APPLICATION OF STATISTICAL MODELING IN INSURANCE PROCESS

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1 УПРАВЛЕНИЕ И УСТОЙЧИВО РАЗВИТИЕ 2/202 (33) MANAGEMENT AND SUSTAINABLE DEVELOPMENT 2/202 (33) APPLICATION OF STATISTICAL MODELING IN INSURANCE PROCESS Riga Techical Uiversity, Riga, Latvia Abstract Risk assessmet is oe of the maor challeges that must be addressed by each isurace compay. To assess risk we eed to kow the value of losses as well as the probability of losses, sice the risk cost is the basic compoet i evaluatig the isurace idemity. Statistical methods should be used for obective evaluatio of isurace processes, but because of complexity i real life processes of isurace, statistical modellig techiques would be preferable. It is particularly importat to develop ad practically apply these methods i Latvia as i recet years (startig from 992) the isurace market i Latvia has experieced steady growth. To improve the competitiveess of the isurace compaies, especially small compaies, it is simply impossible to do without methods allowig us to estimate the parameters of the isurace process. Takig this ito cosideratio it becomes importat to study iformatio systems related to the processes of isurace ad to use moder iformatio techologies for processig the available empirical iformatio ad the dyamic sceario forecastig performace of the isurace process takig ito accout differet assumptios about the factors that could affect the isurace process. The article deals with the various statistical models that assess the risks ad losses of the isurace compay allowig us to simplify the calculatio of isurace premiums, isurace reserves ad assess the fiacial stability of the isurace compay with a sufficietly wide rage of parameters of the real process of isurace. At the preset time trasitio from local iformatio systems to corporate iformatio systems based o etwork techologies is beig accomplished i the Baltic coutries. Therefore, i the future it is importat to iclude such statistical models ito the itegrated Europea iformatio system of processig isurace iformatio. Key words: risk, statistical modelig, isurace compay, fiacial stability, traiig process. Itroductio Risk assessmet is oe the basic tasks to be tackled by ay isurace compay that wats to remai stable i the isurace market. To assess the isurace risk it is ecessary to kow the value ad the probability of losses sice the value of the isurace risk is the key compoet i assessig the isurace idemity. For obective evaluatio of isurace risks mathematical statistical methods ad methods of actuarial mathematics should be used. The complexity of real life processes of isurace substatiates the ecessity to apply the statistical modellig methods which due to the developmet of computer techologies are beig more widely used i modellig isurace processes at all levels, startig with small isurace compaies ad edig with modellig of isurace processes at the level of big isurace compaies. The developmet ad applicatio of statistical methods is of particular importace i Latvia sice i recet years (startig from 992) the isurace market i Latvia has experieced steady growth. I 200 the volume of o-life isurace grew up to 35 millio lats (448 millio Euros). I 200 isurace idemities costituted 93, millio lats (274,8 millio Euros). However, durig the first two moths of 20 the volume of gross premiums totalled up to 3,46 millio lats (44,76 millio euros) which is 35% less tha durig the same period i 200. To a certai extet it may be explaied by fallig prices of the isurace policies as well as by the overall fiacial istability i Europe. I 20 three-quarters of the isurace compay was workig with profit of,5 millio lats (2,3 millio euros). Isurace compaies gross premiums writte by the 20th the first three quarters year-o-year icrease of 27,7% ad was 76,8 millio. lats, as well as the amout of gross claims paid icreased by 2,% ad was 95,3 millio lats. Further improvemet of the competitiveess of the isurace compaies caot be realized without applyig methods which could estimate the parameters of the isurace process to guaratee sufficietly accurate ad adequate decisio-makig process. This research deals with the various statistical models that assess the risks ad losses of the isurace compay ad allow us to accurately eough assess isurace premiums, isurace reserves ad the fiacial stability of the isurace compay for a sufficietly wide rage of parameters of the real process of isurace. 96

