The Prediction Error of Bornhuetter-Ferguson

Transcription

1 The Predcto Error of Borhuetter-Ferguso Thomas Mac Abstract: Together wth the Cha Ladder (CL method, the Borhuetter-Ferguso ( method s oe of the most popular clams reservg methods. Whereas a formula for the predcto error of the CL method has bee publshed already 993, there s stll othg equvalet avalable for the method. O the bass of the reserve formula, ths paper develops a stochastc model for the method. From ths model, a formula for the predcto error of the reserve estmate s derved. Moreover, the model gves mportat advce o how to estmate the parameters for the reserve formula. For example, t turs out that the approprate developmet patter s dfferet from the CL patter. Ths s a ce add-o as t maes a stadaloe reservg method that s fully depedet from CL. The other parameter requred for the reserve s the well-ow tal estmate for the ultmate clams amout. Here the stochastc model clearly shows what has to be meat wth tal. I order to apply the formula for the predcto error, the actuary must assess hs ucertaty about both sets of parameters, about the developmet patter ad about the tal ultmate clams estmates. But for both, much gudace ca be draw from the estmates themselves ad from the ru-off data gve. Fally, a umercal example shows how the resultg predcto error compares to the oe of the CL method. Keywords: Loss reservg, Borhuetter-Ferguso, Stochastc model, Predcto error.. INTRODUCTION For most surace compaes ad ther audtors, the use of the Cha Ladder method (CL ad of the Borhuetter-Ferguso method ( has become a certa stadard or bechmar clams reservg. Ths meas that these methods are appled almost every case, ad oly f they seem to fal, oe loos for other methods. Orgally, these methods gave oly a pot estmate for the clams reserve. But ths was ot satsfactory because the oe could ot decde whether the estmates dffer sgfcatly or ot. Moreover, for the calculato of rs-based captal ad of premum loadgs oe eeds to assess the predcto error of the estmate (.e., the stadard devato of the true clams reserve from the pot estmate. I 993, a formula for the predcto error of the CL reserve estmate was publshed (Mac (993 or the more comprehesve verso Mac (994, whch the mea tme s wdely used. Ths formula gves a aswer to the questo of sgfcat dffereces to other methods ad measures the varablty of the true reserves for busess segmets where CL s acceptable. But for, such a formula s stll mssg. Ths may seem strage because s eve smpler tha CL. But ths smplcty s ust the problem. The predcto error cossts of two compoets, the process error ad the estmato (or parameter error. Whereas the estmato error bascally always ca be calculated va the laws of error propagato, for the process error a stochastc model of the clams Casualty Actuaral Socety E-Forum, Fall 008

2 The Predcto Error of Borhuetter-Ferguso process s requred. The latter was feasble the CL case because the way whch the CL age-toage factors are estmated cotas mplct formato o the uderlyg stochastcs. I the case, o clear procedure o how to estmate the parameters has bee establshed. I such a stuato, may models may seem admssble. The stochastc model for troduced ths paper s very smlar ts structure to the CL model of Mac (993 but adequately reflects the two fudametal dffereces betwee CL ad. The frst dfferece s the fact that the CL reserve s drectly proportoal to the clams amout ow so far whereas the reserve does ot deped at all o the ow clams amout. Ths s reflected a addtoal depedece assumpto of the model. The secod dfferece s the fact that the reserve estmate cludes the full tal of the clams developmet whereas the stadard CL reserve (.e., wthout addtoal tal factor oly cosders the developmet utl a gve last developmet year. The latter fact mples that the parameter estmato for the model has also to cosder the tal of the developmet where there s o data ad some udgmet s requred. Therefore, we do ot gve a uque estmato formula for the tal parameters but dscuss two alteratve ways to cope wth ths problem. I ay case, the developmet patter suggested by the model turs out to be dfferet from the well-ow CL patter. Ths maes to a really stadaloe reservg method. But stll, the actuary may mae hs ow selectos regardg the developmet patter, especally for the tal. I addto to the developmet patter, the reserve formula requres aother elemet, a tal estmate for the ultmate clams amout. Of course, the ucertaty of ths estmate must have a hgh mpact o the predcto error. As ths estmate usually comes from outsde (e.g., from prcg or s smply set by the actuary o the bass of hs owledge of the busess, ts ucertaty must be assessed from outsde of the ru-off tragle, too. Ad a actuary who s able to set (or accept a pot estmate should also be able to quatfy (or as for quatfcato of the ucertaty of ths estmate. Moreover, from the stochastc model mportat advce ca be derved for the assessmet of these estmates ad ther ucertaty. Altogether, ths meas that the predcto error of the reserve estmate depeds largely o the (more or less subectve assessmet of the actuary as t s already the case wth the reserve estmate tself. Secto gves a short revew of the method ad of ts coectos ad dffereces to the CL method. Secto 3 descrbes the approprate stochastc model. Secto 4 shows two ways to estmate or select the model parameters. The estmato of the stadard error of the parameters s dscussed Secto 5 where also the formula for the predcto error ad ts compoets s derved. Secto 6 gves a umercal example ad Secto 7 cocludes. Casualty Actuaral Socety E-Forum, Fall 008 3

