A Loss Reserving Method for Incomplete Claim Data

Transcription

1 A Loss Reserving Method for Incomplete Claim Data René Dahms Bâloise, Aeschengraben 21, CH-4002 Basel June 11, 2008 Abstract A stochastic model of an additive loss reserving method based on the assumption that the claim reserves are good measures for the remaining exposure is presented This model combines a projection of payments and a projection of corresponding reported amounts such that both leads to the same ultimate In addition, the presented method even works for some kind of incomplete triangles We will state estimators for the total necessary reserves and estimators for the corresponding standard error Moreover, we will discuss an example based on motor liability data, where we distinguish between property damage and bodily injury claims and only have proper data to do so for the last six accounting years Keywords: Stochastic Reserving, Loss Development, Mean Squared Error of Prediction 1

2 1 Motivation We want to describe a method which is able to deal with some kind of incomplete triangles of payments and reported amounts As an example you may keep in mind the following situation Assume your company covers motor liability Some years ago one decided to distinguish between bodily injury and property damage claims All new claims, all still open claims and all reopened claims have been migrated to the new claims handling philosophy This implies that for old years only some claims can be identified as bodily injury or property damage claims and therefore one cannot apply the Chain Ladder method on the separated triangles of bodily injury and property damage claims For a more precise definition let S i,k and T i,k denote the incremental payments and incremental reported amounts of accident year i and development year k, for 1 i, k n Here, n denotes the number of observed years By incomplete data we mean that S i,k and T i,k are missing, for i + k Y for some accounting year Y < n Clearly, in this situation the Chain Ladder method doesn t work properly But, if a good measure of the underlying risk for each accident year can be obtained, the Complementary Loss Ratio method, see [3], could be applied to both the incremental triangle of payments and the incremental triangle of reported amounts We will extend the idea of the Complementary Loss Ratio method using the case reserves (outstanding) at the end of the previous development year as risk measure In other words, payments and adjustments to the reported amount during the year are assumed to be proportional to the opening reserves, which are the case reserves at the end of the previous year In order to do this we need all case reserves R i,y i, i < Y We will present unbiased estimators for the total necessary reserves Moreover, we will use the ideas of Mack [2] to present an estimator for the corresponding conditional mean squared error 2 Definition of the model We want to extend the idea of incomplete triangles in the way, that we introduce non-negative weights w i,k, i + k n, which measures our confidence in the observed developing during the year k of accident year i The situation of the incomplete triangles described in the previous section is the special case of weights w i,k = 0, for i + k Y, and w i,k = 1, for Y < i + k n We will use the following basic notations: S i,k the payments during development year k of all claims of accident year i T i,k the adjustments to the reported amount during development year k of all claims of accident year i 2

3 R i,k the outstanding (case reserves) at the end of development year k of all claims of accident year i We consider S i,k, T i,k and R i,k as random variables for which we have observations for i + k n + 1 This variables are linked via R i,k+1 = R i,k S i,k+1 + T i,k+1, for 1 i n and 0 k < n, where we take R i,0 0 As usual we assume the accident years, ie {S 1,k, T 1,k, R 1,k : 1 k n},, {S n,k, T n,k, R n,k : 1 k n}, (21) are independent The minimum and maximum of two real numbers we denote by a b := max(a, b) and a b := min(a, b) Moreover, we define the information D k of the run-off triangles up to development year 1 k n by ( n ) D k := σ B i,k (n+1 i) (22) with i=1 B i,k := σ ({S i,l, T i,l : 1 l k}), (23) the run-off information of accident year i up to development year k The idea of case reserves being a good risk measure for incremental payments and reported amounts can be formalised as follows: Assumption 21 For 1 i n and 1 k < n we assume that Cov [( Si,k+1 T i,k+1 ), E [S i,k+1 B i,k ] = α k R i,k, (24) E [T i,k+1 B i,k ] = β k R i,k, (25) ) Bi,k] ( ) σ 2 = k γ k R i,k =: Σ 2 k R i,k (26) ( Si,k+1 T i,k+1 for some constants α k and β k and some positive definite, symmetric matrices Σ k Assumption (26) ensures that our weighted estimator for the necessary reserves will be of minimal variance under all linear estimators Moreover, (26) also implies that all reserves have to be non-negative and together with (24) and (25) we even get that all reserves have to be positive up to the time where no development is assumed to be taken place Standard calculations using conditional expectations gives us 3 γ k τ 2 k

