Estimation of Tail Development Factors in the Paid-Incurred Chain Reserving Method

Transcription

1 Estimation of Tai Deveopment Factors in the aid-incurred Chain Reserving Method by Michae Merz and Mario V Wüthrich AbSTRACT In many appied caims reserving probems in &C insurance the caims settement process goes beyond the atest deveopment period avaiabe in the observed caims deveopment triange This makes it necessary to estimate so-caed tai deveopment factors which account for the unobserved part of the insurance caims We estimate these tai deveopment factors in a mathematicay consistent way This paper is a modification of the paid-incurred chain (IC) reserving mode of Merz and Wüthrich () This modification then aows for the prediction of the outstanding oss iabiities and the corresponding prediction uncertainty under the incusion of tai deveopment factors KEYwORdS Tai factors caims reserving paid-incurred chain outstanding oss iabiities IC mode caims deveopment triange utimate caim prediction prediction uncertainty MSE VOLUME 7/ISSUE CASUALTY ACTUARIAL SOCIETY 6

2 Variance Advancing the Science of Risk Introduction and mode assumptions Often in &C caims reserving probems the caims settement process goes beyond the atest deveopment period avaiabe in the observed caims deveopment triange This means that there is sti an unobserved part of the insurance caims for which one needs to buid caims reserves In such situations caims reserving actuaries appy so-caed tai deveopment factors to the ast coumn of the caims deveopment triange which account for the settement that goes beyond this atest deveopment period Typicay one has ony imited information for the estimation of such tai deveopment factors Therefore various techniques are appied to estimate these tai deveopment factors Most of these estimation methods are ad hoc methods that do not fit into any stochastic modeing framework opuar estimation techniques for exampe fit parametric curves to the data using the right-hand corner of the caims deveopment triange (Mack 999; Boor 6; Verra and Wüthrich ) In practice one often does a simutaneous study of caims payments and caims incurred data ie incurred-paid ratios are used to determine tai deveopment factors (see Section 3 in Boor 6) In this paper we review the paid-incurred chain (IC) reserving method The og-norma IC reserving mode introduced in Merz and Wüthrich () can easiy be extended so that it aows for the incusion of tai deveopment factors in a natura and mathematicay consistent way Simiar to common practice the tai deveopment factor estimates wi then be based on incurred-paid ratios within our IC reserving framework In the foowing we denote accident years by i { } and deveopment years by { + } Deveopment year refers to the atest observed deveopment year of accident year i = and the step from to + refers to the tai deveopment factors (see Figure ) Cumuative payments in accident year i after deveopment years are denoted by i and the corresponding caims incurred by I i Moreover for the utimate caim we assume i + = I i + with probabiity (see Figure ) This means that we assume that after severa deveopment periods beyond the atest observed deveopment year the cumuative payments and the caims incurred ead to the same utimate caim amount That is utimatey when a caims of accident year i are setted I i + and i + must coincide Mode Assumptions Log-norma IC reserving mode Merz and Wüthrich () Conditionay given parameters Q = (F F + Y Y s s + t t ) we have Figure IC reserving mode Left pane: cumuative payments i deveopment triange; Right pane: caims incurred I i deveopment triange; both eading to the same utimate caim amount i+ = I i+ for accident year i i / + / i i caims payments i i i+ =I i+ I i caims incurred I i i 6 CASUALTY ACTUARIAL SOCIETY VOLUME 7/ISSUE