2 Approach to task modellig The first works o the mathematical theory of isurace were published by F. Ludberg ad X. Cramer who proposed ad ivestigated the socalled classical model of the isurace process. The classical model allows us to calculate the probability of rui ad survival of the isurace compay, the priciples of choice of premium load ad aalyzes the survival time, the probability of isured accidet, isurace rates ad isurace claims. This paper focuses o the developmet of ecoomic ad mathematical models of o-life isurace ad their statistical modellig to estimate the overall losses of the isurace compay. Let us cosider the model of idividual risk, which ca be schematically represeted as follows (see Fig. ). Fig.. Scheme of structure of idividual risk model Leged: umber of cotracts of isurace portfolio; idex of the cliet; q probability of the isured evet; q N idex of the isured evet, i the simplest case N ; N umber of realized isurace evets 0 q N N ; X losses of the cliet with a idex, X, havig the distributio fuctio F(x); Y isurace idemity for the cliet with a idex losses, Z Y Y N X ; Z total compesatio (of losses) of the isurace portfolio, i i i N X ; - level of guaratees of the isurace compay (usually i the rage of 0,8 to 0,95); U0, U value of the iitial isurace fud ad after a certai period of time. It is assumed that for the isurace portfolio the followig coditios are met: - umber of cotracts i the portfolio is costat; - risks to customers are idepedet of each other; - all paymets are made without delay; - fuctio F(x) is equal for all cliets. Isurace portfolio modellig Makig use of the simplest Mote Carlo method whe modellig the isurace portfolio with parameters: =000, q=0,, F(x) - fuctio of a uiform distributio i the iterval (0; 000) whe assessig the average losses, dispersio of losses ad coefficiet of variatio of the isurace portfolio, the relative errors compared with the exact results are as follows: (see Table ). 97

3 APPLICATION OF STATISTICAL MODELING IN INSURANCE PROCESS Table. Isurace portfolio modellig results Model Theory Relative errors M(Z) 49934, e_m(z)= 0,3% D(Z) e_d(z)=,24% Kvar(Z),9%,% e_kvar(z)= 0,75% Isigificat relative errors (less tha 2%) idicate the possibility of a sufficietly accurate aalysis of the simplest isurace portfolio usig the statistical Mote Carlo method. Further the possibility of usig the statistical Mote Carlo method as a alterative to aalytical methods for studyig more complex isurace processes will be show. To a large extet the isurace fud depeds o how well the calculatio of isurace premiums is doe. To state the fiacial stability of the isurace compay it is ecessary to satisfy the followig iequality: U Z U0 P N X 0 () with the give probability (usually =0, or 0,05). Sice U0>0, the iequality () will follow from the iequality: P N X 0 (2) Kowig the distributio F of the variable N. X P variable, we may fid such value C of the, where with probability, the iequality (2) as well as the iequality () hold. The value C shows the required level of aggregate premiums esurig the stability of the isurace compay with probability. C F ( ) (3) I may real life situatios the aalytical solutio of the equatio (3) turs out to be a complex mathematical problem ot always havig precise or sufficietly precise solutio. I this case, a good alterative is the Mote Carlo method. Assume that it is ecessary to evaluate the possibility of reducig the rui, usig the process of reisurace i the followig case: the isurace portfolio cotais N isurace cotracts for year from which the isurace sum of N cotracts is S ad the isurace sum of N2 cotracts is S2. The probability of a claim is equal to q. We assume that the level of deductibles is C. Let us compare the solutio of this problem by a) aalytical method ad b) usig the Mote Carlo methods: a) where N=8000, N=5000, N2=3000, S=0000Ls, S2=20000Ls, p=0,02 due to reisurace whe С=6000Ls, the compay seeks to reduce the probability of rui from 0,4 to 0,3; b) havig applied for modellig the Mote Carlo method, we obtai a fairly accurate (R 2 =,0) regressio depedece of the probability of bakruptcy depedig o the value of losses (see Fig. 2). Fig. 2. Rui probability of the isurace portfolio without reisurace the dotted lie ad polyomial approximatio solid lie 98

