Design of an optimal Bonus-Malus System for automobile insurance & moving between two different scales

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1 Technical Journal of Engineering and Applied Sciences Available online at TJEAS Journal / ISSN TJEAS Design of an optimal Bonus-Malus System for automobile insurance & moving between two different scales Ghodratollah Emamverdi 1*, Melika Firouzi 2, Fatemeh Emdadi 3 1. Assistant Professor of Economics, Islamic Azad University, Central Tehran Branch 2. MSc. student in Actuarial Science, ECO college of Insurance, Allameh Tabataba i University 3. MSc. student in Actuarial Science, ECO college of Insurance, Allameh Tabataba i University Corresponding author Ghodratollah Emamverdi ABSTRACT: This paper focuses on techniques for constructing Bonus-Malus systems in automobile insurance. Specifically, the article presents a practical method for constructing optimal Bonus-Malus scales with reasonable penalties that can be commercially implemented. For this purpose, the symmetry between the overcharges and the undercharges reflected in the usual quadratic loss function is broken through the introduction of parametric asymmetric loss function of exponential type. The resulting system possesses the desirable financial stability property. Key words and phrases: Bonus-Malus system, Markov Chains, exponential loss functions INTRODUCTION BMS, as a kind of experience rating model, has continually been used in automobile insurance market around the world for two main reasons: to cope with adverse selection, forcing high risk policyholders to pay more premiums, in the long run, comparing to the low risk policyholders, and to motivate policyholders to derive more carefully. BMS, in fact, penalizes the policyholders with one or more claims by premium surcharges Maluses) and rewards the claim-free policyholders with premium discounts Bonuses). There are different types of BMS that some of them are based on the scales like the systems being conducted in most European and Asian countries. In these kinds of systems there are several classes, with a relativity assigned to each of them, through which the policyholders move up or down according to the claims they report and the BMS rule during a period one year). Pioneering work in theoretical and practical aspects of bonus-malus systems can be found in Bichsel 196), as well as in Loimaranta 1972). Denuit et al. 2007) gives a comprehensive description of the insurance aspects of bonus-malus systems. The study that led to the Dutch bonus-malus system described in this paper wasdescribed in De Wit et al. 1982). Bonus-malus systems with non-symmetric loss functions are considered in Denuit & Dhaene2001). Central issue Improve car insurance premiums by additionally using the ex post factor claims experience, since otherwise; relatively good clients will take their business elsewhere, leaving only the bad ones. Ex ante factors: Administrative data: type of cover, deductible, Policy holder s data: age, residence, Vehicle data: brand, age, type of fuel, Usage: private or business, It proves that using ex ante risk factors only does not lead to an adequate premium system: some may have been overlooked, others unknown, unobservable or unusable. Remedy: use experience rating to account for the effectof such factors, giving drivers with few claims a bonus and bad drivers a so-called malus. The stochastic process of a driver going up and down on this scale can be described by a Markov system. After the effect of the initial state has vanished, the distribution over the various states is the so-called steady-state distribution. It can be determined using simulation or matrix algebra; it is an eigenvector of the matrix of transition probabilities.one can measure the efficiency of a bonus-malus system by the elasticity of the average premium paid, with respect to the mean number of claimsloimaranta).

2 Modeling Claim Frequencies and Severities All papers on automobile insurance affairs are similar in this point that the number of accidents or claims follows Poisson distribution. In fact, a Poisson distributed random variable is a good choice to represent the number of events that occur in a certain time interval. To express the model mathematically, let N be the number of claims caused by the policyholders, then we will have [ ] Where is called random effect and shows that the policyholders in a portfolio posses different position facing to risk, i.e. the portfolio is heterogeneous with respect to claim frequencies. The random effect, is supposed to have Gamma distribution with parameter that means it has a unit mean. As a result, the unconditional distribution function of N will be Negative Binomial, that is [ ] On the other hand, let aggregate claims will be be the sizes of claims reported by the policyholders, then the are supposed to be independent and identically distribution that two suitable distributions for them are Exponential and Log Normal distribution and also they are assumed to be independent of number of claims. Assume that BMS possess +1 levels numbered from0 to s. Each level l is assigned a relativity that the premium the policyholders have to pay in each level will be the product of these to basic premium which is [ ]. This kind of BM model can be presented by Markov Chains. It means that the next position in the scales will correspond to the current level and the number of claims reported and not past position in the scales. This property induces BMS to posses two main characteristics: firstly, all levels are accessible in a finite number of steps from all other states and secondly, BMS has the Bonus level with a maximal reward that thepolicyholder on that level will remain there after a claim-free year. In order to define BMS relativities and to model a BMS mathematically, we can usethese properties of BMS. As it was mentioned before modeling a BMS depends on thebm rule. This rule is interpreted as Transition Rule and is the rule through which theposition of policyholders change among the scales. If k claims are reported, then { will be the components of a matrix called Transition Matrix Tk ) which is [ ] Further, the probability of moving from level to with annual mean claim frequency will be which is [ ] [ ] The Stationary Probabilities that determine the distribution of BMS then, will be The term, that could be interpreted as a fraction of time a policyholder withexpected annual claim frequency ϑ spends in level l once the steady state has beenreached, is the limit value of the probability that the policyholder is in that level.we can also write ) ) 2893

