A Bayesian Log-normal Model for Multivariate Loss Reserving

Transcription

1 A Bayesan Log-normal Model for Multvarate Loss Reservng November 1, 2011 AUTHOR INFORMATION: Peng Sh, PhD, ASA Dvson of Statstcs Northern Illnos Unversty DeKalb, Illnos USA e-mal: Sanjb Basu, PhD Dvson of Statstcs Northern Illnos Unversty DeKalb, Illnos USA e-mal: Glenn G. Meyers, PhD, FCAS ISO Innovatve Analytcs e-mal: 1

2 Abstract The correlaton among multple lnes of busness plays a crtcal role n aggregatng clams and thus determnng loss reserves for an nsurance portfolo. To accommodate correlaton, most multvarate loss reservng methods focus on the par-wse assocaton between correspondng cells n multple run-off trangles. However, such practce usually reles on the ndependence assumpton across accdent years and gnores the calendar year effects that could affect all open clams smultaneously and nduce dependences among loss trangles. To address above ssue, we study a Bayesan log-normal model n the predcton of outstandng clams for dependent lnes of busness. In addton to the par-wse correlaton, our method allows for an explct examnaton of the correlaton due to common calendar year effects. Further more, dfferent specfcatons of the calendar year trend are consdered to reflect reservng actuares pror knowledge on clam development. In a case study, we analyze an nsurance portfolo of personal and commercal auto lnes from a major US property-casualty nsurer. It s shown that the ncorporaton of calendar year effects mproves model ft sgnfcantly, though t contrbutes substantvely to the predctve varablty. The avalablty of the realzatons of predcted clams permts us to perform a retrospectve test, whch suggests the extra predcton uncertanty s ndspensable n modern rsk management practces. Keywords: Dependent run-off trangles, Calendar year effects, Lognormal model, Bayesan analyss 2

3 1 Introducton In the classc reservng problem for property-casualty nsurers, the prmary goal s to set an adequate reserve to fund losses that have been ncurred but not yet developed. The losses from each sngle lne of busness wrtten by an nsurer are typcally arranged n form of run-off trangles, reflectng the oblgatons ncurred and from year to year. Based on the losses already developed, reservng actuares quantfy the unpad losses and assocated uncertanty for each lne of busness separately, and then aggregate the reserves from multple lnes to determne the company-level reserve. We refer to Taylor (2000), England and Verrall (2002), and Wüthrch and Merz (2008) for excellent revews on loss reservng method for a sngle lne of busness. A recent development n loss reservng lterature s the advances n the multvarate stochastc reservng method, where dependences among multple lnes of busness are ncorporated nto the aggregaton process. The man advantages of examnng multple loss trangles jontly nclude: frst, the dversfcaton effect that arses due to the assocaton among dfferent lnes has a crtcal mplcaton for the loss reserve ndcaton; second, reserve actuares may mprove the accuracy n the predcton of losses for one lne of busness by borrowng nformaton from other correlated lnes. From the perspectve of good rsk management practce, quantfyng the predctve varablty s of more nterest to actuares than forecastng outstandng clams tself. The challenge n multvarate loss reservng s to accommodate the dependence among multple lnes n the determnaton of a suffcent reserve. Ths process could be even more complcated because of the varous sources of dependency. As ponted out by Holmberg (1994), correlatons may appear among losses as they develop over tme or among losses n dfferent accdent years. Other authors have focussed on correlatons over calendar years, thnkng of nflatonary trends as a common unknown factor nducng correlaton. In the multvarate loss reservng lterature, two strands of research have been devoted to ncorporatng dependences among multple lnes nto the measurement of loss reserve varablty. One group has been focusng on a dstrbuton-free setup, where the (condtonal) mean squared predcton error can be derved to measure predcton uncertanty. Studes along ths lne can be found on the multvarate chan-ladder method, such as Braun (2004), Schmdt (2006), and Merz and Wüthrch (2008), and the multvarate addtve loss reservng method, ncludng Hess et al. (2006) and Merz and Wüthrch (2009b) among others. More recently, Merz and Wüthrch (2009a) combned the chan-ladder and addtve loss reservng methods nto one ntegrated framework to account for the heterogenety among multple trangles. Zhang (2010) presented another verson of the multvarate chan-ladder model through the seemngly unrelated regresson technque. The above approaches, decomposng the predcton error nto process error and estmaton error, have demonstrated desrable propertes n the loss reserve ndcaton. The other strand of studes looks more to parametrc methods based on dstrbutonal famles. One advantage of parametrc models s that one could derve a predctve dstrbuton of unpad losses, whch s beleved to be more nformatve to actuares n settng a reasonable reserve range 3

4 than a sngle mean squared predcton error. Varous approaches have been proposed to ncorporate the dependency wthn a parametrc setup. One nvolves modern bootstrappng. For example, Krschner et al. (2002) consdered a synchronzed bootstrap where the correlaton among lnes s reserved through par-wsely resamplng the standardzed resduals from over-dspersed posson models. Taylor and McGure (2007) examned and extended the smlar bootstrap technque n a generalzed lnear model framework. Other works employed emergng copula models to assocate the losses from multple trangles. Both Brehm (2002) and de Jong (2010) consdered the Gaussan copula n the generaton of the jont dstrbuton of losses from dfferent lnes of busness and focused on the correlaton n a model based on normal dstrbutons. In a recent study, Sh and Frees (2010) consdered a copula regresson model that allows one to use a wde range of parametrc famles for the loss dstrbuton and provded a smple alternatve way to vew the dependence among multple loss trangles. Both groups of methodologes (parametrc and non-parametrc) successfully ncorporate the assocaton among multple trangles nto the loss reservng process. However, exstng studes have focused on the par-wse assocaton,.e., each cell n one trangle relates to the equvalent cell n another trangle. As emphaszed by Brehm (2002) and de Jong (2010), most dependences among loss trangles could be arguably drven by certan calendar year effects, such as a court judgment or management decson, that could affect all open clams n an unpad loss portfolo. The dependency among multple trangles due to common calendar year effects s only rarely studed n the loss reservng lterature, though a few researchers have examned the accountng year effects for a sngle lne of busness, for example, see Barnett and Zehnwrth (2000), de Jong (2006), and Wüthrch (2010). We examne calendar year effects n a multvarate loss reservng context through a log-normal model. Specfcally, we use random effects to accommodate the correlaton due to accountng year effects wthn and across run-off trangles. Ths specfcaton s n lne wth de Jong (2010), where the calendar year effects was ntroduced through the correlaton matrx. Note that the calendar year effect wll relax ndependence assumpton among accdent years. Other studes try to ntroduce correlaton across accdent years va Bayesan herarchcal models, for example, see Guszcza (2008) and Zhang et al. (2012). Due to the small sample sze typcally encountered n loss reservng problems, parametrc dstrbutonal methods are subject to the potental model overfttng. One could refer to parametrc bootstrappng to ncorporate parameter uncertanty nto statstcal nference (see, for example, Taylor and McGure (2007) and Sh and Frees (2010)). As an alternatve parametrc approach, we adopt a Bayesan perspectve on the log-normal specfcaton. In a Bayesan framework, one use the data to update pror assumpton and compute the posteror dstrbuton, then the posteror combned wth the samplng dstrbuton s used to compute the predctve dstrbuton of new clams. The Bayesan approach s, n partcular, valuable to reservng actuares, because t offers a natural way to ncorporate pror nformaton and provdes decson maker a predctve dstrbuton of clams n each cell of the lower run-off trangle. Bayesan methods are not new to loss reservng lterature. Earlest efforts are found n Jewell 4

