Recent Development in Claims Reserving

Transcription

1 WDS'06 roceedings o Contributed apers, art, , 2006 SBN MATFZRESS Recent Development in Claims Reserving Jedlička Charles University, Faculty o Mathematics and hysics, rague, Czech Republic Abstract This contribution deals with recent development in the ield o mathematical loss reserving via Chain Ladder that is regarded as the most popular method or setting technical reserves in non lie insurance t could be ormulated deterministically or via a stochastic model However there are some drawbacks o using this method automatically that will be discussed ts generalisation Munich Chain Ladder method will be introduced as a result o it Finally we will present some urther results o more detailed analysis o this method including type o estimates, elasticities o reserve depending on estimate and problems o variability calculation ntroduction Non lie nsurance companies are obliged to set up technical reserves or not yet unpaid claims which occurred in the past calendar years The respective delay until the claim is paid is caused by the time between the date o accident and the date o reports to the insurer and moreover it will take another more time to settle the claim n order to give realistic inancial picture o the overall volume o the claims two types o technical reserves are set up RBNS reserve is set up or Reported But Not Settled claims and BNR reserve deals with the problem o ncurred But Not Reported claims The irst one may be determined by individual estimates or each known not paid claim regarding the experiences and expert opinion o uture compensation that is usually made by employee o claims department The latter reserve could be determined only via mathematical methods using the known development o paid compensation and RBNS reserve RBNS reserve is not set up individually as estimate o uture paid compensation or each and every claim, an actuary can use only data describing the development o paid compensation and estimate the sum o RBNS and BNR reserves together We will mark, i = 0,,n, j = 0,,n i or data o paid claim or incurred where n notiies the dimension o the data sets t is assumed that there is no development i n periods ater accident pass we want to distinguish type o triangle we will add upper indices or data o paid compensation or or incurred data sum o paid compensation and corresponding value o RBNS reserve These data are usually analysed in the so called run-o triangles which could be seen as a matrix where only data in the upper let triangle are known and our aim is to estimate the uture development in the lower right triangle Each row is interpreted as one accident period and each column as a development period ie the variable shows us overall paid or incurred value o all claims occurred in period i and paid or reported until j periods ater the accident happened Thus igures o each diagonal corresponds to one single calendar period Standard Chain Ladder Method That is the most widely used method in loss reserving used or each single run-o triangle t originates rom intuitive deterministic assumptions which were later generalised to obtain stochastic model o chain ladder Standard Chain Ladder - deterministic approach This method is described in the actuarial monographs, eg [ it Cipra, 1999] or [ it Mandl, 1999] and is based on the assumption that ratios o ollowing values in one raw are approximately 118

2 JEDLČKA: RECENT DEVELOMENT N CLAMS RESERVNG constant independently on accident period i but dependent on development period j That is +1 = j, i = 0,,n j = 0,,n 1 1 Regarding the act that we know only data i i+j n estimate j could be based on values +1, i = 0,,n j 1 and, i = 1,,n j only ndividual development actors are deined as F = +1, i = 0, n,, j = 0,,n 1 ntuitive estimate j could be ormulated as arithmetic mean j = 1 F, j = 0,,n 1 n j However the most popular estimate is dierent j 1 +1 j = j 1 2 ts mathematical interpretation could be seen later, based on article [ it Mack, 1993] Our aim is to estimate ultimate values o paid or incurred data which is done according to ollowing ormulae: n 1 Ŷ = i j, i = 1,,n j=n i and corresponding BNR or sum o BNR and RBNS reserves could be gained by subtracting the ultimate and diagonal igures: R i = Ŷ i, i = 1,,n No reserve or accident year 0 is made since we assume that the claim handling is inished ater n periods ater accident Standard Chain Ladder - stochastic model n addition to previous approach we can obtain not only the point estimates o reserves but the variability, mean square error which will imply under normality assumption overall distribution Adequacy o normality assumption should be tested but it does not seem to contradict a reality due to Central limit theorem as run-o development is a sum o individual igures or each claim or policy The stochastic model was irstly presented in the article [ it Mack, 1993] and is based on 3 probabilistic assumptions regarding expectation, variability and inter row independencies t is assumed that or random vector holds E +1, 1,, i,0 = j, i = 0,,n, j = 0, n 1 and or its variability holds that var +1, 1,, i,0 = σ 2 j, i = 0,,n, j = 0, n 1 To simpliy the notation we will deine i j i,0,, We can rewrite this into a linear model or each development period +1 = j + ε, i = 0,,n 3 with notation Eε i j = 0 and varε i j = σ 2 j Moreover it is assumed in [ it Mack, 1993] that loss development between dierent accident years are uncorrelated, that is cov i1,j, i2,j = 0, i 1 i 2 119

