Understanding Operational Risk Capital Approximations: First and Second Orders

Abstract

We set the context for capital approximation within the framework of the Basel II / III regulatory capital accords. This is particularly topical as the Basel III accord is shortly due to take effect. In this regard, we provide a summary of the role of capital adequacy in the new accord, highlighting along the way the significant loss events that have been attributed to the Operational Risk class that was introduced in the Basel II and III accords. Then we provide a semi-tutorial discussion on the modelling aspects of capital estimation under a Loss Distributional Approach (LDA). Our emphasis is to focuss on the important loss processes with regard to those that contribute most to capital, the so called "high consequence, low frequency" loss processes.This leads us to provide a tutorial overview of heavy tailed loss process modelling in OpRisk under Basel III, with discussion on the implications of such tail assumptions for the severity model in an LDA structure. This provides practitioners with a clear understanding of the features that they may wish to consider when developing OpRisk severity models in practice.From this discussion on heavy tailed severity models, we then develop an understanding of the impact such models have on the right tail asymptotics of the compound loss process and we provide detailed presentation of what are known as first and second order tail approximations for the resulting heavy tailed loss process. From this we develop a tutorial on three key families of risk measures and their equivalent second order asymptotic approximations: Value-at-Risk (Basel III industry standard); Expected Shortfall (ES) and the Spectral Risk Measure. These then form the capital approximations.We then provide a few example case studies to illustrate the accuracy of these asymptotic captial approximations, the rate of the convergence of the assymptotic result as a function of the LDA frequency and severity model parameters, the sensitivity of the capital approximation to the model parameters and the sensitivity to model miss-specification.Key words: Basel II/III; Capital Approximation; Loss Distributional Approach; Capital Approximation; Value-at-Risk; Expected Shortfall; Spectral Risk Measure; Subexponential; Regularly Varying. The Changing Landscape of Capital AccordsIn juristictions in which active regulation is applied to the banking sector throughout the world, the modelling of Operational Risk (OpRisk) has progressively taken a prominent place in financial quantitative measurement. This has occurred as a result of Basel II and now Basel III regulatory requirements. For example in the context of banking regulation in Australia, the basic framework of Basel II/III is summarized in Figure 1. In this juristiction one observes that a large amount of the developments in quantitative methodology for estimation of OpRisk capital, development of OpRisk frame- There have been both numerical and simulation based approaches adopted as well as analytical mathematical developments of closed form approximatio...

Risk Margin Quantile Function via Parametric and Non-Parametric Bayesian Approaches

“…Examples of such alternatives that include specifically information on the quantile function include the VaR at some specified quantile level, which is known in Solvency II as the SCR, often then calculated over a predefined time frame, see discussions in Article 101 of the Solvency II Directive discussed in the introduction. Other risk measures that could also be considered and can be obtained from knowledge of the predictive quantile function include the expected shortfall (ES) and spectral risk measures (SRM), see discussions on properties of such quantile based risk measures for capital and reserving in for instance Embrechts et al (1997), Artzner (1999), Dowd et al (2006), Delbaen (2002) and Peters et al (2013) and the references therein. It is also worth noting that the Solvency II Directive clearly defines the risk margin and the SCR as separate concepts.…”

“…for some c ≥ 1. Note, if at least one of the lower triangle losses Y i j is distributed according to a heavy tailed loss distribution, such as sub-exponential, regularly varying or long tailed loss distributions then one can find the precise value for c. For instance, if the total loss is max-sum equivalent, then c = 1, see definitions for regular variation, sub-exponential, long tailed and max-sum equivalence in Bingham et al (1989) and in the context of insurance and quantile function approximations as discussed here, see the recent tutorial and references therein from Peters et al (2013). These conditional predictive distributions can be obtained for any model approximately by solving the integrals using the MCMC samples obtained from the posterior π (θ|D 0 ).…”

“…In such cases, it would be popular to utilize an asymptotic result for the quantile function of the sum, as the quantile level becomes large u → 1. For instance, one could use the first order or second order asymptotic results, see discussions in Peters et al (2013) and Cruz et al (2014). As an example, if the quantile regression was structured such that the distribution of the partial sum Y T = (i, j )∈D l Y i j ∼ F Y T is regularly varying with index ρ ≥ 0 with conditionally independent and identically distributed (i.i.d.)…”

“…see detailed tutorial in Peters et al (2013). This would lead to the approximation of the required quantile asymptotically by the expression…”

A simulation comparison of quantile approximation techniques for compound distributions popular in operational risk

“…Beyond that order the convergence of the series deteriorates. Peters et al (2013) In this paper we conduct a Monte Carlo simulation study to investigate the performance of the closed-form approximation methods using the Poisson as frequency distribution and the Burr, Lognormal and LogNig as severity distributions. In order to limit the scope of the study, we decided to only focus on the Poisson distribution, because of its popularity in practice and the fact that the estimated VaR depends critically on the choice of severity distribution rather than the choice of frequency distribution (see Cope et al 2009).…”