Optimally chosen small portfolios are better than large ones

One of the fundamental principles in portfolio selection models is minimization of risk through diversification of the investment. However, this principle does not necessarily translate into a request for investing in all the assets of the investment universe. Indeed, following a line of research started by Evans and Archer almost fifty years ago, we provide here further evidence that small portfolios are sufficient to achieve almost optimal in-sample risk reduction with respect to variance and to some other popular risk measures, and very good out-of-sample performances. While leading to similar results, our approach is significantly different from the classical one pioneered by Evans and Archer. Indeed, we describe models for choosing the portfolio of a prescribed size with the smallest possible risk, as opposed to the random portfolio choice investigated in most of the previous works. We find that the smallest risk portfolios generally require no more than 15 assets. Furthermore, it is almost always possible to find portfolios that are just 1% more risky than the smallest risk portfolios and contain no more than 10 assets. Furthermore, the optimal small portfolios generally show a better performance than the optimal large ones. Our empirical analysis is based on some new and on some publicly available benchmark data sets often used in the literature

New approximate stochastic dominance approaches for Enhanced Indexation models

In this paper, we discuss portfolio selection strategies for Enhanced Indexation (EI), which are based on stochastic dominance relations. The goal is to select portfolios that stochastically dominate a given benchmark but that, at the same time, must generate some excess return with respect to a benchmark index. To achieve this goal, we propose a new methodology that selects portfolios using the ordered weighted average (OWA) operator, which generalizes previous approaches based on minimax selection rules and still leads to solving linear programming models. We also introduce a new type of approximate stochastic dominance rule and show that it implies the almost Second-order Stochastic Dominance (SSD) criterion proposed by Lizyayev and Ruszczynski (2012). We prove that our EI model based on OWA selects portfolios that dominate a given benchmark through this new form of stochastic dominance criterion. We test the performance of the obtained portfolios in an extensive empirical analysis based on real-world datasets. The computational results show that our proposed approach outperforms several SSD-based strategies widely used in the literature, as well as the global minimum variance portfolio

Portfolio selection problems in practice: a comparison between linear and quadratic optimization models

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

A new family of modified Gaussian copulas for market consistent valuation of government guarantees

This paper deals with a copula-based stochastic dependence problem in the context of financial risks. We discuss the financial framework for assessing the theoretical up-front value of government guarantees on bank liabilities. EU States widely use these contracts to improve the financial system’s stability and manage the banking sector in crisis situations; in Italy, they have also been used to address the consequences of the Covid-19 emergency. From a market viewpoint, we deal with a defaultable guarantee contract where the State-guarantor and the bank-borrower are both subject to default risk, and their risks are interconnected. We show that the classical Gaussian copula is not satisfactory for modeling the dependence among the considered risks. Indeed, using the benchmark market model for credit risk portfolio management, we highlight some contradictory results observed for the up-front values of the guarantee when the default intensity of the guarantor is smaller than that of the borrower. Then, we introduce a new family of modified Gaussian copulas that overcomes the limitations of the standard approach, allowing to determine realistic results in terms of the guarantees “mark-to-model” value when the benchmark market model does not work. Numerical simulations validate the theoretical proposal

A return-diversification approach to portfolio selection

In this paper, we propose a general bi-objective model for portfolio selection, aiming to maximize both a diversification measure and the portfolio expected return. Within this general framework, we focus on maximizing a diversification measure recently proposed by Choueifaty and Coignard for the case of volatility as a risk measure. We first show that the maximum diversification approach is actually equivalent to the Risk Parity approach using volatility under the assumption of equicorrelated assets. Then, we extend the maximum diversification approach formulated for general risk measures. Finally, we provide explicit formulations of our bi-objective model for different risk measures, such as volatility, Mean Absolute Deviation, Conditional Value-at-Risk, and Expectiles, and we present extensive out-of-sample performance results for the portfolios obtained with our model. The empirical analysis, based on five real-world data sets, shows that the return-diversification approach provides portfolios that tend to outperform the strategies based only on a diversification method or on the classical risk-return approach

MAD risk parity portfolios

In this paper, we investigate the features and the performance of the risk parity (RP) portfolios using the mean absolute deviation (MAD) as a risk measure. The RP model is a recent strategy for asset allocation that aims at equally sharing the global portfolio risk among all the assets of an investment universe. We discuss here some existing and new results about the properties of MAD that are useful for the RP approach. We propose several formulations for finding MAD-RP portfolios computationally, and compare them in terms of accuracy and efficiency. Furthermore, we provide extensive empirical analysis based on three real-world datasets, showing that the performances of the RP approaches generally tend to place both in terms of risk and profitability between those obtained from the minimum risk and the Equally Weighted strategies

Portfolio selection problems in practice: a comparison between linear and quadratic optimization models

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

Prevention of venous thrombosis and thrombophlebitis in long-haul flights with pycnogenol.

The aim of this study was to evaluate the occurrence of deep venous thrombosis (DVT) and superficial vein thrombosis (SVT) and its prophylaxis with an oral anti-edema and antithrombotic agent (Pycnogenol®, Horphag, Research Management SA, Geneva, Switzerland) in long-haul flights, in subjects at moderate to high-risk of DVT and SVT. The study pre-included 244 pre-selected subjects; 211 were included (33 were excluded for several reasons due to logistic problems) and 198 completed the study; 13 subjects were lost for follow-up at the end of the flight, all for non-medical problems (i.e., for difficult connections). All subjects were scanned within 90 minutes before the flight and within 2 hours after disembarking. Subjects were supplemented with 100 mg Pycnogenol® per capsule. Treatment subjects received two capsules between 2 and 3 hours before flights with 250 mL of water; two capsules were taken 6 hours later with 250 mL of water and one capsule the next day. The control group received comparable placebo at the same intervals. The flight duration was on average 8 hours and 15 minutes (SD 55 min) (range, 7.45-12.33). In the control group there were five thrombotic events (one DVT and four superficial thromboses) while only nonthrombotic, localized phlebitis was observed in the Pycnogenol®group (5.15% vs. no events; p<0.025). The ITT (intention to treat) analysis detects 13 failures in the control group (eight lost to follow up + five thrombotic events) of 105 subjects (12.4%) vs. five failures (4.7%; all lost, no thrombotic events) in the treatment group (p<0.025). No unwanted effects were observed. In conclusion, this study indicates that Pycnogenol® treatment was effective in decreasing the number of thrombotic events (DVT and SVT) in moderate-to-high risk subjects, during long-haul flights

Computational Finance. MATLAB oriented modeling

Computational Finance is becoming increasingly important in the financial industry. It is the necessary complement to apply the theoretical models to real-world challenges. Indeed, many models used in practice involve complex mathematical problems, for which an exact or a closed-form solution is not available. Consequently, we need to rely on computational techniques and specific numerical algorithms. This book aims at combining theoretical concepts and their practical implementation. Furthermore, the numerical solution of models is exploited both to enhance the understanding of some mathematical and statistical notions and to acquire sound programming skills in MATLAB, which can be useful also in several other programming languages. Most of the content of this book has been taught for several years at a Master’s course in Finance to students with a relatively small background in mathematics, probability and statistics. Hence, the book contains a short description of the fundamental tools needed to address the two main fields of quantitative finance: portfolio selection and derivatives pricing. Both fields are developed here, with a particular emphasis on portfolio selection, where we include recent approaches that have appeared only in the literature. We develop the ability to place financial models in a computational setting. This supports the understanding of theoretical concepts through their practical application

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