Operational risk capital and insurance in emerging markets

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The important advantage of this formulation is that it can condition the frequencydistribution to various control values. In Figure 2 for example Control 1 is set to itsmean value and Control 2 is allowed to vary between 0 and 1 (poor to best-practicelevels). The result is a joint probability distribution of the number of events (n) andControl 2. The lower the quality of controls, the higher is the probability of occurringmore than three loss events. Conversely, when controls are close to best-practice,the probabilities are highly concentrated around zero loss occurrences.Figure 2: Conditional Bivariate Distribution (Frequency and Control 2)Figure 2: Conditional conditional Bivariate Distribution on Control (Frequency 1 and Control 2)conditional on Control 10.80.70.7Join Probability0.60.50.40.30.20.1010.60.50.40.30.20.50.1Control 200 1 2 3 4 5 6 7 8Severity modelingHypothetical loss severity amounts were simulated together with three determiningkey factors: net incomes, the level of complexity of business line operations andlevel of personnel experience in the management of the activity subject to risk. Netincomes (NETY) are positively correlated to the severity of events. The businesscomplexity level (COMPLEX) is a continuous index between 1 and 5. The higherthe business complexity is, the higher the possible severities are. The experiencelevel (EXPR) varies in the continuous interval 1 to 4. The higher the experience levelis, the lower the severities are. Table 3 summarizes the simulated data.Table 3: Descriptive statistics of severity of loss and constraint variablesSBS Revista de Temas Financieros 42
A straightforward way to model the severity loss distributions conditional on therisk indicators outlined in this example is the multivariate Log-Normal. In thisdistribution, the logarithm of loss amounts together with suitable transformationsof key risk indicators can be model as a multivariate normal 20 .Given the conditional nature of this procedure, effects like those presented in Figure3 can be shown. There, the distribution on the left depicts a low-risk situation(benchmark) with NETY = 1 200 US$ thousands, COMPLEX = 2.5 and EXPR= 3.5. The one on the right corresponds to a relatively high-risk situation of thehypothetical bank under study here (NETY = 2 000 US$ thousand, COMPLEX = 4,EXPR = 2). In the low-risk situation, if there is an adverse event, the mean amountof loss is US$ 58 thousand; whereas in the high-risk case the mean increases toUS$ 148 thousand.Figure 3: Conditional loss severity distributionsFigure 3: Conditional loss severity distributionshigh-risk factorslow-risk factors0.020.015Probability0.010.00500204060 80 100 120 140Amount of losses in thounsands of US Dollars160180200Aggregate LossAt this stage it is important to recall the two components of the aggregate lossdistribution modeling:• The frequency of loss events is modeled via a conditional Poisson distributionWith λ conditional to two control variables: Quantity and quality of controls.• The severity of loss events is modeled via a Log-Normal multivariate model.Namely, the logarithm of loss amount x (0) = In(z) is distributed according to anormal distribution conditional on risk indicators such us net income, businesscomplexity levels and personnel experience 21 .20 Of course more interesting multivariate distributions can be used for severity modeling. For the purposes of this paper multivariate normal are enough. To modelall the variables as normals, the original data is monotonically transformed (Appendix B contains the details).21 See Appendix B for details of the log-multivariate modeling.SBS Revista de Temas Financieros 43
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