# Fair Estimation of Capital Risk Allocation

## TL;DR

This paper introduces a new methodology for estimating risk capital allocations based on law-invariant coherent risk measures, focusing on expected shortfall, with explicit formulas, estimators, and backtesting methods, supported by numerical analysis.

## Contribution

It develops explicit formulas and estimators for fair capital allocations under normality and nonparametric settings, advancing risk measure-based allocation estimation methods.

## Key findings

- Explicit formulas for fair allocations under normality.
- Proposed estimators are proven to be fair and asymptotically fair.
- Backtesting methods effectively assess allocation estimation performance.

## Abstract

In this paper we develop a novel methodology for estimation of risk capital allocation. The methodology is rooted in the theory of risk measures. We work within a general, but tractable class of law-invariant coherent risk measures, with a particular focus on expected shortfall. We introduce the concept of fair capital allocations and provide explicit formulae for fair capital allocations in case when the constituents of the risky portfolio are jointly normally distributed. The main focus of the paper is on the problem of approximating fair portfolio allocations in the case of not fully known law of the portfolio constituents. We define and study the concepts of fair allocation estimators and asymptotically fair allocation estimators. A substantial part of our study is devoted to the problem of estimating fair risk allocations for expected shortfall. We study this problem under normality as well as in a nonparametric setup. We derive several estimators, and prove their fairness and/or asymptotic fairness. Last, but not least, we propose two backtesting methodologies that are oriented at assessing the performance of the allocation estimation procedure. The paper closes with a substantial numerical study of the subject.