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

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

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

Minimum risk versus capital and risk diversification strategies for portfolio construction

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

A Quick Tool to Forecast VaR Using Implied and Realized Volatilities

We propose a naive model to forecast ex ante value-at-risk (VaR), using a shrinkage estimator between realized volatility estimated on past return time series as well as implied volatility quoted in the market. Implied volatility is often indicated as the operator's expectation about future risk, while historical volatility straightforwardly represents the realized risk prior to the estimation point, which by definition is backward looking. Therefore, our VaR prediction strategy uses information both on expected future risk and past estimated risk. We examine our model, called shrun volatility VaR, in both the univariate and multivariate cases, empirically comparing its forecasting power with that of four benchmark VaR models. The performance of all VaR models is evaluated using both statistical accuracy and efficiency evaluation tests; this is done according to the Basel II and European Securities and Markets Authority regulatory frameworks, on several major markets, over an out-of-sample period that covers different financial crises. Our results confirm the efficacy of implied volatility indexes as inputs for a VaR model, but only when combined with realized volatilities. Further, due to its ease of implementation, our VaR prediction strategy could be used as a tool for portfolio managers to quickly monitor investment decisions before employing more sophisticated risk management systems