VOL. XLVII,No. X7Part IPROCEEDINGSMay 23-25, 1960TWO STUDIESIN AUTOMOBILEINSURANCERATEMAKINGBYROBERTA. BAILEYand LEROYJ. SIMONSection A, Effectiveness of Merit Rating and Class Rating, uses the Canadianexperience for private passenger automobiles to show (1) that merit rating isalmost as effective as the class plan in separating the better risks from thepoorer risks, (2) that both merit rating and class rating leave unanalyzed aconsiderable amount of variation among risks and (3) that certain availableevidence supports the conclusion that annual mileage, which has long beenfelt to be an important measure of hazard, is a very significant cause of thisunanalyzed variation among risks.Section B, Improved Methods for Determining Classification Rate Relativities, presents a method for obtaining relativities among groups on which amultiple classification system has been imposed. The customary method ofcalculating class relativities uses the total experience for each class with allsubdivisions within the classes added together. With the customary methodit is difficult to make a completely accurate adjustment for different distributionsby territory or merit rating, because any change in the class relativities disturbsthe other sets of relativities and conversely. It is shown that even if such anadjustment were made, the customary method of calculating relativities one setat a time does not reflect the relative credibility of each subgroup and does notproduce the best fit to the actual data. Moreover it produces differences betweenthe actual data and the fitted values which are far too large to be caused bychance. In addition, for private passenger automobile insurance in Canada,it is shown that two sets of relativities which are multiplied together cannotproduce the best fit to the actual data, and some of the consequences of tryingto do so are explained. Some methods are advanced whereby all sets of relativities for classes, merit ratings, territories, and so forth, can be calculatedsimultaneously, which will overcome all the deficiencies in the customarymethod. These improved methods use the technique of minimizing a measure(technically known as the Chi-square test) of the differences between the actualdata and the fitted values. Some applications to other lines of insurance arementioned.1

TWO STUDIES IN AUTOMOBILEINSURANCERATEMAKING2SECTION A: EFFECTIVENESS OF MERIT RATING AND CLASS RATINGIntroductionPrivate passenger automobile insurance uses a multiple classification systern. We classify by age (under or over 25 years) and within each age weclassify by occupation (farm or non-farm). We also classify by use and sex.On top of all this we classify by territory. And now we have begun to classifyby previous accident and conviction record which is popularly called the “meritrating plan.” There is no basic difference between merit rating and class rating if the rates for each merit rating group are based on the subsequent experience of cars previously classified according to their accident and convictionrecord, just as the rates for each class are based on the subsequent experienceof cars previously classified according to the characteristics of the class plan.In actual fact, merit rating is a class rating plan and is part of the multipleclassification system. However, in this paper, as a matter of convenience, andnot implying a basic distinction, we will follow the common usage in the UnitedStates by referring to classification according to previous accident and conviction record as “the merit rating plan,” and to classification according toage, sex, use and occupation as “the class plan.”A class plan which uses age, sex, use and occupation does not preciselyclassify each risk according to its true value. Underwriters have long recognizedthis, and it is further substantiated by the Canadian merit rating experiencewhich shows that risks which have been accident-free for three or more yearshave better experience in the following year than the average for their class.Likewise a class plan which uses only the previous accident record would notprecisely classify each riskl. This is shown by the fact that in the Canadianmerit rating experience, the cars which qualified for the best merit rating havedifferent accident frequencies depending on which class they are in.This means that private passenger automobile risks vary considerably fromeach other and that the class plan and the merit rating plan are both attempts toseparate the better risks from the poorer risks. Neither plan is perfect, but wewould like to discuss the question, “How do merit rating and class ratingcompare with each other in their ability to separate the better risks from thepoorer risks?” After discussing their comparative effectiveness, we shall thendiscuss their absolute effectiveness.Comparative Eflectiveness of Merit Rating and Class RatingTable 1 at the end of this section shows the Canadian automobile experience”arranged to show what it would have looked like if there had been (1) meritrating without class rating and (2) class rating without merit rating. Thepremiums have been adjusted to what they would have been if all the cars hadbeen written at 1B rates, by use of the approximate relativities:ISee also “Some Considerations on Automobile Rating SystemsUtilizing Individual Driving Records” by Lester Dropkin, CAS XLVI, p. 16.5.2The Canadian experience includes that of virtuallyevery insurance company operatingin Canada and is collated by the Statistical Agency (Canadian ing under instructions from the Superintendent of Insurance.

