# Portfolio Optimization For VAR, CVaR, Omega And Utility With General Return Distributions

#### Portfolio Optimization for VAR, CVaR, Omega and Utility with General Return Distributions: A Monte Carlo Approach for Long-Only and Bounded Short Portfolios with Optional Robustness and a Simplified Approach to Covariance Matching by SSRN

William Thornton Shaw

University College London

June 1, 2011

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Abstract:

We develop the idea of using Monte Carlo sampling of random portfolios to solve portfolio investment problems. We explore the need for more general optimization tools, and consider the means by which constrained random portfolios may be generated. DeVroye’s approach to sampling the interior of a simplex (a collection of non-negative random variables adding to unity) is already available for interior solutions of simple fully-invested long-only systems, and we extend this to treat, lower bound constraints, bounded short positions and to sample non-interior points by the method of Face-Edge-Vertex-biased sampling. A practical scheme for long-only and bounded short problems is developed and tested. Non-convex and disconnected regions can be treated by applying rejection for other constraints. The advantage of Monte Carlo methods is that they may be extended to risk functions that are more complicated functions of the return distribution, without explicit gradients, and that the underlying return distribution may be modeled parametrically or empirically based on general distributions. The optimization of expected utility, Omega, Sortino ratios may be handled in a similar manner to quadratic risk, VaR and CVaR, irrespective of whether a reduction to LP or QP form is available. Robustification is also possible, and a Monte Carlo approach allows the possibility of relaxing the general maxi-min approach to one of varying degrees of conservatism. Grid computing technology is an excellent platform for the development of such computations due to the intrinsically parallel nature of the computation. Good comparisons with established results in Mean-Variance and CVaR optimization are obtained, and we give some applications to Omega and expected Utility optimization. Extensions to deploy Sobol and Niederreiter quasi-random methods for random weights are also proposed. Extensions to the value functions of prospect theory are possible. The initial method proposed here is essentially an initial global search which produces a good feasible solution for any number of assets with any risk function and return distribution. This solution is close to optimal in lower dimensions. A by-product of these investigations is a simplified approach – the “double-Cholesky method” – to sampling certain multivariate distributions matching the covariance matrix as well as the mean vector.

#### Portfolio Optimization for VAR, CVaR, Omega and Utility with General Return Distributions: A Monte Carlo Approach for Long-Only and Bounded Short Portfolios with Optional Robustness and a Simplified Approach to Covariance Matching – Introduction

The management of financial portfolios by the methods of quantitative risk management requires the solution of a complicated optimization problem, in which one decides how much of a given investment to allocate to each of N assets – this is the problem of assigning weights. There are many different forms of the problem, depending on how large N is (2 – 2000+), whether short-selling is allowed, investor preferences, and so on. In the simplest form of the problem, which is still relevant to traditional fund management, the weights are non-negative (you cannot go “short”) and are normalized to add to unity. So if we have an amount of cash P to invest and N assets to choose from, the weights are  satisfying This long-only form of the problem is of considerable importance, but not the only one of interest. Additional constraints are often specified, and this will be allowed for in the following discussion. Short-selling may also be considered.

The simplest and very ancient solution to this problem is to take wi = 1=N, which is equal allocation, and has been understood for hundreds of years as a principle. The amount of cash invested in asset i is then, in general, Pwi and in the equal-allocation strategy it is P=N. The usefulness of this robust strategy continues to be explored – see for example the beautiful study by DeMiguel et al  and references therein.

The mathematical theory moved on in the post-WWII work of Markowitz and collaborators, (see for example [15, 16, 17]). My own understanding is that Markowitz was in fact interested in the control of downside risk (the semivariance), but given the technology then available he used variance as a proxy for risk. An up to date summary of his philosophy is given in . When the idea was formulated as (in the simplest version) the minimization of risk subject to a return goal, this resulted in a problem to minimize functions of the form where Cij is the covariance matrix of the assets and Ri are the expected returns. With the summation convention due to Einstein, this is usually written in either of the more compact forms and the minimization is subject to the constraints of equations (1,2) and indeed any other additional conditions (e.g. sector exposure, upper and lower bounds etc.) The quantity  is a Lagrange multiplier expressing an investor’s risk-return preference, and there are other ways of writing down the problem. See full PDF below.