4 After the itroductio of the reisurace process the probability of bakruptcy decreased from 0,5 to 0,3 (see Fig. 3). This agrees well with the result obtaied by the aalytical method, which i this case is rather timecosumig ad requires a good kowledge of actuarial mathematics. Comparig the two methods for solvig this problem coclusio may be made that the applicatio of the Mote Carlo method is simpler tha usig the aalytical method. Modellig esures quicker ad easier adaptatio to various chages i the isurace situatios which is practically very difficult to reach usig aalytical methods. Applicatio of the modellig methods does ot ecessarily require kowledge of the aalytical represetatio of the distributio fuctios, thus kowledge of the empirical distributio fuctios is quite sufficiet (i.e., the existece of empirical iformatio about the isurace situatio i the coutry, about the values of isurace claims, etc.). If there is o empirical iformatio about the losses of the isurace compay which is typical at the iitial stage, cosider usig bechmarkig, fidig ad makig compariso with a more or less similar isurace compay i the give coutry or i the world. Experiece shows that stable isurace compaies i the same cluster of isurace (havig approximately the same volume of services ad providig the same kids of isurace services, havig the same isurace strategy) are similar ad have very similar characteristic parameters. The most complex subects for the study of fiacial stability are the models of collective risk. The theory of collective risk was developed i 909 by a small group of actuaries, maily Scadiavia. I the theory of collective risk a isurace compay is see as a reservoir which produces a cotiuous stream of premiums ad from which paymets are made. The model co- Fig. 3. Rui probabilities of the isurace portfolio without reisurace the dotted lie ad with reisurace the solid lie sists of the followig three elemets:. the flow of premiums P (t) - the total amout of premiums received durig the period (0, t); 2. q (N, t) - the probability that the th paymet will be claimed durig the period (0, t); 3. G (x) - probability with which the paymets are made, ad the amout paid does ot exceed x. From these three elemets it may be derived that the probability for paymet i the iterval (0, t) does ot exceed x what ca be represeted as a fuctio F (x, t): F (x,t) q(, t)g (x), (4) 0 where G (x) for N>0; th covolutio of the fuctio G(x) и G (0) =H(x) (Heaviside fuctio). For the statistical simulatio of correlated radom variables with distributios derived from empirical data, the authors used the method of copulas. The copula for radom values X, X2,, X ca be described by equatio: C(u,u,...,u ) (F (x ),F (x ),...,F (x )), (5)

APPLICATION OF STATISTICAL MODELING IN INSURANCE PROCESS where Fi - margial distributio for radom value Xi, i=,2,,. The algorithm of simulatio of radom - dimesioal vector X=(X, X2,, X) is: I.

.., ( - (- log(v)/x)). Frak, Clayto ad the Gumbel copula ca be simulated usig this procedure. For example, for the Clayto copula simulatio the algorithm is as follows: I.

5 APPLICATION OF STATISTICAL MODELING IN INSURANCE PROCESS where Fi - margial distributio for radom value Xi, i=,2,,. The algorithm of simulatio of radom - dimesioal vector X=(X, X2,, X) is: I. Simulate a variable X with distributio fuctio G such that the Laplace trasform of G is the iverse of the geerator. II. Simulate idepedet variates V,..., V. III. Retur U=( - (-log(v)/x),..., ( - (- log(v)/x)). Frak, Clayto ad the Gumbel copula ca be simulated usig this procedure. For example, for the Clayto copula simulatio the algorithm is as follows: I. Simulate a Gamma variate X~Gamma(/θ,)). II. Simulate idepedet stadard uiform variates V,..., V. III. Retur U = ((-log(v)/x) - /θ,..., (- log(v)/x) - /θ)). The modelig of radom vector X=(X, X2, X3) has bee realised by usig of MatLab programme. The algorithm of simulatio of radom vector X=(X, X2, X3) with kow ρ(rho) correlatio matrix is: MatLab code: =5000; Rho=[ -0,47-0,522; -0,47 0,420; - 0,522 0,42 ]; Z=mvrd([0 0 0], Rho, ); U=ormcdf(Z,0,); X=[logiv(U(:,),4.75,.32) logiv(u(:,2),2.53,0.55) wbliv(u(:,3),0.24,.6)]; plot3(x(:,),x(:,2),x(:,3),'.'); grid o; view([-50, 50]); xlabel('izm'); ylabel('norizm'); zlabel('laiks'); The illustratio of the process of modellig of icidetal value C= (C, C2, C3) is preseted i Fig. 4. Fig. 4. Examples illustratig the process of modellig of icidetal value X=(X, X2, X3) with N=5000 ad Mote-Carlo trials Fig. 4 shows how differet from the usual distributio the real oit distributio of three correlated radom variables ca be. I this case the oparametric method of histograms is the most appropriate. By meas of a histogram, margial distributios are represeted for costructig a copula, presetig a commo distributio of factors. I the simplest case, distributio of each icidetal value may be represeted by meas of a oparametric method a block chart. Coclusio The research of the authors shows that uificatio of actuarial calculatios after the creatio of a software product that implemets the applicatio of Mote Carlo methods of statistical modelig to actuarial problems is possible. The applicatio of Mote Carlo statistical methods is more atural ad easier to deal with whe solvig urget tasks of the isurace process. Settig obectives ca be realized i a laguage close to the descriptio of the real life isurace situatio, which allows greater ad more flexible practical applicatio of methods of actuarial mathematics i real life. Methods for solvig 00