TWOSTUDIESMeritAXYBIN AUTOMOBILEINSURANCE3RATEMAKINGRating DefinitionRelativity-licensed and accident free three or more yearslicensed and accident free two yearslicensed and accident free one yearall others12345-pleasure, no male operator under 25pleasure, non-principal male operator under 25business useunmarried owner or principal operator under 25married owner or principal operator under 25658010”:Class Definitions100165165240165The purpose of any classification plan is to reduce the rates for the betterrisks and to offset this reduction with an appropriate increase in the rates forthe more hazardous risks. We will define “effectiveness” of a classification planin this paper to be the extent to which the plan separates the better risks fromthe overall average. This definition of effectiveness was applied in making anevaluation of a one-year merit rating plan where the better risks would getonly a 1.6% reduction from the average rate if a 15% discount were givenfor a one-year accident-free record. Because the reduction of 1.6% was sosmall, the plan was considered to be ineffective.3Since both merit rating and class rating in Canada include about the sameproportion, 80%) of the cars in the lowest rated class, a measure of the comparative effectiveness of the two is the percentage reduction of the lowest ratedclass from the over-all average.Rating PlanReduction of lowestrated class fromaverageMerit rating aloneClass rating aloneMerit and class rating ofcars in lowestrated class1 15680.9%80.1%66.4%This means that the merit rating plan is 77% as effective as the class plan.The Canadian merit rating plan could be improved by extending it from threeyears to five (which was done during the latter part of 1959) and by includingconvictions. Something also could be gained if the merit rating plan gave extraweight to a loss exceeding, say 1000, since it was noted that there is a positivecorrelation between the loss ratio and the average size of loss. Likewise theCanadian class plan, which is similar to the plans used in the United States,could also be improved. But the point remains that merit rating is almost aseffective as the class plan in separating the better risks from the poorer risksand a substantial improvement is realized when they are used in combination.3 See Muir, J. l , “Principlestems for Liabrhty Insurancepp. 32 and 33.and Practices in Connection With ClassificationAs Applied to Private Passenger Automobiles”,Rating SysCAS XLIV,

4TWOSTUDIESIN AUTOMOBILEINSURANCEKATEMAKINGOur previous paper showed the experience for each class subdivided bymerit rating.* This was a natural format because the class plan was here firstand merit rating was being imposed on top of the already existing class plan.Table 2 at the end of this section shows how the experience would have beenpresented if merit rating had been here first and the class plan was being imposed on top of the already existing merit rating plan. Losses are used this timeinstead of number of claims because there is a much greater difference inaverage claim costs among the classes than among the merit ratings.The relative loss ratio for Class 1 within each merit rating is slightly lowerthan the corresponding ratio shown in our previous paper for merit rating Awithin each class, indicating a greater effectiveness for class rating. The classplan is most effective in the worst merit rating, B, just as merit rating wasshown to be most effective in the worst class, 4.Absolute Efjiectiveness of Merit Rating and Class RatingThus far, this paper has shown, based on the Canadian experience, thatmerit rating is almost as effective as class rating in separating the better risksfrom the poorer risks. But it has not shown in absolute terms just how effective either rating plan is.In order to determine the absolute effectiveness of a rating plan, an analyticalexpression of the distribution of risks according to their “inherent hazard” isneeded. Mr. Dropkin’s paper on the negative binomial distribution” providesa valuable tool for this purpose. His paper shows that inherent hazards ofindividual risks are much more widely distributed than was commonly supposed. The class plan reduces this wide distribution very little. This is illustrated by the fact that merit rating will give the best risks a reduction of 10.5%from the average when there is no class plan and will still give the best riskswithin Class 1 a reduction of 8.9% fi from the average Class 1 rate. This meansthat Class 1 has almost as much variation within it as there is among all classescombined.This demonstrates what has often been recognized, that while merit rating andclass rating are effective tools in a relative sense, in an absolute sense both meritrating and class rating are quite ineffective in separating the better risks from thepoorer risks. There remains a considerable amount of unanalyzed variationamong risks.Cause of the Unanalyzed Variation Among Automobile RisksIt is one thing to show there is variation among risks and another thing to findthe cause of variation.In our previous paper we listed three possible reasons why the empiricalcredibilities discussed there for 1, 2 and 3 years of merit rating were not in theexpected ratio of 1 : 2 : 3. They were:aBailey, Robert A. and Simon, LeRoy J., “An Actuarial Note on the Credibilityence of a single Private Passenger Car”, CAS XLVI; Table 1, p. 162.“Op. Cit.GBailey, RobertA. and Simon, LeRoyJ., Op. Cif., Table4, p. 163.of Experi-

