Dependent Events and Operational Risk

Transcription

1 Dependent Events and Operational Risk Miro R. Powojowski, Diane Reynolds and Hans J.H. Tuenter Most people acknowledge that soe correlation between operational losses exists. Modelling it, however, is a coplex issue. The questions arise: How can this be done? Is it beneficial? As is shown, including correlations in the calculation of operational capital can have a significant effect on the level of reserve capital required to cushion against the risk of operational losses. This article focuses on positively correlated events that arise fro a single root cause. It also provides an exaple of how such events ight be incorporated into operational risk easures. Intuitively, operational loss events are correlated. Consider, for exaple, an electrical failure. This failure ight affect all desks on the trading floor. Each desk would need to file a clai for lost incoe during the downtie and possibly for tie required to ensure that all previous trades have been processed correctly. Clearly, this single event ust be capitalized, but by which profit center? In attepting to allocate capital fairly aong businesses, a ore coplex odel is required one that allows for coon events to affect ultiple business units. In fact, as shall be seen, failure to acknowledge and account for this type of correlation between losses experienced in different businesses and other positive correlations between events leads to an understateent of the capital reserves that are required. One ight argue that this increase in capital charge is undesirable. However, the goal of risk-sensitive capital reserves should be just that: to reflect properly all of the risk assued. Also, it is entirely conceivable that charges related to correlations will becoe part of the regulatory process, as is the case for credit risk. This article presents a siple odel that allows for the incorporation of positive correlations between operational units. However, before the odel can be specified, soe groundwork is required. First, one ust define a set of operational units to be odelled. For a coplete definition of operational units, see Reynolds and Syer (2002a). Coon exaples include the event types and business lines defined by the Basel Coission on Banking Supervision actual organizational structures, and geographical locations. Each operational unit is allocated its related data. Typically, this would be loss data, but other data ay also be used. Second, an analysis of the data is done to deterine whether one operational loss process per operational unit is sufficient, or whether several processes should be used to obtain a better fit to the data. Frequently, however, due to the paucity of data, this analysis results in each operational unit being odelled by a ALGO RESEARCH QUARTERLY 65 VOL. 5, NO. 2 SUMMER 2002

2 single loss process, and independently of the other units. As with any advanced approaches to the easureent of operational risk, an actuarial odel is used. This is described in Reynolds and Syer (2002b). This approach divides the operational loss process into two coponents: frequency, which is the nuber of losses per period, and severity, which is the size of an individual loss. Each line of business, or event type, is assigned its own odel for frequency and severity of losses that occur within an operational unit. By correlating the frequency of loss across operational units, one effectively correlates the total loss across operational units. The odel presented here focuses on the correlation of loss frequency across operational units through an underlying coon cause approach. The operational loss processes are odelled by representing the loss frequency as a cobination of Poisson distributions, while allowing each severity distribution to follow any data-appropriate distribution. A discussion of the Poisson distribution and its principal properties provides essential background to understanding the overall odel and its calibration. Following an introduction to the Poisson distribution, the joint-frequency odel is presented and discussed using a sall exaple. A ethod of calibrating the joint odel is then presented, and the exaple is extended to enrich the discussion. The paper concludes with soe thoughts for future directions of research. Poisson distribution Naed after the French atheatician Siéon Poisson ( ), the Poisson distribution describes the nuber of events that occur in a given interval of tie, when the probability of the event occurring is very sall, but the nuber of trials is very large. The Poisson distribution is forally defined as Pn ( ) ( λt) = n e λt n!, n = 0, 1, 2,, where P(n) is the probability that n events happen over a tie interval of length t, and s the intensity or rate at which events happen per unit of tie. The distribution was studied ore rigorously by the Russian-born statistician Ladislaus von Bortkiewicz ( ), in his onograph, Das Gesetz der kleinen Zahlen. In his book (Bortkiewicz 1898), one can find an early, and perhaps the first, application of the Poisson distribution to operational risk. Von Bortkiewicz analysed data, collected on the nuber of soldiers kicked to death each year by horses in the Prussian ary, and found an extreely good fit with the Poisson distribution. Although today s applications of the Poisson distribution to odel the frequency of operational losses are less draatic, it is still a viable and widely used odel. One reason is that it is a siple odel, since only one paraeter ust be estiated to specify the Poisson distribution its ean. The property that the variance is equal to the ean is iportant for ease of calibration, as is seen in a subsequent section. Another iportant property of the Poisson distribution is that it is a stable distribution: adding two Poisson processes with intensities λ 1 and λ 2 creates another Poisson process with intensity λ 1 + λ 2. The Poisson distribution is well docuented and, hence, one ay draw on a large body of existing knowledge, see for instance, Haight (1967), Johnson and Kotz (1969) and Kingan (1993). Dependent processes with Poisson arginals Consider a set of operational events. Modelling any types of events siultaneously, to account for any types of operational loss processes and any business initiatives, is critical. This section provides a discussion of a bivariate odel, and then generalizes it to a ultivariate forulation. It concludes with a worked exaple based on the event types dictated by the Quantitative ALGO RESEARCH QUARTERLY 66 SUMMER 2002

