Minimum-variance Frontier Research Articles

Financial Models and Portfolio Optimizations in the U.S. Market

Portfolio optimization is an essential task for investors to maximize the profit and minimize the risk in trading, which can be reflected in return and standard deviation respectively. This article selects ten well-known firms and S&P 500 index as the portfolio and calculates their basic descriptive data of annualized return, annualized standard deviation, and correlations. Next, this article applies Markowitz model and Single-index model to the portfolio optimizations, as well as setting five constraints for comparison. After using the above methods, this paper gives the results in tabular and graphic ways. It finds that optimizing with no restriction can generate the biggest permissible regions of the minimum-variance frontier; investors only long stocks under constraint 4 can guarantee a finite risk and a finite positive return, which has a robust performance. Moreover, Markowitz model is flexible in covariances, but the number of estimates could complicate the calculating process; Index model is simple to calculate in linear regression form, but it’s not highly sensitive to the covariance matrix. The purpose of this article is to prove the feasibility of optimization in future portfolio management and helps investors to construct the portfolios with the fundamental theory of financial models.

Analysis of the Energy Company Stock's Portfolio in the Past Twenty Years by Using the Markowitz Model and the Index Model

Setting an optimal portfolio is an important issue to all the people who are in the stock market. Some people use their experience to get profit by buying and selling the energy company’s tocks, but they lack a good portfolio for them to not only decrease the risk but also get more profits. Therefore, the goal of this paper is to decide which one between Markowitz Model and Index Model is best tool for investors to set their energy companies’ portfolio. The research method is to collect data from Yahoo Finance, and then using the data to calculate the minimum variance point, maximum sharp ratio, minimum variance frontier, efficient frontier, inefficient frontier, and capital allocation line for both models. By comparing and analyzing these data, the optimal portfolio can be obtained for investing the energy companies. The result of this research is that both models are good tools for investors to establish their portfolio regard to the energy market.

Classifying the minimum-variance surface of multiple-objective portfolio selection for capital asset pricing models

Portfolio selection is recognized as the birth-place of modern finance; Markowitz emphasizes computing whole efficient frontiers. Moreover, computing efficient sets has long been a topic in multiple-objective optimization. After portfolio selection, an important research direction is capital asset pricing models (CAPM). Black and Fama prove the existence of a unique zero-covariance portfolio on the minimum-variance frontier. By the zero-covariance relationship, Roll then classifies the minimum-variance frontier into a positive-covariance side and a negative-covariance side; Fama and Roll further prove zero-covariance CAPM. Recently, researchers gradually realize additional criteria and extend portfolio selection into multiple-objective portfolio selection. Consequently, the minimum-variance frontier extends to a minimum-variance surface. Moreover, the extension naturally raises questions of classifying the surface for multiple-objective CAPM. There has been no such research until now. In such an area, this paper contributes to the literature by extending the classification. We classify the surface into a positive-covariance side and a negative-covariance side, although the classification depends on specific portfolios and is thus non-uniform. We then analyze classification properties and extend relevant theorems by proposing conjectures, although the conjectures are disproved by theorems and counter-examples. Moreover, we forward the research methodology to general k-objective models, so research arenas for multiple-objective CAPM are opened. This paper brings researchers one step closer to classifying the minimum-variance surface and extending CAPM.

Investor sentiment and portfolio selection

We provide a theoretical framework to examine how investor sentiment impacts the mean-variance tradeoff. We derive a sentiment-adjusted Markowitz efficient frontier in which investor sentiment alters the first two moments of asset returns, the minimum-variance frontier as well as the Capital Market Line. Our theoretical results are consistent with empirical findings that heightened sentiment-related noise trading activity drives perceived prices away from fundamental and increases market volatility. A rational investor neglecting the effect of investor sentiment may end up selecting a sub-optimal portfolio.