3 Estimation of Tai Deveopment Factors in the aid-incurred Chain Reserving Method the random vector (x x + z z ) has a mutivariate Gaussian distribution with uncorreated components given by x i ~ N(F s ) z i ~ N(Y t ) for i { } and { + } and for i { } and { }; cumuative payments i are given by the recursion = + i = i - exp{x i } with initia vaue i - = ; caims incurred I i are given by the (backwards) recursion = I i = I i + exp{-z i } with initia vaue I i + = i + The components of Q are independent and s t > for a (with probabiity ) For an extended mode discussion we refer to Merz and Wüthrich () Basicay the IC Mode Assumptions are a combination of Hertig s (985) og-norma mode (appied to cumuative payments) and Gogo s (993) Bayesian caims reserving mode (appied to caims incurred) In contrast to the IC reserving mode in Merz and Wüthrich () we now add an extra deveopment period from to + This is exacty the crucia step that aows for the consideration of tai deveopment factors and it eads to the study of incurredpaid ratios for the incusion of such tai deveopment factors The IC Mode Assumptions may be criticized because of two restrictive assumptions We briefy discuss how these can be reaxed Assumption i- = for a i { }: If there are known (prior) differences between different accident years i this can easiy be integrated by setting i- = v i with constants v v > describing these prior differences Independence between x i and z i : This is probaby the main weakness of the mode However this assumption can easiy be reaxed in the spirit of Happ and Wüthrich (3) To keep the anaysis simpe we refrain from studying this more compex mode in the present paper Estimation of tai deveopment factors At time one has observed data given by the set D = I : i+ i i i and one needs to predict the utimate caim amounts i + = I i + conditiona on these observations D On the one hand this invoves the cacuation of the conditiona expectations E[ i + D Q] and on the other hand it invoves Bayesian inference on the parameters Q given D (see Theorems 4 and 34 in Merz and Wüthrich ) In this section we discuss how to modify the genera outine of Mode Assumptions to incorporate tai deveopment estimation Utimate caims prediction conditiona on parameters We appy Mode Assumptions to the tai deveopment factor estimation probem Therefore we need to specify the prior distribution of the parameter vector Q Often there is subectivity in caims incurred data I i because the use of different caims adusters with different estimation methods and changing reserving guideines Therefore for the present set-up we have decided to consider caims incurred data I i ony for the estimation of tai deveopment factors ie we work under the assumption of having incompete caims incurred trianges (see aso Dahms (8) and Happ and Wüthrich (3) for caims reserving methods on incompete data) It is not difficut to extend the mode to incorporate a caims incurred information but in the present work this woud detract from the tai deveopment factor estimation discussion VOLUME 7/ISSUE CASUALTY ACTUARIAL SOCIETY 63

4 Variance Advancing the Science of Risk The prediction based on incompete caims incurred data is done as foows Assume there exists * { } such that with probabiity and if * < Ψ τ () τ = τ = * * + =τ τ Ψ = Ψ = =Ψ τ () * * + Note that if * = we simpy assume Y t / These assumptions impy that there is no substantia caims incurred deveopment after caims deveopment period * ie there is no systematic drift in the caims incurred deveopment after * This is seen as foows for {* } E[ exp{ ζ i } ] = E[ E[ exp{ ζi } Θ] ] = E exp Ψ +τ = [ { }] This impies that on average the caims incurred prediction is correct (and we have ony pure random fuctuations around this prediction) ie for {* + + } E[ Ii Ii ]= Ii Vco I I = exp τ ( i i ) { } where Vco( ) denotes the coefficient of variation The fact that we aow t to differ from t corresponds to the difficuty that the tai deveopment factor may cover severa deveopment years beyond the ast observed coumn in the caims deveopment triange and therefore we may aow for standard deviation parameters t > t for the deveopment period from to + (possiby covering more than one period) Remark If there is expert udgment about a drift term in the caims incurred deveopment I i* I i+ this can easiy be integrated by adusting assumptions () () This aso aows one to consider parametric curves as mentioned in Section but in this case it is more appropriate to treat this knowedge as informative as to prior distributions specifying prior uncertainty in this expert udgment simiar to Verra and Wüthrich () Thus assumptions () () impy that there is no systematic drift in {* + + } and under these assumptions we consider tai factor estimation under the restricted observations given by { i k } D* = I : i+ k+ * i k = D I : * In this spirit we consider a cumuative payment observations but ony caims incurred observations from deveopment year * on That is ony the caims incurred I i from the atest - * + deveopment periods * * + are used to estimate tai deveopment factors and the caims reserves We define the foowing parameters η = Φ m and w = σ m m= m= for = + µ = η Ψ and v = w + τ + n n= + n n= Moreover we define the parameters for = * w+ w > for = * β v w = for = * The foowing resut shows that b can be interpreted as the credibiity weight for the caims incurred observations: Theorem Under Mode Assumptions we have conditiona on Q and D* [ ] β E D* Θ = I i β i i + i i i i + ( β i) ( Φ+σ ) +β i Ψ = i+ = i exp 64 CASUALTY ACTUARIAL SOCIETY VOLUME 7/ISSUE

