NCCI S 2007 HAZARD GROUP MAPPING

Transcription

1 NCCI S 2007 HAZARD GROUP MAPPING by John P. Robertson ABSTRACT At the begnnng of 2007, the NCCI mplemented a new seven-hazard-group system, replang the prevous four-hazard-group system. Ths artle desrbes the analyss that led to the assgnment of lasses to the new seven hazard groups. 1. INTRODUCTION A hazard group s a olleton of workers ompensaton lassfatons that have relatvely smlar expeted exess loss fators over a broad range of lmts. At the begnnng of 2007, the Natonal Counl on Compensaton Insurane (NCCI) mplemented a new seven-hazard-group system, replang the prevous four-hazardgroup system. The new hazard groups are not smply a subdvson of the prevous four; they are a substantally dfferent mappng of lasses to hazard group. Ths artle desrbes the analyss that led to the assgnment of lasses to the new seven hazard groups. Under the prevous NCCI four-hazard-group system, the bulk of WC exposure n NCCI states was onentrated n two hazard groups, as an be seen n Table 1. Table 1 Dstrbuton of Classes by Pror Hazard Group NCCI Hazard Group Number of Classes Premum (bllons) % of Total Premum I 38 $ % II 428 $ % III 318 $ % IV 86 $ % In our analyss, we onsdered whether a fner delneaton would be possble, and what mght be the optmal number of hazard groups. Apart from those onsderatons, hazard group assgnments should be revewed perodally beause of hanges over tme n the nsurane ndustry, tehnology, workplaes, and the evoluton of the lassfaton system and Workers Compensaton nfrastruture. The prevous revew had been done n Copyrght 2009 Natonal Counl on Compensaton Insurane, In. All Rghts Reserved. 1

2 Pror Work The pror NCCI hazard groups were developed by frst dentfyng seven varables based on relatve lam frequeny, severty, and pure premum, whh were thought to be ndatve of exess loss potental [3]. These varables were the ratos of lass to statewde weghted average: 1. serous to total lam frequeny rato 2. serous ndemnty severty 3. serous medal severty 4. serous severty, nludng medal 5. serous to total ndemnty pure premum rato 6. serous medal to total medal pure premum rato 7. serous pure premum to total pure premum rato. Beause of the orrelaton between the severty varables and the pure premum varables the above seven varables were redued to just the frst, fourth, and last varables. A prnpal omponents 1 analyss was then done to determne the lnear ombnaton of these varables that maxmzed the proporton of the total varane explaned by ths lnear ombnaton. The lnear ombnaton so dentfed s alled the frst prnpal omponent and s the sngle varable that was used to sort lasses nto hazard groups. Determnaton of the optmal number of hazard groups was outsde the sope of that study and so the number of hazard groups remaned unhanged at four. A very dfferent approah was employed by the W.C. Insurane Ratng Bureau of Calforna (WCIRB) [4]. The WCIRB s objetve was to group lasses wth smlar loss dstrbutons. They used two statsts to sort lasses nto hazard groups. The frst statst was the perentage of lams exess of $150,000. Ths statst was thought of as a proxy for large loss potental. The seond statst measured the dfferene between the lass loss dstrbuton and the average loss dstrbuton aross all lasses. The dfferent hazard groups orresponded to dfferent ranges of these two statsts. The results were heked by usng luster analyss on these two varables. Overvew Our approah owes muh to the pror work on the subjet, yet t s qute dstnt. We sorted lasses nto hazard groups based on ther exess ratos rather than proxy varables. As shown n Corro and Engl [5], a dstrbuton s haraterzed by ts exess ratos and so there s no loss of nformaton n workng wth exess ratos rather than wth the sze of loss densty or dstrbuton funton. Seton 2 desrbes how we omputed lass-spef exess ratos. Seton 3 desrbes how we used luster analyss to group lasses wth smlar exess ratos, and how we determned that seven s the optmal number of hazard groups. 1 See Johnson and Whern [2] for a dsusson of Prnpal Components. Copyrght 2009 Natonal Counl on Compensaton Insurane, In. All Rghts Reserved. 2

3 In seton 4 we ompare the new hazard group assgnments wth the pror assgnments. Followng the analyt determnaton of hazard groups, the tentatve assgnments were revewed by several underwrters, and, based on ths nput, NCCI hanged some assgnments; we desrbe ths n seton 5. Fnally, seton 6 reaps the key deas of ths study and the key features of the new assgnments. 2. CLASS EXCESS RATIOS Gllam [6] desrbes n detal the NCCI proedure for omputng exess ratos by hazard group for ndvdual states. An exess rato or exess loss fator (ELF) s the rato of losses exess of a gven lmt to total losses. 2 In the NCCI proedure, eah ELF for a state and hazard group s a weghted average of ELFs by njury type spef to the state and hazard group. The ELFs for an njury type for a state and hazard group are derved from ELFs for the njury type n the state, adjusted to the estmated mean loss n the hazard group n the state. Injury types used by NCCI are Fatal, Permanent Total, Permanent Partal, Temporary Total, and Medal Only. To put ths n mathematal terms, let X be the random varable gvng the amount of loss for njury type n the state, and let X have densty funton f (x) and mean µ. Let S be the normalzed state exess rato funton for njury type ; that s X S ( r) = E max r, 0 = dt µ r ( t r) g ( t). where g (x)= µ f (µ x) s the densty funton of the normalzed losses X / µ. For a hazard group j, the overall exess rato R j (L) at lmt L s R j ( L) = w, j S ( L / µ, j ), (1) where w,j s the perentage of losses due to njury type n hazard group j, so w = 1 and µ,j s the average ost per ase for njury type n hazard group j., j, In the same way we an ompute ountrywde exess ratos for a gven lass by just knowng the weghts and average osts per ase by njury type for a lass. We based these on the most reent fve years of data as of Aprl, Ths nluded lam ounts and losses by njury type for the states where NCCI ollets suh data. Losses were developed, trended, and brought on-level to reflet hanges n workers ompensaton 2 Another ommon defnton of exess loss fator s as the rato of losses exess of a gven lmt to premum. We are onerned only wth ratos of losses exess of gven lmts to total losses. Copyrght 2009 Natonal Counl on Compensaton Insurane, In. All Rghts Reserved. 3

