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Minimum Rényi entropy portfolios

  • Nathan LassanceEmail author
  • Frédéric Vrins
S.I.: Recent Developments in Financial Modeling and Risk Management
  • 4 Downloads

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.

Keywords

Portfolio selection Shannon entropy Rényi entropy Risk measure Information theory Higher-order moments Risk parity 

Notes

Supplementary material

10479_2019_3364_MOESM1_ESM.pdf (274 kb)
Supplementary material 1 (pdf 273 KB)

References

  1. Abbas, A. (2006). Maximum entropy utility. Operations Research, 54(2), 277–290.Google Scholar
  2. Adcock, C. (2014). Mean-variance-skewness efficient surfaces, Stein’s lemma and the multivariate extended skew-student distribution. European Journal of Operational Research, 234(2), 392–401.Google Scholar
  3. Ardia, D., Bolliger, G., Boudt, K., & Gagnon-Fleury, J. (2017). The impact of covariance misspecification in risk-based portfolios. Annals of Operations Research, 254(1–2), 1–16.Google Scholar
  4. Artzner, P., Delbaen, F., Eber, J., & Heath, D. (1999). Coherent measures of risk. Mathematical Finance, 9(3), 203–228.Google Scholar
  5. Behr, P., Guettler, A., & Miebs, F. (2013). On portfolio optimization: Imposing the right constraints. Journal of Banking and Finance, 37, 1232–1242.Google Scholar
  6. Beirlant, J., Dudewicz, E., Gyofi, L., & van der Meulen, E. (1997). Non-parametric entropy estimation: An overview. International Journal of Mathematical and Statistical Sciences, 6(1), 17–39.Google Scholar
  7. Bera, A., & Park, S. (2008). Optimal portfolio diversification using the maximum entropy principle. Econometric Reviews, 27(4–6), 484–512.Google Scholar
  8. Boudt, K., Peterson, B., & Croux, C. (2008). Estimation and decomposition of downside risk for portfolios with non-normal returns. Journal of Risk, 11, 79–103.Google Scholar
  9. Campbell, L. (1966). Exponential entropy as a measure of extent of a distribution. Z Wahrsch, 5, 217–225.Google Scholar
  10. Carroll, R., Conlon, T., Cotter, J., & Salvador, E. (2017). Asset allocation with correlation: A composite trade-off. European Journal of Operational Research, 262(3), 1164–1180.Google Scholar
  11. Chen, L., He, S., & Zhang, S. (2011). When all risk-adjusted performance measures are the same: In praise of the Sharpe ratio. Quantitative Finance, 11(10), 1439–1447.Google Scholar
  12. Cont, R. (2001). Empirical properties of asset returns: Stylized facts and statistical issues. Quantitative Finance, 1, 223–236.Google Scholar
  13. Cover, T., & Thomas, J. (2006). Elements of information theory (2nd ed.). New Jersey: Wiley.Google Scholar
  14. Daníelsson, J., Jorgensenb, B., Samorodnitskyc, G., Sarma, M., & de Vries, C. (2013). Fat tails, VaR and subadditivity. Journal of Econometrics, 172, 283–291.Google Scholar
  15. DeMiguel, V., Garlappi, L., Nogales, F., & Uppal, R. (2009a). A generalized approach to portfolio optimization: Improving performance by constraining portfolio norms. Management Science, 55(5), 798–812.Google Scholar
  16. DeMiguel, V., Garlappi, L., & Uppal, R. (2009b). Optimal versus naive diversification: How inefficient is the 1/N portfolio strategy? The Review of Financial Studies, 22(5), 1915–1953.Google Scholar
  17. DeMiguel, V., & Nogales, F. (2009). Portfolio selection with robust estimation. Operations Research, 57, 560–577.Google Scholar
  18. Dionisio, A., Menezes, R., & Mendes, A. (2006). An econophysics approach to analyse uncertainty in financial markets: An application to the Portuguese stock market. The European Physical Journal B, 50, 161–164.Google Scholar
  19. Fabozzi, F., Huang, D., & Zhou, G. (2010). Robust portfolios: Contributions from operations research and finance. Annals of Operations Research, 176(1), 191–220.Google Scholar
