Calendar Year Dependence Modeling in Run-Off Triangles

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1 Calendar Year Dependence Modelng n Run-Off Trangles Maro V. Wüthrch 2013 ASTIN Colloquum, May 21-24, The Hague Conference Paper Abstract A central ssue n clams reservng s the modelng of approprate dependence structures n run-off trangles. Most classcal clams reservng models cannot cope wth ths task. We revst the Bayesan multvarate log-normal model of Merz et al. [14] that allows to model calendar year dependence. In ths model closed form solutons are obtaned for clams reserves and the correspondng predcton uncertanty. The numercal example of ths paper provdes new nsghts and conclusons. 1 Introducton and motvaton A central ssue n clams reservng s the modelng of approprate dependence structures n runoff trangles, for example, calendar year dependence. Most classcal clams reservng models cannot cope wth ths task. For nstance, Mack s [13] and Hertg s [7] models assume ndependence between dfferent accdent years. Under these ndependence assumptons predctors and correspondng confdence bounds are derved. Of course, one s aware of the fact that postve correlaton wdens these confdence bounds due to less dversfcaton benefts. Therefore, as a next (and fnal step practtoners do a top level correlaton correcton by smply multplyng the confdence bounds (obtaned from the ndependent case wth a determnstc factor. Only very recently actuares have started to consder more sophstcated (calendar year dependence modelng. Early work on ths topc has been publshed by Barnett-Zehnwrth [1], Brehm [3], de Jong [8], Krschner et al. [10], Köng et al. [11] and Kuang et al. [12]. More recently, the problem has been studed n a Bayesan modelng set-up whch has lead to a consstent uncertanty measurement, see the papers by Donnelly-Wüthrch [5], Merz et al. [14], Sh et al. [16] and Wüthrch [18, 19]. In the present paper we revst the model presented n Merz et al. [14] and we provde an example whch wll gve new nsghts and conclusons. The startng pont of our model s Hertg s [7] log-normal chan-ladder model. Ths clams reservng model s embedded nto a multvarate log-normal framework. Ths multvarate log-normal framework s chosen such that t allows for flexble correlaton modelng, n fact we allow for any correlaton structure between dfferent clams development cells. Parameter uncertanty s modeled wth a multvarate Gaussan ETH Zurch, RskLab, Department of Mathematcs, CH-8092 Zurch. emal: maro.wuethrch@math.ethz.ch 1

2 pror dstrbuton (whch s a conjugate pror to the multvarate log-normal dstrbuton, see Bühlmann-Gsler [4]. Ths combnaton then allows to gve a closed form soluton for the clams reservng problem. Moreover, t allows to derve closed form confdence bounds whch, at the same tme, study process uncertanty and parameter uncertanty. These results are then used to study senstvtes and, moreover, they can be used to calbrate correlaton estmates for solvency purposes. We would lke to hghlght that we get fundamental fndngs from our model (see Conclusons 1-4 below whch explan that the state-of-the-art handlng of correlaton n practce should be changed, because correlaton s often under-estmated (and mss-specfed. Moreover, we also hghlght that the choce of correlaton parameters needs to be done carefully, because effects lke portfolo growth mght completely dstort the correlaton analyss. Organzaton of the paper. In the next secton we ntroduce the Bayesan multvarate log-normal clams reservng model. In Secton 3 we recall mportant propertes of multvarate Gaussan dstrbutons whch are gong to be used n Secton 4 for clams reservng and predcton uncertanty analyss. In Secton 5 we gve a detaled example whch results n the mportant Conclusons 1-4. Fnally, n Secton 6 we brng down our fndngs. 2 Model assumptons Accdent years are denoted by {1,..., I} and development years by j {0,..., J}. We assume that I J + 1 and that all clams are settled after development year J. Cumulatve clams are denoted by C,j > 0. We defne the ndvdual log-lnk ratos ξ,j by (set C, 1 1 for all ξ,j = log (C,j /C,j 1 C,j = C,j 1 exp {ξ,j }. (2.1 We stack the random varables ξ,j nto random vectors whch wll provde a handy notaton. For {1,..., I} and j {0,..., J} we defne the vectors ξ = (ξ,0,..., ξ,j R J+1 and ξ = (ξ 1,..., ξ I R d, wth d = (J + 1I. Next we defne a verson of the Bayesan multvarate log-normal clams reservng model of Merz et al. [14]. Model Assumptons 2.1 (Bayesan multvarate log-normal model Condtonally, gven parameter Θ R J+1, the random vector ξ has a multvarate Gaussan dstrbuton wth fxed postve-defnte covarance matrx Σ R d d and condtonal expected values E [ξ Θ] = Θ, for all {1,..., I}. The parameter Θ has a multvarate Gaussan dstrbuton wth pror mean µ R J+1 and postve-defnte pror covarance matrx T R (J+1 (J+1. 2