4 benefts. Wth some mnor state exeptons, the same lasses apply n all states. As suh, we ould estmate lass exess ratos on a ountrywde bass. Thus for eah lass,, we had a vetor R = (R (L 1 ), R (L 2 ),, R (L n )) of exess ratos at ertan loss lmts L 1, L 2,..., L n. The redblty to assgn to eah lass exess rato vetor s onsdered n the next subseton, and seleton of the loss lmts to use n the analyss s dsussed n Seton 3. Credblty In the pror revew, the redblty gven to a lass was n z = mn 1.5, 1, (2) n + k where n s the number of lams n the lass and k s the average number of lams per lass. Ths gves a lass wth the average number of lams 75% redblty and a lass wth at least twe the average number of lams full redblty. Fgure 1 shows the redblty produed by ths formula by sze of lass. 100% Fgure 1 Class Code Credblty 80% Credblty 60% 40% 20% 0% Number of Class Codes (Ordered by Clam Count) Copyrght 2009 Natonal Counl on Compensaton Insurane, In. All Rghts Reserved. 4

5 The fully redble lasses have over 70% of the total premum as an be seen n Table 2. Table 2 Dstrbuton of Classes by Credblty Clams per Number of % Credblty Range Year Classes Premum 0% z < 10% % 10% z < 20% % 20% z < 30% % 30% z < 40% % 40% z < 50% % 50% z < 60% % 60% z < 70% % 70% z < 80% % 80% z < 90% % 90% z < 100% % z = 100% % Total % That the largest lasses have a dsproportonate share of the lams an be seen n Fgure 2, where the lasses wth the greatest number of lams are to the left. Copyrght 2009 Natonal Counl on Compensaton Insurane, In. All Rghts Reserved. 5

6 100% Fgure 2 Dstrbuton of Classes by Clam Count Cumulatve % of Clams 80% 60% 40% 20% 0% Number of Class Codes (Ordered by Clam Count) Indeed the dstrbuton of lams per lass s very hghly skewed, as an be seen n Fgure 3. Fgure 3 Hstogram of Number of Clams by Class Number of Classes n Interval Number of Clams (thousands) Fgure 4 expands the frst bar n Fgure 3, and shows the perssteny of the skewness. Copyrght 2009 Natonal Counl on Compensaton Insurane, In. All Rghts Reserved. 6

7 Fgure 4 Detal of Hstogram of Number of Clams by Class 500 Number of Classes n Interval ,000 2,000 3,000 4,000 5,000 6,000 7,000 8,000 9,000 10,000 Number of Clams And Fgure 5 further expands the frst bar n Fgure 4 revealng the same general pattern. 180 Fgure 5 Detal of Hstogram of Number of Clams by Class Number of Classes n Interval Number of Clams Copyrght 2009 Natonal Counl on Compensaton Insurane, In. All Rghts Reserved. 7

8 The average number of lams per lass s nearly ten tmes the medan. We thus onsdered usng the medan rather than the mean for k n equaton (2). Ths would have resulted n a very large nrease n redblty as shown n the Fgure 6. Fgure 6 Comparson of Credblty Formulas 100% 80% Credblty 60% 40% 20% 0% k=medan (468) k=mean (3,325) Number of Class Codes (Ordered by Clam Count) We onsdered several other varatons on formula (2) as well. Beause medal only lams have almost no mpat on the ELFs at the publshed lmts, we onsdered exludng all medal only lams. Takng that a step further, we looked at nludng only serous lams. We also onsdered takng k n formula (2) to be the mean number of lams over only those lasses wth some mnmal number of lams. In addton, we onsdered basng redblty on varous square root rules. We onsdered a smple square root rule of the form n z =, 384 where n s the number of lams n a lass, and z s apped at 1. The full redblty standard of 384, gven n Hossak, Pollard, and Zehnwrth [10, page 159], orresponds to a 95% hane of the atual number of lams wthn 10% of the expeted number of lams. For the determnaton of ELFs, serous lams (Fatal, Permanent Total, and major Permanent Partal) are more mportant than non-serous lams, so we looked at the followng varaton on the square root rule above Copyrght 2009 Natonal Counl on Compensaton Insurane, In. All Rghts Reserved. 8

9 where z = N F nf nm + N M + N N + N + N F n F = the number of fatal lams n the lass N F = the number of fatal lams n all lasses n M = the number of permanent total and major permanent partal lams n the lass N M = the number of permanent total and major permanent partal lams n all lasses n m = the number of mnor permanent partal and temporary total lams n the lass N m = the number of mnor permanent partal and temporary total lams n all lasses. M m m n m 384, We also onsdered varyng the full redblty standard by njury type wth the followng redblty formula ns n ns N s + ( N N s ) z = N where n s = the number of serous lams n the lass N s = the number of serous lams n all lasses n = the total number of lams n the lass N = the total number of lams n all lasses. In the end, none of the alternatves onsdered seemed ompellng enough to warrant a hange and the results dd not seem to depend heavly on the redblty formula; onsequently we retaned formula (2) for omputng redblty. For the omplement of redblty we used the exess ratos orrespondng to the urrent hazard group of the lass. More presely, for eah lass we have a vetor of exess ratos R = (R (L 1 ), R (L 2 ),, R (L n )) and a redblty z. We also have a vetor of exess ratos for the hazard group HG ontanng the lass (whh an be determned, as above, as a loss weghted sum over vetors for lasses n HG) R HG = (R HG (L 1 ), R HG (L 2 ),, R HG (L n )). We now assoate to eah lass a redblty-weghted vetor of exess ratos zr + (1-z)R HG. Copyrght 2009 Natonal Counl on Compensaton Insurane, In. All Rghts Reserved. 9