TWOSTUDIESIN AUTOMOBILEINSURANCE5RATEMAKING( 1) new risks entering a class,(2) an individual risk’s chance of having an accident varying fromyear to year, and(3) a markedly skew distribution of risks.With the help of the negative binomial distribution, we can check the thirdalternative. Using the formula derived by Mr. Bailey? for the credibilityZ Ln awhere n number of accident-free years anda is a parameter in the distribution of risks,we find that the relative credibilities for 1, 2 and 3 years should be in the ratioof1:2( ):3(G)By setting the one year credibilityfor Class 1 cars of .055, shown in Table 41of our previous paper,8 equal to , we obtain a 17.2. Therefore thel arelative credibilities for 1, 2 and 3 years should be in the ratio of 1: 1.90: 2.70which are close to 1: 2 : 3 as we had expected. But the actual relative credibilitiesalso shown in Table 4 of our previous paper are in the ratio of 1: 1.38 : 1.62.Therefore while the distribution of risks is definitely skew, it is not skew enoughto account for such large discrepancies, and we may cross out the third alternative listed above.We know that new risks entering the class account for some of the discrepancy, but we do not feel that new risks can account for such large discrepancies.Therefore we feel that the evidence strongly supports the conclusion that theindividual risk’s chance of having an accident does vary significantly from yearto year.Thus far we have shown that merit rating and class rating are of about equaleffectiveness and that a substantial improvement is realized when they are usedin combination. However, both of them leave unanalyzed a considerableamount of variation among risks. In our investigation of the characteristics ofthis unanalyzed variation we have eliminated certain factors from considerationand now feel we have reached the point where we may state that the stillunanalyzed cause (or causes) of variation among individual risks:( 1) has a wide dispersion,(2) varies significantly from year to year for an individual risk, and(3) is measured only to a limited extent by the class plan and themerit rating plan.Annual mileage, which has long been felt to be an important measure of hazard,TBailey, Robert A. Discussion,“Some Considerationson AutomobileUtilizing IndividualDriving Record”, CAS XLVII, p. 152.“Op. cit.RatingSystems

6TWOSTUDIESIN AUTOMOBILEINSURANCERATEMAKINGfits all these requirements better than any other single cause. The distributionof risks according to mileage is widely dispersed.Q Mileage varies significantlyfrom year to year. Farmers, for example, have less mileage than averagelo andbusiness use risks have more mileage than average. The discount for two ormore cars in one family is a reflection of mileage. Accident frequencies (andeven conviction frequencies) are a crude indication of mileage. Mileage iscertainly not the whole story because there is conclusive evidence that newlylicensed drivers and youthful drivers have a higher accident rate per mile thanother drivers and that other things such as drinking and irresponsibility play apart, but the evidence supports the conclusion that mileage is a very significantcause of variation among individual risks.TABLE 1Canada excluding SaskatchewanPolicy Years 1957 & 1958 as of June 30, 1959Private Passenger Automobile Liability- Non FannersMeritEarnedRatinqCar YearsEarned Prem.1B RatesAXYBTotalA XA X 13,118,OOO24,152,OOOgeEitB-wRati LossRatioLossesIncurredat .505,459.4?0RelativeLoss Ratio.8951.1741.2771.6101.000.909.9314,-X, ------Y & B 161,0001A2,75?,520159,108,00033;;9See DeSilva, HarryYork, 1942, p. 12.lolbid., p. 13.84,6W,OOO13,684,OOO6, oo1.146121,421,OOO2,426,OOO.505.5831.00063,1 1,000.39?.?86R. JVhg We Have AutomobileAccidents.John Wiley& Sons, New

