EDUCATION COMMITTEE OF THE SOCIETY OF ACTUARIES ADVANCED TOPICS IN GENERAL INSURANCE STUDY NOTE CREDIBILITY WITH SHIFTING RISK PARAMETERS

EDUCATION COMMITTEE OF THE SOCIETY OF ACTUARIES ADVANCED TOPICS IN GENERAL INSURANCE STUDY NOTE CREDIBILITY WITH SHIFTING RISK PARAMETERS Suar Klugma, FSA, CERA, PhD Copyrgh 04 Socey of Acuares The Educao Commee provdes sudy oes o persos preparg for he examaos of he Socey of Acuares. They are eded o acqua caddaes wh some of he heorecal ad praccal cosderaos volved he varous subjecs. Whle varyg opos are preseed where approprae, lms o he legh of he maeral ad oher cosderaos somemes preve he cluso of all possble opos. These sudy oes do o, however, represe ay offcal opo, erpreaos or edorseme of he Socey of Acuares or s Educao Commee. The Socey s graeful o he auhors for her corbuos preparg he sudy oes. GIADV-0-4 Pred he U.S.A. Socey of Acuares 04 Page

Credbly wh Shfg Rsk Parameers Suar Klugma, FSA, CERA, PhD Iroduco Cosder he followg problem. You have a sequece of observaos ake over several me perods. For example, he observaos mgh be observed pure premums for a large employer s workers compesao coverage. Your goal s o forecas he ex observao he seres. To smplfy he problem, assume ha all exeral flueces have bee removed (such as flaoary red ad busess cycles). Furher suppose he amou of daa ha produced each observao s relavely small. To fx he dea, cosder he followg me seres of 50 pos. 0 5 0 05 00 95 90 85 80 3 5 7 9 35793579333353739443454749 I appears here may be somehg gog o here as he pos are cossely above he average of 03.00 for he frs half, he below, ad he above aga. Perhaps here s some correlao bewee adjace pos. The followg char shows he sample auocorrelaos by lag. Socey of Acuares 04 Page

0.8 0.7 0.6 0.5 0.4 0.3 0. 0. 0-0. 3 4 5 6 7 8 9 0 Ths paer mples ha a auoregressve process of order oe (AR()) may be approprae (more o hs laer). For a problem lke hs here are (a leas) hree approaches. The frs s o assume depedece ad o oher formao. The (assumg equal sample szes) he bes (as mmum varace) ubased lear esmaor of he rue mea s he sample mea, hs case 03.00. If we beleve ha he observaos are fac depede, ad ha a AR() process s approprae, he he bes predco s a weghed average of he mos rece observao ad he sample mea, usg he esmaed lag oe auocorrelao as he wegh. Ths forecas, usg he sample lag oe auocorrelao as he esmae, s 0.69(.64) + 0.3(03.00) = 08.96. The me seres approach s movaed by a assumpo ha observaos are correlaed (ad ypcally ha he correlao depeds oly o he me bewee he observaos). Aoher way of lookg a hs s ha he meas are chagg over me. As a example, f a facory has a culure of safey may perss for a few years, bu as here s urover maageme or workers, he culure may chage. I geeral, f here s such correlao, may make sese o gve more wegh o rece observaos (as was he resul wh he AR() model). Fally, because he sample sze s small, a credbly approach may be approprae. The credbly approach s movaed by wo facors. Frs, wh lmed daa, ay forecas based oly o ha daa may have cosderable varably. Secod, here may be mulple forecass (e.g., for oher sured groups) ad credbly heory says ha aggregae, mulple forecass are more accurae whe he dvdual meas are credbly-weghed wh a overall mea. Ths leads o a reduco wegh o each observao. Ths oe provdes a bref revew of me seres ad credbly approaches whe reaed separaely as boh are reaed elsewhere acuaral educao. I he demosraes a mehod of combg he wo a sgle aalyss (hough o all combaos wll be covered). The oo of combg he wo was explored he paper by Mahler (990). Smlares o hs work wll be oed as he model for combg he wo s developed. The movao for he mehod used for parameer esmao comes from wo Socey of Acuares 04 Page 3

