INSURANCE R A T E M A K I N G
A. E]]ectiveness o] Merit Rating and Class Rating, and
B. Improved Methods [or Determining Classification Rate Relativities
ROBERT A. BAILEY and LERoY J. SIMON
Section A of this paper uses the Canadian experience for private passenger automobiles to show (I) t h a t merit rating is almost as effective as the class plan in separating the better risks from the poorer risks, (2) t h a t both merit rating and class rating leave un- analyzed a considerable amount of variation among risks and (3) t h a t certain available evidence supports the conclusion t h a t annual mileage, which has long been felt to be an important measure of hazard, is a very significant cause of this unanalyzed variation among risks.
Section B presents a method for obtaining relativities among groups on which a multiple classification system has been imposed. The customary method of calculating class relativities uses the total experience for each class with all subdivisions within the classes added together. With the customary method it is difficult to make a completely accurate a d j u s t m e n t for different distri- butions by territory or merit rating, because a n y change in the class relativities disturbs the other sets of relativities and converse- ly. It is shown t h a t even if such an adjustment were made, the customary method of calculating relativities one set at a time does not reflect the relative credibility of each subgroup and does not produce the best fit to the actual data. Moreover it produces differences between the actual data and the fitted values which are far too large to be caused by chance. In addition, for private pas- senger automobile insurance in Canada, it is shown that two sets of relativities which are multiplied together cannot produce the best fit to the actual data, and some of the consequences of trying to do so are explained. Some methods are advanced whereby all sets of relativities for classes, merit ratings, territories, and so forth, can be
calculated simultaneously, which will overcome all the deficiencies in the customary method. These improved methods use the tech- nique of minimizing a measure (technically known as the chi-square test) of the differences between the actual data and the fitted val- ues. Some applications to other lines of insurance are mentioned.
Section A:
E]/ectiveness o/ Merit Rating and Class Rating
Introduction
Private passenger automobile insurance uses a multiple classi- fication system. We classify b y age (under or over 25 years) and within each age we classify b y occupation (farm or non-farm). We also classify b y use and sex. On top of all this we classify b y territory. And now we have begun to classify b y plevious accident and conviction record which is popularly called the "merit rating 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 conviction record, just as the rates for each class are based on the subsequent experience of 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 multiple classification system. However, in this paper, as a matter of convenience, and not im- plying a basic distinction, we will follow the common usage in the United States b y referring to classification according to previous accident and conviction record as "the merit rating plan", and to classifiCation according to age, sex, use and occupation as "the class plan."
A class plan which uses age, sex, use and occupation does not precisely classify each risk according to its true value. Underwriters have long recognized this, and it is further substantiated b y the Canadian merit rating experience which shows that risks which have been accident-free for three or more years have better experience in the following year than the average for their class. Likewise a class plan which uses only the previous accident record would not precisely classify each risk 1. This is shown b y the fact that in the 1) See also "Some Considerations on Automobile R a t i n g Systems Utilizing Individual Driving Records" by Lester Dropkin, Proceedings of the Casualty Actuarial Society (hereafter PCAS) XLVI, p. 16 5.
Canadian merit rating experience, the cars which qualified for the best merit rating have different accident frequencies depending on which class t h e y are in.
This means that private passenger automobile risks v a r y con- siderably from each other and that the class plan and the merit rating plan are both a t t e m p t s to separate the better risks from the poorer risks. Neither plan is perfect, but we would like to discuss the question, " H o w do merit rating and class rating compare with each other in their ability to separate the better risks from the poorer risks ?" After discussing their comparative effectiveness, we shall then discuss their absolute effectiveness.
Comparative E]]ectiveness o] Merit Rating and Class Rating
Table I at the end of this section shows the Canadian automobile experiencO arranged to show what it would have looked like if there had been (I) merit rating without class rating and (2) class rating without merit rating. The premiums have been adjusted to what t h e y would have been if all the cars had been written at I B rates, b y use of the approximate relativities :M e r i t R a t i n g D e f i n i t i o n R e l a t i v i t . y A - l i c e n s e d a n d a c c i d e n t free t h r e e o r m o r e y e a r s 65 X - l i c e n s e d a n d a c c i d e n t free t w o y e a r s 80 Y - l i c e n s e d a n d a c c i d e n t free o n e y e a r 9o B - a l l o t h e r s IOO Class D e f i n i t i o n s 1-pleasure, n o m a l e o p e r a t o r u n d e r 25 IOO 2 - p l e a s u r e , n o n - p r i n c i p a l m a l e o p e r a t o r u n d e r 2 5 I 6 5 3 - b u s i n e s s u s e 16 5 4 - u n m a r r i e d o w n e r o r p r i n c i p a l o p e r a t o r u n d e r 25 240 5 - m a r r i e d o w n e r o r p r i n c i p a l o p e r a t o r u n d e r 25 I65 The purpose of any classification plan is to reduce the rates for the better risks and to offset this reduction with an appropriate increase in the rates for the more hazardous risks. We will define "effectiveness" of a classification plan in this paper to be the extent to which the plan separates the better risks from the overall 1 T h e C a n a d i a n e x p e r i e n c e i n c l u d e s t h a t of v i r t u a l l y e v e r y i n s u r a n c e c o m p a n y o p e r a t i n g i n C a n a d a a n d is c o l l a t e d b y t h e S t a t i s t i c a l A g e n c y ( C a n a d i a n U n d e r w r i t e r s ' A s s o c i a t i o n - - S t a t i s t i c a l D e p a r t m e n t ) a c t i n g u n d e r i n s t r u c t i o n s f r o m t h e S u p e r i n t e n d e n t of I n s u r a n c e .
average. This definition of effectiveness was applied in making an evaluation of a one-year merit rating plan where the better risks would get only a 1.6 % reduction from the average rate if a 15 % discount were given for a one-year accident-free record. Because the reduction of 1.6 % was so small, the plan was considered to be ineffective 1
Since both merit rating and class rating in Canada include about the same proportion, 80 °/o, of the cars in the lowest rated class, a measure of the comparative effectiveness of the two is the percen- tage reduction of the lowest rated class from the over-all average.