TWOSTUDIESIN g SaskatchewanPolicy Years 1957 & 1958 as of June 30, 1959Private Passenger Automobile Liability- Non FarmersEarnedCar YearsEarned Present1B RatesLossesIncurredLosswA - licensed and [email protected] &&in 881,OOORelativeLoss Ratio2 w-mor more xears--63,191,OOO .3974,598,OOO ,6419,589,OOO .6127,964,OOO 1.0351,752,OOO ,54187,094,OOO .452Merit RatiKX - -l.&cpgeg &nd 817,7074,039175,5533zi.i1‘7a-4;45Total 000209,00010,518,OOO@@nAY,J&cg 13A;;1.107.545.593.8781.4181.3542.2901.1971.0002 years-.8651.4871.0941.867.9191.000god accifien t fxes & xesr 686.901.9521.441.82919,633:OoO ,011,0001,281,OCC178,0008,461,OoOge t--wRatinnB ------ all 88,0002,383,OOC3,;poooSECTION B: IMPROVED METHODS FOR DETERMINING CLASSIFICATIONRATE RELATIVITIESMultiple classification systems are quite prevalent in the insurance industry.For example, in fire insurance we classify the simple dwelling risks by town

8TWOSTUDIESIN AUTOMOBILEINSURANCERATEMAKINGgrading as well as by construction, resulting in a 10 x 2 system (typically).Other lines similarly involve multiple classification systems, but automobile isprobably the best example. We have used a class plan and a territorial plan inautomobile and now we have introduced the merit rating plan. It has beencustomary to determine a countrywide set of class relativities. Under the meritrating plan it will be necessary to determine relativities here, too. Assumingthese relativities are to continue to be applied in series as multipliers on a “base”pure premium, the problem then arises as to how to determine the best set ofrelativities. The customary procedure* is to sum over all variables except theone we are interested in and then compute our relativities. For example, to getthe class relativities, get the total mass of experience broken down only byclass. Then the ratio of the experience for each class (usually adjusted in somemanner for differences in the distribution by territory and merit rating) to theoverall average experience will give the individual class relativity. The samesteps would be followed for the merit rating classes and for territories. Thesubdivisions within each class are added together because individually theyare usually not fully credible. Combining them is a means of obtaining a credible volume of experience. This process of combining subgroups results in a lossof some information because any combination yields less information than theaggregate information yielded by the individual subgroups. A method forobtaining relativities which is able to avoid combining the subgroups and isable to use each subgroup individually would produce a better set of relativities.For purposes of illustration we’ll solve the following problem: What is thebest set of class relativities and merit rating relativities to use in Canada? Thedata is presented in Table B in a loss ratio form (all at Class 1B rates) and inTable D as relative loss ratios. We will assume that the territorial factor isproperly reflected in this data because we are dealing with loss ratios. A betterway would be to use pure premiums and to work out territorial relativities atthe same time as class and merit rating relativities. However, such data is notavailable to the authors, but the procedure would be similar in either case. Todetermine what is an acceptable set of relativities we must establish the criteriawhich a set should meet:CriterionCriterionCriterionCriterion1. It should reproduce the experience for each class andmerit rating class and also the overall experience; i.e.,be balanced for each class and in total.2. It should reflect the relative credibility of the variousgroups involved.3. It should provide a minimal amount of departure fromthe raw data for the maximum number of people.4. It should produce a rate for each sub-group of riskswhich is close enough to the experience so that the differences could reasonably be caused by chance.*For example, see“Current Rate Making Proceduresfor Automobile Liability Insurance”,Stern, Phillip K., CAS XLIII, p. 127ff.