4 Corollary 22 Assumption 21 implies the standard Chain Ladder assumptions for the reserve, ie Remark 23 E [R i,k+1 B i,k ] = (1 α k + β k )R i,k := f k R i,k, (27) Var [R i,k+1 B i,k ] = (σ 2 k 2γ k + τ 2 k )R i,k (28) 1) Hence Corollary 22 immediately implies 1 α k + β k 0, for all k 1 2) Looking at cumulative data C i,k := k j=1 S i,j and D i,k := k j=1 T i,j the assumptions (24) and (25) imply that E [C i,k+1 B i,k ] = (1 α k )C i,k + α k D i,k, E [D i,k+1 B i,k ] = (1 + β k )D i,k β k C i,k 3 Estimator of the necessary reserves The main aim of this section it to define weighted estimators for the parameter α k and β k and for the unknown values S i,k, T i,k and R i,k, for i + k > n + 1 Moreover, we will derive some useful conclusions Proposition 31 Under the assumptions stated in Section 2 α k := β k := i=1 w i,ks i,k+1 i=1 w and (31) i,kr i,k i=1 w i,kt i,k+1 i=1 w (32) i,kr i,k are conditionally unbiased estimators for α k and β k, respectively, given D k Moreover, f k := (1 α k + β k ) is the Chain-Ladder estimator for f k, that means f k = i=1 w i,kr i,k+1 i=1 w i,kr i,k (33) The estimators α k, β k and f k are correlated via α k α l Cov β k, β l D l k = 0, f k f l for k l, (34) 4

5 and α k α k σ Cov β k, β k D k 2 γ k γ k σ 2 k n k k = γ k τk 2 τk 2 γ j=1 w2 j,k R j,k k ( f k f k γ k σk 2 τk 2 γ k σk 2 2γ n k ) 2 k + τk 2 j=1 w j,kr j,k (35) Proof: Using (31), (32) and the definition (27) of f k, formula (33) can be easily verified The conditional unbiasedness of the given estimators is a direct consequence of E [ α k D k ] = i=1 w i,ke [S i,k+1 D k ] n k i=1 i=1 w = w i,kα k R i,k i,kr i,k i=1 w = α k i,kr i,k and similar calculations for β k Moreover, for k > l we get t α k α l α t k α l E β k β l D l = E E β k D k β l D l f k f l f k f l t α k α l = β k E β l D l f k f l α t k α l = E β k D l E β l D l, f k f l which proves (34) Finally, in order to prove (35) we will only compute Var[ α k D k ] covariances can be obtained in the same way We get The other Var[ α k D k ] = j=1 w2 i,k Var[S j,k+1 D k ] ( n k j=1 w j,kr j,k ) 2 = σ 2 k j=1 w2 i,k R j,k ( n k j=1 w j,kr j,k ) 2 Now we can look at the random variables S i,k, T i,k and R i,k, for i + k > n + 1 5