3 Under Model Assumptons 2.1, the jont densty of ξ and Θ at poston (ξ, θ s gven by { 1 f(ξ, θ = (2π d/2 det(σ 1/2 exp 1 } 2 (ξ Aθ Σ 1 (ξ Aθ (2.2 { 1 (2π (J+1/2 det(t 1/2 exp 1 } 2 (θ µ T 1 (θ µ, wth matrx A = (1,..., 1 R d (J+1 consstng of I dentty matrces denoted by 1 R (J+1 (J+1 and descrbng the condtonal mean E [ξ Θ] = AΘ = (Θ,..., Θ R d. We summarze some of the remarks of Merz et al. [14] and Wüthrch [19]. Remarks, see also Merz et al. [14], Remarks 2.2. The choce of a multvarate Gaussan model for the ndvdual log-lnk ratos ξ,j gves log-normal dstrbutons for cumulatve clams C,j. For rsk management purposes one mght argue that ths s not a suffcently heavy-taled model and, moreover, ths model does not enjoy tal dependence. Nevertheless, t provdes nterestng nsghts and t may serve as a benchmark model. The covarance matrx Σ wll allow for any correlaton structure between the ndvdual log-lnk ratos ξ,j such as calendar year dependence. We provde explct examples below. Parameter uncertanty s modeled through the choce of a pror dstrbuton for Θ. Thus, a Bayesan nference analyss wll nclude parameter estmaton uncertanty n a natural way through posteror dstrbutons. The model assumptons mply that condtonally, gven Θ, C 1,0 = exp{ξ 1,0 },..., C I,0 = exp{ξ I,0 } are dentcally dstrbuted. That s, there s no specfc accdent year {1,..., I} parameter under Model Assumptons 2.1. Ths model assumpton can easly be relaxed f t s not approprate, for nstance, by settng C, 1 = ν R +. (2.3 Ths s further dscussed below. Crucal for the closed form soluton s the multvarate Gaussan structure (2.2 whch s the key property to all results n the next sectons. Ths s smlar to the results n the addtve model of Sh et al. [16], formulas (1-(4. Relaxaton of ths assumpton may lead to herarchcal generalzed lnear models (HGLM. However, many of the analytcal propertes get lost n these more complex models and ether smulatons or approxmatons need to be used. 3

4 3 Uncondtonal and predctve dstrbutons The Bayesan Model Assumptons 2.1 descrbe the dstrbuton of ξ, gven the parameter Θ. Snce, n general, ths parameter s not known experts specfy a pror dstrbuton for t. In classcal credblty theory one then collects observatons ξ,j and calculates the posteror dstrbuton of Θ, gven these observatons, see e.g. Bühlmann-Gsler [4]. In the present stuaton t s smpler to drectly work wth the uncondtonal dstrbuton of ξ. That s, though we have a classcal Bayesan model ncludng parameter uncertanty, we do not descrbe the posteror dstrbuton of the parameter Θ explctly, but we drectly work on the data havng the parameter dstrbuton of Θ as a latent factor (ths can be done because the parameter Θ acts n a smple way as a locaton parameter, see Proof (3 of Theorem 3.1 n Merz et al. [14]. Theorem 3.1 (uncondtonal dstrbuton Under Model Assumptons 2.1 the random vector ξ has a multvarate Gaussan dstrbuton wth the frst two moments gven by E [ξ] = Aµ and S def. = Cov (ξ = Σ + AT A. Proof. See Theorem 3.1 n Merz et al. [14]. Assume some of the components of ξ are observed: choose a non-empty, real subset D {(, j; = 1,..., I, j = 0,..., J} def. = J,.e. D / {J, }. Let D denote the cardnalty of D. Defne P D to be the projecton R d R D onto the components (, j D of ξ, that s, ξ ξ D = P D ξ, such that ξ D exactly contans the components of ξ whch are n D. Analogously, P D c the projecton onto the components n the complement D c = J \ D of D and ξ Dc correspondng components. Ths provdes the (bjectve decomposton denotes denotes the ξ ( ξ D, ξ Dc, (3.1 whch separates R d nto two dsjont (non-empty subspaces R D and R Dc. predct ξ Dc Our am s to when we have observed ξ D. These projectons only descrbe a permutaton of the components of ξ and the followng corollary s a straghtforward consequence of Theorem 3.1. Corollary 3.2 Under Model Assumptons 2.1 the random vector (ξ D, ξ Dc has a multvarate Gaussan dstrbuton wth the frst two moments gven by µ D = E [ ξ D] = P D Aµ and S D = Cov ( ξ D = P D S P D, µ D c = E [ ξ Dc ] = PD c Aµ and S D c = Cov ( ξ Dc = PD c S P D c. The covarance matrx between the components ξ D and ξ Dc s gven by S D c,d = S D,D c = Cov ( ξ D, ξ Dc = PD S P D c. 4

