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INTRODUCTION
In the framework of modern portfolio theory (Markowitz, 1952), a portfolio of assets is characterised by a desired property, its 'reward' and something undesirable, its 'risk'. Markowitz identified these two properties with the expectation and the variance of returns, respectively, hence the expression mean-variance optimisation (MVO). Though undoubtedly the cornerstone of portfolio selection, MVO has been criticised over the years for both theoretical and practical reasons. From the financial economist's perspective, investors should maximise some utility function of future wealth. As the latter is a function of the portfolio returns, this approach incorporates all relevant parts of the whole return distribution, not just the first two moments. As, from a more practical viewpoint, this creates an additional problem (how does the utility function look like?), many academic writers and even more so practitioners agree that utility optimisation is not the goal. The choice criterion should, however, take in account the empirical characteristics of actual return series like asymmetry or fat tails. In essence then, variance does not appear to be a good measure of 'risk'. (Taking the mean for the 'reward' is rarely criticised.) In recent years, a large number of alternative specifications has been suggested (see, for instance, Rachev et al (2005)). This development has been driven by increased theoretical interest in the properties of risk measures (Pedersen and Satchell, 1998, 2002; Artzner et al , 1999) and the growth of alternative instruments (hedge funds, derivatives) with strongly non-symmetric distributions. Unfortunately, there exists little research so far that investigates the empirical performance of these new risk and performance measures when applied to portfolio optimisation; the few existing studies (for example, Biglova et al (2004); Farinelli et al (2008)) are usually based on only a small number of assets (often indices).
When applied naïvely to historical data, MVO often leads to 'overfit' portfolios with poor out-of-sample performance (Cohen and Pogue, 1967; Jobson and Korkie, 1980; Michaud, 1989; Best and Grauer, 1991). Though this is not a problem of MVO per se , but rather arises as a result of often highly correlated data, it certainly presents a drawback for empirical applications. Different procedures have been suggested to mitigate this overfitting, usually resulting in a constraining of either input parameters or obtained portfolio weights...