R a t i n g Plan Merit rating alone Class rating alone Merit and class rating
combined
Reduction of lowest rated class
from average Io.5 % I3.7 % 2t.4 % Relative Reduction 77 I 0 0 156 Proportion of cars in lowest rated class 80.9 % 8o.I % 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 b y extending it from three years to five (which was done during the latter part of 1959) and b y including convictions. Something also could be gained if the merit rating plan gave extra weight to a loss exceeding, say $ iooo, since it was noted that there is a positive correlation between the loss ratio and the average size of loss. Likewise the Canadian class plan, which is similar to the plans used in the United States, could also be improved. B u t the point remains that merit rating is almost as effective as the class plan in separating the better risks from the poorer risks and a substantial improvement is realized when t h e y are used in combination.
Our previous paper showed the experience for each class subdi- vided b y merit rating 2. This was a natural format because the class plan was here first and merit rating was being imposed on top of the x See Muir, J. M., "Principles a n d Practices in Connection W i t h Classi- fication R a t i n g Systems for Liability Insurance As Applied to P r i v a t e Passenger Automobiles", PCAS XLIV, pp. 32 and 33.
2) Bailey, Robert A. and Simon, LeRoy J., " A n Actuarial Note on the Credibility of Experience of a Single Private Passenger Car", PCAS; X L V I ; Table i, p. i62. This table is reproduced here for convenient reference as Table 3.
already existing class plan. Table 2 at the end of this section shows how the experience would have been presented 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 time instead of number of claims because there is a much greater differ- ence in average claim costs among the classes than among the merit ratings.
The relative loss ratio for Class I within each merit rating is slightly lower t h a n the corresponding ratio shown in our previous paper for merit rating A within each class, indicating a greater effectiveness for class rating. The class plan is most effective in the worst merit rating, B, just as merit rating was shown to be most effective in the worst class, 4-
Absolute E[/ectiveness o/ Merit Rating and Class Rating
Thus far, this paper has shown, based on the Canadian expe- rience, t h a t merit rating is almost as effective as class rating in separating the better risks from 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 analytical expression of the distribution of risks according to their "inherent h a z a r d " is needed. Mr. Dropkin's paper on the negative binomial distribution 1 provides a valuable tool for this purpose. His paper shows t h a t inherent hazards of individual 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 t h a t merit rating will give the best risks a reduction of lO. 5 % from the average when there is no class plan and will still give the best risks within Class I a reduction of 8. 9 % ~ from the average Class I rate. This means t h a t Class I has almost as much variation within it as there is among all classes combined. This demonstrates what has often been recognized, t h a t while merit rating and class rating are effective tools in a relative sense, in an absolute sense both merit rating and class rating are quite ineffective in separating the better risks from the poorer risks.
l) Op. Cit.
*) Bailey, R o b e r t A. a n d S i m o n , L e R o y J., Op. Cit., T a b l e 4, P. I63. T h i s t a b l e is r e p r o d u c e d h e r e for c o n v e n i e n t r e f e r e n c e as T a b l e 4.
T h e r e r e m a i n s a c o n s i d e r a b l e a m o u n t of u n a n a l y z e d v a r i a t i o n a m o n g risks.
Cause o/ the Unanalyzed Variation among Automobile Risks
I t is one t h i n g t o s h o w t h e r e is v a r i a t i o n a m o n g risks a n d a n o t h e r t h i n g t o find t h e c a u s e of v a r i a t i o n .
I n o u r p r e v i o u s p a p e r we listed t h r e e possible r e a s o n s w h y t h e e m p i r i c a l credibilities discussed t h e r e f o r I, 2 a n d 3 y e a r s of m e r i t r a t i n g were n o t in t h e e x p e c t e d r a t i o of i : 2 : 3. T h e y were :
(I) n e w risks e n t e r i n g a class,
(2) a n i n d i v i d u a l r i s k ' s c h a n c e of h a v i n g a n a c c i d e n t v a r y i n g f r o m y e a r to y e a r , a n d (3) a m a r k e d l y s k e w d i s t r i b u t i o n of risks. W i t h t h e h e l p of t h e n e g a t i v e b i n o m i a l d i s t r i b u t i o n , we c a n c h e c k t h e t h i r d a l t e r n a t i v e . Using t h e f o r m u l a d e r i v e d b y Mr. B a i l e y 1 f m t h e c r e d i b i l i t y n Z - - n + a w h e r e n = n u m b e r of a c c i d e n t - f r e e y e a r s a n d a is a p a r a m e t e r in t h e d i s t r i b u t i o n of risks,
we find t h a t t h e r e l a t i v e credibilities for I, 2 a n d 3 y e a r s s h o u l d be in t h e r a t i o of
/ l + a /
" 3
B y s e t t i n g t h e one y e a r c r e d i b i l i t y for Class I cars of .055, s h o w n I
in T a b l e 4 of o u r p r e v i o u s p a p e r 3, e q u a l t o - - , we o b t a i n a I + a
= 17.2. T h e r e f o r e t h e r e l a t i v e credibilities for I, 2 a n d 3 y e a r s should be in t h e r a t i o of I • 1.9o • 2.70 which are close to I " 2 ' 3 as we h a d e x p e c t e d . B u t t h e a c t u a l r e l a t i v e credibilities also s h o w n in T a b l e 4 of o u r p r e v i o u s p a p e r are in t h e r a t i o of I • 1.38 • 1.62. T h e r e f o r e while t h e d i s t r i b u t i o n of risks is d e f i n i t e l y skew, it is n o t 1) Bailey, R o b e r t A. Discussion, " S o m e C o n s i d e r a t i o n s o n A u t o m o b i l e R a t i n g S y s t e m s U t i l i z i n g I n d i v i d u a l D r i v i n g R e c o r d s " , P C A S , X L V I [ .
skew enough to account for such large discrepancies, and we m a y cross out the third alternative listed above.
We know t h a t new risks entering the class account for some of the discrepancy, but we do not feel t h a t new risks can account for such large discrepancies. Therefore we teel t h a t the evidence strongly supports the conclusion t h a t the individual risk's chance of having an accident does vary significantly from year to year.
Thus far we have shown t h a t merit rating and class rating are of about equal effectiveness and t h a t a substantial improvement is realized when t h e y are used in combination. However, both of them leave unanalyzed a considerable a m o u n t of variation among risks. In our investigation of the characteristics of this unanalyzed variation we have eliminated certain factors from consideration and now feel we have reached the point where we m a y state t h a t the still unanalyzed cause (or causes) of variation among individual risks :
(I) 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 the merit rating plan.