3 The Predcto Error of Borhuetter-Ferguso. THE METHOD Let C, deote the cumulatve clams amout (ether pad or curred of accdet year after years of developmet,,, ad v be the premum volume of accdet year where deotes the most recet accdet year. The C,- deotes the curretly ow clams amout of accdet year. Let further S, C, C,- deote the cremetal clams amout (wth C,0 0 ad U the (uow ultmate clams amout of accdet year. The R U C,- s the (uow true clams reserve for accdet year. Let fally S, U C, be the cremetal clams amout after developmet year (tal developmet. Borhuetter-Ferguso (97 troduced ther method to estmate R order to cope wth a maor weaess of the CL method. Therefore we frst cosder ths weaess. CL uses l ratos (age-toage factors fˆ ad a tal factor fˆ order to proect the curret clams amout C,- to ultmate, CL.e., t estmates Û C, fˆ... fˆ fˆ, ad therefore the CL reserve s Rˆ ( fˆ... CL CL Û C, C,. fˆ Ths meas that the reserve strogly depeds o the curret amout C,-, whch ca, for example, lead to a osese reserve Rˆ CL 0 for accdet years where curretly o clams are pad or reported, whch s ot uusual excess-of-loss resurace for the most recet accdet year(s. The reserve estmate avods ths depedecy from the curret clams amout C,-. It s Rˆ ( ẑ Û where Û vq ˆ wth a pror estmate ˆq for the ultmate clams rato q U /v of accdet year, ẑ [0, ] s the estmated percetage of the ultmate clams amout that s expected to be ow after developmet year. The term ˆq s called pror (or tal as opposed to the posteror estmate (C,- Rˆ /v for the ultmate clams rato, whch s based o the pror ˆq ad s dfferet ff C,- ẑ vqˆ,.e., f the curret clams amout devates from ts estmated expectato. The percetages z, z,... costtute the expected cumulatve developmet patter ad ẑ s therefore a estmate for the percetage of the expected outstadg clams of accdet year. Havg already a estmate Û, the questo may arse why does ot smply use Rˆ Û C, as reserve estmate. I that case, the reserve estmate would become the hgher, the smaller the Casualty Actuaral Socety E-Forum, Fall 008 4

4 The Predcto Error of Borhuetter-Ferguso curret amout C,- s ad would aga strogly deped o C,-. Wth CL, the reserve estmate behaves ust the opposte way,.e., s the smaller, the smaller C,- s. Here taes a eutral posto: It does ot care about the sze of C,- at all,.e., t cosders the devato betwee the observed amout C,- ad the expected amout ẑ Û as purely radom ad by o meas dcatve for the future developmet. Altogether, the essetal feature of the method s to avod ay depedecy betwee C,- ad Rˆ. I order to apply the method, the actuary has to estmate the parameters q ad z for all ad. I practce, the ultmate clams ratos q are estmated varous ways, maly based o addtoal prcg ad maret formato such a way that ay expected dffereces betwee the accdet years are reasoably reflected. The z are usually derved from the (selected CL l ratos fˆ,..., fˆ together wth a selected tal factor fˆ the followg way: ẑ ( fˆ fˆ,..., ẑ ( fˆ... fˆ fˆ, ẑ fˆ. The systematc use of the CL l ratos assumes that the outstadg clams part s a drect multple of the already ow part at each pot of the developmet. Ths cotradcts the basc dea of the depedece betwee C,- ad Rˆ,.e., betwee past ad future clams, whch was fudametal for the org of the method. At least, wth the use of the CL patter, the method caot really clam to be a stadaloe reservg method. Moreover, the followg we wll see that the stochastc model suggests a dfferet way to estmate the developmet patter. 3. A STOCHASTIC MODEL UNDERLYING THE METHOD From the reserve formula t s clear that the approprate model for has to be cross-classfed of the type E(C, x z or equvaletly E(S, x y for ad. Because of x y (x a(y /a for ay a > 0, x ad y are oly uque up to a costat factor. Thus we ca wthout loss of geeralty mpose the restrcto y y y. Ths yelds E(U E(S, S, x ad shows that x ca be cosdered to be a measure of volume for accdet year. We therefore wll assume addto that Var(U s proportoal to x or Var(U /x proportoal to /x. Ths s the usual assumpto for the fluece of the volume o the varace. Furthermore, the fudametal property of depedece betwee past ad future clams suggests to assume that all cremets S, of the same accdet year are depedet the depedece of the accdet years themselves beg a stadard assumpto ayway. Note that the depedece wth the accdet years does ot hold the CL model of Mac (993. Casualty Actuaral Socety E-Forum, Fall 008 5

5 The Predcto Error of Borhuetter-Ferguso Thus we wor wth the followg model for the cremets S,,, : ( All cremets S, are depedet. ( There are uow parameters x, y wth E(S, x y ad y y. (3 There are uow proportoalty costats s wth Var(S, x. s From these assumptos, we deduce E(R x (y - y x ( z - wth z : y y, whch shows that the expected clams reserve has the same form as the reserve estmate. Furthermore, we have Var(U Var(S S, ( s... s x, whch shows that Var(U s proportoal to x as teded. Ths model s thought to be the most geeral model fttg to the phlosophy of the method. Le wth the CL model ad as suggested by havg a ow parameter y for the expectato each colum, t here, too, maes sese to assume that the varablty costat s s the same for all S, wth each colum but dffers from colum to colum. The smpler assumpto Var(S, cx y for all, seems to cotradct to realty as has already bee metoed by Taylor (00 because the the coeffcet of varato of the clam sze s versely related to the mea clam sze, whch s opposte of what oe observes. Moreover, ths last varace assumpto s ust a specal case of (3 ad thus less geeral. Fally, ths varace assumpto would mply that all y be > 0, whch s ot the case wth (3, ad whch would prevet usg the model for curred clams amouts where egatve cremetal clams are ot ucommo. Le wth the CL model of Mac (993, ths model s heavly parametrzed, especally for the late developmet years. But, of course, the actuary may depedg o the data apply addtoal regresso assumptos order to reduce the umber of parameters ad to stablze the estmates. Ths s show the umercal example below. From the above model, we deduce further Var(R ( s... s x. As bacgroud for the ext secto, we ote that wth x,, x ow, Casualty Actuaral Socety E-Forum, Fall 008 6