5 Estimation of Tai Deveopment Factors in the aid-incurred Chain Reserving Method For the conditiona variance we obtain [ ] Var D* Θ = E D* Θ i + i + ( β σ ) + {( i) } = i+ exp For i > - * there hods b -i = and therefore we obtain a purey caims payment based prediction [see aso Hertig s mode (985) presented in Section of Merz and Wüthrich ()] + i i Φ+σ = i+ exp For i - * there hods b -i > and therefore we obtain a correction term to the purey caims payment based prediction which is based on the caims incurred-paid ratio I i -i / i -i ie for a arge incurredpaid ratio we get a higher expected utimate caim as can be seen from β β I = exp ( β ) og +β og I i i i i i i i i i i i i Ii i = i iexp β i og i i arameter estimation the genera case The ikeihood function of the restricted observations D* is given by [see aso (35) in Merz and Wüthrich ()] ( Θ) D* = * exp σ σ Φ og exp v w v w i i ( i i) I µ i η i og i i i i Ii exp Ψ + τ og Ii + i i = * where means that ony reevant terms dependent on Q are considered The first ine describes the caims τ payment deveopment the ast ine describes the caims incurred deveopment and the midde ine describes the gap between the diagona caims incurred and the diagona caims payment observations In order to perform a Bayesian inference anaysis on the parameters we need to specify the prior distribution of Q Mode Assumptions IC tai deveopment factor mode We assume Mode Assumptions hod true with positive constants s s + t * = = t - = t Y * = = Y - = t / and Y = t / Moreover it hods Φ N( φ s ) for m { + } m m m with prior parameters f m R and s m > Under Mode Assumptions the erior distribution u(f D* ) of F = (F F + ) given D* is given by + u( Φ D* ) D* Θ ( Φm φm) exp (3) m= sm This immediatey impies the foowing theorem: Theorem 3 Under Mode Assumptions the erior u(f D* ) of F is a mutivariate Gaussian distribution with erior mean (f f + ) and erior covariance matrix S(D* ) Define the erior standard deviation by s = s + + σ for ( ) = + Then the inverse covariance matrix S(D* ) - = (a nm ) n m + is given by ( n ) ( m ) anm = ( sn ) n m + ( vi wi ) { = } { nm * + } * The erior mean (f f + ) is obtained by ( φ φ ) = ( D + ) ( c c + ) VOLUME 7/ISSUE CASUALTY ACTUARIAL SOCIETY 65

6 Variance Advancing the Science of Risk with vector (c c + ) given by c = φ + s σ + * + v og w i i i i I og i i i i i + τ +τ { * + } Note that the ast term in the definition of a nm and in the definition of c corresponds to the deveopment years in D* where we have both caims payments and caims incurred information Theorem 3 immediatey impies the foowing coroary: Coroary 4 Under Mode Assumptions the erior u(f D* ) of F is a mutivariate Gaussian distribution with F F * (F *+ F + ) being independent with ) N s Φ * { D ( φ =γφ+ γ φ } (4) for * and credibiity weight and empirica mean defined by γ + = φ = + + σ s and + og i i for = * Henceforth Coroary 4 shows that for deveopment years * we obtain the we-known credibiity weighted average between the prior mean f and the average observation f The case > * is more invoved: one basicay obtains a weighted average between the prior mean f the average observation f and the incurred-paid ratios og I i -i / i -i i - + Remark Mode Assumptions specify a Bayesian mode with mutivariate Gaussian distributions This setup aows for cosed-form soutions For other distributiona assumptions the probem can ony be soved numericay using Markov chain Monte Caro methods Bayesian statistics ike the Bayesian information criterion BIC woud then aow for mode testing and mode seection If one restricts to inear credibiity estimators see Bühmann and Giser (5) then f given in (4) corresponds to the inear credibiity estimator in more genera modes 3 arameter estimation specia case * = We consider the specia case * = that is ony the caims incurred observation I is considered in the tai deveopment factor anaysis This immediatey provides: Coroary 5 Choose * = Under Mode Assumptions the erior distribution u(f D* ) of F is a mutivariate Gaussian distribution with F F + being independent For m * = the erior distribution of F m is given by (4) The erior of F + is given by Φ D N + = + * ~ φ γ + + og with inverse variance given by a I τ + γ = s + ( σ +τ ) and credibiity weight given by γ + = + ( σ +τ + ) s+ a φ This means that in the case * = we obtain a credibiity-weighted average between the prior tai deveopment factor f + and the observation og I Henceforth in this case ony the atest incurredpaid ratio is considered for the estimation of the tai deveopment factor 3 osterior caims prediction and prediction uncertainty 3 Genera case In view of Theorems and 3 we can now predict the utimate caim i+ conditiona on the restricted observations D* under Mode Assumptions 66 CASUALTY ACTUARIAL SOCIETY VOLUME 7/ISSUE