6 Theorem 32 Let the assumptions of Section 2 be fulfilled Then k 2 Ŝ i,k := α k 1 Ri,k 1 = α k 1 R i,n+1 i T i,k := β k 1 Ri,k 1 = β k 2 k 1 R i,n+1 i k 1 R i,k := R i,n+1 i f l, (36) f l, (37) f l (38) (39) are conditionally unbiased estimators for E[S i,k D n+1 i ], E[T i,k D n+1 i ] and E[R i,k D n+1 i ], for i + k > n + 1, respectively Moreover, if there is no tail development, that means R 1,n = 0, then Ŝ i,k = R i,n+1 i + T i,k (310) Proof: The unbiasedness of the estimators Ŝi,k, T i,k and R i,k are direct consequences of Proposition 31 and Corollary 22 Now, let us assume that R 1,n = 0 Then we get f n 1 = 0, which yields R i,n = 0, for 2 i n Therefore, it follows (Ŝi,k T i,k ) = R i,n+1 i R i,n+2 i + ( R i,k 1 R i,k ) this completes the proof = R i,n+1 i R i,n = R i,n+1 i, k=n+3 i Equation (310) shows that there is no gap between the estimations of the total necessary reserves using payments S i,k and using reported amounts T i,k This is an advantage over the standard Chain Ladder method, where one may have a systematically gap between the estimator using payments and the one using reported amounts, see Braun [1] Therefore, the presented extension of the Complementary Loss Ratio method is in this respect comparable with the Munich Chain Ladder method, which extends the Chain Ladder method in order to eliminate this gap, see Mack and Quarg [4] Unfortunately, for the Munich Chain Ladder method an estimator for the mean squared error of the estimated necessary reserves is still missing 6

7 4 Estimator of the mean squared error In this section we want to derive an estimator for the conditional mean squared error ( ) 2 mse( R i ) := E (S i,k Ŝi,k) Dn (41) of the estimated necessary reserves R i := under the knowledge D n of the whole run-off triangle, see also (310) We want to skip problems of dealing with tail factors That s why we assume Ŝ i,n R 1,n = 0 (42) In order to get further on we need estimators for σk 2 and τ k 2, for 1 k < n, and for γ k, for 1 k < n 1 We take the following weighted conditionally unbiased estimators: n k ( σ k 2 := Z 1 Si,k+1 k w i,k R i,k R i,k i=1 n k ( τ k 2 := Z 1 Ti,k+1 k w i,k R i,k R i,k i=1 n k ( γ k := Z 1 Si,k+1 k w i,k R i,k R i,k i=1 α k β k ) 2, (43) ) 2, (44) ) ( ) Ti,k+1 α k β k, (45) R i,k where n k n k i=1 Z k := w i,k w2 i,k R i,k i=1 w, i,kr i,k i=1 for 1 k < n 1 Note that E[ σ k 2 D k] = σk 2 and analogously for τ k 2 and γ2 k We still lack estimators for σn 1 2 and τ n 1 2 If there is even no development after year n 1 we set σ n 1 2 = τ n 1 2 = 0 7

8 Otherwise, one can extrapolate σ n 1 2 and τ n 1 2 in the same way as described by Mack [2] In our examples we take ( σ σ n := min n 2 Now, fix n + 1 i < k n and let Then we obtain [ ( ) 2 ] E (S i,k Ŝi,k) Dn = Var = Var = [ [ k 1,k 2 =n+2 i σ n 3 2, σ n 3, 2 σ n 2 2 ) ( τ τ n := min n 2 τ n 3 2, τ n 3, 2 τ n 2 2 S i,k Dn ] + S i,k B i ] + Cov[S i,k1, S i,k2 B i ] + B i := B i,n+1 i ( ( k 1,k 2 =n+2 i ), (E[S i,k D n ] Ŝi,k) (E[S i,k B i ] Ŝi,k) ) 2 ) 2 (46) (E[S i,k1 B i ] Ŝi,k 1 )(E[S i,k2 B i ] Ŝi,k 2 ) Note, the first term corresponds to the process variance and the second term to the parameter estimation error, see, eg, Wüthrich-Merz [5] We split the analysis of the addends of the first sum of the last line into the two parts: k 1 = k 2 and k 1 < k 2 For k := k 1 = k 2 we get [ ] [ ] Var[S i,k Bi ] = E Var[S i,k B i,k 1 ] + Var E[S i,k B i,k 1 ] B i = σ 2 k 1 E[R i,k 1 B i ] + α 2 k 1 Var[R i,k 1 B i ] B i 8