Factor reversal in the euro zone stock returns: Evidence from the crisis period

The adoption of the euro led to a shift in importance from country to industry effects in euro zone stock returns. For the first time, this paper shows that country effects have regained importance in the recent spate of crises. This euro-wide factor reversal is driven by countries with poor economic fundamentals, comprising Portugal, Italy, Ireland, Greece, and Spain (PIIGS). The results imply that a more traditional country portfolio approach provides greater diversification benefits during crisis periods and the minimum-variance frontier of industry portfolios in PIIGS countries can be improved by adjusting country weights.

Fast algorithm for the Markowitz critical line method

The critical line method developed by the Nobel Prize winner H. Markowitz is a classical technique for the construction of a minimum-variance frontier within the paradigm of “the expected return-risk” (mean-variance) and finding minimum portfolios. Considerable interest has recently been attracted to the development of a fast algorithm for the construction of the minimum-variance frontier. In some works, such algorithms have been used to find statistically stable optimal portfolios. An algorithm based on the critical line method has recently been proposed by Andras Niedermayer and Daniel Niedermayer. Its testing showed that it is faster than all similar algorithms known before by several orders of magnitude. In this paper, we present an algorithm for constructing the minimumvariance frontier for the Markowitz problem with the condition of nonnegativity of the optimal portfolio, which requires about half as many operations as the Niedermayers’ algorithm. To this end, we had to perform a more thorough analytical and geometric study of the Markowitz problem.

The Distribution of the Sample Minimum-Variance Frontier

In this paper, we present a finite sample analysis of the sample minimum-variance frontier under the assumption that the returns are independent and multivariate normally distributed. We show that the sample minimum-variance frontier is a highly biased estimator of the population frontier and we propose an improved estimator of the population frontier. In addition, we provide the exact distribution of the out-of-sample mean and variance of sample minimum-variance portfolios. This allows us to understand the impact of estimation error on the performance of in-sample optimal portfolios.

The Distribution of the Sample Minimum-Variance Frontier

In this paper, we present a finite sample analysis of the sample minimum-variance frontier under the assumption that the returns are independent and multivariate normally distributed. We show that the sample minimum-variance frontier is a highly biased estimator of the population frontier, and we propose an improved estimator of the population frontier. In addition, we provide the exact distribution of the out-of-sample mean and variance of sample minimum-variance portfolios. This allows us to understand the impact of estimation error on the performance of in-sample optimal portfolios.

Are Euro Area Small Cap Stocks an Asset Class? Evidence from Mean‐Variance Spanning Tests

Abstract In this paper we perform regression‐based tests for mean‐variance spanning in order to detect the effect of investing in euro area small capitalisation stocks on the minimum variance frontier, and apply different measures to assess the extent of diversification gains. Empirical analysis shows that euro area small and mid cap stocks, as classified by size quartile and quintile rankings, arise as truly autonomous asset classes. This result is robust to different methodologies used to form size‐based portfolios, and holds relative to both euro area large cap stocks and other international asset classes, US small capitalisation stocks included.

Positively Weighted Minimum-Variance Portfolios and the Structure of Asset Expected Returns

In this paper, we derive simple, directly computable conditions for minimum-variance portfolios to have all positive weights. We show that either there is no minimum-variance portfolio with all positive weights or there is a single segment of the minimum-variance frontier for which all portfolios have positive weights. Then, we examine the likelihood of observing positively weighted minimum-variance portfolios. Analytical and computational results suggest that: i) even if the mean vector and covariance matrix are compatible with a given positively weighted portfolio being mean-variance efficient, the proportion of the minimum-variance frontier containing positively weighted portfolios is small and decreases as the number of assets in the universe increases, and ii) small perturbations in the means will likely lead to no positively weighted minimum-variance portfolios.

