Efficient portfolios and extreme risks: a Pareto-Dirichlet approach (Q6546994)

Classic portfolio theory, as established by Markowitz, defines risk primarily as variance. However, recent economic crises have led academics and practitioners to question this narrow definition, suggesting that higher-order moments such as skewness and kurtosis should be considered in risk assessments. Studies by \textit{R. C. Scott} and \textit{P. A. Horvath} [J. Finance 35, No. 4, 915--919 (1980; \url{doi:10.2307/2327209})] and \textit{C. R. Harvey} and \textit{A. Siddique} [J. Finance 55, No. 3, 1263--1295 (2000; \url{doi:10.1111/0022-1082.00247})] emphasize that investors have preferences for skewness, which can explain variations in stock returns. This has prompted researchers to augment the traditional mean-variance (MV) optimization with higher-order moments, resulting in approaches like mean-variance-skewness-kurtosis (MVSK) optimization.\N\NThe authors develop a Pareto-Dirichlet structure to generate the MVSK efficient set. They describe MVSK efficient portfolios as Pareto efficient within the feasible portfolio space. In particular, a portfolio is considered MVSK inefficient if another portfolio demonstrates a superior profile across the first four moments. This method addresses moment estimation errors and integrates techniques like the \textit{F. Black} and \textit{A. B. Litterman} [J. Fixed Income 1, No. 2, 7--18 (1991; \url{doi:10.3905/jfi.1991.408013})] market-implied equilibrium returns or the Bayesian framework of \textit{C. R. Harvey} et al. [Quant. Finance 10, No. 5, 469--485 (2010; Zbl 1195.91181)] for adjusting portfolio return moments.\N\NWith the MVSK efficient set in place, optimal portfolios can be established based on individual preferences. To achieve this, the authors introduce a generalized Sharpe ratio. By using indifference results, they calibrate the parameters of this ratio and demonstrate how to derive optimal portfolios. Their findings show that this method, which leverages Pareto-Dirichlet simulations in conjunction with the generalized Sharpe ratio, generates optimal portfolios that outperform other optimization methods, as measured by all traditional performance indicators. The work concludes with further possible future directions of research.