5 We are now ready to gve the crucal statement provdng the predctve dstrbuton of ξ Dc, condtonally gven observatons ξ D. Theorem 3.3 (predctve dstrbuton Under Model Assumptons 2.1 we have the followng statement: the condtonal dstrbuton of ξ Dc, gven ξ D, s a multvarate Gaussan dstrbuton wth condtonal mean gven by µ post D c = E [ ξ Dc ξ D] = µ D c + S D c,d (S D 1 ( ξ D µ D, and condtonal covarance matrx gven by S post D c = Cov ( ξ Dc ξ D = S D c S D c,d (S D 1 S D,D c. Proof. See Result 4.4 n Johnson-Wchern [9] and Theorem 3.4 n Merz et al. [14]. 4 Ultmate clam predcton and predcton uncertanty 4.1 Total run-off uncertanty We apply Theorem 3.3 to the clams reservng problem. Assume that we are at tme t I (wth t < I + J. Then, we have observed the cumulatve clams C,j wth + j t. Note that we choose t I to ensure that the frst column {(, j; 1 I, j = 0} of J has been observed. Thus, the data s determned by the ndces and the resultng σ-feld at tme t s gven by D t = {(, j J ; + j t}, (4.1 F t def. = σ {C,j ; (, j D t } = σ {ξ,j ; (, j D t } = σ { ξ Dt}. The Bayesan predctor for the ultmate clam C,J, t J + 1, at tme t I s gven by J Ĉ (t,j = E [C,J F t ] = C,t E exp ξ,j. (4.2 j=t +1 Assume that 1 t + 1 J. Then we can defne the set of ndces ξdt D c t = D c t {(l, j J ; l = } = {(, t + 1,..., (, J}, whch provdes the projecton P D c t : R d R J (t. Thus, we have e t ξdc t def. = (1,..., 1 P D c t ξ = J j=t +1 ξ,j. Note that ths sum exactly consders all un-observed components of ξ Dc t for a gven accdent year. The followng statements are straghtforward consequences of (4.2 and Theorem

6 Theorem 4.1 (ultmate clam predcton Under Model Assumptons 2.1 the ultmate clam predctor Ĉ(t,J gven by of accdent year {t J + 1,..., I} s at calendar year t {I,..., I + J 1} Ĉ (t,j = C,t { exp e t µpost Dt c + 1 } 2 e t Spost Dt c e t. The correspondng clams reserves are then at calendar year t gven by R (t { = Ĉ(t,J C,t = C,t (exp e t µpost Dt c + 1 } 2 e t Spost Dt c e t 1. (4.3 In order to analyze predcton uncertanty one typcally studes the volatlty of the dfferences Ĉ (t,j C,J = R (t (C,J C,t, (4.4.e. how much the true ultmate clam C,J may devate from ts predcton Ĉ(t,J at tme t. Theorem 4.1 allows for a closed form calculaton of the ultmate clam predctor Ĉ(t,J and Theorem 3.3 allows for a smulaton based analyss of any rsk measure on the dfference (4.4. Snce we do not want to rely on smulatons we choose a partcular rsk measure whch can be calculated n closed form. The condtonal mean square error of predcton (MSEP s defned by ( ( msep C,J F t Ĉ (t,j = E C,J 2 Ĉ (t,j F t. In vew of the Bayesan predctor (4.2,.e. takng condtonal expectatons, we have ( ( msep C,J F t Ĉ (t,j = Var C,J F t = Cov (C,J, C l,j F t. (4.5,l Thus, for the condtonal MSEP we need to calculate these condtonal covarances. Theorem 4.2 (total predcton uncertanty Under Model Assumptons 2.1 we obtan for calendar year t {I,..., I + J 1} msep C,J F t ( Ĉ (t,j =,l Ĉ (t,j Ĉ(t l,j ( { } exp e t Spost Dt c e t l 1, where the summaton runs over, l {t J + 1,..., I}. Theorems 4.1 and 4.2 gve closed form solutons for the ultmate clam predcton and the condtonal MSEP analyss. The key to these results s Theorem 3.3. Moreover, Theorem 3.3 provdes the full predctve dstrbuton of the nexperenced part of the clams development table whch would allow to analyze any other rsk measure usng Monte Carlo smulatons. 6

