The Chain Ladder Reserve Uncertainties Revisited

Transcription

1 The Chain Ladder Reserve Uncertainties Revisited by Alois Gisler Paper to be presented at the ASTIN Colloquium 6 in Lisbon Abstract: Chain ladder () is still one of the most popular and most used reserving method for the insurance practice. In 993 Mack [7] presented the distribution-free -model and derived a formula for the uncertainty of the -reserves, which refers to the ultimate prediction uncertainty measured by the mean square error of prediction. For calculating the reserve risk and the cost-of-capital loading in solvency (SST and solvency II) one also needs estimators for the one-year prediction uncertainty of all future accounting years until nal development. In a recent paper [] Merz and Wüthrich considered the di erent prediction uncertainties, that is the ultimate prediction uncertainty as well as the one year prediction uncertainties for all future accounting years until nal development within the framework of a speci c Bayesian- model. Taking a non-informative prior and after a rst Taylor approximation they received the already existing result of Mack for the ultimate prediction uncertainty and formulas for the one-year run-o uncertainties for all future accounting years until nal settlement of the run-o. However the Bayesian- model and the distribution free -model of Mack are two di erent pairs of shoes. Thus the results derived in [] are results with regard to a di erent model and we do not know, whether they are also appropriate in the classical chain ladder model of Mack. In this paper we derive the di erent kinds of prediction uncertainties strictly within the framework of the distribution-free model of Mack. By doing so, we gain more insight into the di erences between the two model approaches and nd the following main results: a) the formulas for the one-year prediction uncertainty in the classical Mack-model are di erent to the Merz-Wüthrich formulas, b) the Merz-Wüthrich formulas are obtained by a rst order Taylor expansion, c) the Mack formula as well as the Merz-Wüthrich formulas for the total over all accident years can be written in a simpler way, d) we can see "behind the formulas", as they can be interpreted in an intuitive and understandable way. Keywords: claims reserving, distribution free chain-ladder model, Bayesian chain-ladder model, conditional mean square error of prediction, ultimate run-o uncertainty, one-year run-o uncertainties, Mack s formula, Wuethrich-Merz formulas, cost of capital loading, market value margin.

2 Introduction Accurate claims reserves are essential for an insurance company. It is by far the most important item on the liability side of the balance sheet and has a big impact on the pro t and loss (P&C) account. A change of the reserves by a small percentage might well turn a positive year result into a negative one and vice versa. When non-life insurance companies went bankrupt, insu cient reserves were mostly one of the main reasons. Chain Ladder () and Bornhuetter Ferguson (BF) are still the most used and most popular reserving methods in the insurance practice. In this paper we concentrate on the -method. has been used for decades for calculating reserves. In its origin it is a pure pragmatic method without an underlying mathematical model. As long as one is only interested in the reserve estimate, there is no need for a mathematical model. But as soon as one is also interested in the accuracy of the -reserves, one needs an underlying stochastic model. Under run-o risk we understand the risk of an adverse claims development. It can be de ned as minus the claims development result (). As common in the actuarial literature (see for instance [9]), we will take the conditional mean square error of prediction (msep) as a measure of the reserve uncertainties. For best estimate reserves this msep is equal to the conditional expectation of the square of the. Thereby we distinguish between the ultimate run-o risk referring to the ultimate claims development result () and the one-year run-o risks referring to the resulting in one accounting year. It was only in 993 when Mack [7] presented a stochastic model and derived a formula for estimating the msep of the ultimate run-o risk (see Theorem A. in appendix A). With the emergence of the new solvency regulation (Swiss solvency test and solvency II), there arose the need to assess another kind of reserve risk. The risk considered is the change of risk bearing capital within the next year (one-year time horizon). Hence the reserve risk relevant for solvency purposes is the one-year run-o risk in the next accounting year instead of the ultimate run-o risk considered in Mack. This risk is re ected under the position "claims development result" or "loss experience previous years" in next year s P&L account. A formula for estimating this one year uncertainty of the next accounting year was rst published in a paper by Merz and Wüthrich [] in. As the new solvency regulations are based on a market consistent valuation, the best estimate reserves have to be complemented by a market value margin corresponding to the discounted costs of capital needed for the entire run-o. For this purpose one also needs estimators of the uncertainty of the one-year run-o risk in later accounting years until the end of the claims development. In a recent paper [] of end 4 Merz and Wüthrich reconsidered all the di erent reserve uncertainties within a speci c Bayesian model (with Gamma-priors for the factors and with conditionally Gamma distributed observations (observed individual -factors)), which is very similar to a model which was earlier considered in [4]. In this model they derived formulas for all the di erent kind of -uncertainties inclusive the one-year run-o uncertainties in future accounting years. For the distribution-free case, the authors derive the formulas for the reserve uncertainties obtained by taking a