3 Ipact Study (QIS) of the Basel Coittee on Banking Supervision (BCBS). See the study by the Bank of International Settleents (2001). Two operational loss processes First, consider the events originating fro two operational loss processes Y 1 and Y 2, and odel these events by two separate Poisson distributions. For a fixed period of tie, over which the events are recorded (typically one onth or one year), let N 1 and N 2 be the nuber of events of the operational processes Y 1 and, respectively. Y 2 Under these assuptions, N 1 will follow a Poisson distribution with intensity µ 1, so that the ean and the variance of the nuber of events are both equal to µ 1. The sae applies to N 2, with corresponding intensity. To coplete the specification of the odel, it is necessary to specify the dependence between N 1 and N 2. The siplest assuption is, of course, to assue that they are independent. But, what if experience indicates that the events in Y 2 tend to coincide with the events in Y 1? Fro a risk anageent perspective, the ability to odel siultaneous events in the processes Y 1 and Y 2 then becoes paraount. There is a need for ultivariate arrival processes that are capable of odelling the joint events in any processes. Individually, the events in each process reain Poisson distributed with a separate occurrence rate. One way to do this is to create three underlying loss processes. Assue X 1, X 2 and X 3 are independent Poisson processes with intensities λ 1, λ 2 and λ 3, respectively. If Y 1 = X 1 + X 3, Y 2 = X 2 + X 3 µ 2 (1) each have Poisson arginal distributions, with ean µ 1 = λ 1 and µ 2 = λ 2, respectively. However, the two coponents are now dependent. Their covariance and correlation coefficient are given by: and N 1 = M 1 + M 3, N 2 = M 2 + M 3 Cov( N 1, N 2 ) = Var( M 3 ) = λ 3, ρ Fro this one sees that, by introducing a set of underlying, abstract variables, each following a Poisson distribution, one can create dependent distributions for the frequencies of two loss processes, while aintaining their arginal distributions as Poisson. Multiple operational loss processes λ 3 = ( λ 1 )( λ 2 ) The exaple easily generalizes to the ultidiensional case, where there are several processes. Suppose one has n observed processes Y j, j = 1,, n, for which one wants to odel a dependent structure. To do this, underlying Poisson processes are considered, with intensities,,,, and a corresponding nuber of events M i. Each of these underlying processes can be assigned to one or ore of the observed processes, which can be captured by introducing the indicator variables δ ij : N j = δ ij M i. The nuber of events N j of process Y j then follows a Poisson distribution with intensity (2) is taken as the underlying relationship, a twodiensional process that exhibits dependence is obtained. Denote the nuber of events of type X 1, X 2 and X 3 by M 1, M 2 and M 3, respectively. The distributions of µ j = δ ij. This dependence odel has an intuitive interpretation: it postulates the existence of (3) ALGO RESEARCH QUARTERLY 67 SUMMER 2002

4 events that affect ore than one operational unit. This type of event ay be thought of as a coon-cause event, affecting all processes Y j. The covariance and correlation coefficients are readily deterined as: and, Cov( N j, N k ) = δ ij δ ik ρ jk = δ ij δ ik δ ij δ ij (4) For the sake of copleteness, an eighth category is included, Other Risks, to allow for anything that does not fall into one of the QIS categories. Independent events First, assue that there are eight, independent, underlying loss processes: one specific to each risk category, and that all processes have a Poisson distribution with ean λ = 0.5. Thus, the expected frequency of events is once every two years. Given an event, let the resulting losses be independent and norally distributed with a ean of US $4M and a standard deviation of US $0.5M for all processes. This situation is depicted in Figure 1. The covariance structure is ore easily described in atrix notation as C = Λ T, (5) where C is the covariance atrix, is an n incidence atrix, describing the relationship between the observed and the underlying processes, and Λ = diag( ) is a diagonal atrix with the intensities as the eleents. Note that all eleents of the covariance atrix are nonnegative (and, hence, also all correlations), because all the eleents of the atrices and Λ are nonnegative. Monte-Carlo siulations To illustrate the odel, we eploy the sae classification schee used by the BCBS in collecting data for QIS. It divides risk into seven broad categories: Internal Fraud External Fraud Eployent Practices Business Services Physical Assets Business Disruption, and Process Manageent. Figure 1: Independent loss processes; each process follows a Poisson distribution with intensity λ = 0.5 To gain insight into the shape of the distribution of the copany-wide losses, a Monte-Carlo siulation with 10,000 scenarios was perfored. The resulting, epirical distribution is depicted in Figure 2. The 99% quantile of this distribution is US $37.82M. Dependent events To create dependencies between the processes, take the previous odel, but with intensities 0.4 ALGO RESEARCH QUARTERLY 68 SUMMER 2002