6 The Predcto Error of Borhuetter-Ferguso ŷ S, x, ( s a lear mmum varace ubased estmate of y,, ad ŝ ( S x ŷ, x ( s a ubased estmate of s, PARAMETER ESTIMATION FOR THE MODEL From the model above we clearly see what s meat wth callg Û a pror or tal estmate: It has to be a estmate xˆ for the ucodtoal ( pror, tal expectato x E(U ad ot for the posteror expectato E(U C,-, gve C,-. Ths shows that the clams amout C,- S, S,- ow so far should ot be the ma bass for the estmate xˆ. For example, t ( would be wrog to use for xˆ the posteror estmate ˆ C, R of last year s reservg because ths s a estmate for E(U C,- ad ot for E(U. Eve a very large radom clam that happeed accdet year ad s already ow must ot chage the estmate xˆ as log as t fts the radomess assumed the prcg model. As a extreme example, we mght have a accdet year where xˆ < C,-. Thus, the estmate Û should be pror to mag the ow clams experece C, of accdet year a decsve bass of the estmate. But ths does ot mea that the pror estmate xˆ caot chage durg the clams developmet. To fx deas, let us assume that xˆ orgally stems from prcg (whch has tae place before the ed of developmet year. Usually, the prcg s based o the (treded clams experece of the precedg accdet years (.e., o the years -, -, ad o assumptos o the future clams cost flato. Ths basc formato develops from year to year because the clams experece of the precedg years develops as well as the relevat flato dex. Thus, we ca reprce the busess of accdet year every later year ad thus arrve at updated estmates for x E(U. We may eve clude the clams experece of the accdet years,, to ths reprcg of accdet year as log as t ca be traslated to the portfolo of accdet year. I ay case, the ow clams experece C,- should oly have a margal fluece o xˆ otherwse we would rather estmate E(U C,. Thus, the estmate xˆ may chage over the years but ormally ot to a large extet, at least f the frst estmate for x came from a soud prcg. Whe the actuary does ot have the result of a complete reprcg avalable, he has at least the data {v, C } of the ru-off tragle. O bass of ths data ad some rather geeral formato o rate Casualty Actuaral Socety E-Forum, Fall 008 7

7 The Predcto Error of Borhuetter-Ferguso level chages, he may follow the procedure outled Mac (006 whch s ot a full reprcg but brgs all accdet years o about the same clams rato level as bass for the calculato of the tal ultmate clams rato qˆ. After these clarfyg remars, we assume that the tal estmate Û of Secto fulflls the requremets for beg a estmate of x E(U. Thus we wrte Û stead of xˆ the followg. Havg ow a estmate Û for E(U, we are oly left wth the tas to estmate y ad s. The ma problem here s the fact that we have oly very few observatos for the late developmet years. As we do ot have ay observatos beyod developmet year, we caot estmate the tal rato y wthout further assumptos. A outsde estmate may be gaed from smlar portfolos wth more accdet years where the clams experece of later developmet years tha year s avalable. Wthout such formato, the actuary may arrve at a estmate ŷ by extrapolato from ŷ,...,ŷ (whch are ot avalable yet. Smlarly, a estmate for s caot be obtaed from the oly avalable observato of colum aloe but may be obtaed by extrapolato, too. Therefore, order to fx deas for a teratve procedure, we frst cosder the stuato where we have already reasoable estmates ŷ, ŝ,..., ŝ. The we ca get a weghted least squares estmate (.e., wth the weghts versely proportoal to the varaces for y,, y by mmzg Q ( S Û ŷ, Û ŝ uder the costrat ŷ... ŷ ŷ. As startg values for the mmzato we ca use ŷ~ S, Û, (3 (see ( but these wll usually ot fulfll the costrat. I most cases the data wll ot be so stable that the resultg least squares estmates ŷ,...,ŷ seem relable eough to leave them as they are (especally for large. Therefore, the actuary wll apply a smoothg procedure to select hs ow fal ŷ,...,ŷ,ŷ (.e., cludg a possble revso of the tal rato vew of the other ŷ wth ŷ... ŷ ŷ. O the bass of the fact that the actuary wll ay case mae some ow selectos due to the few data, he ca dspese wth the above exact mmzato ad ust proceed as follows: He starts wth the raw estmates ŷ~,, as gve (3 ad apples some maual smoothg ad extrapolatg order to arrve at hs fal selecto for ŷ,...,ŷ,ŷ fulfllg ŷ... ŷ ŷ. I vew of (, he the estmates s by Casualty Actuaral Socety E-Forum, Fall 008 8