  20. Favre, L., & Galeano, J. (2002). Mean-modified value-at-risk optimization with hedge funds. Journal of Alternative Investments, 5(2), 21–25.Google Scholar
  21. Flores, Y., Bianchi, R., Drew, M., & Trück, S. (2017). The diversification delta: A different perspective. Journal of Portfolio Management, 43(4), 112–124.Google Scholar
  22. Hampel, F., Ronchetti, E., Rousseeuw, P., & Stahel, W. (1986). Robust statistics: The approach based on influence functions. New York: Wiley.Google Scholar
  23. Harvey, C. R., Liechty, J. C., Liechty, M. W., & Müller, P. (2010). Portfolio selection with higher moments. Quantitative Finance, 10(5), 469–485.Google Scholar
  24. Hegde, A., Lan, T., & Erdogmus, D. (2005). Order statistics based estimator for Rényi entropy. In IEEE workshop on machine learning for signal processing (pp. 335–339).Google Scholar
  25. Hyvärinen, A., Karhunen, J., & Oja, E. (2001). Independent component analysis. New York: Wiley.Google Scholar
  26. Johnson, O., & Vignat, C. (2007). Some results concerning maximum Rényi entropy distributions. Annales de l’Institut Henri-Poincaré (B) Probab. Statist, 43(3), 339–351.Google Scholar
  27. Jorion, P. (1986). Bayes–Stein estimation for portfolio analysis. Journal of Financial and Quantitative Analysis, 21(3), 279–292.Google Scholar
  28. Jose, V., Nau, R., & Winkler, R. (2008). Scoring rules, generalized entropy, and utility maximization. Operations Research, 56(5), 1146–1157.Google Scholar
  29. Jurczenko, E., & Maillet, B. (2006). Multi-moment asset allocation and pricing models. West Sussex: Wiley.Google Scholar
  30. Kolm, P., Tütüncü, R., & Fabozzi, F. (2014). 60 years of portfolio optimization: Practical challenges and current trends. European Journal of Operational Research, 234(2), 356–371.Google Scholar
  31. Koski, T., & Persson, L. (1992). Some properties of generalized exponential entropies with application to data compression. Information Sciences, 62, 103–132.Google Scholar
  32. Learned-Miller, E., & Fisher, J. (2003). ICA using spacings estimates of entropy. Journal of Machine Learning Research, 4, 1271–1295.Google Scholar
  33. Ledoit, O., & Wolf, M. (2003). Improved estimation of the covariance matrix of stock returns with an application to portfolio selection. Journal of Empirical Finance, 10, 603–621.Google Scholar
  34. Ledoit, O., & Wolf, M. (2004a). A well-conditioned estimator for large-dimensional covariance matrices. Journal of Multivariate Analysis, 88, 365–411.Google Scholar
  35. Ledoit, O., & Wolf, M. (2004b). Honey, I shrunk the sample covariance matrix. Journal of Portfolio Management, 30, 110–119.Google Scholar
  36. Levy, H., & Levy, M. (2014). The benefits of differential variance-based constraints in portfolio optimization. European Journal of Operational Research, 234, 372–381.Google Scholar
  37. Li, D., & Ng, W. L. (2000). Optimal dynamic portfolio selection: Multiperiod mean-variance formulation. Mathematical Finance, 10(3), 387–406. Google Scholar
  38. Maillard, S., Roncalli, T., & Teiletche, J. (2010). On the properties of equally-weighted risk contributions portfolios. Journal of Portfolio Management, 36(4), 60–70.Google Scholar
  39. Markowitz, H. (1952). Portfolio selection. The Journal of Finance, 7(1), 77–91.Google Scholar
  40. Martellini, L., & Ziemann, V. (2010). Improved estimates of higher-order comoments and implications for portfolio selection. Review of Financial Studies, 23(4), 1467–1502.Google Scholar
  41. Ormos, M., & Zibriczky, D. (2014). Entropy-based financial asset pricing. Plos One, 9(12), e115742.Google Scholar
  42. Pézier, J. (2004). Risk and risk aversion. In C. Alexander & E. Sheedy (Eds.), The professional risk managers’ handbook. Wilmington: PRMIA Publications.Google Scholar
