## The quantitative modeling of operational risk: between g-and-h and EVT (2007)

Venue: | ASTIN Bulletin |

Citations: | 14 - 8 self |

### BibTeX

@ARTICLE{Degen07thequantitative,

author = {Matthias Degen and Paul Embrechts and Dominik D. Lambrigger},

title = {The quantitative modeling of operational risk: between g-and-h and EVT},

journal = {ASTIN Bulletin},

year = {2007},

volume = {38}

}

### OpenURL

### Abstract

Operational risk has become an important risk component in the banking and insurance world. The availability of (few) reasonable data sets has given some authors the opportunity to analyze operational risk data and to propose different models for quantification. As proposed in Dutta and Perry [12], the parametric g-and-h distribution has recently emerged as an interesting candidate. In our paper, we discuss some fundamental properties of the g-and-h distribu-tion and their link to extreme value theory (EVT). We show that for the g-and-h distribution, convergence of the excess distribution to the generalized Pareto dis-tribution (GPD) is extremely slow and therefore quantile estimation using EVT may lead to inaccurate results if data are well modeled by a g-and-h distribution. We further discuss the subadditivity property of Value-at-Risk (VaR) for g-and-h random variables and show that for reasonable g and h parameter values, super-additivity may appear when estimating high quantiles. Finally, we look at the g-and-h distribution in the one-claim-causes-ruin paradigm.

### Citations

608 |
Regular variation
- Bingham, Goldie, et al.
- 1987
(Show Context)
Citation Context ...nalytically precise. A standard theory for describing heavy-tailed behavior of statistical models is Karamata’s theory of regular variation. For a detailed treatment of the theory, see Bingham et al. =-=[6]-=-. Embrechts et al. [13] contains a summary useful for our purposes. Recall that a measurable function L : R → (0, ∞) is slowly varying (denoted L ∈ SV ) if for t > 0: L(tx) lim = 1. x→∞ L(x) 7s0.1 0.0... |

569 |
Modeling Extremal Events for Insurance and Finance
- Embrechts, Klüppelberg, et al.
- 1997
(Show Context)
Citation Context ...ese findings with theory. We expect the reader to have studied Moscadelli [25] and Dutta and Perry [12] in detail. A basic textbook for EVT in the context of insurance and finance is Embrechts et al. =-=[13]-=-; see also Chapter 7 in McNeil et al. [21]. Before we start our discussion, we find it worthwhile to put the record straight on EVT: papers like Diebold et al. [10] and Dutta and Perry [12] highlight ... |

406 |
Extreme values, regular variation, and point processes
- Resnick
- 1987
(Show Context)
Citation Context ...tion deteriorates further for the g-and-h where the convergence is extremely slow. Note that one can always construct dfs with arbitrary slow convergence of the excess df towards the GPD; see Resnick =-=[29]-=-, Exercise 2.4.7. This result is in a violent contrast to the rate of convergence in the Central Limit Theorem which, for finite variance rvs, is always n −1/2 . From a theoretical point of view this ... |

105 | Estimation of tail-related risk measures for heteroscedastic financial time series: an extreme value approach
- McNeil, Frey
- 2000
(Show Context)
Citation Context ...l methods yield rather similar results, whereas for very high quantiles, the results differ substantially. Of course for the S&P data a more dynamic modeling, as for instance given in McNeil and Frey =-=[20]-=- including careful backtesting, would be useful. In the case of the Danish data backtesting to find the better fitting procedure is not really available. Once more, these results are in no way conclus... |

88 |
Ruin Probabilities
- Asmussen
- 2000
(Show Context)
Citation Context ...stribution of the total loss Sd is determined by the tail distribution of the maximum loss. We are in the so-called “one-claim-causes-ruin” regime; see Embrechts et al. [13], Section 8.3, or Asmussen =-=[1]-=-. More generally, consider (Xi)i≥0 a sequence of iid g-and-h rvs, independent of a counting process (Nt)t≥0 and St = � Nt i=1 Xi. Hence we have Gt(x) := P[St ≤ x] = ∞� P[Nt = n]F n∗ (x), where F n∗ de... |

