The Credibility of the Overall Rate Indication: Making the Theory Work

Transcription

1 The Creblty of the Overall Rate Incaton: Makng the Theory Work by Joseph Boor ABSTRACT Actuares have use the so-calle square root rule for the creblty for many years, even though the F value can take any value, an ts assumpton that the ata recevng the complement of creblty s stable s often volate. Best estmate creblty requres fewer or no assumptons, but often requres certan key constants. Ths paper proves a varety of methos for estmatng the key constants neee to mplement best estmate creblty formulas, especally those arsng from the Gerber-Jones formula. As such, ths paper proves the tools neee to mplement key theoretcal formulas n practcal actuaral work. KEYWORDS Creblty, best estmate creblty, overall rate ncaton, Brownan moton, process varance, parameter varance VOLUME 9/ISSUE CASUALTY ACTUARIAL SOCIETY 67

2 Varance Avancng the Scence of Rsk. Introucton It s well establshe that the lmte fluctuaton or square root creblty has lmtatons. Snce t s esgne to prouce stable estmates, not best estmates, t oes not prove the most accurate rates. Further, snce any concevable combnaton of the fluctuaton that may be acceptable an probablty of a chance volaton of the accepte fluctuaton s a pror no fferent than any other, t s challengng to show that any partcular full creblty stanar s better than any other. Lastly, the square root rule reles on an assumpton that the statstc recevng the complement of creblty s stable. When the complement of creblty s, say, three years of 5% tren, that assumpton s clearly volate. So there s a strong nee for best-estmate creblty. Some tme ago (967) Hans Bühlmann evelope a formula 3 for the best estmate creblty of a sngle rsk or a sngle class when the complement of creblty s assgne to the large group that the rsk or class s part of. Hs P/(P + K) formula 4 s well known an represents a truly optmal (n the sense of makng the best prectons) creblty formula. But a formula s also neee for the creblty of the overall rate change for a prouct or lne of busness. It s qute common n actuaral work to evelop a rate ncaton for such a group, realze that supplemental ata s neee, an creblty weght the overall ncate change wth somethng such as the nflatonary tren snce the last rate change. 5 Conserng that the overall rate change affects every rate for every class an every rsk, ths author beleves that the creblty of the overall rate A logcal resoluton of ths woul be to nvoke best estmate prncples. Whle the approach of ths paper s that of best estmate creblty, t s mportant to note that ths approach nvolves much more analyss than lmte fluctuaton creblty. So, there may certan cases where cost lmtatons support the use of lmte fluctuaton creblty. If the complement of creblty nvolves small rate changes, lmte fluctuaton creblty may serve the goal of rate stablty. If the best estmate creblty oes not serve that goal effectvely, there may be some crcumstances where lmte fluctuaton creblty s preferable. 3 See Bühlmann (967). 4 To avo confuson between the varable names use n varous formulas, ths wll generally be reference wth U use n leu of P wthn ths artcle. The reaer s avse to be aware of the alternate notaton. 5 See Boor (996). ncaton eserves as much attenton as the creblty of the class ata wthn t. A sol theoretcal backgroun has been la for the creblty of ths overall rate ncaton. Creblty s by nature a process that s esgne to upate an estmate of loss costs. A paper by Jones an Gerber (975) proves formulas for the weghts n upatng formulas (to be scusse later) n terms of the covarances of the hstorcal ata ponts. 6 Ths formula, n fact, proves the optmum lnear estmate of future costs gven all the pror ata, not just the ata use n the current rate upate. Nevertheless, knowng the mathematcal form of the creblty s not the same thng as beng able to compute the creblty. As wll be shown, stanar creblty formulas erve from the Gerber-Jones approach use values for the Brownan moton varance n year-to-year tren, plus values for the observaton error varances between observe ata ponts an the true expecte costs that unerle them. 7 To compute the creblty, t s necessary to estmate those varance parameters. Ths paper proves technques esgne to o just that.. The theory Key creblty formulas for the overall rate ncaton In ths secton the key theoretcal results from the Jones an Gerber (975) paper are presente. Ths shoul prove the practtoner a summary of the key formulas that create best-estmate creblty. Lkely none of the materal s new... The general Gerber-Jones formulas The goal s to apply the Gerber-Jones formulas to a realstc moel (ultmately, geometrc Brownan moton for tren, an observaton error wth a constant coeffcent of varaton) of the relatonshp between hstorcal ata an the unknown future loss 6 Other relevant papers nclue those by Mahler (998) an Leolter, Klugman, an Lee (99). 7 In ths context, both varances are ntene to have meanng n a broa sense rather than the mathematcally narrow efnton of varance. 68 CASUALTY ACTUARIAL SOCIETY VOLUME 9/ISSUE

