Francesco Cesarone

A new method for mean-variance portfolio optimization with cardinality constraints

Several portfolio selection models take into account practical limitations on the number of assets to include and on their weights in the portfolio. We present here a study of the Limited Asset Markowitz (LAM) model, where the assets are limited with the introduction of quantity and cardinality constraints.We propose a completely new approach for solving the LAM model based on a reformulation as a Standard Quadratic Program, on a new lower bound that we establish, and on other recent theoretical and computational results for such problem. These results lead to an exact algorithm for solving the LAM model for small size problems. For larger problems, such algorithm can be relaxed to an efficient and accurate heuristic procedure that is able to find the optimal or the best-known solutions for problems based on some standard financial data sets that are used by several other authors. We also test our method on five new data sets involving real-world capital market indices from major stock markets. We compare our results with those of CPLEX and with those obtained with very recent heuristic approaches in order to illustrate the effectiveness of our method in terms of solution quality and of computation time. All our data sets and results are publicly available for use by other researchers.

On exact and approximate stochastic dominance strategies for portfolio selection

One recent and promising strategy for Enhanced Indexation is the selection of portfolios that stochastically dominate the benchmark. We propose here a new type of approximate stochastic dominance rule which implies other existing approximate stochastic dominance rules. We then use it to find the portfolio that approximately stochastically dominates a given benchmark with the best possible approximation. Our model is initially formulated as a Linear Program with exponentially many constraints, and then reformulated in a more compact manner so that it can be very efficiently solved in practice. This reformulation also reveals an interesting financial interpretation. We compare our approach with several exact and approximate stochastic dominance models for portfolio selection. An extensive empirical analysis on real and publicly available datasets shows very good out-of-sample performances of our model.

A linear risk-return model for enhanced indexation in portfolio optimization

Enhanced indexation (EI) is the problem of selecting a portfolio that should produce excess return with respect to a given benchmark index. In this work, we propose a linear bi-objective optimization approach to EI that maximizes average excess return and minimizes underperformance over a learning period. Our model can be efficiently solved to optimality by means of standard linear programming techniques. On the theoretical side, we investigate conditions that guarantee or forbid the existence of a portfolio strictly outperforming the index. On the practical side, we support our model with extensive empirical analysis on publicly available real-world financial datasets, including comparison with previous studies, performance and diversification analysis, and verification of some of the proposed theoretical results on real data.

Real-world datasets for portfolio selection and solutions of some stochastic dominance portfolio models

A large number of portfolio selection models have appeared in the literature since the pioneering work of Markowitz. However, even when computational and empirical results are described, they are often hard to replicate and compare due to the unavailability of the datasets used in the experiments. We provide here several datasets for portfolio selection generated using real-world price values from several major stock markets. The datasets contain weekly return values, adjusted for dividends and for stock splits, which are cleaned from errors as much as possible. The datasets are available in different formats, and can be used as benchmarks for testing the performances of portfolio selection models and for comparing the efficiency of the algorithms used to solve them. We also provide, for these datasets, the portfolios obtained by several selection strategies based on Stochastic Dominance models (see “On Exact and Approximate Stochastic Dominance Strategies for Portfolio Selection” (Bruni et al. [2])). We believe that testing portfolio models on publicly available datasets greatly simplifies the comparison of the different portfolio selection strategies.

Linear vs. quadratic portfolio selection models with hard real-world constraints

Several risk–return portfolio models take into account practical limitations on the number of assets to be included in the portfolio and on their weights. We present here a comparative study, both from the efficiency and from the performance viewpoint, of the Limited Asset Markowitz (LAM), the Limited Asset mean semi-absolute deviation (LAMSAD), and the Limited Asset conditional value-at-risk (LACVaR) models, where the assets are limited with the introduction of quantity and of cardinality constraints.The mixed integer linear LAMSAD and LACVaR models are solved with a state of the art commercial code, while the mixed integer quadratic LAM model is solved both with a commercial code and with a more efficient new method, recently proposed by the authors. Rather unexpectedly, for medium to large sizes it is easier to solve the quadratic LAM model with the new method, than to solve the linear LACVaR and LAMSAD models with the commercial solver. Furthermore, the new method has the advantage of finding all the extreme points of a more general tri-objective problem at no additional computational cost.We compare the out-of-sample performances of the three models and of the equally weighted portfolio. We show that there is no apparent dominance relation among the different approaches and, in contrast with previous studies, we find that the equally weighted portfolio does not seem to have any advantage over the three proposed models. Our empirical results are based on some new and old publicly available data sets often used in the literature.Several portfolio selection models take into account practical limitations on the number of assets to include and on their weights in the portfolio. We present here a study of the Limited Asset Markowitz (LAM), of the Limited Asset Mean Absolute Deviation (LAMAD) and of the Limited Asset Conditional Value-at-Risk (LACVaR) models, where the assets are limited with the introduction of quantity and cardinality constraints. We propose a completely new approach for solving the LAM model, based on reformulation as a Standard Quadratic Program and on some recent theoretical results. With this approach we obtain optimal solutions both for some well-known financial data sets used by several other authors, and for some unsolved large size portfolio problems. We also test our method on five new data sets involving real-world capital market indices from major stock markets. Our computational experience shows that, rather unexpectedly, it is easier to solve the quadratic LAM model with our algorithm, than to solve the linear LACVaR and LAMAD models with CPLEX, one of the best commercial codes for mixed integer linear programming (MILP) problems. Finally, on the new data sets we have also compared, using out-of-sample analysis, the performance of the portfolios obtained by the Limited Asset models with the performance provided by the unconstrained models and with that of the official capital market indices.