8 The Predcto Error of Borhuetter-Ferguso ŝ~ ( S Û ŷ, Û, -, (4 ad aga apples some smoothg order to select hs fal ŝ,...,ŝ ad a extrapolato to obta ŝ. Note that ŝ caot be obtaed ths way because t usually has to cover several developmet years as s the case for ŷ, too. Therefore, ŝ may be arrved at by terpolatg a regresso of ŝ agast ŷ at the pot ŷ. (Note that some ŷ may be egatve. The whole estmato procedure s show the umercal example. A more formal way to estmate the parameters y, s ( case of rather stable data would be as follows: O the bass of ŷ~,, accordg to (3, we decde o the formula for a smoothg regresso, e.g., l( ŷ β α for above some < (assumg y > 0 there, whch the s extrapolated utl some fal developmet year >. The we calculate ŝ~ (accordg to (4 but usg the smootheed ŷ for >. The resultg values ŝ~,...,ŝ ~ are ow ept fxed ad used the above costraed mmzato of Q to obta better values for ŷ,...,ŷ,α, β uder the costrat ( β... exp( α ŷ α β.... ŷ exp ( Note that Q we have to leave out the term for (, (, because ow we do ot yet have a value for ŝ. Ths mmzato yelds our selectos for all ŷ : The values for,, are obtaed drectly, those for,, are tae from the smoothg regresso ad ŷ s obtaed by addg up the extrapolated values of the regresso up to developmet year. Usg these ŷ, we calculate ew values ŝ~ accordg to (4 ad plot l ( ŝ~ for > agast ŷ or l ( ŷ order to select approprate values for ŝ, especally for (over ŷ ad (over ŷ. Of course, we could ow apply aother costrat mmzato wth these ew values of ŝ, but usually ths wll ot chage much. Note that the values of ŝ for > wll be overestmated a lttle as we dd ot chage the degrees of freedom formula (4 for ŝ~ whch would have bee possble as the regresso employs fewer parameters. As the result of each of these two estmato procedures we have selected ŝ,...,ŝ,ŝ from whch we estmate the clams reserve by Rˆ ( ŷ... ŷ Û ( ẑ Û wth ẑ ŷ... ŷ. ŷ,...,ŷ,ŷ ad ŝ,...,ŝ,ŝ wll be eeded for the predcto error. The propertes of the above estmators ca be setched as follows: Casualty Actuaral Socety E-Forum, Fall 008 9

9 The Predcto Error of Borhuetter-Ferguso (a ŷ,...,ŷ,ŷ are parwse (slghtly egatvely correlated as they have to add up to uty. (b ŷ,...,ŷ,ŷ ad therefore also ẑ,...,ẑ are practcally depedet from Û,...,Û as the latter do ot really fluece the sze of ay y ˆ because these have to add up to uty ay case ad because of selectos ad regressos used. (c Rˆ ad R are depedet (due to. (d E( Û E( U x,. (e E( ŷ y,, ad therefore E( ẑ z,. (f E( ŝ s,. I (d (f we have smply assumed that the actuary s selectos are ubased. The ubasedess of the reserve estmate Rˆ follows drectly from these propertes: ( Rˆ E( Û E( ẑ x ( z E( R E. Note that the raw estmates ŷ~ accordg to (3 are detcal to the estmates ˆβ Mac (006 whch were show there as beg suggested drectly by the reserve formula tself. I ay case ad eve wthout ay smoothg of ŷ~, the resultg developmet patter wll tur out to be dfferet from the CL patter (see also the umercal example below. Now we are prepared to derve the formula for the predcto error. 5. THE PREDICTION ERROR OF THE METHOD As oe s terested the future varablty oly, gve the data observed so far, the mea squared error of predcto of ay reserve estmate Rˆ s defed to be ( Rˆ E ( Rˆ R S,..., S msep,,. Accordg to (, R S,- S, s depedet from S,,, S,-. Also, the reserve estmate Rˆ ca be tae as beg depedet from S,,, S,- (as these play at most a margal role whe selectg Û ad ŷ,..., ŷ, more precsely, R ad Rˆ are tae to be commoly depedet from S,,, S,-. Thus we have msep ( ( Rˆ E ( Rˆ R Casualty Actuaral Socety E-Forum, Fall

10 The Predcto Error of Borhuetter-Ferguso Var ( Rˆ ( ( ( R E Rˆ E R ( Rˆ Var( R Var,.e., the mea squared error of predcto s the sum of the (squared estmato error ( of the (squared process error Var ( R. For the process error we smply have ( R Var( S... Var( S x ( s... s Var,,, Var Rˆ ad whch wll be estmated by ( R Û ( ŝ... ŝ Vˆ ar. For the estmato error of Rˆ Û ( ẑ, we use the geeral formula ( XY ( E( X Var( Y Var( X Var( Y Var( X ( E( Y Var for depedet radom varables X ad Y ad obta Var ( ˆ R ( E( Uˆ Var( zˆ Var( Uˆ Var( zˆ Var( Uˆ ( E( zˆ ( x Var( Uˆ Var( zˆ Var( Uˆ ( z. Whereas we have already estmators Û for x ad ẑ for z -, we stll eed estmates for Var ad Var( ẑ,.e., we have to quatfy the precso of Û ad ẑ. ( Û The stadard error s.e.û (,.e., a estmate for Var ( Û ( error.e.( Rˆ C,-, whch has to be cluded to s.e.û (. Le Û tself,.e.û (, caot be obtaed from the estmato s of last year s reservg because ths would gore the varablty of s s best be obtaed from a reprcg of the busess. But oe has to be cautous there. For example, the varablty of the post post posteror clams rato estmates Û v,...,û v would uderestmate s.e.û ( v because these estmates are postvely correlated va the commo estmates ẑ. Smlarly, also the tal estmates Û,, Û wll usually be postvely correlated. Thus the formula v ( s.e.û ( Û v v qˆ wth qˆ Û v (5 Casualty Actuaral Socety E-Forum, Fall 008 3