papers by Frees, Youg, ad Luo (999 ad 00). I he remader of hs oe hey wll smply be referred o as Mahler ad Frees respecvely. As oed he smple example, s mpora o udersad wha s o mea by chage over me. Ths does o refer o red (here are oher credbly mehods for red). I also does o refer o abrup chages a specfc me pos (such as a chage markeg, uderwrg, or regulao) or o ecoomc cycles. All of hese should be accoued for wh daa adjusmes before a me seres or credbly approach s appled. Mahler uses he losg perceage of Major League Baseball eams as a llusrao. There s o overall red as oal he colleco of eams always loses half of s games. Wha s lef s uaccoued for radom flucuaos he mea wh some emporal coeco. I baseball, may players reur for he ex seaso ad hus here s lkely a posve correlao bewee records of oe year ad he ex. Bu over me here wll be sgfca chages ad laer years are lkely o be esseally depede of earler years. A smlar example s used here o llusrae he varous approaches. For he baseball example used here, o keep he calculaos smple, he daa are oly from he sxee years 998-03. The las expaso (o 30 eams) occurred 998 so he umber of years of observao s he same for all eams (hs s o a requreme ay of he formulas ha wll be preseed). The observao s he umber of ws each seaso. Whle exposures are o exacly 6 for each eam each year, he dffereces are small eough ha hey ca be gored. I all he calculaos ha follow, exposures are ake as raher ha 6. Ths has o effec o ay of he esmaes oher ha he process varace esmae. The credbly formulas preseed allow for exposures o dffer for each observao. As a llusrao of he paers, he followg graph has he ws each year for he eam wh he fewes oal ws (Kasas Cy Royals), he eam closes o a average of 8 ws per year (Arzoa Damodbacks), he eam wh he mos oal ws (New York Yakees), ad a eam wh a eresg paer (Tampa Bay Rays). I s eresg o oe ha he average sample lag oe auocorrelao over he 30 eams s 0.6 (hough he rage s from 0.3 o 0.66) ad he average sample auocorrelaos for oher lags are close o zero (hough aga here s cosderable varao amog eams). Auocorrelaos have a hgh sadard error (a rough approxmao s he square roo of he rao of ad he sample sze, hs case sqr(/6) = 0.5) ad so hs varao s o surprsg. For example, 03 Tampa Bay ad Texas had a playoff game for he fal Amerca League wld card spo. Ths 63 rd game coued he fal sadgs. Or, f a raed ou game has o bee rescheduled by he ed of he seaso ad s oucome does o affec playoff elgbly, he game s o played. Durg hs perod here was a chage owershp ad maageme ha had a drec effec o he playg feld (Ker, 0). I hs oe s examples, forecass wll be gve for each of hese four eams. Socey of Acuares 04 Page 4

0 0 00 90 80 70 60 KCR ARI NYY TBR 50 40 998 003 008 03 Tme Seres Approach Ths seco s o a exhausve dscusso of me seres modelg. Raher, hree commoly used models wll be dscussed. Readers should cosul oe of he may exs o me seres modelg o lear abou he rage of models ad more formal mehods of model seleco ad valdao. I all cases hs oe s assumed ha he observaos have bee rasformed o become weakly saoary. Tha meas he ucodoal mea s he same a all mes ad ha he covarace bewee wo observaos depeds oly o he me ha separaes hem. Oe way o hk abou hs s ha based o frs ad secod momes, f you are lookg a a sequece of observaos s o possble o defy he me perod from whch hey orgaed. Oe model, he auoregressve model of order has already bee meoed. The defo s ha he observao a me s relaed o he prevous observao by x = ρx + ( ρ) μ+ ε where < ρ < ad { ε } s a sequece of depede radom varables wh a mea of zero ad a sadard devao of σ ε. Ofe, a ormal dsrbuo s assumed, bu ha s o ecessary o esure ha s hs process s saoary. Wh hese assumpos, Var( x) = σx = σε /( ρ ) ad Corr( x, x s) = ρ. Parameer esmao ca be doe a ad hoc maer (for example, usg he sample mea for μ, he sample lag oe correlao for ρ, ad he sample sadard devao of he resduals, ˆ ε ˆ ˆ ˆ = x ρx ( ρ) μ, o esmae σ ε ). For he 50 observaos preseed he roduco, hose esmaes are ˆ μ = 03.00, ˆ ρ = 0.697, ad ˆ σ ε = 4.559. A secod ad hoc approach s o oe ha he AR() model s a regresso formula. The leas squares esmae s Socey of Acuares 04 Page 5

x = 0.7746 + 9.030 wh 47 degrees of freedom from whch a mea of 0.75 ca be ferred. The sadard error s 4.555. A hrd approach s o make a dsrbuoal assumpo ad use maxmum lkelhood. Rug hose same 50 observaos hrough he fpp package R (whch defauls o he ormal dsrbuo ad MLE) produces x ˆ μ = 03.94, ˆ ρ = 0.738, ad ˆ σ ε = 4.50. The forecas for fuure observaos s a smple maer of applyg he AR() model formula wh he error erm for fuure values se equal o zero (s expeced value). The forecas error s also easy o oba (og ha hs verso reflecs oly process varably ad o esmao error). Usg he R esmaes, xˆ ˆ ˆ 5 = ρx ˆ 50 + ( ρ) μ = 0.738(.64) + 0.69(03.94) = 09.6 Var( x x ) = Var[ ρx + ( ρ) μ + ε x )] = σε Var ˆ ( x ) 0.34. 5 50 50 5 50 5 x ˆ 50 = σ ε = The formulas are codoal upo he mos rece observao because s kow ad o radom. The baseball daa s more dffcul o aalyze ha 6 observaos are oo few o oba relable esmaes of he auocorrelaos. For hs aalyss we wll use he average sample lag oe auocorrelao preseed earler ad beg wh he AR() model. Usg he average sample lag oe auocorrelao of 0.655, 3 he resuls for he four eams (where he forecas s 0.655 mes he 03 ws plus 0.73845 mes he average ws) are: Team 03 ws Average ws Forecas Kasas Cy 86 69.00 73.45 Arzoa 8 80.69 80.77 Tampa Bay 9 74.69 79. New York Yakees 85 96.8 93.7 Whe oly he lag oe auocorrelao s esseally ozero a MA() model may be approprae. Ths model s I he follows ha x,. = μ + ε θε < θ < Var( x) = Var( ε θε ) = Var( ε) + Var( θε ) = ( + θ ) σε Cov( x, x ) = Cov( ε θε, ε θε ) = Cov( θε, ε ) = θσ θ Corr( x, x ) =. + θ ε 3 A alerave would be o use each eam s ow sample lag oe auocorrelao for s forecas. The approach ake here s more le wh he models preseed laer hs oe. Socey of Acuares 04 Page 6

