Modeling operational risk data reported above a time-varying threshold

Transcription

1 Modeling operational risk data reported above a tie-varying threshold Pavel V. Shevchenko CSIRO Matheatical and Inforation Sciences, Sydney, Locked bag 7, North Ryde, NSW, 670, Australia. e-ail: Pavel.Shevchenko@csiro.au Grigory Tenov * Edgeworth Centre for Financial Matheatics, School of Matheatical Sciences, University College Cork, Ireland, e-ail: g.tenov@ucc.ie This version 0 une 2009 This is a preprint of an article published in The ournal of Operational Risk 4(2), pp. 9-42, Abstract Typically, operational risk losses are reported above a threshold. Fitting data reported above a constant threshold is a well known and studied proble. However, in practice, the losses are scaled for business and other factors before the fitting and thus the threshold is varying across the scaled data saple. A reporting level ay also change when a bank changes its reporting policy. We present both the axiu likelihood and Bayesian Markov chain Monte Carlo approaches to fitting the frequency and severity loss distributions using data in the case of a tie varying threshold. Estiation of the annual loss distribution accounting for paraeter uncertainty is also presented. Keywords: operational risk, truncated data, copound Poisson distribution, loss distribution approach, extree value theory, Bayesian inference, Markov chain Monte Carlo * A significant part of the work was ade within PRisMa Lab in Vienna University of Technology Final parts of the paper, fro the side of the second author, were copleted at the Edgeworth Centre for Financial Matheatics, SFI research grant 07/MI/008, in University College Cork

2 . Introduction The Basel II Accord requires banks to eet a capital requireent for operational risk as part of an overall risk-based capital fraework; see BIS (2006). To estiate the capital charge, any banks adopt the Loss Distribution Approach (LDA) under the Basel II Advanced Measureent Approaches (AMA). The LDA is based on estiation of the severity and frequency distributions of the loss events for each risk cell in a bank over a one year tie horizon. The industry usually refers these risk cells as risk nodes in the Basel II regulatory atrix of eight business lines by seven risk types. The capital charge for operational risk is then based on the quantile of the annual loss distribution. Accurate odelling of the severity and frequency distributions is the key to estiating a capital charge. There are various iportant aspects of operational risk odeling discussed in the literature, e.g. Chavez-Deoulin et al. (2006), Cruz (2002, 2004) and Shevchenko (2009) to ention a few. One of the challenges in odelling operational risk is the lack of coplete data often a bank s internal data are not reported below a certain level (typically of the order of 0,000). These data are said to be left-truncated. Generally speaking, issing data increase uncertainty in odelling. Soeties a threshold level is introduced to avoid difficulties with collection of too any sall losses. Industry data are available through external databases fro vendors (e.g. Algo OpData provides publicly reported operational risk losses above US$illion) and consortia of banks (e.g. ORX provides operational risk losses above 20,000 reported by ORX ebers). Several Loss Data Collection Exercises (LDCE) for historical operational risk losses over any institutions were conducted and their analyses reported in the literature. In this respect, two papers are of high iportance: Moscadelli (2004) analysing 2002 LDCE and Dutta and Perry (2006) analysing 2004 LDCE where the data were ainly above 0,000 and US$0,000 respectively. Often, odelling of issing data is done assuing a paraetric distribution for losses below and above the threshold. Then fitting is accoplished using losses reported above the threshold via the axiu likelihood ethod (see e.g. Frachot, Moudoulaud and Roncalli (2004)) or the Expectation Maxiization algorith (see e.g. Bee (2005)). The effect of data truncation in operational risk was studied in Baud, Frachot and Roncalli (2003), Chernobai et al. (2005), Mignola and Ugoccioni (2006), and Luo et al. (2007). Typically the case of a constant threshold is discussed in research studies. In this paper we consider the case of a threshold level varying across observations. One of the reasons for varying threshold in operational risk loss data is that the losses are scaled for inflation and other factors before fitting, to reflect changes in risk over tie. The reporting level ay also change fro tie to tie within a bank when reporting policy is changed. The proble with ultiple thresholds also appears when different copanies report losses into the sae database using different threshold levels; see Baud et al. (2002). One coon practice in calculating the annual loss distribution is to ignore the uncertainty in the fitted paraeters. That is, the distribution conditional on the fitted paraeters is used to estiate quantiles and final capital charge. Ignoring this uncertainty, which is always present in loss data odelling, ay lead to a significant underestiation of capital charge. The uncertainty of paraeter estiates can be treated efficiently using a Bayesian fraework; see Shevchenko and Wüthrich (2006), and Shevchenko (2008) for application of Bayesian inference in the operational risk context. In this paper we use the Bayesian Markov chain Monte Carlo (MCMC) procedure that estiates not only paraeter point estiators but also the distribution of the paraeter errors. This allows for quantification of the posterior distribution for the paraeters and annual loss distribution accounting for the process and paraeter uncertainties. Typically, practitioners in operational risk regard MCMC ethods as difficult to ipleent and use. Here, for illustrative 2

