The likelihood ratio test statistic of mean-variance efficiency without a riskless asset

The question whether a given porfolio is mean-variance efficient is a basic problem of investment analysis. Mean-variance efficiency is also the basis of the Capital Asset Pricing Model. This paper presents the explicit form of the likelihood ratio test of the hypothesis that a given portfolio, or a particular market index, is ex-ante mean-variance efficient in the case where there is no riskless asset. Geometric relations are illustrated to provide intuition about the constrained maximum likelihood estimators and the test statistic, and two simple economic interpretations of the test are given.     

On the Exclusion of Assets from Tests of the Mean Variance Efficiency of the Market Portfolio

This paper presents an analysis of the testability of the mean variance efficiency of a market index when the returns on some components of the index itself are not perfectly observable. The results are basically not supportive of the notion that mean variance efficiency is testable on a subset of the assets. Bounding the market share of the missing asset and its expected return is not sufficient to produce a valid test. When the variance of the missing asset is bounded, and the amount of wealth that might be missing is small, it is possible, in principle, to reject correctly the mean variance efficiency of a market index.     

Exact distribution-free tests of mean-variance efficiency

This paper develops exact distribution-free tests of unconditional mean-variance efficiency. These new tests allow for unknown forms of non-normalities, conditional heteroskedasticity, and other non-linear temporal dependencies among the absolute values of the error terms in the asset pricing model. Exactness here rests on the assumption that the joint temporal error density is symmetric around zero. This still leaves open the possibility of return distribution asymmetry via coskewness with the benchmark portfolio. A simulation study shows that the new tests have very good power relative to that of many commonly used tests. The inference procedures developed are further illustrated by tests of the mean-variance efficiency of a market index using a 42-year sample of monthly returns on ten U.S. equity portfolios.     

THE TESTING OF AUSTRALIAN STOCK MARKET INDICES FOR MEAN-VARIANCE EFFICIENCY

Three different market indices are tested for mean-variance efficiency using monthly data for leading Australian securities, and following the methodologies suggested in Roll (1979). The balance of the evidence is against index efficiency and against the two-parameter asset pricing theory. However, this could be influenced by imperfections in the tests, inadequate data, and sampling errors in the betas.     

A critique of the asset pricing theory's tests Part I: On past and potential testability of the theory

Testing the two-parameter asset pricing theory is difficult (and currently infeasible). Due to a mathematical equivalence between the individual return/‘beta’ linearity relation and the market portfolio's mean-variance efficiency, any valid test presupposes complete knowledge of the true market portfolio's composition. This implies, inter alia, that every individual asset must be included in a correct test. Errors of inference inducible by incomplete tests are discussed and some ambiguities in published tests are explained.     

The Geometry of the Maximum Likelihood Estimator of the Zero-Beta Return

This paper explores geometric relations, in mean-variance space, among the sample frontier, the maximum likelihood estimator, and two other estimators of the zerobeta return. It is also demonstrated that a partition of the portfolio space is determined by a family of parabolas; the zeros of each parabola are the maximum likelihood estimators associated with all portfolios on the parabola. This observation is the basis for an additional interpretation of the statistic of the Likelihood Ratio Test of portfolio efficiency without a riskless asset.     

Testing the CAPM with broader market indexes: A problem of mean-deficiency

This study addresses a problem that can arise when a broader market index is used to test the CAPM: a return series used in the index can exclude part of an asset's return. If the excluded return is constant, then a test of mean-variance efficiency can be constructed, but an additional parameter must be estimated. This point is illustrated in tests with both broader market indexes and stocks-only indexes. The broader indexes exclude the rental return on real estate and durables. The excluded rental return is estimated under the assumption that the index portfolio is mean-variance efficient.     

Should Investors Avoid All Actively Managed Mutual Funds? A Study in Bayesian Performance Evaluation

This paper analyzes mutual-fund performance from an investor's perspective. We study the portfolio-choice problem for a mean-variance investor choosing among a risk-free asset, index funds, and actively managed mutual funds. To solve this problem, we employ a Bayesian method of performance evaluation; a key innovation in our approach is the development of a flexible set of prior beliefs about managerial skill. We then apply our methodology to a sample of 1,437 mutual funds. We find that some extremely skeptical prior beliefs nevertheless lead to economically significant allocations to active managers.     