Annual mileage, which has long been felt to be an important measure of hazard, fits all these requirements better t h a n a n y other single cause. The distribution of risks according to mileage is widely dispersed 1. Mileage varies significantly from year to year. Farmers, for example, have less mileage t h a n average 2 and business use risks have more mileage t h a n average. The discount for two or more cars in one family is a reflection of mileage. Accident frequen- cies (and even conviction frequencies) are a crude indication of mileage. Mileage is certainly not the whole story because there is conclusive evidence t h a t newly licensed drivers and youthful drivers have a higher accident rate per mile t h a n other drivers a n d t h a t other things such as drinking and irresponsibility play a part, but the evidence supports the conclusion t h a t mileage is a very signif- icant cause of variation among individual risks.
1) See D e S i l v a , H a r r y R. W h y We Have Automobile Accidents. J o h n W i l e y & Sons, N e w Y o r k , 1942, p. 12.
A X Y B Total A + X A + X + Y T A B L E I Canada excluding S a s k a t c h e w a n Policy Years 1957 & 1958 as of J u n e 3 o, 1959 P r i v a t e Passenger Automobile L i a b i l i t y - Non F a r m e r s
E a r n e d Prem.
Merit E a r n e d a t Present [ Losses Loss Relative
Incurred R a t i o Loss
R a t i n g Car Years I B R a t e s [
R a t i o I
Classes I, 2, 3, 4 & 5 Combined
3,356,480 I92,88I,ooo 87,094,00o .452 .895
175'553 1°,518,°°° 6,233,000 .593 1.174
219,597 I3,I 18,ooo 8,461,ooo .645 1.277
398,445 24,152,ooo I9,633,ooo .813 1.61o
4,I5°,°75 24o,669,ooo 121,421,ooo .5o5 I.OOO
3'532'°33 203,399,°°° 93,327,°°° -459 .909
3,751,63o 216,517,0oo IOI,788,ooo .47 ° .931
Merit Ratings A, X, Y & B Combined Class 1 2 3 4 5 Total 1 A 3,325,714 168,998 321,327 252,397 81,639 4, I5o,o75 2,757,520 I94, io6,ooo 9,385,o0o 2o,627,ooo I2,39o, ooo 4,I6I,OOO 24o,669,ooo 159,to8,ooo 84,607,000 6,505,000 13,684,ooo 14,199,000 2,426,ooo I21,42I,OOO 63,191 , O O O • 436 .693 .663 1.146 .583 .505 .397 .863 1.372 1.313 2.269 1.154 1 . 0 0 0 .786 I 2 3 4 5 Total T A B L E 2 C a n a d a excluding Saskatchewan Policy Years 1957 & 1958 as of J u n e 3 ° , I959 P r i v a t e Passenger Automobile L i a b i l i t y - Non F a r m e r s
Earned Prem.[ [ R e l a t i v e
E a r n e d a t Present [ Losses Loss
I
LossClass Car Years I B Rates i[ Incurred R a t i o ,[
R a t i o
Merit R a t i n g A - licensed and accident-free 3 or more years
2'757'520 159,1°8,°°° 63,191,°°o .397 .878
13o,535 7,I75,ooo 4,598,000 .641 1.418
247,424 15,663,ooo 9,589,0o0 .612 1.354
156,87I 7,694,0o0 7,964,ooo 1.o35 2.29 o
64,13o 3,241,ooo 1,752,ooo .541 1.197
I 2 3 4 5 T o t a l Class M e r i t R a t i n g X 13o,7o6 7,233 15,868 17,7o7 4,039 175,553 E a r n e d E a r n e d P r e m . Losses L o s s Car Y e a r s a t P r e s e n t I n c u r r e d R a t i o I B R a t e s - - l i c e n s e d a n d a c c i d e n t - f r e e 2 y e a r s 7,9IO,OOO 4,o55,ooo .513 431,ooo 380,000 .882 1,o8o,ooo 7 o i , o o o .649 888,000 983,ooo 1.1o 7 2o9,ooo 114,ooo .545 IO,5X8,ooo 6,233,ooo .593 R e l a t i v e Loss R a t i o .865 1.487 1.o94 1.867 .919 I . O O O I 2 3 4 5 T o t a l M e r i t R a t i n g Y - - l i c e n s e d a n d a c c i d e n t - f r e e I y e a r 163,544 9~862,ooo 5,552,ooo .563 9,726 572,ooo 439,ooo .767
2o,369 1,382,ooo I,OlI,OOO .732 2I,O89 1,o52,ooo 1 , 2 8 i , o o o ][.218 4,869 25o,ooo 178,ooo .712 219,597 I 3 , I 1 8 , o o o 8,461,ooo .645 .873 1.189 1.135 1.888 1.1o 4 I . O O O I 2 3 4 5 T o t a l M e r i t 273,944 21,5o4 37,666 56,73 o 8,6Ol 398,445 R a t i n g B 17,226,000 1,207,000 2,502,000 2,756,000 461,000 24,152,OOO - - all o t h e r 11,8o9,ooo 1,o88,ooo 2,383,0o0 3,971,ooo 382,0oo I 9 , 6 3 3 , o o o .686 .9Ol • 952 1.441 .829 .813 .844 i . I O 8 1.171 1.772 1.o2o i . o o o T A B L E 3 C a n a d a e x c l u d i n g S a s k a t c h e w a n P o l i c y Y e a r s 1957 & 1958 as of J u n e 3 ° , 1959 P r i v a t e P a s e n g e r A u t o m o b i l e L i a b i l i t y - N o n - F a r m e r s M e r i t R a t i n g A X Y B T o t a l A + X A + X + Y E a r n e d P r e m . E a r n e d a t P r e s e n t C a r Y e a r s B R a t e s Class i - - P l e a s u r e - - n o m a l e o p e r a t o r I59,IO8,OOO 217,151 7,91o, ooo 13,792 9,862,000 19,346 17,226,ooo 37,73 o I 9 4 , 1 o 6 , o o o 2 8 8 , o i 9 I 6 7 , o i 8 , o o o 23o,943 I 7 6 , 8 8 o , o o o 25o,289 2,757,52o 13o,7o6 163,544 273,944 3,325,714 2,888,226 3,o51,77 ° C l a i m No. of Freq. p e r C l a i m s $ i o o o I n c u r r e d of P r e m . u n d e r 25 1.365 1.744 1.962 2.19o 1.484 1.383 1.415 R e l a t i v e C l a i m F r e q . .920 1.175 1.322 1.476 I .ooo -932 -954
Claim
IEarnad Prem. No. of Freq. per Relative
Merit Earned [ at Present Clatms Claim
Incurred
R a t i n g Car Years I B Rates $ IOOO [ of Prem. ] Freq.