9 And for k 1 < k 2 we compute Cov[S i,k1, S i,k2 Bi ] [ = E Cov[S i,k1, S i,k2 B i,k2 1] = α k2 1Cov[S i,k1, R i,k2 1 B i ] B i = α k2 1f k2 2 f k1 Cov[S i,k1, R i,k1 B i ] ] [ + Cov E[S i,k1 B i,k2 1], E[S i,k2 B i,k2 1] = α k2 1f k2 2 f k1 Cov[S i,k1, R i,k1 1 S i,k1 + T i,k1 Bi ] ) = α k2 1f k2 2 f k1 (α k1 1f k1 1Var[R i,k1 1 Bi ] + (γ k1 1 σk )E[R i,k 1 1 Bi ] Because of Corollary 22, we can use the same arguments as in Mack [2] and get E[R i,k B i ] = R i,n+1 i Var[R i,k B i ] = R i,n+1 i k 1 k 1 f l, for all k > n + 1 i Using the notation σ l 2, for k 1 = k 2 = l + 1, â k1,k 2,l := α 2 l B i f n+1 i f l 1 (σ 2 l 2γ l + τ 2 l )f 2 l+1 f 2 k 1, γ l σ 2 l α l fl, for k 1 > k 2 = l + 1 or k 2 > k 1 = l + 1, ] (47) σ 2 l 2 γ l + τ 2 l f 2 l, for k 1 k 2 > l + 1 or k 2 k 1 > l + 1, and replacing all unknown variables by their estimators we get for the addends of the first sum of the last line of (46) (k 1 k 2 ) 1 Ĉov[S i,k1, S i,k2 Bi ] = Ŝi,k 1 Ŝ i,k2 â k1,k 2,l R i,l, where we used the notation R i,k := R i,k, for i + k n + 1 In order to estimate the addends of the second sum of the last line of (46) we start with ) f l ( ( ) E[S i,k Bi ] Ŝi,k = R i,n+1 i α k 1 k 2 9 k 2 f l α k 1

10 Using the abbreviation { (α F i,k,l := k 1 α k 1 ) f k 2 f n+1 i, for l = k 1, α k 1 f k 2 f l+1 (f l f l ) f l 1 f n+1 i, otherwise, (48) we can proceed with (E[S i,k1 B i ] Ŝi,k 1 )(E[S i,k2 B i ] Ŝi,k 2 ) = R 2 i,n+1 i k 1 1 l 1 =n+1 i k 2 1 l 2 =n+1 i F i,k1,l 1 F i,k2,l 2 Now as in Mack [2] we approximate F i,k1,l 1 F i,k2,l 2 by E[F i,k1,l 1 F i,k2,l 2 D l1 l 2 ] This means we take the average over as little data as possible keeping as much as possible values S i,k and T i,k from the observed data fixed Using the same arguments as in the proof of Proposition 31 we get and E[F i,k1,l 1 F i,k2,l 2 D l1 l 2 ] = 0, for l 1 l 2, E[F i,k1,lf i,k2,l D l ] Var[ α k1 1 D k1 1] f k f n+1 i 2, for k 1 = k 2 = l + 1, αk f k f l+1 2 Var[ f l D l ] f l 1 2 f n+1 i 2, for k 1 = k 2 > l + 1, α = k1 1f k1 2 f k2 Cov[ f k2 1, α k2 1 D k2 1] f k f n+1 i 2, for k 1 > k 2 = l + 1, α k1 1f k1 2 f k2 1α k2 1f 2 k 2 2 f 2 l+1 Var[ f l D l ] f 2 l 1 f 2 n+1 i, for k 1 > k 2 > l + 1 Now we use (35) and definition (47) and replace all unknown variables by their estimators This and similar arguments for T i,n with the definition τ l 2, for k 1 = k 2 = l + 1, β l 2 τ l 2 γ l bk1,k 2,l :=, for k 1 > k 2 = l + 1 or k 2 > k 1 = l + 1, β l fl σ 2 l 2 γ l + τ 2 l f 2 l, for k 1 k 2 > l + 1 or k 2 k 1 > l + 1, yield 10