3 purpose, we deonstrate the use of a Rando Walk Metropolis Hastings (RWMH) within Gibbs algorith which is very efficient for estiation of the distribution paraeters when the likelihood can be calculated efficiently. If the likelihood is difficult to evaluate as in the cases of odeling any risks that follow Poisson processes with dependent intensities (see Peters et al. (2009b)) or severity densities that do not have closed for (e.g. g-and-h distribution), ore advanced MCMC ethods such as Slice Sapler or Approxiate Bayesian Coputation can be used; see Peters and Sisson (2006) in the operational risk context. Also, in general, questions of Bayesian odel choice ust be addressed; see Peters and Sisson (2006). We do not consider real data. The ain obective of the paper is to deonstrate how to odel data reported above a tie varying threshold and illustrate the use of the RWMH within Gibbs algorith. The organisation of this paper is as follows. Section 2 describes the odels of constant and varying in tie thresholds. Section 3 and Section 4 describe Bayesian fraework and MCMC procedures. Section 5 presents nuerical results. Discussions and conclusions are presented in Section Model Hereafter, we consider a single risk cell only. To clarify notation, we shall use upper case sybols to represent rando variables, lower case sybols for their realizations and bold for vectors. The coonly used LDA for operational risk assues that loss events follow soe point process so that the annual loss in a risk cell in year is Z N = i= X ( ). () i Here, N is the nuber of events (frequency) and X i (), i =,..., N are the severities of the events in year. Typically it is assued that the loss events are odelled by a hoogeneous Poisson process with the intensity paraeter λ. Thus, N, =,2,... are independent and identically distributed (iid) rando variables (rvs) fro the Poisson distribution, Poisson (λ), with n λ Pr[ N = n] = p( n λ ) = exp[ λ], λ > 0, n = 0,,... (2) n! and the event inter-arrival ties δ T = T T, =,2,... (where T 0 < T < T2 <... are the event ties and T 0 = t0 is the start of the observation period) are iid rvs fro an exponential distribution with the pdf and cdf g ( τ λ) = λ exp( λτ ) and G ( τ λ) = exp( λτ ) (3) respectively. General properties of Poisson processes are discussed in any textbooks, see e.g. Daley and Vere-ones (988). The severities of the events X, =,2,... which occur at T, =,2,... respectively, are odelled as iid rvs fro a continuous distribution F ( x β), 0 < x <, whose density is denoted as f ( x β). Here, β are the severity distribution paraeters. Hereafter, it is assued that the severities and frequencies of the events are 3

4 independent. If convenient, we ay index severities X and event ties T, =,2,... as X i () and T i (), i =,..., N, =,2,... respectively. It is iplicitly assued that considered data for the severities and frequencies correspond to the losses before insurance policy (if any) is applied. If details of the insurance policies are known (e.g. top cover liit, excess aount, etc), see Shevchenko (2009), then it should not be difficult to account for insurance recoveries, when the annual loss distribution over next reporting year is estiated via siulation of the frequencies and severities. Often, event ties are needed to calculate the insurance recoveries and these are easily siulated for Poisson processes considered in this paper. 2.. Constant threshold: likelihood and MLEs If the losses originating fro f ( x β) and p ( k λ) are recorded above a known reporting level (truncation level) L, then the density of the losses above L is left-truncated f ( x β) f L ( x β ) = ; L x <. (4) F( L β) The events of the losses above L follow a Poisson process with the intensity θ ( γ, L) = λ ( F( L β)), (5) a so-called thinned Poisson process, and the annual nuber of events above the threshold is distributed as Poisson (θ ). Hereafter, γ = ( λ, β) is a vector of all distribution paraeters Consider a rando vector Y of the events recorded above the threshold L over a period of M years consisting of the annual frequencies N ~, =,..., M and severities X ~, =,...,, ~ ~ = N N M. The oint density (likelihood) of Y at N ~ = n~ and X ~ = ~ x can be written as = M l ( y γ) = f ( ~ x β) p( n ~ θ ( γ, L)). (6) L = Then the axiu likelihood estiators (MLEs) γˆ can be found as a solution of lnl( y γ) = [ F( L β)] λ lnl( y γ) = β = ln f β L M = ( ~ x lnp( n ~ θ F( L β)] β) λ β θ ( γ, L)) = 0, M = lnp( n ~ θ θ ( γ, L)) = 0. (7) It is easy to see that the MLEs βˆ for the severity paraeters can be found arginally (independently fro frequency) by axiizing = ln f L ( ~ x β ) (8) 4

5 and then the first equation in (7) gives the MLE for the intensity ˆ λ =. (9) [ M n~ F( L βˆ)] M = Siilarly, if the data Y of the events above a constant threshold over the tie period [ t 0, t E ] ~ ~ ~ consist of the event inter-arrival ties δ T = T T, =,..., (where T ~, =,2,.. are the ~ event ties and T 0 = t0 ) and the severities X ~, =,...,, then the oint density (likelihood) of Y at δ T ~ ~ ~ = τ and X = ~ x l( y γ) = [ G( t E ~ t θ ( γ, L))] = λ exp( θ ( γ, L)( t E t 0 = )) f L = ( ~ x f ( ~ x β) g( ~ τ θ ( γ, L)) β), (0) where [ G( t ~ t θ ( γ, L))] is the probability that no event will occur within ~ t, t ]. Then it E ( E is easy to see that the MLEs βˆ for the severity paraeters are obtained by axiizing = ln f L ( ~ x β ) with respect to β and the intensity MLE is ˆ λ = [, () F( L βˆ)]( t ) E t 0 which is equivalent to (9) if the start and end of the observation period correspond to the beginning and end of the first and last years respectively Threshold varying in tie: likelihood and MLEs Often, in practice, before fitting a specific severity distribution, a odeller scales the losses by soe factors (inflation, business factors, etc). The reporting threshold should be scaled correspondingly and thus the losses in the fitted saple will have different threshold levels. To odel this situation consider the following set up. In the absence of threshold, the events follow a hoogeneous Poisson process with the intensity λ and the severities X are iid fro F (. β). The losses are reported above the known tie dependent level L (t). Denote the severities and arrival ties of the reported losses as start of the observation period. X ~ and T ~, =,..., respectively and t 0 is the Under the above assuptions, the events above L (t) follow a non-hoogeneous Poisson process with the intensity θ ( γ, t)) = λ ( F( t) β)). (2) 5

