Expectiles

Diversification quotient based on expectiles ArXiv ID: 2411.14646 “View on arXiv” Authors: Unknown Abstract A diversification quotient (DQ) quantifies diversification in stochastic portfolio models based on a family of risk measures. We study DQ based on expectiles, offering a useful alternative to conventional risk measures such as Value-at-Risk (VaR) and Expected Shortfall (ES). The expectile-based DQ admits simple formulas and has a natural connection to the Omega ratio. Moreover, the expectile-based DQ is not affected by small-sample issues faced by VaR-based or ES-based DQ due to the scarcity of tail data. The expectile-based DQ exhibits pseudo-convexity in portfolio weights, allowing gradient descent algorithms for portfolio selection. We show that the corresponding optimization problem can be efficiently solved using linear programming techniques in real-data applications. Explicit formulas for DQ based on expectiles are also derived for elliptical and multivariate regularly varying distribution models. Our findings enhance the understanding of the DQ’s role in financial risk management and highlight its potential to improve portfolio construction strategies. ...

Higher order measures of risk and stochastic dominance ArXiv ID: 2402.15387 “View on arXiv” Authors: Unknown Abstract Higher order risk measures are stochastic optimization problems by design, and for this reason they enjoy valuable properties in optimization under uncertainties. They nicely integrate with stochastic optimization problems, as has been observed by the intriguing concept of the risk quadrangles, for example. Stochastic dominance is a binary relation for random variables to compare random outcomes. It is demonstrated that the concepts of higher order risk measures and stochastic dominance are equivalent, they can be employed to characterize the other. The paper explores these relations and connects stochastic orders, higher order risk measures and the risk quadrangle. Expectiles are employed to exemplify the relations obtained. ...

A return-diversification approach to portfolio selection ArXiv ID: 2312.09707 “View on arXiv” Authors: Unknown Abstract In this paper, we propose a general bi-objective model for portfolio selection, aiming to maximize both a diversification measure and the portfolio expected return. Within this general framework, we focus on maximizing a diversification measure recently proposed by Choueifaty and Coignard for the case of volatility as a risk measure. We first show that the maximum diversification approach is actually equivalent to the Risk Parity approach using volatility under the assumption of equicorrelated assets. Then, we extend the maximum diversification approach formulated for general risk measures. Finally, we provide explicit formulations of our bi-objective model for different risk measures, such as volatility, Mean Absolute Deviation, Conditional Value-at-Risk, and Expectiles, and we present extensive out-of-sample performance results for the portfolios obtained with our model. The empirical analysis, based on five real-world data sets, shows that the return-diversification approach provides portfolios that tend to outperform the strategies based only on a diversification method or on the classical risk-return approach. ...

Optimal Transport Divergences induced by Scoring Functions ArXiv ID: 2311.12183 “View on arXiv” Authors: Unknown Abstract We employ scoring functions, used in statistics for eliciting risk functionals, as cost functions in the Monge-Kantorovich (MK) optimal transport problem. This gives raise to a rich variety of novel asymmetric MK divergences, which subsume the family of Bregman-Wasserstein divergences. We show that for distributions on the real line, the comonotonic coupling is optimal for the majority of the new divergences. Specifically, we derive the optimal coupling of the MK divergences induced by functionals including the mean, generalised quantiles, expectiles, and shortfall measures. Furthermore, we show that while any elicitable law-invariant coherent risk measure gives raise to infinitely many MK divergences, the comonotonic coupling is simultaneously optimal. The novel MK divergences, which can be efficiently calculated, open an array of applications in robust stochastic optimisation. We derive sharp bounds on distortion risk measures under a Bregman-Wasserstein divergence constraint, and solve for cost-efficient payoffs under benchmark constraints. ...