Empirical Issues in Portfolio Selection

2.3.4 Mean-Conditional Value at Risk (M-CVaR)

Whilst the statistical deficiencies of M-VaR continue to draw criticism from scholars, Rockafellar and Uryasev (2000, 2002) have developed the mean-CVaR (M-CVaR) portfolio framework as a viable alternative.¹⁶ The scholarly contributions of Rockafellar and Uryasev (2000, 2002) demonstrate that the convex properties of M-CVaR provide more efficient portfolios than the M-VaR framework. Subsequent studies by Alexander and Baptista (2004) and Topaloglou, Vladimirou and Zenios (2002) also examine M-CVaR and their findings were found to be consistent with Rockafellar and Uryasev (2000, 2002).¹⁷ Other studies by Krokhmal, Uryasev and Zrazhevsky (2002) reveal that portfolio rebalancing methods in a mean-CVaR portfolio setting provide better out-of-sample performance in comparison to alternative portfolio frameworks. Although CVaR is not a standard risk metric in global finance, the literature demonstrates that it is a viable alternative to VaR. In summing up, the discovery of M-CVaR as a meaningful substitute for M-VaR motivates Chapter 5 of this thesis which compares the portfolio decisions of MVA and M-CVaR investors.

2.3.5 Empirical Issues in Portfolio Selection

The development of the portfolio selection literature has also motivated researchers to consider the empirical implementation of these frameworks. Scholars have identified statistical problems which confront investors when optimal portfolio choice is implemented in an empirical setting. The review of literature in this section summarises the statistical problems identified in the literature and the solutions that have been

16 Although Rockafellar and Uryasev (2000, 2002) refer to this risk metric as CVaR, Acerbi (2002) independently developed it under the name of Expected Shortfall (ES).

17 Other studies employ CVaR but digress from a typical portfolio selection framework. For instance, Alexander, Coleman and Li (2006) develop a new function to more efficiently minimise CVaR in a portfolio of derivatives. Rockafellar, Uryasev and Zabarankin (2006) employ CVaR to demonstrate the theoretical limitations of portfolio selection when derivatives such as options are introduced.

developed to transform modern portfolio theory into practice. This strand of literature is important due to the emphasis on empirical portfolio selection in the third empirical study of this thesis.

The literature concerned with the empirical implementation of portfolio selection originates from Brown (1976), Jobson, Korkie and Ratti (1979), Jobson and Korkie (1980, 1981a, 1981b, 1982), Klein and Bawa (1976) and Michaud (1989, 1998). These bodies of work highlight the fact that MVA portfolio optimisation treats input parameters as true parameters which are known with certainty. However, in reality, the MVA input parameters are prone to sampling and estimation error. To address these problems in portfolio optimisation, scholars have proposed various methods to develop more meaningful inputs for portfolio selection. This section reviews the literature which focuses on the development of improved portfolio input parameters and the biases which stem from autocorrelated returns.

The estimate of the expected mean return is probably the most important input in portfolio selection. This is highlighted by Merton (1980) who shows that the estimation risk associated with expected mean returns are high. In another work, Chopra and Ziemba (1993) estimate that the error effects from sample mean returns are a magnitude higher than errors from variance and covariance inputs. In short, the estimation risk associated with expected mean returns contributes more error to portfolio selection than estimation risk of the covariance matrix.

To address estimation risk in expected mean returns, the literature has adopted Bayesian methods to improve portfolio selection inputs in an empirical setting.¹⁸ One of the earliest tools to estimate expected mean returns originate from Jobson, Korkie and Ratti (1979), Jobson and Korkie (1980, 1981a, 1981b, 1982) and Jorion (1985, 1986) who employ the James-Stein (1961) estimator which shrinks the estimated mean returns towards a global mean of asset returns. In an alternative approach, Frost and Savarino (1986) propose another Stein estimator which shrinks the estimated mean returns towards an equal-weighted market portfolio. In a related strand of literature, Black and Litterman (1992) and Polsen and Tew (2000) employ semi-Bayesian and traditional Bayesian methods in conditional asset allocation frameworks. To sum up, these academic contributions and the practitioner based works of Michaud (1998) and Scherer (2002) provide compelling evidence to demonstrate that Bayesian methods in estimating an asset’s first moment improve the posterior estimation of inputs in empirical portfolio selection.

The second portfolio selection parameter which is subject to estimation risk is the covariance matrix. Although Merton (1980) and Chopra and Ziemba (1993) demonstrate that the estimation risk of the covariance matrix is less significant than expected means, Jobson and Korkie (1980) and Michaud (1989) find that the covariance matrix exhibits a high degree of error in the presence of many similarly related securities such as an equities portfolio. As stated in Ledoit and Wolf (2003), the problem of the sample covariance matrix is that it imposes too little structure.

To overcome the statistical deficiencies of the sample covariance matrix, scholars have developed various methods to minimise the estimation risk in the covariance matrix in portfolio selection. One of the earliest methods to estimate a more efficient covariance matrix in portfolio selection originates from Elton and Gruber (1973) who developed the

18 The early work of Stein (1955) demonstrated that the sample mean return is inadmissible under general conditions. To address the issue of estimating efficient sample means, James and Stein (1961) developed a simple Bayesian shrinkage operator which shrinks the estimated mean returns towards a global mean.

Studies by Efron and Morris (1977) and Copas (1983) have shown that the James and Stein (1961) shrinkage procedure provides efficient estimates of sample means.

constant correlation approach which imposes structure on the covariance matrix. This technique reduces sampling error at the cost of specification error. Other studies such as Frost and Savarino (1986) employ a Bayesian estimator which shrinks the variances and covariances towards an identical set of parameters. More recently, Ledoit and Wolf (2003, 2004) propose Stein procedures for the estimation of large sets of similarly related securities such as stocks. Ledoit and Wolf (2003) shrinks the covariance matrix towards a single-factor Sharpe (1963) based matrix while Ledoit and Wolf (2004) shrink the covariance matrix towards a constant correlation model. The conclusion to be drawn from this literature is that Bayesian techniques are the method of choice to more readily estimate the covariance matrix in portfolio selection.