10 It s these redblty-weghted vetors of exess ratos that we use n the luster analyss desrbed n the next seton. 3. ANALYTIC DETERMINATION OF THE NEW HAZARD GROUPS The fundamental analyt method used to determne the new hazard groups s Cluster Analyss. Ths s desrbed n more detal below, but s a way to group lasses wth smlar ELFs. Seleton of Loss Lmts The lass exess rato s a funton of the loss lmt, so t was neessary to selet the attahment ponts to use n the analyss. We used lmts of 100, 250, 500, 1000, and 5000, n thousands of dollars. Beause exess ratos at dfferent lmts were hghly orrelated, fve lmts were thought to be suffent. We onsdered usng fewer lmts but deded that t was better to use fve lmts to over the range ommonly used for retrospetve ratng. We began by onsderng the 17 lmts for whh NCCI publshed exess loss fators before These lmts, n thousands of dollars, were: 25, 30, 35, 40, 50, 75, 100, 125, 150, 175, 200, 250, 300, 500, 1000, 2000, and We modfed ths lst by droppng $300,000 and addng $750,000. We redued ths to the fve seleted attahment ponts based prmarly on two onsderatons: ELFs at any par of exess lmts are hghly orrelated aross lasses, espeally when the rato of the lmts s lose to 1. Lmts below $100,000 are heavly represented n the lst of 17 lmts. The orrelatons were omputed usng only the 162 lasses wth at least 75% redblty. Classes wth small redblty have estmated ELFs lose to those for the pror overall hazard group. Inludng the low-redblty lasses would skew the orrelatons towards those of the overall hazard groups. Even among the fve seleted lmts, orrelatons between ELFs for pars of lmts are very hgh, as an be seen n Table 3. Table 3 Correlatons Among Exess Ratos at Seleted Lmts Lmt 100, , ,000 1,000,000 5,000, , , , ,000, ,000,000 Copyrght 2009 Natonal Counl on Compensaton Insurane, In. All Rghts Reserved. 10

11 Eah of the 12 lmts not used has a orrelaton oeffent of at least wth a lmt that was used, as an be seen n Table 4. Table 4 Correlatons of ELFs for Pars of Lmts Lmts not Seleted Most Correlated Lmt of the Fve Seleted Correlaton Coeffent 25, , , , , , , , , , , , , , , , , , , , ,000 1,000, ,000,000 1,000, Although we ultmately used fve lmts, we expermented by lusterng wth dfferent lmts. We found that the hazard group assgnments resultng from fve lmts were qute smlar to those resultng from 17. When mappng the lasses to seven hazard groups, only 68 out of 870 lasses were assgned to dfferent hazard groups and these aounted for just 5.5% of the total premum. To see whether fve lmts were more than needed for the analyss, we tred lusterng the lasses usng only a sngle lmt. In one nstane we used $100,000 and n another we used $1,000,000. Fgures 7 and 8 ompare those sngle lmt assgnments wth lusterng usng the fve-lmt approah. In both ases, the results dffered from the fve-lmt ase, markedly so when $1,000,000 was used. Copyrght 2009 Natonal Counl on Compensaton Insurane, In. All Rghts Reserved. 11

12 Fgure 7 Clusterng usng $100,000 Lmt Compared to Fve Seleted Lmts (The number of lasses that moved s shown above eah bar.) 16% Perent of Premum 12% 8% 4% 0% Up 1 HG Down 1 HG Up 2 HGs Movement of Class Codes Fgure 8 Clusterng usng $1,000,000 Lmt Compared to Fve Seleted Lmts (The number of lasses that moved s shown above eah bar.) 16% 58 Perent of Premum 12% 8% 4% 0% Up 1 HG Down 1 HG Up 2 HGs Movement of Class Codes Ths ndates that too muh nformaton s lost by droppng down to one lmt. Retrospetvely rated poles are purhased over a range of attahment ponts and no sngle lmt aptures the full varablty n exess ratos. We used prnpal omponents analyss to enhane the lusterng nvestgaton. The frst two prnpal omponents of the fve lmts retaned over 99% of the varaton n the Copyrght 2009 Natonal Counl on Compensaton Insurane, In. All Rghts Reserved. 12

13 data. Whle ths mght suggest that fewer lmts ould have been used, we deded to use fve lmts n order to over the range of lmts ommonly used n retrospetve ratng. The dstane between two lasses n prnpal omponents spae does not have the same smple nterpretaton as t does n exess rato spae. However prnpal omponents analyss allows one to projet a fve-dmensonal plot onto two dmensons. Clusterng usng the fve lmts and plottng the resultng hazard group assgnments usng the frst two prnpal omponents showed that the lusters were well separated and that outlers were easly dentfed. In our vew, ths onfrmed the suess of the fve-dmensonal lusterng. Metrs The objetve of assgnng lasses to hazard groups s to group lasses wth smlar vetors of exess ratos. Ths rases the queston of how to determne how smlar or lose two vetors are. The usual approah s to measure the dstane between the vetors. If x = (x 1, x 2,, x n ) and y = (y 1, y 2,, y n ) are two vetors n R n, then the usual Euldean, or L 2, dstane between x and y s spefed as x y 2 = n ( x y ) = 1 2. Ths metr s used extensvely n statsts and s what we used. In lnear regresson ths metr penalzes large devatons. That s, one bg devaton s seen to be worse than many small devatons. There are many other metrs. Perhaps the seond most ommon dstane funton s the L 1 metr whh spefes x y = n x y. 1 = 1 Here a large devaton n one omponent gets no more weght than many small devatons. The ntutve ratonale for usng ths metr s that t mnmzes the relatve error n estmatng exess premum. If R (L) s the hypothetally orret exess rato at a lmt of L for a lass and the premum on the poly s P then the exess premum s gven by P PLR R (L), where PLR denotes the permssble loss rato. But n prate the lass exess rato s approxmated by the hazard group exess rato R HG (L). The relatve error n estmatng the exess premum s then Copyrght 2009 Natonal Counl on Compensaton Insurane, In. All Rghts Reserved. 13

