Approaches to modeling operational risks of frequency and severity in insurance

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1 IOSR Joural of Ecoomics ad Fiace (IOSR-JEF) e-issn: , p-issn: Volume 4, Issue 4. (Jul-Aug. 2014), PP Approaches to modelig operatioal risks of frequecy ad severity i isurace Fatima Zahra El Arif 1, Nadia Lamchichi 2, Amie Dafir 3 1 PhD studet Ceter of Doctoral Studies, Faculty of Ecoomic Scieces, Uiversity Mohammed V-Souissi, Rabat, Morocco. 2 PhD studet Ceter of Doctoral Studies, Faculty of Ecoomic Scieces, Uiversity Mohammed V-Souissi, Rabat, Morocco. 3 PhD Ceter of Doctoral Studies, Faculty of Ecoomic Scieces, Uiversity Mohammed V-Souissi, Rabat, Morocco. Abstract: Oe of the great ew of Solvecy II is the requiremet for isurace compaies to raise some of their ow fuds to cover their exposure to operatioal risks. To calculate this amout of capital, the regulator proposes two approaches : a stadard approach ad a advaced approach. The stadard approach is a simplified approach calculated as a percetage of premiums or reserves. The advaced approach is a iteral model where the risk actually correspods to the situatio of the compay. EIOPA ecourages isurace compaies to opt for iteral model by makig the stadard approach much more cosumig i stockholders' equity. Thus, we propose i this paper a modelig by advaced approach that distiguishes frequecy risks of severity risks. Frequecy risks are defied as potetial loss of low but frequet amouts : they are modeled by the Loss Distributio Approach. Severity risks are risks of losses of very importat but very few amouts : they are modeled by Bayesia etworks. Keywords: Operatioal risk, stadard approach, advaced approach, frequecy risks, severity risks, Loss Distributio Approach, Bayesia etworks, isurace. I. Itroductio The defiitio of operatioal risk is a challege. This risk has a atypical character as far as it cocers all the activities of the isurace. It is also ofte difficult to estimate it ad quatify it idepedetly of other risks which characterizes the activity of isurace. Ideed, the maagemet practices ad measure of this risk are based very heavily o what happes o the market risk, the maturity of which is much more importat. However, approaches (such as Loss Distributio Approach or LDA, ad Bayesia etworks) are becomig icreasigly importat to maage ad quatify this risk, especially sice the establishmet of the systems of collectio of losses. To idetify the methodology to be used, the type of risk is idetified durig the risk mappig ad should be reviewed periodically. Frequecy risks are defied as potetial loss of low but frequet amouts : they are modeled by the Loss Distributio Approach. Severity risks are risks of losses of very importat but very few amouts : they are modeled by Bayesia etworks. However, the risks of importat ad frequet losses will ot be modeled, because whe a severity risk arises, the isurace compay will establish checks ad other actio plas, that will reduce or elimiate this risk (black box) [Fig. 1]. II. Modelig Of Frequecy Risk : Loss Distributio Approach Loss Distributio Approach is to adjust statistical laws to data loss, specifically modeled o the oe had, the frequecy of operatioal icidets, ad secodly, their severity, ad the combie them to obtai the distributio of total losses. This approach is ofte used to model the total cost of claims i the cotext of pricig ad provisioig. The essetial coditio for applyig this method will be the availability of historical losses data to calibrate the model. II.1. Step 1: selectio ad calibratig of frequecy ad severity laws Modelig the distributio of losses will for each risk k : S k = j=1 Where N k is the radom variable represetig the umber of losses for risk K, N k X k (j) 49 Page

