Abstract
Accounting for the non-normality of asset returns remains one of the main challenges in portfolio optimization. In this paper, we tackle this problem by assessing the risk of the portfolio through the “amount of randomness” conveyed by its returns. We achieve this using an objective function that relies on the exponential of Rényi entropy, an information-theoretic criterion that precisely quantifies the uncertainty embedded in a distribution, accounting for higher-order moments. Compared to Shannon entropy, Rényi entropy features a parameter that can be tuned to play around the notion of uncertainty. A Gram–Charlier expansion shows that it controls the relative contributions of the central (variance) and tail (kurtosis) parts of the distribution in the measure. We further rely on a non-parametric estimator of the exponential Rényi entropy that extends a robust sample-spacings estimator initially designed for Shannon entropy. A portfolio-selection application illustrates that minimizing Rényi entropy yields portfolios that outperform state-of-the-art minimum-variance portfolios in terms of risk-return-turnover trade-off. We also show how Rényi entropy can be used in risk-parity strategies.
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Notes
\(L^p(\varOmega )\) is the space of random variables defined on the support set \(\varOmega \) having finite pth moment.
We are grateful to the authors of the aforementioned paper for discussions about the provided counter-examples.
Minimizing \(H_\alpha ^{\exp }(P)\) or \(H_\alpha (P)\) is equivalent as \(\exp (x)\) is a monotonically increasing function.
When applied to our empirical data in Sect. 5, the Parzen estimator (with Gaussian kernel) achieves a worse risk-adjusted performance than the m-spacings estimator considered here for a wide range of values of the bandwidth parameter.
Specifically, the results in Table 2 for \(m=18\) and \(m=35\) yield a very similar performance. The only changes are in terms of turnover, which is higher for \(m=18\) and nearly identical for \(m=35\).
For completeness, we have checked the results for \(\alpha \in \{0.05,0.1,0.2\}\) as well. The turnover barely decreases compared to \(\alpha =0.3\), and the performance measures remain nearly identical.
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The authors are grateful to Kris Boudt, Victor DeMiguel, Mikael Petitjean and two anonymous reviewers for their comments and suggestions. The authors also thank participants of the Actuarial and Financial Mathematics 2018 Conference, the 35th Annual Conference of the French Finance Association (AFFI) and the 2018 Belgian Financial Research Forum for their feedback. This work was supported by the Fonds de la Recherche Scientifique (F.R.S.-FNRS) [Grant Number FC 17775]
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Lassance, N., Vrins, F. Minimum Rényi entropy portfolios. Ann Oper Res 299, 23–46 (2021). https://doi.org/10.1007/s10479-019-03364-2
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DOI: https://doi.org/10.1007/s10479-019-03364-2