7 Estimation of Tai Deveopment Factors in the aid-incurred Chain Reserving Method roposition 3 Bayesian utimate caims predictor Under Mode Assumptions we predict the utimate caim i+ given D* by β i β i E D* I exp i + = i i i i + β i + τ iτ ( i) + β + ( β i ) = i+ σ + {( i) exp β ( D* ) e e i + i + = i+ φ where e = ( ) R + with the first components equa to Next we determine the prediction uncertainty Mode Assumptions and Theorem 3 constitute a fu distributiona mode which aows for the cacuation of any risk measure (using Monte Caro simuations) under the erior distribution given D* Here we use the most popuar measure for the prediction uncertainty in caims reserving the socaed conditiona mean square error of prediction (MSE) The conditiona MSE has the advantage that we can cacuate it anayticay Anaytica soutions have the advantage that they aow for more basic sensitivity anaysis The conditiona MSE is given by (see aso Section 3 in Wüthrich and Merz (8)) msep E i D ( + * ) i + * D = E i E i D D + + * * = Var ( i+ D* ) ie in this Bayesian setup the conditiona MSE is equa to the erior variance This erior variance aows for the usua decouping into average processes error and average parameter estimation error; see (A3) The conditiona MSE satisfies Var( i + D* ) = Cov ( i + k + D* ) ik = We obtain the foowing theorem: Theorem 3 Under Mode Assumptions the conditiona MSE of the Bayesian predictor E[S i = i+ D* ] for the aggregate utimate caim S i = i+ is given by msep E i D ( * ) * + i + D i = = ( β i)( β k) e i Σ ( D) e k + i k ( β i ) Σ i ( * { = } = + σ e ) ik [ i + ] [ k + ] E D* E D* 3 Specia case * = with non-informative priors We revisit the specia case * = and we aso assume non-informative priors meaning that s In that case we obtain that the erior distributions of F F + are independent Gaussian distributions with Φ * { N φ D =φ = } + for and ( s ) σ = + I Φ N φ D = + τ + { } + og * ( s ) = a = σ +τ i og i This impies for the utimate caim prediction for i > + ( s ) E [ i D] = i i φ + σ + * exp + = i+ = i i = i+ with chain-adder factors fˆ ( ut) fˆ (3) + { } fˆ σ = exp φ (3) VOLUME 7/ISSUE CASUALTY ACTUARIAL SOCIETY 67

8 Variance Advancing the Science of Risk Tabe Observed caims payments data i i < I fˆ ( ut) + = exp { σ + +τ} (33) That is the first terms in the product on the righthand side of (3) are the cassica chain-adder factors for Hertig s og-norma mode (985); see aso (5) (5) in Wüthrich and Merz (8) The ast term in (3) however describes the tai deveopment factor (adusted for the variance) For i = we have E D = fˆ ( ut) * = I exp σ +τ (34) [ + ] + { + } 4 Exampe In this section we provide an exampe We assume that = 9 and that the caims payment data i and the caims incurred data I i for i + are given by Tabes and respectivey We first need to determine * We choose the vaue * such that there is no substantia caims incurred deveopment (no systematic drift) after deveopment period * This choice is made based on actuaria udgment We therefore ook at the individua chain-adder factors I i+ /I i and i + + These are provided in Tabe 3 In the upper right triange in Tabe 3 (with the individua chain adder factors for years 6 7 8) we see no further systematic deveopment so we concentrate on possibe choices * {6 9} The standard deviation parameters s s and t shoud be determined with prior knowedge ony In our exampe we assume that we have noninformative priors which means that we set s = For s and t we take an empirica Bayesian point of Tabe Observed caims incurred data I i i < CASUALTY ACTUARIAL SOCIETY VOLUME 7/ISSUE

