Paper: arXiv 2408.03147

Authors: Peng Liu, Andreas Tsanakas, Yunran Wei

Abstract

In this paper, we study the risk sharing problem among multiple agents using Lambda Value-at-Risk as their preference functional, under heterogeneous beliefs, where beliefs are represented by several probability measures. We obtain semi-explicit formulas for the inf-convolution of multiple Lambda Value-at-Risk measures under heterogeneous beliefs and the explicit forms of the corresponding optimal allocations. To show the impact of belief heterogeneity, we consider three cases: homogeneous beliefs, conditional beliefs and absolutely continuous beliefs. For those cases, we find more explicit expressions for the inf-convolution, showing the influence of the relation of the beliefs on the inf-convolution. Moreover, we consider, in a two-agent setting, the inf-convolution of one Lambda Value-at-Risk and a general risk measure, including expected utility, distortion risk measures and Lambda Value-at-Risk as special cases, with differing beliefs. The expression of the inf-convolution and the form of the optimal allocation are obtained. In all above cases we demonstrate that trivial outcomes arise when both belief inconsistency and risk tolerance are high. Finally, we discuss risk sharing for an alternative definition of Lambda Value-at-Risk.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 1.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous theoretical exploration of risk sharing and inf-convolution for Lambda Value-at-Risk under heterogeneous beliefs, backed by sophisticated probabilistic derivations. However, it is entirely theoretical with no empirical backtesting, market data validation, or empirical analysis.

Research Flowchart

  flowchart TD
    A[Research Goal: Analyze Risk Sharing & Inf-Convolution with Lambda VaR under Heterogeneous Beliefs] --> B[Inputs: Multiple Probability Measures & Lambda VaR Risk Functionals]
    
    B --> C[Methodology: Derive Inf-Convolution & Compute Optimal Risk Allocations]
    
    C --> D[Case Analysis: Evaluate Belief Scenarios<br>Homogeneous, Conditional, & Absolutely Continuous]
    C --> E[Two-Agent Extension: Combine Lambda VaR with General Risk Measures]
    
    D --> F[Key Outcomes: Semi-explicit & Explicit Formulas for Optimal Allocations]
    E --> F
    
    F --> G[Core Finding: High Belief Inconsistency + High Risk Tolerance Yields Trivial Outcomes]