5 (1989; 1990), Verrall (1990), and Haastrup and Aras (1996). Wth the development of Markov chan Monte Carlo (MCMC) method, Bayesan methods have found more applcatons when studyng loss reserves for ndvdual lnes of busness. Some recent work nclude de Alba (2002; 2006), Ntzoufras and Dellaportas (2002), Verrall (2004), England and Verrall (2006), de Alba and Neto-Barajas (2008), Peters et al. (2009) and Meyers (2009). However, the lterature on Bayesan multvarate loss reservng methods s sparse. Merz and Wüthrch (2010) s one example. We employ a Bayesan log-normal model to examne the dependences among multple run-off trangles. Our focus s the common calendar year effects that affect all open clams smultaneously. The correlaton ntroduced by calendar year effects as well as the correlaton between equvalent cells wll be dentfed and examned separately. We also consder varous specfcatons of the calendar year effects to reflect actuares pror nformaton. The artcle s structured as follows: Secton 2 ntroduces the Bayesan log-normal model for dependent loss reservng. Dfferent specfcatons on calendar year effects are consdered and Bayesan model selecton methods are dscussed. A case study s performed n Secton 3, whch shows the sgnfcant calendar year trend as well as ts effects on loss reserves and rsk margns. Based on realzed future payments, Secton 4 assesses dfferent log-normal models and compares ther predctons wth exstng approaches. Secton 5 concludes the paper. 2 Modelng 2.1 Notatons In the followng we assume that a run-off portfolo of a property-casualty nsurer conssts of L run-off trangles of the same sze, each of whch corresponds to a sngle lne of busness or a homogeneous subportfolo. Consder a trangle wth I accdent years and J( I) development lags. Let (= 1,, I) and j(= 1,, J) ndcate the th accdent year and jth development lag, ( respectvely, and t = +j ndcate the correspondng calendar year. Defne Z = Z (1) ),, Z(L) to be the vector of ncremental pad losses at the th accdent year and the jth development lag. Then by calendar year I + 1, all clams n the upper trangle wll be realzed. Let Z R and Z P denote the set of varables n the upper and lower trangles, respectvely,.e., Z R = {Z : + j I + 1} and Z P = {Z : + j > I + 1}. We further assume that all clams wll be closed wthn J years, that s, Z (l) = 0 for l = 1,, L and j > J. Under these assumptons, our goal s to forecast the outstandng payment n the lower trangle Z P, based on the observed payment n the upper trangle Z R. Instead of drectly modelng losses, current lterature looks more to normalzed ncremental losses where losses are normalzed by dvdng by an exposure varable. The exposure, measurng the volume of the busness, can be the number of polces or the amount of premums wrtten. Consderng the exposure vector ( ω = ω (1) ),, ω(l) for accdent year and development year j, we defne the vector normalzed 5

6 ( ncremental losses as Y = Z /ω = Z (1) /ω(1),, Z(L) /ω (L) ). Smlarly, Y R and Y P are used to denote the normalzed ncremental losses that have been realzed n the upper trangle and the normalzed ncremental losses that are to be predcted n the lower trangle, respectvely. In a Bayesan framework, predctons are provded by means of a predctve dstrbuton: f(y P y R ) = f(y P θ)f(θ y R )dθ, where θ denotes the set of parameters, f(y P θ) s specfed by a samplng dstrbuton, and f(θ y R ) represents the posteror dstrbuton whch s determned by the lkelhood and a pror densty. The MCMC methods are commonly used to derve approxmatons for the predctve dstrbuton, especally when the dstrbuton s not a known type or does not have a closed form (see, for example, Robert and Casella (2004)). Once we have predctve dstrbutons for normalzed ncremental losses, the predcton for reserves are straghtforward to derve. 2.2 Bayesan Log-normal Model Because of the small sample sze n run-off trangles, we focus on parametrc methods based on dstrbutonal famles. Among parametrc dstrbutonal famles, log-normal s one that have been extensvely studed for ncremental clams n loss reservng lterature. The log-normal model, frst employed by Kremer (1982) for clams reservng, examnes the logarthm of ncremental losses and uses a multplcatve structure for the mean. Another approach based on a log-normal dstrbuton s the Hoerl curve (see England and Verrall (2002)). Usng a log lnk functon, the Hoerl curve replaced the chan-ladder type systematc component n the log-normal model wth one that s lnear n development lag and ts logarthm. An alternatve perspectve on the log-normal model nvolves Bayesan methods. Incorporatng clam count uncertanty, de Alba (2002) and Ntzoufras and Dellaportas (2002) employed a Bayesan log-normal model to forecast outstandng clams. de Alba (2006) consdered a three-parameter log-normal model to accommodate negatve values n the run-off trangle. Our focus n ths study s the dependences among multple trangles. Extendng the above lterature, we assume a multvarate log-normal dstrbuton for the normalzed ncremental losses ( Y = Y (1),, Y (L) ),.e. wth 1 f(y ; µ, Σ) = (2π) L/2 Σ 1/2 ( L y = y (1). y (L), µ = n=1 y(n) µ (1). µ (L) ( 1 ) exp, and Σ = ) 2 (log y µ ) Σ 1 (log y µ ), (1) σ (1,1) σ (1,L)..... σ (L,1) σ (L,L) The covarance matrx Σ captures the par-wse correlaton between a cell from one trangle and. 6