7 4.2 Clams development result (one-year uncertanty Crucal n solvency consderatons s the so-called clams development result (CDR. The CDR descrbes the changes n the predctors Ĉ(t,J f we update the nformaton from tme t to t + 1,.e. F t F t+1. The CDR at tme t + 1 for accdent year s defned by the dfference CDR (t+1 = Ĉ(t,J Ĉ(t+1,J. Because our successve Bayesan predctons (4.2 are (F t t -martngales we obtan the dentty [ ] E F t = 0. CDR (t+1 Ths explans that CDR (t+1 s typcally predcted by 0 at tme t, see also Merz-Wüthrch [15]. For the condtonal MSEP of ths predcton we obtan the dentty (CDR predcton uncertanty 2 msep CDR(t+1 ( (0 = E F t = Var Theorem 4.1 mples n a frst step {( Ĉ (t+1,j = C,t+1 exp = C,t exp ( CDR (t+1 0 Ĉ (t+1,j F t F t =,l e t+1 µpost D c t e t+1 Spost D c t+1 { ξ,t+1 + = Var ( CDR (t+1 (Ĉ(t+1 Cov,J, Ĉ(t+1 l,j e t+1 ( e t+1 µpost D c t e t+1 Spost D c t+1 F t F t. (4.6 } 1 {+J>t+1} (4.7 } 1 {+J>t+1}, e t+1 where 1 { } denotes the ndcator functon. The crucal observaton s that only the frst two terms n the above exponent depend on the observatons ξ D t+1 at tme t + 1, and ths dependence s lnear. In analogy to (3.1 we decouple (bjectvely these observatons n D t+1 as follows: ξ D t+1 (ξ Dt, ξ D t+1\d t. The frst term descrbes the components that are observable at tme t and the second term the components that are observed n calendar year t + 1, whch also means D t+1 \ D t D c t. Thus, for the latter components we also need to apply Theorem 3.3, condtonally gven F t. The frst random term on the rght-hand sde of (4.7, gven F t, s ξ,t+1. We defne a lnear map by choosng b t R Dc t such that b t ξdc t = ξ,t+1. The second random term, gven F t, on the rght-hand sde of (4.7 s gven by µ post D c t+1 = µ D c t+1 + S D c t+1,d t+1 ( SDt+1 1 (ξ D t+1 µ Dt+1. We decouple ξ D t+1. We defne the lnear functon B t+1 : R D t+1 R D t+1 wth B t+1 ξ D t+1 = ( ξ,j 1 {+j=t+1} (,j D t+1, 7

