month. The SRI constraint imposes large costs on investors whose beliefs allow a substantial amount of fund-manager skill, i.e., investors who rely heavily on individual funds' track records to predict future performance.

socially responsible investing, mutual funds, portfolio selection

Abstract:
This study investigates whether market-wide liquidity is a state variable important for asset pricing. We find that expected stock returns are related cross-sectionally to the sensitivities of returns to fluctuations in aggregate liquidity. Our monthly liquidity measure, an average of individual-stock measures estimated with daily data, relies on the principle that order flow induces greater return reversals when liquidity is lower. Over a 34-year period, the average return on stocks with high sensitivities to liquidity exceeds that for stocks with low sensitivities by 7.5% annually, adjusted for exposures to the market return as well as size, value, and momentum factors.

Asset pricing, liquidity risk, expected returns

Abstract:
This study investigates whether market-wide liquidity is a state variable important for asset pricing. We find that expected stock returns are related cross-sectionally to the sensitivities of returns to fluctuations in aggregate liquidity. Our monthly liquidity measure, an average of individual-stock measures estimated with daily data, relies on the principle that order flow induces greater return reversals when liquidity is lower. Over a 34-year period, the average return on stocks with high sensitivities to liquidity exceeds that for stocks with low sensitivities by 7.5% annually, adjusted for exposures to the market return as well as size, value, and momentum factors.

Abstract:
This study investigates whether market-wide liquidity is a state variable important for asset pricing. We find that expected stock returns are related cross-sectionally to the sensitivities of returns to fluctuations in aggregate liquidity. Our monthly liquidity measure, an average of individual-stock measures estimated with daily data, relies on the principle that order flow induces greater return reversals when liquidity is lower. Over a 34-year period, the average return on stocks with high sensitivities to liquidity exceeds that for stocks with low sensitivities by 7.5% annually, adjusted for exposures to the market return as well as size, value, and momentum factors.

Asset pricing, liquidity risk, expected returns

Abstract:
The standard regression approach to modeling return predictability seems too restrictive in one way but too lax in another. A predictive regression models expected returns as an exact linear function of a given set of predictors but does not exploit the likely economic property that innovations in expected returns are negatively correlated with unexpected returns. We develop an alternative framework - a predictive system - that accommodates imperfect predictors and beliefs about that negative correlation. In this framework, the predictive ability of imperfect predictors is supplemented by information in lagged returns as well as lags of the predictors. Compared to predictive regressions, predictive systems deliver different and substantially more precise estimates of expected returns as well as different assessments of a given predictor's usefulness.

Expected stock return, predictability, predictive regression, predictive system, state space model

Abstract:
We develop a framework for estimating expected returns---a predictive system---that allows predictors to be imperfectly correlated with the conditional expected return. When predictors are imperfect, the estimated expected return depends on past returns in a manner that hinges on the correlation between unexpected returns and innovations in expected returns. We find empirically that prior beliefs about this correlation, which is most likely negative, substantially affect estimates of expected returns as well as various inferences about predictability, including assessments of a predictor's usefulness. Compared to standard predictive regressions, predictive systems deliver different and more precise estimates of expected returns.

Abstract:
We develop a framework for estimating expected returns - a predictive system - that allows predictors to be imperfectly correlated with the conditional expected return. When predictors are imperfect, the estimated expected return depends on past returns in a manner that hinges on the correlation between unexpected returns and innovations in expected returns. We find empirically that prior beliefs about this correlation, which is most likely negative, substantially affect estimates of expected returns as well as various inferences about predictability, including assessments of a predictor's usefulness. Compared to standard predictive regressions, predictive systems deliver different and more precise estimates of expected returns.

predictability, expected stock return, state space model, predictive system, predictive regression, imperfect predictors, Bayesian

Abstract:
We construct optimal portfolios of equity funds by combining historical returns on funds and passive indexes with prior views about asset pricing and skill. By including both benchmark and nonbenchmark indexes, we distinguish pricing-model inaccuracy from managerial skill. Even modest confidence in a pricing model helps construct portfolios with high Sharpe ratios. Investing in active mutual funds can be optimal even for investors who believe active managers cannot outperform passive indexes. Optimal portfolios exclude hot-hand funds even for investors who believe momentum is priced. Our large universe of funds offers no close substitutes for the Fama-French and momentum benchmarks.

mutual funds, portfolio selection

Abstract:
Our framework for evaluating and investing in mutual funds combines observed returns on funds and passive assets with prior beliefs that distinguish pricing-model inaccuracy from managerial skill. A fund's "alpha" is defined using passive benchmarks. We show that returns on non-benchmark passive assets help estimate that alpha more precisely for most funds. The resulting estimates generally vary less than standard estimates across alternative benchmark specifications. Optimal portfolios constructed from a large universe of equity funds can include actively managed funds even when managerial skill is precluded. The fund universe offers no close substitutes for the Fama-French and momentum benchmarks.

