De patria Pomponii Melae et in duo priora capita libri I ad And Schottum

The simplicity and the intuitive appeal of the mean-variance approach has attracted significant attention from academia and industry. However, contrary to its theoretical reputation, Markowitz’s classical framework has not been used extensively by practitioners as a tool for optimising a large-scale portfolio, due to its numerous implementation difficulties.

The impracticality is that extreme weights or corner solutions from the mean- variance model may be inconvenient in asset allocation, since the investor can neither assign unrealistic weights to each asset, nor diversify risk by investing different assets. Imposing constraints on portfolio weights could alleviate this problem and enable the portfolio to perform better (Frost and Savarino, 1988; Grupa and Eichhorn, 1998; Grauer and Shen, 2000). Discussions regarding the non-short selling constraints can be found in the literature (Jagannathan and Ma, 2003). Additionally, the sensitivity of portfolio weights (Best and Grauer, 1991; Best and Grauer, 1992; Black and Litterman, 1992; Broadie, 1993) is an annoying problem for practitioners as well, as they have to pay significant amounts of transaction costs to buy and sell stocks with weights dramatically changed. The main reason for these problems is the estimation errors in the inputs of the mean-variance model.The accuracy of the estimation of input data will heavily affect the weights allocated to each asset in the mean-variance optimisation, called ‘estimation-error maximisers’ (Michaud, 1989). Michaud argues that the optimised portfolios tend to overweight (underweight) assets

M i M f M f i Cov R r E R r E( ) ( ( )₂ ) _,     _(2.13) ( ₂, )( ( _M) _f) M M i f E r R Cov R     (2.14) R_f _i(E(r_M)R_f) (2.15)

35 with large (small) expected returns, negative (positive) correlations and small (large) variances. Merton (1980) demonstrates that historical returns are bad proxies for expected returns. He also demonstrates evidence that the estimated variance and covariance from the historical data will be much more accurate than the corresponding expected return estimates. Similarly, Chopra and Ziemba (1993) verify that the impact of estimation errors on the expected returns on portfolio choice dominates that of estimation errors in variances and covariance. Therefore, they suggest paying attention to estimate, ‘less noisy’ expected returns, followed by a good estimation of variance. To address these problems, robust estimates of the input parameters for optimisation problems become an important research issue. It is advisable to use the Bayes-Stein shrinkage estimator (Jorion, 1985) or the Bayesian estimator (Frost and Savarino, 1986) as alternative estimators of expected return to reduce estimation risk and improve out-of-sample portfolio performance. However, except for estimation error, Green and Hollifield (1992) explain that the high correlation among assets result from the dominance of a single factor in the covariance of asset returns triggering the extreme weights. Therefore, it cannot ignore the impact of correlations on portfolio weights. Fabozzi et al. (2008) suggest using a factor model to model covariance and correlations and therefore deal with the issue of highly correlated assets.

Another significant problem is the computational difficulty associated with inputs of the expected returns and the expected variance-covariance structure for all assets in the investment universe. For example, if there were 100 assets, it would be burdensome for a practitioner to compute 4,950 parameters in the covariance matrix. In practice, it is impossible for portfolio managers to estimate reliable returns for all assets. Estimation errors exist when they anticipate an expected return by using a simple average of historical sample returns. In addition, it is widely agreed that financial asset return volatilities and correlations are time-varying, with persistent dynamics. Asset return volatilities become an important ingredient in many applications, such as portfolio optimisation and market risk measurement. The most popular approach to modelling the conditional covariance matrix of returns is the multivariate GARCH class of models. These models include the Vech and Diagonal Vech models (Bollerslev et al., 1988), the BEKK model (Engle and Kroner, 1995), the

36 Constant Correlation model (Bollerslev, 1990), the Factor ARCH model (Engle et al., 1990), and the Dynamic Conditional Correlation or DCC model (Engle and Sheppard, 2001). However, the Vech model and the BEKK model suffer from the curse of large dimensionality, and the Diagonal Vech models, the Constant Correlation model and the Factor ARCH model have cross-equation restrictions on the elements of the covariance matrix (Harris et al., 2007). Other approaches, such as rolling estimators of the sample covariance matrix, exponentially weighted estimators and multivariate stochastic volatility models (Harvey et al., 1994) can also be used to estimate the conditional covariance matrix. In the portfolio optimisation world, a portfolio manager would usually work on a large number of assets to diversify the unsystematic risk; the relationship and the co-movement among those assets will directly affect the performance of the whole portfolio. The choice of volatility models is an art. In addition, from the perspective of investor perception against risk and distribution of asset returns, investors usually prefer a larger profit to a small or negative profit and, obviously, their perception of risk is not symmetric around the mean. The use of variance as a measure of risk becomes a critical weakness of the mean-variance approach. Besides, it is well known that the distribution of asset returns is not normal (Mandelbrot, 1963; Fama, 1965; Müller et al., 1998; Rachev and Mittnik, 2000; Rachev et al., 2008). It is not appropriate to consider only the first and second moment of distribution in portfolio optimisation. Both academics and practitioners focus their attention on meeting the requirement of alternative risk measures for portfolio optimisation, such as the ‘safety first’ strategy (Roy, 1952), semivariance (Markowitz, 1959), lower partial moment (Bawa, 1975), mean absolute deviation (Konno and Yamazaki, 1991), VaR (Jorion, 1997; Ahn et al., 1999; Basak and Shapiro, 2001), and CVaR (Rockafellar and Uryasev, 2000; Rockafellar and Uryasev, 2002). Because of the shortcomings of mean-variance optimisation, several researchers have introduced VaR ( Huisman et al., 1999; Campbell, 2001; Favre and Galeano, 2002) and CVaR (Souza and Gokcan, 2004; Agarwal and Naik, 2004) to extend portfolio optimisation techniques under fat-tailed distributions. Alexander and Baptista (2001, 2002, 2004) thoroughly study the implications of VaR and CVaR constraints on the mean-variance model based

37 on theoretical work. I will provide a detailed introduction of VaR and CVaR risk measures in Section 2.2.