8 that s, all components of ξ D t+1 are set equal to 0 f they are F t -measurable. Ths mples ξ D t+1 = (1 B t+1 ξ D t+1 + B t+1 ξ D t+1, where the frst term s F t -measurable and the second term exactly corresponds to observatons n calendar year t + 1,.e. to components n D t+1 \ D t Dt c. To smplfy notaton we defne a lnear map by choosng p t R Dc t such that p t ξdc t = b t ξdc t + 1 {+J>t+1} e t+1 S D c t+1,d t+1 ( SDt+1 1 Bt+1 ξ D t+1. Ths then allows to rewrte (4.7 as follows Ĉ (t+1,j = g t (F t exp { } p t ξdc t, for an approprate F t -measurable varable g t (F t, see also Lemma 5.1 n Merz et al. [14]. Theorem 4.3 (CDR predcton uncertanty Under Model Assumptons 2.1 we have for t {I,..., I + J 1},, l {t J + 1,..., I} Cov F t (Ĉ(t+1,J, Ĉ(t+1 l,j = Ĉ(t,J ( { } Ĉ(t l,j exp p t Spost Dt c p t l 1. Proof. See Theorem 5.2 n Merz et al. [14]. Theorem 4.3 now allows to calculate the MSEP of the CDR gven n (4.6. In Lemma 5.3 of Merz et al. [14] there are dfferent representatons of B t+1, b t and p t whch make the calculatons more operatonal. 5 Examples and senstvtes In ths secton we study the nfluence of dfferent choces of the covarance structure Σ R d d on the resultng clams reserves R (t (Theorem 4.1 and the correspondng condtonal MSEP s for the total predcton uncertanty (Theorem 4.2 and for the CDR predcton uncertanty (Theorem 4.3, n partcular, we choose calendar year correlaton. Therefore, we revst the motor thrd party lablty (MTPL nsurance run-off data of Braun [2], see Table 4 below. Ths data s partcularly dffcult because the portfolo undergoes strong growth. We make dfferent choces for the covarance matrx Σ. Frst we rewrte ths covarance matrx Σ n order to get a better understandng. Smlar to the pror mean µ we assume that the condtonal standard devaton of ξ,j does not depend on the accdent year {1,..., I}, thus Var (ξ,j Θ 1/2 = σ j for all (, j J. We defne the standard devaton vectors σ = (σ 0,..., σ J R J+1 of ξ and σ = Aσ R d for ξ. Wth ths notaton at hand we can rewrte the covarance matrx Σ as Cov (ξ Θ = Σ = dag(σ Λ dag(σ, where Λ R d d denotes the condtonal correlaton matrx of ξ. The numercal values of the chosen standard devaton parameter σ are gven n Table 4 below. Our am n the next sectons s to examne dfferent (postve-defnte choces of the correlaton matrx Λ. For the pror uncertanty n Θ we choose T = τ 2 1 wth τ = 1 (ths s a rather non-nformatve choce. 8

9 accdent year clams reserves msep 1/2 (Ĉ(I C,J F I,J msep 1/2 (0 CDR (I+1 F I R(I absolute n % reserves absolute n % reserves % % , % 1, % 4 1,221 2, % 1, % , % 2, % 6 3,470 4, % 3, % 7 3,993 6, % 4, % 8 10,318 8, % 5, % 9 21,807 11, % 7, % 10 53,919 19, % 13, % ,247 30, % 20, % ,398 47, % 30, % ,756 97, % 76, % , , % 140, % cov 1/2 71,658 50,826 total 2,017, , % 172, % Table 1: Independent case Λ = 1 for MTPL portfolo provded n Table Independent choce for Λ As benchmark model we consder Λ = 1,.e. all components of ξ are condtonally ndependent, gven Θ (note that n the multvarate Gaussan case uncorrelatedness and ndependence concde. The resultng model s a Bayesan verson of Hertg s [7] log-normal clams reservng model. It corresponds to the model typcally analyzed n practce where one assumes that payments n dfferent accdent years are ndependent, condtonally gven the model parameters (see also Mack [13]. Under these model assumptons we obtan the results provded n Table 1. In ths table we gve the clams reserves and the correspondng condtonal MSEP s, based on the nformaton F I, for sngle accdent years {1,..., 14} and on the lne total for aggregated accdent years 14 =1. We observe that the calculated total clams reserves (I R (aggregated over all accdent years {1,..., 14} result n the value of 2,017,675. Our frst am s to study the condtonal MSEP s n ths benchmark model. We have, n vew of formula (4.5, msep C,J F I ( Ĉ (I,J = (Ĉ(I msep C,J F I,J + Cov (C,J, C l,j F I. (5.1 l The frst term on the rght-hand sde (rhs of (5.1 s the condtonal MSEP for sngle accdent years (whch s gven on lnes = 1,..., 14 n Table 1. The second term on the rhs of (5.1 corresponds to the mpled covarance from the smultaneous parameter uncertanty n Θ over 9