5 Independent Events Frequency Copany-wide losses ($M) Mean Sdev 8.15 Skewness 0.51 Kurtosis % Figure 2: Epirical distribution of copany-wide losses under the independent odel, using a Monte Carlo siulation with 10,000 scenarios; the 99% quantile is $37.82M for all the independent loss processes, and introduce a single enterprise-wide source of loss, odelled by a Poisson distribution with intensity λ 9 = 0.1. The nuber of losses, experienced in each risk class, is now a cobination of the overall nuber of losses experienced due to fir-wide issues and the losses specific to the risk class. This odel is depicted in Figure 3. Note that the arginal distributions of the events are still Poisson with one event expected every two years. However, the losses are now dependent with a pairwise correlation of 0.2, as calculated using Equation 4. This iplies that differences in the loss distribution are due solely to the effect of correlations and not to changes in the arginal distribution. Another Monte-Carlo siulation with 10,000 scenarios was perfored, and the resulting epirical distribution is shown in Figure 4. The 99% quantile of this distribution is US $59.19M, while the expected losses reain the sae as in the independent case. Coparison It is evident that the overall budget requireents, as easured by the expected losses, are not affected by the correlation between events. However, the risk, as easured by the unexpected losses, has increased considerably by the inclusion of these fir-wide events. A cursory inspection of Figures 2 and 4 shows that the latter distribution has a considerably fatter tail. The exaple also shows that the effect of dependence between processes on risk easures, such as Value-at-Risk (VaR), ay be considerable even when the correlation between events is relatively low. Paraeter estiation As with any odel designed for practical applications, the value of the odel is influenced significantly by one s ability to calibrate and ipleent it as well as its ability to represent reality. In this case, the odel is easily ipleented, once the paraeters are known, and it is possible to estiate the paraeters using coon ethods. ALGO RESEARCH QUARTERLY 69 SUMMER 2002

6 Figure 3: Dependent loss processes; loss processes specific to a risk class follow a Poisson distribution with intensity λ = 0.4. The fir-wide loss process follows a Poisson process with intensity λ = 0.1 In this section, the calibration ethod is discussed and illustrated by using the Monte-Carlo siulation results for the two odels considered previously. It is well known that the axiu likelihood estiate for the intensity of a Poisson process is the saple ean of the observed values. Ideally, one would like to use the axiu likelihood ethod to obtain estiates for the paraeters in the ultivariate processes. However, the expressions for the likelihood function are not always that easily derived, and the resulting expressions ay be difficult to use in practice. The approach used here is to utilize the axiu likelihood estiates for the intensities of the observed processes, and then incorporate these estiates into an optiization proble. Let µˆ j be the saple ean for the events of the j-th observed process, and thus the axiu likelihood estiate for the intensity µ j. For ease of exposition, assue, for the tie being, that the syste λ = µˆ, λ 0 (6) is solvable. Note that this approach ensures that the observed intensities are atched. Given the above syste of equations, one now wants to deterine the paraeter set that ost closely atches the higher oents, as captured by the covariance atrix. This is done by forulating it as an optiization proble over a distance function If the Frobenius nor is chosen as the distance nor the su of squared distances of the atrix eleents a routine quadratic optiization proble is obtained, which is readily solved by standard optiization software. Illustrative applications Now the calibration ethod is applied to the odels discussed previously. Two processes in Ĉ Λ T s.t. λ = µˆ, λ 0. For the structure given in Equations 1 and 2, the incidence atrix is given by (7) ALGO RESEARCH QUARTERLY 70 SUMMER 2002