3 Relativities with a Quadratic Loss Function Relativities The relativity associated with level l is denoted as ; the meaning is that an insured occupying that level pays an amount of premium equal to % of the a priori premium determined on the basis of his observable characteristics. The determination of the s given the a priori classification implemented by the insurer is the main task of the actuary. The idea is to make as close as possible to the risk facto of a policyholder picked at random from the portfolio. The closeness is usually measured by the expected square difference between Θ and, but other loss functions can be used, too. Bayesian Relativities Predictive accuracy is a useful measure of the efficiency of a bonus-malus scale. The idea behind this notion is as follows: A bonus-malus scale is good at discriminating among the good and the bad drivers if the premium they pay is close to their true premium. According to Norberg 1976), once the number of classes and the transition rules have been fixed, the optimal relativity associated with level l is determined by maximizing the asymptotic predictive accuracy. Let us pick at random a policyholder from the portfolio. Both the a priori expected claim frequency and relative risk parameter are random in this case. Let us denote as Λthe random) a priori expected claim frequency of this randomly selected policyholder, and as Θ the residual effect of the risk factors not included in the ratemaking. The actual unknown) annual expected claim frequency of this policyholder is then ΛΘ. Since the random effect Θ represents residual effects of hidden covariates, the random variables Λ and Θ may reasonably be assumed to be mutually independent. Let be the weight of the kth risk class whose annual expected claim frequency is. Clearly, [Λ ]. Let L be the level occupied by this randomly selected policyholder once the steady state has been reached. The distribution of L can be written as [ ] Here, [ ] represents the proportion of the policyholders in level l. Our aim is to minimize the expected squared difference between the true relative premium Θand the relative premium applicable to this policyholder after the steady state has been reached), i.e. the goal is to minimize [ ] [ ] [ ] [ ] The solution is given by [ ] [ [ Λ ] ] [ Λ ] [Λ ] [ ] [ ] [ ] [ ] It is easily seen that [ ] [ [ ]] [ ] resulting in financial equilibrium once steady state is reached. If the insurance company does not enforce any a priori ratemaking system, all the s are equal to [ ] and 3.3) reduces to the formula that has been derived in NORBERG 1976). 289

4 Change of Scale The aim of the present section is to show how to develop rules allowing the transfer of a policyholder to a bonus-malus scale knowing his level in his previous bonus-malus scale. The a posteriori probability density function of given the level L occupied in the bonus-malus scale is given by [ ] [ ] Because we want to move a policyholder from one scale to the other, we should try to put the policyholder at a level which is as close as possible to his level in his original bonusmalus scale. By close, we mean here having the a posteriori random effect as close as possible. Total Variation Distance The total variation metric is often used to measure the distance to the stationary distribution. Recall that the total variation distance between two random variables X and Y, denoted as, is given by For counting random variables M and N,.10) obviously reduces to [ ] [ ] There is a close connection between and the standard variation distance, which considers the supremum of the difference between the probability masses given to some random events. Specifically, given two random variables X and Y, can be represented as Kolmogorov Distance In addition to the total variation distance used previously, we also need the Kolmogorovdistance. The Kolmogorov or uniform) metric based on the well-known Kolmogorov-Smirnov statistic associated with the goodness-of-fit test with that name), is definedas follows: The Kolmogorov distance between the random variables X and Y isgiven by Given two random variables X and Y, we have that. This result is an immediate consequence of.), since [ ]. Distances between the Random Effects A first measure may be to compare the expected a posteriori random effect, which actually is the relative premium at level _ in the bonus-malus scale: [ ] So the closest level in scale 2 to the level occupied in scale 1 is given by This rule simply amounts to placing the policyholder in the new scale at the level with the closest relativity to the one applicable in the previous scale. Because most commercial scales are normalized to associate a unit relativity with the entry level), this means that theinsurer has to compute the relativities for both scales. The implicit assumption is that the new entrant has the same characteristics as the policyholders in the portfolio no adverse selection being allowed). Another measure of discrepancy consists of comparing the distribution functions of the a posteriori random effects. This can be done by using the Kolmogorov distance or the total variation distance : ) [ ] [ ] ) Summarizing, a policyholder being at level in scale 1 and moving to scale 2 will be put in the level of scale 2 that minimizes one of the following distances: ) ) ) 2895