10 all accdent years, that s, Cov (C,J, C l,j F I = E [Cov (C,J, C l,j F I, Θ F I ] l l + Cov (E [C,J F I, Θ], E [C l,j F I, Θ] F I. l The frst term on the rhs s equal to zero (due to the uncorrelated choce gven by Λ = 1. The square-root of the second term s gven by 71, 658 (and for the CDR by 50, 826, see Table 1. Ths allows to defne the mpled average correlaton between accdent years ψ a.y. (0 by ψ a.y. (0 def. l = Cov (C,J, C l,j F I l Var (C,J F I 1/2 = 4.3%. (5.2 1/2 Var (C l,j F I Thus, the mpled average correlaton between accdent years of the jont parameter uncertanty n Θ s 4.3% n our example. Bascally, ths means that, see (5.1, ( msep C,J F I Ĉ (I,J = (Ĉ(I msep C,J F I,J + ψ a.y. (0 l Var (C,J F I 1/2 Var (C l,j F I 1/2, wth ψ a.y. (0 = 4.3%. Completely analogously we obtan for the mpled average correlaton between accdent years for the CDR predcton uncertanty ψ CDR a.y. (0 = 3.8%. These mpled correlatons play an mportant role n practce because typcally they are specfed/adjusted by experts. We are now tempted to choose ψ a.y. (0 [0, 1], however, we wll see that ths noton can by qute controversy and even msleadng! 5.2 Calendar year correlaton for Λ We ntroduce calendar year dependence. The correlaton matrx Λ s chosen such that for a fxed ρ [0, 1 we have Cov (ξ,j, ξ l,k Θ = σ j σ k ρ 1 {+j=l+k} for (, j (l, k. (5.3 The results for these choces (5.3 are gven n Table 2. The frst observaton s that the clams reserves are strongly decreasng n ρ. Ths strkng observaton s gong to be dscussed n the next secton. A very mportant observaton s that the condtonal MSEP s are an ncreasng functon n ρ. For example, a correlaton parameter of ρ = 20% leads to a substantal ncrease of the square-rooted uncertanty of 276, , %! The full curve s gven n Fgure 1(a, lhs, where we see that ths ncrease s a non-lnear functon that needs to be analyzed n a careful bottom-up approach. 10

11 ( calendar year clams reserves msep 1/2 C,J F I Ĉ(I,J msep 1/2 (0 CDR(I+1 F I (I correlaton ρ R absolute n % reserves absolute n % reserves 0% 2,017, , % 172, % 10% 1,952, , % 196, % 20% 1,913, , % 216, % 30% 1,883, , % 234, % 40% 1,859, , % 251, % 50% 1,837, , % 266, % 60% 1,818, , % 280, % 70% 1,801, , % 293, % 80% 1,785, , % 305, % 90% 1,769, , % 317, % Table 2: Calendar year correlaton ρ accordng to ( % 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% 0% 20% 40% 60% 80% 100% 120% 100% 80% 60% 40% 20% 0% 0% 20% 40% 60% 80% 100% ultmate msep^(1/2 CDR msep^(1/2 lnear functon correlaton ultmate correlaton CDR Fgure 1: Calendar year correlaton ρ accordng to (5.3: (a lhs: relatve ncrease of square-rooted ( uncertantes as a functon of the calendar year correlaton ρ [0, 1 for msep 1/2 C,J F I Ĉ(I,J and msep 1/2 (0; CDR(I+1 F I (b rhs: average mpled correlatons between accdent years ψ a.y. (ρ and ψ CDR a.y. (ρ as a functon of the calendar year correlaton ρ [0, 1. Concluson 1 The ntroducton of calendar year dependence may substantally ncrease the predcton uncertanty. Ths relatve ncrease s non-lnear n the correlaton parameter ρ [0, 1 (Fgure 1(a, lhs. Next we calculate the average mpled correlatons between accdent years ψ a.y. (ρ and ψ CDR a.y. (ρ as a functon of the calendar year correlaton ρ [0, 1 (see also (5.2. We defne these average mpled correlatons between accdent years by ψ a.y. (ρ def. = l Cov(ρ (C,J, C l,j F I l Var(0 (C,J F I 1/2, (5.4 Var (0 1/2 (C l,j F I where Cov (ρ (, F I corresponds to the choce ρ [0, 1 and Var (0 ( F I to the choce ρ = 0. The results are presented n Fgure 1(b, rhs. We see that ψ a.y. (ρ and ψ CDR a.y. (ρ are non-lnearly 11