3 The Creblty of the Overall Rate Incaton: Makng the Theory Work cost. So, to facltate the reaer s unerstanng, the key Gerber-Jones formulas are shown below. The frst statement that must be mae s that the Gerber-Jones formula, an, unless state otherwse, all other formulas, assume that any necessary tren an current level ajustments have alreay been mae to the ata. For example, although the pror ata use n a creblty formula nvolves trenng an current level ajustments, those ajustments are assume to have been one 8 n the backgroun, so all that s nvolve s etermnng the optmum creblty weghts for the prevous years. Wth that backgroun, a creblty formula 9 an ata pattern s of the upatng type 0 through the n + st projecton (e.g., the optmum estmate of future loss costs P n+ s a creblty weghte average P n+ Z n S n + ( Z n )P n of the prevous estmate of loss costs P n an the new ata S n ) f there s a constant m an sequences V, V,..., V n an W, W,..., W n such that ES [ ] m Cov[ S, Sj] V+ W for each of the S, (.) for each case where j,an (.) W f < j. (.3) Further, when the creblty formula an ata pattern are of that upatng type, then the optmum crebltes are W W + Z V Z W W + Z V + V (.4) an Z W W + V. (.5).. The lnear upatng-type formulas As a frst step towars unerstanng the notaton, t s helpful to ntrouce the creblty uner a stanar lnear Brownan moton wth a rft (T), varance parameter for the Brownan moton, an a constant error varance s between each trene ata pont S S* + (n + )T an the trene unerlyng expecte cost at pero, or L L* + (n + )T. Logcally, the actual evatons from the expecte loss (S L E per ths lnear moel) coul be expecte to be nepenent from both each other an the L s. Of note, ths treatment s not new, but s presente so that the reaer may unerstan the process. Then, f we take m to be the true mean expecte loss at tme 3 n +, m L n+ E[P n+ ] then the unerlyng pror expecte loss follows a Brownan moton. Further, snce Cov[A + ab, C + bb] abvar[b] when A, B, an C are mutually nepenent, [, j] [ +, + ( j ) + j] Cov S S Cov L E L L L E Var[ L ] ( < j) (.6) (notng that L j s further along n the Brownan moton than L, the ranom moton between L an L j s nepenent of L ). Further, Cov[ S, S] + s (.7) 8 A scusson of the ratonale for separatng the tren estmaton from the varance parameter estmaton wll be presente later. 9 Ths presentaton of the Gerber-Jones formulas s slghtly weakene to smplfy the presentaton for a more general auence. The ntereste reaer s encourage to revew the orgnal artcle for the broaer result. 0 Any creblty process on any set of ata ponts can be esgne so that the rate ncaton s the optmum combnaton of the new ata pont an the current rate. Creblty formulas of the upatng type note that the current rate s merely a pror creblty-weghte combnaton (n practce, usually wth trenng) of the pror ata ponts. For a creblty formula to be of the upatng type, the creblty-weghte combnaton of the new pont an the current rate s also the optmum combnaton of all the pror ata ponts f they were consere nvually. Least square error, conserng all possble lnear combnatons of the ajuste pror year ata. So, n the Gerber-Jones formula V s ;an (.8) W. (.9) Here m s not use n the same sense as n the Jones an Gerber (975) paper, where t s L 0. 3 The resultng creblty s entcal f some L n+dt s esre nstea of L n+, as one may affrm by pluggng n the alternate covarance structure. W n W n s unaffecte. VOLUME 9/ISSUE CASUALTY ACTUARIAL SOCIETY 69

4 Varance Avancng the Scence of Rsk Hence, per formula (.4), + Z s Z + Z s + s, (.0) where each Z s the optmum creblty to use when combnng the new ata (S ) wth the pror estmate (P ) to prouce the optmum estmate P + of L +. Further, the resultng combnaton of all the pror ata ponts S j<+ that P + represents s the optmum estmate of L + gven the avalable ata. Jones an Gerber (975) also show that the successve Z s converge to a lmt (whch coul concevably be use as a proxy for the creblty Z when s large). In ths scenaro, settng Z Z n formula (.0) an solvng for Z gves Z s + 4. (.) s.3. The geometrc Brownan moton formulas The lnear moel has a key weakness t assumes that the growth n losses s lnear. In fact, t s wellestablshe that most nsurance lnes of busness suffer nflaton that causes loss costs to grow exponentally rather than lnearly. That realty requres an ajustment to the Brownan moton moel. Instea of havng E[L L ] 0 for each, we shoul expect zero growth (E[L /L ] ). Instea of expectng the L L s to have entcal an nepenent normal strbutons, one woul expect the L /L s to have nepenent entcal lognormal strbutons, wth the aforementone mean of unty an some common varance of. So f one begns wth unajuste ata ponts, each enote as S*, the ponts use to estmate L n+ E[P n+ ] are the nflate values S* ( + T ) n+ S s. Lastly, a moel for the fferences between the observe S s an the true expecte costs, the L s, must be nclue. In ths moel, the ratos S /L are assume to have nepenent, entcal lognormal strbutons wth a mean of unty an a constant varance of s. These strbutons are also expecte to be nepenent from those of the year-to-year rfts (L /L s). The common observaton varance of the trene values assumpton s consstent wth roughly equal numbers of clams from year to year wth severty nflaton affectng the loss szes. It woul be less proper for an ncreasng book of busness that encompasses more an more expecte clams from year to year wth consequent reuctons n the coeffcent of varaton of the process varance. In any event, the covarance structure, usng the entty 4 Cov[AB, CB] E[A] E[C] Var[B], s 5 [, j] [, ( j ) j] E[ Lj] E[ Lj Lj],..., E[ L L ] Cov S S Cov L E L L L E [ ] [ ]( ) E E Var L > j,..., Var[ L ] ( ) ( j) + >, (.) Further, by the entty Var[AB] Var[A]Var[B] + E[A] Var[B] + E[B] Var[A], ( ) ( ) Cov[ S, S ] s (.3) So, the key values for the Gerber-Jones formula n ths case are V W ( ) + ; (.4) ( ) s + ;an, (.5) + Z s Z + s + Z s + s. (.6) A comparson to equaton (.0) shows that ths s entcal to the formula for the lnear case, except for the atonal s term n the enomnator. 4 Ths hols when A, B, an C are nepenent. 5 The last step n ths calculaton, computng the fnal varance, nvolves propertes of the lognormal strbuton. Detals of the mathematcs are not presente, as there s an opportunty for confuson between the s use n ths paper to enote observaton varance an the sgma parameter use n specfyng lognormal strbutons. 70 CASUALTY ACTUARIAL SOCIETY VOLUME 9/ISSUE