2 X k (j) is the radom variable represetig the cost of the loss j for risk k, S k is the sum of the losses of risk k. The mai laws used for frequecy are usually Poisso, Biomial ad Negative Biomial. There is a ear cosesus place to use the method of maximum likelihood to estimate the parameters of the law. The choice of model is the validated by statistical tests. II.2. Step 2 : costructio of total losses distributio It is possible to obtai a good approximatio of the distributio fuctio of the distributio of losses S k, with the method of Mote-Carlo simulatio [Fig. 2]. II.2.1. Practical problem : the collectio threshold Geerally, operatioal will ot report all losses: it will the establish a reportig threshold. The observatios which we will have will be losses that exceed this threshold. It will thus take ito accout this trucatio i our adjustmets ad chages i estimates of parameters ad fit testig accordigly. Ideed, if we do ot take ito accout the fact that the data is trucated, we will uderestimate the frequecy, ad adjust the law of the idividual costs icorrectly. II.2.2. Law of idividual cost The data below the threshold are ot ski, but must be take ito accout i the estimatio of parameters : it will estimate the parameters of the law takig ito accout the lack of data o the left of the histogram as we watch i the graphic [Fig. 3]. Let U be the collectio threshold losses. Let X be the radom variable represetig the cost of o-trucated losses (that is to say losses, the amout may be less tha U) ad F its distributio fuctio. This coditioal law is the law of X trucated to the left of U. We ca explai the distributio fuctio of the trucated law kowig that the observatios are beyod the threshold U : For all x > U P X < x X U = For all x U P X < x X U = 0 P(U X < x) P(X U) = F x F(U) 1 F(U) The graph [Fig. 4] shows the deformatio of a log-ormal law l (2, 1), depedig o the level of collectio threshold 1. Let (x 1, x 2,..., x ) a sample of losses that exceed the threshold U, the coditioal likelihood i threshold U ca be writte : f(x i ) P(X i U) = f(x i ) 1 F(U) The log-likelihood ca be the writte : l f(x i) 1 F(U) = l(f(x i) ) l(1 F(U)) The parameters are estimated i a covetioal maer by maximizig the log-likelihood. II.2.3. Law of frequecy Where there is a collectio threshold, the frequecy of observed losses is uderestimated ad must be adjusted to accout for ureported losses. For each observatio period i, the umber of claims oted i is icreased by the estimated umber of claims below the threshold oted m i : i r = i + m i 1 Note : Some distributios are stable by a trucatio to the left as the Expoetial ad Pareto laws. The method of maximum likelihood allows us to calibrate our laws, but does ot give us a aalytical solutio. 50 Page

3 The ratio betwee the umber of losses below the threshold ad the umber of losses observed, is the same as the ratio betwee the area below the curve of the desity of the idividual cost situated to the left of the threshold, ad that was situated to the right of the threshold : m i i = F(U) 1 F(U) Where F is the fuctio of the trucated distributio of X, whose parameters have bee estimated with the previous method. From where : i r = i + m i = i + i F(U ) 1 F(U) = i 1 F U III. Modelig Of Severity Risk : Bayesia Method The Bayesia approach is to coduct a qualitative risk aalysis by experts ad tur it ito a quatitative aalysis. A Bayesia etwork is a probabilistic causal graph represetig the structure of kowledge i a certai field. It cosists of discrete radom variables coected by directed arcs, these variables are called odes. Distributio is attached to each ode. Arcs are liks that represet causal depedece. We studied i particular the method XOS (exposure, occurrece, severity), which is to defie ad model the three characteristic parameters of risk : the exposure, the occurrece ad the severity. These three variables are iflueced by variables called Key Risk Idicator (KRI). We are goig to preset the methods to estimate the various elemets of the Bayesia etwork. Assessmet of exposure : The exposure is all the elemets of the compay which are exposed to the risk. She must be defied so that the risk ca arise oly oce i most i the year. Assessmet of occurrece : The item of expositio beig selected so that it ca be stukch oly i most oly oce, the occurrece will be built by a biomial distributio B (, p), where is the umber of exhibits ad p the probability which we shall have to estimate. Assessmet of severity : It is ecessary to take place i the situatio where the occurrece of the loss is prove, ad to idetify the quatifiable variables (KRI) occurig i the calculatio of the gravity. The structure of the Bayesia etwork is defied by the experts through scearios. Bayesia etwork parameters ca be determied empirically or by experts. Oce the Bayesia etwork built, it remais to defie the algorithm of calculatio. Let (X 1, X 2,..., X ) are exposed objects at the studied operatioal risk. Let P i = P(Exposure = X i ) the probability that exposure let be objects X i. Let PS i = P (Occurrece = "yes" Exposure = X i ) the probability of the risk occurrig give that the exposure is X i. These two probabilities are kow (they were estimated as see above). Let PG i is = P (Severity Occurrece = "yes" ad Exposure = X i ) the distributio of the severity kowig that the risk occurred o X i objects. The algorithm cosists i realizig successively the followig steps : 1. positio exposure to X i, the occurrece to Yes i the Bayesia etwork ad read the distributio of severity PG i = P (Severity Occurrece = "yes" ad Exposure = X i ). 2. sample the umber of losses F i accordig to biomial law B (b(x i ) ; PS i ). 3. for each icidet of 1 to F i, sample the severity accordig to the PG i distributio. 4. sum the severities F i. By repeatig these 4 steps a large umber of time by keepig the sums of the severity every time, we so obtai a distributio of total losses. IV. Example Of Applicatio Of The Bayesia Method We ow apply the Bayesia model to the risk of error i the passage of orders, o a isurace compay. This risk is treated very simply because our idea is to show how the Bayesia approach applies. The first step i the Bayesia approach is to create the etwork usig a graph, ad defie the distributios correspodig to each factor. These parameters ca be reviewed i a phase of "back-testig". A aalysis of this risk has allowed us to build the graph [Fig. 5]. Recall that the Bayesia approach proposed for operatioal risk is to defie three objects : the exposure, the occurrece ad the severity 2. 2 The XOS method we described above. 51 Page

