Estimation risk and optimal combined portfolio strategies

Abstract

The traditional Mean-Variance (MV) framework of Markowitz(1952) has been the foundation of numerous research works for many years, benefiting from its mathematical tractability and intuitive clarity for investors. However, a significant limitation of this framework is its dependence on the mean vector and covariance matrix of asset returns, which are generally unknown and have to be estimated using historical data. The resulting plug-in portfolio, which uses these estimates instead of the true parameter values, often exhibits poor out-of-sample performance due to estimation risk. A considerable amount of research proposes various sophisticated estimators for these two unknown parameters or introduces portfolio constraints and regularizations. In this thesis, however, we focus on an alternative approach to mitigate estimation risk by utilizing combined portfolios and directly optimizing the expected out-of-sample performance. We review the relevant literature and present essential preliminary discussions in Chapter 1. Building on this, we introduce three distinct perspectives in portfolio selection, each aimed at assessing the efficiency of combined portfolios in managing estimation risk. These perspectives guide the detailed examination of research projects presented in the subsequent three chapters of the thesis.

Chapter 2 discusses the Tail Mean-Variance (TMV) portfolio selection with estimation risk. The TMV risk measure has emerged from the actuarial community as a criterion for risk management and portfolio selection, with a focus on extreme losses. The existing literature on portfolio optimization under the TMV criterion relies on the plug-in approach, which introduces estimation risk and leads to significant deterioration in the out-of-sample portfolio performance. To address this issue, we propose a combination of the plug-in and 1/N rules and optimize its expected out-of-sample performance. Our study is based on the Mean-Variance-Standard-deviation (MVS) performance measure, which encompasses the TMV, classical MV, and Mean-Standard-Deviation (MStD) as special cases. The MStD criterion is particularly relevant to mean-risk portfolio selection when risk is assessed using quantile-based risk measures. Our proposed combined portfolio consistently outperforms the plug-in MVS and 1/N portfolios in both simulated and real-world datasets.

Chapter 3 focuses on Environmental, Social, and Governance (ESG) investing with estimation risk taken into account. Recently, there has been a significant increase in the commitment of institutional investors to responsible investment. We explore an ESG constrained framework that integrates the ESG criteria into decision-making processes, aiming to enhance risk-adjusted returns by ensuring that the total ESG score of the portfolio meets a specified target. The optimal ESG portfolio satisfies a three-fund separation. However, similar to the traditional MV portfolio, the practical application of the optimal ESG portfolio often encounters estimation risk. To mitigate estimation risk, we introduce a combined three-fund portfolio comprising components corresponding to the plug-in ESG portfolio, and we derive the optimal combination coefficients under the expected out-of-sample MV utility optimization, incorporating either an inequality or equality constraint on the expected total ESG score of the portfolio. Both simulation and empirical studies indicate that the implementable combined portfolio outperforms the plug-in ESG portfolio.

Chapter 4 introduces a novel Winning Probability Weighted (WPW) framework for constructing combined portfolios from any pair of constituent portfolios. This framework is centered around the concept of winning probability, which evaluates the likelihood that one constituent portfolio will outperform another in terms of out-of-sample returns. To ensure comparability, the constituent portfolios are adjusted to align with their long-term risk profiles. We utilize machine learning techniques that incorporate financial market factors alongside historical asset returns to estimate the winning probabilities, which then taken as the combination coefficients for the combined portfolio. Additionally, we optimize the expected out-of-sample MV utility of the combined portfolio to enhance its performance. Extensive empirical studies demonstrate the superiority of the proposed WPW approach over existing analytical methods in terms of certainty equivalent return across various scenarios.

Finally, Chapter 5 summarizes the thesis and outlines potential directions for further research.