9 Estimation of Tai Deveopment Factors in the aid-incurred Chain Reserving Method Tabe 3 Individua chain adder factors I i /I i for > and i < average view and estimate them from the data For = - we set i σˆ = og φ i Unfortunatey s and s + cannot be estimated from the data because we do not have sufficient observations Therefore we make the ad hoc choice σ ˆ = σ ˆ = min σˆ σˆ σˆ σˆ + We estimate the parameter t = t* = = t - with the empirica standard deviation of og I i+ /I i for i + + and 6 (because we assume that there is no systematic caims incurred deveopment after deveopment period 6; see Tabe 3) Finay for t we do the ad hoc (expert) choice t = 3t This suggests that we have (approximatey) another three uncorreated deveopment periods beyond = 9 unti a caims are finay setted Of course additiona information about t (if avaiabe) shoud be used here These choices provide the standard deviation parameters given in Tabe 4 Now we are ready to cacuate the caims reserves and the corresponding prediction uncertainty in our mode according to roposition 3 and Theorem 3 We do this for * {6 9} The resuts are provided in Tabe 5 Interpretations The anaysis shows that in the presence of tai deveopment Hertig s mode (985) may substantiay underestimate the outstanding oss iabiities compared to the IC tai deveopment factor modes for * = Ony the IC tai deveopment factor mode for * = 6 gives simiar reserves This comes from the fact that the incurred deveopment factors sti give a downward trend to incurred osses in deveopment periods 6 and 7 (see average in Tabe 3) which contradicts our mode assumptions () () and suggests to choose * = 8 or 9 Of course as mentioned above this expert choice is based on the rationae that there is no systematic drift after * and statistica methods coud ustify this hypothesis/ choice Incuding tai deveopment factors for * = 8 9 aso gives a higher prediction uncertainty msep / compared to Hertig s mode (985) without tai deveopment factors This finding is in ine with the Tabe 4 Estimated ˆ for and ˆ for ŝ tˆ 37 VOLUME 7/ISSUE CASUALTY ACTUARIAL SOCIETY 69

10 Variance Advancing the Science of Risk Tabe 5 Estimated caims reserves and corresponding prediction standard deviation in the IC tai deveopment factor mode (Mode Assumptions ) for * {6 9} and the estimated caims reserves according to Hertig s mode (985) [see Section 3 in Merz and Wüthrich ()] without tai deveopment factor reserves msep / reserves msep / reserves msep / reserves msep / reserves msep / Hertig s mode [6] no tai factor * = 9 IC tai factor * = 8 IC tai factor * = 7 IC tai factor * = 6 IC tai factor tot ones in Verra and Wüthrich () and shows that prediction uncertainty needs a carefu evauation in the presence of tai deveopment Note that for * = 9 we simutaneousy consider caims payments and caims incurred information for accident year i = For * = 8 we simutaneousy consider caims payments and caims incurred information for accident years i = This resuts in a much ower prediction uncertainty in these accident years (above the horizonta ine in the corresponding coumns of Tabe 5) The reason is that the caims incurred information has ony itte uncertainty (since we assume Y to be constant for *) This substantiay reduces the prediction uncertainty We may question whether there is so much information in these ast caims incurred observations If this is not the case we shoud either increase t and t or we shoud use ess informative priors in () () The atter woud bring us back to the mode of Merz and Wüthrich () and Happ and Wüthrich (3) with the additiona assumption that there is no systematic drift after * Moreover this atter mode woud aso aow us to consider more information than ust the restricted one given by D* In the present work we have decided to work with the restricted information D* ony because then we can fuy concentrate on tai factor estimation Otherwise tai factor estimation woud be more hidden in the data and anaysis 5 Concusion We have modified the IC reserving mode from Merz and Wüthrich () so that it aows for the incorporation of tai deveopment factors These tai deveopment factors are estimated considering caims incurred-paid ratios in an appropriate way This extends the ad hoc methods used in practice and because we perform our anaysis in a mathematicay consistent way we aso obtain formuas for the prediction uncertainty These are obtained anayticay for the conditiona MSE and these can be obtained numericay for other uncertainty measures using Monte Caro simuations (because we work in a Bayesian setup) The case study highights the need to incorporate tai deveopment factors in the presence of tai deveopment since otherwise both the outstanding oss iabiities and the prediction uncertainty are underestimated 7 CASUALTY ACTUARIAL SOCIETY VOLUME 7/ISSUE