3 JEDLČKA: RECENT DEVELOMENT N CLAMS RESERVNG Using Aitken estimate or model 3 we obtain j as j = j 1 +1 j 1 4 since rom theory o linear models j =,j V 1,j 1,j V 1,j+1 = σ 2 j 1 1 σ 2 j 1 +1 t is now easy to obtain 4, using notation,j = 0,j,,,j,j+1 = 0,j+1,,,j+1 and V = Varε,j = σ 2 j,j or covariance matrix n this univariate case there is no need o σ 2 j estimate or computation o j However it is used or computing mean square error o the reserve Mack in his article [ it Mack, 1993] suggested ollowing straight orward estimate o variability o development actors σ 2 j = 1 n j 1 k=n i +1 We can apply this ormula to compute mean square error o the overall reserve mse ˆR N σ i = Ŷ 2 k k Ŷ n k i,k j=1 5 Munich Chain Ladder We could apply now stochastic model o Chain Ladder separately to aid and ncurred data and we would expect that the ultimates o both triangles should be comparable since ater suiciently long development all claims are paid and no RBNS reserve should be booked Since it does not hold in practice, paper [ it Quarg, 2004] introduced method analysing both triangles and their interdependencies simultaneously We remind that we use upper right indices to distinguish values and parameters o each type o triangle, eg, j,σ j etc nter triangular dependencies are modelled via ratios o paid and incurred values Q = / = Average ratio or development period j is later deined as q j = / j = Standard Chain Ladder method SCL is used instead o Munich Chain Ladder method MCL the problem with inconsistence exists or known data as well as or prediction More accurately, it can be proved that i paid incurred ratio is under average in the time o estimate it will persist in the prediction or this accident year and vice versa MCL solves this problem very elegantly adjusting the developments actors This adjustment is based on thought that i current paid incurred ratio is low ie below average it means that it is not paid enough or reserved more than enough comparing to another accident years So it is expected that the amount o payments will be increased in uture period which implies that the corresponding paid development actor should be increased and corresponding incurred actor should be lower than usual oppositely paid and incurred ratio is above average it may j 2 120

4 JEDLČKA: RECENT DEVELOMENT N CLAMS RESERVNG be interpreted that the uture payment will be lower or increase o incurred will be substantially higher These types o dependencies are modelled or all development period ater standardisation Thus we use residual values with mean 0 and standard deviation 1 since ResX C = X EX C σx C We ormulate two regression models which inally produce ollowing estimates o development actors and +1 E Res B i s E Res = λ ResQ 1 i j +1 B i s = λ ResQ i j t was switched rom paid incurred ratio Q to incurred paid ratio Q 1 to obtain positive correlation in both cases B i s notiies two dimensional process i s, i s o both data types in the time o reserve estimates resp σ +1 E B i s = s + λ E +1 B i s σ = s + λ +1 i s σq 1 is +1 i s σq 1 is Q 1 EQ 1 is 6 Q EQ i s Moreover we assume that vectors B i1 s and B i2 s are stochastically independent i i 1 i 2 Let us assume that Q is deined as arameters λ a λ determine then the adjustment o SCL development actors For practical implementation we have to obtain urther estimates o σq 1 is, σq i s and σq 1 is Estimate o EQ i s is ormulated as q s = s / s Estimate o variability o paid incurred ratio σq i s is suggested as ρ s / using 2 ρ s = 1 s n s Q q s 2 n the same way we can obtain that q 1 s = s / s estimates EQ 1 and also ρ 2 s / is estimate o σq 1 ρ using s = 1 n s Comparing o parameters estimates is s Q 1 q s 1 2 Estimate o regression parameters λ and λ was originally in the article [ it Quarg, 2004] obtained by ordinary least square method n our opinion this method is not the most suitable one since the result is strongly aected by the outliers which may occur in this kind o situation ndeed we analysed three dierent portolio including original data used in the article [ it Quarg, 2004] and two another portolios We can thus compare original ordinary least squares estimates o λ parameters with estimates obtained by some robust methods We decided to use Huber s robust regression approach, bisquare methods and Least trimmed squares LTS methods Generally speaking the irst two methods evaluate each observation and the outliers receive lower weight Apart rom this approach LTS method directly cut o the outlying observation which does not correspond with probabilistic model Dierences between LTS1 and LTS2 are based on numbers o observations that are assumed not to contradict the model t is about 60% in irst situation and 75% approximately in the latter case 121