7 the equvalent cell from another trangle. The correlaton nduced by common calendar year effects s ntroduced through the mean specfcaton. We consder three factors n the mean structure of ncremental losses. For the lth (l = 1,, L) run-off trangle, we specfy µ (l) = µ(l) + α (l) + β (l) j + γ t=+j, where α (l) and β (l) j ndcate the accdent year and development year effects that are unque to the lth trangle, respectvely. As dscussed n Secton 1, correlaton among multple run-off trangles often arses due to common calendar year effects that nfluence all open clams smultaneously. Motvated by ths fact, we employ term γ t to capture the calendar year effect that s dentcal for all trangles. One could thnk of γ t as a calendar year specfc random effect and thus t ntroduces correlaton among cell n the same calendar year wthn and across trangles. To complete a Bayesan specfcaton, one needs to specfy pror dstrbutons for all the structural varables. Followng ( the lterature, ) we assume ( that pror) dstrbutons for parameters are ndependent. Let α = α (1),, α (L) and β = β (1),, β (L), we specfy: Here Σ α ( = dag ), and Q s a sem-postve defnte L L matrx. α N (0, Σ α ), for = 1,, I 1, and α = 0 for = I β j N ( 0, Σ βj ) for j = 1,, J 1, and βj = 0 for j = J τ Σ = Σ 1 Wshart(Q, L). ) (, Σ β = dag σ 2,, σ 2 σ 2,, σ 2 α (1) α (L) β (1) β (L) Note that accdent year and development year effects are treated as fxed effects n ths analyss. We choose to present the model n a more general way to emphasze one advantage of Bayesan analyss,.e., the prvate nformaton could be ncorporated nto the pror specfcaton. We assgn dffuse prors on the fxed effects so that they are estmated based on data, for example, α (l) N(0, 10 3 ). Wthout any prelmnary knowledge on the calendar year trend, one natural choce s a random effect specfcaton: γ t N(0, σ 2 γ), for t = 2,, I + J, τ γ = 1/σ 2 γ Gamma(κ γ, ν γ ), The prors on hyperparameters are specfed for precson parameters τ γ and τ Σ. Then the posteror dstrbuton can be expressed as: I f(θ y R ) =1 I+1 j=1 I J f(y µ, Σ) p(α ) =1 j=1 I+1 p(β ) t=2 ( p(γ t ) p(σ γ )p(σ) ). (2) We are partcularly nterested n the assocaton of the clams wthn and across run-off trangles nduced by the common calendar year effects. Based on the above model, we wll explctly examne 7

8 three types of condtonal correlaton. Note that all the correlatons are are condtonal measures and calculated based on log-scale payments. The correlaton between payments of the same calendar year wthn a sngle trangle. For t = + j = + j and, we consder: ( ) ρ 1 = Corr y (l), y(l) α j, β j, η = σ 2 γ σ 2 γ + σ (l,l). The correlaton between payments that are of the same calendar year but dfferent accdent year and are from dfferent trangles. For t = + j = + j, and l l, we consder the followng measure: ( ) ρ 2 = Corr y (l), ) y(l α j, β j, η = σγ 2. σγ 2 + σ (l,l) σγ 2 + σ (l,l ) The correlaton between losses that are of the same calendar year and the same accdent year and are from dfferent trangles. For l l, we examne the correlaton: ( ) ρ 3 = Corr y, l y (l ) α, β j, η = ρ 31 + ρ 32, where ρ 31 = σγ 2, and ρ 32 = σγ 2 + σ (l,l) σγ 2 + σ (l,l ) σ (l,l ) σ 2 γ + σ (l,l) σ 2 γ + σ (l,l ) The frst two types of correlaton (ρ I and ρ II ) are nduced by the common calendar year effects. The thrd type could be decomposed nto two components (ρ III,1 and ρ III,2 ), whch corresponds to the correlaton ntroduced by the calendar year effect and the cell-specfc effect, respectvely. The sze of above correlatons s determned by the volatlty of calendar year effects, the volatlty of clams, as well as the resdual assocaton between lnes of busness. 2.3 Modelng Calendar Year Trend Secton 2.2 ntroduced a basc Bayesan log-normal model for dependent loss reservng where the calendar year effects are captured by a random effect term γ t. For the predcton purposes, further assumptons need to be made on term γ t. Wthout any pror knowledge on the calendar year trend, t s natural to treat γ t as ndependently and dentcally dstrbuted as n equaton (2). However, when one has certan expectatons on the calendar year trend, the ndependence assumpton on γ t could be volated. Ths secton examnes alternatve formulatons for the calendar year trend that relax the ndependence assumpton. Specfcally, we consder a random walk model and a auto-regressve model for γ t. Under the random walk specfcaton, the calendar year trend n next year s equal to the current trend plus 8