11 The Predcto Error of Borhuetter-Ferguso (whch s aalogous to (, ( for 3 s applcable oly f the tal estmates Û ca be assumed to be ucorrelated. But eve the, usg the real premums v would clude the maret cycle of premum adequacy to (.e.û s.e.û those stuatos where we ca predct the maret cycle rather well. Thus, we should remove the fluece of the maret cycle from s, whch would overestmate ( (5 by usg o-level premums v~. I addto, we should correct for ay postve correlato betwee the Û s by replacg the term of (5 wth for example, for a costat correlato coeffcet ρ ˆ U betwee Û ad Û or wth (approxmately for a decreasg correlato coeffcet ρˆ U ( wth v v. ; the precse formula beg ρ Usually, these stadard errors s.e.( Û wll ot chage much over the years. Of course, we wll have slght chages as log as the Û chage. But eve at the ed of the developmet, we wll ot ow E(U much more precsely tha at the begg. The actuary should exame the plausblty of the resultg values of s.e.( Û, for stace the followg way: If we assume a ormal dstrbuto, the the terval ( Û s.e. ( Û,Û s.e. ( Û wll cota the true E(U wth 95% probablty. Thus, f the sze of the terval s plausble, the s.e.( Û s plausble, too. Next, we have to decde o how to estmate Var ( ( ẑ Var( ŷ... ŷ Var( ŷ... ŷ ẑ Var. From property (a we see that we wll be o the safe sde whe we replace Var( ŷ... ŷ Var( ŷ... Var( ŷ the case wth ŷ... ŷ as fally Var( ŷ... ŷ ( ẑ Var( ẑ for small wth Var ( ŷ... Var( ŷ Var( ŷ... Var( ŷ. More precsely, we replace stll beg o the safe sde Var ( wth ( Var( ŷ... Var( ŷ, Var( ŷ... Var( ŷ., U v v v v wth. But whereas the latter sum creases wth each addtoal term, ths s ot Var( 0. Therefore we replace ẑ ad for large wth m Due to ŷ ŷ~ S, x, we ca assume that s Var( ŷ Var S, x x,. Casualty Actuaral Socety E-Forum, Fall 008 3

12 The Predcto Error of Borhuetter-Ferguso Var by ŷ Therefore we estmate ( (.e.( ŷ ŝ s Û,. (6 But the value of s.e. ( ŷ s.e. ( ŷ 0. 5ŷ,.e., a coeffcet of varato c.v. ( wth 95% probablty wth the terval (0; y. must come from outsde. Wthout ths, a plausble choce s ofte ˆ ŷ 50%, assumg a ormal dstrbuto Altogether, our estmate ( ( ẑ s.e. for Var ( ẑ s ( (.e. ẑ m ( s.e. ( ŷ... ( s.e. ( ŷ, ( s.e. ( ŷ... ( s.e. ( ŷ ( s. (7 s.e. ẑ s.e. I ay case, we have ( ( 0.e.( ẑ. Of course, the actuary wll chec the plausblty of s smlarly as s.e.( Û ad, f ecessary, maually adust some of the resultg values. Thus we fally obta the followg estmator for the mea squared error of predcto: mˆ sep ( s.e. ( ẑ ( Rˆ Û ( ŝ... ŝ Û ( s.e.û ( ( ( s.e.û ( ( ẑ. Ths s the formula oe eeds for rs-based captal ad premum loadg calculatos as well as for the costructo of a cofdece terval for R. I order to chec the sgfcace of dffereces betwee alteratve reserve estmates or to costruct a cofdece terval for E(U oe oly eeds the pure estmato error ( s.e. ( ẑ ( s.e. ( Rˆ Û ( s.e.û ( ( ( s.e.û ( ( ẑ. A closer aalyss of ths formula shows that s.e. ( Rˆ Û s.e. ( ẑ for ( Rˆ Û s.e.û ( Û s.e. for ẑ close to, ẑ close to 0,.e., for the very gree accdet years, the ucertaty of the tal ultmate clams estmate s drectly trasferred to the reserve estmate. For the overall reserve R R R, we have the ubased estmate Rˆ Rˆ... Rˆ. Its mea squared error of predcto s msep( Rˆ Var( Rˆ Var(R. For the process error we have Casualty Actuaral Socety E-Forum, Fall

13 The Predcto Error of Borhuetter-Ferguso Var(R Var(R Var(R due to the depedece of the accdet years ( ad thus get the estmate ( ŝ... ŝ Û Vˆar( R. The estmato error Var( Rˆ s more volved because Rˆ,..., Rˆ are postvely correlated va the commo parameter estmates ŷ (ad addto va the Û s. We have Var ( Rˆ Var( Rˆ Cov( Rˆ,Rˆ < For Cov( Rˆ,Rˆ Cov( Û ẑ,û ( ẑ. ( we use the geeral formula Cov(XY, WZ Cov(X, W E(Y E(Z Cov(X, W Cov(Y, Z E(X E(W Cov(Y, Z for radom varables X, Y, W, Z where the sets {X, W} ad {Y, Z} are depedet. We omt the term the mddle, whch s of lower order, ad obta Cov ( Û ẑ,û ( ẑ ( ρ ( Û Var Û E ẑ E ẑ ρ Var ẑ Var ẑ E Û E Û U z ( ( ( ( ( ( ( Var wth the correlato coeffcets z U (,Û Var Û Var( Û ρ Cov Û, ( ( ẑ, ẑ Var( ẑ Var( ẑ ρ Cov. Thus, we oly have to estmate these correlato coeffcets as we have estmates for all the other terms. If the actuary does ot has the possblty to obta data-based estmates for ρ U (e.g., from z U reprcg ad ρ, he may smply use oe of the two estmates ˆρ as gve above (after (5 ad ρˆ z ẑ ẑ ( ẑ ( ẑ for < ad ẑ... ẑ. The latter estmate stems from assumg a Drchlet dstrbuto (whch s a geeralzato of the Beta dstrbuto for ŷ,...,ŷ. Thus we fally get Casualty Actuaral Socety E-Forum, Fall