  43. Pham, D., Vrins, F., & Verleysen, M. (2008). On the risk of using Rényi’s entropy for blind source separation. IEEE Transactions on Signal Processing, 56(10), 4611–4620.Google Scholar
  44. Philippatos, G., & Wilson, C. (1972). Entropy, market risk, and the selection of efficient portfolios. Applied Economics, 4(3), 209–220.Google Scholar
  45. Pun, C. S. (2018). Time-consistent mean-variance portfolio selection with only risky assets. Economic Modelling, 75, 281–292.Google Scholar
  46. Qi, Y., Steuer, R., & Wimmer, M. (2017). An analytical derivation of the efficient surface in portfolio selection with three criteria. Annals of Operations Research, 251(1–2), 161–177.Google Scholar
  47. Rényi, A. (1961). On measures of entropy and information. In Fourth Berkeley symposium on mathematical statistics and probability (pp. 547–561).Google Scholar
  48. Rockafellar, R., Uryasev, S., & Zabarankin, M. (2006). Generalized deviations in risk analysis. Finance and Stochastics, 10, 51–74.Google Scholar
  49. Sbuelz, A., & Trojani, F. (2008). Asset prices with locally constrained-entropy recursive multiple-priors utility. Journal of Economic Dynamics and Control, 32(11), 3695–3717.Google Scholar
  50. Scutellà, M., & Recchia, R. (2013). Robust portfolio asset allocation and risk measures. Annals of Operations Research, 204(1), 145–169.Google Scholar
  51. Shannon, C. (1948). A mathematical theory of communication. Bell Systems Technical Journal, 27, 379–423.Google Scholar
  52. Tasche, D. (2008). Capital allocation to business units and sub-portfolios: The Euler principle. Pillar II in the new basel accord: The challenge of economic capital. Risk Books. In Resti, A (pp. 423–453).Google Scholar
  53. Ugray, Z., Lasdon, L., Plummer, J., Glover, F., Kelly, J., & Martí, R. (2007). Scatter search and local NLP solvers: A multistart framework for global optimization. INFORMS Journal on Computing, 19(3), 328–340.Google Scholar
  54. van Es, B. (1992). Estimating functionals related to a density by a class of statistics based on spacings. Scandinavian Journal of Statistics, 19(1), 61–72.Google Scholar
  55. Vanduffel, S., & Yao, J. (2017). A stein type lemma for the multivariate generalized hyperbolic distribution. European Journal of Operational Research, 261(2), 606–612.Google Scholar
  56. Vasicek, O. (1976). A test for normality based on entropy. Journal of the Royal Statistical Society Series B (Methodological), 38(1), 54–59.Google Scholar
  57. Vermorken, M., Medda, F., & Schroder, T. (2012). The diversification delta: A higher-moment measure for portfolio diversification. Journal of Portfolio Management, 39(1), 67–74.Google Scholar
  58. Vrins, F., Pham, D., & Verleysen, M. (2007). Mixing and non-mixing local minima of the entropy contrast for blind source separation. IEEE Transactions on Information Theory, 53(3), 1030–1042.Google Scholar
  59. Wachowiak, M., Smolikova, R., Tourassi, G., & Elmaghraby, A. (2005). Estimation of generalized entropies with sample spacing. Pattern Analysis and Applications, 8, 95–101.Google Scholar
  60. Yang, J., & Qiu, W. (2005). A measure of risk and a decision-making model based on expected utility and entropy. European Journal of Operational Research, 164(3), 792–799.Google Scholar
  61. Zhou, R., Cai, R., & Tong, G. (2013). Applications of entropy in finance: A review. Entropy, 15, 4909–4931.Google Scholar
  62. Zografos, K., & Nadarajah, S. (2003). Formulas for Rényi information and related measures for univariate distributions. Information Sciences, 155(1–2), 119–138. Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Louvain FinanceUCLouvain (BE)Louvain-la-NeuveBelgium
  2. 2.Louvain Finance, Center for Operations Research and EconometricsUCLouvain (BE)Louvain-la-NeuveBelgium

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