6 Denote t +h Λ( t, h) = θ ( γ, x)) dx. (3) t Then, given that (-)th event occurred at ~ t, the inter-arrival tie for the -th event ~ ~ ~ δ T T T is distributed fro = with the density g G ( τ γ ) exp( Λ( t, τ )) (4) = ( = τ τ γ ) θ ( γ, t + τ ))exp( Λ( t, )). (5) The nuber of events in year is Poisson( Λ ( s,)) distributed, where s is the tie of the beginning of year. Also note that the nuber of events fro a non-hoogeneous Poisson process over non-overlapping periods are independent. The oint likelihood of the data Y of the events above L (t) over the tie period ~ ~ ~ [ t 0, t E ], consisting of the inter-arrival ties δ T = T T and severities X ~, =,..., above L (t) can be written as Using (4) and (5), it is siplified to l ( y γ) = [ G ( t ~ t γ)] f ( ~ ) ( ~ ~ x β g τ γ). (6) E = t ) l ( y γ) = λ exp( Λ( t0, te t0)) f ( ~ x β). te Here, explicitly, Λ( t0, te t0) = λ [ F( x) β)] dx. Then, the axiu likelihood equations are t0 = lnl( y γ) = λ λ te t0 lnl( y γ) = Λ( t β β [ F( x) β)] dx = 0, 0, t E t 0 ) + = ln f ( ~ x β β) = 0. (7) The first equation gives ˆ λ =. (8) t E [ F( x) βˆ)] dx t0 6

7 This can be substituted into (6) and axiization will be required in respect to β only. The likelihood contains an integral over the severity distribution. If integration is not possible in closed for then it can be calculated nuerically (that can be done efficiently using standard routines available in any nuerical packages). For convenience, one can assue that a threshold is constant between the reported events: L ( t) = t ), ~ t t ~ < t and L ( t ) = L ( t E ) ~ for t < t t, so that E t E t0 [ F( x) β )] dx = [ F( t ) β)]( t ~ t ) + [ F( ~ t ) β)] τ. (9) E E = Of course this assuption is reasonable if the intensity of the events is not sall. Typically scaling is done on the annual basis and one can assue piecewise constant threshold per annu and the integral is replaced by a siple suation. The MLEs for severity paraeters calculated arginally, i.e. by siply axiizing ln f ( ~ t ) x β), do not differ aterially fro the results of the oint estiation if the variability of the threshold is not extreely fast. The results of the estiation for the siulated data in the case of exponentially varying threshold, presented in section 5, confir that intuitive observation, although the difference can still be significant if the intensity is sall. Also, arginal estiation does not allow for quantification of the covariances between frequency and severity paraeters required to account for paraeter uncertainty. If a data vector Y of the events above the reporting threshold consists of the annual counts N ~, =,..., M and severities X ~ ~ ~, =,..., ( = N N M ) then the oint likelihood of observations is M t ) = l ( y γ) = f ( ~ x β) p( n ~ Λ( s,)). (20) = Usually, in practice, scaling is done on an annual basis. Thus we can consider the case of piecewise constant threshold per annu such that for year : L ( t) = L, θ = θ ( γ, t)) = λ( F( L β)), s t < s +, where s is the tie of the beginning of year. The likelihood in this case = M l ( y γ) = f ( ~ ) ( ~ x β) p( n ~ θ )) (2) L t = and equations to find MLEs are lnl( y γ) = λ lnl( y γ) = β M = = [ F( L ln f β ~ t ) β)] θ ( ~ x lnp( n ~ β) λ M = θ ) = 0, F( L β)] β θ lnp( n ~ θ ) = 0. (22) 7

8 The first equation gives = M n~ ˆ = λ = M. (23) [ F( L This can be substituted into the likelihood function (2), so axiization to obtain the MLEs is required in respect to severity paraeters only. The MLEs of the severity paraeters should be estiated ointly with the intensity. However, given that the intensity MLE can be expressed in ters of the severity paraeters MLEs via (8) or (23), the axiization of the likelihood can be done effectively in respect to severity paraeters only by substituting (8) into (6), or respectively substituting (23) into (2). βˆ)] 2.3. Non-hoogeneous Poisson The results of Section 2.2 can easily be extended to the case when the event process before truncation is a non-hoogeneous Poisson process with the tie dependent intensity λ (t). In this case, after truncation, the events above L (t) follow a non-hoogeneous Poisson with the intensity λ ( t)( F( t) β)). One has to siply change λ to λ (t) in (2-5), then the expressions for the likelihoods (6) and (20) are still valid. A paraetric for can be assued for λ (t), for exaple: λ; λ ( t) = λ2; t < t, c t t, c where t c could be a tie of a new policy/control introduced; or one can consider the intensity as a function of soe explanatory variables; etc. Then all unknown paraeters can be fitted by the axiu likelihood or MCMC as described in Section 4. A particular paraetric for is proble specific and our nuerical exaples below consider the case of constant intensity only. If before the truncation, the process is a non-hoogeneous Poisson λ (t) and the truncation level is unknown function of tie L (t), then it ight be a proble to distinguish the change in the intensity versus the change in the threshold. Though the iportance of this case in practice is questionable. Certainly it depends on specific paraetric odels chosen for λ (t) and L (t) and will not be considered in this paper. Significant for practical applications is the case when not ust the threshold but also the losses theselves are subect to a certain trend with possibly unknown or isspecified paraeters (so that the assuption of i.i.d. observations is violated). The likelihood of such a odel would then include, besides the paraeters of the severity and frequency, also the paraeters associated with the trend. In that case, the optiization in the axiu likelihood ethod (or the MCMC odelling) should be ade with respect to all the paraeters of the odel, including the trend paraeters. Naturally, techniques for optiization and odelling 8