14 The Predcto Error of Borhuetter-Ferguso ( s.e. ( Rˆ ( s.e. ( Rˆ Ĉov( Rˆ,Rˆ < wth Ĉov U z ( Rˆ,Rˆ ρˆ s.e. ( Û s.e. ( Û ( ẑ ( ẑ ˆ ρ s.e. ( ẑ s.e. ( ẑ Û Û. 6. NUMERICAL EXAMPLE The pad tragle of Exhbt A of Mac (006, see also Table 0 below, wth 3 s used as example ad we eep the tal ultmate clams estmates Û from there (Exhbt C, colum (I, see Table below, secod colum. I a frst approach, we also eep the developmet patter ẑ ( b of Exhbt C, row (9, of Mac (006, see the row selected z the frst bloc of Table below. Ths patter ca also be obtaed except for roudg dffereces from the raw estmates ŷ~ accordg to (3 by maually smoothg wth the selectos ŷ 8 8%, ŷ 9 5%, ŷ 0 3.7%, ŷ.%, ŷ.5%, ŷ 3.4% ad a tal rato ŷ 4 3.5%, see the secod ad thrd row of Table below. I Mac (006, ths tal rato was based o the calculato for the curred data. From the patter ad the tal Û the reserve estmates Rˆ Û ( ŷ... ŷ Û ( ẑ are calculated. These reserves, see the fourth colum of Table, are thus the same as Mac (006 except for roudg dffereces. ŝ For the predcto error, we frst select. For ths purpose, we calculate the raw ŝ~ accordg to (4 ad plot l ( ŝ~ agast ŷ for the decreasg part 4. We see that the plot loos reasoably smooth. Crucal cases are always ŝ~ ad ŝ~, whch rely o very few data. Here (3, accordg to the plot, ŝ~.8 ad ŝ~ 9.5 seem to be rather small. Thus, we adust these to ŝ ŝ 30, ŝ ŝ 5, leave ŝ~, 0, as they are,.e., ŝ ~ ŝ, ad maually select from the plot the mssg values ŝ 3 0 (over ŷ 3.4% ad ŝ 4 35 (over ŷ 4 3.5%. Wth these selectos for ŝ, we calculate s.e.( ŷ for 3 accordg to (6 ad fd the resultg values ad ther coeffcets of varato plausble. The, we have to quatfy our ucertaty o ŷ 4 3.5% ad select t to be s.e. ( ŷ4.5% assumg a 95%-rage from 0.5% up to 6.5%. Ths fts well to the s.e. of ŷ 0, whch s close to s.e. ẑ accordg to (7. All estmates ad selectos are show the frst bloc of Table, where a bold umber dcates a pure selecto or a chage from the raw estmate. ŷ 4. Now we calculate ( Fally, we have to select s.e.( Û. I ths example, we have a extreme premum cycle: The ultmate clams ratos Û /v frst decrease to 63%, the crease to 77%, the decrease aga to 69% (see Mac (006. Thus, a applcato of equato (5 does ot mae sese. I Mac (006, o-level Casualty Actuaral Socety E-Forum, Fall

15 The Predcto Error of Borhuetter-Ferguso premum factors r were estmated whch brg all accdet years o about the same clams rato Casualty Actuaral Socety E-Forum, Fall

16 The Predcto Error of Borhuetter-Ferguso level. The, the pror Û were chose to be ( ŷ~... Û ~ mˆ ŷ vr wth ŷ~ accordg to (3 ad a certa costat factor mˆ. We ca assume that the varablty of r mˆ s small compared to the oe of ŷ~... ŷ~. The we have Var ( Û ( v r mˆ Var( ŷ~... ŷ~ ( v r mˆ ( Var ŷ~... Var( ( ~ ŷ because the ŷ~ s are fully depedet due to as they do ot have to add up to uty. As the dervato of (6, we have Var ( ŷ~ s Û ( ŷ~ Û.,.e., we tae ( s.e. ( s.e. ( ŷ ŝ Fally, order to get rd of the factor c.v.û v r mˆ, we cosder the coeffcet of varato ad obta s.e.û ( ( ( s.e. ( ŷ~... ( s.e. ( ŷ~ Û ŷ~... ŷ~ 6. 7%. As we have gored the varablty of r mˆ ad have elmated the full premum cycle (whch probably would ot have bee acheved a pror, we delberately crease ths c.v. to c.v.û ( 0% for all accdet years. Ths s cosdered to be a rather hgh ucertaty for a estmate of E(U for classcal surace busess because, e.g., for Û /v 90%, ths correspods to a wde 95% cofdece rage of (7%; 08% ote that ths s the rage for E(U ad ot for U! Note further that ths approach oly wors for pror estmates Û that were obtaed ths specfc way. It caot be appled to estmates Û obtaed dfferetly, e.g., va reprcg, because each approach to Û has ts ow ucertates. Normally, c.v.(û wll ot be the same for all accdet years but wll be lower for years wth hgher volume. I our example, we leave c.v.(û 0% costat (see the thrd colum of Table assumg the varyg volume has essetally bee caused by wrtg varyg shares of the same treates. Wth these selectos, we obta the error estmates show the bloc Borh/Ferg of Table. We also may apply the alteratve estmato procedure descrbed Secto 4: The, we do ot use the patter of Mac (006 but start wth the orgal raw ŷ~ accordg to (3 (see secod row of Table ad select as last paymet year 0. Loog at the plot of l ( ŷ~ agast, we select 3 ad tae a tal smoothg regresso l( ŷ α β wth α ad β Casualty Actuaral Socety E-Forum, Fall