12 ncreasng functons n ρ. More surprsngly, we see that ψ a.y. (ρ > 1 for ρ > 85% whch seems to be an error at the frst sght. However, these fgures are correct and there results a counterntutve correlaton bgger than 1! The reason for ths value bgger than 1 s that (5.4 does not gve a well-specfed correlaton measure because enumerator and denomnator belong to dfferent models (specfcatons of ρ. For dfferent ρ s also the condtonal MSEP s for sngle accdent years (gven n Table 1 for ρ = 0 wll change; and thus Table 1 does not allow to reconstruct the results for ρ > 0 usng an average mpled correlaton factor (bounded by 1. We conclude agan that only a bottom-up calculaton can provde the full flavor of the results, because f we need to reconstruct Table 1 for ρ > 0 then we drectly obtan the full pcture of the overall uncertantes. Concluson 2 The ntroducton of an average mpled correlaton between accdent years gven by (5.4 does not allow to construct the condtonal MSEP s for ρ > 0 from the condtonal MSEP s wth ρ = 0 n an ntutve way, because ψ a.y. (ρ can take arbtrary and counter-ntutve values (Fgure 1(b, rhs. 3000, , , , , ,000,0 0% 20% 40% 60% 80% 100% clams reserves confdence ultmate confdence CDR 3000, , , , , ,000,0 0% 20% 40% 60% 80% 100% clams reserves confdence ultmate confdence CDR Fgure 2: Clams reserves and 2 standard devatons confdence bounds for total uncertanty and CDR uncertanty as a functon of ρ [0, 1: (a lhs: model (5.3; (b rhs: model (5.5. Fnally, n Fgure 2 we plot the resultng clams reserves as a functon of calendar year correlaton ρ [0, 1 and the correspondng confdence ntervals of 2 standard devatons for the total uncertanty and the one-year CDR uncertanty. Fgure 2(a, lhs, shows model (5.3. As already mentoned, we observe a strong decrease n clams reserves whch we are gong to dscuss next. 5.3 Calendar year correlaton for Λ, modfed verson As mentoned above, we observe a strong decrease n clams reserves as a functon of calendar year correlaton ρ [0, 1, see Table 2. The reason for ths decrease s that the pcture s heavly dstorted by the strong growth of the portfolo, see frst column n Table 4, below. Ths strong growth mples that the model has dffcultes to dstngush (and allocate growth and calendar year effects,.e. to assgn volatlty to the accdent year axs (vertcal axs and the calendar year axs (dagonal axs. Ths s a common phenomenon n multvarate tme-seres analyss. There 12

13 ( clams reserves msep 1/2 C,J F I Ĉ(I,J msep 1/2 (0 CDR(I+1 F I (I correlaton ρ R absolute n % reserves absolute n % reserves 0% 2,017, , % 172, % 10% 1,987, , % 197, % 20% 1,972, , % 219, % 30% 1,962, , % 238, % 40% 1,954, , % 256, % 50% 1,948, , % 273, % 60% 1,943, , % 288, % 70% 1,939, , % 303, % 80% 1,936, , % 317, % 90% 1,933, , % 330, % Table 3: Calendar year correlaton ρ accordng to (5.5. are dfferent ways to crcumvent ths dffculty. The most obvous one s to ntroduce an accdent year parameter ν > 0 as mentoned n (2.3. The dffculty wth ths approach s that ν needs to be estmated n an approprate way whch requres more (external knowledge, smlar to the Bornhuetter-Ferguson clams reservng method. We propose to proceed dfferently: observe that the frst column n Table 4 only plays the role of a volume measure (or a denomnator for the chan-ladder method (we have a multplcatve structure n C,j. Therefore, we exclude the frst column from the correlaton analyss. Ths s acheved by settng the followng model assumptons for Λ. We choose Σ such that for ρ [0, 1 Cov (ξ,j, ξ l,k Θ = σ j σ k ρ 1 {+j=l+k and j,k 1} for (, j (l, k. (5.5 Assumpton (5.3 s modfed by the addtonal restrcton that we only consder correlaton for j, k 1. The results for these choces (5.5 are gven n Table 3 and Fgure 2(b, rhs. We observe that we obtan a much more reasonable pcture now whch s not dstorted by portfolo growth. The clams reserves only slghtly decrease as a functon of ρ and the uncertanty ncreases n a smlar magntude n both models (5.3 and (5.5. Ths brng us to a next concluson. Concluson 3 Strong portfolo growth or decrease needs a careful analyss because change of volume often conceals other effects. We close wth the followng general statement. Concluson 4 An overall top correlaton as specfed n Solvency 2 (see QIS5 [6], SCR.9.34 or n the Swss Solvency Test (see SST [17], Secton 8.4 typcally under-estmates the predcton uncertanty f one starts wth the benchmark model ρ = 0. 6 Conclusons We summarze the fndngs and gve a bref outlook, we also refer to the conclusons n Merz et al. [14] and n Wüthrch [19]: 13