11 Estimation of Tai Deveopment Factors in the aid-incurred Chain Reserving Method A Appendix: roofs In this appendix we prove a the statements We start with a we-known resut for mutivariate Gaussian distributions see eg Appendix A in osthuma et a (8) and ohnson and Wichern (988): Lemma A Assume (X X n ) is mutivariate Gaussian distributed with mean (m m n ) and positive definite covariance matrix S Then we have for the conditiona distribution: X Xn X { } N m + ( X m ) ( where X () = (X X n ) is mutivariate Gaussian with mean m () = (m m n ) and positive definite covariance matrix S S is the variance of X and S = S is the covariance vector between X and X () roof of Theorem We first consider the case i > - * that is I ik D* for k = - i henceforth for accident years i > - * we do not consider caims incurred information Using the conditiona independence of accident years given the parameters Q we obtain [ i + ] [ i + i i ] E D* Θ = E Θ Furthermore i > - * impies b -i = Therefore the caim foows from Mode Assumptions as in () in Merz and Wüthrich () and because b = for < * Simiary we obtain for the conditiona variance + i + [ i + ] { } = i+ Var D* Θ = E D* Θ exp σ ) The case i - * is more invoved Using again the independence of accident years conditiona on Q we obtain [ i + ] [ i + i i i i * i i ] E D* Θ = E I I Θ henceforth we now have both caims payments and caims incurred observations for accident year i - * We set = - i then using Lemma A we obtain competey anaogous to Theorem 4 and Coroary 5 in Merz and Wüthrich () [ ] ( i ) ( i ) } Ei + D* Θ = exp{ η + + β og η +β ogi µ + β ( w w ) + + i i { } = + = β β = I exp β ( Φ +σ ) +β Ψ Anaogousy Theorem 4 from Merz and Wüthrich () impies for the variance [ ] Var i + D* Θ = Ei + D* Θ + exp ( β) σ This proves the theorem = + roof of Theorem 3 and Coroary 4 We first write a the reevant terms of the ikeihood of F given D* They are given by u( Φ D* ) * i exp ( Φ φ ) Φ s σ i og i = = i exp ( Φ φ ) Φ s σ i og i = * + = ( Φ φ ) s * exp + + exp + ( v i w i) + i Ii i Φm τ +τ m i i i og (A) = + From this we easiy see that the erior distribution of F given D* is again mutivariate Gaussian and there ony remains to determine the erior mean and covariance matrix If we square out a terms in (A) for obtaining the F and the F F n terms we find the covariance matrix S(D* ) First of a we observe that the deveopment periods with * are a on the first ine of (A) which proves the independence statement on F F * (F *+ VOLUME 7/ISSUE CASUALTY ACTUARIAL SOCIETY 7

12 Variance Advancing the Science of Risk F + ) Moreover we see for * that the erior variance of F given D* is given by s = s + ( + ) σ ( ) which provides a nm for n m = * The erior mean is given by φ =( s ) φ + s σ i og i Next we square out a terms for > * to get the covariance matrix We obtain + n= * + n * n n m ( v i w i) s σ Φ+ ΦΦ nm = * + ( n+ ) ( m+ ) n n + n m n = ( ) ( ) n n m ( vi wi ) n s σ Φ+ ΦΦ = * + nm = * + * n n This provides a nm for n m = * + + The erior mean is obtained by soving the erior maximum ikeihood functions for F * + They are given by ogu( Φ D* ) = φ + Φ s σ iτ +τ I + og * + + v w i i i i i i Henceforth this impies og i i +! Φ a = (A) m= * + m m ( c c ) = ( D* ) ( Φ Φ ) + + from which the caim foows roof of Coroary 5 The coroary foows from Theorem 3 and Coroary 4 roof of roposition 3 From Theorem we obtain [ ] [ [ ] ] β i β i Ei + D* = EEi + D* Θ D* = i i Ii i + iτ +τ E exp ( β i) ( Φ+σ ) +β i = i+ D* + β i i β σ τ +τ i = i i Ii i {( β i) +β i } exp = i+ + E exp ( β i) Φ D * = i+ From Theorem 3 we know that given D* I F = (F F + ) has a erior mutivariate Gaussian distribution with erior mean (f f + ) and erior covariance matrix S(D* ) Henceforth the + erior distribution of S =-i+ F is Gaussian with + mean S =-i+ f and variance e -i+ S(D* ) e -i+ This proves the proposition roof of Theorem 3 We obtain with the tower property of conditiona expectations [ ] ( [ i + ] [ k + ] ) Cov i+ D* k+ = E Cov i + k D* + Θ D* + Cov E D* Θ E D* Θ D* (A3) This is the usua decomposition into average process (co-)variance and average parameter error The first term in (A3) is equa to for i k because accident years i are independent conditionay given Q Henceforth there remains the case i = k Using Theorems and 3 we obtain [ ( i + ) ] + [ [ i + ] ] { i } = i+ + E Var D* Θ D* = E E D* Θ D* exp ( β ) σ { } ( β i) β i = I exp ( β ) σ +β ( i τ +τ ) i i i i i i = i+ ( β σ ) + E exp ( β i) Φ D * = i+ + {( i) } = i+ exp From Theorem 3 we know that given D* I F = (F F + ) has a erior mutivariate Gaussian distribution with erior mean (f f + ) and erior covariance matrix S(D* ) Henceforth the + erior distribution of S =-i+ F is Gaussian with + mean S =-i+ f and variance e -i+ S(D* )e -i+ This impies for the first term (A3) 7 CASUALTY ACTUARIAL SOCIETY VOLUME 7/ISSUE