9 regarding this type of odel becoe significantly ore coplicated; consideration of this odel is a coplex proble and is beyond the scope of the present paper MLE asyptotic properties Often, as a saple size increases, the MLEs γˆ have the following useful asyptotic properties: under the ild regularity conditions, γˆ is a consistent estiator of the true paraeter γ, i.e. γˆ converges to γ in probability; under the stronger regularity conditions, γ γˆ γ has a Noral distribution with zero ean and covariance atrix I ( ), where Ii, ( γ) = E[ lnl ( Y γ) / γ i γ ] is the Fisher inforation atrix. If I (γ) can not be found in closed for, then (for a given realization y ) typically it is estiated by the observed inforation atrix ˆ, ˆ I i, ( γ) = 2 i 2 ln l( y γ) γ γ Both I i ( γ) and I i ( ) depend on the unknown true paraeter γ, which is estiated by γˆ in the final estiate of the covariances between MLEs, γ ( Iˆ (ˆ) γ ) i i,. cov( ˆ γ, ˆ γ ). (24) If closed for is not available then the required second order derivatives are calculated nuerically using finite difference ethod. Precise regularity conditions required and proofs can be found in any textbooks; see e.g. Lehann (983) Theore Rearks The required regularity conditions for the above asyptotic theore are conditions to ensure that the density is sooth with regard to paraeters and there is nothing unusual about the density, see Lehann (983). These include that: the true paraeter is an interior point of the paraeter space; the density support does not depend on the paraeters; the density differentiation with respect to the paraeter and the integration over y can be swapped; third derivatives with respect to the paraeters are bounded; and few others. Though the required conditions are ild, they are often difficult to be proved. Here, we ust assue that these conditions are satisfied. Whether a saple size is large enough to use the asyptotic results is another difficult question that should be addressed in real applications. 3. Bayesian Estiation The paraeters fitted using real data are estiates that have statistical fitting errors due to a finite saple size. The true paraeters are not known. In our experience with banks, typically, uncertainty in fitted paraeters is ignored when capital is quantified. That is paraeters are fixed to their point estiates (e.g. axiu likelihood estiates) when the annual loss distribution and its quantile are calculated. The Bayesian fraework is convenient to 9

10 account for paraeter uncertainty on quantile estiates, see Shevchenko and Wüthrich (2006) for an application of the Bayesian fraework to operational risk. Consider odel () with a rando vector of data (severities and frequencies) Y over M years. Given γ = ( λ, β), denote the density of the annual loss as h ( z γ). In the Bayesian approach the paraeters γ are odelled as rando variables. Then the density of the full predictive distribution for the next year annual loss Z M + is h ( z y) = h( z γ) π ( γ y) dγ, (25) where π ( γ y) is the oint posterior density of the paraeters given data Y. Fro Bayes rule π ( γ y) l( y γ) π ( γ), (26) where l ( y γ) is the likelihood of observations and π (γ) is a prior distribution for the paraeters (the prior distribution can be specified by an expert or fitted using external data or can be taken as uninforative so that inference is iplied by data only). The ode of the posterior distribution γ ˆ = ode( γ) can be used as a point estiator for the paraeters. For large saple size (and continuous prior distribution), it is coon to approxiate lnπ ( γ y) by a second-order Taylor series expansion around γˆ. Then π ( γ y) is approxiately a ultivariate Noral distribution with the ean γˆ and covariance atrix calculated as the inverse of the atrix: ~ (I) i, = 2 lnπ ( γ y) γ γ i γ= γˆ. (27) In the case of iproper constant priors, i.e. π ( γ y) l( y γ), this approxiation copares to the Gaussian approxiation for the MLEs (24). Also, note that in the case of constant priors, the ode of the posterior and MLE are the sae. This is also true if the prior is unifor within a bounded region, provided that the MLE is within this region. Soeties it is possible to find the posterior distribution π ( γ y) of the paraeters in closed for. However, in general, π ( γ y) should be estiated nuerically, e.g. using one of the Markov chain Monte Carlo ethods. 3.. The quantile of the full predictive annual loss distribution B The quantile Q of the full predictive distribution h ( z y) can be easily calculated, for exaple, by the following MC procedure: Step. Siulate λ and β fro the oint distribution π ( γ y). 0

11 Step 2. Given λ and β, calculate the annual loss z = n x i i= by siulating n fro the frequency distribution p (. λ) and x i, i =,..., n fro the loss severity distribution F (. β). Step 3. Repeat Step to Step 2 K ties to get realisations of total annual loss z, =,..., K. B Step 4. Estiate the quantile, Qˆ , of the total annual loss and its standard error (due to finite nuber of siulations) using siulated saple z, =,..., K in the usual way, see e.g. Shevchenko (2008). In the above the total annual loss accounts for both the process uncertainty (severity and frequencies are rando variables) and the paraeter uncertainty (paraeters are siulated fro their posterior distribution). The paraeter uncertainty coes fro the fact that we do not know the true values of the paraeters Distribution of the quantile of the annual loss distribution Another approach to account for paraeter uncertainty is to consider a quantile Q 0.999( γ) of the conditional annual loss distribution h ( z γ). Then, given that γ is distributed fro π ( γ y), one can find the distribution for Q 0.999( γ) and for a one-sided or two-sided predictive interval to contain the true value of the quantile with soe probability q. Then one can argue that the conservative estiate of the capital charge should be based on the upper bound of the constructed confidence interval. Calculation of the predictive interval for the quantile Q ) of h ( z γ) can be accoplished as follows: 0.999( γ Step. Siulate γ = ( λ, β) fro π ( γ y). Step 2. Given λ and β fro the Step, calculate Q 0.999( γ). Conceptually this can be accoplished via Monte Carlo: a) siulating n fro p (. λ) and x i, i =,..., n fro f (. β) ; n b) calculating the annual loss z = i = x i ; c) repeating steps a) and b) any ties to build a saple of annual losses used to estiate Q 0.999( γ) in a usual way. Step 3. Repeat Steps -2, K ties to build a saple of possible realizations of the quantile and use the saple to find the distribution of Q 0.999( γ). Then a predictive interval to contain the true value of Q with soe probability q can be constructed using the saple in the usual way. The use of Monte Carlo in Step 2 to copute Q 0.999( γ) for a given paraeter vector γ is too coputationally deanding and it is ore efficient to use deterinistic ethods. One can choose well known Paner recursion, Fast Fourier Transfor or direct inversion of characteristic function ethods. The analysis of these techniques and coparison with Monte Carlo ethod in the context of operational risk (see Tenov and Warnung (2008) and Luo and Shevchenko (2009)) shows that these ethods are efficient for estiation of the quantile, considering both the accuracy and the speed of calculation. In a recent paper, Peters, ohansen and Doucet (2007) proposed a hybrid Monte Carlo approach utilizing Paner recursion, iportance sapling and trans-diensional Markov chain Monte Carlo. Note that this estiation requires a specification of a confidence level q, which could be difficult. One could choose q = as a conservative estiate for the quantile, as