66 |
Statistics of extremes: Theory and applications
- Beirlant, Goegebeur, et al.
- 2004
(Show Context)
Citation Context ...ion of the underlying issues, see Embrechts and Nešlehová [14]. 3.2 Threshold choice There exists a huge literature on the optimal threshold selection problem in EVT; see for instance Beirlant et al. =-=[3]-=- for a review. Within a capital charge calculation problem, the choice of threshold u above which EVT fits well the tail of the underlying df may significantly influence the value estimated. We stress... |

44 |
Pitfalls and Opportunities in the Use of Extreme Value Theory in Risk Management
- Diebold, Schuermann, et al.
- 2000
(Show Context)
Citation Context ...nce and finance is Embrechts et al. [13]; see also Chapter 7 in McNeil et al. [21]. Before we start our discussion, we find it worthwhile to put the record straight on EVT: papers like Diebold et al. =-=[10]-=- and Dutta and Perry [12] highlight weaknesses of EVT when it comes to some real applications, especially in finance. In Embrechts et al. [13] these points were already stressed very explicitly. Like ... |

43 |
The modelling of operational risk: experience with the analysis of the data collected by the Basel Committee. Banca D’Italia, Termini di discussione No
- Moscadelli
(Show Context)
Citation Context ...nt will no doubt leave its footprint on the LDA platform is Bühlmann and Gisler [5]. For the present paper, two fundamental papers, which are center stage to the whole LDA controversy, are Moscadelli =-=[25]-=- and Dutta and Perry [12]. Both are very competently written papers championing different analytic approaches to the capital charge problem. Whereas Moscadelli [25] is strongly based on EVT, Dutta and... |

29 | The peaks over thresholds method for estimating high quantiles of loss distributions
- McNeil, Saladin
- 1997
(Show Context)
Citation Context ...n Dutta and Perry [12], which are in a range around g = 2, h = 0.2, the SRMSE is close to 400%. The numbers reported in Table 2 are somewhat counterintuitive. Indeed in papers like McNeil and Saladin =-=[22]-=- and Dutta and Perry [12] it is stated that heavier tailed models require higher thresholds and likewise a larger sample size to achieve a similar error bound. Table 2 on the other hand indicates, tha... |

26 |
Exploring Data Tables, Trends, and Shapes
- Hoaglin, Mosteller, et al.
- 1985
(Show Context)
Citation Context ...per, we restrict our attention to the basic case where g and h are constants. The parameters g and h govern the skewness and the heavy-tailedness of the distribution, respectively; see Hoaglin et al. =-=[16]-=-. In the case h = 0, equation (1) reduces to X = a + b egZ − 1 , which is referred g to as the g-distribution. The g-distribution thus corresponds to a scaled lognormal distribution. In the case g = 0... |

25 |
A Course in Credibility Theory and its Applications
- Bühlmann, Gisler
- 2005
(Show Context)
Citation Context ... Chapter 10 carries the title “Operational Risk and Insurance Analytics”. Another recent actuarial text that at some point will no doubt leave its footprint on the LDA platform is Bühlmann and Gisler =-=[5]-=-. For the present paper, two fundamental papers, which are center stage to the whole LDA controversy, are Moscadelli [25] and Dutta and Perry [12]. Both are very competently written papers championing... |

23 |
2001): “An Academic Response to
- Daníelsson, Embrechts, et al.
(Show Context)
Citation Context ...ince the early discussion around Basel II and Solvency 2, the pros and cons of a quantitative (Pillar I) approach to operational risk have been widely put forward. Some papers, like Daníelsson et al. =-=[7]-=-, have early on warned against an over optimistic view that tools from market (and to some extent credit) risk management can easily be transported to the Basel II framework for operational risk. Also... |

21 |
A tale of tails: an empirical analysis of loss distribution models for estimating operational risk capital.’, Working Paper N˚. 06-13, Federal Reserve Bank of Boston
- Dutta, J
- 2007
(Show Context)
Citation Context ...e availability of (few) reasonable data sets has given some authors the opportunity to analyze operational risk data and to propose different models for quantification. As proposed in Dutta and Perry =-=[12]-=-, the parametric g-and-h distribution has recently emerged as an interesting candidate. In our paper, we discuss some fundamental properties of the g-and-h distribution and their link to extreme value... |