5 JEDLČKA: RECENT DEVELOMENT N CLAMS RESERVNG The results o our calculation o parameters estimate and ultimate values can be seen in the ollowing table The results o Bi Square method are not presented or third portolio since they did not dier rom Hueber method type o CL SCL MCL regression est no regression OLS method Hueber Bi Square LTS 1 LTS 2 p 1 λ ,77 Ŷ λ Ŷ / 95% 99% 99% 99% 98% 99% p 2 λ ,45 Ŷ λ Ŷ / 123% 91% 91% 91% 91% 91% p 3 λ Ŷ λ Ŷ / 91% 101% 101% 101% 101% The dierences in the ultimate values depending on applied regression estimate lead us to urther sensitivity study o relationship between these two variables The derivation will be perormed only or aid data The principles or ncurred are analogous We will start rom ormula 6 to deine estimate o development actor used in reserve calculation as i,k = k + λ σ k 1 Qi,k 1 qk ρ k t is straightorward that ultimate value o paid amount due to claims occurred in accident period i is calculated as Ŷ = i,ai n 1 j=ai using notation ai = n i we inspect the value o paid ultimate estimate Ŷ as a unction o λ we can derive how strongly the ultimate values and thus also reserve since reserve is dierent only by a constant diagonal value are aected by the choice o appropriate estimate o λ We can write all derivative are understood with respect to λ : Ŷ = n 1 j=ai i,ai i,ai 1 = Ŷ n 1 j=ai Using ormula i,k = k + λ i,k we can make inal adjustment o the above mentioned ormula Ŷ = 1λ n 1 j 1 j=ai We urther derived rather surprising result that E Ŷ B i ai = 0 i the expectation exists That could be interpreted there is no systematical inluence o varying the regression 122

6 JEDLČKA: RECENT DEVELOMENT N CLAMS RESERVNG estimates onto the ultimates values t is rational that we do not see regression estimates as random variable since we are interested in the sensitivity only t is easy to prove that E B i s, λ = 0 since the model assumptions imply that EQ B i s, λ = q s independently on accident period i Using again ormula i,k = k + λ i,k we get E i,k B ik, λ = E k rovided that both expectations exist we later obtain E i,k B i k, λ = E k + λ i,k B i k, λ = 1 + λ E i,k B i k, λ = 1 k This proves the ormula E Ŷ k B i ai = 0 Munich Chain Ladder gave us so ar only ormula or E +1 k B i s or E +1 B i s and no inormation about the variability o development actors We will drive this starting rom regression model o residual data t is again suicient to perorm the derivation or paid triangle only The standard linear model theory implies that σr 2 Res2 i s var Res +1 i s B i s = i j,i+j n Res2 Rearranging this ormula we obtain +1 var B i s = varλ σ 2 +1 i s i s = varλ Res 2 Res 2 / i s t is straightorward to substitute the theoretical parameters by their estimates similarly as in ormula or expectation This potentially enables us to calculate the mean square error as or Munich Chain Ladder, see 5, similarly as or Standard Chain Ladder Conclusion and implication or urther research The recent developments o the most popular method o actuarial reserving in non lie insurance were described and widely discussed in this paper t has been shown that standard method o estimates are not the best solution in its recently published generalisation We succeeded in deriving some more properties o that Munich Chain Ladder method that looks as useul especially related to various parameter estimates Moreover we discussed ormula or variability o development actors that could be used or means square error calculation similarly as in Mack s model Acknowledgments The author thanks his supervisor, pro Tomas Cipra or valuable comments, remarks and overall help with the research Reerences Cipra, T, ojistná matematika: teorie a praxe Ekopress, raha 1999 Jedlička,, Kočvara, J, Strnad, J, Techniky výpočtu BNR rezerv a jejich aplikace v pojištěn i odpovědnosti z provozu vozidla, Seminář z aktuárských věd, Univerzita Karlova, raha, 2004 Mack, T, Distribution ree Calculation o the Standard Error o Chain Ladder Reserves Estimates, ASTN Bulletin, Vol 23, No 2, 1993 Mandl,, Mazurová, L, Matematické základy neživotn iho pojištěn i, Matyzpress, raha 1999 Quarg, G, Mack, T, Munich Chain Ladder, Munich Re Blatter, Munich, 2004 i s 123