12 ost of statistical estiates use the 95% confidence level. However, in the context of operational risk the preferable confidence level should be in agreeent with the requireents fro regulator. Unless this level is set exactly, estiating the full predictive distribution for the annual loss and using its quantile Q B for quantification of the capital charge as described in section 3. would be ore appealing. 4. MCMC procedure Siulation of paraeters γ = ( λ, β) fro the posterior distribution π ( γ Y) required in procedures in Sections 3. and 3.2 can be accoplished using MCMC techniques. Here, we briefly outline Rando Walk Metropolis Hastings (RWMH) within Gibbs schee used for estiating the posterior distribution of paraeters. In the context of operational risk, the RWMH within Gibbs is entioned in Peters and Sisson (2006) and used in Peters et al. (2009a) for a siilar proble in insurance. For other references concerning this algorith, see Gelan et al. (997), Bedard and Rosenthal (2008), Roberts and Rosenthal (200), and Robert and Casella (2004). The algorith creates a reversible Markov chain with a stationary distribution corresponding to our target posterior distribution. It should be noted that the Gibbs sapler creates a Markov chain where each iteration involves scanning either deterinistically or randoly over the variables that coprise the target stationary distribution of the chain. This process involves sapling each proposed paraeter update fro the corresponding full conditional posterior distribution. In our calculations we assue that all paraeters are independent under the prior density π (γ) and distributed uniforly with γ i ~ U ( ai, bi ) on a wide ranges so that inference (k) is ainly iplied by data only (also see reark at the end of this section). Denote by γ the ( k =0) state of the chain at iteration k (usually the initial state γ is taken as MLEs). Then the ( k ) MCMC sapler proceeds by proposing to ove the ith paraeter fro a state γ i to a new proposed state γ * i. The latter is sapled fro a MCMC proposal transition kernel. As a proposal transition kernel, we use the Gaussian distribution truncated below a i and above b i, with the density f ( T ) N * ( k ) ( * ( k ) f N ( γ i ; γ i, σ i ) γ i ; γ i, σ i ) = ( k ) ( k F ( b ; γ, σ ) F ( a ; γ ), (28), σ ) N i i i N i i i where f N ( x; μ, σ ) and F N ( x; μ, σ ) are the Noral density and its distribution respectively with the ean μ and standard deviation σ. Then the proposed ove is accepted with the probability * ( T ) ( k ) * ( k) * π ( γ y) f N ( γ i ; γ i, σ i ) p( γ, γ ) = in, ( k) ( T ) * ( k ), (29) π ( γ y) f N ( γ i ; γ i, σ i ) where y is the vector of observations and π ( γ * y) is the posterior distribution. Also, here γ * ( k) ( k) * ( k ) = ( γ,..., γ i, γ i, γ i+ updated while i +, i + 2,... are not updated yet. *,...), i.e. γ is a new state, where paraeters,2,..., i are already 2

13 The acceptance probability is a function of the transition kernel and the posterior density. Note that, a noralization constant for the posterior density is not needed here. If under the reection rule one accepts the ove then the new state of the ith paraeter at iteration ( k ) * ( k ) ( k ) k is given by γ i = γ i, otherwise the paraeter reains in the current state γ i = γ i and an attept to ove that paraeter is repeated at the next iteration. In following this procedure, one builds a set of correlated saples fro the target posterior distribution which have several asyptotic properties. One of the ost useful of these properties is the convergence of ergodic averages constructed using the Markov chain saples to the averages obtained under the posterior distribution. The chain has to be run until it has sufficiently converged to the stationary distribution (posterior distribution) and then one obtains saples fro the posterior distribution. General properties of this algorith, including convergence results can be found in e.g. Robert and Casella (2004) and application to a siilar task in clais reserving is given in Peters et al. (2009a). Rearks The RWMH algorith is siple in nature and easy to ipleent. However, if one does not choose the proposal distribution carefully, then the algorith only gives a very slow convergence to the stationary distribution. There have been several studies regarding the optial scaling of proposal distributions to ensure optial convergence rates see e.g. Gelan et al. (997), Bedard and Rosenthal (2007) and Roberts and Rosenthal (200). According to the listed works, the asyptotic acceptance rate optiizing the efficiency of the process is independent of the target density (for ultivariate target distributions with i.i.d. coponents). In this case it is recoended that σ i are chosen to ensure that the acceptance probability is close to To obtain this acceptance rate, one is required to perfor soe tuning of the proposal variance prior to final siulations. Recently, advanced odifications of MCMC algoriths were developed which allow on-line adaptations of optial proposal distributions for specified ixing criteria. These algoriths are known as adaptive MCMC algoriths, and their use ay significantly iprove the effectiveness of the MCMC schee. Adaptive MCMC algoriths can be used in our fraework as well. For detailed descriptions and exaples, we refer to Atchade and Rosenthal (2005) and Roberts and Rosenthal (2006). For practical use of the RWMH algorith, it is iportant that the likelihood can be evaluated efficiently. This is usually the case if the severity density has a closed for (e.g. Pareto distribution considered in exaples below, Lognoral distribution and any other standard distributions). In particular, it should not be a proble to use RWMH in the case of four-paraeter GB2 distribution used in data analysis Dutta and Perry (2006). If the density can not be easily evaluated, while a siulation fro the density is efficient (as in the case of another four-paraeter distribution g-and-h used in Dutta and Perry (2006)), then ore advanced MCMC ethods such as Approxiate Bayesian Coputation can be used; see Peters and Sisson (2006). Suarizing this section, we list the ost iportant quantities provided by MCMC and playing a role in the estiation of the aggregate loss quantiles: the ean of the posterior distribution (can be used as a point estiator for the paraeters); the nuerical standard error of the MCMC estiates (this quantity reflects the error in the point estiates of a paraeter due to the finite nuber of chain iterations); the standard deviation of the posterior distribution (this uncertainty is associated with the finite size of the observed data saple). 3