6 problems cosidered i this research ca be applied i the traiig process for studets of ecoomic ad egieerig specialties. It is obvious that aalytical study of isurace processes described by fuctios of such kid may be performed oly uder some specific assumptios. It should be oted that i the real life isurace process the character of distributios of radom variables is ofte ot described by ay kow closed aalytical distributios. I this case the Mote Carlo method ca also be used to ivestigate the collective risk. Researches of the collective risk models coducted for educatioal purposes showed good agreemet with those obtaied by aalytical methods (with the umber of Mote Carlo trials the relative error costitutes <5% -0%). Refereces. Cleme, R., Reilly, T. Correlatios ad copulas for decisio ad risk aalysis. Maagemet Sciece vol. 45. pp Embrechts, P., Lidskog, F., McNeil, A. Modellig depedece with copulas ad applicatios to risk maagemet. I: Hadbook of Heavy Tailed Distributios i Fiace. Ed. E. Rachev Elsevier. pp Glaz, J. Approximatios for the Multivariate Normal Distributio with Applicatios i Fiace ad Ecoomics. I: Applied Stochastic Models ad Data Aalysis. G. Govaert, J. Jasse ad N. Limios, eds. Uiversite de Techologie de Compiege, Compiege, Frace Volume Harrigto, S., Niehaus, G. Risk Maagemet ad Isurace. New York p. 5. Jasos, V., Dideko, K., Jureoks, V., Isurace as a tool for steady developmet of agriculture. VIII Iteratioal scietific coferece Maagemet ad Sustaiable Developmet. Bulgaria p Jasos, V., Jureoks, V., Dideko, K. Ivestigatio of Ecoomic Systems usig Modellig Methods with Copula. The 3 th Iteratioal Coferece o Harbor Maritime Multimodal Logistics Modellig & Simulatio HMS 200. October 3-5. Fez, Morocco. pp Jureoks, V., Jasos, V., Dideko, K., Applicatio of Bechmarkig ad Idex Method i Research of Ecoomic Systems. X Iteratioal Scietific Coferece Maagemet ad Sustaiable Developmet. March 2-23, Yudola, Bulgaria pp Jureoks, V., Jasos, V., Dideko, K., Modellig of Stability of Ecoomic Systems Usig Bechmarkig ad Dyamic Programmig. X Iteratioal Coferece o Computer Modellig ad Simulatio EUROSIM/UKSim. -3 April, Cambridge, Uited Kigdom pp Jureoks, V., Jasos, V., Dideko, K., Ivestigatio of Ecoomic Systems usig Modellig Methods with Copula. XI Iteratioal Coferece o Computer Modellig ad Simulatio UKSim. March 25-27, Cambridge, Uited Kigdom pp Lidskog, F. Modellig Depedece with Copulas, Master Thesis-MS , Departmet of Mathematics, Royal Istitute of Techology, Stockholm, Swede Par ieguldīumu pārvaldes sabiedrbu u ieguldīumu fodu darbbas rādītāiem 20. gada 3. Ceturksī. [ pazioumi_masu_iformacias_l/202/ _ par_ieguldiumu_parvalde/full_ver/]. 2. Pettere, G., Jasos, V. Stochastic aalysis of isurace liabilities. 9-th Iteratioal Vilius Coferece o Probability Theory ad Mathematical Statistics Ray, P. A Practical Guide to Multi-risk Crop Isurace for Developig Coutries. U.S.A. Sciece Publishers Ic p. 4. Sklar, A., Radom variables, oit distributios, ad copulas. Kyberetica pp Корнилов, И. Основы страховой математики. М. Юнити-Дана

Subject CT1 Financial Mathematics Core Technical Syllabus

Subject CT1 Financial Mathematics Core Technical Syllabus Subject CT1 Fiacial Mathematics Core Techical Syllabus for the 2018 exams 1 Jue 2017 Subject CT1 Fiacial Mathematics Core Techical Aim The aim of the Fiacial Mathematics subject is to provide a groudig