Wh he sample lag oe auocorrelao of 0.655, ca be ferred ha θ = 0.84. You may recall ha ay MA model ca be rewre as a AR model, bu wh a fe umber of erms. I hs case, s x = μ θ ( x μ) θ ( x μ) + ε. Ths places geomercally decreasg weghs o he devaos from he mea. Because he parameer esmae s egave, he sgs of he weghs alerae. Whe appled o he four eams, he resuls are: Team 03 ws Average ws Forecas Kasas Cy 86 69.00 73.6 Arzoa 8 80.69 8.3 Tampa Bay 9 74.69 78.59 New York Yakees 85 96.8 93.65 These forecass are o much dffere. Wh hs model he average ws receves a wegh of 0.7798, slghly more ha wh he AR() model. The mos rece seaso also ges more wegh, a 0.84. The ma dfferece s ha he secod mos rece seaso receves a wegh of 0.0798. Isead of he acual observaos havg a MA() model, le he dffereces of he observaos have hs model ( me seres ermology, hs s a ARIMA(0,,) model) wh o cosa erm. Havg o cosa makes sese as oherwse a eam would have a ever creasg (or decreasg) expeced umber of ws. The model s y,. = x x = ε θε < θ < For our baseball daa, he average sample lag oe correlao of he dffereces s 0.945 whch ca be solved for ˆ θ = 0.3577. Ths model ca also be wre as a pure AR model: x = ( θ)[ x + θx + θ x + ] + ε. 3 Ths model s called smple expoeal smoohg ad s oe of he models examed by Mahler (hough he approached as a reasoable way o mpose decreasg emphass o older observaos raher ha from a ARIMA modelg perspecve). I he case of baseball forecasg, he weghs are 0.674, 0.96, 0.076,, 0.00000003. Because he fe seres eds a 6 erms, he weghs do o sum o exacly (hough hs case hey come exremely close). I pracce, he weghs should be rescaled. Cosder predcg he 04 w oal for he Tampa Bay Rays. Ther ws for he 6 years were 63, 69, 69, 6, 55, 63, 70, 67, 6, 66, 97, 84, 96, 9, 90, ad 9. Applyg he weghs leads o a forecas of 9.50. The oher eams have forecased values of KCR 8.0, ARI 8.48, ad NYY 88.45. Noe ha all ARIMA models, forecass more ha oe perod ahead ca be made by repeaed applcaos of he formula wh fuure error erms se equal o zero. Socey of Acuares 04 Page 7

Icorporag Credbly We frs ur o credbly mehods ha do o ake o accou correlao over me. Oly greaes accuracy credbly wll be dscussed. Beg model-based, hs mehod ca be exeded o he mevaryg suao a formal maer. I hs seco we revew he Bühlma-Sraub emprcal Bayes approach ad apply o he baseball daa. To keep hs seco smple, oly he case of equal exposures wll be covered. The geeralzao o arbrary exposures wll be provded he ex seco. To develop he model, le X be he radom observao from group a me. For hs smple seg we wll assume k groups ad me perods for each group. Here ad hroughou, he erm group wll be used o descrbe he ey for whch we wa o make a forecas. I he baseball example, a group s a eam, k = 30, ad = 6. The model does o specfy a dsrbuo, oly he frs wo momes. The model has wo levels. The op level s a model for a uobservable radom quay, he mea for group. Le ξ be ha mea ad le E( ξ ) = μ ad Var( ξ ) = τ. I s furher assumed ha hese k radom varables are depede. Ths process assgs he meas o each group. I appears ha for baseball eams hs process has assged a hgh mea o he New York Yakees ad a low mea o he Kasas Cy Royals. The secod level provdes he momes of he acual observaos. Gve he (uobservable) mea, he momes are EX ( ξ ) = ξ ad Var( X ξ ) = σ. The objecve s o use he avalable daa o esmae he k ukow meas. The emprcal Bayes approach deermes he esmaes wo seps. The frs s o assume ha he op level mea ad he wo varaces are kow ad ha he esmaor mus be a lear combao of he observaos from he group. The esmae s he he value ha mmzes mea squared error. The formula wll o be derved here. I s produced all credbly exbooks ad s a specal case of he formula derved he ex seco. The soluo s ˆ ξ ( ),, = ZX + Z μ X = X Z =. = σ + The secod sep s o use he daa o esmae he hree parameers. The usual esmaes are derved by posg reasoable esmaors ad he makg adjusmes o make hem ubased. The formulas are ˆ μ = X = k X k = = k k = X X = X X k ( ) = = k = ˆ σ ( ), ˆ τ ( ) ˆ σ. τ The formulas are easly modfed whe here are dfferg umbers of me perods per group or dfferg exposures per observao. Whe appled o he baseball daa, he resuls are Socey of Acuares 04 Page 8