14 Let us reark on a role of the choice of prior distributions in the Bayesian odelling. As we consider a basic odel in the present work, we assue priors to be siply constant priors defined on soe large region. This allows to get all inferences ainly iplied by observations only. The constant priors used in this approach ay be called non-inforative priors, in contrast to a different approach associated with the choice of the so-called inforative priors. In the latter approach, the choice of eligible inforative priors plays an iportant role, as it has to be based on statistical properties of the data associated with priors. Inforative priors can be used if external data and expert opinions are taken into account, see e.g. Labrigger et al (2007). Strictly speaking, constant priors on a large region can be quite inforative in soe situations. In our nuerical exaples, we checked that the ipact of chosen constant priors is not aterial by changing the region bounds. 5. Results To illustrate the above procedures and calculations we perfor a siulation experient assuing a Poisson (λ) process with Pareto distribution, Pareto ( α, β ), for the severities α F ( x α, β ) = ( + x / β ), x 0, α > 0, β > 0, (30) where α and β are the shape and the scale paraeters respectively. Then, for a reporting threshold varying in tie L (t), the distribution of the severity above the threshold occurred at tie t is a left-truncated Pareto distribution with pdf f α ( + x / β ) ( + t) / β ) L ( t) ) α α ( x α, β ) = ; t x <. (3) β The log-likelihood of the reported events (6) is lnl ( y γ) = ln( α / β ) ( + α) ln( + x / β ) + lnλ λ ( + t) / β ) dt. (32) = t E t0 α te t0 In soe special cases the integral λ ( + t) / β ) dt in (32) can be calculated explicitly but in general it will be calculated nuerically. The latter ight decrease the overall speed of the whole MCMC procedure as the integral has to be recalculated for each MCMC iteration. If the threshold is piecewise constant function of tie (as assued in our experients below) then the integral is replaced by a siple suation. Also, in cases when the intensity of events is high enough, one can assue that the threshold is constant between events. α 5.. Siulation set up We consider the observation period 0 t M. Assue that the change of the threshold in tie follows the exponential law, as this choice of the law for the varying threshold relies on the assuption that one of the ost significant scaling factors for the losses is the inflation. If the real inflation rate is known, all the losses are scaled with respect to it, but this scaling cannot recover the issing data, which fell below the threshold and were not reported. Moreover, we assue that the scaling of the data is done per annu, therefore the value of the threshold stays 4

15 constant each year t) = L = L0 exp( r ), t <, =,2,.... Then, we siulate the loss events using the following procedure: Step. Siulate the Poisson process event ties t, =,2... covering the period of M years by siulating iid inter-arrival ties τ = t t fro the exponential distribution with the paraeter λ. Find the nuber of events n occurred during each year =,..., M and total nuber of events n = n n M. Step 2. Siulate iid severities x for the event ties t, =,..., n respectively fro Pareto ( α, β ). Step 3. Reove the events fro the siulated saple when the event loss is below a varying threshold L. Step 4. Given truncated saple of severities and event ties, estiate the paraeters ( α, β, λ) via the MLE and MCMC procedures using likelihood (32). True paraeter values. We consider various observation periods M and use the following paraeters (unless indicated otherwise) Inflation rate r = ; Pareto distribution shape and scale paraeters α = 2. 0 and β = 3. 0 respectively; Poisson yearly intensity λ = 50 ; Initial threshold value L 0 = We choose the threshold L 0 and scale paraeter β values so that after scaling by a coon large factor they correspond to ore or less typical values observed with real data. The standard deviations of the RWMH proposal transition kernel, adusted to keep the RWMH acceptance probabilities close to 0.234, are σ = 0. 2, σ = 0. 3, σ = 5. α Prior structure for MCMC. The unifor prior distributions are used for all paraeters, with the following bounds: α [0., 6], β [0., 8] and λ [0., 500]. In the calculations below we checked that increasing the bounds does not lead to a aterial change in the estiates and thus the inferences are iplied by data only. β λ 5.2. MCMC and MLE results For each of the chains obtained with MCMC, the first 000 chain iterations are discarded, and all the results are obtained using the rest of the chain only. This is quite a coon practice in MCMC odelling, used to increase the accuracy of results (as the chain needs to be run for a certain length before it starts to converge to the true values). Table presents results for the posterior ean and standard deviation for each paraeter using different chain lengths K. Figure shows the histogras and scatter plots of the severity paraeters α, β and intensity λ obtained via the MCMC procedure for one of the data realisations over 5 years. Histogras were obtained fro MCMC chains of length 5000 for each paraeter. Note that the paraeters are utually dependent. The standard deviations of the posterior distributions for the paraeters reflect the paraeter uncertainty due to the finite nuber of data points in the fitted saple while the finite nuber of iterations K in the chain results in the nuerical error in the MCMC estiates. The nuerical standard error can be decreased by increasing K and can be estiated as follows: split the chain into the non-overlapping bins; calculate the MCMC estiator using 5