Abstract:
Our framework for evaluating and investing in mutual funds combines observed returns on funds and passive assets with prior beliefs that distinguish pricing-model inaccuracy from managerial skill. A fund's "alpha" is defined using passive benchmarks. We show that returns on non-benchmark passive assets help estimate that alpha more precisely for most funds. The resulting estimates generally vary less than standard estimates across alternative benchmark specifications. Optimal portfolios constructed from a large universe of equity funds can include actively managed funds even when managerial skill is precluded. The fund universe offers no close substitutes for the Fama-French and momentum benchmarks.

Abstract:
Estimates of standard performance measures can be improved by using returns on assets not used to de?ne those measures. Alpha, the intercept in a regression of a fund's return on passive benchmark returns, can be estimated more precisely by using information in returns on non-benchmark passive assets, whether or not one believes those assets are priced by the benchmarks. A fund's Sharpe ratio can be estimated more precisely by using returns on other assets as well as the fund. New estimates of these performance measures for a large universe of equity mutual funds exhibit substantial dierences from the usual estimates.

Performance evaluation; Mutual funds; Bayesian analysis

Abstract:
Equity costs of capital for individual firms are estimated using several models that relate expected returns to betas on one or more pervasive factors. A Bayesian approach incorporates prior uncertainty about an asset's mispricing as well as uncertainty about betas and factor means. Substantial prior uncertainty about mispricing results in an estimated cost of equity close to that obtained with mispricing ruled out. Uncertainty about which pricing model to use appears to be less important, on average, than within-model parameter uncertainty. In the absence of mispricing uncertainty, uncertainty about factor means is generally the most important source of overall uncertainty about a firm's cost of equity, although uncertainty about betas is nearly as important.

Abstract:
A long return history is useful in estimating the current equity premium even if the historical distribution has experienced structural breaks. The long series helps not only if the timing of breaks is uncertain but also if one believes that large shifts in the premium are unlikely or that the premium is associated, in part, with volatility. Our framework incorporates these features along with a belief that prices are likely to move opposite to contemporaneous shifts in the premium. The estimated premium since 1834 fluctuates between four and six percent and exhibits its sharpest drop in the last decade.

Abstract:
This study explores multivariate methods for investment analysis based on a sample of return histories that differ in length across assets. The longer histories provide greater information about moments of returns, not only for the longer-history assets, but for the shorter-history assets as well. To account for the remaining parameter uncertainty, or estimation risk,' portfolio opportunities are characterized by a Bayesian predictive distribution. Examples involving emerging markets demonstrate the value of using the combined sample of histories and accounting for estimation risk, as compared to truncating the sample to produce equal-length histories or ignoring estimation risk by using maximum-likelihood estimates.

Abstract:
This study explores multivariate methods for investment analysis based on a sample of return histories that differ in length across assets. The longer histories provide greater information about moments of returns, not only for the longer-history assets, but for the shorter-history assets as well. To account for the remaining parameter uncertainty, or "estimation risk," portfolio opportunities are characterized by a Bayesian predictive distribution. Examples involving emerging markets demonstrate the value of using the combined sample of histories and accounting for estimation risk, as compared to truncating the sample to produce equal-length histories or ignoring estimation risk by using maximum-likelihood estimates.

Abstract:
The predictability of monthly stock returns is investigated from the perspective of a risk-averse investor who uses the data to update initially vague beliefs about the conditional distribution of returns. The optimal stocks-versus-cash allocation of the investor can depend importantly on the current value of a predictive variable, such as dividend yield, even though a null hypothesis of no predictability might not be rejected at conventional significance levels. When viewed in this economic context, the empirical evidence indicates a strong degree of predictability in monthly stock returns.

Abstract:
Sample evidence about the predictability of monthly stock returns is considered from the perspective of a risk-averse Bayesian investor who must allocate funds between stocks and cash. The investor uses the sample evidence to update prior beliefs about the parameters in a regression of stock returns on a set of predictive variables. The regression elation can seem weak when described by usual statistical measures, but the current values of the predictive variables can exert a substantial influence on the investor's portfolio decision, even when the investor's prior beliefs are weighted against predictability.

Abstract:
When a rate of return is regressed on a lagged stochastic regressor, such as a dividend yield, the regression disturbance is correlated with the regressor's innovation. The OLS estimator's finite-sample properties, derived here, can depart substantially from the standard regression setting. Bayesian posterior distributions for the regression parameters are obtained under specifications that differ with respect to (i) prior beliefs about the autocorrelation of the regressor and (ii) whether the initial observation of the regressor is specified as fixed or stochastic. The posteriors differ across such specifications asset allocations in the presence of estimation risk exhibit sensitivity to those differences.

Abstract:
A long return history is useful in estimating the current equity premium even if the historical distribution has experienced structural breaks. The long series helps not only if the timing of breaks is uncertain but also if one believes that large shifts in the premium are unlikely or that the premium is associated, in part, with volatility. Our framework incorporates these features along with a belief that prices are likely to move opposite to contemporaneous shifts in the premium. The estimated premium since 1834 fluctuates between four and six percent and exhibits its sharpest drop in the last decade.