ˆ μ = 80.9646 ˆ σ = 04.53 ˆ τ = 35.385. Wh hs model, ad daa, he same credbly facor s used for all 30 eams. For Tampa Bay, he calculao s 6 Z = = 0.84396 04.53 6 + 35.385 ˆ ξ = 0.84396(74.6875) + 0.5604(80.9646) = 75.67. 7 Applyg he same weghs o he oher eams yelds KCR 70.87, ARI 80.73, ad NYY 94.34. There s a alerave way o ge o he same resul. The advaage s ha ulke he emprcal Bayes approach, whch reles o mapulaos of arbrarly seleced sums of squares, hs approach leads drecly o he soluo. As such, s easer o geeralze, as wll be doe he ex seco. The alerave s o se up he problem as a radom effecs lear model. More deals abou radom effecs models ad acuaral applcaos ca be foud Frees ad also Klker (0). The model s X N N = μ + α + ε, α ~ (0, τ ), ε ~ (0, σ ). Wha dsgushes hs model from he more radoal fxed effecs model s ha he mddle erm s a radom varable. Frs oe ha f we dd use a fxed effecs regresso model, he esmaor for each group would be he sample mea ad hece o apply credbly. As Klker oes, he movao ha led o he radom effecs model s o drecly applcable o surace segs. However, as we shall see, does exacly wha we wa, shrkg he esmaes oward he overall mea. Aoher key po s ha ulke he Bühlma-Sraub approach, a dsrbuoal assumpo has bee corporaed. Whle we may o beleve ha he ormal dsrbuo s he correc model, we do oe ha here s a correspodece bewee he ormal dsrbuo ad leas squares esmao. So wll o be surprsg f hs alerave approach produces smlar resuls. We wll ow derve he esmaes for hs model. Because he dsrbuos are specfed, maxmum lkelhood esmao could be used. However, varace esmaes ed o be based (recall ha for a sample from he ormal dsrbuo, he MLE of he varace has he deomaor raher ha he ha produces a ubased esmaor). A echque kow as resrced maxmum lkelhood (REML) has bee developed o address hs problem. Raher ha derve he REML loglkelhood fuco (whch s doe he 999 Frees paper), we prese he regular loglkelhood fuco ad he po ou he dfferece. Wrg he regular loglkelhood fuco s a challege because he observaos wh a group are depede (hough observaos from dffere groups are depede). For observaos group, we have Socey of Acuares 04 Page 9

EX ( ) = E( μ + α + ε ) = μ Var( X ) = Var( μ + α + ε ) = τ + σ Cov X X Cov Var s (, s ) = ( μ + α + ε, μ + α + εs ) = ( α ) = τ,. The he vecor of observaos from group has he mulvarae ormal desy fuco f( x,, x μσ,, τ ) = exp ( ) ( ) / / ( π ) [de( )] x μ V x μ V where x = ( x,, x ) s a vecor of he observaos from group, μ = ( μ,, μ ) s a vecor coag he mea, ad V s a marx wh σ + τ o he dagoal ad τ everywhere else. The loglkelhood for he ere se of observaos s he k k k ll = l( π ) l[de( V)] ( x μ) V ( x μ ). = Takg he dervave wh respec o he mea s relavely smple. The soluo s k k sum( V x) x = = = ˆ μ = = ksum [ ( V )] k due o he fac ha he verse of V has he same srucure as V, wh a cosa value o he dagoal ad a dffere, bu cosa, value off he dagoal. The sum fuco adds he elemes of he marx or vecor o whch s appled. I ca be show ha maxmzg he lkelhood wh respec o he wo varace erms resuls a formula smlar o he emprcal Bayes formula, bu he esmaor of τ has a dffere deomaor. The REML esmaor chages he loglkelhood fuco o k k k l L= l( π ) l[de( V)] l[ k sum( V )] ( x ˆ) ( ˆ μ V x μ ). = The bass of he esmaor s o oba a lkelhood fuco ha does o deped o he mea (oe ha he mea s replaced by s esmae here). The lkelhood s esseally based o he resduals afer subraco of he esmaed mea. I urs ou ha he values ha maxmze hs fuco exacly mach he emprcal Bayes esmaes. I should be oed ha hs exac mach occurs oly whe he oal exposures are decal across groups. Whe oal exposures dffer amog groups, he REML esmaes wll be based. Wha we lose bas we ga by havg a approach ha s easy o geeralze. Tha s he subjec of he ex seco. Before movg o, here s oe more eleme o address ad ha s how he ukow meas are esmaed whe usg a radom effecs model. I urs ou hs s doe usg he same credbly formula as used earler ad so o ew formulas are eeded. Socey of Acuares 04 Page 0

A Geeral Model A model ha allows for boh me depedece ad credbly wll ow be developed. As he prevous seco, wll be doe wo dffere ways. Oe s a radoal credbly approach. Ths wll provde he formula for he credbly facors. The secod s a radom effecs model approach. Ths wll provde he mehod for esmag he parameers. To beg he credbly dscusso, he group subscrp wll be dropped. The oher groups are oly eeded for parameer esmao. The followg dscusso mrrors ha preseed Klugma, e al. (0). Le X, X,, X be he radom varables ha represe observaos a mes hrough. The goal s o use hese observaos o esmae a fuure radom observao, X +d. Oly lear fucos wll be used. Thus ˆ + = d 0 + + + X Z Z X Z X. The values ha mmze he expeced squared error are he soluos o he equaos EX [ ] = Z + ZEX ( ) + d 0 = Cov( X, X ) = Z Cov( X, X ), =,,. + d s s s= The soluo requres a model wh suffce specfcao o eable calculao of he requred expeced values ad covaraces. Oe such model wll ow be specfed. I parcular, corporaes auocorrelaos. I s a hree-level model. The op level saes ha here s a mea for he group ha s sable over me. Le ξ deoe ha mea. Tha does o mea each year has hs mea bu ha whou addoal formao (such as from earby years) hs mea s he bes represeao of a gve year s expeced value. Tha mea s draw from a dsrbuo wh mea μ ad varace τ. Ths s a sadard feaure of credbly models. I allows each group o have s ow mea. The secod level reflecs he depedecy srucure of he meas over me. Le he meas for he observaos a mes hrough be ( θ,, θ ). These values are draw from a mulvarae dsrbuo where each value has a mea of ξ. To allow for depedece, le Cov( θ, θs) = δs. For ow, o resrcos wll be placed o hese values (oher ha ha he covarace marx mus be posve defe). The dea s ha he mea a me s radom, bu relaed o he mea a oher mes. So f oe perod has a large mea, may perss for several fuure perods. The hrd level s he observao self. The value X s draw from a dsrbuo wh mea ad varace σ / w where w s a kow wegh. Ths par s smlar o he Bühlma-Sraub model. The ex sep s o deerme he elemes of he equaos for he credbly weghs. I hese formulas, ad s ca be ay of he values,,,, + d. Socey of Acuares 04 Page