14 There s no reasonable way (short-cut to choose correlaton on a top level because often ntuton fals as Conclusons 1-4 show. Overall correlatons can only be determned n a bottom-up approach. We provde the whole tool kt (Theorems 4.2 and 4.3 for ths bottom-up approach n the multvarate log-normal model. Other dstrbutonal models can often only be solved numercally and then the senstvty analyss becomes much ntransparent, see for nstance Wüthrch [18]. Other rsk measures than the condtonal MSEP (such as Value-at-Rsk or Tal-Value-at- Rsk can only be calculated numercally (n our case smple Monte Carlo smulatons can be used and both process and parameter uncertanty are consdered. In addton, t would be nterestng to analyze the behavor of these rsk measures under dependence structures wth tal dependence. The choce of the (postve-defnte covarance matrx Σ needs a thorough dscusson. Sometmes data s helpful but we beleve that expert opnon s as mportant as data n order to get reasonable choces for Σ. Often calendar year dependence s correlated wth nflaton that depends on fnancal market developments. Smlar to Donnelly-Wüthrch [5], Köng et al. [11] and Sh et al. [16] these dependence structures between nsurance clams and fnancal market movements should be studed n order to enhance the qualty of the predcton. Often an auto-regressve model s approprate to model such busness cycles. Theorem 3.3 on predctve dstrbutons apples to any multvarate Gaussan set-up, thus, t can be appled to any multvarate Gaussan clams reservng model, see for nstance the addtve model of Sh et al. [16]. However, the latter s more general n the sense that t also chooses a pror dstrbuton for the covarance matrx Σ. The dsadvantage of ths more general set-up s that t no longer allows for closed form solutons and requres Markov chan Monte Carlo smulatons, see also Wüthrch [18, 19]. References [1] Barnett, G., Zehnwrth, B. (2000. Best estmates for reserves. Proc. CAS LXXXVII, [2] Braun, C. (2004. The predcton error of the chan ladder method appled to correlated runoff trangles. Astn Bulletn 34/2, [3] Brehm, P.J. (2002. Correlaton and the aggregaton of unpad loss dstrbutons. CAS Forum, [4] Bühlmann, H., Gsler, A. (2005. A Course n Credblty Theory and ts Applcatons. Sprnger. [5] Donnelly, C., Wüthrch, M.V. (2012. Bayesan predcton of dsablty nsurance frequences usng economc factors. Annals of Actuaral Scence 6/2, [6] European Commsson (2010. QIS5 Techncal Specfcatons, Annex to Call for Advce from CEIOPS on QIS5. 14

15 [7] Hertg, J. (1985. A statstcal approach to the IBNR-reserves n marne nsurance. Astn Bulletn 15/2, [8] Jong, de P. (2006. Forecastng runoff trangles. North Amercan Actuaral Journal 10/2, [9] Johnson, R.A., Wchern, D.W. (1988. Appled Multvarate Statstcal Analyss. 2nd edton. Prentce-Hall. [10] Krschner, G.S., Kerley, C., Isaacs, B. (2008. Two approaches to calculatng correlated reserves ndcatons across multple lnes of busness. Varance 2/1, [11] Köng, B., Weber, F., Wüthrch, M.V. (2011. Predcton of dsablty frequences n lfe nsurance. Zavarovalnšk horzont, Journal of Slovensko aktuarsko združenje 7/3, [12] Kuang, D., Nelsen, B., Nelsen, J.P. (2008. Forecastng wth the age-perod-cohort model and the extended chan-ladder model. Bometrka 95, [13] Mack, T. (1993. Dstrbuton-free calculaton of the standard error of chan ladder reserve estmates. Astn Bulletn 23/2, [14] Merz, M., Wüthrch, M.V., Hashorva, E. (2013. Dependence modelng n multvarate clams run-off trangles. To appear n Annals of Actuaral Scence. [15] Merz, M., Wüthrch, M.V. (2008. Modellng the clams development result for solvency purposes. CAS E-Forum, Fall 2008, [16] Sh, P., Basu, S., Meyers, G.G. (2012. A Bayesan log-normal model for multvarate loss reservng. North Amercan Actuaral Journal 16/1, [17] Swss Solvency Test (2006. FINMA SST Technsches Dokument, Verson 2. October [18] Wüthrch, M.V. (2010. Accountng year effects modellng n the stochastc chan ladder reservng method. North Amercan Actuaral Journal 14/2, [19] Wüthrch, M.V. (2012. Dscusson of A Bayesan log-normal model for multvarate loss reservng by Sh-Basu-Meyers. North Amercan Actuaral Journal 16/3,

16 / j , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,997 µ σ Table 4: Observed motor thrd party lablty (MTPL data C,j wth + j I = 14. The vector µ = (µ0,..., µj denotes the pror mean of parameter vector Θ and σ = (σ0,..., σj denotes the standard devaton parameters mpled from Σ, and for the pror covarance matrx we choose T = τ 2 1 wth τ = 1. 16