5 RESULTS In this part we will apply a BMS rule being conducted in one European country, the data derived from Dana Insurance Company and the equations proposed in the last section in order to obtain a BMS for Iran Insurance Market. The distribution function of number of claims is supposed to be mixing Poisson distribution that under the assumption of Gamma distribution for average claim frequencies leads to a Negative Binomial distribution as unconditional distribution of number of claims. The observed number of claims derived from Dana Insurance Company data, Automobile collision insurance data of year2010, is as follows The observed number of policies with different number of claims of Dana insurance company during year 2010_ Table 1) Number of claim Number of policies with claim Table , % 1 12, % 2 2, % % % Total Percentage of policies with claim Claim Valid Nlistwise) N Range Table2. Descriptive statistics Minimum Maximum 0 Mean Std Deviation Variance If we assumed a Poisson distribution for above data, both the moment and the maximum likelihood estimators for parameter would be the mean of sample which is = In the case of Poisson distribution and. Furthermore, computing the probabilities of the Poisson distribution with this parameter and multiplying them the sample size n=79116, lead to the following table for expected number of policies and residuals Expected number of policies and residual for each number of claims under the Poisson distribution- Table 2) Number of claim 0 Table3. Expected number Residual of policies 61, , , , , Total Referring to for -square goodness of fit statistic, we will obtained which leads us to reject the Poisson distribution for observed data. This is confirmed when looking at the descriptive table of observed number of claims. The mean of the data is for different from its variance, whereas this is not the case for Poisson data. However, the Negative Binomial distribution results to a better fit. Using 2896

6 Tech J Engin & App Sci., 3 21): , 2013 for observed data, we will have mass function for each point it is enough to use. To obtain the probability where. Performing this probability mass function structure and multiplying the probabilities by the sample size, the expected number of claims will be as follow Expected number of policies and residual for each number of claims under the Negative Binomial-Table 3) Number of claim 0 Table. Expected number of Residual policies 63, , , Total Referring to for -square goodness of fit statistic, we will obtained, which is much better than in the case of Poisson distribution. However, the Inverse Gaussian distribution results to a better fit. Using for observed data, we will have mass function for each point it is enough to use where. To obtain the probability [ ] and. Performing this probability mass function structure and multiplying the probabilities by the sample size, the expected number of claims will be as follow Expected number of policies and residual for each number of claims under the Inverse-Gaussian -Table ) Number of claim Table5. Expected number of Residual policies 0 63, , , Total Referring to for -square goodness of fit statistic, we will obtained, which is much better than in the case of Poisson distribution and a little better than in the case of Negative Binomial distribution. 2897

7 Bonus-Malus Relatives In this part we will apply the simple Bonus-Malus System, the scaled-based systems, to our data to obtain the relatives explained in the previous section. Two systems will be discussed, on the being conducted in Iran and other the system being conducted in Taiwan. The Iranian Bonus-Malus System, rewards the policyholders with no claim in year going down one level in the scale whereas send them to the first level, the level with the largest premium, as penalty. In the system being performed in Taiwanese also the claim-free policyholder are transformed to one level down but the different occurs when penalizing the policyholder with one or more claims in a year. The system, in fact, transforms the policyholder with one claim to two levels up whereas it transforms the others to the first level, the level with the largest premium. The rules of these two systems have been shown in two following tables Iranian Bonus-Malus Rules Table 5) class In this case the transition probability matrix will be Table6. Class after 0 claim Class after 1 or more claims And steady-state distribution is ) ) ) ) Taiwanese Bonus-Malus Rules- Table 6) ) class Class after 0 claim In this case the transition probability matrix will be Table7. Class after 1 claim Class after 2 claim Class after 3 or more claims 2898