EX [ ] = EEX [ ( θ )] = E[ θ ] = EE [ ( θ ξ)] = E[ ξ] = μ Var X E Var X Var E X E w Var ( ) = [ ( θ)] + [ ( θ)] = [ σ / ] + ( θ) = σ / w + E[ Var( θ ξ)] + Var[ E( θ ξ)] = σ / w + E( δ ) + Var( ξ) = σ / w + δ + τ CovX (, X) = EXX [ ] μ = EEXX [ ( θ, θ )] μ = E[ θθ ] μ The equaos are he s s s s = E E = E + = + + = + s [ ( θθ s ξ)] μ [ δs ξ ] μ δs τ μ μ δs τ,. s μ = Z + μ 0 = Z, + d s s s= δ + τ = Z σ / w + Z ( δ + τ ), =,,. Oce he secod se of equaos s solved, he frs equao s easly solved as Z0 = Z μ whch = apples he compleme of credbly o he overall mea, f kow, or s esmae. The secod se of equaos s lear ad so has he marx soluo σ / w + δ + τ δ + τ δ + τ δ, + d+ τ δ + τ σ / w δ + δ + τ δ + τ, + d+ τ Z Z =. Z δ + τ δ + τ σ / w δ + δ + τ, + d + τ The marx he mddle ca be dvded o hree compoes. Le D be he frs compoe, a dagoal marx wh σ / w,, σ / w o he dagoal. The secod compoe s Σ, a marx wh he δ s values. The fal marx s T, whch every eleme s τ. The, wh S deog he rgh-had vecor, Z= ( D+ + ) S= V S Σ Τ. I wll be useful o have he mea squared error boh as a fuco of he geeral value of Z ad for he value ha mmzes. The developme s (subsug Z0 = Z μ ): = Socey of Acuares 04 Page

EX+ d Z0 ZX = EX+ d μ + Zμ ZX = EX+ d μ Z( X μ) = = = = = Var( X ) + Z Z Cov( X, X ) Z Cov( X, X ) + d s s + d = s= = / w+ d + d, + d ZZs( s + Zσ w Z δ+ d, + τ = s= = = = σ + δ + τ + δ + τ ) / ( ) = σ / w + δ + τ + Z ( D+ Σ + T) Z ZS + d + d, + d = σ / w + δ + τ + ZVZ ZS. + d + d, + d A he opmal value of Z he value s σ / w + δ + τ + ZVZ ZS = σ / w + δ + τ + SV VV S SV S + d + d, + d + d + d, + d = σ / w + δ + τ SV S. + d + d, + d The Bühlma-Sraub model s he same bu wh Σ = 0 ad all elemes of S equal o τ. I should also be oed ha he curre model s overspecfed (addg a cosa o τ ad subracg he same amou from each δ produces he same soluo). Ths model has far oo may parameers o be useful. I fac, here would ever be eough daa pos o esmae all of hem. For he remader of hs oe, he followg smplfcaos wll be mposed: All groups have he same parameers. The covarace erms deped oly o he dfferece bewee he subscrps. Ths s cosse wh he saoary requreme whe performg me seres aalyss. As a resul, whe he model s exeded o mulple groups, here wll be commo values for δ0 δ δ δ+ d + τ δ δ0 δ δ+ d τ σ, τ, + Σ =, ad S =. δ δ δ δ + τ 0 d The oal umber of parameers s + + + d = + d +. Ths model s sll overspecfed. There are a mos + d + free parameers. Cosder he baseball example wh 30 groups ad 6 years of observao. We wa o forecas he value year ahead. There are 480 daa pos ad oly 8 parameers o esmae. Bu here s a problem. The frs erm of S requres δ 6, bu here s o daa avalable o esmae he covarace of observaos 6 years apar. There are wo possble soluos:. Base he credbly esmae o oly he mos rece 5 years, usg he frs year s observaos oly o esmae he eeded covarace. Socey of Acuares 04 Page 3