9 a whte nose: γ t = γ t 1 + η t, t = 2,, I + J η t are..d. and η t N(0, σ 2 η). The specfcaton s completed by an ntal value γ 1 = 0 and a pror on hyper parameter τ η = 1/σ 2 η Gamma(κ η, ν η ). The random walk model s sutable f one expects an ncreasng uncertanty on calendar year trend when projectng nto the future. dstrbuton n equaton (2) becomes I f(θ y R ) =1 I+1 j=1 I J f(y µ, Σ) p(α ) =1 j=1 Under above assumptons, the posteror I+1 p(β ) t=2 ( p(γ t γ t 1 ) p(σ γ )p(σ) ). (3) An alternatve approach that relates the calendar year effects n one year to other years s through an auto-regressve model. Here we consder frst-order auto-regressve (AR1) formulaton, though t s straghtforward to extend to the case of hgher orders: γ t = ϕγ t 1 + η t, t = 2,, I + J wth γ 1 N(0, σ 2 γ), η t beng..d. and η t N(0, σ 2 η). To guarantee the statonary condton, we further assume σ 2 η = (1 ϕ 2 )σ 2 γ. The AR1 model s fnalzed wth a flat pror ϕ Unform( 1, 1) and a conjugate pror τ γ = 1/σ 2 γ Gamma(κ γ, ν γ ). The posteror dstrbuton for the AR1 model can be derved as: I f(θ y R ) =1 I+1 j=1 2.4 Model Selecton I J f(y µ, Σ) p(α ) =1 j=1 I+1 p(β ) t=2 ( p(γ t γ t 1, ϕ) p(σ γ )p(σ)p(ϕ) The emphass of ths work s to present a smple approach for examnng dependent trangles and forecastng outstandng clams, whle takng nto account the common calendar year effects. Lke any statstcal modelng, one mght be nterested n the performance of the proposed model verses alternatve specfcatons. To emphasze the role of correlaton, we consder the followng formulatons: α (l) Model 1: Assume ndependence among run-off trangles,.e., Σ = I L and µ (l) = µ(l) +α (l) +β (l) j. Model 2: Jontly model multple trangles wthout calendar year effects,.e., µ (l) = µ (l) + + β (l) j. Model 3.1: Jontly model multple trangles wth dentcally and ndependently dstrbuted calendar year effects,.e., µ (l) = µ(l) + α (l) + β (l) j + γ t=+j. Model 3.2: Jontly model multple trangles wth random walk calendar year effects,.e., µ (l) = µ(l) + α (l) + β (l) j + γ t=+j and γ t = γ t 1 + η t, t = 2,, I + J. 9 ). (4)

10 Model 3.3: Jontly model multple trangles wth AR1 calendar year effects,.e., µ (l) = µ (l) + α (l) + β (l) j + γ t=+j and γ t = γ t 1 + η t, t = 2,, I + J. In general, the model performance could be evaluated from two perspectves: the goodness-offt of the tranng data and the predctve ablty usng a hold-out sample. Recall that n a loss reservng context, a model s estmated usng the observatons n the upper trangle and the goal s to predct the outstandng payments n the lower trangle. Thus the data s naturally dvded nto a tranng set (the upper trangle) and a valdaton set (the lower trangle). To assess the goodness-of-ft of a Bayesan model, we look at both the devance measure and the logarthm of the pseudo-margnal lkelhood (LPML) statstc. The devance for a gven model M s equal to mnus twce log-lkelhood (see Gelman et al. (2003)): D(θ) = 2 log f(y R θ) = 2 L l I =1 I+1 j j=1 log f(y (l) θ). (5) Snce the devance s based on the lkelhood functon, we expect to observe a smaller devance for a better ft. As a sngle measure, we also calculate the average devance accordng to: E[f(y R θ)] = 2 L l I =1 I+1 j j=1 log f(y (l) θ)f(θ yr )dθ. (6) In addton, we employ a more formal statstc, condtonal predctve ordnate (CPO), to compare dfferent models. The CPO s a leave-one-out cross valdaton technque and s a wdely used model assessment tool n varous Bayesan analyss (see, for example, Gelfand et al. (1992)). To adapt ths concept to the multvarate loss reservng, a vector of observatons s dropped n the calculaton of CPO. Specfcally, we defne CPO for the vector y = (y (1),, y(l) ) as CP O = f(y y ) = f(y θ)f(θ y )dθ, (7) where y represents the data set wth vector y taken out. Further detals on the calculaton of CPO can be found n Gelfand and Dey (1994) and Gelfand (1996). The CPO for a model M s calculated as CP O M = I =1 I+1 j j=1 log CP O. (8) The above statstcs s also known as LPML (see, for example, Gesser and Eddy (1979), Gelfand and Mallck (1995) and Snha and Dey (1997) among others). A larger LPML ndcates a better model performance. To compare the predctve performance of alternatve models, we rely on the retrospectve tests that are based on the valdaton set. As wll be seen, the avalablty of realzatons of the future payments n the lower trangle allow us to perform such retrospectve tests. The frst measure that we examne s the L-crteron (see, for example, Laud and Ibrahm (1995). In the loss reservng 10

11 lterature, the L-crteron s dscussed by Ntzoufras and Dellaportas (2002). Assumng that the ncremental losses n both upper and lower trangles are observed, we use observatons n the upper trangle to ft the model and to predct outstandng payments n the lower trangle. The L-crteron can be expressed as: L measure = 1 S b S s=b+1 L l I J =1 j=i+2 ( ) [z (l) ] s z (l) 2, (9) where [z (l) ] s ndcates the ncremental losses of cell (, j) of trangle l n the sth teraton by the MCMC method, b s the number of teratons dscarded as burn-n, and S denotes the total number of teratons. The L-crteron selects the model that mnmzes equaton (9). In addton, we calculate a quas-lpml statstc usng the realzatons n lower trangles: Quas LP ML = I J log f(z z R ). (10) =1 j=i+2 As above, a better predctve performance s ndcated by a larger LPML statstc. 3 Emprcal Analyss 3.1 Data Characterstcs The run-off trangle data are from the Schedule P of the Natonal Assocaton of Insurance Commssoners (NAIC) database. The Schedule P ncludes frm level run-off trangles of aggregated clams for major busness lnes of property-casualty nsurers. And the trangles are avalable for both ncurred and pad losses. We consder an nsurance portfolo of personal auto and commercal auto lnes of busness from a major property-casualty nsurer n US. We use the trangles of pad losses n Schedule P of year Each trangle contans losses for accdent years and at most ten development years. Ths portfolo has been consdered n Sh and Frees (2010). To make ths artcle self-contaned, we represent the trangle data for personal auto lne and commercal auto lne n Table A.1 and Table A.2 n Appendx, respectvely. The realzatons of future payments (lower trangle) are from Schedule P of subsequent years. For example, the losses of accdent year 1989 s from Schedule P of year 1998, the losses of accdent year 1990 s from Schedule P of year 1999, and so on. The observatons n overlappng years are used to valdate the qualty of tranng data. We normalze the payment by dvdng by the net premums earned n the correspondng accdent year. Fgure 1 exhbts the the tme seres plot of loss ratos for personal and commercal auto lnes by accdent year. The loss ratos of the two lnes present dfferent patterns n early development years, though they algn wth each other n later development years. The comparson n Fgure 1 also shows that the development of the commercal automoble lne s more volatle than that of the personal automoble lne. The loss ratos decrease to a small value, suggestng that 11