5 The Creblty of the Overall Rate Incaton: Makng the Theory Work But, one shoul conser that when at least one of the values an s s very small, the combnaton term s shoul be a small part of the enomnator. Thus, one mght say that, for the case of geometrc Brownan moton, + Z s Z + Z s + s. (.7) Further, the steay-state creblty may be approxmate as Z s + 4 s (.8) As a relevant se note, the summans nvolve n equatons (.3) an (.4) woul nflate unformly as the losses are projecte ahea more than one year, to some n + t nstea of to tme n +, an the creblty equaton woul reman unchange Mult-year formulas an best estmate creblty for the overall rate ncaton The approach outlne earler nvolves upatng a rate wth a sngle new year of ata. But t s very common to see rate ncatons that upate a rate wth, say, the weghte average of the ata from the last fve years. The role of ths mult-year ata n a best estmate creblty formula merts scusson. 3.. Reasons not to reuse oler years Upatng formulas that use multple years reuse ata from pror estmates. So, the reuse of ata shoul be evaluate. The frst pont to be mae s that usng multple years s perfectly approprate when lmte fluctuaton creblty s nvolve. Lmte fluctuaton creblty eals solely wth the extent to whch the boy of ata recevng creblty can be rele on to not create unwarrante ncreases or ecreases 6 For reference, ths s also true for the lnear case. of some specfe sze. It oes not purport to create a best estmate of the future costs. It has been state, though, by the well-respecte Howar Mahler n 986 that ths metho often prouces future loss estmates that are comparable to those of best estmate creblty. To state t smply, re-usng pror years n a Gerber- Jones formula unuly complcates the computatons. For example, assume an estmate has been contnually upate over 4 years from P an S to P 5 wth rollng fve-year averages 7 Q,..., Q 4 of the ata ponts S,..., S 4. Logcally, the step s to prouce the estmate P 6 usng Q 5. Note, though, that the covarance between Q 5 an Q 4 s farly hgh, snce they have the ponts S, S, S 3, an S 4 n common. However, Q 5 an Q have no common components. 8 Generally, 9 Cov[Q 5, Q 4 ] Cov[Q 5, Q ]. Therefore, the Gerber-Jones formula cannot be use when multple years are combne. 0 Therefore, the practce of combnng multple years of ata n ths context s suboptmal. That concluson has a very relevant corollary. If the exposures most useful for lmte fluctuaton creblty stem from fve or even ten years, but best estmate creblty s only base on the most recent year, the resultng crebltes shoul by nature be fferent. Therefore, there are crcumstances where lmte fluctuaton creblty s not a goo substtute for best estmate creblty. 3.. Correctng the pror estmate for changes n ultmate loss estmates There s, however, one respect n whch the use of multple years coul mprove the estmate. The exstng rate s base on the ata avalable earler, when the varous years losses were less mature than they are at the tme of the upate rate ncaton. 7 Or an average of the avalable years, where fewer than fve are avalable. 8 Lkely they o have some nrect common components stemmng from the epenence of the backwars rft per a Brownan moton of some type as per sectons. an.3. 9 A more etale analyss of ths may be foun n Appenx B. 0 It may, alternately, work when rates are mae bannually an two years of upatng ata are combne. VOLUME 9/ISSUE CASUALTY ACTUARIAL SOCIETY 7

6 Varance Avancng the Scence of Rsk So, t makes sense to upate the exstng rate for the atonal evelopment before usng t n the creblty formula. Of course, the exstng rate s a multple creblty weghte average of many years. Further, t s not just an average of many years of loss ratos or pure premums, t s rather ether an average of trene loss ratos brought to the current rate level or trene pure premums. So, some calculatons must be one to nclue ths atonal loss evelopment n the pror rate that s use as the complement of creblty. Due to the requrement to use current level ata, the correcton process for loss rato ratemakng s slghtly more complex than that of pure premum ratemakng. Therefore, Table shows how the calculatons neee to upate a loss rato at present rates for loss evelopment mght flow. The references to Pror an Last Pror refer to the ata use n computng the loss rato estmate that was use n the last rate change. The Frst Assgne values refer to what was use the frst tme the specfc year of ata was use. Also, note that although the loss ratos of many years are lkely embee n the pror loss rato, only the last fve were revse. That s because more mature years see fewer year-to-year revsons n ultmate losses, an contrbute a mnshng porton after creblty (see column 7). Table. Sample upate of pror rate revew loss rato nformaton for ultmate loss changes Year () (Data) Loss Rato at Charge () (Data) Loss Rato at Charge (3) () () Absolute Loss Rato Change (4) (Data) Last Pror Current Level Factor % 74% % % 90% 8% % 40% 8% % 70% 5% % 5% %.00 Year (5) (Data) Creblty Frst Assgne (6).0 (5) Complement of Creblty (7) (6) [Next (7)] Complement of Creblty (8) (5) [Next (7)] Creblty n Last Pror % 55% 9% 7% 007 3% 68% 6% 8% % 6% 4% 5% % 65% 39% % 00 40% 60% 60% 40% Year (9) (Data) Tren Rate Frst Assgne (0) [.0+(5)] [Next (0)] Total Tren Factor n Last Pror () (3) (7) (0)/(4) Change to Pror Estmate () (Selecte) Change to be Reflecte 006 6% % 0.08% 007 7% % 0.66% 008 8%.95.7%.7% 009 9%.99.09%.09% 00 0% % 4.80% A. Total Change to Pror B. Pror Loss Rato for Ratemakng C. Last Rate Change Taken D. Tren Factor for ths Flng E. (B.+A.)8D./(.0+C.) New Pror Value to whch Complement of Creblty s Apple 5.36% 65.7% 5.00%. 7.6% 7 CASUALTY ACTUARIAL SOCIETY VOLUME 9/ISSUE

7 The Creblty of the Overall Rate Incaton: Makng the Theory Work It s also worth mentonng that n ths example the current level factors coul be upate for the next rate revew by smply multplyng column (4) by unty plus tem C. Smlar ajustments coul be mae for the Creblty n Last Pror an Total Tren Factor n Last Pror columns. Of course, ths example mrrors the calculatons n the theoretcal lterature the ata s assume to be collecte at mnght of December 3, 0, then use to make rates that are effectve at :0 a.m. of January, 0. However, the correctons neee to reflect practcal realtes woul appear to be straghtforwar Upate ultmate losses an upatng-type creblty It coul be expecte that the process of upatng pror year ultmate losses coul stort the optmum creblty. In lnes such as excess casualty rensurance, the ultmate loss estmates S n, S n, etc., for the most recent years coul have a very hgh observaton error, an those fve or so years back coul be much closer estmates of the true expecte loss L s wthn ther respectve years. So, on that bass the true optmum creblty coul be expecte to be larger for some of the oler years than the most recent year. However, that woul clearly not create an upate. Some perspectve can be prove about ths stuaton. Frst, when pror year estmates are not correcte, the formulas of secton o prove the optmum creblty. Further, upatng the pror year ultmate losses can only be expecte to mprove the accuracy of the resultng loss precton. So, ths approach can be expecte to prouce a hgh qualty estmate of future costs, up to any storton ue to lengthy loss evelopment. If loss evelopment uncertanty s expecte to sgnfcantly stort the creblty, t may well be preferable to smply start from scratch each year wth the ultmate loss estmates for, say, the last twenty years. One may then compute estmates of the process varance n each year, estmates of the loss evelopment error varance n each year, an the Brownan moton- type varance parameter. It s not ffcult to see that, uner the lnear moel (possbly the geometrc as well), an upatng formula can be erve for the assgnment of weghts to the varous years. It shoul be clear that the resultng creblty weghts may ffer greatly between years. However, t oes not nvolve the sort of upatng of the pror rate that s part of the typcal actuaral applcaton. Rather t nvolves smply computng a rate from scratch. Snce the focus of ths paper s on upatng an exstng rate wth new ata, ths stuaton wll not be analyze further n ths paper. 4. Estmatng the parameters: Z, K, B, an The secton wll gve the reaer some tools for creatng estmates of the key varances, an thus help create better loss cost projectons. It s not ntene to be a survey on the subject. Rather t s ntene to gve the practtoner the tools neee to mplement best estmate ratemakng. The ntereste reaer may revew some of the eas n De Vlyer 98 an Hayne 985, to get two other perspectves on ths subject. Frst, a few quck notes are n orer: Note. In many stuatons, t s not necessary to estmate both an s. Key formulas can be converte to a functon of K /s, so K s all one nees to estmate. Note. When estmatng an s for geometrc Brownan moton, note that they are functons of an s from the logarthmc transform to a lnear Brownan moton, exp( ), an exp(s ) s. So, once one etermnes how to estmate the constants of varance (or even just ther rato) n a lnear Brownan moton, one may estmate the creblty for the geometrc Brownan moton. Note 3. The observaton errors (wth varance s ) consst logcally of a combnaton of the sample varance (.e., the lmtatons of the law of large A further escrpton s beyon the scope of ths paper, but once the concept of an upatng formula s abanone, t may be preferable to use a moel such as ntegrate Brownan moton to better mrror realty. See Appenx B for a eeper scusson of the ntal creblty Z. VOLUME 9/ISSUE CASUALTY ACTUARIAL SOCIETY 73