Class 2 - - Pleasure --- Non-principal A X Y B Total A + X A + X + Y 13o,535 7,233 9,726 21,5o4 168,998 137,768 147,494 11,84o,ooo 712,OOO 944,000 1,992,00o 15,488,ooo 1 2 , 5 5 2 , 0 0 0 I3,496,ooo
male operator under 25 14,5o6 i , o o i 1,43 ° 3,421 20,358 15,5o7 16,937 1.225 1.4o6 1.515 1.717 1.314 1.235 1.255 .932 i.o7 o 1.153 1.3o7 I.OOO .94 ° .955 A X Y B Total A + X A + X + Y
Class 3 - Business use 247,424 15,868 2o,369 37,666 321,327 263,292 283,661 25,846,ooo 31,964 1,783,ooo 2,695 2, 281 ,ooo 3,546 4,I29,ooo 7,565 34,o39,ooo 45,77 ° 27,62%o0o 34,659 29,9IO,OOO 38,2o5 1.237 1.511 1.555 1.832 1.345 1.254 1.277 .920 1.123 1.156 1.362 I .OOO .932 .949
Class 4 - - U n m a r r i e d owner or principal operator under 25 A X Y B Total A + X A + X + Y 156,871 17,7o7 21,o89 56,73 ° 252,397 174,578 195,667 18,45o,ooo 2,I3O,OOO 2,523,000 6,6o8,oo0 29,7It,OOO 20,580,000 23,IO3,OOO 22,884 3,054 3,618 11,345 4o,9ol 25,938 29,556 I. 240 1.434 1.434 1.717 1.377 1.26o .279 .9Ol i .o41 i . o 4 1 1.247 I .OOO .915 .929
Class 5 - Married owner or principal operator under 25 A X Y B Total A + X A + X + Y 64,13 ° 4,039 4,869 8,6Ol 81,639 68,169 73,038 5,349,000 345,000 413,ooo 761 ,ooo 6,868,ooo 5,694,ooo 6, lO7,OOO 6,560 487 613 1,291 8,951 7,o47 7,660 1 . 2 2 6 1 . 4 1 2 1.484 i .696 1.3o3 1.238 t.254 • 9 4 t 1 .o84 1.139 1.3o2 1 . 0 0 0 .950 .962
T A B L E 4
Canada excluding Saskatchewan
Policy Years 1957 & 1958 as of June 3 o, 1959 Private Passenger Automobile L i a b i l i t y - - N o n - F a r m e r s
Merit Rating Earned Pre- miums at Present B Rates Incurred Losses Loss Relative Ratio Loss Ratio Total A + X A + X + Y
Class i - - P l e a s u r e - - n o male operator tmder 25
A I59,IO8,OOO 63,191,ooo -397 .911
X 7,91o,ooo 4,o55,ooo .513 1.177
Y 9,862,000 5,552,000 .563 1.291
B 17,226,ooo 11,8o9,ooo .686 1.573
194,1o6,ooo 84,6o7,ooo .436 i.ooo
I67,oi8,ooo 67,246,ooo .403 .924 176,88o,ooo 72,798,000 .412 .945 Credibility Class t I year I 2 y e a r s t 3 y e a r s I .055 .076 .089 Relative Credibility Class I year I 2 y e a r s I 3 y e a r s I I.OOO 1.38 1.62
Section B:
Improved Methods [or
Determining Classification Rate Relat~vities
Multiple classification systems are quite prevalent in the in- surance industry. For example, in fire insurance we classify the simple dwelling risks by town grading as well as by construction, resulting in a IO × 2 system (typically). Other lines similarly involve multiple classification systems, but automobile is probably the best example. We have used a class plan and a territorial plan in automobile and now we have introduced the merit rating plan. It has been customary to determine a countrywide set of class relativities. Under the merit rating plan it will be necessaly to determine relativities here, too. Assuming these 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 of relativities. The customary procedure * is to sum over all variables except the one we are interested in and then compute our relativities. For example, to get the class relativities, get the total mass of experience broken down only by class. Then the ratio of the experience for each class (usually adjusted in some manner for differences in the distribution by territory and merit rating) to the overall average experience will give the individual class relativity. The same steps would be followed for the merit rating classes and for territories. The subdivisions within each class are added together because individually t h e y are usually not fully credible. Combining them is a means of obtaining a credible volume of experience. This process of combining subgroups results in a loss of some information because a n y combination yields less information t h a n the aggregate information yielded by the individual subgroups. A method for obtaining relativities which is able to avoid combining the sub- groups and is able to use each subgroup individually would produce a better set of relativities.
For purposes of illustration we'll solve the following problem: W h a t is the best set of class relativities and merit rating relativities to use in Canada ? The data is presented in Table B in a loss ,,atio form (all at Class i B rates) and in Table D as relative loss ratios. We will assume t h a t the territorial factor is properly reflected in this data because we are dealing with loss ratios. A better way would be to use pure premiums and to work out territorial relativities at the same time as class and merit rating relativities. However, such d a t a is not available to the authors, but the procedure would be similar in either case. To determine what is an acceptable set of relativities we nmst establish t h e criteria which a set should meet: Criterion I. It should reproduce the experience for each class and merit rating class and also the overall experience; i.e., be
balanced
for each class and in total.Criterion 2. It should reflect the relative
credibility
of the various groups involved.Criterion 3. It should provide a minimal a m o u n t of
departure
from the raw d a t a for the m a x i m u m number of people.
*) For example, see "Current Rate Making Procedures for Automobile Liability Insurance", Stern, Phillip K., PCAS X L I I I , p. x27 ff.
Criterion 4. It should produce a rate for each sub-group of risks which is close enough to the experience so t h a t the differences could reasonably be caused by chance.
A set which meets these four criteria will be judged to be a " b e s t " set of relativities. If more t h a n one set satisfactorily meets the four criteria, the choice among sets m a y be made on a non-mathe- matical basis such as (a) simplicity of application, (b) similarity to existing sets, (c) ease of explanation to non-technical personnel or (d) the a c t u a r y ' s personal preference.
