Paper: arXiv 2306.16351
Authors: Viktor Kuzmenko, Anton Malandii, Stan Uryasev
Abstract
The paper explores the concept of the \emph{expectile risk measure} within the framework of the Fundamental Risk Quadrangle (FRQ) theory. According to the FRQ theory, a quadrangle comprises four stochastic functions associated with a random variable: error'', regret’’, risk'', and deviation’’. These functions are interconnected through a stochastic function known as the ``statistic’’. Expectile is a risk measure that, similar to VaR (quantile) and CVaR (superquantile), can be employed in risk management. While quadrangles based on VaR and CVaR statistics are well-established and widely used, the paper focuses on the recently proposed quadrangles based on expectile. The aim of this paper is to rigorously examine the properties of these Expectile Quadrangles, with particular emphasis on a quadrangle that encompasses expectile as both a statistic and a measure of risk.
Complexity vs Empirical Score
- Math Complexity: 7.5/10
- Empirical Rigor: 2.5/10
- Quadrant: Lab Rats — theoretically deep, empirically untested
Why this score: The paper provides a rigorous mathematical exploration of the Expectile Quadrangle within Rockafellar and Uryasev’s Fundamental Risk Quadrangle framework. While strong on theoretical proofs and convex analysis, it lacks empirical backtesting and real-world financial data applications.
Research Flowchart
flowchart TD
A[Research Goal: Rigorously examine Expectile Quadrangles in FRQ Theory] --> B[Input: Stochastic Random Variables & Risk Definitions]
B --> C[Methodology: Connect Error, Regret, Risk, and Deviation via Expectile Statistic]
C --> D[Computational Analysis: Compare Expectile Quadrangle with VaR/Quantile & CVaR/Superquantile]
D --> E[Key Finding: Formulation of Quadrangle with Expectile as both Statistic & Risk Measure]
E --> F[Outcomes: Enhanced Mathematical Framework for Cross-Asset Risk Management]