14 P PLR R HG ( L) P PLR R P ( L) = PLR R HG ( L) R ( L). If we assume that eah loss lmt s equally lkely to be hosen by the nsured, then the expeted relatve error n estmatng the exess premum s gven by n = 1 PLR n R HG PLR ( L ) R ( L ) = RHG R n 1, whh s proportonal to the L 1 dstane between the two exess rato vetors. Our analyss was not very senstve to whether the L 1 or L 2 metr was used and we preferred the more tradtonal L 2 metr. Standardzaton When lusterng varables measured n dfferent unts, standardzaton s typally appled to prevent a varable wth large values from exertng undue nfluene on the results. Standardzaton ensures that eah varable has a smlar mpat on the lusters. Duda and Hart [12] pont out that standardzaton s approprate when the spread of values n the data s due to normal random varaton, however t an be qute napproprate f the spread s due to the presene of sublasses. Thus, ths routne normalzaton may be less than helpful n the ases of greatest nterest. We onsdered two ommon approahes to standardzaton. The usual approah s to subtrat the mean and dvde by the standard devaton of eah varable. For example, f x 1, x2, K, x n are the sample values of some random varable, wth sample mean x, and sample standard devaton s, then the standardzed values are gven by x x z =. s An alternatve standardzaton method depends on the range of observatons. Under ths approah we would take x mn x z =. max x mn x We onduted two luster analyss trals n whh we standardzed aordng to the approahes desrbed above. In eah ase we lustered the lasses nto seven hazard groups. Both trals resulted n hazard groups that were not very dfferent from those produed wthout standardzaton. Further, two ssues were apparent wth regard to standardzng n our partular analyss. Frst, exess ratos at dfferent lmts have a smlar unt of measure, whh s dollars of exess loss per dollar of total loss. That s, exess ratos share a ommon Copyrght 2009 Natonal Counl on Compensaton Insurane, In. All Rghts Reserved. 14

15 denomnator. Any attempt to standardze would have resulted n new varables wthout a ommon unt nterpretaton. Seond, all exess ratos are between zero and one. Some standardzaton approahes would have resulted n standardzed observatons outsde ths range. Another onsderaton s that exess ratos have a greater range at lower lmts. Wthout standardzaton, the exess ratos at lower loss lmts have more nfluene on the lusters than do those at hgher lmts. Ths s not undesrable beause exess ratos at lower lmts are based more on observed loss experene than on ftted loss dstrbutons (see Corro and Engl [5]). Even on a natonal bass, there are few lams wth reported losses above $5,000,000, but there are many more lams above $100,000. Greater onfdene an be plaed n the relatve auray of exess ratos at lower lmts beause they are based on a greater volume of data. In summary, the determnaton was made not to standardze beause standardzaton would have elmnated the ommon denomnator and t would have led to nreased emphass on hgher lmts. Our lusterng algorthm used the L 2 metr and unstandardzed redblty-weghted lass exess ratos at the fve seleted loss lmts: $100,000, $250,000, $500,000, $1,000,000 and $5,000,000. Premum weghts were used to luster the lasses, as wll be dsussed n the next seton. Cluster Analyss Gven a set of n objets, the objetve of luster analyss s to group smlar objets. In our ase, we wanted to group lasses wth smlar vetors of exess ratos, where smlarty s determned by the L 2 metr. At ths stage the number of lusters s taken as gven. Typally parttons of the objets nto 1, 2, 3,, n lusters are onsdered. Nonherarhal luster analyss smply seeks the best partton for any gven number of lusters. In herarhal luster analyss the partton wth k + 1 lusters s related to the partton wth k lusters n that one of the k lusters s smply subdvded to get the k + 1 element partton. Thus f two objets are n dfferent lusters n the k luster partton then they wll be n dfferent lusters n all parttons wth more than k elements. Ths plaes a restrton on the lusters that an be sensble n some ontexts. Our approah was non-herarhal. Optmalty of k-means The lusterng tehnque we used s alled k-means. For a gven number, k, of lusters, k-means groups the lasses nto k hazard groups so as to mnmze where the entrod k = 1 HG R = R 1 HG R 2 HG 2, (3) R Copyrght 2009 Natonal Counl on Compensaton Insurane, In. All Rghts Reserved. 15

16 s the average exess rato vetor for the th hazard group and HG denotes the number of lasses n hazard group. Theoretally there s a dfferene between the hazard group exess rato vetor, R HG, omputed usng (1), and the hazard group entrod, R, but n prate ths dfferene s very small. Hazard groups determned by k-means have several desrable optmalty propertes. Frst, they maxmze the followng statst where k = 1 HG 1, (4) 2 R R R 1 R = C R 2 R 2 2 s the overall average exess rato vetor, wth C = HG beng the total number of lasses. Formula (4) s analogous to the R 2 statst n lnear regresson. It gves the perentage of the total varaton explaned by the hazard groups. A seond way to evaluate hazard groups s based on the tradtonal onepts of wthn and between varane. We would lke the hazard groups to be homogeneous and well separated. Thus we would lke to mnmze the wthn varane and maxmze the between varane. We show now that usng k-means aomplshes both. Instead of onsderng a sngle exess rato for eah lass, we have a vetor of exess ratos. Thus we have not a sngle random varable orrespondng to a sngle loss lmt, but rather a random vetor, wth one random varable for eah loss lmt, from whh we get a varane-ovarane matrx. If X s the random varable for the exess rato funton at the th loss lmt, L, aross lasses, then the observed values are the R (L ). The varane-ovarane matrx of the random vetor X = (X 1, X 2,, X n ) s gven by where σ 11 σ 21 Σ = M σ n1 σ k σ σ σ 12 M 22 n2 L L O L σ 1n σ 2n, M σ nn [ X µ )( X )] = E µ ( k k s the ovarane of X and X k and = E[ ] µ. If we regard X as a 1 x n matrx then X Copyrght 2009 Natonal Counl on Compensaton Insurane, In. All Rghts Reserved. 16

17 T Σ = E[( X µ ) ( X µ )], where µ = µ, µ, K, µ ) and (X µ) T s the transpose of (X µ). ( 1 2 n In prate the varane-ovarane matrx s not known, but must be estmated from the data,.e., the vetors R = (R (L 1 ), R (L 2 ),, R (L n )). Let 1 x j = R ( L j ), C where C s the total number of lasses, and let x = x, x, K, x ). ( 1 2 n Then the sample ovarane of the ELFs at L and L k s s k 1 = k ) C ( R ( L ) x )( R ( L x ) k, and the sample varane-ovarane matrx s gven by S s11 s12 L s1n s s s n = = L 2 M M O M C sn1 sn2 L snn T ( R x) ( R x). One way to generalze the onept of varane to the multvarate ontext s to onsder the trae of S trae ( S ) = s + s + L+. 11 Ths s just the sum of the sample varanes of eah varable and s alled the total sample varane. We let 22 s nn T ( R x) ( R x) T = CS =. Copyrght 2009 Natonal Counl on Compensaton Insurane, In. All Rghts Reserved. 17

