The theoretical portfolio concepts outlined in the previous section are unfortunately not directly applicable in practice. So far the population moments have been em- ployed in the analysis, but these entities are unknown. In empirical applications these unknown parameters must be replaced by estimates. The locus of the set for the fea- sible portfolios is below the efficient frontier. At first glance, the sample mean and the unbiased estimator of the variance-covariance matrix of the returns seem to be appropriate candidates for replacing the population moments. In practice, however, potential estimation errors exert a direct impact on the portfolio weights such that, for instance, the desired properties of an efficient and/or a minimum variance port- folio are no longer valid, in general. Ultimately, estimation errors are mirrored by a higher portfolio risk compared to the case of population moments. This is regardless of whether the estimates have been generated from historic data orex anteforecasts for these parameters are employed. For portfolio optimizations that are based on esti- mators for the expected returns and the variance-covariance matrix, these should have a greater effect compared to optimization approaches that only rest on estimates for the return dispersion,ceteris paribus(see Chopra and Ziemba 1993; Merton 1980).
Hence, mean-variance portfolio optimizations should suffer more severely from esti- mation error than minimum variance ones. In empirical simulations and studies it was found that the weights of the former kind of portfolio optimizations are characterized
k k by wide spans and erratic behavior over time (see, for example, DeMiguel et al. 2007;
Frahm 2008; Jagannathan and Ma 2003; Kempf and Memmel 2006; Ledoit and Wolf 2003). From a normative point of view both characteristics are undesired. The effect of “haphazardly” behaving weights is ameliorated for minimum variance portfolios, and hence this portfolio design is to be preferred with respect to the potential im- pact of estimation errors. The sensitivity of the optimal solutions for mean-variance portfolios with respect to the utilized expected returns is per se not a flaw of the ap- proach proposed by Markowitz, but rather an artifact of the quadratic optimization for deriving the portfolio weights.
The errors of the estimates for the expected returns and the variance-covariance matrix could be quantified heuristically beforehand by means of Monte Carlo simu- lations. This portfolio resampling was proposed by Michaud (1998), and a detailed description with a critique is given in Scherer (2002) and Scherer (2010, Chapter 4).
In a first step, the estimates ̂𝜇0andΣ̂0for the theoretical moments𝜇andΣfor given sample sizeTare calculated andmpoints on the empirical efficient frontier are com- puted. Next,Krandom samples of dimension(T×N)are generated and from these the sample moments are determined, giving in totalKpairs(̂𝜇i, ̂Σi),i=1,…,K. Each of these pairs is then used to computempoints on the respective efficient frontiers.
The locus of these simulated efficient frontiers is below the efficient frontier for ̂𝜇0
andΣ̂0. To assess the impact of the estimation error with respect to the mean, the above procedure is repeated, but now the random pairs(̂𝜇i, ̂Σ0),i=1,…,K, are used.
That is, the estimation error is confined to the expected returns. Likewise, the impact of the estimation error for the return dispersion can be evaluated by generating ran- dom pairs(̂𝜇0, ̂Σi),i=1,…,K. To conclude the exposition of portfolio resampling, it should be noted that randomized efficient portfolios can be retrieved by averaging the weights overmandK. This approach should not be viewed as a panacea for cop- ing with estimation errors. The main critique against this approach is the propagation of errors. The initial values(̂𝜇0, ̂Σ0)from which the random samples are generated are estimates themselves and are hence contaminated by estimation errors. Therefore, the initial errors are replicated and mirrored by applying the Monte Carlo analysis.
Furthermore, for the case of constrained portfolio optimizations the above procedure can yield unintuitive results (see Scherer 2010, Chapter 4).
Further, recall from Chapter 3 that asset returns are in general not multivariate normally distributed. This implies that, in addition to estimation errors, a model er- ror often exists. A non-stationary return process would be modelled according to a distribution for stationary processes. Hence, there is a trade-off between using a dis- tribution assumption for stationary processes on the one hand, thereby committing a model error, and using a longer sample span by which the stationarity assumption is more likely to be violated but the estimation error diminishes.
The above-stated consequences of estimation errors could in principle be amelio- rated beforehand by imposing restrictions on the weights. The following example elucidates the effect of placing constraints on the portfolio weights. Consider two independent investment opportunitiesAandBwith an expected return of 3% and a volatility of 10%. A return-maximizing agent would be indifferent to all linear com- binations between these two assets. However, an estimation error as high as one basis
k k point would result in a very different outcome. Suppose that the estimates for the
expected returns are ̂𝜇A=3.01 and ̂𝜇B=2.99, respectively. This would imply an infinitely high long position in assetAthat is financed by an equal-sized short po- sition in assetB. Matters are rather different if long-only and/or bound constraints are included in the optimization. It was found that these kinds of restrictions yield a favorable out-of-sample performance (see, for instance, Frost and Savarino 1988) or are associated with a reduced portfolio risk (see, for instance, Gupta and Eichhorn 1998; Jagannathan and Ma 2003). Both of these empirical findings can be traced back to a smaller implied estimation error if restrictions are imposed on the weights.
It is worth mentioning that the locus of portfolios in the(𝜇, 𝜎)plane are inferior to efficient portfolios to a greater degree as the restrictions become more binding. How- ever, in general an investor is eager to achieve a portfolio allocation that comes as close as possible to the efficient frontier. But the implementation of long-only con- straints is undesirable for other reasons. For instance, the implementation of most of the hedge-fund type strategies requires short positioning to be allowed. In summary, the imposition of constraints is not a panacea for all kinds of portfolio strategies and optimizations. Part III of this book will address these issues in more detail and also offer examples of how these more recent advances in portfolio construction can be explored withR.
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Part II