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 Shrinkage Approach to Model Uncertainty and Asset Allocation

This article takes a shrinkage approach to examine the empiricalimplications of aversion to model uncertainty. The shrinkageapproach explicitly shows how predictive distributions incorporatedata and prior beliefs. It enables us to solve the optimal portfoliosfor uncertainty-averse investors. Aversion to uncertainty aboutthe capital asset pricing model leads investors to hold a portfoliothat is not mean-variance efficient for any predictive distribution.However, mean-variance efficient portfolios corresponding toextremely strong beliefs in the Fama–French model areapproximately optimal for uncertainty-averse investors. Theempirical Bayes approach does not result in optimal portfoliosfor investors who are averse to model uncertainty.     

Owner-Occupied Housing and the Composition of the Household Portfolio: The Case of France

This paper investigates the impact of housing demand on the composition of the optimal portfolios of homeowners in France, following the methodology developed by Flavin and Yamashita (NBER Working Paper 6389, 2002). We use historical data on housing prices and financial assets returns to estimate the mean return and covariance matrix of a set of assets including housing. We then calculate mean-variance efficient frontiers associated to various levels of the housing-to-net wealth ratio, corresponding to the average ratios observed for different age groups in the 1998 French Wealth Survey sample. Our numerical results fit the average portfolios in different age brackets quite well. Also, returns of housing and its covariance with the other assets indicate there is room in France for housing price derivatives.     

The Opportunity Cost of a Mean-Variance Efficient Choice

The mean-variance criterion is one of the most frequently used methods for selecting investment portfolios. Yet, because it is an approximation of an investor's maximum expected utility choice, some theoreticians and practitioners have criticized the approach. This paper examines the investment loss that different investors experience by accepting a mean-variance efficient portfolio. Simulated security returns with extreme distributional characteristics are used to determine the extent of an investor's loss. The results indicate that even under very unreasonable investment distributional assumptions, an investor's loss by accepting a mean-variance efficient choice rarely exceeds a small fraction of one percent per invested dollar.     

On the characteristics of dynamic correlations between asset pairs

Recent research provides considerable evidence that correlations between assets change significantly over time and diversification benefits of correlations may vary substantially based on the time-varying measure of correlation used for different asset types. Our study evaluates and compares alternative time-series correlation modeling techniques according to both statistical and economic metrics, focusing specifically on individual asset pairs. We identify the moving correlation structure that best tracks the dynamic conditional correlation estimates using a large set of different financial time series encompassing 467 asset pairs in nine different asset classes. Results from our direct, statistical loss function based, and indirect, portfolio mean-variance based, forecast evaluations provide optimal window-length ranges for 36 asset-class pairs which should help in portfolio construction as well as risk management. Furthermore for robustness tests, we implement the model confidence set approach which, without a benchmark specification, produces a set of models constructed to contain the best models with a given level of confidence among competing forecast evaluations.     

Solutions and Simulations of Some One-Dimensional Stochastic Differential Equations

We consider a one dimensional SDE dX _t  = μ(X _t )dt + σ(X _t )dB _t . We give a new general formula for solutions that involves solving an associated ordinary differential equation. Explicit solutions are obtained in cases where the ODE has such. This recovers linear case but also some non-linear cases. In any case our approach leads to a new simulation scheme that returns positive values for processes on I R⁺{{\hskip 0.02in \hbox{\rm I}\hskip -.02in \hbox{\rm R}{^+}}}, which is advantageous when modelling prices or rates.     

Stochastic dominance and portfolio analysis

The principle of stochastic dominance is used to characterize the optimal efficient sets when the distributions of the random prospects belong to a family. Most of the well-known distributions are considered. In each case, the optimal efficient sets are characterized by easily verifiable conditions on the parameters of the distributions. These optimal efficient sets are then compared with the corresponding mean-variance (MV) efficient set. It is often found that the optimal efficient sets are proper subsets of the MV efficient set. Thus, the MV criterion is a proper efficiency criterion, but the MV efficient set can be excessively large compared to the optimal efficient set.     

