Measuring the risk of a nonlinear portfolio with fat tailed risk factors through probability conserving transformation

This paper presents a new heuristic for fast approximation of VaR (Value-at-Risk) and CVaR (conditional Value-at-Risk) for financial portfolios, where the net worth of a portfolio is a non-linear function of possibly non-Gaussian risk factors. The proposed method is based on mapping non-normal marginal distributions into normal distributions via a probability conserving transformation and then using a quadratic, i.e. Delta–Gamma, approximation for the portfolio value. The method is very general and can deal with a wide range of marginal distributions of risk factors, including non-parametric distributions. Its computational load is comparable with the Delta–Gamma–Normal method based on Fourier inversion. However, unlike the Delta–Gamma–Normal method, the proposed heuristic preserves the tail behaviour of the individual risk factors, which may be seen as a significant advantage. We demonstrate the utility of the new method with comprehensive numerical experiments on simulated as well as real financial data

Portfolio Diversification Under Local, Moderate and Global Deviations From Power Laws

This paper focuses on the analysis of portfolio diversification for a wide class of nonlinear transformations of heavy-tailed risks. We show that diversification of a portfolio of nonlinear transformations of thick-tailed risks increases riskiness if expectations of these functions are infinite. In addition, coherency of the value at risk measure is always violated for such portfolios. On the contrary, for nonlinearly transformed heavy-tailed risks with finite expectations, the stylized fact that diversification is preferable continues to hold. Moreover, in the latter setting, the value of risk is a coherent measure of risk. The framework of transformations of long-tailed random variables includes many models with Pareto-type distributions that exhibit local, moderate and global deviations from power tails in the form of additional slowly varying or exponential factors. This leads to a refined understanding of under what distributional assumptions diversification increases riskiness.

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

Relative Robust Portfolio Optimization

Considering mean-variance portfolio problems with uncertain model parameters, we contrast the classical absolute robust optimization approach with the relative robust approach based on a maximum regret function. Although the latter problems are NP-hard in general, we show that tractable inner and outer approximations exist in several cases that are of central interest in asset management

Expected Utility Maximization and Conditional Value-at-Risk Deviation-based Sharpe Ratio in Dynamic Stochastic Portfolio Optimization

In this paper we investigate the expected terminal utility maximization approach for a dynamic stochastic portfolio optimization problem. We solve it numerically by solving an evolutionary Hamilton-Jacobi-Bellman equation which is transformed by means of the Riccati transformation. We examine the dependence of the results on the shape of a chosen utility function in regard to the associated risk aversion level. We define the Conditional value-at-risk deviation ($CVaRD$) based Sharpe ratio for measuring risk-adjusted performance of a dynamic portfolio. We compute optimal strategies for a portfolio investment problem motivated by the German DAX 30 Index and we evaluate and analyze the dependence of the $CVaRD$-based Sharpe ratio on the utility function and the associated risk aversion level

Optimizing Omega

"The original publication is available at www.springerlink.com " Copyright Springer. DOI: 10.1007/s10898-008-9396-5This paper considers the Omega function, proposed by Cascon, Keating & Shadwick as a performance measure for comparing financial assets. We discuss the use of Omega as a basis for portfolio selection. We show that the problem of choosing portfolio weights in order to maximize Omega typically has many local solutions and we describe some preliminary computational experience of finding the global optimum using a NAG library implementation of the Huyer & Neumaier MCS method.Peer reviewe

A recursive algorithm for multivariate risk measures and a set-valued Bellman's principle

A method for calculating multi-portfolio time consistent multivariate risk measures in discrete time is presented. Market models for $d$ assets with transaction costs or illiquidity and possible trading constraints are considered on a finite probability space. The set of capital requirements at each time and state is calculated recursively backwards in time along the event tree. We motivate why the proposed procedure can be seen as a set-valued Bellman's principle, that might be of independent interest within the growing field of set optimization. We give conditions under which the backwards calculation of the sets reduces to solving a sequence of linear, respectively convex vector optimization problems. Numerical examples are given and include superhedging under illiquidity, the set-valued entropic risk measure, and the multi-portfolio time consistent version of the relaxed worst case risk measure and of the set-valued average value at risk.Comment: 25 pages, 5 figure