. Propose a furher smplfcao smlar o hose see he me seres models. The exrapolao ca be used. The frs soluo may be approprae f here are eough years of daa so ha he correlaos bewee dsa observaos are very low. Droppg he frs year may make lle dfferece. The secod soluo eforces a paer o he covaraces ha ca be exeded. Now ha we ca oba he forecass gve he parameer values, s me o ur o parameer esmaes. For ha we use he radom effecs model formulao. For group : X X = = μ+ α + γ + ε, wh all vecors depede of each oher X μ α γ ε μ=, ~ N(, ), ~ N(, ), α = 0T γ = 0Σ ε = ~ N( 0D, ) μ α γ ε The wo mddle erms ca be combed: X X = = μ+ θ + ε X μ θ ε μ=, ~ N(, ), θ = 0 Σ+ Τ ε = ~ N( 0, D) μ θ ε I Frees he geeral model wh radom effecs s preseed as Our model fs hs specfcao wh X = Uθ + Wβ + ε, θ ~ N( 0, R), ε ~ N( 0, D ). U = I, he dey marx W =, a colum vecor of s β = μ, a scalar R = Σ + T D = D. The REML esmaes are foud by maxmzg he followg fuco, frs saed wh he geeral radom effecs model, where V = U RU + D : Socey of Acuares 04 Page 4

k k k ˆ l L = 0.5 l de( ) l de( ) ( ) ( ˆ V + W V W + X W β V X Wβ) = = = k k WV W WV X = = ˆ β =. For he parcular model hs oe, where V = Σ + T+ D ad sum meas o add all he elemes of he gve marx or vecor: k k k ll= 0.5lde( V) l ( ) ( ˆ) ( ˆ + sum V + X μ V X μ) = = = k sum( V X) = ˆ =. k sum( V ) = μ All he formulas are place o make forecass ha combe me seres ad credbly. Forecasg wh boh Tme Seres ad Credbly The ulmae goal s o make a forecas ha s a weghed average of observaos from he group plus wegh o a exeral value. I a deal suao, he weghs are deermed usg hese wo formulas: Σ Τ Z0 = Z μ. = Z= ( D+ + ) S= V S The formulas deped o kowg he correlao srucure ad havg esmaes of he mea ad he parameers ha defe he correlao srucure. The esmaes ca be obaed va REML usg he formulas he prevous seco. The challege whe usg me seres models s ha wh lmed daa (ad ofe acuaral aalyses here are few years of daa) may ARIMA models wll f well. Hece here may be a varey of correlao srucures ha make sese. Judgme may be requred selecg a model ha leads o a se of weghs ha makes sese. Mahler offers a soluo ha separaes he wo seps. He separaely poses a correlao srucure (usg oe ha s as geeral as possble whle sll beg ameable o esmao) ad a paer of weghs. The weghs are he seleced o mmze he leas squares crero σ / w + d + δ + d, + d + τ + ZVZ ZS where he weghs are cosraed o follow he se paer. Ths wll lead o a subopmal resul wh regard o he leas squares crero, bu may come close o beg opmal. I has he advaage of producg weghs ha are easer o expla. Ths approach wll be llusraed wh he baseball example. Socey of Acuares 04 Page 5

Baseball Example We beg wh a aalog of he AR() model. We kow he paer of correlaos for ha model. Thus g he addoal specfcao s δg = δρ where g s he umber of perods separag he wo observaos. Ths adds wo parameers o he Bühlma-Sraub model. The REML maxmzao resuls are 4 ˆ σ = 30.49, ˆ τ = 4.77, ˆ δ = 95.80, ˆ ρ = 0.667, ˆ μ = 80.97. Because he exposures are he same for all observaos, he credbly weghs are he same for each eam. The vecor of weghs s: Year 998 999 000 00 00 003 004 005 Wegh 0.085 0.00 0.0084 0.0080 0.0079 0.0079 0.0079 0.0079 Year 006 007 008 009 00 0 0 03 Wegh 0.0079 0.0079 0.008 0.0090 0.07 0.0300 0.085 0.4664 The remag credbly s 0.78 o be appled o he overall mea. Whe appled o he four eams, he 04 forecass are: KCR 80.4, ARI 8.03, TBR 86., ad NYY 87.07. Noe ha he credbly weghs do o exacly reflec a AR() model for each eam wh he resulg esmae credbly-weghed wh a overall mea of 8. Dog so would have cosa weghs for he frs 5 years ad a hgher wegh for 03. Oher ha he addoal wegh o 0, he resuls are prey close. Ths brgs us o he approach ake by Mahler. Havg posed a correlao srucure ad esmaed he requred varaces ad correlaos, ay credbly srucure ca be mposed. Ths ca be doe by reurg o he mea squared error formula. We kow s mmzed a he values he prevous able. However, suppose we accep a subopmal resul ha has he cer AR() paer. Tha s, mmze he error bu wh he cosras Z = Z = = Z = a, Z = b, a, b 0, 5a+ b. 5 6 Before dog ha, oe ha he mea squared error whe opmzed s 94.47. The opmal value wh he cosras occurs a a = 0.0380 ad b = 0.574. Ths was obaed by seg up he vecor of Z values as a fuco of a ad b. The a mmzao program ca fd he (a, b) par ha acheves he mmum mea squared error (all oher values he formula were prevously esmaed). The mea squared error creases o 95.53. Ths appears o be a good radeoff for a cleaer resul. The forecass are ow KCR 80.86, ARI 80.9, TBR 85.4, ad NYY 86.50. The ma dfferece s ha he frs verso places addoal wegh o he 0 value. We could ry opmzg wh uque values for he las wo weghs raher ha he las oe. Of course, hs mus produce a smaller mea squared error. 4 No opmzao mehod ca be assured o provde perfec resuls. The calculaos hs oe were doe boh Excel ad R. A mes here were dffereces he fourh or ffh sgfca dg. Socey of Acuares 04 Page 6