Coherent and convex monetary risk measures for unbounded càdlàg processes

Assume that the random future evolution of values is modelled in continuous time. Then, a risk measure can be viewed as a functional on a space of continuous-time stochastic processes. In this paper we study coherent and convex monetary risk measures on the space of all càdlàg processes that are adapted to a given filtration. We show that if such risk measures are required to be real-valued, then they can only depend on a stochastic process in a way that is uninteresting for many applications. Therefore, we allow them to take values in ( −∞, ∞]. The economic interpretation of a value of ∞ is that the corresponding financial position is so risky that no additional amount of money can make it acceptable. The main result of the paper gives different characterizations of coherent or convex monetary risk measures on the space of all bounded adapted càdlàg processes that can be extended to coherent or convex monetary risk measures on the space of all adapted càdlàg processes. As examples we discuss a new approach to measure the risk of an insurance company and a coherent risk measure for unbounded càdlàg processes induced by a so called m-stable set.Due to errors during the typesetting process, this article was published incorrectly in Finance Stoch 9(3):369–387 (2005). The address of the first author was printed incorrectly, and in the whole paper the angular brackets were misprinted as [ ]. The complete corrected article is given here. The online version of the original paper can be found at: http://dx.doi.org/10.1007/s00780-004-0150-7     

Characteristic-based mean-variance portfolio choice

We study empirical mean-variance optimization when the portfolio weights are restricted to be direct functions of underlying stock characteristics such as value and momentum. The closed-form solution to the portfolio weights estimator shows that the portfolio problem in this case reduces to a mean-variance analysis of assets with returns given by single-characteristic strategies (e.g., momentum or value). In an empirical application to international stock return indexes, we show that the direct approach to estimating portfolio weights clearly beats a naive regression-based approach that models the conditional mean. However, a portfolio based on equal weights of the single-characteristic strategies performs about as well, and sometimes better, than the direct estimation approach, highlighting again the difficulties in beating the equal-weighted case in mean-variance analysis. The empirical results also highlight the potential for ‘stock-picking’ in international indexes using characteristics such as value and momentum with the characteristic-based portfolios obtaining Sharpe ratios approximately three times larger than the world market.     

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.     

Mean-variance vs. stochastic dominance some empirical findings on efficient sets

The study attempts to shed additional light on the issue of the costs and benefits of using the mean-variance criterion as opposed to stochastic dominance criteria for investment decisions. Relevant probabilities which facilitate measurement of these costs and benefits are identified. The mean-variance criterion is shown to be useful to some extent in identifying potentially optimal portfolios. However, it is shown that the informationally less demanding mean-variance criterion admits two types of errors: (i) including portfolios that no expected utility maximizing risk averters would choose, and (ii) excluding portfolios which some risk averters would find optimal. The empirical investigation also indicates that although the composition of the efficient sets appears to be unstable over time, the relationships between the efficient sets are persistent over time.     

A relaxation algorithm for building undominated portfolios

Stochastic dominance methods which have been developed in recent years are generally more valid than mean-variance (EV) and higher moment methods for selecting a portfolio from a given finite set of possible portfolios. One of the limitations of these methods is the lack of procedures for building portfolios from a given set of securities and the probability distribution of their returns. Markowitz has developed an algorithm based on the restriction method in linear programming to build undominated portfolios. In this paper a more efficient method based on the relaxation method of linear programming is developed and tested for efficiency. Computational results justify its use as a practical tool for portfolio building.     

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.     

Multivariate Lagrange Multiplier Tests for Fractional Integration

We introduce a multivariate Lagrange multiplier (LM) test forfractional integration. We derive and analyze the LM statisticand show that it is asymptotically noncentral chi-squared distributedunder local alternatives, and that, under Gaussianity, the LMtest is asymptotically efficient against local alternatives.It is shown that the regression variant in Breitung and Hassler(2002, Journal of Econometrics 110, 167–185) is not equivalentto the LM test in the multivariate case, although it is in theunivariate case. A generalization of the LM test that explicitlyallows for different integration orders for each variable isalso introduced. The finite sample properties of the LM testare evaluated by Monte Carlo experiments which demonstrate thatit is superior to the Breitung and Hassler (2002) test. An applicationto multivariate time series of real interest rates for six countriesis offered, demonstrating that more clear-cut evidence can bedrawn from multivariate tests compared to conducting severalunivariate tests.     

