A Loss Reserving Method for Incomplete Claim Data Or how to close the gap between projections of payments and reported amounts?

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1 A Loss Reserving Method for Incomplete Claim Data Or how to close the gap between projections of payments and reported amounts? René Dahms Baloise Insurance Switzerland July 2008, International ASTIN Colloquium Manchester

2 Projected Gap xposure measure Incomplete data Outlook Questions Three motivations 1 The gap between projections of payments and reported amounts

3 Projected Gap xposure measure Incomplete data Outlook Questions Three motivations 1 The gap between projections of payments and reported amounts 2 Case reserves as exposure measure

4 Projected Gap xposure measure Incomplete data Outlook Questions Three motivations 1 The gap between projections of payments and reported amounts 2 Case reserves as exposure measure 3 Reserving based on incomplete data (an example)

5 Projected Gap xposure measure Incomplete data Outlook Questions Three motivations 1 The gap between projections of payments and reported amounts 2 Case reserves as exposure measure 3 Reserving based on incomplete data (an example) 4 Summary and outlook

6 Projected Gap xposure measure Incomplete data Outlook Questions The typical outcome of projections in % of ultimate Ultimate Payments Reported amounts Development year

7 Projected Gap xposure measure Incomplete data Outlook Questions The typical outcome of projections in % of ultimate Ultimate Payments Reported amounts 100 } projected gap Development year

8 Projected Gap xposure measure Incomplete data Outlook Questions The typical outcome of projections in % of ultimate Ultimate Payments Reported amounts } projected gap Development year

9 Projected Gap xposure measure Incomplete data Outlook Questions How to close the gap? Munich-Chain-Ladder tries to deal with the systematic gap of Chain-Ladder-projections, but still leads to different estimates for payments and reported amounts.

10 Projected Gap xposure measure Incomplete data Outlook Questions How to close the gap? Munich-Chain-Ladder tries to deal with the systematic gap of Chain-Ladder-projections, but still leads to different estimates for payments and reported amounts. Weighting of ultimates like in Bornhuetter-Ferguson or Cape Cod (or using credibility theory). We often use L P U P + L A U A L P + L A, where U P and U A are the projected ultimates based on payments and reported amounts, respectively, and L P and L A are the corresponding lag factors or their reciprocal values, depending on which are smaller.

11 Projected Gap xposure measure Incomplete data Outlook Questions How to close the gap? Munich-Chain-Ladder tries to deal with the systematic gap of Chain-Ladder-projections, but still leads to different estimates for payments and reported amounts. Weighting of ultimates like in Bornhuetter-Ferguson or Cape Cod (or using credibility theory). We often use L P U P + L A U A L P + L A, where U P and U A are the projected ultimates based on payments and reported amounts, respectively, and L P and L A are the corresponding lag factors or their reciprocal values, depending on which are smaller. Making assumptions on the joint distribution of payments and reported amounts.

12 Projected Gap xposure measure Incomplete data Outlook Questions How to close the gap? Munich-Chain-Ladder tries to deal with the systematic gap of Chain-Ladder-projections, but still leads to different estimates for payments and reported amounts. Weighting of ultimates like in Bornhuetter-Ferguson or Cape Cod (or using credibility theory). We often use L P U P + L A U A L P + L A, where U P and U A are the projected ultimates based on payments and reported amounts, respectively, and L P and L A are the corresponding lag factors or their reciprocal values, depending on which are smaller. Making assumptions on the joint distribution of payments and reported amounts. Is there a distribution free stochastic model which combines the information of payments and reported amounts?

13 Projected Gap xposure measure Incomplete data Outlook Questions What is the correct exposure for the payments of the next development year? At first some definitions for accident year i and development year j: C i,j - cumulative payments S i,j - incremental payments (C i,j = jk=1 S i,k)

14 Projected Gap xposure measure Incomplete data Outlook Questions What is the correct exposure for the payments of the next development year? At first some definitions for accident year i and development year j: C i,j - cumulative payments S i,j - incremental payments (C i,j = jk=1 S i,k) D i,j - reported amounts T i,j - changes of reported amounts (D i,j = jk=1 T i,k)

15 Projected Gap xposure measure Incomplete data Outlook Questions What is the correct exposure for the payments of the next development year? At first some definitions for accident year i and development year j: C i,j - cumulative payments S i,j - incremental payments (C i,j = jk=1 S i,k) D i,j - reported amounts T i,j - changes of reported amounts (D i,j = jk=1 T i,k) R i,j - case reserves (R i,j = D i,j C i,j )

16 Projected Gap xposure measure Incomplete data Outlook Questions What is the correct exposure for the payments of the next development year? At first some definitions for accident year i and development year j: C i,j - cumulative payments S i,j - incremental payments (C i,j = jk=1 S i,k) D i,j - reported amounts T i,j - changes of reported amounts (D i,j = jk=1 T i,k) R i,j - case reserves (R i,j = D i,j C i,j ) C k, D k and k are the σ-algebras containing all the information of the payment triangle, the reported amount triangle and both triangles, respectively, up to development period k.

