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Article Excerpt
1. Introduction
Since the development of portfolio theory by Markowitz (1952), mean-variance efficient portfolios have received considerable attention, playing a key role in a variety of financial fields from investment analysis and asset pricing to topics in corporate finance. It is well known that the often-disappointing performance of mean-variance optimized portfolios, in large part, stems from the use of past performance to estimate the unknown parameters of the assets' probability distribution. Even when a statistical estimator is unbiased, bias can emerge when the estimator is used as an input to a nonlinear optimization process. Finding a solution to this combined estimation/optimization problem has been pursued in various ways, both theoretically, primarily using Bayesian methods, and empirically. Yet the current literature does not provide theoretical guidance as to the exact size of the impact of estimation error, or as to when and how mean-variance techniques should be used. This paper gives that analysis, by proving asymptotic non-Bayesian closed-form formulas for portfolio performance while accounting for estimation risk. This is the first theoretical paper in the existing literature to quantify the impact of estimation error without relying upon prior assumptions on the unknown parameters, and to give an adjustment for statistical noise that will asymptotically reflect the actual performance, within the Markowitz mean-variance framework.
We perform the following thought experiment. Suppose that an investor forms classical sample estimates of asset means, variances, and covariances and uses them as if they were the true assets' distribution parameters to form mean-variance efficient portfolios and to judge their anticipated performance. We refer to these classical sample performance estimates as "naive" estimates because they do not take into account the estimation error stemming from using a sample rather than the population in determining these parameters. If the estimation error wrongly suggests that an asset will have a high expected return, then an optimized portfolio heavily invested in this asset will be disappointing. In particular, the naive portfolio mean estimates tend to be biased upwards and the naive portfolio variance estimates tend to be biased downwards, resulting in nominally efficient portfolios that are "over-optimistic," in the sense that such an investor will believe she can achieve a higher mean and lower variance than is actually available from the performance of her portfolio. We quantify this "over-optimism" bias in closed-form asymptotic formulas, which help in two ways when portfolio weights are influenced by statistical noise. First, there may be a systematic component that moves the mean performance away from the target mean; this effect is captured by our mean adjustment. Second, to the extent that the portfolio weights inherit the variability of the noise, the variance of the portfolio performance will increase; this is precisely the effect that is picked up by our standard deviation adjustment.
Our contributions are derived within a new theoretical framework, giving the investor an adjustment to the performance of naively-estimated efficient portfolios that will more accurately reflect actual portfolio performance by accounting for estimation error distortions. Achieving this closed-form bias adjustment for the mean and the risk is not an easy task because the exact functional forms of the mean and standard deviation of next-period performance of a naively-formed portfolio are very complex once estimation error has been used by the nonlinear multivariate optimization process. We use the method of statistical differentials to find Taylor-series approximations to expectations of random variables, obtaining results that are asymptotically correct when the number of time periods is large and that remain statistically consistent when estimated values are substituted for unknown parameters. In effect, we use perturbation analysis to discover how estimation errors are misused by the mean-variance optimization technology in its attempt to improve performance while wrongly believing that the estimated parameters are correct.
A number of approaches have been proposed to study and resolve the problem of bias resulting from estimation error. Our theoretical adjustment is consistent with the empirical findings in the literature regarding the bias induced by estimation error on mean-variance efficient portfolios, including empirical studies of the estimation risk problem by Frost and Savarino (1986b), who observe through simulation that the magnitude of the problem depends on the ratio of the number of assets to the number of observed time periods. Results on diversification and mean-variance efficiency by Green and Hollifield (1992) are motivated by the observation that, when using sample moments, the resulting portfolios are often highly nondiversified. Frankfurter et al. (1971) perform an experiment in which the impact of estimation error is so strong that the usefulness of mean-variance approaches is questioned. Muller (1993) shows that optimized portfolios tend to be more risky ex post than predicted ex ante. Chopra and Ziemba (1993) find (for perturbations of individual covariance matrix elements) and Merton (1980) shows (for the one-asset case) that the influence of the estimation error in the mean is more critical than the error in the variance. Dhingra (1980) shows that the uncertainty in optimal portfolio selection increases with the target return. Clark-son et al. (1996) show that estimation risk has a meaningful and measurable nondiversifiable component. Inference methods for the estimated weights of mean-variance portfolios may be found in Britten-Jones (1999). Solutions to the estimation risk problem include Jorion (1986, 1991), who shows how a James-Stein shrinkage estimator outperforms the sample mean, and Fomby and Samanta (1991), who study a non-Bayesian Stein-Rule approach. For an agent with quadratic utility, ter Horst et al. (2002) show how to adjust the level of risk aversion to compensate for estimation risk of the asset expected returns. The effects of estimation risk on market prices and returns are studied by Lewellen and Shanken (2002). Michaud (1998) reviews the problems associated with mean-variance efficient portfolios and presents a number of estimation techniques.
An estimated portfolio might be compared to the ideal portfolio that could be formed in the absence of statistical noise. Jobson and Korkie (1980) study statistical properties of the estimated Sharpe-ratio maximizing portfolio, as compared to the ideal portfolio that maximizes the Sharpe ratio for the true asset means and covariance matrix, which are unobservable to investors with noisy data. In contrast, we examine the step-ahead performance of estimated portfolios and, although this step-ahead performance depends on unknown parameters, we show how to obtain consistent estimators (with statistically significant bias reduction) using only the sample data.
A Bayesian model with a noninformative prior fails to account for bias in the mean. For example, a theoretical Bayesian approach to estimation risk (Bawa et al. 1979) incorporates estimation risk directly into the decision problem using predictive distributions. Using Jeffreys' noninformative invariant priors, they find that no adjustment in the mean of the predictive distribution is needed due to estimation risk. They do, however, find an estimation-risk adjustment to...
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