17 The Predcto Error of Borhuetter-Ferguso for >. Wth the resultg tal values for ŷ, tal values for ŝ~,, ŝ~ are calculated accordg to (4, whch the are ept fxed durg the followg mmzato of Q (wthout the term for ad 3. The mmum s obtaed at ŷ 0.65%, ŷ 4.7%, ŷ 3 3.0%, α , ad β 0.90, whch leads to ŷ 4 3.9% by addg up the extrapolated values for ŷ from 4 to 0. For the other ŷ (from the ew regresso ad the resultg ẑ see the bloc Alteratve Estmates of Table. The the correspodg ew ŝ~ are calculated accordg to (4 ad the resultg values l ( ŝ~ are plotted agast ŷ for >. I vew of ths plot, we chage ŝ~ 8.7 to ŝ 5 ad select ŝ 3 3 ad ŝ Fally, we calculate s.e.( ŷ accordg to (6 ad select c.v. ( ŷ4 50% whch gves s.e. ( ŷ4.93. The resultg reserves Rˆ, see Table, bloc Borh/Ferg, are slghtly hgher tha Rˆ for the old years ad slghtly lower for the ew oes. The amouts (ot the percetages of the predcto error (usg c.v.(û 0% as before are all a lttle bt hgher. Usg ˆ U ρ (, the overall reserve s Rˆ 875,497 wth a predcto error of 7,940 cosstg of a estmato error of 6,770 ad a process error of 37,5. As comparso we apply the Cha Ladder method, too. All parameters used are gve the last bloc of Table. We have replaced the last four raw age-to-age factors wth.04,.03,.0,.05, ad selected a tal factor of.04. The latter s accordace wth the tal rato of 3.5% - 3.9% used above. From the age-to-age factors we ca derve the correspodg cumulatve developmet patter ẑ as descrbed Secto. The resultg values show Table are close to the z- estmates of the two approaches but ot detcal. The mplemetato of the tal factor to the formulae for the predcto error has bee doe accordg to Mac (999. The raw sgmaparameters (see Mac (993 or Mac (999 have bee ept ad were supplemeted wth ˆσ 8 ad ˆσ 40 o bass of a plot of l( ˆ σ agast l ( fˆ. Fally, for the tal factor, s.e.( fˆ 0.0 was assumed,.e., a 95%-rage from.00 to.08. Ths yelds the results show the last bloc of Table. The CL reserves are close to the oes of except for the most recet years 003 ad 004: I 003, the CL reserve s about half of the reserve, whereas 004 the CL reserve s more tha twce the reserve. Ths hgher volatlty s reflected the maredly hgher predcto errors for 999, caused by a much hgher process error. The CL ad reserve estmates for are ot sgfcatly dfferet (.e., ot dfferet by more tha s.e. ( Rˆ. But the reserves for 003 are udged as beg dfferet by ether method; the 004 reserves are oly dfferet from the vewpot whereas the CL estmato error s so large that the reserve s ot udged to be dfferet although t s less tha 50% of the CL reserve. Ths s a good example for the fact that CL ofte caot be reasoably appled the stadard way for ew accdet years Excess busess where almost othg s pad the frst developmet year(s. Casualty Actuaral Socety E-Forum, Fall

18 The Predcto Error of Borhuetter-Ferguso CONCLUSION O the bass of the reserve formula, ths paper has developed a stochastc model for the method that corporates the fudametal property of the depedece betwee past ad future clams amouts (see model assumpto. Model assumpto s a drect cosequece of the reserve formula too. Oly assumpto 3 s ot forced by the method tself but ths assumpto s rather geeral ad s oly eeded to derve the formula for the predcto error. Already from assumptos ad mportat cosequeces for a soud applcato of the method ca be draw. Oe s the fact that the approprate developmet patter should ot be derved from the CL age-to-age factors but be calculated depedetly o bass of formula (3. Ths maes a fully stadaloe reservg method. Moreover, the stochastc model gves mportat advce o how to arrve ad how ot to arrve at the tal estmate for the ultmate clams amout. For example, t shows that a procedure that s ofte used automated reservg systems s rather questoable: It s the use of last year s posteror estmate as tal estmate for ths year s reservg. O the other had, the model shows that the tal estmate for a dvdual accdet year may chage over tme as the formato that has led to the estmate develops. The depedece assumpto may seem more restrctve tha the correspodg assumpto of the CL model of Mac (993. The requred depedece betwee the cremetal amouts wth every accdet year may be volated, e.g., by chages the reservg process or the reportg behavor. I the CL model, ths depedece s ot requred, but a smlar requremet ca be deduced from the CL model: It s the fact that the dvdual developmet factors C, /C must be ucorrelated wth every accdet year. Ths eeds ot be fulflled the model but ca be volated by the same chages as metoed before. As a cosequece, we obta a way of how to decde whch model better suts the data by checg these depedece/ucorrelatedess propertes. Here we see the ma advatage of havg a model: It gves some gudace o how to estmate the parameters ad allows varous procedures (e.g., tests, plots to see whch model better suts the data. Ad, last but ot least, t gves the possblty to quatfy the reserve varablty. Especally for the model, the gudace metoed leaves eough room for the actuary to brg hs specfc owledge of the busess as t was always the case wth the method. He has to select the parameters (as before ad, addto, must assess hs ucertaty about hs selectos. The gudace gve by the model maes ths crucal tas feasble. Ad as a reward, the actuary usually wll obta less volatle reserve results tha wth CL, especally for the most recet accdet years (see the example above. Ths s a bg advatage regardg rs modelg ad premum loadg calculatos. Altogether, ths paper gves a stochastc foudato equvalet to the oe already avalable for CL. Casualty Actuaral Socety E-Forum, Fall