16 iterations for each bin; find the standard deviation of the obtained saple of estiators and divide by the square root of the nuber of bins. Table shows the MCMC estiates and their nuerical standard errors obtained using 00 bins. For our siulation paraeters, the 6 nuerical standard errors are less then 2% when K =0 which is sall enough to be 6 neglected. For all subsequent results, we use the chain length K =0. Table 2 copares ML estiates for arginal and oint estiations of the severity and frequency paraeters. Maxiization of the likelihood was perfored using a nuerical SPlus procedure based on a quasi-newton ethod using the double dogleg step with the BFGS secant update to the Hessian. We also show the results for the case of a is-specified MLE that corresponds to the MLE fitting under a wrong assuption of constant threshold t) L0. One observes fro Table 3, that MCMC and ML estiators and corresponding standard deviations are in good agreeent while is-specified MLE leads to significantly different results. As the data size increases, the paraeter posterior ean converges to the true value and posterior standard deviation approaches zero. The sae is observed for the MLE and its standard deviation. This convergence is suggested by the results in Table 4. One observes fro Table 4, that MCMC and ML estiators are in good agreeent. Note that standard deviations for MLEs are calculated under the Gaussian approxiation for large data saples, so in general it is better to use MCMC. The MCMC algorith was ipleented in C and coputing tie for 0 6 iterations fro a Markov chain in the case of a 5 year data saple was approxiately 2 inutes on Intel Dual Xeon 2.4 GHz, 2 GB RAM. Full predictive distribution and quantile estiation (conditional on data) Using the siulation odel specified above and applying the ethodology described in Section 3., we estiate the full predictive distribution (conditional on data), see Figure 2, and B calculate its Qˆ quantile. In Figure 2, the second histogra indicates the case of the data left truncated with the constant threshold (corresponding to the axiu threshold over the period of observations). The full predictive distribution (conditional on data) accounts for process uncertainty (that coes fro the fact that we odel losses using rando process) and paraeter uncertainty (that coes fro the fact that the true paraeter values are not known). We assue that nuerical error due to finite nuber of siulations is negligibly sall, as indicated by previous results. The estiated value of the quantile of the full predictive distribution is 805.2, while the true quantile of the aggregate loss distribution with the specified paraeters is equal to (the latter is calculated via FFT with advanced aliasing reduction techniques, the ethod described in Schaller and Tenov (2008)). The difference is due to paraeter uncertainty for a finite saple size. Relying on the data left truncated with the constant threshold, the estiated value of the quantile of the full predictive distribution is increased to because a saller data set is used leading to a larger paraeter uncertainty. The ipact of paraeter uncertainty on the quantile can be also assessed by calculating the distribution (conditional on data) of Q ( γ), where γ = ( λ, β) is fro the posterior ( k ) ( k ) ( k ) distribution π ( γ y), see Section 3.2. Using the saple γ = ( λ, β ) fro the posterior ( k ) distribution obtained via MCMC we calculate a saple of the quantile Q0.999( γ ) via FFT and its histogra is presented on Figure 3. The epirical characteristics of the distribution for Q ( γ) are: ean = 26.5; ode = 788.0; edian = 29.7; standard deviation = 764.4; 0.9- quantile = 3780; 0.95-quantile = 4697; 0.99-quantile = One can observe that the 6

17 conditional distribution of the quantile is considerably skewed, though its ean is less than the quantile of the full predictive distribution. The nuerical error (due to the finite nuber of siulations K) of the quantile estiates can be calculated by foring a conservative confidence interval using the fact that the nuber of saples not exceeding the quantile has a Binoial distribution that can be approxiated by the Noral distribution, see e.g. Shevchenko (2008). In the case of our 6 siulation set up and K = 0, the nuerical standard error is of the order of -2%. Unconditional distribution characteristics As the observation period increases the paraeter posterior variance (the paraeter B uncertainty) should converge to zero and the quantile Qˆ should converge to the true value 824.4, see Figure 4. For each selected observation period in Figure 4, 20 independent siulations were perfored. The ean values of Qˆ B for each observation period are connected by solid line deonstrating the convergence. 6. Conclusions In this paper, we considered odelling of a single operational risk in the case of data reported above a threshold varying in tie. Of course, the assuption in this study is that issing losses and reported losses are realizations fro the sae distribution. Thus the approach should be used with caution especially if a large proportion of data is issing. In the latter case, it ight be better to ignore issing data copletely. We have also deonstrated how the oint posterior distribution of the paraeters can be estiated using MCMC and used to estiate the annual loss distribution accounting for both process and paraeter uncertainties. In particular, we advocate the use of the Rando walk Metropolis-Hastings within Gibbs algorith which appeared to be efficient for the proble considered. Though, if the likelihood for the severity/frequency odel can not be evaluated efficiently (as in the case of g-and-h distribution), then ore advanced MCMC ethods, such as Approxiate Bayesian Coputation should be used. Basically, cobining the ethodology proposed in the present paper with Approxiate Bayesian Coputation according to Peters and Sisson (2006), the generalization of the whole ethodology to the case of distributions that do not allow an explicit likelihood is quite straightforward. There are no conceptual probles to introduce tie-varying threshold features into the odel of any loss processes with dependence considered in Peters et al (2009b). However, while this ultivariate odel was presented for a general case involving frequencies and severities, the odel calibration and required posterior densities were considered for the case of dependent frequencies only. In the case of threshold, the severity and frequency paraeters should be estiated ointly. Deriving all required posteriors and ipleentation for this case are challenging tasks that will be addressed in future research. In general, the following rearks can be ade: estiation of the MLE uncertainties relies on asyptotic Gaussian approxiation in the liit of large data saples; severity and frequency paraeters should be estiated ointly; it is beneficial to use all available inforation in the fitting procedure. In the present study we assued constant non-inforative priors so that inferences are iplied by data only. It is iportant to note that inforative priors can be used to incorporate external data and expert opinions into the odel, see e.g. Labrigger et al (2007). For a closely related credibility theory application in the context of operational risk, see Bühlann et al (2007). 7