8 Varance Avancng the Scence of Rsk numbers ue to the hgh skew n nsurance statstcs an nablty of small clam samples to fully estmate the true expecte losses each year) an the loss evelopment uncertanty between the early ata we base our projectons on an the fnal actual clams costs n each year. Further, the sample varance an evelopment varance are nepenent an so may be ae to etermne s. Note 4. (Subtracton of Two Estmate Quanttes) If we subtract one hghly uncertan large number from another large number, an the fference s small, the result has a large varance most of the tme. When estmatng a small number, that large varance typcally overwhelms the true small value one seeks to estmate. Note 5. (Common Atve Error n all the Data) If all the hstorcal ata ponts are affecte equally an smultaneously by a common error that s nepenent of all the other error terms (for example, all the ata s base by aton of a sngle, unform, unknown, amount e from some strbuton wth a zero mean), then the optmal soluton may be estmate by sregarng ths error. Logcally, ths may be converte algebracally to a stuaton where one s estmatng a future value that contans e, wth e remove from all the hstorcal ata. Snce the varance of e s nepenent of all aspects of varance n the hstorcal ata, the e component of the costs beng precte s not susceptble to estmaton usng the hstorcal ata. Hence, t may be sregare n optmzng the estmate of future costs. A smlar result hols when e s a constant error multpler wth a mean of one wthn the ata, except that one must conser that the mean of the nverse of e may not be unty. Wth those concerns n mn, a few methos for estmatng the key parameters follow. 4.. Metho : The creblty that woul have worke n the past. Ths approach actually nvolves no estmaton of or s ; rather, t estmates Z rectly. Snce estmatng Z rectly removes the barrers to mplementng best estmate creblty for the overall rate ncaton, t merts scusson (even though t oes not nvolve an s ). The basc methoology nvolves assumng some creblty value Z, then usng all the ata but the last year to estmate the last year gven. Assume that one has, say, ten years of on-level, approprately trene 3 loss ratos. Then, one coul note that the ffth year s value coul be estmatng by frst applyng some unknown creblty factor Z to the fourth 4 year s ata, Z( Z) to the thr year s ata, Z( Z) to the secon year s ata, etc., then vng by the sum of the crebltes, ( Z) 4, to correct for the off-balance. In effect, a sngle creblty value s assume to have been proper for all four upates. Once that equaton s establshe, one coul vary Z n orer to fn whch Z mnmzes the square fference between the ffth year s ata an the creblty-weghte average. Most moern spreasheet programs contan soluton-generatng capabltes that make t straghtforwar to fn such a soluton. Then, one may also construct smlar equatons to solve for a common creblty of Z that use the frst fve values to prect the sxth, the frst sx values to prect the seventh, etc. The last step nvolves replacng the nvual solutons of Z that each mnmze the square error of a sngle prectve step wth a soluton of a sngle Z that mnmzes the sum of all the square errors of all the prectve steps smultaneously. The resultng Z s arguably the best estmator of the creblty n the ata, at least as long as a sngle creblty s approprate for all the years. Table llustrates how ths process woul work wth ten years of essentally ranom sample ata. The shae boxes show the nputs an outputs to the soluton process (note that the Target box pulls 3 Ths may nvolve, snce the tme pero s so long, usng fferent trens for fferent pror peros. 4 Ths seemngly contemplates the same stroke of mnght ratemakng ssue scusse earler. However, note that snce the ranom varaton between say, the ffth year an the sxth year affects all the hstorcal year estmate errors entcally; note 5 ncates that the Z best sute to estmate the sxth year from the frst four years s the same Z that s best sute to estmate the ffth year from the frst four years. 74 CASUALTY ACTUARIAL SOCIETY VOLUME 9/ISSUE

9 The Creblty of the Overall Rate Incaton: Makng the Theory Work Table. Sample calculaton of Z from ntal reporte ata an fnal cost of ten years of ata when ata has zero tren Part. Data an Estmaton of Oler Years Accent Year () Data Intal Data Values () Data Fnal Ultmate Value Input/Output for Soluton Functon Value to mnmze Value to vary to mnmze Target s (3) Z(( Z) k) All Estmatng Weghts (4) [5 Later (3)] Weghts for Estmatng 995 Target (5) () (4) 995 Estmate Z (6) [4 Later (3)] Weghts for Estmatng 996 (7) () (5) 996 Estmate A. Column Sums B. (A./[A. n Prev. col.] Loss Rato Est C. (from ()) Actual Loss Rato Values D. (B-C.) Square Error n Estmate Part. Estmaton of Remanng Years an Total Precton Error (Target) Accent Year (8) [3 Later(3)] () 997 Estmate (9) [ Later (3)] () 998 Weghts (0) [Next Row(3)] () 999 Estmate () (3) () 000 Estmate A. (as above) B. (as above) C. (as above) Sum of Est. Errors Target D. (as above) VOLUME 9/ISSUE CASUALTY ACTUARIAL SOCIETY 75