12 clams n both lnes are lkely to be closed wthn ten years. Commercal Auto Personal Auto Loss Rato Development Lag Fgure 1: Tme seres plots of loss rato by accdent year. Each panel corresponds to an accdent year, and contans two curves ndcatng commercal and personal auto lnes. To further motvate our model specfcaton, we examne the dependency between the two runoffs. We frst look nto the correlaton between loss ratos whch s found to be hghly correlated. In fact, the Pearson correlaton coeffcent s Ths observaton s not surprsng and s consstent wth Fgure 1. However, to gan some ntutve knowledge on the correlaton structure, one needs to cast out the trangle-specfc effects over accdent years and development years. To do so, we ft a log-normal model wthout calendar year effects for each run-off trangle ndependently, and examne relatonshp between the Pearson resduals (ϵ (l) = (log y(l) µ(l) )/ σ (l) ). We provde the scatter plot of resduals from personal auto and commercal auto lnes n Fgure 2, whch shows the par-wse assocaton among the two lnes,.e., the correlaton between a cell n one trangle and the correspondng cell n another trangle. After removng the accdent year and development year effects, there s consderable negatve correlaton (wth correlaton coeffcent equal to 0.38) between loss ratos. Ths observaton suggests that some form of correlaton structure s needed when jontly modelng the two trangles. It s worth commentng on the negatve correlaton here. As explaned n Secton 2.2, calendar year effect wll only ntroduce postve correlaton. So the negatve correlaton motvates the specfcaton of the correlaton matrx Σ, whch could capture both postve and negatve resdual correlatons when excludng the calendar year effect. Thus the observe negatve correlaton n Fgure 2 represents a combned result of 12

13 2 1 Personal Auto Commercal Auto Fgure 2: Resduals of personal auto lne versus commercal auto lne. Each dot n the plot corresponds to a par of losses from the same accdent year and development lag n the two trangles. calendar year effect and resdual correlaton mpled by Σ. More dscusson on ths wll be provded n Secton 3.2. We move ths analyss one step further by breakng down the correlaton accordng to calendar years. In a run-off trangle, the cells along each dagonal corresponds to a sngle calendar year. Compared wth Fgure 2, the negatve relatonshp becomes less promnent for each calendar year. The decayng correlaton when examned by calendar year foreshadows the exstence of common calendar year effects that could affect clams n all accdent years smultaneously, because the calendar year effects could nduce potental postve assocaton among multple trangles and thus offset the underlyng negatve relatonshp among resduals. The effect wll be demonstrated n the followng analyss. We fnalze ths secton by explorng the calendar year effects for the two lnes of busness. The box plots of resduals from the two ndependent log-normal model are dsplayed over calendar years n Fgure 3. The comparson of box plots shows smlar calendar year trend for the personal and commercal auto lnes, though there s a slght dvergence n later calendar years. For a smple presentaton and easy motvaton for the correlaton caused by common calendar year effects, we consder a ntutve formulaton by specfyng an dentcal calendar year trend for the two trangles, and assumes that the heterogenety wll be absorbed by the correlaton matrx Σ. Dscussons on more flexble calendar year effects are found n Secton 5. 13

14 Resdual Personal Auto Commercal Auto Calendar Year Fgure 3: Box plots of resduals for personal and commercal auto lnes by calendar year. Each par of box plots corresponds to a calendar year wth the lght one for personal auto and the dark one for commercal auto. 3.2 Estmaton Results of Model 1, Model 2 and Model 3.1 Ths secton examnes the estmaton results of Model 1, Model 2 and Model 3.1, and ther mplcatons on reserve ndcaton. To recap, Model 1 assumes that the two lnes of busness are ndependent and the calendar year effects could be accommodated by the accdent year and development year factors. In contrast, Model 2 allows for a par-wse correlaton among trangles, relatng each cell n one run-off to the correspondng cell n the other. Extendng Model 2, Model 3.1 captures the common calendar year effects explctly through a calendar year random effect and thus ntroduces dependences among clams across accdent years. The estmated parameters and goodness-of-ft statstcs are summarzed n Table 1. Note that we only report the posteror mean and 95% estmates for the common locaton parameters n the three models. All estmates are based on 100,000 MCMC teratons wth the frst 50,000 teratons dscarded as burn-n sample. Though not reported here, we generate multple chans from dfferent ntal values and the convergency for each parameter s confrmed wth Gelman-Rubn statstc. There s no sgnfcant dfference n the estmates of the posteror mean for locaton parameters. However, when comparng the 95% estmates, we observe narrower nterval estmates for more complcated models. Ths could be explaned by the extra parameters n more advanced models that render more flexblty to accommodate the underlyng data characterstcs. We confrm ths 14