19 The Predcto Error of Borhuetter-Ferguso Acowledgemet I am grateful to Dave Clar, Alos Gsler, ad Gerhard Quarg for helpful dscussos the cocepto phase of ths paper ad to the CAS revewers Joh Gbso ad Mar Weger for ther commets that helped to mprove the text. REFERENCES [] Borhuetter, R.L., ad R.E. Ferguso, The Actuary ad IBNR, Proceedgs of the Casualty Actuaral Socety 97, 59(:8-95, [] Mac, T., Dstrbuto-free Calculato of the Stadard Error of Cha Ladder Reserve Estmates, ASTIN Bullet 993, 3(:3-5, [3] Mac, T., Measurg the Varablty of Cha Ladder Reserve Estmates, Casualty Actuaral Socety Forum, Sprg 994:0-8, [4] Mac, T., The Stadard Error of Cha Ladder Reserve Estmates: Recursve Calculato ad Icluso of a Tal Factor, ASTIN Bullet 999, 9(:36-366, [5] Mac, T., Parameter Estmato for Borhuetter-Ferguso. Casualty Actuaral Socety Forum, Fall 006:4-57, [6] Taylor, G.C., Wrtte dscusso of Stochastc Clams Reservg Geeral Isurace by P.D. Eglad ad R.J. Verrall, Brtsh Actuaral Joural 00, 8(3: Casualty Actuaral Socety E-Forum, Fall

20 Table 0, Part A. Cumulatve Pad Loss Amouts Acc.Year Premum Dev.Yr Table 0, Part B. Icremetal Pad Loss Amouts Acc.Year Premum Dev.Yr

21 Table. Parameter Estmates devel. year raw y 0.6% 4.4%.8% 9.0% 5.0% 0.6%.4% 6.5% 4.0%.%.5%.9% 0.0% selected y 0.6% 4.5%.7% 9.0% 5.0% 0.6%.4% 8.0% 5.0% 3.7%.%.5%.4% 3.5% selected z 0.6% 5.% 7.8% 36.8% 5.8% 6.4% 74.8% 8.8% 87.8% 9.5% 93.6% 95.% 96.5% 00.0% raw s² selected s² selected s.e.(y 0.7% 0.79% 0.75%.70%.39%.4%.50%.54%.30%.6%.63%.86%.49%.50% s.e.(y accumulated --> 0.7% 0.83%.%.03%.46%.76% 3.7% 4.03% 4.3% 4.53% 4.8% 5.6% 5.73% 5.9% s.e.(y accumulated <-- 5.9% 5.86% 5.8% 5.56% 5.38% 5.4% 4.60% 4.34% 4.4% 3.8% 3.45%.9%.50% 0.00% s.e.(z 0.7% 0.83%.%.03%.46%.76% 3.7% 4.03% 4.4% 3.8% 3.45%.9%.50% 0.00% Alteratve Estmates selected y 0.7% 4.7% 3.0% 0.8% 5.6%.6% 8.7% 6.5% 4.8% 3.6%.7%.0%.5% 3.9% selected z 0.7% 5.3% 8.3% 39.% 54.7% 66.3% 75.0% 8.5% 86.3% 89.9% 9.6% 94.6% 96.% 00.0% selected s² selected s.e.(y 0.7% 0.79% 0.75%.80%.4%.30%.9%.40%.7%.56%.57%.86%.67%.93% s.e.(z 0.7% 0.83%.%.%.55%.86% 4.09% 4.3% 4.38% 4.09% 3.78% 3.9%.93% 0.00% Cha Ladder Estmates ATA factor f cumulatve patter z 0.7% 5.3% 8.% 37.5% 5.4% 63.% 76.0% 8.6% 86.7% 90.% 9.9% 94.7% 96.% 00.0% sgma² s.e.(f Table. Reserves ad Errors Accdet Ital Ult. Coeff. of Var. Borh/Ferg Coeff. of Var. Borh/Ferg Cha Ladder Year Clams U c.v.(u Reserve R Pred. Error Pred. Error Estm. Error Proc.Error Reserve R Pred. Error Estm. Error Proc.Error Reserve R Pred. Error Estm. Error Proc.Error % % % % % % % % % % % % % % % % % % % % % % % % % %