Let us define xi as the class relativity for the #h class ( i = 1,2,3, 4, 5) and Yi as the class relativity for the 1 'th merit rating class (j =: I, 2, 3, 4 representing A, X, Y and B respectively). Let rii be the actual relative loss ratio for persons classified as class i and merit rating class i; r.i is the relative loss ratio of the j'th merit rating class where all i classes are combined; r i. is the relative loss ratio of the i th class where all j merit rating classes are combined; and finally r.. is the relative loss ratio for all classes and merit rating classes combined and thus equals I.OO. Let us also define n~ i as the number of earned car years of exposure. The nii are shown in Table A.
Relativities calculated by the c u s t o m a r y method, which we will call "Method I " , are as follows"
xi - - ri. (I1
and Yi = r.i i and are shown in Table C.
The estimated relative loss ratio is then x~yi, and, if multiplied by the overall loss ratio, will produce the estimated loss ratio for the i, j class. Or, if x~yj is multiplied by the overall pure premium, it would produce the estimated pure premium for the i, j" class. The estimated relative loss ratios, xiYi obtained b y Method i are shown in Table D. When compared with the actual relative loss ratios,
rii, also shown in Table D, it it is evident t h a t there are some
undesireably large differences. Moreover, all xiyl are too low and all
xiy4 are too high.
To test the balance (Criterion I above) we calculate
X n i i x ~ y i / Z niirii (2)
A set of relativities is balanced if equation (2) equals i.ooo. The balance as determined b y equation (2) is shown in Table E. Method I is out of balance in total and far out of balance for the individual classes. If the off-balance in the total is corrected, the classes will still be far out of balance. The reason w h y the classes are so far out of balance is that in our calculation of x~ and Yi, no adiustment was made for differences in the distribution b y class or merit rating class. This illustrates what happens if a merit rating plan is imposed on an already existing class plan without any adjustment in the class relativities. If we had made some tentative adjustment, the off-balance b y class and merit rating class would have been reduced. To make a completely accurate adjustment in the class relativities is difficult, however, because any adjustment in the class relativities disturbs the relativities for the merit rating classes and conversely, thus requiring an adjustment process which zig-zags back and forth. However, even if such an adjustment were made so that Criterion I would be satisfied, Method i would still not satisfy Criteria, 2, 3 and 4, as will be shown later.
Again speaking in general, in order to reflect the relative credi- bility of the various groups involved (Criterion 2), the indicated proportional departure of each group
actual loss r a t i o - expected loss ratio expected loss ratio
should 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 each group should be given a weight inversely proportional to the standard deviation of the indication. The standard deviation of the indication is inversely proportional 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 b y the customary relativities (Method i) because when all the data is added together for, say, class i, to obtain ri. each subgroup rii is given a weight approxi- mately proportional to the expected number of claims instead of the square root of the expected number of claims. This is one of the reasons why Method i does not satisfy Criteria 3 and 4. Moreover,
if each e n t r y in a row of rij is of low credibility, the resulting ri. will not be too trustworthy. Nevertheless, the resulting ri. will be treated as IOO % credible by Method I in the determination of xi. Methods 2, 3 and 4 developed below will remove these defects. Each rii will contribute to the final set in proportion to its relative credibility in relationship to all other rq in the table and not just in relationship to the other members of its row or column, and conversely each xi and Yi will be influenced by all the rii and not just by one row or column of rq.
Thereis no assurance t h a t Criteria 3 and 4 are met by the custom- ary relativities (Method I). In the paragraphs t h a t follow we will show clearly t h a t this set of relativities results in an average depar- ture t h a t is far from minimal and further, t h a t 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 us calculate how much error the average policyholder will have in his estimated relativity by calculating,
nii l rij - - xiYi l / ~ niirii (3)
i,~ i,~
The result of this calculation is shown in Table E.
Equation (3) endeavors to measure how much " i n e q u i t y " the set has. The farther a policyholder's rate is from the indications of the raw data, the more " i n e q u i t y " is involved. Anyone who has dealt directly with insureds at the time of a rate increase, knows t h a t you can be much more positive .when the rate for his class is very close to the indications of experience. The more persons in- volved in a given sized inequity, the more important it is.
To test a set of relativities for compliance with Criterion 4 (dif- ferences between the raw d a t a and the estimated relativities should be small enough to be caused by chance), the Chi-square test is appropriate. It is shown in the Appendix t h a t in terms of relative loss ratios, exposures and relativities,
X* = K E nij (rij - - xiyj)* (4)
iO x~yj
where K is a constant dependent on the d a t a and for the Canadian data, K equals approximately 1/2oo. The values of ~(* are shown in Table E.
It Should be noticed that the X * formula (4) is equivalent to giving the square of the indication a weight proportional to the expected number of claims.
~ = g Z nil ( r i i - - x i Y i ) 2 = g Z nqxiyi('r'~--x~Y~t
i,j x~3~ ~ i,j \ x~y~ ]
This means that a set of xi, Yi, which is specifically designed to produce a minimum X ~ will automatically reflect the relative credibility of each group involved (Criterion 2). This is accom- plished without a credibility weighting process involving tabular credibilities. Moreover, since a set of xi, Yi, which produces a minimum X 2 will very likely also satisfy Criterion 3 (minimal average amount of departure) and will come v e r y close to satisfying Criterion I (balance), it seems evident that the best set of relativities will be those which are designed specifically to produce a minimum X z. These relativities can be obtained b y setting the partial deri- vatives of X ~ equal to zero.
~X2 -- K Z niiYi - - K Z nq r~q _ o (5)
~x, ~ ~ x~yj
Solving for xi, we obtain
x, = [~ n q r ~ ' / ~ n,iy,] ½ (6)
-3,-/,
and similarly,• ,
]t
[~ n " r i ' / ~ niixi (7) Y J = [ i x---~-l iThis gives us nine equations in nine unknowns. Since the equations are not of a simple, rational form, the easiest w a y to arrive at a numerical solution is b y a method of iteration, as follows :
I. 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 first estimates of Yi.
3. Use the first estimates of Yi in the right hand side of (6) to obtain the second estimates of xi.
4. Repeat this process until two consecutive sets of solutions are identical (or substantially so).
Notice t h a t t h e r e are an infinite n u m b e r of solutions for xi a n d Yi, all of which, however, p r o d u c e t h e same set of x i y j . This is t r u e because each xi m a y be m u l t i p l i e d b y a c o n s t a n t if each yj is d i v i d e d b y t h e s a m e c o n s t a n t . T h e results of this m e t h o d , which we shall call " M e t h o d 2 " are s h o w n in T a b l e C. T h e e s t i m a t e d r e l a t i v e loss ratios, x i Y i , are s h o w n in T a b l e D a n d the tests of Criteria I, 3 a n d 4 are s h o w n in T a b l e E.