15 by examnng the two model performance crtera reported at the bottom of the table. It s shown that Model 1 has the hghest average devance and Model 3.1 has the lowest average devance. We note that the average devance s a not suffcent evdence to show the goodness-of-ft of a model. We also look nto the complete hstory of devance that s not ncluded here. The trace of devance suggests a strong correlaton between personal and commercal auto lnes as well as a sgnfcant calendar year effect. The LPML statstc mples a consstent concluson wth Model 3.1 outperformng the other two nested models, emphaszng the crtcal role played by the calendar year trend. To demonstrate the effects of the calendar year trend explctly, we examne the followng seven correlaton coeffcents: ρ 1 : the correlaton between two cells n the same calendar year for personal auto lne,.e., t s calculated accordng to ρ I defned n Secton 2.2; ρ 2 : the correlaton between two cells n the same calendar year for commercal auto lne,.e., t s calculated accordng to ρ I defned n Secton 2.2; ρ 3 : the correlaton between two cells, one from personal auto lne and the other from commercal auto lne. The two cells are of the same calendar year but dfferent accdent year. Ths coeffcent corresponds to ρ II defned n Secton 2.2; ρ 4 : the par-wse correlaton between one cell from personal auto lne and the correspondng cell from commercal auto lne. Ths coeffcent corresponds to ρ III defned n Secton 2.2; ρ 5 : the resdual correlaton between two correspondng cells from the two trangles after controllng for accdent year effects, development year effects, as well as calendar year effects,.e., the correlaton coeffcent mpled by the covarance matrx Σ; ρ 6 : the correlaton between two correspondng cells from the two trangles, ntroduced by the calendar year effect. That s the coeffcent ρ III,1 n Secton 2.2; ρ 7 : the correlaton between two correspondng cells from the two trangles, ntroduced by the cell-specfc effect. That s the coeffcent ρ III,2 n Secton 2.2. As defned n Secton 2.2, we treat all above correlaton coeffcents as random varables n the Bayesan context. Thus one could derve ther posteror dstrbutons based on MCMC as other model parameters. The estmates of above correlaton measures are dsplayed n Fgure 4. The parallel horzontal lnes represent the 95% estmate nterval and the dots represent the posteror mean. The common calendar year effects lead to a strong correlaton wthn the trangle of personal auto lne but a weak correlaton wthn the trangle of commercal auto lne. Ths s explaned by the fact that the volatlty of calendar year trend s comparable to the ntrnsc volatlty of personal auto lne, whle t s far smaller than that of commercal auto lne. The calendar year random effect also contrbutes to the moderate correlaton between two trangles ndcated by ρ 3. In terms of parwse correlaton, the two trangles are negatvely assocated when calendar year effects are excluded (see ρ 5 ). Incorporatng calendar year effects, we fnd that the postve assocaton ntroduced theren offsets the underlyng negatve relatonshp. To be more specfc, ρ 4 represents a combned results of ρ 6 and ρ 7. Ths analyss provdes evdence of calendar year effects. 15

16 Table 1: Estmated parameters for Model 1, Model 2, and Model 3.1 Model 1 Model 2 Model 3.1 Parameter Mean 95% Interval Mean 95% Interval Mean 95% Interval µ (1) (-7.544,-6.977) (-7.531,-7.004) (-7.576,-6.934) α (1) (-0.010,0.416) (0.016,0.402) (-0.029,0.410) α (1) (-0.042,0.381) (-0.018,0.368) (-0.061,0.361) α (1) (-0.038,0.385) (-0.014,0.374) (-0.056,0.354) α (1) (-0.142,0.285) (-0.118,0.272) (-0.144,0.252) α (1) (-0.188,0.242) (-0.163,0.229) (-0.167,0.218) α (1) (-0.189,0.245) (-0.166,0.234) (-0.145,0.229) α (1) (-0.192,0.252) (-0.169,0.238) (-0.122,0.246) α (1) (-0.245,0.204) (-0.225,0.194) (-0.170,0.199) α (1) (-0.279,0.196) (-0.258,0.186) (-0.231,0.142) β (1) (5.713,6.126) (5.719,6.117) (5.683,6.131) β (1) (5.490,5.904) (5.493,5.891) (5.458,5.889) β (1) (4.664,5.081) (4.671,5.069) (4.641,5.053) β (1) (4.067,4.485) (4.074,4.473) (4.056,4.459) β (1) (3.456,3.878) (3.462,3.864) (3.466,3.855) β (1) (2.695,3.120) (2.701,3.107) (2.729,3.107) β (1) (2.034,2.464) (2.040,2.454) (2.091,2.465) β (1) (1.206,1.649) (1.212,1.640) (1.269,1.644) β (1) (0.776,1.241) (0.785,1.230) (0.812,1.195) µ (2) (-7.007,-4.968) (-6.880,-4.886) (-7.146,-4.977) α (2) (-0.632,0.916) (-0.651,0.853) (-0.614,0.991) α (2) (-0.763,0.794) (-0.768,0.729) (-0.746,0.849) α (2) (-0.802,0.746) (-0.812,0.688) (-0.790,0.800) α (2) (-0.947,0.618) (-0.952,0.561) (-0.915,0.669) α (2) (-0.934,0.637) (-0.948,0.579) (-0.895,0.705) α (2) (-0.958,0.625) (-0.967,0.581) (-0.890,0.706) α (2) (-0.879,0.736) (-0.890,0.691) (-0.789,0.838) α (2) (-0.990,0.664) (-1.000,0.620) (-0.901,0.771) α (2) (-0.848,0.892) (-0.853,0.837) (-0.798,0.946) β (2) (3.495,4.973) (3.420,4.967) (3.462,5.008) β (2) (3.652,5.133) (3.588,5.134) (3.617,5.147) β (2) (3.396,4.881) (3.332,4.882) (3.359,4.883) β (2) (3.011,4.504) (2.946,4.506) (2.976,4.507) β (2) (2.333,3.844) (2.273,3.846) (2.316,3.866) β (2) (1.940,3.454) (1.869,3.454) (1.944,3.494) β (2) (1.296,2.838) (1.217,2.829) (1.322,2.895) β (2) (0.484,2.066) (0.418,2.065) (0.509,2.120) β (2) (0.226,1.906) (0.170,1.901) (0.216,1.911) Devance LPML

17 ρ 7 ρ 6 ρ 5 ρ 4 ρ 3 ρ 2 ρ Fgure 4: Posteror estmates of correlaton coeffcents. For each coeffcent, the sold dot ndcates the posteror mean and the horzontal lne ndcates the 95% estmate nterval. In the reservng problem, one s more nterested n the effect of dependences among trangles on reserve ndcatons. We present the reserve estmates for personal and commercal auto lnes as well as the total reserve for the nsurance portfolo n Table 2. One advantage of Bayesan method s that a predctve dstrbuton of outstandng payments could be derved rather than a pont predcton wth confdence ntervals. Table 2 compares the predctons from Model 1, Model 2 and Model 3.1. For llustraton purposes, only predctve mean and standard devaton of the predctve dstrbuton are reported. Table 2: Reserve estmatons from Model 1, Model 2, and Model 3.1 Model 1 Model 2 Model 3.1 Reserve Mean StdDev mean StdDev mean StdDev Personal Auto 6,528, ,600 6,493, ,900 6,555, ,600 Commercal Auto 488, , , , , ,600 Aggregate 7,016, ,500 6,985, ,400 7,042, ,500 We make several observatons from the comparson n Table 2: Frst, the posteror means from the three models are comparable to each other, whch agrees wth the close estmates of locaton parameters as shown n Table 1. Second, because of dversfcaton effects, the varance of aggregated reserves are much smaller than the sum of varances from both trangles. Note that dversfcaton occurs when two busness lnes are not perfectly correlated. Fnally, the extra parameters on the calendar year random effect contrbute extensvely to the predctve varablty on loss reserves. Ths s supported by the fact that the predctve dstrbuton of outstandng 17

