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Efficient portfolios and extreme risks: a Pareto–Dirichlet approach

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Abstract

This paper solves the mean variance skewness kurtosis (MVSK) portfolio optimization problem by introducing a general Pareto–Dirichlet method. We approximate the feasible portfolio set with a calibrated Dirichlet distribution, where a portfolio is MVSK efficient if its profile in terms of the first four moments is not dominated by any other portfolio. Compared to existing higher order portfolio optimization methods, the Pareto–Dirichlet approach cannot misclassify inefficient portfolios as efficient and produces the efficient set in a very quick way. Coupling the Pareto–Dirichlet approach with a new criterion that generalizes the Sharpe ratio, we are able to produce optimal portfolios in a quick way also. We illustrate our approach with Fama-French 30 Industry Portfolios, where we show that the optimal portfolios derived with our method are preferred to those derived with other optimization schemes by all tested classic performance measures.

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Notes

  1. Another avenue to extend the classic MV optimization is about mean semi-variance optimization programs. See Markowitz (1959), Markowitz et al. (1993), Ogryczak and Ruszczyński (1999), Mansini and Speranza (1999), and Estrada (2008), among others.

  2. Other related studies include Sun and Yan (2003), Ryoo (2007), Mencía and Sentana (2009), Kerstens et al. (2011a), Adcock (2014), Chen et al. (2017), and Landsman et al. (2020), for example.

  3. Briec and Kerstens (2010) and Kerstens et al. (2011a) further illustrate the application of the shortage function method in the MVS portfolio selection problem.

  4. See Jurczenko et al. (2006), Lai et al. (2006), Jurczenko and Maillet (2012), and Dokov et al. (2017).

  5. The details of industry portfolios are accessible via the following link: https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/Data_Library/det_30_ind_port.html.

  6. While market implied equilibrium returns are not “true” returns, they are less subject to noise than raw returns. The adjustment shown here is simple and we use it mainly for illustrative purposes. Because adjusted return moment profiles are inputs to the Pareto–Dirichlet approach, investors can conveniently adopt any standard moment adjustment method and combine it with the Pareto–Dirichlet approach.

  7. Specifically, we follow Ingersoll et al. (2007, Equation 20) to calculate market implied risk aversion as follows:

    $$\begin{aligned} \lambda =\frac{\text {ln}(E[1+r_{mkt}+r_f])-\text {ln}(E[1+r_f])}{\text {var(ln}(1+r_{mkt}+r_f))}. \end{aligned}$$

    We see that the market implied risk aversion is negative as long as the market factor has a negative average return.

  8. More discussions of the input uncertainty for portfolio selection are provided by Harvey et al. (2008) and Michaud and Michaud (2008), among others.

  9. It takes less than one minute to produce the figure from scratch in Matlab on a 64 bits personal laptop with an i7 processor and 16GB of RAM.

  10. When employed for efficient frontier construction, the shortage function method has to be loaded with a reasonably large set of evaluated portfolios. Assume, for instance, that each dimension of the MVSK space is discretized into fifty points, thus there are \(50^4=6.25\) million starting points over the investment universe. Consider that running the shortage function optimization program for each of these starting points takes one second. The total computation time would be 1,736 h, and a more refined discretization design like 100 points per dimension requires skyrocketing computation time.

  11. An alternative formulation that could be explored in future studies can be

    $$\begin{aligned} \text {GSR}=\frac{\mu + b \cdot s }{a \cdot \sigma + c \cdot \kappa }, \end{aligned}$$

    where the preferred odd moments are at the numerator and the disliked even moments are at the denominator.

  12. Using excess asset returns or raw asset returns makes trivial difference to variance-covariance matrix and to other cross-moment matrices. Investors can easily adapt the line from the point of riskfree rate through the maximum GSR portfolio constructed with raw returns.

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  1. Emlyon Business School, 23, Avenue Guy de Collongue, 69134, Ecully Cedex, France

    Olivier Le Courtois

  2. ESSCA School of Management, 4, Pont Pasteur, 69007, Lyon, France

    Xia Xu

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  1. Olivier Le Courtois
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Le Courtois, O., Xu, X. Efficient portfolios and extreme risks: a Pareto–Dirichlet approach. Ann Oper Res 335, 261–292 (2024). https://doi.org/10.1007/s10479-023-05507-y

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