TWOSTUDIESIN AUTOMOBILEINSURANCERATEMAKING9A set which meets these four criteria will be judged to be a “best” set of relativities. If more than one set satisfactorily meets the four criteria, the choiceamong sets may be made on a non-mathematical basis such as (a) simplicity ofapplication, (b) similarity to existing sets, (c) ease of explanation to non-technical personnel or (d) the actuary’s personal preference.Let us define xi as the class relativity for the ith class (i 1,2,3,4,5) and yjas the class relativity for the jth merit rating class (j 1,2,3,4 representingA, X, Y and B respectively). Let rij be the actual relative loss ratio for personsclassified as class i and merit rating class j; r.j is the relative loss ratio of the jtbmerit rating class where all i classes are combined; ri. is the relative loss ratio ofthe ith class where all j merit rating classes are combined; and finally r. is therelative loss ratio for all classes and merit rating classes combined and thusequals 1 .OO. Let us also define nij as the number of earned car years of exposure. The n! j are shown in Table A.Relativities calculated by the customary method, which we will call “Methodl”, are as follows:Xi ri.and(1)yj f.j)and are shown in Table C.The estimated relative loss ratio is then Xiyj, and, if multiplied by the overallloss ratio, will produce the estimated loss ratio for the i, j class. Or, if xiyj ismultiplied by the overall pure premium, it would produce the estimated purepremium for the i, j class. The estimated relative loss ratios, Xiyj obtained byMethod 1 are shown in Table D. When compared with the actual relative lossratios, rij, also shown in Table D, it is evident that there are some undesirablylarge differences. Moreover, all xiyl are too low and all xiyl are too high.To test the balance (Criterion 1 above) we calculateZIlijXiyj/ZIlijrij(2)summing over each i, each j and total.A set of relativities is balanced if equation (2) equals 1.000. The balance asdetermined by equation (2) is shown in Table E. Method 1 is out of balance intotal and far out of balance for the individual classes. If the off-balance in thetotal is corrected, the classes will still be far out of balance. The reason why theclasses are so far out of balance is that in our calculation of xi and yj, no adjustment was made for differences in the distribution by class or merit rating class.This illustrates what happens if a merit rating plan is imposed on an alreadyexisting class plan without any adjustment in the class relativities. If we hadmade some tentative adjustment, the off-balance by class and merit rating classwould have been reduced. To make a completely accurate adjustment in theclass relativities is difficult, however, because any adjustment in the class relativities disturbs the relativities for the merit rating classes and conversely, thusrequiring an adjustment process which zig-zags back and forth. However, evenif such an adjustment were made so that Criterion 1 would be satisfied, Method1 would still not satisfy Criteria 2, 3 and 4, as will be shown later.Again speaking in general, in order to reflect the relative credibility of the

10TWOSTUDIESINAUTOMOBILEINSURANCERATEMAKINGvarious groups involved (Criterion 2)) the indicated proportionalof each groupactual loss ratio - expected loss ratioexpected loss ratiodepartureshould be given a weight proportional to the square root of the expected number of losses for the group. This is based on the fact that the indication of eachgroup should be given a weight inversely proportional to the standard deviation of the indication. The standard deviation of the indication is inverselyproportional to the square root of the expected number of losses for the group.An equivalent credibility procedure would be to give the square of the indication a weight proportional to the expected number of losses.Criterion 2 (Credibility)is not met by the customary relativities (Method1) because when all the data is added together for, say, class i, to obtain ri,each subgroup rij is given a weight approximately proportional to the expectednumber of claims instead of the square root of the expected number of claims.This is one of the reasons why Method 1 does not satisfy Criteria 3 and 4.Moreover, if each entry in a row of ri j is of low credibility, the resulting ri willnot be too trustworthy. Nevertheless, the resulting ri. will be treated as 100%credible by Method 1 in the determination of xi. Methods 2, 3 and 4 developedbelow will remove these defects. Each rij will contribute to the final set inproportion to its relative credibility in relationship to all other rij in the tableand not just in relationship to the other members of its row or column, andconversely each xi and yj will be influenced by all the ri j and not just by onerow or column Of ri j .There is no assurance that Criteria 3 and 4 are met by the customary relativities (Method 1). In the paragraphs that follow we will show clearly thatthis set of relativities results in an average departure that is far from minimaland further, that the individual departures are too large to be caused by chance.As a test of a set of relativities for compliance with Criterion 3, let uscalculate how much error the average policyholder will have in his estimatedrelativity by calculating,znijI,3I rij-Xiyj1 /Z nijriji,j(3)The result of this calculation is shown in Table E.Equation (3) endeavors to measure how much “inequity” the set has. Thefarther a policyholder’s rate is from the indications of the raw data, the more“inequity” is involved. Anyone who has dealt directly with insureds at thetime of a rate increase, knows that you can be much more positive when therate for his class is very close to the indications of experience. The more persons involved in a given sized inequity, the more important it is.To test a set of relativities for compliance with Criterion 4 (differencesbetween the raw data and the estimated relativities should be small enough tobe caused by chance), the Chi-square test is appropriate. It is shown in theAppendix that in terms of relative loss ratios, exposures and relativities, K nij(lij-XiYj)2i,j(4)Xiyj