An Examination of Alternative Factor Models in UK Stock Returns

This paper examines the mean-variance efficiency of a number offactor models in UK stock returns. The paper also explores, using theapproach of MacKinlay (1995), whether missing risk factors ornonrisk-based explanations best explain the pricing errors of thedifferent factor models. The evidence in the paper suggests that themean-variance efficiency of each factor model is rejected and missing riskfactors are unable to explain the pricing errors of any of the models.Some nonrisk-based explanations, which posit a wide spread in abnormalreturns, may be a more plausible source of explaining the pricing errorsof the factor models.     

Multifactor Asset Pricing Models

Since the early 1960s, the mean-variance Capital Asset Pricing Model (CAPM) has been a dominant paradigm in modern finance. Recently, the accumulation of anomalous evidence, and a realisation that empirical tests of the model are tautologically related to the efficiency of the market index, have pushed that paradigm to a point of crisis. This paper reviews alternative asset pricing models which coexisted with the CAPM and may provide plausible substitutes. The major distinguishing feature of these models is that they predict multiple risk factors and, with the exception of the Arbitrage Pricing Theory (APT), are extensions of the CAPM.     

Relative Performance Evaluation of Mutual Funds: A Non-Parametric Approach

We propose an alternative mutual fund performance index which addresses the benchmark problem and controls for economies of scale in managing mutual funds. We advance a new concept of 'return-cost' efficiency as another important element in evaluating portfolio management, in addition to the mean-variance efficiency concept. Our index based on a non-parametric estimation is shown to be similar to the Sharpe index with multiple slopes (or factors). We have shown that all fund categories, except income funds, have similar average efficiency scores after controlling for economies of scale. Most funds operate in increasing returns to scale and seem to be successful in holding mean-variance efficient portfolios, but unsuccessful in allocating transaction costs efficiently, evidenced by excessive turnovers and loads.     

Econometrical analysis of the sample efficient frontier

The efficient frontier is a parabola in the mean-variance space which is uniquely determined by three characteristics. Assuming that the portfolio asset returns are independent and multivariate normally distributed, we derive tests and confidence sets for all possible arrangements of these characteristics. Note that all of our results are based on the exact distributions for a finite sample size. Moreover, we determine a confidence region of the whole efficient frontier in the mean-variance space. It is shown that this set is bordered by five parabolas.     

A mean-variance framework for tests of asset pricing models

This article presents a mean-variance framework for likelihood-ratiotests of asset pricing models. A pricing model is tested byexamining the position of one or more reference portfolios insample mean-standard-deviation space. Included are tests ofboth single-beta and multiple-beta relations, with or withouta riskless asset, using either a general or a specific alternativehypothesis. Tests with a factor that is not a portfolio returnare also included. The mean-variance framework is illustratedby testing the zero-beta CAPM, a two-beta pricing model, andthe consumption-beta model.     

Is there a real estate factor premium?

In this paper, we study the variation of expected returns on five different asset portfolios in a multi-factor model. We found the presence of a real estate factor, in addition to both a stock factor and a bond factor in asset pricing. This suggests that mutual fund managers should seriously consider including real estate assets in their portfolios, since one cannot capture the real estate factor premium without having some kind of real estate exposure. Another result is that the market segmentation found in previous studies disappears in a more general model of asset pricing in which we allow for multi-factors other than the market factor to affect asset returns. This implies that real estate assets can be treated just like other assets as far as mean-variance efficient asset allocations are concerned. We also have some preliminary evidence that equity REITs and the Russell-NCREIF index are driven by the same underlying real estate factor.     