4 The exposure : the exposure has to correspod to the objects of expositio of the compay which ca be touched by the risk oly oce for the period. We chose the orders. Ideed, a order ca be erroeous oly oce. We do ot aticipate icrease i the umber of orders for year to come ad the umber of observed orders is o average 25 OOO a year. The occurrece : The occurrece is defied as the error o a order. O average, we have a umber of aual losses of 18.2, is a probability of error o the order of 18.2 / = %. The severity : severity is defied as the sum of the loss due to shiftig ad trasactio costs. We make the followig assumptios about the factors ivolved i the calculatio of the severity. The amout of the error (expressed i millios of euros) : The amout of the error (i millios of euros) 5 66% 15 18% 50 16% The duratio of error correctio : the period from the date of occurrece of the error to the date of correctio by the passage of a ew order : The duratio of correctio (i days) 0,125 66% 1 33% 90 1% The trasactio costs : we assume that trasactio costs are equal to 0.25% of the trasactio amout. The amout of error is passed first time ad a secod time durig the correctio. The fees will be calculated by multiplyig the amout of the error by 0.5%. The rate shifts : it is about the variatio of the rates observed durig the duratio of correctio of the error. The loss due to shift : it is calculated by multiplyig the amout of the error by the rate shifts. The rate shifts The duratio of correctio (i days) 0, % 66% 62% 60% 5% 34% 37,99% 38% 30% 0% 0,01% 12% We ow study the algorithm that we described i the theoretical part : Here, the exposed objects are of the same type, that are orders of the same ature : X i = X, which implies that P i = P (Exposure = X i ) = 1. We deduce the probability from it that the risk arises kowig that the exposure is X i : PS i = P (Occurrece = "yes" Exposure = X i ) = P (Occurrece = "yes") = %. The algorithm cosists i realizig successively the followig steps : 1. calculate the distributio of severity PG i = P (Severity Occurrece = "yes" ad Exposure = X i ). 2. sample the umber of losses F i accordig to biomial distributio B (b(x i ) ; PS i ) = B (25000; 0,0728%). Ideed, orders are idepedet ad each order follows a Beroulli distributio. 3. for each icidet of 1 to F i, sample the severity accordig to the PG i distributio. 4. sum the severities F i. We repeat these 4 stages times by keepig the sums of the severity every time. We so obtai a distributio of total losses ad we ca deduct the VaR from it i 99,5 %, [Fig. 6]. We realized tests of sesibility i the various parameters of the Bayesia etwork : i every set of tests, we vary the parameters of a factor by fixig the parameters of all factors. Bayesia model has may advatages : - It allows to take ito accout both quatitative factors but also qualitative factors, that do ot make most of the models ; - It allows to visualize the causal liks betwee the variables : risk aggregatio is performed by the costructio of etworks, thus avoidig the estimatio of correlatios ; - It allows to detect factors of reductio of the risks through iferece ; - The Risk Maagemet ca use it to develop actio plas ad see the effectiveess of these. 52 Page

Matrix of Prouty for the modellig of frequecy ad severity risks Fig 2.