17 Projected Gap xposure measure Incomplete data Outlook Questions What is the correct exposure for the payments of the next development year? Mack s model of the Chain-Ladder-Method assumes that [S i,k+1 C k ] = (f k 1) C i,k. Therefore, if you believe in Chain-Ladder you have to believe in the cumulative payments as exposure measure for the payments of the next development year.

18 Projected Gap xposure measure Incomplete data Outlook Questions What is the correct exposure for the payments of the next development year? Mack s model of the Chain-Ladder-Method assumes that [S i,k+1 C k ] = (f k 1) C i,k. Therefore, if you believe in Chain-Ladder you have to believe in the cumulative payments as exposure measure for the payments of the next development year. Additive method (Complementary Loss Ratio Method) assume that [S i,k+1 C k ] = q k P i, where P i is an external given risk measure for accident year i, for instance the risk premium. Therefore, if you believe in this method you have to believe that the risk measure P i is the correct one.

19 Projected Gap xposure measure Incomplete data Outlook Questions Assumptions for the presented method We assume that [S i,k+1 k ] = α k R i,k and [T i,k+1 k ] = β k R i,k, for some constants α k and β k.

20 Projected Gap xposure measure Incomplete data Outlook Questions Assumptions for the presented method We assume that [S i,k+1 k ] = α k R i,k and [T i,k+1 k ] = β k R i,k, for some constants α k and β k. This means, that we take the opening case reserves R i,k as exposure measure for the development during the year.

21 Projected Gap xposure measure Incomplete data Outlook Questions Assumptions for the presented method We assume that [S i,k+1 k ] = α k R i,k and [T i,k+1 k ] = β k R i,k, for some constants α k and β k. This means, that we take the opening case reserves R i,k as exposure measure for the development during the year. It follows that the case reserves meet a Chain-Ladder like assumption: [R i,k+1 k ] = f k R i,k, where f k = 1 α k + β k.

22 Projected Gap xposure measure Incomplete data Outlook Questions Assumptions for the presented method We assume that [S i,k+1 k ] = α k R i,k and [T i,k+1 k ] = β k R i,k, for some constants α k and β k. This means, that we take the opening case reserves R i,k as exposure measure for the development during the year. It follows that the case reserves meet a Chain-Ladder like assumption: [R i,k+1 k ] = f k R i,k, where f k = 1 α k + β k. All accident years are independent.

23 Projected Gap xposure measure Incomplete data Outlook Questions Assumptions for the presented method We assume that [S i,k+1 k ] = α k R i,k and [T i,k+1 k ] = β k R i,k, for some constants α k and β k. This means, that we take the opening case reserves R i,k as exposure measure for the development during the year. It follows that the case reserves meet a Chain-Ladder like assumption: [R i,k+1 k ] = f k R i,k, where f k = 1 α k + β k. All accident years are independent. [ (Si,k+1 ) ( Cov T i,k+1, Si,k+1 ) ] T k i,k+1 = R i,k Σ 2 k for some positive definite, symmetric matrices Σ k.

24 Projected Gap xposure measure Incomplete data Outlook Questions Derived estimators One estimator for the reserves of accident year i: n 1 Reserve i = R i,n+1 i k=n+1 i α k k 1 l=n+1 i fl

25 Projected Gap xposure measure Incomplete data Outlook Questions Derived estimators One estimator for the reserves of accident year i: n 1 Reserve i = R i,n+1 i k=n+1 i = R i,n+1 i (1 + α k n 1 k=n+1 i k 1 l=n+1 i β k fl k 1 l=n+1 i ) fl,

26 Projected Gap xposure measure Incomplete data Outlook Questions Derived estimators One estimator for the reserves of accident year i: n 1 Reserve i = R i,n+1 i with estimators k=n+1 i = R i,n+1 i (1 + α k n 1 k=n+1 i k 1 l=n+1 i β k fl k 1 l=n+1 i fl ), α k = n k i=1 S i,k+1 n k i=1 R, βk = i,k n k i=1 T i,k+1 n k i=1 R i,k and fk = 1 α k + β k.