17 | Infinite mean models and the LDA for operational risk
- Neˇslehová, Embrechts, et al.
- 2006
(Show Context)
Citation Context ... arrived at. The overall α = 15% coefficient in the BIA is corroborated. The information coming through from individual banks with respect to the use of EVT is mixed. As explained in Nešlehová et al. =-=[26]-=-, the statistical properties of the data are no doubt a main fact underlying this diffuse image. When it comes to high quantile estimation (and 99.9% is very high) EVT emerges as a very natural key me... |

13 |
Operational Risk: Modeling Analytics
- Panjer
- 2006
(Show Context)
Citation Context ...he moment questionable. By now, numerous papers, reports, software, textbooks have been written on the subject. For our purposes, as textbooks we would like to mention McNeil et al. [21] 2sand Panjer =-=[27]-=-. Both books stress the relevance of actuarial methodology towards a successful LDA; it is no coincidence that in McNeil et al. [21], Chapter 10 carries the title “Operational Risk and Insurance Analy... |

12 | Subadditivity re-examined: the case for value-at-risk
- Danelsson, Samorodnitsky, et al.
- 2005
(Show Context)
Citation Context ...g large enough) for levels �α = 99.9% and higher, such that subadditivity of Value-at-Risk fails for all α < �α. This should be viewed in contrast to the following 24sproposition by Daníelsson et al. =-=[8]-=-. See also that paper for a definition of bivariate regular variation. Proposition 4.1 Suppose that the non-degenerate vector (X1, X2) is regularly varying with extreme value index ξ < 1. Then VaRα is... |

7 |
M.Wüthrich, The quantification of operational risk using internal data, relevant external data and expert opinion
- Lambrigger, Shevchenko
(Show Context)
Citation Context ...are required to augment internal data modeling with external data and expert opinion. An approach that allows for combining these sources of information is for instance discussed in Lambrigger et al. =-=[17]-=-. The fact that current data—especially at the individual bank level—are far from being of high quality or abundant, makes a reliable LDA for the moment questionable. By now, numerous papers, reports,... |

7 | Developing Scenarios for Future Extreme Losses Using
- McNeil, Saladin
- 1998
(Show Context)
Citation Context ...r Statistics Figure 4: Hill plot for g = 4, h = 0.2 and n = 10 6 . To confirm our findings of the previous section we performed a quantile estimation study along the lines of McNeil and Saladin [22], =-=[23]-=-. Instead of applying sophisticated optimal threshold selection procedures we likewise concentrated on an ad-hoc method by taking into account only a certain percentage of the highest data points; see... |

6 | Multivariate models for operational risk
- Böcker, Klüppelberg
- 2010
(Show Context)
Citation Context ... relatively few claims cause the major part of the total operational risk loss. Papers highlighting this phenomenon in an operational risk context are Nešlehová et al. [26] and Böcker and Klüppelberg =-=[2]-=-. Though these publications contain the relevant results, for matter of completeness we reformulate the main conclusions in terms of the g-and-h distribu27stion. We concentrate on the iid case, change... |

6 |
The inverse of the cumulative standard normal probability function
- Dominici
(Show Context)
Citation Context ...−1/x) (Φ −1 (1 − 1/x)) h (1 − e−gΦ−1 (1−1/x) ). 16sUsing the following approximation for the quantile function of the normal, Φ −1� 1 − 1 � � ∼ log x x2 x2 − log log , x → ∞, 2π 2π (see e.g. Dominici =-=[11]-=-, Proposition 21) we arrive at log l(tx) l(x) = g � Φ −1 (1 − 1/(tx)) − Φ −1 (1 − 1/x) � = +h log Φ−1 (1 − 1/x) 1 − e−gΦ−1 (1−1/x) � g h − (2 log x) 1/2 2 log x − � g 1 + o (2 log x) 3/2 (log x) 3/2 �... |

5 |
Adaptive threshold selection in tail index estimation
- Matthys, Beirlant
- 2000
(Show Context)
Citation Context ...hen the asymptotic mean square error (AMSE) of the Hill estimator satisfies AMSEHk,n := (ABiasHk,n) 2 + AVarHk,n � � � b (n + 1)/(k + 1) = 1 − ρ � 2 + ξ2 ; (2) k see for instance Matthys and Beirlant =-=[19]-=-. Applying this result to the regularly varying g-and-h df F with index 1/ξ = 1/h ∈ (0, ∞), we get l(x) = = 1 g(2π) h/2 egΦ−1 (1−1/x) − 1 (Φ−1 (1 − 1/x)) h 1 g(2π) h/2 e gΦ−1 (1−1/x) (Φ −1 (1 − 1/x)) ... |