5 The major drawback of Bayesia etworks is that they are slow to implemet because they require a detailed aalysis of each risk. V. Figures Ad Tables Fig 1. Matrix of Prouty for the modellig of frequecy ad severity risks Fig 2. Costructio of the distributio of the total losses Fig 3. Cost of idividual losses ad the adequacy of the theoretical law adjusted To accout for trucated data 53 Page

6 Fig 4. Deformatio of a log-ormal law l (2, 1) depedig o the threshold level collectios Fig 5. Applicatio of the Bayesia model at the risk of error i the passage of orders o a isurace compay Fig 6. Distributio of total losses VI. Coclusio The quatificatio of operatioal risk ca ot ad should ot be see as a ed i itself. Methodological aspects related to the use of differet approaches available (LDA, Bayesia etworks) ad iterpretatio of parameters that follows, allow a more successful risk maagemet. This is evidet also i terms of both the regulator ad withi isurace compaies. The geeral idea of the LDA method is to model the loss of operatioal risk for a give period (eg, oe year), ad deduce the value-at-risk (VaR). Thus, like most models for measurig operatioal risk, the LDA is based o a very old actuarial approach (frequecy / severity), ad widely used i the field of isurace to model similar problems. Bayesia etworks are tools that have may advatages : they allow to visualize the causal liks betwee variables, they ca take ito accout qualitative variables, they ca maage risks by idetifyig variables levers. This model will be useful to Risk Maagemet to cotrol the risks. The major drawback of Bayesia etworks is that they are slow to implemet. Aother faster solutio to model the risks of severity would be the law of Extremes. 54 Page

7 Refereces [1] Alexader, C., (2003). Operatioal Risk : Regulatio, Aalysis ad Maagemet. Ed. FT Pretice Hall, 336 p. [2] Arzac, R. E., (1976). Profits ad Safety i the Theory of the firm uder Price Ucertaity. Iteratioal Ecoomic Preview, 17, [3] Bee, M., (2006). The Advaced Measuremet Approach to Operatioal Risk. Lodo, Ed. E. Davis, Risk Books, 350 p. [4] Berouilli, D. (1954). Expositio of New Theory of the Measuremet of Risk. Ecoometrica, 22, [5] Chavez-Demouli, V., P. Embrechts ad, J. Nešlehová, (2006). Quatitative Models for Operatioal Risk: Extremes, Depedece ad Aggregatio. Joural of Bakig ad Fiace, 30, 10, [6] Cherobai, A., C. Me, S. Trûck ad S. T. Rachev, (2005a). A Note o the Estimatio of the Frequecy ad Severity Distributio of Operatioal Losses. Mathematical Scietist 30, 2, [7] Coles S., Powell E., (1996). Bayesia methods i extreme value modellig: a review ad ew developmets. Iterat. Statist. Rev. 64, [8] Codami, L., Louisot, J.-P. ; Naîm, P., (2007). Risk Quatificatio. Ed. Wiley, 286 p. [9] El Sayyad, G. M. (1973). Bayesia ad Classical aalysis of Poisso Regressio. Joural of the Royal Statistical Society, Series B, 35, [10] Frachot, A., O., Moudoulaud, T., Rocalli, (2004). Loss Distributio Approach i Practice. The Basel Hadbook : A Guide for Fiacial Practioers. Micheal Og, Risk Books. Electroic copy available at: [11] Naïm P. ; Wuillemi P.-H., Leray P., Pourret O., Becker A., (2007). Réseaux Bayésies. Ed. Eyrolles, Collectio Algorithmes, 424 p. [12] Partrat C., Besso J.-L., (2004), Assurace o-vie, Modélisatio et Simulatio. Ecoomica, 820 p. 55 Page

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