18 The matrx T s proportonal to the varane-ovarane matrx for the whole data set. It s alled the dsperson matrx, and s the matrx of sums of squares and ross produts. We an proeed smlarly wthn eah hazard group and defne If we let W B = HG T ( R x ) ( R x ) T = HG ( x x) ( x x),. then t an be shown (see Späth [7]) that T ( R x) ( R x) = B + W HG. We then let W = k W. = 1 Ths s the pooled wthn group dsperson matrx. For the between varane we let B = k B = 1 Ths s the weghted between group dsperson matrx. We then have. T = B + W. Ths means, roughly that the total varane s the sum of the between varane and the wthn varane. Takng the trae we get trae(t) = trae(b) + trae(w). Thus the total sample varane s the sum of the between and wthn sample varane. Beause trae(t) s onstant, maxmzng trae(b) s equvalent to mnmzng trae(w), whh s what k-means luster analyss aomplshes. Weghted k-means As observed n seton 2, some lasses are muh larger than others. To avod lettng the small lasses have an undue nfluene on the analyss, we weghted eah lass by ts premum. In smplest terms, ths amounts to ountng a lass twe f t has twe as muh premum as the smallest lass. So nstead of mnmzng the expresson n (3), we nstead mnmzed Copyrght 2009 Natonal Counl on Compensaton Insurane, In. All Rghts Reserved. 18

19 k = 1 HG w R R 2 2, where w s the perentage of the total premum n lass. We used the premumweghted entrods as well, that s Optmal Number of Hazard Groups w HG R R =. w HG So far, we have dsussed the task of determnng lusters when the number of lusters s gven. We now address how to tell whether one number of lusters performs better than another, e.g., whether seven lusters works better than sx or eght. Varous test statsts an be used to help determne the optmal number of lusters. The proedure s to ompute the test statst for eah number of lusters under onsderaton and then dentfy the number of lusters at whh the hosen statst reahes an optmal value (ether a mnmum or a maxmum, dependng on the partular test statst beng used). Mllgan and Cooper ([8], [9]) tested suh proedures to determne whh statsts were the most relable. Mllgan and Cooper [8] performed a smulaton to test 30 proedures. The smulated lusters were well separated from eah other and they dd not overlap. For eah smulated data set, the true number of lusters was known, and they omputed the number of lusters ndated by eah method of determnng the optmal number of lusters. The methods were ranked aordng to the number of tmes that they suessfully ndated the orret number of lusters. They noted that ther smulaton was dealzed but that It s hard to beleve that a method that fals on the present data would perform better on less defned strutures ([8], page 161). Hene, although the hazard group data had both nose and overlap, t was useful to refer to Mllgan and Cooper [8] to determne whh methods to rule out. In a later study, Cooper and Mllgan [9] onduted tests that were more relevant to our applaton beause random errors were added to the smulated data. That study found that the two best performng methods n the error-free senaro were also the best wth errors ([9], page 319). The best performng method s due to Calnsk and Harabasz. Mllgan and Cooper ([8], page 163) defne the Calnsk and Harabasz statst as trae trae ( B) / ( k 1) ( W )/ ( n k) Copyrght 2009 Natonal Counl on Compensaton Insurane, In. All Rghts Reserved. 19

20 where n s the number of lasses and k s the number of hazard groups, B s the between luster sum of squares and ross produt matrx, and W s the wthn luster sum of squares and ross produt matrx. A hgher value of ths statst ndates better lusters beause that orresponds to hgher between lusters dstanes (the numerator) and lower wthn luster dstanes (the denomnator). Ths test s also known as the Pseudo-F test due to ts resemblane to the F-test of regresson analyss, often used to determne whether the explanatory varables as a group are statstally sgnfant. Another test that ranked hgh n the Mllgan and Cooper testng was the Cub Clusterng Crteron (CCC). Ths test ompares the amount of varane explaned by a gven set of lusters to that expeted when lusters are formed at random based on data sampled from the mult-dmensonal unform dstrbuton. If the amount of varane explaned by the lusters s sgnfantly hgher than expeted then a hgh value of the CCC statst wll result, ndatng a hgh-performng set of lusters. An optmum number of lusters s dentfed when the test statst reahes a maxmum (Mllgan and Cooper [8], page 164). Mllgan and Cooper [8] found that the Calnsk and Harabasz test produed the orret number of lusters for 390 data sets out of 432. The CCC test produed the orret value 321 tmes. We ould not use some of the other methods that ranked hgh beause they were only applable to herarhal lusterng, or for other reasons. In a SAS Insttute tehnal report, Sarle [11] noted that the CCC s less relable when the data s elongated (.e., varables are hghly orrelated). Exess ratos are orrelated aross lmts, so we gave the CCC results less weght than the Calnsk and Harabasz results. We performed luster analyses for four to nne hazard groups. There were four hazard groups n the pror NCCI system, and we saw no reason to onsder any smaller number. Implementng ten or more hazard groups would be substantally more dffult than mplementng nne or fewer, beause havng 10 or more requres an addtonal dgt fro odng hazard groups. Testng up to nne was approprate beause the Workers Compensaton Insurane Ratng Bureau of Calforna uses nne hazard groups [4]. In the frst phase of our luster analyss, we assgned lasses and alulated the two test statsts for eah number of groups under onsderaton. Fgure 9 shows that the Calnsk and Harabasz statst ndated that the best number of hazard groups was seven. Fgure 10 shows that the CCC statst suggested nne hazard groups. Copyrght 2009 Natonal Counl on Compensaton Insurane, In. All Rghts Reserved. 20