Abstract:
We investigate the portfolio choices of mean-variance-optimizing investors who use sample evidence to update prior beliefs centered on either risk-based or characteristic-based pricing models. With dogmatic beliefs in such models and an unconstrained ratio of position size to capital, optimal portfolios can differ across models to economically significant degrees. The differences are substantially reduced by modest uncertainty about the models' pricing abilities. When the ratio of position size to capital is subject to realistic constraints, the differences in portfolios across models become even less important, nonexistent in some cases.

Abstract:
Costs of equity for individual firms are estimated in a Bayesian framework using several factor-based pricing models. Substantial prior uncertainty about mispricing often produces an estimated cost of equity close to that obtained with mispricing precluded, even for a stock whose average return departs significantly from the pricing model's prediction. Uncertainty about which pricing model to use is less important, on average, than within-model parameter uncertainty. In the absence of mispricing uncertainty, uncertainty about factor premiums is generally the largest source of overall uncertainty about a firm's cost of equity, although uncertainty about betas is nearly as important.

Abstract:
A Bayesian approach is used to investigate a sample's information about a portfolio's degree of inefficiency. With standard diffuse priors, posterior distributions for measures of portfolio inefficiency can concentrate well away from values consistent with efficiency, even when the portfolio is exactly efficient in the sample. The data indicate that the NYSE-AMEX market portfolio is rather inefficient in the presence of a riskless asset, although this conclusion is justified only after an analysis using informative priors. Including a riskless asset significantly reduces any sample's ability to produce posterior distributions supporting small degrees of inefficiency.

Abstract:
A Bayesian approach is used to investigate a sample's information about a portfolio's degree of inefficiency. With standard diffuse priors, posterior distributions for measures of portfolio ineffciency can concentrate well away from values consistent with efficiency, even when the port- folio is exactly efficient in the sample. The data indicate that the NYSE-AMEX market portfolio is rather inefficient in the presence of a riskless asset, although this conclusion is justified only after an analysis using informative priors. Including a riskless asset significantly reduces any sample's ability to produce posterior distributions supporting small degrees of inefficiency.

Abstract:
Asymptotic variance of estimated parameters in models of conditional expectations are calculated analytically assuming a GARCH process for conditional volatility. Under such heteroskedasticity, OLS estimators or parameters in single-period models can posses substantially larger asymptotic variances the GMM estimators employing additional multiperiod moment conditions - an approach yielding no efficiency gain under homoskedasticity. In estimating models of long- horizon expectations, the VAR approach provides an efficiency advantage over long-horizon regressions under homoskedasticity, but that ordering can reverse under heteroskedasticity, especially when the conditional mean and variance are both persistent. In such cases, the VAR approach maintains a slight efficiency advantage if the OLS estimator is replaced by an alternative GMM estimator. Heteroskedasticity can increase dramatically the apparent asymptotic power advantages of long-horizon regressions to reject constant expectations against persistent alternatives.

Abstract:
No abstract is available for this paper.

Abstract:
This study investigates whether marketwide liquidity is a state variable important for asset pricing. We find that expected stock returns are related cross-sectionally to the sensitivities of returns to fluctuations in aggregate liquidity. Our monthly liquidity measure, an average of individual-stock measures estimated with daily data, relies on the principle that order flow induces greater return reversals when liquidity is lower. From 1966 through 1999, the average return on stocks with high sensitivities to liquidity exceeds that for stocks with low sensitivities by 7.5 percent annually, adjusted for exposures to the market return as well as size, value, and momentum factors. Furthermore, a liquidity risk factor accounts for half of the profits to a momentum strategy over the same 34-year period.

Abstract:
Estimates of standard performance measures can be improved by using returns on assets not used to define those measures. Alpha, the intercept in a regression of a fund's return on passive benchmark returns, can be estimated more precisely by using information in returns on nonbenchmark passive assets, whether or not one believes that those assets are priced by the benchmarks. A fund's Sharpe ratio can be estimated more precisely by using returns on other assets as well as the fund. New estimates of these performance measures for a large universe of equity mutual funds exhibit substantial differences from the usual estimates.

performance evaluation, mutual funds, Bayesian analysis

Abstract:
The Capital Asset Pricing Model implies (i) the market portfolio is efficient and (ii) expected returns are linearly related to betas. Many do not view these implications as separate, since either implies the other, but we demonstrate that either can hold nearly perfectly while the other fails grossly. If the index portfolio is inefficient, then the coefficients and R squared from an ordinary least squares regression of expected returns on betas can equal essentially any values and bear no relation to the index portfolio's mean variance location. That location does determine the outcome of a mean beta regression fitted by generalized least squares.

Abstract:
Sample evidence about the predictability of monthly stock returns is considered from the perspective of an investor allocating funds between stocks and cash. A regression of stock returns on a set of predictive variables might seem weak when described by usual statistical measures, but such measures can fail to convey the economic significance of the sample evidence when it is used by a risk-averse Bayesian investor to update prior beliefs about the regression relation and to compute an optimal asset allocation. Even when those prior beliefs are weighted substantially against predictability, the current values of the predictive variables can exert a strong influence on the portfolio decision.