Haezendonck-Goovaerts capital allocation rules

Abstract

This paper deals with the problem of capital allocation for a peculiar class of risk measures, namely the Haezendonck-Goovaerts (HG) ones (Bellini and Rosazza Gianin, 2008; Goovaerts et al., 2004). To this aim, we generalize the capital allocation rule (CAR) introduced by Xun et al. (2019) for Orlicz risk premia (Haezendonck and Goovaerts, 1982) as well as for HG risk measures, using an approach based on Orlicz quantiles (Bellini and Rosazza Gianin, 2012). We therefore study the properties of different CARs for HG risk measures in the quantile-based setting. Finally, we provide robust versions of the introduced CARs, considering ambiguity both over the probabilistic model and over the Young function, following the scheme of Bellini et al. (2018).

Introduction

In this paper, we focus on the problem of capital allocation in the context of a well known class of risk measures, namely the Haezendonck-Goovaerts (HG) ones. Roughly speaking, a capital allocation problem consists in, given a risk measure ρ and a set of financial positions $\mathcal{X}$, finding a suitable way (that is, satisfying some desirable properties) of sharing the risk capital $\rho (X)$ (interpreted as a buffer against default) among the sub-units of X, for each $X\in \mathcal{X}$.

The capital allocation problem has been investigated for general risk measures with different approaches and broad scopes (see, among others, Denault, 2001; Kalkbrener, 2005; Delbaen, 2000; Dhaene et al., 2012; Centrone and Rosazza Gianin, 2018; Tsanakas, 2009). Formally, following (Delbaen, 2000; Denault, 2001; Kalkbrener, 2005), a capital allocation rule (CAR) for a risk measure $\rho :L^{\infty }\to \mathbb{R}$ is a map $\mathrm{\Lambda }:L^{\infty }\times L^{\infty }\to \mathbb{R}$ such that $\mathrm{\Lambda }(X,X)=\rho (X)$ for every $X\in L^{\infty }$, where $\mathrm{\Lambda }(X,Y)$ is interpreted as the risk contribution of a sub-position X to the risk of the whole position Y.

At the same time, Haezendonck-Goovaerts risk measures have been studied in the last decades both from a mathematical point of view and from an actuarial one (see, among others, Bellini et al., 2018; Bellini and Rosazza Gianin, 2008, Bellini and Rosazza Gianin, 2012; Bellini et al., 2014; Goovaerts et al., 2004; Haezendonck and Goovaerts, 1982). This class of risk measures, based on the so called Orlicz premium introduced by Haezendonck and Goovaerts (1982), has become popular also because it generalizes the well-known Conditional Value at Risk (CVaR).

Among the different methods of capital allocations, the quantile-based approach is the more natural when focusing on HG risk measures since this family of risk measures intrinsically depends on generalized quantiles. Many works on quantile-based capital allocation are present in the literature and face the problem both from a theoretical and from an empirical standpoint. For instance, Kalkbrener (2005) and Tasche (2004) study and derive explicit formulas, by using VaR and CVaR as underlying risk measures, for the popular gradient allocation. In this regard, the reader is also referred to the recent works of Asimit et al. (2019) and Gómez et al. (2021), among others.

A specific capital allocation method, which is “tailored” for Orlicz premia and works beyond the special case of CVaR, has been recently introduced by Xun et al. (2019), generalizing the contribution to shortfall, provided by Overbeck (2000) for CVaR. However, despite its desirable properties, the method works only for $X\in L_{+}^{\infty }$ and depends on the quantile (or VaR) of the aggregated risk $Y\in L_{+}^{\infty }$. Therefore, it is still somehow connected to CVaR and excludes the possibility of allocating capital to positions not representing just a loss, which is instead financially meaningful e. g. for internal purposes, when assessing the performances of various business lines.

Starting from the approach of Xun et al. (2019) and motivated by the fact that HG risk measures are meaningful beyond the case of non-negative random variables, our main goal is to introduce a capital allocation method for HG risk measures which is defined for any pair $(X,Y)\in L^{\infty }\times L^{\infty }$ (not only positive) and overcomes the special case of CVaR, by maintaining some of the properties required for a significant capital allocation rule.

The aim of the paper is, therefore, to generalize the CAR proposed by Xun et al. (2019) for Orlicz premia in two directions. First, inspired by Bellini and Rosazza Gianin (2012), we extend the work of Xun et al. (2019) by providing capital allocation rules both for Orlicz risk premia and for HG risk measures, not only in terms of VaR but also of Orlicz quantiles (defined in Bellini and Rosazza Gianin, 2012) that are more appropriate when the involved Young function is not necessarily linear. We show indeed that such CARs satisfy most of the usually required properties and are also reasonable from a financial point of view. A comparison among the approaches here introduced and two popular capital allocation rules, that is, the gradient method and the Aumann-Shapley one (Centrone and Rosazza Gianin, 2018; Kalkbrener, 2005), is also provided. Since a deep analysis on the gradient approach has been recently provided by Gómez et al. (2021) for higher moment risk measures, corresponding to HG risk measures for power Young functions, we also extend one of their results on Orlicz quantiles to the case of general Young functions.

Second, inspired by robust Orlicz premia and robust HG risk measures recently introduced by Bellini et al. (2018), we provide some extensions of the proposed methods of capital allocation to cover ambiguity over the probabilistic model and over the risk perception of the decision-maker. In particular, we first introduce robust Orlicz quantiles and study their properties, obtaining results similar to the non-robust case. By using robust Orlicz quantiles, we then provide robust versions of the presented methods to account for ambiguity over the probability measure and for ambiguity over the utility/loss function. We find out that the robust versions work well for the quantile-based methods, providing results very close to the non-robust case.

The paper is organized as follows: in Section 2 we briefly recall some known facts about HG risk measures and capital allocation rules; in Section 3 we present the capital allocation methods based on Orlicz quantiles and study their properties. Section 4 is instead devoted to the robust versions.