21 Fgure 9 Indated Number of Hazard Groups Calnsk and Harabasz 3500 CH Statst Number of Hazard Groups Fgure 10 Indated Number of Hazard Groups Cub Clusterng Crteron 125 CCC Statst Number of Hazard Groups But nne hazard groups produed rossover, meanng that the hazard group exess rato for a hgher hazard group was lower than that for a lower hazard group at some hgh loss lmt. In our opnon, ths suggested that more lusters were beng used than ould aurately be dstngushed. Copyrght 2009 Natonal Counl on Compensaton Insurane, In. All Rghts Reserved. 21

22 As an be seen n Table 2, most of the premum s onentrated n the largest lasses wth the hghest redblty. We were onerned that the ndated number of hazard groups n the analyss above ould have been dstorted by the presene of hundreds of non-redble lasses. In the seond phase of our luster analyss, we appled the tests to determne the optmal number of lusters usng large lasses only. In one senaro, we appled the Calnsk and Harabasz and CCC tests usng only those lasses wth redblty greater than or equal to 50 perent. In a seond senaro, we appled the tests usng only fully redble lasses. As shown n Fgure 11, the ndated number of hazard groups was seven for both tests n both senaros. Fgure 11 Statsts for Varous Numbers of Hazard Groups Only Classes wth at Least 50 Perent Credblty 65 CH Statst CCC Statst Number of Hazard Groups Number of Hazard Groups At Least 50% Credblty Full Credblty In summary, we used two test statsts n three senaros for a total of sx tests. Seven hazard groups was the ndated optmal number n fve of these sx tests. The exepton was the senaro n whh all lasses were nluded, where the CCC test ndated that nne hazard groups were optmal. But there are four reasons why ths exepton reeved lttle emphass: Mllgan and Cooper ([8], [9]) found that the Calnsk and Harabasz proedure outperformed the CCC proedure. The CCC proedure deserves less weght when orrelaton s present, whh was the ase n all of our senaros. The seleton of the optmal number of lusters ought to be drven by the large lasses where most of the experene s onentrated. The large lasses have the hghest redblty and so the most onfdene an be plaed n ther exess ratos. There s rossover n the nne hazard groups, and we don t thnk the data supports hazard groups wth rossover. Copyrght 2009 Natonal Counl on Compensaton Insurane, In. All Rghts Reserved. 22

23 We onluded that seven hazard groups were optmal. These are denoted A to G, wth Hazard Group A havng the smallest ELFs and Hazard Group G havng the largest. Alternate Mappng to Four Hazard Groups We reognzed that some nsurers would not be able to adopt the seven hazard group system mmedately beause they needed addtonal tme to make the neessary systems hanges. Therefore we produed a four hazard group alternatve to supplement the seven hazard group system. We hose to ollapse the seven hazard groups nto four by ombnng Hazard Groups A and B to form Hazard Group 1, ombnng C and D to form 2, ombnng E and F to form 3, and lettng Hazard Group 4 be the same as G. Havng an alternate mappng to four hazard groups smplfes omparsons between the pror and new mappngs as well. Pror to hoosng ths smple sheme we onsdered other alternatves. We tred usng k-means luster analyss to map the seven hazard group entrods nto four. Ths approah resulted n a hazard group premum dstrbuton that was not homogeneous enough. Another approah we onsdered was usng luster analyss to group the lasses dretly nto four hazard groups. That approah yelded reasonable results, but t resulted n a non-herarhal ollapsng sheme,.e., the seven hazard groups were not a result of subdvdng the four hazard groups. The herarhal ollapsng sheme we hose has ths feature, whh allows users to know whh of the four hazard groups a lass s n based on knowng that lass assgnment n the seven hazard group system. The new four hazard group system s ntended to be temporary. The four hazard group system s n plae only to ensure that all arrers have suffent tme to make the transton to seven hazard groups. 4. COMPARISON OF NEW MAPPING WITH OLD Dstrbuton of Classes and Premum The bulk of the exposure was onentrated n two of the hazard groups pror to our revew. Hazard Groups I and IV ontaned a small perentage of the total premum. Hazard Groups II and III, on the other hand, ontaned 97 perent of the total premum (see Table 1). We knew that a more homogeneous dstrbuton of premum by hazard group would mprove prng auray. When dsussng the new hazard groups n ths seton we wll fous on the mappng that resulted dretly from the statstal analyss. Later on, as wll be dsussed n the underwrtng revew subseton, numerous lasses were reassgned among the groups based on feedbak gathered n our survey of underwrtng experts. These hanges are not refleted n Fgures 12 to 20. Fgures 12 and 13 ompare the pror mappng to the ollapsed new mappng based on the dstrbuton of lasses and premum. Copyrght 2009 Natonal Counl on Compensaton Insurane, In. All Rghts Reserved. 23

24 Fgure 12 Pror Mappng vs. Collapsed New Mappng Number of Classes per Hazard Group Pror Mappng HG IV 86 HG I 38 New Mappng HG 4 88 HG HG III 318 HG II 428 HG HG Fgure 13 Pror Mappng vs. Collapsed New Mappng Perent of Premum by Hazard Group Pror Mappng HG IV 2.5% HG I 0.9% New Mappng HG 4 4.8% HG % HG III 51.1% HG II 45.6% HG % HG % Hazard Group 1 has a large number of lasses and a substantal porton of total premum n ontrast to Hazard Group I. Hazard Groups 2 and 3 have beome slghtly smaller than before although they are stll large. In the pror mappng Hazard Groups II and III eah had over 45 perent of the premum, but n the new mappng, none of the four groups has as muh as 40 perent. Ths refnement allows for mproved homogenety of lasses wthn eah hazard group. Hazard Group 4 has retaned a smlar number of lasses but t has more premum than Group IV. Fgure 14 shows that most of the lasses and premum remaned n the same hazard group when assgned to the new four Hazard Groups. Copyrght 2009 Natonal Counl on Compensaton Insurane, In. All Rghts Reserved. 24

25 Perent of Premum 70% 60% 50% 40% 30% 20% 10% 0% Fgure 14 Comparson of Old wth New Assgnment to Four Hazard Groups (The number of lasses that moved s shown above eah bar.) No Movement Up 1 HG Down 1 HG Down 2 HGs Movement 3 Among those lasses that dd move, the great majorty (300 lasses and 37 perent of the premum) moved down one hazard group. Most of ths movement was from Hazard Group II to 1. The movements of lasses and premum are detaled n Table 5. Copyrght 2009 Natonal Counl on Compensaton Insurane, In. All Rghts Reserved. 25