Dominance of a Class of Stein type Estimators for Optimal Portfolio Weights When the Covariance Matrix is Unknown

For the estimation problem of mean-variance optimal portfolio weights, several previous studies have proposed applying Stein type estimators. However, few studies have addressed this problem analytically. Since the form of the loss function used in this problem is not of the quadratic type commonly used in statistical studies, there have been some difficulties in showing analytically the general dominance results. However, dominance results are given here of a class of Stein type estimators for the mean-variance optimal portfolio weights when the covariance matrix is unknown and is estimated. The class of estimators is broader than the one given in a previous study. The results we have obtained enable us to clarify conditions for some previously proposed estimators in finance to have smaller risks than the estimator which we obtain by plugging in the sample estimates.     

On the Cross-sectional Relation between Expected Returns and Betas

There is an exact linear relation between expected returns and true “betas” when the market portfolio is on the ex ante mean-variance efficient frontier, but empirical research has found little relation between sample mean returns and estimated betas. A possible explanation is that market portfolio proxies are mean-variance inefficient. We categorize proxies that produce particular relations between expected returns and true betas. For the special case of a zero relation, a market portfolio proxy must lie inside the efficient frontier, but it may be close to the frontier.     

Fair valuation of insurance liabilities via mean-variance hedging in a multi-period setting

A general class of fair valuations which are both market-consistent (mark-to-market for any hedgeable part of a claim) and actuarial (mark-to-model for any claim that is independent of financial market evolutions) was introduced in Dhaene et al. [Insurance: Mathematics & Economics, 76, 14–27 (2017)] in a single period framework. In particular, the authors considered mean-variance hedge-based (MVHB) valuations where fair valuations of insurance liabilities are expressed in terms of mean-variance hedges and actuarial valuations. In this paper, we generalize this MVHB approach to a multi-period dynamic investment setting. We show that the classes of fair valuations and MVHB valuations are equivalent in this generalized setting. We derive tractable formulas for the fair valuation of equity-linked contracts and show how the actuarial part of their MVHB valuation decomposes into a diversifiable and a non-diversifiable component.     

The Sampling Error in Estimates of Mean-Variance Efficient Portfolio Weights

This paper presents an exact finite-sample statistical procedure for testing hypotheses about the weights of mean-variance efficient portfolios. The estimation and inference procedures on efficient portfolio weights are performed in the same way as for the coefficients in an OLS regression. OLS t - and F -statistics can be used for tests on efficient weights, and when returns are multivariate normal, these statistics have exact t and F distributions in a finite sample. Using 20 years of data on 11 country stock indexes, we find that the sampling error in estimates of the weights of a global efficient portfolio is large.     

Optimal rules for ordering uncertain prospects

In this paper, we obtain the optimal selection rule for ordering uncertain prospects for all individuals with decreasing absolute risk averse utility functions. The optimal selection rule minimizes the admissible set of alternatives by discarding, from among a given set of alternatives, those that are inferior (for each utility function in the restricted class) to a member of the given set. We show that the Third Order Stochastic Dominance (TSD) rule is the optimal rule when comparing uncertain prospects with equal means. We also show that in the general case of unequal means, no known selection rule uses both necessary and sufficient conditions for dominance, and the TSD rule may be used to obtain a reasonable approximation to the smallest admissible set. The TSD rule is complex and we provide an efficient algorithm to obtain the TSD admissible set. For certain restrictive classes of the probability distributions (of returns on uncertain prospects) which cover most commonly used distributions in finance and economics, we obtain the optimal rule and show that it reduces to a simple form. We also study the relationship of the optimal selection rule to others previously advocated in the literature, including the more popular mean-variance rule as well as the semi-variance rule.     

On statistical inference on the mean-variance efficient set

A proof of the asymptotic distribution of the estimated mean-variance frontier is given. A Bayesian prediction interval is derived for the capital asset pricing model. Numerical illustrations show that the prediction intervals for the CAPM are smaller than those for the constant mean model, if the fit of the CAPM is better than that of the constant mean model.