Regularizing Portfolio Optimization

The optimization of large portfolios displays an inherent instability to estimation error. This poses a fundamental problem, because solutions that are not stable under sample fluctuations may look optimal for a given sample, but are, in effect, very far from optimal with respect to the average risk. In this paper, we approach the problem from the point of view of statistical learning theory. The occurrence of the instability is intimately related to over-fitting which can be avoided using known regularization methods. We show how regularized portfolio optimization with the expected shortfall as a risk measure is related to support vector regression. The budget constraint dictates a modification. We present the resulting optimization problem and discuss the solution. The L2 norm of the weight vector is used as a regularizer, which corresponds to a diversification "pressure". This means that diversification, besides counteracting downward fluctuations in some assets by upward fluctuations in others, is also crucial because it improves the stability of the solution. The approach we provide here allows for the simultaneous treatment of optimization and diversification in one framework that enables the investor to trade-off between the two, depending on the size of the available data set

Large-scale volatility models: theoretical properties of professionals' practice

This article examines the way in which GARCH models are estimated and used for forecasting by practitioners in particular using the highly popular Riskmetrics-super-TM approach. Although it permits sizable computational gains and provide a simple way to impose positive semi-definitiveness of multivariate version of the model, we show that this approach delivers non-consistent parameter' estimates. The novel theoretical result is corroborated by a set of Monte Carlo exercises. A set of empirical applications suggest that this could cause, in general, unreliable forecasts of conditional volatilities and correlations. Copyright 2008 The Author

A new Fourier transform algorithm for value-at-risk

In this paper, we introduce a new Fourier method for computing value-at-risk for a portfolio with derivatives and for return models with fat tails. The new method does not assume that the characteristic function for the return model is known explicitly. We define a class of admissible models for returns and present statistical evidence that supports our approach. We discuss the details of the algorithm. The paper concludes with two applications of value-at-risk. Both examples illustrate the effect that changes in the models for portfolio value and for risk factor returns have on the value-at-risk surface