Statistically Efficient Construction of α-Risk-Minimizing Portfolio

We propose a semiparametrically efficient estimator for α-risk-minimizing portfolio weights. Based on the work of Bassett et al. (2004), an α-risk-minimizing portfolio optimization is formulated as a linear quantile regression problem. The quantile regression method uses a pseudolikelihood based on an asymmetric Laplace reference density, and asymptotic properties such as consistency and asymptotic normality are obtained. We apply the results of Hallin et al. (2008) to the problem of constructing α-risk-minimizing portfolios using residual signs and ranks and a general reference density. Monte Carlo simulations assess the performance of the proposed method. Empirical applications are also investigated. 1. Introduction Since the first formation of Markowitz’s mean-variance model, portfolio optimization and construction have been a critical part of asset and fund management. At the same time, portfolio risk assessment has become an essential tool in risk management. Yet there are well-known shortcomings of variance as a risk measure for the purposes of portfolio optimization; namely, variance is a good risk measure only for elliptical and symmetric return distributions. The proper mathematical characterization of risk is of central importance in finance. The choice of an adequate risk measure is a complex task that, in principle, involves deep consideration of the attitudes of market players and the structure of markets. Recently, value at risk (VaR) has gained widespread use, in practice as well as in regulation. VaR has been criticized, however, because as a quantile is no reason to be convex, and indeed, it is easy to construct portfolios for which VaR seriously violates convexity. The shortcomings of VaR led to the introduction of coherent risk measures. Artzner et al. [1] and F？llmer and Schied [2] question whether VaR qualifies as such a measure, and both find that VaR is not an adequate measure of risk. Unlike VaR, expected shortfall (or tail VaR), which is defined as the expected portfolio tail return, has been shown to have all necessary characteristics of a coherent risk measure. In this paper, we use -risk as a risk measure that satisfies the conditions of coherent risk measure (see [3]). Variants of the -risk measure include expected shortfall and tail VaR. The -risk-minimizing portfolio, introduced as a pessimistic portfolio in Bassett et al. [3], can be formulated as a problem of linear quantile regression. Since the seminal work by Koenker and Bassett [4], quantile regression (QR) has become more widely used to describe the conditional