10 Varance Avancng the Scence of Rsk up the Target value compute at the bottom of the spreasheet). Ths metho has goo utlty as long as an s are stable over tme an the ata s not prone to very rare large losses. 5 It s reasonable to expect to be stable as long as the average tren factor s stable, but often that oes not occur. Further, t woul be reasonable to expect s to be farly stable as long as the premum volume n the lne, ajuste for tren, s stable. What must be sa. Ths approach has nothng to o wth the formulas state earler. However, t oes aress the key queston n ths paper, etermnng the optmum creblty. Further, snce Z has a formula n an s, t may also use to etermne a secon varance constant once a frst varance constant s known. Then, one mght possbly revse the estmate of s (erve from Z an ) to better account for process varance ue to large losses, an consequentally revse the estmate of Z. 4.. Metho : Fttng K an B across a large number of smlar atasets In ths case, one mght assume that the ratemaker s computng rates for a sngle lne of busness n 50 U.S. states, or some other stuaton where there s a farly large number of segments, an all the segments have approxmately the same tren an observatonerror-varance-per-unt-of-exposure characterstcs. One woul also have to assume that the complement of creblty s stll suppose to be assgne to the exstng rate plus tren, not some amalgam of all the segments. One must also assume that the ol premum/exposure an loss ata usng n prcng the last, say, twelve years of rates are avalable for each of the segments. An lastly, t woul help f the secon-to-last ata pont for each segment, possbly the last ata pont, s evelope enough that each value L n+,s, for each class (s) s as close an estmate of the expecte costs E n+,s as s reasonably possble. 5 On the other han, f one converts to basc lmts ratemakng wth (necessarly) relable ncrease lmts factors, then that problem may be sgnfcantly mtgate. Just lke the estmaton of Z n the prevous subsecton, K an B may be estmate from the ata by solvng for the values that woul prouce the best estmates of the most recent costs n the varous segments. In the prevous subsecton the total square fferences between the creblty-weghte average of varous sets of years an the future years they project were mnmze. In ths case, for each segment s, one must construct the creblty-weghte average P n,s of the last n ( 0, or 5, or whatever s most feasble) years of ata (the S,s s) n orer to estmate each L n+,s. In ong so, the crebltes shoul be compute usng formula (A.7) Z s, ( ) ( )( ) Us, + Z, s K + BUs, U + + Z K + BU s,, s s,. (4.) Per the soluton routne, K an B shoul then be mofe so that the square errors the resultng P n+,s s make n estmatng the L n+,s s are mnmze. Crucally, K an B are not to vary from segment to segment. Rather, a sngle par of K an B that mnmze the sum of all the square precton errors s to be foun va the soluton algorthm. So the weght assgne to the year n ata for the lne s ata, S n,s, s ( )( ) ( ) M Z Z... Z Z. n, s n, s n, s n +, s n, s (4.) The resultng prectons 6 of the L n+,s s are then the varous values of n ( ) P M S + Z S (4.3) n+, s s, s, s, (where each S 0.s represents the rate or ratng nformaton n effect just before the experence pero). As before, the sum across all the s s of the square estmatng errors Σ s (P n,s L n,s ), or perhaps a premum 6 The frst creblty formula (.5) from Jones an Gerber (975) of Z W U was use for the frst step. For convenence,, s, s W + V U + K + BU s,, s the remanng creblty after conserng all fve years was assgne to the frst year of ata. In practce, that woul be assgne to the rate n effect when the pero began, ajuste for tren an to the present rate level. n 0. s 76 CASUALTY ACTUARIAL SOCIETY VOLUME 9/ISSUE

11 The Creblty of the Overall Rate Incaton: Makng the Theory Work or exposure weghte average Σ s W n,s (P n,s L n,s ) coul be compute n the spreasheet. The resultng value coul be calle the Target an the soluton routne or feature coul be use to vary K an B untl the lowest value of the Target s foun. A sample spreasheet llustratng ths approach wth ata segments an common tren, process, an parameter varance constants, but fferent samples from those constants among the segments, s shown n Table 3. The expecte loss ratos for each segment were smulate usng a geometrc Brownan Table 3. Sample calculaton of K an B from ata for twelve separate segments subject to a common K an B (alternate annotaton style for matrx ata) Part : Dstrbuton Propertes Tren Varance ( ) Values to Vary n Solver True Values Process Varance (t to be ve by premum) K (solver) 30.0 (K t / ) Parameter Varance.0009 (l ) B.473 (solver).565 (B l / ) Part : Premums {U year,class U,s (Data)} Year Class Class Class 3 Class 4 Class 5 Class 6 Class 7 Class 8 Class 9 Class 0 Class Class Target Part 3: Loss Ratos {L,s (Data); n ths example, generate usng orgnal means (.6,.65,.55 for each group of three) an unty mean/lognormal rft t an observaton error wth rft varance, observaton varance + l } Us, 65.9% 65.% 48.% 60.% 6.3% 55.0% 60.% 66.% 53.9% 6.8% 67.% 50.% 6.3% 6.0% 58.3% 57.5% 66.0% 56.% 57.7% 67.9% 54.3% 63.6% 66.7% 5.0% % 64.8% 53.8% 59.% 6.8% 59.6% 55.% 65.8% 5.3% 6.5% 63.6% 49.4% % 74.3% 54.6% 56.3% 64.8% 55.3% 54.9% 6.0% 55.0% 6.3% 66.% 50.7% % 80.8% 58.3% 50.3% 6.7% 5.7% 5.7% 59.0% 55.9% 60.% 68.9% 45.3% Target % 73.6% 6.3% 53.% 67.% 5.0% 54.% 65.8% 58.8% 65.4% 63.5% 43.6% Part 4: Crebltes Z s, [ Us, + Z, s( K + BUs, )] ;except Z [ U + ( + Z )]( K + BU ) s,, s s,, s U, s U + K + BU, s, s Part 5: Weghts Compute Usng Crebltes {W 5,s Z 5,s ; W,s Z,s ( Z +,s )... ( Z 5,s ); W pror,s 00% (Z,s + Z,s + Z 3,s + Z 4,s + Z 5,s )} Pror (contnue on next page) VOLUME 9/ISSUE CASUALTY ACTUARIAL SOCIETY 77