8 And steady-state distribution is ) ) ) ) ) ) ) ) ) The relatives could be obtained through the formula mentioned in previous section for computing the stationary probabilities and relatives. To solve the integrals applied in relative construction, formula ) ), we have to use some numerical methods. The IML package of SAS program helps here, for relevant commands see Appendix. The relatives for our data with =.2819 and for Iranian system will be Relatives associated to each level for Dana Insurance Company data under -1/top Rule Table7) Class 0 Table8. [ ] The relatives for our data with and for Iranian system will be Relatives associated to each level for Dana Insurance Company data under Taiwan Bonus-Malus Rule Table 8) 2899

9 Class Table9. [ ] de L0 L1 L2 L3 L L5 L L L E L E Table 9. Distances between and for the two metrics dtv L0 L1 L2 L3 L L5 L0.9306E L E L E L E w=.25 dnew L0 L1 L2 L3 L L5 L L L L W=.75 L0 L1 L2 L3 L L5 dnew L L L L Table 9) provides the distances between the conditional random effect for our two metrics. On the basis of these distances, we see that the minima are attained for the following transition rules. Transition rules from scale -1/Top to Taiwan scales are: Table W=.25 W=0.75 CONCLUTION This paper is the Bonus-Malus systems in Automobile insurance. The first part of this paper was spent to an introduction of BMS,systems based on the scales and systems not based on the scales. The paper ended with comparing the System being conducted in Iranian Insurance Companies. In numerical illustrations part, per claim were computed for Dana Insurance Company data via SAS/SPSS/@RISK package and focuses on techniques for constructing Bonus-Malus systems in automobile insurance. Specifically, the article presents a practical method for constructing optimal Bonus-Malus scales with reasonable penalties that can be commercially implemented. For this purpose, the symmetry between the overcharges and the undercharges reflected in the usual quadratic loss function is broken through the introduction of parametric asymmetric loss function of exponential type. The resulting system possesses the desirable financial stability property. 2900

10 REFERENCES A Bayesian Approach to Bonus-malus system an application in Automobile insurance/ Fatemeh Atatalb/February-2012 A COMPARATIVE ANALYSIS OF 30 BONUS-MALUS SYSTEMSBY JEAN LEMAIRE AND HONGMIN Zl A Review on Bonus-Malus systems & the introduction to Dynamic Bonus-Malus system by the use of Hidden Markov models/atefeh Kanani Dizaii/ February-2011 Bonus and Malus in Principal-Agent Relationswith Fixed Pay and Real Eff ort/annette Kirstein Bonus-Malus scales in segmented Tariffs: Gide & Sundt s work Revisited /S piterbois/m Denuit/J-F Walhin BONUS-MALUS SCALES USING EXPONENTIAL LOSS FUNCTIONS/MICHEL DENUIT/JAN DHAENE BONUS-MALUS SYSTEMS WITH VARYING/DEDUCTIBLES/SANDRA PITREBOISz, JEAN-FRANC_OIS/WALHIN&MICHEL DENUITy/August 17, 200/Revision: January 2, 2005 Design of an optimal Bonus-Malus system for Iran Automobile insurance market / Mahdieh Amini Bayat /January Design of an Optimal Bonus-Malus System for IranAutomobile Insurance Market/Farhad Khorrami/Ghadir Mahdavi/Mahdieh Amini Bayat Designing optimal bonus-malus systems from di erent types of claims/jean Pinquet/Mai 1998 EXPONENTIAL BONUS-MALUS SYSTEMSINTEGRATING A PRIORI RISKCLASSIFICATION/LLU_IS BERM_UDEZ/MICHEL DENUIT/JAN DHAENE Fitting the Belgian Bonus-Malus System/S. Pitrebois1, M. Denuit2 and J.F.Walhin Measuring sensitivity in a bonus malus system/e. Gómez, A. Hernández, J.M. Pérez, F.J. Vázquez-Polo/ Received 1 October 2001; received in revised form 1 March 2002; accepted 8 April 2002 NO-CLAIM DISCOUNT CR BONUSJMALUS SYSTEMS IN EUROPE/GUY HWHITEHEAD ON A BONUS-MALUS SYSTEM WHERE THE CLAIM FREQUENCY DISTRIBUTION IS GEOMETRIC AND THE CLAIM SEVERITY DISTRIBUTION IS PARETO/Mehmet Mertand Yasemin Saykan OPTIMAL CLAIM DECISIONS FOR A BONUS-MALUS SYSTEM: A CONTINUOUS APPROACH/NELSON DE PRIL The ASTIN Bulletin lo 1979) The Bonus-Malus System in Tunisia:/An Empirical Evaluation/Georges Dionne, HEC Montréal/Olfa Ghali, Université de Paris X- Nanterre Appendix proc iml; /* the function in numerator of c)*/ start fun1t); alpha= ;r1=t#exp-lambda#t)##)#1/gammaalpha))#alpha##alpha)#t##alpha-1))#exp- alpha#t));tgraton of function* returnr1); /*taking intgration of function*/ a1={0.p}; call quadz1,"fun1",a1); print z1[format=e21.1]; /*the function in denominator of c)*/ start fun2s); alpha=; r2=exp-lambda#s)##)#1/gammaalpha))#alpha##alpha)#s##alpha-1))#exp-alpha#s)); returnr2); /*taking intgration of function*/ a2={0.p}; call quadz2,"fun2",a2); print z2[format=e21.1]; r=z1/z2; print r; proc iml; /*the function in numerator of c)*/ start fun1t); r1=t#exp-lambda#t)##3)#1-exp #t))#1/gammaalpha))#alpha##alpha)#t##alpha-1))#exp- alpha#t)); returnr1); /*taking intgration of function*/ a1={0.p}; call quadz1,"fun1",a1); 2901