13 Estimation of Tai Deveopment Factors in the aid-incurred Chain Reserving Method [ ] [ ] E Var i + D* Θ D* = Ei + D* exp ( β ) e ( D* ) e { i i+ i+ } + {( β i) σ } = i+ exp Finay we consider the ast term in (A3) Appying Theorems and 3 we obtain Cov ( E [ D* i + Θ] E[ k + D* Θ] D* ) + β σ i τ +τ i i i i Ii i {( i) i } β = exp β +β = i+ + β σ k τ +τ k β k k k Ik kexp ( β k) +β k = k+ ( Cov exp ( β i) Φ = i+ + exp{ ( β k) Φ} D* ) + { } = k+ Henceforth we need to cacuate this ast covariance term Due to Theorem 3 the oint distribution of the exponents is a mutivariate Gaussian distribution with covariance ( - b -i )( - b -k ) e -i+ S(D* ) e -k+ This impies Cov E D* Θ E D* D* i + Θ k + = E D* E D* i + k + exp{ β β i e + D* e k i k + } ( ( ) ) which is the we-known covariance formua for og-norma distributions Coecting the terms for i k gives the off-diagona terms For i = k we obtain the terms [ ] { } E D* exp β e D* e ( exp β σ ) i + i + i + i + {( i) } = i+ [ i + ] ( { i + i + i } ) + E D* exp ( β ) e ( D* ) e [ ] ( { = E D* exp ( β ) e ( D* ) e i + i + i + i + i } ) = i+ + ( β ) σ This competes the proof References Boor Estimating Tai Deveopment Factors: What to Do When the Triange Runs Out Casuaty Actuaria Society Forum Winter 6 pp Bühmann H and A Giser A Course in Credibiity Theory and its Appications New York: Springer 5 Dahms R A Loss Reserving Method for Incompete Caim Data Buetin of the Swiss Association of Actuaries 8 pp 7 48 Gogo D Using Expected Loss Ratios in Reserving Insurance: Mathematics and Economics 993 pp Happ S and M V Wüthrich aid-incurred Chain Reserving Method with Dependence Modeing ASTIN Buetin 43 3 pp Hertig A Statistica Approach to the IBNR-Reserves in Marine Insurance ASTIN Buetin pp 7 83 ohnson R A and D W Wichern Appied Mutivariate Statistica Anaysis (nd ed) Engewood Ciffs N: rentice-ha 988 Mack T The Standard Error of Chain Ladder Reserve Estimates: Recursive Cacuation and Incusion of a Tai Factor ASTIN Buetin pp Merz M and M V Wüthrich aid-incurred Chain Caims Reserving Method Insurance: Mathematics and Economics 46 pp osthuma B E A Cator W Veerkamp and E W van Zwet Combined Anaysis of aid and Incurred Losses Casuaty Actuaria Society E-Forum Fa 8 pp 7 93 Verra R and M V Wüthrich Reversibe ump Markov Chain Monte Caro Method for arameter Reduction in Caims Reserving North American Actuaria ourna 6 pp 4 59 Wüthrich M V and M Merz Stochastic Caims Reserving Methods in Insurance Hoboken N: Wiey 8 VOLUME 7/ISSUE CASUALTY ACTUARIAL SOCIETY 73