I t is e v i d e n t t h a t M e t h o d 2, which derives all sets of relativities s i m u l t a n e o u s l y , solves t h e difficult p r o b l e m of o b t a i n i n g relativities which are b a l a n c e d in t o t a l a n d b y class. I t a u t o m a t i c a l l y satisfies Criterion 2 (credibility) a n d it also reduces s u b s t a n t i a l l y t h e a v e r a g e e r r o r a n d X 2 (Criteria 3 a n d 4). B u t in spite of this i m p r o v e m e n t , t h e a v e r a g e e r r o r of .o317 still does not c o m p a r e v e r y f a v o r a b l y w i t h a p r o f i t m a r g i n of .050 or t h e r e a b o u t s , especially for a com- p a n y t h a t writes a d i s p r o p o r t i o n of business in one class. Moreover, Z ~, a l t h o u g h m u c h less t h a n for M e t h o d I, is still too high to be t h e result of chance. This m e a n s t h a t a set of factors which are multi- plied t o g e t h e r , x i Y i , c a n n o t s a t i s f a c t o r i l y r e p r e s e n t t h e a c t u a l d a t a for C a n a d i a n p r i v a t e passenger automobiles, a l t h o u g h it m a y be s a t i s f a c t o r y for o t h e r lines or t y p e s of data.
T u r n i n g t o t h e a c t u a l d a t a , s h o w n in T a b l e D, it can be seen t h a t the p e r c e n t a g e difference b e t w e e n t h e lowest a n d the highest m e r i t r a t i n g decreases as t h e r a t e for the class increases, r a n g i n g f r o m 73 % for class I down to 39 % for class 4. W i t h these conditions p r e s e n t in the basic C a n a d i a n data, it is little w o n d e r t h a t the m u l t i p l i c a t i v e relativities do n o t fit satisfactorily.
A possible m e t h o d , which we will call " M e t h o d 3", is to let t h e e s t i m a t e d r e l a t i v e loss r a t i o be x~ + Yi, where the relativities are a d d e d instead of multiplied. T h e Z ~ f o r m u l a becomes
X a : K E n i i ( r i i - - x i - - y i ) 2 (8) i,j x , + y j
A n d setting t h e p a r t i a l d e r i v a t i v e s of Z 2 equal to zero we h a v e :
~Z2 . 2
- - K Z n i j - - K E n # r ~ - - o (9)
x, , j (x, + y~)~
F o r convenience let us write (9) as ](xi) = o. If we first o b t a i n an e s t i m a t e of x~, we can o b t a i n a correction, A xi, to be a d d e d to xi b y the use of N e w t o n ' s m e t h o d ; t h a t is,
± xi l(xi)
l'(x,)
w h e r e 1' (x~) is the d e r i v a t i v e of / ( x i ) . Using this p r o c e d u r e we o b t a i n
n i i [ rii - - ~ nii
~ x, + yj
xi
=(io)
2 E [ rij \~ I
T h e expression for A Yi is t h e same as for A xi e x c e p t t h a t t h e s u m m a t i o n s are t a k e n o v e r i i n s t e a d of i. T h e x~ a n d Yi are d e r i v e d as follows:
I. Select a set of first e s t i m a t e s of xi a n d Yi.
2. Use these values in (io) to o b t a i n A xi a n d A y i.
3. A d d A x~ to xi a n d A Yi t o Yi to o b t a i n t h e second e s t i m a t e s of xi a n d Yi.
4. R e p e a t this process u n t i l all A xi a n d A Yi are e q u a l to zero. I t s h o u l d be n o t e d here again t h a t t h e r e are an infinite n u m b e r of solutions for xi a n d Yi, all of which, h o w e v e r , p r o d u c e t h e same set of xi + Yi. This is t r u e because a c o n s t a n t m a y be a d d e d to all t h e xi if t h e same c o n s t a n t is s u b t r a c t e d f r o m all t h e Yi, a n d this will n o t c h a n g e a n y of t h e e s t i m a t e d r e l a t i v e loss ratios, x~ + y;. In fact, if we let t h e e s t i m a t e d r e l a t i v e loss ratios be (xi + y~ - - I) a n d a l t e r f o r m u l a (io) accordingly, we can use t h e values of xi a n d
Yl o b t a i n e d b y M e t h o d I as o u r first e s t i m a t e s to be used in (IO). I t m a y be well at this p o i n t to e m p h a s i z e h o w little absolute m e a n i n g c a n be a t t a c h e d t o a given set of relativities. W h e t h e r t h e y were b a s e d on a m i n i m u m X * or not, or w h e t h e r t h e y were m u l t i p l i c a t i v e or additive, a simple t r a n s f o r m a t i o n c a n c h a n g e t h e i r i n d i v i d u a l v a l u e s and, of course, the h a p p e n s t a n c e of o u r choice of initial values in solving e i t h e r (6) or (IO) will p r o d u c e one solution i n s t e a d of a n o t h e r . I t is q u i t e n a t u r a l us to a t t e m p t to a t t a c h a special m e a n i n g t o a d e v e l o p e d set of relativities; t h a t is, to i m p a r t to t h e m some special q u a l i t y in a n d of themselves. H o w e v e r , t h e y can o n l y be r e g a r d e d in r e l a t i o n s h i p to t h e c o o r d i n a t e s y s t e m in which t h e y find themselves.
The values of xi and yf obtained b y Method 3 are shown in T a b l e C, the estimated relative loss ratios, xi + yj, are shown in Table D and the tests of Criteria I, 3 and 4 are shown in Table E.
I t is evident t h a t Method 3 not only satisfies Criteria I and 2 (balance and credibility) but it also reduces the average error to .0098 which is much b e t t e r than Methods I and 2, and it produces a X ~ which could v e r y easily be the result of chance. This means t h a t while the actual data cannot be represented satisfactorily b y a set of relativities which are multiplied, x i y i , the actual data can be satisfactorily represented b y a set of relativities which are added, x ~ + yj.