11 Estimator 41 Let the assumptions of Section 2 be fulfilled Then the conditional mean squared errors of the estimated necessary reserves R i and of the estimated IBNR Î i := can be estimated, respectively, by mse( R i ) := mse(îi) := k 1,k 2 =n+2 i k 1,k 2 =n+2 i (k 1 k 2 ) 1 Ŝ i,k1 Ŝ i,k2 (k 1 k 2 ) 1 T i,k1 Ti,k2 T i,k â k1,k 2,l bk1,k 2,l 1 + R i,l 1 R i,l + n l wj,l 2 R j,l ) 2, w j,l R j,l j=1 j=1 ( n l n l j=1 ( n l wj,l 2 R j,l ) 2 w j,l R j,l j=1 Corollary 42 Let the assumptions of Section 2 be fulfilled and assume that there does not exist any tail development Then mse( R i ) and mse(îi) are estimators for the conditional mean squared error of the ultimate reserves mse( R i ) Proof: Since we assume that there does not exists any tail development, it follows from Theorem 32 that R i = k=n+1 i Ŝ i,k = R i,n+1 i + k=n+1 i T i,k = R i,n+1 i + Îi This and the measurability of R i,n+1 i with respect to D n proves mse( R i ) = mse(îi), which implies our statement 11

12 Often an estimator for the mean squared error of the estimated total necessary reserve R := R i is of interest, too Since R i and R j, i j, depend on the common parameter estimates α k and β k, we cannot simply take the sum of all mse( R i ) But using the same techniques as for the derivation of the result of Estimator 41 we obtain: Estimator 43 Assume the assumptions of Section 2 are fulfilled Then the conditional overall mean squared error of the reserves can be estimated by mse( R i ) = mse( R i ) i 1 <i 2 n k 1 =n+2 i 1 k 2 =n+2 i 2 Ŝ i1,k 1 Ŝ i2,k 2 (k 1 k 2 ) 1 n l j=1 â k1,k 2,l ( n l 1 j=1 w 2 j,l R j,l w j,l R j,l ) 2 Moreover, the conditional overall mean squared error of the IBNR can be estimated by mse( Î i ) = mse(îi) i 1 <i 2 n k 1 =n+2 i 1 Derivation: We start with mse( R i ) = mse( R i ) i 1 <i 2 n k 1 =n+2 i 1 k 2 =n+2 i 2 (k 1 k 2 ) 1,k 1 Ti2,k 2,k 2,l k 2 =n+2 i 2 Ti1 1 bk1 Similar to the derivation of Estimator 41 we substitute E[S i,k D n ] Ŝi,k = n l j=1 ( n l j=1 w 2 j,l R j,l w j,l R j,l ) 2 (E[S i1,k 1 D n ] Ŝi 1,k 1 )(E[S i2,k 2 D n ] Ŝi 2,k 2 ) 12 k 1 F i,k,l,

13 where F i,k,l is defined by (48) This leads to mse( R i ) = mse( R i ) i 1 <i 2 n k 1 =n+2 i 1 k 2 =n+2 i 2 k 1 1 l 1 =n+1 i 1 k 2 1 l 2 =n+1 i 2 F i1,k 1,l 1 F i2,k 2,l 2 We proceed with the product F i1,k 1,l 1 F i2,k 2,l 2 in the same way as with F i,k1,l 1 F i,k2,l 2 in the derivation of Estimator 41 and obtain our claimed estimator for the overall mean squared error of the reserves by replacing all unknown parameters with their estimators Moreover, the same procedure leads to the claimed estimator for the overall mean squared error of the IBNR Finally, the same arguments as in the proof of Corollary 42 yield Corollary 44 Let the assumptions of Section 2 be fulfilled and assume that there does not exist any tail development Then mse( R i ) and mse( Î i ) are estimators for the conditional overall mean squared error of the ultimate reserves mse( R i ) Note, there are also other methods that lead to an estimate for the estimation error (second term in (46)) An alternative would be to apply the conditional resampling approach, see Wüthrich-Merz [5] 5 Two examples As first example we choose some complete triangles and no weights in order to compare the standard Chain Ladder method, see Mack [2], with our method The triangles are shown in Table 4 and Table 5 As estimators for the parameters we get the values shown in Table 1, Table 2 and Table 3 13