12 Varance Avancng the Scence of Rsk Table 3. Sample calculaton of K an B from ata for twelve separate segments subject to a common K an B (alternate annotaton style for matrx ata) (contnue) Part 6: Projectons {P s [Sum of W,s L,s ) for all years ] + W pror,s L,s }; an Square Estmaton Errors per Year 6 Observe Data {R s (P s Target L 6,s ) } Projecton to year 6 {P s } 65.5% 75.% 56.3% 53.8% 6.8% 54.4% 53.6% 6.% 55.% 60.8% 67.4% 47.6% Square Error vs. Target 6 {R s } Straght Sum of Square Errors {R + + R } 7.6E-04.4E-04.5E-03 3.E-05.9E-03.E-03.E-05.E-03.4E-03.E-03.5E-03.6E-03 Weghte (wth U 5,s for each R s ) Sum of Square Errors Soluton routne vare K an B n prevous gray area to mnmze ths value Part 7: Valaton wth Actual Year 6 Unerlyng Expecte Loss Ratos {E s (Data)} Expecte Loss Rato at Year 6.54E-0.46E % 73.8% 57.0% 49.6% 66.4% 50.5% 53.7% 66.4% 57.4% 6.5% 64.9% 43.8% Sum of Errors Projectng Expecte Loss wth K, B n Gray {S sum of (P s E s ) } Sum of Error w/true K,B {T; same as S only true K, B use throughout process}.06e-0.07e-0 Rato Error w/est. K, B to True K, B {S/T} 00% motons wth the varance specfe n Part. The actual loss ratos are also affecte by the parameter varance an the process varance (a common factor, ve by the premum per the Law of Large Numbers) lste there. The actual values of K an B are on the very left of Part. Lastly, the K an B values that mnmze the sum of premum-weghte sum of square errors n projectng the sxth year s smulate value (usng the creblty weghts 7 efne by K, B, an the premum ata) are hghlghte n gray. Note that the loss ratos for year were eeme to have projecton errors smlar to the rate pror to the experence pero, so they were use for the S 0.s s. What must be sa. In testng ths metho, t appears that t may requre a substantal number of ata ponts to relably estmate of K an B usng ths process. In partcular, twelve classes o not appear to be suffcent for the test case above. However, the fact that K an B are combne as K + BU n the equaton means that they act together to mpact the creblty. The only fference s that the B term reacts to exposure 7 For example, the most recent year has creblty Z n, the former year has ( Z n )Z n, an then ( Z n )( Z n )Z n, etc., as creblty has been apple at successve upates. or premum volume, whereas K oes not. In ths case, at a premum of about 0 the estmate value of K + BU s about equal to the true unerlyng value. Next, the actual qualty of the estmaton, the errors n estmatng the true (unaffecte by process or parameter varance) expecte loss ratos for year 6 (as shown at the top of Part 7) were compute. As one may see, the fference between the precton error usng the estmate K an B an the actual K an B s neglgble. Ths suggests that, as long as the sample sze (number of s values) s small an the fference n premums, exposures, etc., s small, t may be more helpful to smply replace K + BU wth K n the creblty formula Metho 3: Estmatng an from the hstorcal ata Ths metho nvolves usng fferent lnear combnatons of square fferences between values. As such, t s orente towars stanar, lnear, Brownan moton. However, note that the logs of values from a geometrc Brownan moton form a lnear Brownan moton. So, one may convert geometrc Brownan moton ata to lnear ata, estmate the values of an s that work n the lnear context, then convert those to comparable rft varance an process/ 78 CASUALTY ACTUARIAL SOCIETY VOLUME 9/ISSUE

13 The Creblty of the Overall Rate Incaton: Makng the Theory Work parameter varance values. For example, the geometrc Brownan moton varance parameter woul be e when s the varance n the corresponng lnear Brownan moton an the mean of the geometrc Brownan moton steps s specfe to be unty (no change n the multplcatve context). So, the goal s to fn functons of the S s that prove nsght nto the values of an s. For example, the square fference between the begnnng an enng values (S n S ) reflects two samples of parameter/process error at the two enponts an n samples from the Brownan moton varance. So, f the two types of varance are smlarly sze, the square fference between the two enponts shoul be omnate by a multple of the Brownan moton varance. Smlarly, f one as the square fferences between ajacent ponts Σ n (S + S ) one woul expect the result to be omnate by a multple of the process 8 varance s. Further, one mght expect that more precse approxmatons mght be mae by usng lnear combnatons of those two values. So, one mght begn by computng the expecte values of (S n S ) an Σ n (S + S ). Frst, note that, snce the mean expecte change n values from the Brownan moton (after tren correcton) s zero, an the expecte process rsk s zero. ( ) [ ] E Sn S Var Sn S (4.4) However, S n S may be expresse as a sum of nepenent varables, each wth mean zero, as (S n L n ) + (L n L ) + (L S ). So, t s compose of a process error, a Brownan moton of length n, an the negatve of a process error. Therefore, ( ) E Sn S [ n n] [ n ] [ ] Var S L + Var L L + Var L S ( n ) ( n ) s + + s + s. (4.5) 8 To avo the cumbersome phrase parameter/process varance, the smpler phrase process varance shoul be unerstoo to have the same meanng throughout ths subsecton. Smlarly, n E S S ( + ) n Var[ L+ L] Var[ S L] + [ n n] [ ] + Var S L + Var S L ( n ) ( n ) + s + s + s ( n ) ( n ) n + s. (4.6) Knowng those values, t s possble to construct estmators for an s. One may realy see that, by the lnearty of expectatons, ( + ) ( n ) ( n ) E S S S S an n ( ) ( ) ( n ) (( ) ) n + n s n + s n ( )( n ) ( + ) ( n )( n ) E n S S S S { } s, (4.7) ( n ) ( n ) + ( n ) + ( n ) ( n )( n ) s s. (4.8) So, by creatvely usng the fferences between the frst an last pont, an the fferences between ajacent ponts, one may estmate the values of an s. An example of the use of equatons (4.7) an (4.8) s shown n Table 4. The actual observable ata over 5 years n column was generate ranomly over 5 years, usng the actual values 3% an s 7%. The values of an s were then estmate from the ata. As one may see, the estmates are farly close. But they nonetheless sgnfcantly overestmate the creblty. VOLUME 9/ISSUE CASUALTY ACTUARIAL SOCIETY 79