Another m e t h o d of obtaining relativities which we will call " M e t h o d 4" is a compromise between Methods 2 and 3. Let the estimated relative loss ratios be a x i y i - - (a - - I) and then minimize Z 2 - ~ K ~ n ~ i [ r ~ i - ( a x i y i - - a + I)~ 2 (II)
,,~ a x , y , - (a - - I)
If a = I, (II) reduces to (4) which is Method 2. With the proper selection of a , greater t h a n I, results can be produced which are v e r y similar to Method 3. It seems t h a t the only practical w a y to obtain the o p t i m u m value of a is b y judgment. Basing our j u d g m e n t on the four corner values of r n, r14, r41 and r44, we selected a = 3. For c o m p u t a t i o n a l purposes, equation (II) was translated to the form of equation (4) b y adding 2 to each rij and dividing the results b y 3. The relativities, xi and Yi, were then obtained b y the iterative process described for Method 2 and are shown in Table C. T h e estimated relative loss ratios, 3 x ~ Y i - 2, are shown in TaMe D and the tests of Criteria I, 3 and 4 are shown in Table E. It can be seen t h a t Method 4 produces results v e r y similar to Method 3, and for the Canadian d a t a t h a t both Methods 3 and 4 satisfy all four criteria listed at the beginning of this section. Moreover, t h e y both are methods of calculating all sets of rela- tivities simultaneously.
We have developed only a two dimensional problem (x b y y) here, but the methods can easily be extended to include more dimensions such as farm versus non-farm and territorial relativities. A small computer would be very useful in performing the tedious calcu- lations which would be involved.
Consequences o] Using Multiplicative Rdativities
When the a t t e m p t is made to fit a set of relativities which are multiplied,
xiYi,
to a set of d a t a that should be fitted b y a set of relativities which are added,xi + Yi,
rates are produced for the lowest rated class that are too high in the lowest merit rating and too low in the highest merit rating and rates are produced for the highest rated class that are too low in the lowest merit rating and too high in the highest merit rating. This can be seen in Table D b y comparing Method 2 with the actual d a t a or with Method 3.It is evident that the same difficulty occurs in private passenger automobile insurance in the United States when a countrywide set of class relativities is multiplied b y a set of territory relativities. Several attempts have been made to correct this difficulty. Two sets of countrywide class relativities are used, one for large cities and one for all other territories *. The relativities for large cities have a smaller spread between the lowest and the highest rated classes. This is quite likely not caused b y a difference in classification experience between high-rated territories and low-rated territories b u t it is the result of trying to use two sets of relativities which are multiplied when it is quite likely that the two sets of relativities should be added instead of multiplied. Another example of an a t t e m p t to correct this situation is the fact that in New York City, which is about the highest rated territory in the United States and where, in addition, the experience has enough volume to be credible, a special set of class relativities is used which has much less spread between the lowest and highest rated classes than the sets of relativities used elsewhere.
The reason this difficulty has not become more noticeable in other territories is that very few territories have sufficient volume to be credible for each class. B u t it is very likely that multiplying countrywide class relativities b y territory relativities has produced and is producing rates which are too high for Class I in very low rated territories and too low for Class i in very high rated terri-
tories. This situation will become worse if three sets of relativities, *) See Stern,
Op. cit.,
p. 154 and Livingston, G. R., & Carlson, T. O., discussion of "Principles and Practices in Connection with Classification R a t i n g Systems for Liability Insurance as Applied to P r i v a t e Passenger Automobiles". PCAS XLV, p. 23o.for territories, classes and merit rating classes, are all multiplied together, xiyizk. The introduction of merit rating makes it all the more i m p o r t a n t to use a method of obtaining relativities which will satisfy the four criteria listed at the beginning of this section.
The methods developed in this paper, designated Methods 2, 3 and 4, have possibilities of wide application in m a n y lines of in- surance. For example, the non-reviewed workmen's compensation classes could be treated on a nationwide basis with relativities established by class and state. General Liability classes, which often involve a limited amount of exposure, could similarly be treated on a nationwide basis with relativities by class and territory. A & H involves m a n y relativities. In automobile insurance itself the excess limits tables could be tested to determine whether the limits charges are, in fact, multiplicative with the basic rates or are more properly included as some other function. One can also visualize Homeowners rate making on a pure premium basis per homeowner with relativities for protection grading, construction and policy size (this latter item is a quantitative characteristic of the experience and would introduce an interesting facet into the problem). A multitude of similar practical problems could also be solved through this technique.
T A B L E A A r r a y of N u m b e r of E a r n e d Car Y e a r s of E x p o s u r e ** no w i t h ooo o m i t t e d Class * I 5 3 2 4 T o t a l M e r i t R a t i n g Class A I 2758 64 247 131 157 3357 I x ~ Y B 2 3 4 164 274 5 9 2o 38 I O 2 2 2i 57 22o 400 131
it
I76 T o t a l 3327 82 321 17o 253 4153 *) T h e s e c l a s s i f i c a t i o n s h a v e b e e n r e a r r a n g e d so t h a t t h e " T o t a l " c o l u m n in T a b l e 13 is in a s c e n d i n g o r d e r . **) S o u r c e : T a b l e 2 a t e n d of S e c t i o n A.T A B L E B A r r a y of L o s s R a t i o s a t i B R a t e s ~ M e r i t R a t i n g Class Class z 5 3 2 4 T o t a l A I .397 • 541 .612 .641 i .035 .452 X 2 .513 .545 .649 .882 1.1o7 .593 Y 3 .563 .712 • 732 .767 1.218 -645 B 4 .686 .829 .952 . 9 o i 1.441 .813 T o t a l • 436 .583 .663 .693 1.146 .505 Class M e r i t R a t i n g Class X 1 X~ X 3 X t X 4 T A B L E C R e l a t i v i t i e s M e t h o d i C u s t o m a r y 8) .863 1.154 1.313 1.372 2.269 .895 1 . I 7 4 1.277 1.61o M e t h o d 2 (Min X ~ o n x, yj) .881 1.161 1.3o9 1.367 2.125 .906 1.113 1.215 1.462 M e t h o d 3 (Min X z o n x~ + Yt) .869 1.1,45 1.291 1.352 2 . I 7 2 - . 0 8 3 .135 .237 .512 M e t h o d 4 (Min X 2 o n 3x, Y~-2) .958 1.049 i .099 1.118 1.384 • 97 I i.o4o i .o76 I.I67 1) S o u r c e : T a b l e 2 a t e n d of S e c t i o n A. I) T h e s e c l a s s i f i c a t i o n s h a v e b e e n r e a r r a n g e d so t h a t t h e , , T o t a l " c o l u m n i n T a b l e B is i n a s c e n d i n g o r d e r . a) S o u r c e : T o t a l c o l u m n a n d T o t a l r o w i n T a b l e B d i v i d e d b y .5o5.