7 Dependent Events Frequency Copany-wide losses ($M) Mean Sdev Skewness 1.85 Kurtosis % Figure 4: Epirical distribution of copany-wide losses under the dependent odel, using a Monte-Carlo siulation with 10,000 scenarios; the 99% quantile is $59.19M = and the covariance atrix by (8) Dependent exaple In the dependent odel, shown in Figure 3, the incidence atrix is given by C = λ 1 λ 3 λ 3 λ 2 The optiization proble becoes: in ( Ĉ 11 λ 1 λ 3 ) 2 + 2( Ĉ 12 λ 3 ) 2 + ( Ĉ 22 λ 2 λ 3 ) 2. (9) = , (10) s.t. λ 1 = µˆ 1, λ 2 = µˆ 2, λ 1, λ 2, λ 3 0. and the covariance atrix by C = diag( ) + λ 9 E, where E is a atrix consisting of all ones. Because of the equality constraints, this particular instance is an optiization proble over λ 3 only, and is readily solved. Independent exaple In the independent odel, shown in Figure 1, the incidence atrix is the identity atrix, and the solution to the optiization proble is given by the saple eans of the observed events. The optiization proble becoes: 8 in ( Ĉ ij λ 9 ) 2 + ( Ĉ ii λ 9 ) 2 j i 8 s.t. + λ 9 = µˆ i,,, 8, 0, for all i. ALGO RESEARCH QUARTERLY 71 SUMMER 2002

8 As before, by virtue of the equality constraints, this is an optiization proble over λ 9 only, and is readily solved nuerically. Nuerical exaples To illustrate the optiization approach, 50 scenarios were taken that were generated under the dependent event odel given in Figure 3. The saple covariance atrix was coputed, and the optiization perfored to deterine estiates for the intensities. This resulted in The procedure was repeated with 50 scenarios generated under the independent event odel given in Figure 1, and the following result was obtained: Note that, even though the dependent odel was applied to scenarios generated fro the independent odel, the correct conclusion that there is no coon cause of event is obtained. Conclusions ( λ 1, λ, 9 ) = ( 0.38, 0.48, 0.40, 0.56, 0.44, 0.38, 0.26, 0.38, 0.12 ). ( λ 1, λ, 9 ) = ( 0.32, 0.32, 0.42, 0.58, Correlations have a significant ipact on capital calculations. In order to ensure that risk-sensitive capital allocations are fair to all businesses, special odels are required. One ust be able to calculate correlations accurately and defensibly, and to use this inforation in deterining and allocating capital. This paper presents a siple and easily calibrated odel for including positive correlations. It deonstrates, by eans of an exaple, the significant influence that positive correlations can have on required capital. Models such as these, which allow intuitive relationships to be odelled to provide ore risksensitive capital allocation, need to be exained in greater detail if the quantitative easures of operational capital are to achieve credibility within the industry. Acknowledgeents 0.50, 0.50, 0.44, 0.52, 0.00 ). The authors would like to thank Thoas Eigl and Helut Mausser for coents and helpful discussions. References Bank of International Settleents, 2001, Quantitative Ipact Study of the Basel Coittee on Banking Supervision, Bortkiewicz, L. von, 1898, Das Gesetz der kleinen Zahlen, Leipzig: B.G. Teubner. Haight, F.A. 1967, Handbook of the Poisson distribution, New York, John Wiley & Sons. Johnson, N.L. and S. Kotz, 1969, Distributions in Statistics: Discrete Distributions, Boston, MA: Houghton Mifflin Copany. Kingan, J.F.C., 1993, Poisson processes, Oxford Studies in Probability, Vol. 3, New York, NY: Clarendon Press. Reynolds, D. and D. Syer, 2002a, Mark to future and operational risk, Algo Research Quarterly, 5(2): (This issue) Reynolds, D. and D. Syer, 2002b, Algo Acadey Notes, Algo Research Quarterly, 5(2): (This issue) Endnotes 1. The feasibility of the syste of equations in Equation 6 can be verified by an application of Farkas lea. In the case that this syste does not have a solution, we can incorporate the first constraint in the objective function, using Lagrangian ultipliers or a penalty ethod. These odifications also result in a quadratic optiization proble. 2. The optiization forulation easily allows one to incorporate preferences regarding the relative iportance of eleents of the saple covariance atrix by introducing weights into the eleents of the objective function. Further, the optiization forulation allows one to exaine the effect of different configurations of the underlying processes by taking different incidence atrices. This is a feature that a axiu likelihood ethod does not have. 3. To be able to use the odel, one has to specify the nuber,, of underlying processes, and decide how the underlying processes influence ALGO RESEARCH QUARTERLY 72 SUMMER 2002

9 the observed processes. Although potentially there are 2 n possible ways of doing this, one needs to exercise caution. The paucity of data in the setting of operational risk does not allow for the estiation of too any paraeters, and, hence, there is the need to tri down the odel. In ost cases, the business analysis will help to deterine the nuber of underlying processes and which processes they influence. ALGO RESEARCH QUARTERLY 73 SUMMER 2002