26 Table 5 Comparson of Dstrbutons of Classes Between Pror and New Hazard Group Assgnments Pror Mappng Hazard Group I II III IV Total Number of Classes % Premum 0.9% 45.6% 51.1% 2.5% 100% New Mappng Hazard Group % 25.4% 0.5% 0.0% 26.7% % 19.6% 11.8% 0.0% 31.4% % 0.6% 36.3% 0.2% 37.1% % 0.0% 2.6% 2.2% 4.8% The table an be read vertally. For nstane, among the 428 lasses n Hazard Group II, 255 were mapped nto Hazard Group 1, 164 nto Hazard Group 2, nne nto Hazard Group 3, and none nto Hazard Group 4. The 255 lasses that moved from Hazard Group II nto Hazard Group 1 omprsed 25.4% of the total premum. A sgnfant number of lasses and amount of premum moved from Hazard Group III to 2. Three lasses moved from III to 1. Just 15 lasses moved up by one hazard group, makng up three perent of the premum. Hazard Group 1 s so large prmarly beause of lasses that entered t from Hazard Group II. Hazard Group 2 s qute dfferent than Hazard Group II beause many of the lasses n 2 orgnated n III and many of the lasses that were n II have moved nto 1. The new seven hazard group assgnment has a farly homogenous dstrbuton of lasses and premum, as shown n Fgure 15. Copyrght 2009 Natonal Counl on Compensaton Insurane, In. All Rghts Reserved. 26

27 Fgure 15 Number of Classes and Perent of Premum n Eah Hazard Group Number of Classes per HG Perent of Premum per HG % 9.33% % 18.44% 10.14% 17.37% 21.25% HG A HG B HG C HG D HG E HG F HG G Ths s a marked mprovement over the pror mappng. In terms of premum, Hazard Group A s 11 tmes larger than Hazard Group I was. Hazard Group G s twe as large as Hazard Group IV was. Table 6 shows the dstrbuton of lasses to hazard groups based on ther level of redblty. Overall there were 162 lasses wth at least 75 perent redblty and 708 lasses wth lower redblty. Generally, wthn eah hazard group most of the premum s due to hghly redble lasses but most of the lasses have lower redblty. Hazard Groups D and G are exeptons. Hazard Group D has nearly equal numbers of hgh and low-redblty lasses. In Hazard Group G, hgh and low-redblty lasses have smlar premum perentages. Copyrght 2009 Natonal Counl on Compensaton Insurane, In. All Rghts Reserved. 27

28 Table 6 Number of Classes wth Gven Credblty by Hazard Group 162 Classes wth 708 Classes wth Credblty 75% Credblty < 75% Number of Number of Hazard Group % Premum % Premum Classes Classes A % % B % % C % % D % % E % % F % % G 4 2.4% % Total % % Although Hazard Groups B and E have far more lasses than the other hazard groups, they do not have far more premum. The reason that they have the most lasses wth redblty less than 75 perent s that the omplement of redblty s the pror hazard group exess rato. For nstane, the exess rato of Hazard Group III at $100,000 was whh s lose to the exess rato of Hazard Group E. Gven a small lass n Hazard Group III, the redblty-weghted exess rato was lkely to be lose to the exess rato of Hazard Group E. Range of Exess Ratos In Fgure 16 eah horzontal bar represents the range of redblty-weghted exess ratos wthn a partular hazard group. The vertal lne wthn eah bar represents the overall exess rato for the hazard group. Copyrght 2009 Natonal Counl on Compensaton Insurane, In. All Rghts Reserved. 28

29 Fgure 16 Pror Mappng Exess Rato Ranges at $100K IV Hazard Group III II I Credblty-Weghted Class Exess Ratos at $100K Among the lasses n Hazard Group I, the exess ratos at $100,000 ranged from to In Hazard Group II, the exess ratos at $100,000 ranged from to Thus the range of Hazard Group I exess ratos was ontaned wthn that of Hazard Group II, ndatng that Hazard Groups I and II were not as well-separated as mght be desred. The same behavor was observed at $1,000,000 as well. As shown n Fgure 17, k-means lusterng resulted n well-separated hazard groups. Beause fve dmensons were used, we ould not avod overlap n eah dmenson, but the exess rato dstrbuton s a noteable mprovement over the pror mappng. Copyrght 2009 Natonal Counl on Compensaton Insurane, In. All Rghts Reserved. 29

30 Fgure 17 New Mappng Exess Rato Ranges at $100K G F Hazard Group E D C B A Credblty-Weghted Class Exess Ratos at $100K The new mappng also shows a well-separated exess rato dstrbuton at $1,000,000 as shown n the Fgure 18. Fgure 18 New Mappng Exess Rato Ranges at $1M G F Hazard Group E D C B A Credblty-Weghted Class Exess Ratos at $1M Most of the exposure s onentrated n the largest lasses, and so the hazard group exess ratos are hghly senstve to the plaement of large lasses. In Fgures 16-18, the range of exess ratos for eah hazard group s alulated usng all of the lasses n that hazard group. Copyrght 2009 Natonal Counl on Compensaton Insurane, In. All Rghts Reserved. 30

31 Fgures 19 and 20 show that f ranges are omputed usng only those lasses wth at least 75 perent redblty, then the separaton of hazard groups by exess ratos s qute strong at both $100,000 and $1,000,000. G F Fgure 19 New Mappng Exess Rato Ranges at $100K Classes wth at least 75% Credblty Hazard Group E D C B A Credblty-Weghted Class Exess Ratos at $100K Fgure 20 New Mappng Exess Rato Ranges at $1M Classes wth at least 75% Credblty G F Hazard Group E D C B A Credblty-Weghted Class Exess Ratos at $1M Copyrght 2009 Natonal Counl on Compensaton Insurane, In. All Rghts Reserved. 31