18 payments from Model 3.1 s much wder than those from Model 1 and Model 2. When comparng to the frst two nested models, the larger standard devaton of Model 1 mght be attrbuted to the potental model msspecfcaton. Ths s because the sgnfcant negatve correlaton between the two trangles s gnored n Model Estmaton Results of Model 3.1, Model 3.2 and Model 3.3 After llustratng the substantve calendar year effects, ths secton compares alternatve formulatons accordng to Model 3.1, Model 3.2 and Model 3.3. As dscussed n Secton 2.2, the correlaton structure reles on the volatlty of both calendar year effect and cell specfc effect. Thus, nstead of locaton parameters, we exhbt n Table 3 the estmates of precson parameters. The posteror means and 95% estmates are based on 100,000 teratons wth 50,000 burn-n. The mean devance and LPML statstc are provded to evaluate model performance. Table 3: Estmated parameters for Model 3.1, Model 3.2, and Model 3.3 Model 3.1 Model 3.2 Model 3.3 Parameter Mean 95% Interval Mean 95% Interval Mean 95% Interval τ ( , ) ( , ) ( , ) τ (2.857,38.150) (2.616,38.490) (2.515,38.130) τ (2.857,38.150) (2.616,38.490) (2.515,38.130) τ (4.658,14.520) (4.580,14.190) (4.596,13.910) τ γ (63.810, ) (54.320, ) (10.800, ) Devance LPML The three dfferent specfcatons of the calendar year trend γ t lead to close estmates of the precson parameters n Σ and slghtly dfferent estmate of τ γ. Though t s a rough comparson, the relatvely small τ 22 explans the weak correlaton wthn commercal auto lne (ρ 2 n Fgure 4), and the comparable sze of τ γ and τ 11 accounts for the strong correlaton wthn personal auto lne (ρ 1 n Fgure 4). The mean devance, reported for demonstraton purposes, does not provde much nformaton n ths case. The trace plot of devance also suggests a close performance of the three models. The LPML statstc seems to support the correlated formulaton on calendar year effects, especally the random walk specfcaton. However, t s not our ntenton to select the best model at ths pont. As dscussed prevously, we ntend to present varous specfcatons that could reflect the actuary s pror knowledge on the calendar year trend. To vsualze the common calendar year effects n loss reservng ndcaton, we exhbt ther dstrbutons over years n Fgure 5. The 95% nterval estmates are shown as parallel vertcal lnes wth a dot ndcatng the correspondng posteror mean. The sold lne corresponds to the calendar year effects n upper trangle (years ), and the dashed lne corresponds to the calendar year effects n lower trangle (years ). The calendar year trend n Fgure 3 s well captured by all three models, though the..d assumpton (Model 3.1) produces the lowest volatlty. As expected, we observe an ncreasng volatlty n the random walk specfcaton (Model 18

19 3.2) when projectng nto the future. In contrast, the volatlty of calendar year effects n Model 3.1 and Model 3.3 are relatvely stable over years. Model 3.1 Model 3.2 Model 3.3 γ t γ t γ t Calendar Year Calendar Year Calendar Year Fgure 5: Dstrbuton of the calendar year trend over years for the three dfferent specfcatons. The 95% nterval estmates are shown as parallel vertcal lnes wth a dot ndcatng the correspondng posteror mean. The sold lne corresponds to the calendar year effects n upper trangle, and the dashed lne corresponds to the calendar year effects n lower trangle. The followng presents the reserve predctons from the above three models. In addton to total reserve projecton, reservng actuares mght also be nterested n accdent year and calendar year reserves from the accountng or rsk management perspectves. Accdent year reserve represents a projecton of the unpad losses for accdents occurred n a partcular year, and calendar year reserve represents a projecton of outstandng payments for a certan calendar year. Current multvarate stochastc clams reservng methods emphasze the predcton of total and accdent year reserves based on the ndependence assumpton over accdent years. However, the extenson to calendar year reserves s not always straghtforward. Bayesan methods have the advantage of provdng the predctve dstrbuton for each sngle cell n the lower trangle. Thus calendar year reserves can be easly derved. Table 4 and Table 5 present the mean and standard devaton of the predctve dstrbuton of accdent year and calendar year reserves, respectvely. The standard devaton n the predctve dstrbuton s an analogy to the predcton error of non-parametrc methods. Not reported here, there are sgnfcant dversfcaton effects among accdent year and calendar year reserves. Consstently, the three models produce comparable pont estmates for outstandng payments. The ndependence model results n the smallest predctve uncertanty due to the most parsmonous specfcaton. The random walk model allows for an ncreasng volatlty n calendar year effects, whch contrbutes to the largest predcton error for loss reserves. The predctve varablty has mportant mplcatons n the determnaton of a reasonable reserve range and thus rsk margn n prudent rsk management practce. Typcally, reserve actuares fnd t more helpful to have a predctve dstrbuton of unpad losses rather than a pont estmaton wth assocated predcton error, because the advanced quanttatve rsk management tools are easer to be mplemented when a complete dstrbuton s avalable. Ths secton computes rsk 19