11 print z1[format=e21.1]; /*the function in denominator of c)*/ start fun2s); r2=exp-lambda#s)##3)#1-exp #t))#1/gammaalpha))#alpha##alpha)#s##alpha-1))#exp- alpha#s)); returnr2); /*takingintgration of function*/ a2={0.p}; call quadz2,"fun2",a2); print z2[format=e21.1]; r=z1/z2; print r; proc iml; /*the function in numerator of c)*/ start fun1t); r1=t#exp-lambda#t)##2)#1-exp #t))#1/gammaalpha))#alpha##alpha)#t##alpha-1))#exp- alpha#t)); returnr1); /*taking intgration of function*/ a1={0.p}; call quadz1,"fun1",a1); print z1[format=e21.1]; /*the function in denominator of c)*/ start fun2s); r2=exp-lambda#s)##2)#1-exp #t))#1/gammaalpha))#alpha##alpha)#s##alpha-1))#exp- alpha#s)); returnr2); /*takingintgration of function*/ a2={0.p}; call quadz2,"fun2",a2); print z2[format=e21.1]; r=z1/z2; print r; proc iml; /*the function in numerator of c)*/ start fun1t); r1=t#exp-lambda#t)##1)#1-exp #t))#1/gammaalpha))#alpha##alpha)#t##alpha-1))#exp- alpha#t)); returnr1); /*taking intgration of function*/ a1={0.p}; call quadz1,"fun1",a1); print z1[format=e21.1]; /*the function in denominator of c)*/ start fun2s); 2902

12 r2=exp-lambda#s)##1)#1-exp #t))#1/gammaalpha))#alpha##alpha)#s##alpha-1))#exp- alpha#s)); returnr2); /*takingintgration of function*/ a2={0.p}; call quadz2,"fun2",a2); print z2[format=e21.1]; r=z1/z2; print r; proc iml; /*the function in numerator of c)*/ start fun1t); r1=t#exp-lambda#t)##0)#1-exp #t))#1/gammaalpha))#alpha##alpha)#t##alpha-1))#exp- alpha#t)); returnr1); /*taking intgration of function*/ a1={0.p}; call quadz1,"fun1",a1); print z1[format=e21.1]; /*the function in denominator of c)*/ start fun2s); r2=exp-lambda#s)##0)#1-exp #t))#1/gammaalpha))#alpha##alpha)#s##alpha-1))#exp- alpha#s)); returnr2); /*takingintgration of function*/ a2={0.p}; call quadz2,"fun2",a2); print z2[format=e21.1]; r=z1/z2; print r; proc iml; start f1c); s=0.9797; m=8.9585; e=c/c#sqrt2#3.1#s))#exp-0.5#1/s)#logc)-m)##2)); returne); start fqad); a={0 d}; call quad z1,"f1",a); returnz1); start f2c); s=0.9797; m=8.9585; p=1/c#sqrt2#3.1#s))#exp-0.5#1/s)#logc)-m)##2)); returnp); start fqbd); b={d.p}; call quadz2,"f2",b); returnz2); 2903