Recall ha for he baseball daa oly he average sample lag oe auocorrelao coeffce s o close o zero. Ths mples a MA() model may be approprae. The correlao srucure s ha δ 0 ad δ have uque values ad all oher covaraces are zero. Because he sample szes are equal, hs formulao σ ad δ 0 always appear ogeher as σ + δ 0. The REML esmaes are The resulg weghs are: σ + δ = 04.5, ˆ τ = 3.55, ˆ δ = 3.4, ˆ μ = 80.97. 0 Year 998 999 000 00 00 003 004 005 Wegh 0.0459 0.0305 0.0357 0.0339 0.0345 0.0343 0.0344 0.0344 Year 006 007 008 009 00 0 0 03 Wegh 0.034 0.0348 0.039 0.0387 0.03 0.0733 0.089 0.38 The remag credbly s 0.80, o be appled o he overall mea. Whe appled o he four eams, he 04 forecass are: KCR 76.87, ARI 8.4, TBR 80.30, ad NYY 90.9. The ARIMA(0,,) model used earler has geomercally decreasg weghs (whch may be more sasfacory) bu does o exed o a credbly coex. Afer akg dffereces, all 30 eams ow have a expeced value of zero. There s o way o model he varao he hypohecal meas whe hey are kow o be he same. However, he Mahler approach suggess a way o ge he resul we are lookg for (geomercally decreasg weghs wh wegh also beg placed o he overall mea). The cosras ca be chaged o become b Z6 = a Z = ab = a> b a b 6 6,,,,5, 0,0,. Because hs paer does o relae o a usable ARIMA model, cosder a arbrary correlao srucure (as Mahler dd). Raher ha force a paer o he correlaos, hey wll oly be resrced o be he same for a gve lag. Two prevously meoed ssues eed o be addressed. The frs s ha o esmae of he correlao for observaos 6 years apar ca be obaed. If urs ou ha he esmaed correlaos a lags 3, 4, ad 5 are very small, he oe a lag 6 ca be ake as zero. The secod ssue s ha hs geeral model here s a lack of specfcy. Recall ha he key marces are σ + δ0 + τ δ + τ δ5 + τ δ6 + τ δ + τ σ + δ0 + τ δ4 + τ δ5 + τ V =, ad S=. δ5 + τ δ4 + τ σ + δ0 + τ δ + τ Noe ha here are 6 uque eres V bu 8 parameers. Le he jh ery he marx be j. I s o possble o separaely esmae he hree corbuors o he marx. The REML esmaes are geerally γ Socey of Acuares 04 Page 7

decreasg. The las value s egave, whch dcaes ha seg γ6 = δ6 + τ he frs eleme of S o zero may o be approprae. Isead, s se equal o he esmae of γ 5. The credbly weghs are a mxure of posve ad egave values, whch may o be reasoable. The mea squared error (whch s mmzed, gve hs model) s 83.. The forecass are KCR 84.50, ARI 79.09, TBR 9.4, ad NYY 83.0. Eforcg geomercally decreasg credbly weghs creases he mea squared error from 83. o 95.39. Because hs s based o a model wh a large umber of esmaed parameers, hs crease may o be meagful. The revsed Z values are (oly values ha are a leas 0.000 are preseed): Year 006 007 008 009 00 0 0 03 Wegh 0.000 0.0003 0.000 0.0033 0.03 0.0388 0.334 0.4583 The overall mea receves a wegh of 0.3534. The forecass here are KCR 8.47, ARI 8.8, TBR 87.8, ad NYY 85.56. The approach jus ake s jus oe way o explo he fac ha he varace parameers ca be esmaed separaely from he deermao of he credbly weghs. Thus, ay correlao srucure could have bee proposed ad he geomercally decreasg credbly weghs derved from ha srucure. Smlarly, for ay correlao srucure, ay paer of credbly weghs ca be mposed. Fal Commes A varey of approaches was employed o aalyze a sgle problem. For a parcular problem s mpora o udersad whch (or boh) of he wo facors eeds aeo (he me varyg mea ad he credbly adjusme). Baseball ws are a case where boh are mpora. I 983, Bll James posulaed wo forces ha affec eams from year o year. He amed oe he Plexglass Prcple, og If a eam mproves oe seaso, wll lkely decle he ex. The magude of hs effec ca be calbraed wh a me varyg mea. The oher he amed he Whrlpool Prcple, og All eams are draw forcefully oward he ceer. Ths effec ca be calbraed wh a credbly adjusme. 5 As wh ay model buldg exercse, resuls should combe exper kowledge of he pheomeo ad sascal aalyss. Suppose you had he ask of forecasg Tampa Bay s w oal for 04. The varous esmaes produced hs oe are preseed alog wh commeary. 5 James (983), p. 0. The clear acrylc subsace s spelled wh boh oe ad wo s s. James lkely chose o ame hs prcple Plexglas(s) because eams ha mproved see a clear pahway o furher success, bu more lkely ha o bouce backward he ex year. Socey of Acuares 04 Page 8