Minimum risk versus capital and risk diversification strategies for portfolio construction

Abstract In this paper, we propose an extensive empirical analysis on three categories of portfolio selection models with very different objectives: minimization of risk, maximization of capital diversification, and uniform distribution of risk allocation. The latter approach, also called Risk Parity or Equal Risk Contribution (ERC), is a recent strategy for asset allocation that aims at equally sharing the risk among all the assets of the selected portfolio. The risk measure commonly used to select ERC portfolios is volatility. We propose here new developments of the ERC approach using Conditional Value-at-Risk (CVaR) as a risk measure. Furthermore, under appropriate conditions, we also provide an approach to find a CVaR ERC portfolio as a solution of a convex optimization problem. We investigate how these classes of portfolio models (Minimum-Risk, Capital-Diversification, and Risk-Diversification) work on seven investment universes, each with different sources of risk, including equities, bonds, and mixed assets. Then, we highlight some strengths and weaknesses of all portfolio strategies in terms of various performance measures.

Equal Risk Bounding is better than Risk Parity for portfolio selection

Risk Parity (RP), also called equally weighted risk contribution, is a recent approach to risk diversification for portfolio selection. RP is based on the principle that the fractions of the capital invested in each asset should be chosen so as to make the total risk contributions of all assets equal among them. We show here that the Risk Parity approach is theoretically dominated by an alternative similar approach that does not actually require equally weighted risk contribution of all assets but only an equal upper bound on all such risks. This alternative approach, called Equal Risk Bounding (ERB), requires the solution of a nonconvex quadratically constrained optimization problem. The ERB approach, while starting from different requirements, turns out to be strictly linked to the RP approach. Indeed, when short selling is allowed, we prove that an ERB portfolio is actually an RP portfolio with minimum variance. When short selling is not allowed, there is a unique RP portfolio and it contains all assets in the market. In this case, the ERB approach might lead to the RP portfolio or it might lead to portfolios with smaller variance that do not contain all assets, and where the risk contributions of each asset included in the portfolio is strictly smaller than in the RP portfolio. We define a new riskiness index for assets that allows to identify those assets that are more likely to be excluded from the ERB portfolio. With these tools we then provide an exact method for small size nonconvex ERB models and a very efficient and accurate heuristic for larger problems of this type. In the case of a common constant pairwise correlation among all assets, a closed form solution to the ERB model is obtained and used to perform a parametric analysis when varying the level of correlation. The practical advantages of the ERB approach over the RP strategy are illustrated with some numerical examples. Computational experience on real-world and on simulated data confirms accuracy and efficiency of our heuristic approach to the ERB model also in comparison with some state-of-the-art local and global optimization codes.

Does Greater Diversification Really Improve Performance in Portfolio Selection

One of the fundamental principles in portfolio selection models is minimization of risk through diversification of the investment. This seems to require that in a given working universe, or market, the investment should be spread among all (or almost all) the available assets. Indeed, this is what some classical investment strategies, like Equally-Weighted portfolios, or more recent and refined ones, like Risk Parity, actually recommend. The purpose of this work consists in giving some empirical evidence of the fact that diversifying through the use of larger portfolios is not the best way to achieve an improvement in out-of-sample performance. More precisely, we investigate the role of the restriction on the number of assets in a portfolio (a cardinality constraint) on the in-sample and out-of-sample outcomes of the Equally-Weighted approach and of some well-known portfolio selection models that minimize risk through the use of Variance, Semi-Mean Absolute Deviation, and Conditional Value-at-Risk. Our empirical analysis is based on some new and on some publicly available benchmark data sets often used in the literature.

A Linear Risk-Return Model for Enhanced Indexation

Enhanced Indexation is the problem of selecting a portfolio that should produce excess return with respect to a given benchmark index. In this work we propose a linear bi-objective optimization approach to Enhanced Indexation that maximizes average excess return and minimizes underperformance over a learning period. Our model can be efficiently solved to optimality by means of standard Linear Programming techniques. On the theoretical side, we investigate conditions that guarantee or forbid the existence of a portfolio strictly outperforming the index. We also support our model with extensive empirical analysis on publicly available real-world financial datasets, including comparison with previous studies, performance and diversification analysis, and verification of some of the proposed theoretical results on real data.

A Quick Tool to Forecast VaR Using Implied and Realized Volatilities

We propose here a naive model to forecast ex-ante Value-at-Risk (VaR) using a shrinkage estimator between realized volatility estimated on past return time series, and implied volatility extracted from option pricing data. Implied volatility is often indicated as the operators expectation about future risk, while the historical volatility straightforwardly represents the realized risk prior to the estimation point, which by definition is backward looking. In a nutshell, our prediction strategy for VaR uses information both on the expected future risk and on the past estimated risk.We examine our model, called Shrinked Volatility VaR, both in the univariate and in the multivariate cases, empirically comparing its forecasting power with that of two benchmark VaR estimation models based on the Historical Filtered Bootstrap and on the RiskMetrics approaches.The performance of all VaR models analyzed is evaluated using both statistical accuracy tests and efficiency evaluation tests, according to the Basel II and ESMA regulatory frameworks, on several major markets around the world over an out-of-sample period that covers different financial crises.Our results confirm the efficacy of the implied volatility indexes as inputs for a VaR model, but combined together with realized volatilities. Furthermore, due to its ease of implementation, our prediction strategy to forecast VaR could be used as a tool for portfolio managers to quickly monitor investment decisions before employing more sophisticated risk management systems.