Disentangling the role of variance and covariance information in portfolio selection problems

The covariation among financial asset returns is often a key ingredient used in the construction of optimal portfolios. Estimating covariances from data, however, is challenging due to the potential influence of estimation error, specially in high-dimensional problems, which can impact negatively the performance of the resulting portfolios. We address this question by putting forward a simple approach to disentangle the role of variance and covariance information in the case of mean-variance efficient portfolios. Specifically, mean-variance portfolios can be represented as a two-fund rule: one fund is a fully invested portfolio that depends on diagonal covariance elements, whereas the other is a long-short, self financed portfolio associated with the presence of non-zero off-diagonal covariance elements. We characterize the contribution of each of these two components to the overall performance in terms of out-of-sample returns, risk, risk-adjusted returns and turnover. Finally, we provide an empirical illustration of the proposed portfolio decomposition using both simulated and real market data.     

Benefits of international diversification with investment constraints: An over-time perspective

This paper investigates the benefits and asset allocation of the optimal international diversification for the U.S.A. investor while considering various portfolio constraints. Although the global financial market is becoming more integrated, the findings suggest that adding lower and upper weighting bounds reduces, but does not completely eliminate, the potential economic value of international investment. The addition of investment constraints makes asset allocation more feasible and decreases the volatility in portfolio return. The time-variation in the optimal asset allocation implies that fund managers should rebalance international portfolios dynamically. The out-of-sample test suggests that the Markowitz model with constraints realizes trivial improvement in mean-variance efficiency but still demonstrates significant reduction in risk.     

The performance of model based option trading strategies

This paper analyzes returns to trading strategies in options markets that exploit information given by a theoretical asset pricing model. We examine trading strategies in which a positive portfolio weight is assigned to assets which market prices exceed the price of a theoretical asset pricing model. We investigate portfolio rules which mimic standard mean-variance analysis is used to construct optimal model based portfolio weights. In essence, these portfolio rules allow estimation risk, as well as price risk to be approximately hedged. An empirical exercise shows that the portfolio rules give out-of-sample Sharpe ratios exceeding unity for S&P 500 options. Portfolio returns have no discernible correlation with systematic risk factors, which is troubling for traditional risk based asset pricing explanations.     

The Ex Post Performance of Four Portfolio Selection Algorithms

This paper explores the ex post performance of four widely cited (and sometimes applied) normative portfolio selection models. Each is supposed to solve the same portfolio selection problem relative to the same mean-variance efficiency criteria. It has been shown elsewhere, and this paper confirms, that the models result in vastly different ex ante stock selection strategies. However, the acid test of normative theory is ex post performance relative to a set of efficiency criteria or some other standard. The empirical results reported here show that, with one exception, the ex post performance of the models is consistent with the same mean-variance efficiency criteria, and, over a predictable range of outcomes, consistently outperform the index portfolio based on Standard & Poor's 500 Stock Index.     

Benchmark Portfolio Inefficiency and Deviations from the Security Market Line

This paper theoretically evaluates the robustness of the Security Market Line relationship when the market proxy employed is not mean-variance efficient. The analysis focuses on the behavior of the “benchmark errors,” the deviations of assets and portfolios from the Security Market Line. First, we characterize how the location of an asset in mean-variance space determines its benchmark error. Then the continuity properties of the benchmark errors are studied. The results indicate that the magnitudes of the errors exhibit continuous but not uniformly continuous behaviors. The relative rankings based on deviations from the Security Market Line, however, exhibit some severe discontinuities. In fact, these can be exactly reversed for two proxies arbitrarily close in mean-variance space.     

Equilibrium Underdiversification and the Preference for Skewness

We develop a one-period model of investor asset holdings whereinvestors have heterogeneous preference for skewness. Introducingheterogeneous preference for skewness allows the model's investors,in equilibrium, to underdiversify. We find support for our model'sthree key implications using a dataset of 60,000 individualinvestor accounts. First, we document that the portfolio returnsof underdiversified investors are substantially more positivelyskewed than those of diversified investors. Second, we showthat the apparent mean-variance inefficiency of underdiversifiedinvestors can be largely explained by the fact that investorssacrifice mean-variance efficiency for higher skewness exposure.Furthermore, we show that idiosyncratic skewness, and not justcoskewness, can impact equilibrium prices. Third, the underdiversificationof investors does not appear to be coincidentally related toskewness. Stocks most often selected by underdiversified investorshave substantially higher average skewness—especiallyidiosyncratic skewness—than stocks most often selectedby diversified investors.