4 |
A diagnostic for selecting the threshold in extreme value analysis
- Guillou, Hall
- 2001
(Show Context)
Citation Context ...aracteristic pattern which essentially remains the same for other threshold selection procedures. We confirmed this by implementing the optimal threshold selection method proposed by Guillou and Hall =-=[15]-=- and by applying an ad-hoc selection method, using a fixed percentage of exceedances of 5%. Further, we applied a method based on a logarithmic regression model provided by Beirlant et al. [4], where ... |

3 | Heavy-tailed distributions and rating
- Beirlant, Matthys, et al.
(Show Context)
Citation Context ...d Hall [15] and by applying an ad-hoc selection method, using a fixed percentage of exceedances of 5%. Further, we applied a method based on a logarithmic regression model provided by Beirlant et al. =-=[4]-=-, where the authors try to handle the case ϱ = 0. They analyze slowly varying functions of the following form, with C, β > 0. L(x) = C (log(x)) β (1 + o(1)), (3) If data come from a loggamma distribut... |

3 |
Extreme value theory and high quantile convergence
- Makarov
- 2006
(Show Context)
Citation Context ...estimator for the g-and-h distribution is much worse—in terms of high standardized bias and SRMSE—than 20sfor any of the distributions used in that paper. From a methodological point of view, Makarov =-=[18]-=- is also relevant in this respect. In that paper, the author shows that uniform relative quantile convergence in the Pickands-Balkema-de Haan Theorem necessarily needs a slowly varying function L whic... |

3 |
Rate of convergence for the generalized Pareto approximation of the excesses
- Raoult, Worms
- 2003
(Show Context)
Citation Context ... this, define d(u) := sup x∈(0,x0−u) � �Fu(x) − Gξ,β(u)(x) � � V (t) := (1 − F ) −1 (e −t ) A(t) := V ′′ (log t) V ′ − ξ, (log t) for some F ∈ MDA(Hξ). The following proposition (see Raoult and Worms =-=[28]-=-, Corollary 1) gives insight into the behavior of the rate of convergence to 0 of d(u) in cases including, for example, the g-and-h distribution with ξ = h > 0. Proposition 3.1 Let F ∈ MDA(Hξ) be a df... |

2 | EVT-based estimation of risk capital and convergence of high quantiles. Advances in Applied Probability 40, 696–715. of high-quantile estimators 27
- Degen, Embrechts
- 2008
(Show Context)
Citation Context ...tile estimation for power tail data very much depends on the second order behavior of the underlying (mostly unknown) slowly varying function L; for further insight on this, see Degen 9sand Embrechts =-=[9]-=-. Below we derive an explicit asymptotic formula for the slowly varying function L in the case of the g-and-h distribution. For g, h > 0 we have and hence leading to L(x) = L(x) = F (x)x 1/h = (1 − Φ(... |

1 |
Extreme value theory. Copulas. Two talks on
- Embrechts, Nešlehová
- 2006
(Show Context)
Citation Context ...d “optimal”) remains the Achilles heel of any high quantile estimation procedure based on EVT. For a more pedagogic and entertaining presentation of the underlying issues, see Embrechts and Nešlehová =-=[14]-=-. 3.2 Threshold choice There exists a huge literature on the optimal threshold selection problem in EVT; see for instance Beirlant et al. [3] for a review. Within a capital charge calculation problem,... |

1 |
Fitting quantiles: doubling, HR, HH and HHH distributions
- Morgenthaler, Tukey
- 2000
(Show Context)
Citation Context ...1 − Φ(u))(ge gu + hu(e gu − 1)) Remark: In a similar way, one shows that also the h-distribution (h > 0) is regularly varying with the same index. This was already mentioned in Morgenthaler and Tukey =-=[24]-=-. The g-distribution (g > 0) however is—as a scaled lognormal distribution—subexponential but not regularly varying. At this point the reader is advised to have a look at Section 1.3.2 and Appendix A ... |