14 k f k σ 2 k Table 1: Chain Ladder parameters for payments of Example 1 k f k σ 2 k Table 2: Chain Ladder parameters for reported amounts of Example 1 k α k β k σ 2 k τ 2 k γ k Table 3: Extended Complementary Loss Ratio parameters of Example 1 The resulting estimated reserves are shown in Table 6 Table 7 and Table 8 list the corresponding standard errors and the standard errors in percent of the reserves In this example the extended Complementary Loss Ratio method leads to reserves which lie for almost all accident years between the two corresponding Chain Ladder reserves Chain Ladder Ext Compl Paid Incurred Loss Ratio Total Table 6: Estimated reserves of Example 1 14

15 Table 4: Cumulated triangle of payments of Example Table 5: Cumulated triangle of reported amounts of Example 1 15

16 1 Chain Ladder Ext Compl Loss Ratio Paid Incurred Paid Incurred Total Table 7: Estimated standard errors of the reserves of Example 1 1 Chain Ladder Ext Compl Loss Ratio Paid Incurred Paid Incurred 2 784% 08% 01% 46% 3 595% 163% 68% 83% 4 420% 28% 29% 35% 5 374% 11% 37% 39% 6 262% 30% 39% 40% 7 241% 30% 36% 37% 8 228% 83% 91% 91% 9 209% 147% 126% 126% % 126% 125% 125% Total 149% 43% 44% 44% Table 8: Estimated standard errors in percent of the reserves of Example 1 It is remarkable that although we have to estimate 44 parameters for the extended Complementary Loss Ration method compared to 18 for the Chain Ladder method, we don t get much higher and for some years even smaller standard errors This may be caused by the estimated negative correlations γ k, for k > 3 And because you combine two sources of information, that is, you have more parameters to estimate but you also have more observations to do this parameter estimation 16

17 Table 9: Incremental triangle of payments of Example Table 10: Incremental triangle of reported amounts of Example 2 17

18 For the second example we take some part of a motor liability portfolio We took all small bodily injury claims Because of the lack of good information, we can trust the split into bodily injury and property damage claims only for the last 6 business years The corresponding incremental triangles are shown in Table 9 and Table 10 The opening reserves R i,6 i, 1 i 5, are given by Table 11 We choose weights i R i,6 i Table 11: Opening reserves for Example 2 w i,k := { 0, for i + k 5, 1, otherwise Moreover, since we only see 10 years of claim development, we have to decide how to deal with the reserves remaining after 10 years In this example we assume that only fifty percent of those remaining reserves will be paid Note, this decision does not affect the estimated standard errors Applying our method we get the following estimators for the parameters: k α k β k σ 2 k τ 2 k γ k Table 12: Estimated parameters of Example 2 This leads to the following estimated reserves and their corresponding standard errors (se): 18

19 Paid Reported Amount Reserves IBNR se in % Res se in % Res % % % % % % % % % % % % % % % % % % Total % % Table 13: Estimated reserves and standard errors of Example 2 If you try to apply the Chain Ladder method to the last 6 years, you would have to estimate a tail factor for the payments of about 117 and for the reported amount of about 087 in order to get similar results for the total necessary reserves Acknowledgement: I would like to thank Benedetto Conti for introducing me to non life insurance and reserving Working with him pushed me to develop the presented reserving method Authors address: René Dahms, Bâloise, Aeschengraben 21, CH-4002 Basel References [1] Braun, C (2004) The Prediction Error of the Chain Ladder Method Applied to Correlated Run-off Triangles Astin Bulletin 342, , [2] Mack, T (1993) Distribution-free calculation of the standard error of Chain Ladder reserve estimates Astin Bulletin 232, [3] Mack, T (1997) Schadenversicherungsmathematik Schriftenreihe Angewandte Versicherungsmathematik 28, [4] Quarg, G, Mack, T (2004) Munich Chain Ladder BDGVFM XXVI (4), [5] Wüthrich, M, Merz, M (2008) Stochastic Claims Reserving methods in Insurance John Wiley & Sons Ltd 19