32 5. UNDERWRITING REVIEW After ompletng the luster analyss, we onduted a survey of underwrters to solt ther omments on the proposed new mappng. The survey was sent to all members of NCCI s Underwrtng Advsory Lst (UAL), and nluded the draft mappng that resulted from the analyt determnaton of the hazard groups. The survey asked the underwrters to judge the hazardousness of eah lass based on the lkelhood that a gven lam would be a serous lam. We also ponted out that f the mx of operatons n two lasses was very smlar then the two lasses should probably be n the same hazard group. Members of the UAL reommended hanges n the hazard group assgnment for a thrd of the lasses. We also reeved feedbak from two underwrters on staff at NCCI. After the survey omments were ompled, a team onsstng of NCCI atuares and underwrters revewed the omments from UAL members and deded on the fnal assgnment for eah lass. When dedng whether to reassgn a lass, we onsdered whether the feedbak on that lass was onsstent. We onsdered the redblty of eah lass and plaed more weght on the luster analyss results for those lasses wth a large volume of loss experene. For eah lass we ompared the exess ratos to the overall hazard group exess ratos and dentfed the nearest two hazard groups. Class 0030 llustrates the proess used at NCCI to dede on the hazard group for eah lass. Ths s the lass for employees n the sugar ane plantaton ndustry and s only applable n a small number of states. Ths lass had 12% redblty, was n Hazard Group III under the pror mappng, and was assgned to Hazard Group E under the luster analyss. An underwrter ponted out that Class 0030 has operatons smlar to Class 2021, whh s for employees who work at sugar ane refnng. Class 2021 apples natonally, had 31% redblty, was n Hazard Group II under the pror mappng, was assgned to Hazard Group C under the luster analyss, and pror to redblty weghtng had exess ratos lose to the overall exess ratos for Hazard Group D. Credblty weghtng had redued Class 2021 s exess ratos so that they were between the overall exess ratos of Hazard Groups C and D, beause the pror assgnment of Class 2021 had been to Hazard Group II. We onluded that Hazard Group D was the best hoe for 2021 based on ts exess ratos pror to redblty weghtng and ts mx of operatons. We determned that 0030 should be assgned to the same hazard group as 2021, so we also assgned Class 0030 to Hazard Group D. Underwrters made several other types of omment besdes those omparng one lass to another. For nstane, they ommented on the degree to whh employees n a gven Copyrght 2009 Natonal Counl on Compensaton Insurane, In. All Rghts Reserved. 32

33 lass are prone to rsk from automoble adents. They ommented on the extent to whh heavy mahnery s used n varous oupatons and how muh exposure there s to dangerous substanes. Fgure 21 dsplays the movements of premum and lasses durng the underwrtng revew under the ollapsed new mappng. It shows that the overall effet of the underwrtng revew was to move a sgnfant number of lasses up to a hgher hazard group. Fgure 21 Perent of Premum That Moved Durng the Underwrtng Revew (The number of lasses that moved s shown above eah bar.) Perent of Premum 100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% 732 No Movement Up 1 HG Down 1 HG Up 2 HGs Up > 2 HGs Movement The majorty of the lasses that moved up one hazard group, 78 of them, moved from Hazard Group 1 to 2, whle 20 lasses moved from Hazard Group 2 to 3, and 23 lasses moved from Hazard Group 3 to CONCLUSION Our approah to remappng the hazard groups was founded on three key deas: 1. Computng exess ratos by lass The data s too sparse to dretly estmate exess ratos by both lass and state. But ountrywde exess ratos an be omputed by lass n the same way that hazard group exess ratos are omputed. Ths does not requre separate loss dstrbutons for eah lass. The exstng loss dstrbutons by njury type an be used along wth the usual sale assumpton. Thus all that s needed s average Copyrght 2009 Natonal Counl on Compensaton Insurane, In. All Rghts Reserved. 33

34 osts per ase by njury type and njury type weghts for eah lass. 2. Sortng lasses based on exess ratos Rather than usng ndret varables to apture the amorphous onept of exess loss potental, we used exess ratos dretly beause hazard groups are ndeed used to separate lasses based on exess ratos. Beause a loss dstrbuton s n fat haraterzed by ts exess loss funton, ths approah nvolves no loss of nformaton. By sortng lasses based on exess ratos we aheve the goal of sortng lasses based on ther loss dstrbutons as well. 3. Cluster Analyss Problems nvolvng sortng objets nto groups are not unque to atuaral sene. We were thus able to make use of a large statstal lterature on luster analyss. Ths provded an objetve rteron for determnng the hazard groups as well as the optmal number of hazard groups. Our approah to determnng the seven hazard groups was non-herarhal beause we wanted the best seven group partton and beause hypothetal parttons nto sx hazard groups are not relevant n ths ontext. As a result of our analyss the number of NCCI hazard groups was nreased from four to seven. The dstrbuton of both premum and lasses s muh more even n the new hazard groups. The hghest hazard group s stll relatvely small. The new seven hazard groups ollapse naturally and herarhally nto four hazard groups. Comparng the new four hazard groups wth the old, over two-thrds of the lasses, wth nearly 60% of the premum, dd not move at all. Ths stablty was largely a result of the fat that we used the old hazard group as a omplement of redblty and there were a large number of lasses wth very lttle premum. Of the lasses that dd move, the overwhelmng majorty moved down one hazard group. The new mappng was fled n md-2006 to be effetve wth the frst rate or loss ost flng n eah state on or after January 1, The flng (Item Flng B-1403) was approved pror to the end of 2006 n all states n whh NCCI fles rates or loss osts. ACKNOWLEDGMENTS Many staff at NCCI ontrbuted to ths paper, nludng Greg Engl, Ron Wlkns, and Dan Corro. We also thank the NCCI Retrospetve Ratng Workng Group and the NCCI Underwrtng Advsory Lst for ther nput. REFERENCES [1] Mahler, Howard C., Workers Compensaton Exess Ratos: An Alternatve Method of Estmaton, PCAS LXXXV, 1998, pp , proeed/proeed98/ pdf. [2] Johnson, Rhard A. and Dean W. Whern, Appled Multvarate Statstal Analyss, Prente Hall, Copyrght 2009 Natonal Counl on Compensaton Insurane, In. All Rghts Reserved. 34