Data-driven robust mean-CVaR portfolio selection under distribution ambiguity

Abstract

In this paper, we present a computationally tractable optimization method for a robust mean-CVaR portfolio selection model under the condition of distribution ambiguity. We develop an extension that allows the model to capture a zero net adjustment via a linear constraint in the mean return, which can be cast as a tractable conic programme. Also, we adopt a nonparametric bootstrap approach to calibrate the levels of ambiguity and show that the portfolio strategies are relatively immune to variations in input values. Finally, we show that the resulting robust portfolio is very well diversified and superior to its non-robust counterpart in terms of portfolio stability, expected returns and turnover. The results of numerical experiments with simulated and real market data shed light on the established behaviour of our distributionally robust optimization model.

Keywords:

• Portfolio selection
• Distributionally robust optimization
• Zero net adjustment
• Bootstrap
• Conic programmes

AMS Subject Classifications:

• G11
• C14
• C52
• C61
• C63

Acknowledgements

The authors wish to thank the reviewers and the editor for providing feedback which helped improve the final version of this paper.

Notes

No potential conflict of interest was reported by the authors.

1 In general, decision-makers are exposed not only to risk (refers to events for which the probabilities of the future outcomes are known) but also to ambiguity (refers to events for which the probabilities of the future outcomes are unknown) when making investment decisions. The distinction between risk and ambiguity was first made by Knight (1921) and latter supported by the empirical experiments of Ellsberg (1961), whose findings have shown that agents are not always able to derive a unique probability distribution over the reference state space. After Ellsberg’s seminal paper, uncertain environment has become better known as ambiguity and the general dislike for it as ambiguity aversion. Ellsberg (1961) argues that most people are ambiguity-averse, that is, they prefer a lottery with known probabilities to a similar lottery with unknown probabilities. In recent years, a rapidly growing literature on ambiguity aversion is emerging; see, among others, Garlappi et al. (2007) and Ma et al. (2008) for optimal portfolio choice, Cao et al. (2005) and Ui (2011) for non-participation or selective participation in markets, and Faria and Correia-da-Silva (2014) for European call option pricing.

2 Unlike classical approaches to decision-making such as von Neumann-Morgenstern paradigm of expected utility maximization that neglect an agent’s preference on the choice among multiple probability models, Gilboa and Schmeidler (1989) provide a system of axioms under which an agent’s preference on the choice of the models can be characterized by the worst-case approach. However, this approach does not distinguish between ambiguity and aversion to ambiguity, and hence is sometimes criticized because it apparently implies extreme ambiguity aversion. A few studies overcoming this issue have been proposed. For example, Klibanoff et al. (2005) provide an axiomatic foundation for the smooth ambiguity model. This model allows us to separate ambiguity from ambiguity attitudes and allows for smooth indifference curves, avoiding the infinite ambiguity aversion implied in the MEU approach. A recent paper by Izhakian (2017) provides an axiomatic foundation for a model of decision-making under ambiguity, where the preference representation is referred to as sign-dependent expected utility under uncertain probabilities (EUUP). However, there is still a debate in the literature about the axiomatic foundations of this line of models (see Epstein 2010, Klibanoff et al. 2012). Because of this, the approach of Gilboa and Schmeidler (1989) is still seen to be the main reference in the literature.

3 Ghirardato et al. (2004) axiomatize a model termed $\alpha$-maxmin expected utility ($\alpha$-MEU) wherein it is possible in a certain sense to distinguish ambiguity attitude from ambiguity. The $\alpha$-MEU model is$$\alpha \underset{\mathrm{P}\in \mathbb{D}}{inf}{\text{E}}_{\mathrm{P}}[U(X)]+(1-\alpha )\underset{\mathrm{P}\in \mathbb{D}}{sup}{\text{E}}_{\mathrm{P}}[U(X)],$$

where $\alpha \in [0,1]$ is a parameter, X is a random payoff, U is a general utility function, and $\mathbb{D}$ is a set of prior probability measures. A key feature of $\alpha$-MEU is that it differentiates the level of ambiguity aversion, specified by $\alpha$, and the level of ambiguity, specified by the range of $\mathbb{D}$. There is more flexibility with $\alpha$-MEU in capturing the ambiguity attitude (parameterized by $\alpha$) of the decision-maker ($\alpha$=1, 0 represents, respectively, extremely ambiguity-averse and extremely ambiguity-loving attitudes). Similar to the $\alpha$-MEU criterion, we can develop a so-called $\alpha$-maxmin mean-CVaR criterion. This paper just considers the extreme case of this criterion when $\alpha =1$. The more general cases are left for our future research.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China [grant number 71721001], [grant number 71471180]; the Natural Science Foundation of Guangdong Province of China [grant number 2014A030312003]; the Hong Kong RGC [grant number 15209614], [grant number 15224215], [grant number 15255416].