Mehod Forecas Commes AR() usg ow daa 79. Too much wegh o he average over pas 6 years MA() usg ow daa 78.59 Smlar o AR() plus egave wegh o 0 ARIMA(0,,) usg ow daa 9.50 Perhaps oo much emphass o rece resuls Bühlma-Sraub 75.67 Weghs all pror years equally AR() wh radom effecs 86. Wegh s shfed o overall mea of 8 raher ha frs 5 years AR() wh Zs forced o be 85.4 Smlar o prevous cosa for years -5 MA() wh radom effecs 80.30 Negave wegh o 0 Arbrary radom effec 9.4 Uusual paer of credbly weghs correlaos Arbrary correlaos wh geomercally decreasg weghs 87.8 Follows a smooh paer whou resrcg he correlao srucure The mos reasoable mehod may be he fal oe preseed. Decreasg posve weghs places more emphass o rece performace ad weghg he resul wh he overall mea of 8 allows for he ofe observed decle afer a successful seaso or ru of seasos. Thus a forecas of 88 ws s a reasoable cocluso. However, he rage of esmaes dcaes ha resuls may vary. As a fal comme, whle hs oe was beg prepared, Tampa Bay decded o rea s bes pcher for 04, sged a free-age relef pcher, ad execued some rades. Thus, whle he forecas of 88 ws s a good sarg po, hs addoal kowledge may be releva whe seg a fal forecas. Refereces Frees, E., Youg, V., ad Luo, Y., 999, A Logudal Daa Aalyss Ierpreao of Credbly Models, Isurace: Mahemacs ad Ecoomcs, Vol. 4:3, 9-47. Frees, E., Youg, V., ad Luo, Y., 00, Case Sudes Usg Pael Daa Models, Norh Amerca Acuaral Joural, Vol. 5:4, 4-4. James, B., 983, The Bll James Baseball Absrac 983, Ballae. Ker, J., 0, The Exra %: How Wall Sree Sraeges Took a Major League Baseball Team from Wors o Frs, ESPN Books ad Ballae Books. Klker, F., 0, Geeralzed Lear Mxed Models for Raemakg: A Meas of Iroducg Credbly o a Geeralzed Lear Model Seg, Casualy Acuaral Socey E-Forum, Wer 0, -5. Klugma, S., Pajer, H., ad Wllmo, G., 0, Loss Models: From Daa o Decsos, 4 h edo, Joh Wley ad Sos. Mahler, H., 990, A Example of Credbly ad Shfg Rsk Parameers, Proceedgs of he Casualy Acuaral Socey, Vol. LXXVII, 5-308. Socey of Acuares 04 Page 9

Daa Ses The ffy po me seres s:.8 05.8 04.4 98.5 0.60 07.5 07.03 07.78 0.46 03.8 08.7 99.4 04.47 05. 08.3.96.6 4.0 4.0.83 03.83 07.3 05.00 07.7 04.99 05.98 95.3 89.6 9.83 0.44 93.66 93.56 90.98 94.79 9.5 94.48 04.47 00. 0.73 97. 98.6 00.63 99.75 99.00 95.50 98.69 0.75 06.86 0.98.64 The baseball daa was obaed from baseball-referece.com. Year ARI ATL BAL BOS CHC CHW CIN CLE COL DET HOU KCR ANA LAD FLA 998 65 06 79 9 90 80 77 89 77 65 0 7 85 83 54 999 00 03 78 94 67 75 96 97 7 69 97 64 70 77 64 000 85 95 74 85 65 95 85 90 8 79 7 77 8 86 79 00 9 88 63 8 88 83 66 9 73 66 93 65 75 86 76 00 98 0 67 93 67 8 78 74 73 55 84 6 99 9 79 003 84 0 7 95 88 86 69 68 74 43 87 83 77 85 9 004 5 96 78 98 89 83 76 80 68 7 9 58 9 93 83 005 77 90 74 95 79 99 73 93 67 7 89 56 95 7 83 006 76 79 70 86 66 90 80 78 76 95 8 6 89 88 78 007 90 84 69 96 85 7 7 96 90 88 73 69 94 8 7 008 8 7 68 95 97 89 74 8 74 74 86 75 00 84 84 009 70 86 64 95 83 79 78 65 9 86 74 65 97 95 87 00 65 9 66 89 75 88 9 69 83 8 76 67 80 80 80 0 94 89 69 90 7 79 79 80 73 95 56 7 86 8 7 0 8 94 93 69 6 85 97 68 64 88 55 7 89 86 69 03 8 96 85 97 66 63 90 9 74 93 5 86 78 9 6 Socey of Acuares 04 Page 0

Year MIL MIN NYM NYY OAK PHI PIT SDP SFG SEA STL TBR TEX TOR WSN 998 74 70 88 4 74 75 69 98 89 76 83 63 88 88 65 999 74 63 97 98 87 77 78 74 86 79 75 69 95 84 68 000 73 69 94 87 9 65 69 76 97 9 95 69 7 83 67 00 68 85 8 95 0 86 6 79 90 6 93 6 73 80 68 00 56 94 75 03 03 80 7 66 95 93 97 55 7 78 83 003 68 90 66 0 96 86 75 64 00 93 85 63 7 86 83 004 67 9 7 0 9 86 7 87 9 63 05 70 89 67 67 005 8 83 83 95 88 88 67 8 75 69 00 67 79 80 8 006 75 96 97 97 93 85 67 88 76 78 83 6 80 87 7 007 83 79 88 94 76 89 68 89 7 88 78 66 75 83 73 008 90 88 89 89 75 9 67 63 7 6 86 97 79 86 59 009 80 87 70 03 75 93 6 75 88 85 9 84 87 75 59 00 77 94 79 95 8 97 57 90 9 6 86 96 90 85 69 0 96 63 77 97 74 0 7 7 86 67 90 9 96 8 80 0 83 66 74 95 94 8 79 76 94 75 88 90 93 73 98 03 74 66 74 85 96 73 94 76 76 7 97 9 9 74 86 Socey of Acuares 04 Page