Liu, Hao

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#### Three Essays on Robust Optimization of Efficient Portfolios

2013, Liu, Hao

The mean-variance approach was first proposed by Markowitz (1952), and laid the foundation of the modern portfolio theory. Despite its theoretical appeal, the practical implementation of optimized portfolios is strongly restricted by the fact that the two inputs, the means and the covariance matrix of asset returns, are unknown and have to be estimated by available historical information. Due to the estimation risk inherited from inputs, desired properties of estimated optimal portfolios are dramatically degraded. This problem has been addressed by empirical research and is well known by both practitioners and academics for many years. However, only quite recently, some studies such as Kan & Zhou (2007) and Frahm & Memmel (2010) tried to provide analytical insights into the real-world portfolio choice problems which help us to understand key aspects of the empirical portfolios and to find the possible way to improve the portfolio performance. This dissertation is a collection of three stand-alone papers and contributes to the recent literature by taking some important issues into account such as the investor’s risk preference, non-negativity constraints as well as the presence of structural breaks.

The first chapter analyzes the estimation risk of efficient portfolio selection. We use the concept of certainty equivalent as the basis for a well-defined statistical loss function and a monetary measure to assess estimation risk. For given risk preferences we provide analytical results for different sources of estimation risk such as sample size, dimension of the portfolio choice problem and correlation structure of the return process. Our results show that theoretically sub-optimal portfolio choice strategies turn out to be superior once estimation risk is taken into account. Since estimation risk crucially depends on risk preferences, the choice of the estimator for a given portfolio strategy becomes endogenous depending on sample size, number of assets and properties of the return process. We show that a shrinkage approach accounting for estimation risk is generally superior to simple theoretically subopti- mal strategies. Moreover, focusing on just one source of estimation risk, e.g. risk ￼reduction in covariance estimation, can lead to suboptimal portfolios.

Imposing portfolio constraints is one of the most effective ways to improve plug-in estimates of mean-variance portfolios. Jagannathan & Ma (2003) show that the non-negativity constraint in construction of the global minimum variance portfolio has a shrinkage interpretation and could improve the portfolio performance even if the constraint is wrong in population. The second chapter generalizes the theoretical result of Jagannathan & Ma (2003) to the efficient portfolio case where the investor’s risk preference plays a crucial role in portfolio construction. We show that imposing the non-negativity constraint on efficient portfolios is equivalent to using a modified covariance matrix which depends on asset expected returns and the risk preferences of investors. We conduct a simulation study with realistic inputs to demonstrate the trade-off between the theoretical and empirical losses of the constrained portfolio with respect to the investor’s risk preferences. In addition, different constrained and unconstrained portfolio strategies are compared in both simulation and empirical studies. We find that conservative but unconstrained portfolio strategies proposed by recent studies could outperform constrained portfolios even in the small sample case where the mean and the covariance matrix are estimated with large estimation errors.

The third chapter of the thesis takes possible structural breaks into account and analyzes the estimation risk of different mean-variance portfolio strategies with and without the adding-up constraint. Building upon as idea from Pesaran & Timmermann (2007), we provide an analytical comparison of empirical portfolios estimated by including pre-break data with pure post-break portfolio strategies. It is shown that portfolios incorporating pre-break information can be dominating with respect to their certainty equivalents and the dominance relationship is consistent for dif- ferent risk aversion levels. Although the theoretical result is obtained under the assumption that there is only a unique structural break whose date is known, our approach combining portfolios estimated from pre- and post-break data can be easily generalized to the multiple break case with unknown break points. In addition, un- der the normality assumption, we provide an unbiased way to estimate the difference of certainty equivalents between combined portfolios and pure post-break portfolios which allows us to identify the benefit of using pre-break information in portfolio construction.