18 One of the features of the described Bayesian approach is that the variance of the posterior distribution π ( γ y) will converge to zero for a large nuber of observations. This eans that the true value of the paraeters will be known exactly. However, there are any factors (for exaple, political, econoical, legal, etc.) changing in tie that should not allow precise knowledge. One can odel this by liiting the variance of the posterior distribution by soe lower levels (e.g. 5%). This has been done in any solvency approaches for the insurance industry, see e.g. FOPI (2006) forulas (25)-(26). This can also be odelled by allowing the paraeters be truly stochastic and possibly dependent as in Peters et al (2009b). Acknowledgeents The fist author thanks PRisMa Lab, Vienna University of Technology for financial travel assistance whilst copleting aspects of this work at PRisMa Lab. The authors thank the anonyous referee whose useful coents and rearks allowed iproving the paper. References Atchade Y. and Rosenthal,. (2005) On adaptive Markov chain Monte Carlo algoriths. Bernoulli (5), Baud, N., Frachot, A. and Roncalli, T. (2003). How to avoid over-estiating capital charge for operational risk? Operational Risk Risk s Newsletter, February. Baud, N., Frachot, A. and Roncalli, T. (2002). Internal Data, External Data and Consortiu Data for Operational Risk Measureent: How to Pool Data Properly? Working Paper, Crédit Lyonnais, Groupe de Recherche Opérationnelle. Bee, M. (2005). On Maxiu Likelihood Estiation of Operational Loss Distributions. Discussion paper No.3, Dipartiento di Econoia, Universita degli Studi di Trento. Bedard, M. and Rosenthal,.S. (2008). Optial scaling of Metropolis algoriths: heading towards general target distributions. To appear in The Canadian ournal of Statistics 36(4), BIS (2006). International Convergence of Capital Measureent and Capital Standards. Basel Coittee on Banking Supervision, Bank for International Settleents, Basel. Bühlann, H., Shevchenko, P.V. and M. V. Wüthrich (2007). A Toy Model for Operational Risk Quantification using Credibility Theory. The ournal of Operational Risk 2(), 3-9. Chavez-Deoulin, V., Ebrechts, P. and Nešlehová,. (2006). Quantitative Models for Operational Risk: Extrees, Dependence and Aggregation. ournal of Banking and Finance 30(0), Chernobai, A., Menn, C., Trück, S. and Rachev, S.T. (2005). A note on the estiation of the frequency and severity distribution of operational losses. The Matheatical Scientist 30(2). Cruz, M.G. (2002). Modelling, Measuring and Hedging Operational Risk. ohn Wiley & Sons, UK. Cruz, M.G., editor (2004). Operational Risk Modelling and Analysis: Theory and Practice. Risk Books, London. 8

19 Daley, D.. and Vere-ones, D. (988). An Introduction to the Theory of Point Processes. Springer-Verlag, New York Dutta, K. and Perry,. (2006). A Tale of Tails: An Epirical Analysis of Loss Distribution Models for Estiating Operational Risk Capital. Working papers No. 06-3, Federal Reserve Bank of Boston. FOPI (2006). Swiss Solvency Test, technical docuent, version October 2. Federal Office of Private Insurance, Bern. Frachot, A., Moudoulaud, O. and Roncalli, T. (2004). Loss distribution approach in practice. The Basel Handbook: A Guide for Financial Practitioners, edited by M. Ong, Risk Books. Gelan, A., Gilks, W.R., Roberts, G.O. (997). Weak convergence and optial scaling of rando walks etropolis algorith. Annals of Applied Probability 7, Labrigger, D.D., Shevchenko, P.V. and Wüthrich, M.V. (2007). The Quantification of Operational Risk using Internal Data, Relevant External Data and Expert Opinions. The ournal of Operational Risk 2(3), Lehann, E.L. (983). Theory of Point Estiation. Wiley & Sons, New York. Luo, X., Shevchenko, P.V. and Donnelly,.B. (2007). Addressing the Ipact of Data Truncation and Paraeter Uncertainty on Operational Risk Estiates. The ournal of Operational Risk 2(4), Luo, X. and Shevchenko, P.V. (2009). Coputing Tails of Copound Distributions using Direct Nuerical Integration. To appear in the ournal of Coputational Finance. Mignola G. and Ugoccioni, R. (2006). Effect of a data collection threshold in the loss distribution approach. The ournal of Operational Risk (4), Moscadelli M. (2004). The odelling of operational risk: experience with the analysis of the data collected by the Basel Coittee. Preprint, Banca d Italia, Tei di discussione No. 57. Peters, G.W., ohansen, A.M. and Doucet, A. (2007). Siulation of the annual loss distribution in operational risk via Paner recursions and Volterra integral equations for value-at-risk and expected shortfall estiation. The ournal of Operational Risk 2(3), Peters, G.W. and Sisson, S.A. (2006). Monte Carlo sapling and operational risk. The ournal of Operational Risk (3), Peters, G.W., Shevchenko, P.V. and Wüthrich, M.V. (2009a). Model uncertainty in clais reserving within Tweedie's copound Poisson odels. ASTIN Bulletin 39(), -33. Peters, G.W., Shevchenko, P.V. and Wüthrich, M.V. (2009b). Dynaic operational risk: odeling dependence and cobining different data sources of inforation. The ournal of Operational Risk 4(2), Robert, C.P. and Casella, G. (2004). Monte Carlo Statistical Methods. 2nd Edition. Monte Carlo Statistical Methods. 2nd Edition. New York. Roberts, G.O. and Rosenthal,.S. (200). Optial scaling for various Metropolis-Hastings algoriths. Statistical Science 6, Roberts, G.O. and Rosenthal,.S. (2006). Exaples of Adaptive MCMC. To appear in. Cop. Graph. Stat. Available at 9

20 Schaller, P. and Tenov, G. (2008). Fast and efficient coputation of convolutions: applying FFT to heavy tailed distributions. Coputational Methods in Applied Matheatics. 8(2), Shevchenko, P.V. and Wüthrich, M.V. (2006). The Structural Modelling of Operational Risk via the Bayesian Inference: Cobining Loss Data with Expert Opinions. The ournal of Operational Risk (3), Shevchenko, P.V. (2008). Estiation of Operational Risk Capital Charge under Paraeter Uncertainty. The ournal of Operational Risk 3(), Shevchenko, P.V. (2009). Ipleenting Loss Distribution Approach for Operational Risk. Preprint arxiv: available on Tenov, G. and Warnung, R. (2008). A coparison of loss aggregation ethods for operational risk. The ournal of Operational Risk 3(),