TWOSTUDIESINAUTOMOBILEINSURANCERATEMAKING11where K is a constant dependent on the data and for the Canadian data, Kequals approximately l/200. The values of x2 are shown in Table E.It should be noticed that the x2 formula (4) is equivalent to giving the squareof the indication a weight proportional to the expected number of claims.nij(rij-Xiyj)’,‘ I ;Xiyj KFjnijxiyj( ri’[;iyi)This means that a set of xi, yj, which is specifically designed to produce a minimum x2 will automatically reflect the relative credibility of each group involved(Criterion 2). This is accomplished without a credibility weighting processinvolving tabular credibilities. Moreover, since a set of xi, yj, which produces aminimum x2 will very likely also satisfy Criterion 3 (minimal average amount ofdeparture) and will come very close to satisfying Criterion 1 (balance), it seemsevident that the best set of relativities will be those which are designed specifically to produce a minimum x2. These relativities can be obtained by setting thepartial derivatives of x2 equal to zero.- 2x2 KZnijyji3Xijllij?ij- KX jXHYjSolving for xi, we obtainXi [F2. L/Fnijyj]' o(5)(6)and similarly,yj [F21! Z/Fnijxi]'(7)This gives us nine equations in nine unknowns. Since the equations are not ofa simple, rational form, the easiest way to arrive at a numerical solution is bya method of iteration, as follows:1. Take ri. (the customary method of obtaining Xi) as the first estimates Of Xi.2. Use these values in the right hand side of (7) to obtain the firstestimates of yj.3. Use the first estimates of yj in the right hand side of (6) toobtain the second estimates, of xi.4. Repeat this process until two consecutive sets of solutions areidentical (or substantially so).Notice that there are an infinite number of solutions for xi and yj, all of which,however, produce the same set of xiyj. This is true because each xi may bemultiplied by a constant if each yj is divided by the same constant. The resultsof this method, which we shall call “Method 2” are shown in Table C. The estimated relative loss ratios, xiyj, are shown in Table D and the tests of Criteria 1,3 and 4 are shown in Table E.

TWO12STUDIESINAUTOMOBILEINSURANCERATEMAKINGIt is evident that Method 2, which derives all sets of relativities simultaneously, solves the difficult problem of obtaining relativities which are balancedin total and by class. It automatically satisfies Criterion 2 (credibility)and italso reduces substantially the average error and x2 (Criteria 3 and 4). But inspite of this improvement, the average error of .0317 still does not comparevery favorably with a profit margin of .050 or thereabouts, especially for a company that writes a disproportion of business in one class. Moreover, x2, althoughmuch less than for Method 1, is still too high to be the result of chance. Thismeans that a set of factors which are multiplied together, xiyj, cannot satisfactorily represent the actual data for Canadian private passenger automobiles,although it may be satisfactory for other lines or types of data.Turning to the actual data, shown in Table D, it can be seen that the percentage difference between the lowest and the highest merit rating decreases asthe rate for the class increases, ranging from 73% for class 1 down to 39% forclass 4. With these conditions present in the basic Canadian data, it is littlewonder that the multiplicative relativities do not fit satisfactorily.A possible method, which we will call “Method 3”, is to let the estimatedrelative loss ratio be xi yj, where the relativities are added instead of multiplied. The x2 formula becomesnij(rij-Xi-yj)’X2 K zi,jXi yjAnd setting the partial derivatives of x2 equal to zero we have:lli jr:j(xi yj)23X2-rK nij-K aXi OFor convenience let us write (9) as f (xi ) 0. If we first obtain an estimateof xi, we can ob

Section A, Effectiveness of Merit Rating and Class Rating, uses the Canadian experience for private passenger automobiles to show (1) that merit rating is almost as effective as the class plan in separating the better risks from the poorer risks, (2) that both merit rating and class rating leave unanalyzed a .