T A B L E D A r r a y s of R e l a t i v e Loss R a t i o s A c t u a l , r~ ( T a b l e B d i v i d e d b y .505) M e r i t R a t i n g Class I 5 Class 3 2 4 A X Y i 2 3 .786 i . o i 6 1.1i 5 1.o71 i . o 7 9 1.41o 1.212 1.285 1.45o 1.269 1.747 1.519 2.o5o 2.192 2.412 B 4 1.358 1.642 1.885 1.784 2.853
X
I 5 3 2 4 M e t h o d I, x, yj ( C u s t o m a r y ) • 772 1.o33 1.175 i .228 2.031 1.Ol 3 1.355 1.541 1.611 2.664 1.102 I "474 1.677 1.752 2.898 i .389 I 1.858 5 2 . I I 4 3 2.2o9 2 3.653 4 M e t h o d 2, x~y 1 ( M i n i m u m X z o n x, yj) 3 • 798 .981! 1.o7o 1.o52 1.292 I 1.411 1.186 1.457 1.59o 1.239 1.521 1.661 1.925 2.3651 2.582 1.288 i .697 1.914 i .999 3.1o7X
i 5 3 2 4 M e t h o d 3, x, + y~ M i n i m u m X 2 o n x, + yj) M e t h o d 4, 3 x ~ > q - 2 ( M i n i m u m X a o n 3x, y j - 2) 1 .786 I.O62 1.2o8 1.269 2.089 2 1.004 1.28o 1.426 1.487 2.307 3 4 1.1o6 1.381 I 1.382 1.657 5 1.528 1.8o 3 3 1.589 1.864 2 2 . 4 ° 9 2.684 4 I .787 I.O57 1.198 1.255 2.o29 2 .988 1.276 1.429 1.489 2.320 3 1.o9o 1.387 1.543 1.606 2.464 1.354 1.675 1.846 1.915 2.845T A B L E E T e s t s of C r i t e r i a I, 3 a n d 4 C r i t e r i o n I, B a l a n c e Class M e r i t R a t i n g Class T o t a l T o t a l X 2 D e g r e e s of f r e e d o m f o r ;(~ P r o b a b i l i t y ~ s > o b s e r v e d ] M e t h o d x a .9886 x 5 1.oo99 x a 1.o195 x~ 1.o23 ° x 4 1.1o67 Yl .98o6 Yt 1.°589 Ya 1.°536 Y4 I . I I 2 2 1.O103 I M e t h o d 2 M e t h o d 1.ooo 7 i.OOli 1.OOl 4 1.oo24 1.ooo6 .9993 1.oo27 1.oo27 1.oo27 .9974 1.ooo6 1.OOl 5 i , o o 2 6 1.oo83 1.OOl 5 I.OO2O 1.oo25 .9931 I.OOli 1.ooo6 C r i t e r i o n 3, A v e r a g e E r r o r ,o4oi .o317 .0098 I O 1 2 .6o less t h a n . .OOl C r i t e r i o n 4, C h i - S q u a r e 98 34 I 2 1 2 a b o u t . o o i 3 [ M e t h o d 4 -9979 I , O O O 8 ,9982 1.ooo 5 .9994 .9978 -9996 .9986 1 . 0 0 0 2 .9983 . o I I I 8 I I • 7 ° A P P E N D I X
H a r a l d Cram6r in his book, " M a t h e m a t i c a l M e t h o d s of S t a t i s t i c s " , pages 233 a n d 234, shows t h a t if 41 . . . ~n are n i n d e p e n d e n t r a n d o m variables each of which is n o r m a l w i t h a m e a n of o a n d a v a r i a n c e of I, t h e n
is d i s t r i b u t e d according to t h e well-known Chi-square d i s t r i b u t i o n . F o r t h e C a n a d i a n data,
a c t u a l r e l a t i v e loss r a t i o - e x p e c t e d r e l a t i v e loss r a t i o s t a n d a r d d e v i a t i o n of t h e a c t u a l r e l a t i v e loss r a t i o
has a m e a n of o a n d a v a r i a n c e of I a n d is v e r y close to n o r m a l w h e n t h e a c t u a l r e l a t i v e loss r a t i o is b a s e d on t h e a v e r a g e of a
large number of car years. The actual relative loss ratio is ri i, the expected relative loss ratio is xiyj. The variance of the actual re- lative loss ratio is approximately
follows: Letting C =
S : - -
rnt = m c - ~ m p Then S: =2°°xiYi and is developed as
nf.f value of claim
number of car years
variance of the pure premium for one car year variance of the claim cost
mean claim frequency per car year mean claim cost
mean pure premium per car year E C 2 / n - - (Z C / n) ~ by definition
rntE(~
C~ nml) ~ + X; C~/ nm! - - (Z C~ nml) ~] - - (X; C /,~)~
m I (m: + S:) - - m~m: (12) Equation (12) agrees with Mr. R. E. Beard, "Analytical Expressions of the Risks Involved in General Insurance", in Transactions XVth International Congress of Actuaries, 1957, Vol.II,
p. 233.If we let the time interval be less t h a n one year a n d approach zero as a limit, which is appropriate if S** is based on a distribution of claims where each claim is listed separately regardless of how close in time t h e y m a y have occurred, the second term in equation (12) becomes insignificant and we obtain
S'p = m! (m: + S:) (I3)
which agrees with Mr. A. L. Bailey, "Sampling Theory in Casualty Insurance", PCAS X X l X , p. 60.
Formula (13) can be written
s ; = " i + m : ~ (14)
Equation (14) is the variance of the pure premium for one car year. The variance of the mean pure premium per car year based on a group of n car years, where each mean is divided by the overall mean pure premium, P, is therefore
_
Since P = M c M t, where M is the overall mean, and mp is approxi- Snp
mately equal to M c m t, and xiYi ~ ~-, equation (15) can be written
"~ \ n,, ] \ M t l I + m'fl
I t is estimated t h a t for the Canadian data, which is total limits for
B I a n d P D combined, I + equals approximately 2o. This is only
• ~ ' n . !
a rough estimate based on the limited d a t a available to the authors. M! = .o97. Therefore for the Canadian d a t a
X2 = I .. Ig n q ( r i i - - xiYi)~, approximately.
200 ,,~ Xly ~
Notice t h a t the same constant, K, is produced regardless of whether