Exact Arbitrage Pricing and the Minimum-Variance Frontier

The author examines the relationship between the Arbitrage Pricing Theory of Ross and mean-variance analysis. In particular, conditions are derived on the vector of the factor risk premia that are equivalent to the existence of a strictly positively weighted portfolio on the minimum-variance frontier. Also, a sufficient condition is given under which the existence of a positive minimum-variance portfolio of all the assets in the economy will imply the existence of a positive minimum-variance portfolio on a subset. This means that rejection of the hypothesis of the existence of a positive minimum-variance portfolio on a subset satisfying this condition implies rejection for the whole set. A GROWING BODY OF literature explores the relationship between the Arbitrage Pricing Theory (APT) of Ross [13, 14] and the mean-variance model. Among the early work in this direction were the papers of Chamberlain and Rothschild [2] and Chamberlain [1]. These papers examine a sequence of economies and characterize the relationship in the limit between the mean and portfolio cost functionals and the minimum-variance frontier. Chamberlain [1] also gives a condition under which exact arbitrage pricing will hold. Dybvig [4] and Grinblatt and Titman [7] examine equilibrium models in an APT context for a finite economy. They derive sharp bounds on the deviation from exact arbitrage pricing. Recent papers by Huberman, Kandel, and Stambaugh [9] and Grinblatt and Titman [8] explore the relationship between the APT and the mean-variance model by studying portfolios that can be used as proxies for the factors and examining the relationship these portfolios bear to the minimum-variance frontier. These two papers also deal with equilibrium models. In spite of the number and variety of equilibrium-APT models, the question of the pricing of factor risk in equilibrium remains largely unexplored. This paper makes some progress in addressing that issue by looking at a given equilibrium restriction and studying the constraints it imposes on the vector of factor premia itself. The most familiar equilibrium restriction we might study is the central implication of the Capital Asset Pricing Model (CAPM), that the market portfolio is efficient. In particular, no portfolio with the same expected return as the market portfolio has a smaller variance. The restriction we actually study is weaker than

Exact Arbitrage Pricing and the Minimum-Variance Frontier

Exact Arbitrage Pricing and the Minimum-Variance Frontier

Mean-Variance Spanning

The authors propose a likelihood-ratio test of the hypothesis that the minimum-variance frontier of a set of K assets coincides with the frontier of this set and another set of N assets. They study the relation between this hypothesis, exact arbitrage pricing, and mutual fund separation. The exact distribution of the test statistic is available. The authors test the hypothesis that the frontier spanned by three size-sorted stock portfolios is the same as the frontier spanned by thirty-three size-sorted stock portfolios.

Positively Weighted Portfolios on the Minimum-Variance Frontier

Duality theory is employed to provide necessary and sufficient conditions for portfolios on the minimum-variance frontier to have positive investment proportions in all assets. These conditions involve the feasibility of portfolios that have non-negative correlation with all assets and positive correlation with at least one. Using these results, several qualitative results concerning the signs of investment proportions in efficient portfolios are proved. It is argued that the conditions that ensure all-positive weights in efficient portfolios are intuitively compelling and are not unique to the CAPM. With large numbers of assets, however, the signs of weights in minimum-variance portfolios can be very sensitive to slight departures from these conditions due to, for example, sampling error. THE BEHAVIOR OF THE minimum-variance frontier is of general concern to portfolio theory and plays a central role in the study of asset pricing. An unresolved question in the theory of the minimum-variance frontier involves when it will contain portfolios with positive weights on all assets. This paper addresses that question. We show that the conditions that ensure positive investment proportions in efficient portfolios have, by Farkas' Lemma, a simple dual. This dual, or alternative, system of inequalities puts conditions directly on the variances, covariances, and means of the returns, and these conditions can be interpreted intuitively. This contrasts with simply considering the first-order conditions for variance minimization, since these involve the inverted covariance matrix. With a large number of assets, it can be difficult to translate one's intuitive sense about how assets covary into statements about the inverse of the covariance matrix. The duality conditions we derive involve the existence of portfolios that are positively (or negatively) correlated with all assets. Indeed, we show that a necessary and sufficient condition for no portfolio on the frontier to have all-positive weights is the feasibility of an arbitrage or hedge portfolio, with weights that sum to zero, that has an expected payoff of zero and has non-negative correlation with all assets. The duality conditions also make it easy to construct examples of the types of covariability that permit or preclude positive investment proportions on all assets in frontier portfolios. There are at least three apparent reasons for interest in the signs of weights