14 Varance Avancng the Scence of Rsk Table 4. Sample estmaton of an from hstorcal ata Part : Data Brownan S.D. 3% process S.D. 7% Imple K Part : Data an Analyss () Year () Brownan Expecte Loss (3) Process Error (4) ()+(3) Inclung Process Error (Observe Data) (5) (4) Prevous (4) Annual Change (6) (5) (5) Square Annual Changes Total 0.5 A. C. Estmate of process varance: [A. B.]/[(5 )] Assocate stanar evaton D. Estmate of varance parameter for Brownan moton: [(n )B. A.]/[(5 )(5 )] Assocate stanar evaton (7) (4)(en) (4)(begn) Total Change from Begnnng to En (8) (7) (7) Square Total Change B % % E. Value of K C./D..93 F. Estmate Steay-State Creblty (equaton (.) formula usng C. an D.) 48% G. True Steay-State Creblty (equaton (.) formula usng values at top) 35% A note about tren The theory unerlyng ths paper assumes that the expecte loss, a pror, s the same for all years. That generally requres that hstorcal losses have been trene (an premums ajuste to the current rate an exposure level) before the calculatons commence. Of course, f the tren s compute usng the same ata as the calculatons, the calculate value of may be suppresse. For example, f the ranom movement began wth a large upwar jump early n the pero, an another jump later, because the value of s hgh, the analyss of tren may ncorrectly nfer that t s hgh tren rather than a hgh Brownan moton varance. Of course, f the tren s clearly much larger than, t may well be less of an ssue. Further, as note later, the problem of estmatng an s s relatvely ll-contone. 9 So reucng 9 For example, a bg outler coul have arsen from ether process varance or rft varance. Thus t woul be ffcult to nfer whch type of varance s large from the observe ata ponts. 80 CASUALTY ACTUARIAL SOCIETY VOLUME 9/ISSUE

15 The Creblty of the Overall Rate Incaton: Makng the Theory Work the egrees of freeom of the approxmaton by estmatng tren smultaneously, gven a small number of ata ponts, may not be relable. However, one mght be avse to use some relate ata, such as calenar year reporte loss frequency an calenar year close clam severty, to estmate the tren. On the other han, f there are a large number of ata ponts relatve 30 to the volatlty n the ata, then the mpact of the ranom observaton error n the ntal an enng ponts on the tren estmate shoul be mnmal. A thr aspect of tren eserves menton as well. Wthout a correcton, the ranom lognormal aspect of geometrc Brownan woul prouce a mean above one at all ponts after t begns. In effect, the ranomness of the strbuton combne wth the skew of the lognormal tens to generate ts own tren. So, the transforme (nto a lnear verson) verson of the ata ponts, rather than havng a normal-type 3 strbuton wth mean zero, must have a lognormal strbuton wth mean. That means that external tren must often be correcte, especally tren compute by averagng several year-to-year growth rates. To complcate matters, s then unknown, so the value neee for the correcton s unknown. However, some crue ntal estmate of the value of may be use when estmatng tren, an then, once the tren s estmate, the estmate may be refne, etc. The process may be contnue teratvely untl a consstent tren an are compute. Conser that f the estmate s prouce by loglnear regresson of ata wth smlar geometrc Brownan moton varance, shoul alreay be subsume nto the tren. Further, f qualty surrogate ata s avalable for trenng, that opton eserves serous conseraton. What must be sa. There are some specal conseratons that shoul help explan why the approxma- 30 The author s not aware of any specfc measure that woul realy efne ths, so t woul lkely nee to be assesse jugmentally. 3 The author recognzes that use of the normal strbuton s an mplct assumpton, but Boor (0) shows t s generally a reasonable approxmaton n the nsurance lne of busness context. tons are not more precse. Frst, t may be ffcult to stngush say, whether a very hgh last pont s ue to a very hgh uptck n the Brownan moton because s large, or a large process error because s s hgh. So, the basc problem of approxmatng an s may often be ll-contone. Secon, t s mportant to revew Note 4 at the begnnng of ths secton. At ts core, Note 4 says that the error varance n computng the quanttes above coul be as much as the sum of the varances of the two tems you are subtractng. Whle the error oes not qute reach the sum of the varances (ue to nter-correlaton of the two quanttes), one shoul stll be extremely cautous f the fference (the estmate of or s ) s much smaller than each of the values nvolve n the subtracton. Nevertheless, even though the creblty etermne usng ths metho sometmes only has moerate precson, t s moerately close to the best estmate creblty. Therefore, t stll has the potental to create more accurate estmates than the stablty-centere classcal creblty Metho 4: Estmatng structurally from loss ata an by subtracton Gven the formulas n equatons (4.5) an (4.6), t s clear that, once one of an s s relably estmate, the other may be estmate. It shoul also be clear that equaton (4.5) has relatvely more content n than equaton (4.6). So, f one has a qualty estmate of s, the formula ( S S ) n s n (4.9) may be use to estmate. Some estmate of s s requre to use that formula, though. One metho for estmatng s nvolves what may be escrbe as a structural analyss. Such a process nvolves ecomposng the process/parameter rsk nto ts components an then estmatng each component separately. The process rsk s some ways better represente n hstorcal creblty formulas (such as the P/(P + K), or U/(U + K) n the notaton of ths paper), so t wll be VOLUME 9/ISSUE CASUALTY ACTUARIAL SOCIETY 8