27 Projected Gap xposure measure Incomplete data Outlook Questions Derived estimators One estimator for the reserves of accident year i: n 1 Reserve i = R i,n+1 i with estimators k=n+1 i = R i,n+1 i (1 + α k n 1 k=n+1 i k 1 l=n+1 i β k fl k 1 l=n+1 i fl ), α k = n k i=1 S i,k+1 n k i=1 R, βk = i,k n k i=1 T i,k+1 n k i=1 R i,k and fk = 1 α k + β k. Two estimators for the (conditional) mean squared error of the estimated reserves. One depends more on the payments and the other more on the reported amounts.

28 Projected Gap xposure measure Incomplete data Outlook Questions Separating accident damage from bodily injury claims The bodily injury flag has been introduced some time ago, but has been applied to new, still open or reopened claims, only.

29 Projected Gap xposure measure Incomplete data Outlook Questions Separating accident damage from bodily injury claims The bodily injury flag has been introduced some time ago, but has been applied to new, still open or reopened claims, only. This means, the incremental triangles are of the form development years accident years 0

30 Projected Gap xposure measure Incomplete data Outlook Questions Separating accident damage from bodily injury claims The bodily injury flag has been introduced some time ago, but has been applied to new, still open or reopened claims, only. This means, the incremental triangles are of the form development years accident years 0 The same kind of data you may get if your company acquires another company and only migrates all open (and reopened) claims.

31 Projected Gap xposure measure Incomplete data Outlook Questions Separating accident damage from bodily injury claims The bodily injury flag has been introduced some time ago, but has been applied to new, still open or reopened claims, only. This means, the incremental triangles are of the form development years accident years 0 The same kind of data you may get if your company acquires another company and only migrates all open (and reopened) claims. Most of the standard reserving methods will not work on such data.

32 Projected Gap xposure measure Incomplete data Outlook Questions Separating accident damage from bodily injury claims The bodily injury flag has been introduced some time ago, but has been applied to new, still open or reopened claims, only. This means, the incremental triangles are of the form development years accident years 0 The same kind of data you may get if your company acquires another company and only migrates all open (and reopened) claims. Most of the standard reserving methods will not work on such data. The presented method will work if case reserves are available.

33 Projected Gap xposure measure Incomplete data Outlook Questions Benefits of the presented method Combines information of payments and reported amounts in a natural way to estimate the reserves.

34 Projected Gap xposure measure Incomplete data Outlook Questions Benefits of the presented method Combines information of payments and reported amounts in a natural way to estimate the reserves. Works for some kind of incomplete data.

35 Projected Gap xposure measure Incomplete data Outlook Questions Benefits of the presented method Combines information of payments and reported amounts in a natural way to estimate the reserves. Works for some kind of incomplete data. asy to compute.

36 Projected Gap xposure measure Incomplete data Outlook Questions Drawbacks of the presented method Does not work if the case reserves are too small or even non positive. For instance, in cases with a huge impact by late reported or reopened claims.

37 Projected Gap xposure measure Incomplete data Outlook Questions Drawbacks of the presented method Does not work if the case reserves are too small or even non positive. For instance, in cases with a huge impact by late reported or reopened claims. No statement about the quality of the estimator of the reserves. We only know that it is conditionally mean preserving. For the estimator of the mean squared error we do not even know that.

38 Projected Gap xposure measure Incomplete data Outlook Questions Drawbacks of the presented method Does not work if the case reserves are too small or even non positive. For instance, in cases with a huge impact by late reported or reopened claims. No statement about the quality of the estimator of the reserves. We only know that it is conditionally mean preserving. For the estimator of the mean squared error we do not even know that. More difficult to handle than two separate projections.

39 Projected Gap xposure measure Incomplete data Outlook Questions Outlook There is a paper in preparation (M. Merz, M. Wüthrich and R. D.) that gives an estimator for the one year solvency risk.

40 Projected Gap xposure measure Incomplete data Outlook Questions Outlook There is a paper in preparation (M. Merz, M. Wüthrich and R. D.) that gives an estimator for the one year solvency risk. A model based on assumptions on the joint distribution of payments and reported amounts which leads to the same estimators as in the presented model would be convenient. Such a model may give us a better understanding of the method itself and may lead to further stochastic statements about the distribution of the estimated reserves.

41 Projected Gap xposure measure Incomplete data Outlook Questions Questions?

42 Projected Gap xposure measure Incomplete data Outlook Questions Questions? njoy your meal.

Modelling the Claims Development Result for Solvency Purposes

Modelling the Claims Development Result for Solvency Purposes Mario V Wüthrich ETH Zurich Financial and Actuarial Mathematics Vienna University of Technology October 6, 2009 wwwmathethzch/ wueth c 2009