20 Table 4: Predcton of reserves by accdent year Model 3.1 Model 3.2 Model 3.3 Accdent Year Mean StdDev mean StdDev mean StdDev , , , ,540 2,583 22,020 2,777 21,840 2, ,060 4,608 45,300 5,733 44,870 5, ,420 10, ,700 13, ,600 11, ,200 21, ,200 29, ,400 25, ,200 47, ,100 65, ,000 55, ,600 94, , , , , ,691, ,600 1,754, ,700 1,730, , ,577, ,500 3,669, ,600 3,617, ,700 Table 5: Predcton of reserves by calendar year Model 3.1 Model 3.2 Model 3.3 Calendar Year Mean StdDev mean StdDev mean StdDev ,478, ,400 3,536, ,300 3,508, , ,713, ,900 1,765, ,500 1,742, , , , , , , , ,500 66, , , ,700 86, ,800 34, ,200 63, ,300 47, ,000 18, ,800 36, ,900 25, ,790 8,786 59,610 18,540 56,880 12, ,330 4,654 28,400 9,821 26,960 6, ,811 1,497 7,659 2,946 7,229 2,021 margn usng the proposed Bayesan log-normal model and llustrates the mplcaton of predctve varablty on rsk margn calculaton. Rsk margn, also known as rsk captal, s the amount of fund that property-casualty nsurers set asde as a buffer aganst potental unexpected losses that s defned as the amount of actual losses over expected losses. The value-at-rsk (VaR) and condtonal tal expectaton (CTE) are two extensvely used quanttatve rsk measures to determne the rsk margn n property-casualty nsurance ndustry. The VaR( ) s smply the 100(1- )th percentle of the loss dstrbuton. The CTE( ) s the expected losses condtonal on exceedng the VaR( ). We calculate rsk margn based on both rsk measures for the nsurance portfolo that conssts of personal auto and commercal auto lnes. Table 6 summarzes the estmaton results for the three models wth dfferent specfcatons on calendar year effects. The rsk margn s calculated as the dfference between the rsk measure and expected losses, and the calculaton s carred out based on 90th, 95th, and 99th percentles of the loss dstrbuton respectvely. For example, the nsurer needs at least $877,992 to cover the potental unexpected losses wth a 90% confdence level, f the rsk margn s calculated usng Model 3.1 and based on VaR. Apparently one observes a larger rsk margn for a hgher percentle, snce larger losses are expected n the further rght tal of the loss dstrbuton. As antcpated, 20

21 rsk margns based on CTE s greater than those based on VaR because CTE consders all possble losses above VaR. As already been foreshadowed by the reserve estmates n Table 4 and Table 5, the narrower predctve dstrbuton of Model 3.1 leads to smaller rsk margns and the wder predctve dstrbuton of Model 3.3 produces larger rsk margns. It s nconclusve to comment on the suffcency of rsk margn wthout the realzatons on unpad losses. We leave ths topc for further dscusson n the next secton. Another mportant mplcaton of multvarate reservng modelng s the dversfcaton effect when aggregatng clams from dependent lnes of busness. We refer nterested readers to Sh and Frees (2010) for detaled dscusson. Table 6: Estmated rsk margns for Model 3.1, Model 3.2, and Model 3.3 Tal Probablty 90% 95% 99% Based on VaR Model ,922 1,197,922 1,937,922 Model 3.2 1,118,482 1,548,482 2,578,482 Model 3.3 1,008,040 1,358,040 2,128,040 Based on CTE Model 3.1 1,342,504 1,661,602 2,411,662 Model 3.2 1,759,842 2,211,242 3,369,742 Model 3.3 1,504,966 1,842,796 2,630,060 4 Model Valdaton 4.1 Comparson wth Exstng Methods Ths secton compares the reserve predctons from Bayesan log-normal models wth varous exstng approaches. The frst comparson s performed wth parametrc methods that provde a predctve dstrbuton of total outstandng clams. Among them, Brehm (2002) employed a log-normal model for the unpad losses of each lne and a normal copula for the aggregaton of losses from multple lnes. The predctve dstrbuton of outstandng payments s generated through Monte Carlo smulaton. As an alternatve to Bayesan method, parametrc bootstrappng ncorporates parameter uncertanty through resamplng technques and uses the updated parameters to compute predctve dstrbuton. Taylor and McGure (2007) and Krschner et al. (2008) are two examples wthn ths lne of work. Extendng the bootstrap over-dspersed Posson model n England and Verrall (2002), both studed a synchronous bootstrappng that reserves the par-wse assocaton among trangles n the resamplng process. Both gamma and over-dspersed posson process error are appled n the synchronous bootstrappng method. Fgure 6 dsplays the predctve dstrbuton of total unpad losses of the nsurance portfolo from varous parametrc methods. The frst panel compares the Bayesan log-normal model wth the above parametrc methods. The estmates of expected losses are close, however, the predctve varablty vares consderably across dfferent methods. As expected, the log-normal model by Brehm (2002) gves tghter predctve dstrbuton snce the parameter uncertanty s not ncor- 21

22 porated nto the statstcal nference. The bootstrap methods n Taylor and McGure (2007) and Krschner et al. (2008) suggest smlar reserve ndcaton. Overcomng the lmtaton of tradtonal parametrc models, the bootstrappng provdes relatvely wder predcton. Buldng on the exstng studes, the proposed Bayesan log-normal model captures the correlaton due to calendar year effects n addton to the par-wse correlaton between multple trangles. The result suggests that ncorporatng calendar year effects contrbutes to a great extent to the predctve varablty. The second panel reproduces predctve dstrbutons of outstandng payments from the three formulatons of the Bayesan log-normal model. In agreement wth prevous analyss, the..d specfcaton provdes the smallest predcton uncertanty and the random walk specfcaton renders the largest predcton uncertanty. However, the dfference among the three s gnorable when compared to models wthout calendar year effects. Densty 0.0e e e e 06 Model 3.1 Brehm(2002) Taylor(2007) Krschner(2008) Densty 0.0e e e e 06 Model 3.1 Model 3.2 Model e e e e e e e e+07 Unpad Losses Unpad Losses Fgure 6: Predctve dstrbuton of outstandng clams of varous methods. The left panel compares the proposed method wth exstng technques. The rght panel compares the proposed method under varous specfcatons for calendar year effect. The second comparson nvolves non-parametrc methods. Besdes the above parametrc approaches, we apply three non-parametrc methods to the nsurance portfolo: the multvarate chan-ladder method n Merz and Wüthrch (2008), the combned chan-ladder method n Merz and Wüthrch (2009a), and generalzed chan-ladder method n Zhang (2010). The dstrbutonfree methods have desrable propertes, usng the mean square error of predctons to measure predctve varablty. We summarze the estmated portfolo reserves and the correspondng predcton error from varous approaches n Table 7. Note that the reported predcton error represents the standard devaton of the predctve dstrbuton for parametrc models, and the mean square error of predcton for non-parametrc models. The predctons from the non-parametrc approaches are n lne wth the observatons n Fgure 6. All parametrc and non-parametrc methods n the 22