16 Varance Avancng the Scence of Rsk analyze frst. Thankfully, as long as there are enough clams n the ata to relably estmate the upper en of the severty strbuton, one may use the collectve rsk equaton to calculate the process varance (whch may be labele a ). Then, a E[ # clams] Var[ severty] + Var[ # clams] E[ severty], (4.0) or n the loss rato or pure premum context, a [# ] [ ] [# ] [ ] ( premumor exposures) E clams Var severty + Var clams E severty. (4.) So, as long as the proper ata s avalable, 3 the process varance s realy estmable. The other porton that must be estmate s the parameter varance, whch wll smlarly be enote b. Note that any year-to-year varatons n the tren are subsume nto. So, n most cases the only parameter-type varance that nee be consere s the uncertanty n loss evelopment to ultmate. That varance has two parts: uncertanty about what the correct expecte loss evelopment factor s; an varance of the ultmate loss n each year, as estmate usng loss evelopment, aroun the actual ultmate loss. It s not har to see that the uncertanty about the expecte loss evelopment factor can be essentally gnore per Note 5 at the begnnng of ths secton. The varance n future loss emergence 33 on the varous years requres some analyss, though. Estmatng the remanng ranom b, gven appropr- 3 The formulas are beyon the scope of ths paper, but correctons for ncomplete large loss samples an correctons to prove measures of the fferng severtes of evelope losses may be one when they are neee. 33 Ths approach assumes the key volatlty n loss emerge les n what has emerge to ate, rather than that the future evelopment may be heavly ranom ue to fortutous late, larger losses. If the latter s the key ssue, t woul be more accurate to vew the process, gven losses of maturty m, as creblty-rven process of estmatng the future reporte losses at maturty m, then apply the loss evelopment factor to the result. In such a case, the clam counts an severty strbuton use n equatons (4.0) an (4.) shoul just use the loss ata through m months. ate volume n the trangle, can be one usng some farly well establshe proceures. For example, a paper by Hayne (985) etals one approach. The result of ths approach woul be a multplcatve strbuton wth a mean of unty an a varance of some b. Of course, t s then necessary to combne a an b. Frst, a shoul be converte to a multplcatve strbuton to use wth the multplcatve loss evelopment strbuton. Such a strbuton woul represent expecte loss +processerror the rato, whch has a expecte loss a mean of one an varance. The multplcatve combnaton of these two clearly nepenent ( expecte loss ) strbutons gves Varanceofprocess parameter varancengeometrc Brownan motonspace a b + + a b ( expecte loss) ( expecte loss). (4.) So, when that s converte to a parameter n the lnear moel 34, one may show that a a b + ( expecte loss) log. (4.3) a b + + ( expecte loss) Then, that estmate may be combne wth equaton (4.9) to obtan an estmate of Metho 5: Estmatng usng a larger ataset an by subtracton Just as s may be estmate usng alternate approaches, may often be estmate n solaton as well. If a larger proxy ataset (for example, the 34 Conser the value of the varance of a lognormal strbuton of mean one, compare to ts normal mean, strbuton varance parameter. 8 CASUALTY ACTUARIAL SOCIETY VOLUME 9/ISSUE

17 The Creblty of the Overall Rate Incaton: Makng the Theory Work countrywe prvate passenger auto experence of a major carrer when rates are beng mae a low volume state) s avalable, an that ataset has very mnmal process/parameter rsk, then the formula (4.8) from subsecton 4.3 shoul prouce a very hgh qualty estmate of. Then, usng equaton (4.6), s may be estmate va n ( S + S ) ( n ) s 4.6. All or many of the above (4.4) Several methos were presente above. They all have fferent strengths an weaknesses. Whenever possble, t may be helpful to revew the results of more than one metho. Note that that the creblty formula s not a formula n s an per se, t s actually a formula n ether the rato K or n K an B. So, a when fferent values for s an result from fferent approaches, but the rato K s smlar, the methos funamentally agree. Also, note that what may look lke large changes n K may have a very mnor effect on the creblty when K s very large. Lastly, shoul the methos sagree t creates an opportunty to evaluate the strengths an weaknesses of each one. Summary The square root or classcal creblty process has been n use for many years. Nevertheless, that metho has sgnfcant a flaw n that the statstcal assumptons (confence level an falure threshol) may be chosen arbtrarly. Further, t assumes that whatever ata receves the complement of creblty s stable an relable, even when that ata s, say, four years of a 0% tren rate. It s hope that ths avancement, by provng a relable creblty process that uses mnmal assumptons, wll restructure the creblty processes use by casualty actuares. Then, the professon can be comfortable that rate ncatons that use the resultng creblty values are as accurate as possble. References Boor, J. A., Creblty Base on Accuracy, Proceengs of the Casualty Actuaral Socety 79, 99, pp Boor, J. A., The Complement of Creblty, Proceengs of the Casualty Actuaral Socety 83, 996, pp. 40. Boor, J. A., An Analytc Approach to Estmatng the Requre Surplus, Benchmark Proft, an Optmum Rensurance Retenton for an Insurance Enterprse Usng Moments of the Severty Dstrbuton an Key Frequency Dstrbuton Values, Electronc Theses, Treatses an Dssertatons, Paper 476 (0). Bühlmann, H., Experence Ratng an Creblty, ASTIN Bulletn 4, 967, pp De Vlyer, F., Practcal Creblty Theory wth Emphass on Optmal Parameter Estmaton, ASTIN Bulletn, 98, pp Jones, D. A. an H. U. Gerber, Creblty Formulas of the Upatng Type, Transactons of the Socety of Actuares 7, 975, pp Hayne, R. M., An Estmate of Statstcal Varaton n Development Factor Methos, Proceengs of the Casualty Actuaral Socety 7, 985, pp Leolter, J., S. Klugman, an C.-S. Lee, Creblty Moels wth Tme-Varyng Tren Components, ASTIN Bulletn, 99, pp Mahler, H. C., An Actuaral Note on Creblty Parameters, Proceengs of the Casualty Actuaral Socety 73, 986, pp. 6. Mahler, H. C., Creblty wth Shftng Rsk Parameters, Rsk Heterogenety, an Parameter Uncertanty, Proceengs of the Casualty Actuaral Socety 85, 998, pp Appenx A B, K an Upatng Creblty Uner Bühlmann-Type Assumptons an Mofe Bühlmann-Type Assumptons The assumptons use above are farly broa. s a farly general tren volatlty. More mportant, all the errors that come between each raw ata pont S an the true mean for that year L are assume to have (log) normal strbutons wth mean 0 (or for the lognormal case) an varance s. So, each s woul logcally contan some process varance, PrV, an some parameter-type varance ue to ssues such as uncertanty of how the remanng losses wll evelop g. Bühlmann (967) analyze the class creblty, notng that the process varance ecrease as the VOLUME 9/ISSUE CASUALTY ACTUARIAL SOCIETY 83