3 non-informative prior followed by a rst order Taylor approximation. These formulas can be found in appendix A. However the Bayesian -model is di erent to the distribution free model of Mack. Thus we do not know whether the results derived in [] are also appropriate for the classical Mack model. For instance, in the model considered in [], the mean square error of prediction (msep) does only exist, if the observed triangle ful ls certain properties (see Theorem 3. and condition (4.3) in []). But in the Mack-model the msep does always exist. In this paper we derive the di erent kinds of prediction uncertainties strictly within the framework of the distribution-free model of Mack. By doing so, we gain more insight into the di erences between the two model approaches and nd the following main results: a) The formulas derived for the one-year prediction uncertainty in the classical Mackmodel are di erent to the Merz-Wüthrich formulas. b) The Merz-Wüthrich formulas are obtained by a rst order Taylor expansion. c) The Mack formula as well as the Merz-Wüthrich formulas for the total over all accident years can be written in a much simpler way. d) We can see "behind the formulas", as they can be interpreted in an intuitive and understandable way. e) The derivation of the formulas is straightforward. The mathematics is simple and easy to understand and makes use of two basic tools, the telescope formula 3.5 and the estimation principle 3.6. In this connection it is also worthwhile to remember a discussion in 6 with regard to the estimation error in the Mack formula. In [] the authors suggested a di erent estimator, which we call the BBMW estimator. There was quite some discussion about these estimators (see [], [], [4]). Reconsidering this discussion we come to the conclusion that the Mack formula for the estimation error is the appropriate one, whereas the BBMW estimator is the adequate formula in the -Bayes model, but not in the Mack-model. More details about this side result of the paper can be found in appendix B At this point we should also mention the most recent paper [] of Ancus Röhr. He starts with a rst order Taylor approximation of the claims development result () and calculates the prediction uncertainty of this modi ed. He then also obtains the Mack as well as the Merz-Wüthrich results. Organisation of the paper: In section we introduce some notation and the data structure. In section 3 we review the -reserving method and the stochastic -model of Mack. At the end of this section the telescope formula and the estimation principle are presented. In section 4 we derive the formula for the ultimate run-o uncertainty (Mackformula). In section 5 we consider the run-o uncertainty of the next accounting year, whereas in section 6 the formulas for the one-year uncertainties for all future accounting years until nal claims development are derived. These results can be compared with the Wüthrich-Merz formulas. Finally, a numerical example is presented in section 7. 3

4 Notation and Data Structure The starting point of claims reserving are observations D I from accident periods i i ; : : : ; I and development periods ; : : : ; J arranged in a table with i on the vertical axes and on the horizontal axes. In the following we will call D I a data triangle, also in the case, where the shape is a trapezoid. The data in the triangle are denoted by C i; > and represent the cumulative claim gures (usually claim payments or incurred losses) of accident year i fi ; : : : ; Ig at the end of development year f ; : : : ; J): We further assume that the number of accident years is bigger or equal to the number of development years, that is J > I i. The index is introduced because in the actuarial literature the rst development year is sometimes denoted by zero and sometimes by, hence is either zero or one. devolopment years J i i I accident years i claims development triangle At time I; the data C i; D I in the upper left part are known, whereas the data C i; DI c in the lower right part are future observations we want to predict: We assume that all claims are settled after development year J and that therefore C denotes the ultimate claim of accident year i. Some notations: - diagonal functions To simplify notation and as already done by Ancus Röhr in [] it is convenient to introduce the diagonal functions De nition. diagonal functions i : maxf such that C i; D I g; () i : maxfi such that C i; D I g: () Note that C i;i is the diagonal element in row i and that C i ; is the diagonal element in column. - The set of observations at time I is given by D I fc i; : i i ; : : : ; I; i g: 4

5 - Later we will also need the set of observations known at the end development year ; which we denote by B : fc i;k : C i;k D I ; k g: (3) - coe cient of variation We denote the coe cient of variation of a random variable (r.v.) X by CoV (X) : p Var (X) E [X] - In this paper empty sums and empty products are de ned by De nition. (empty sum and empty product) X u l x k : if u < l; Y u kl x k : if u < l: - We will later in the paper use the following weights: De nition.3 The weights w i; are de ned by < C i; ; if C i; D I w i; : : bc i; ; if C i; DI c : (4) 3 The chain ladder method 3. The chain ladder method The method is a pragmatic method which has been used for decades for estimating reserves. The basic assumption behind is that, up to random uctuation, the columns in the development triangle are proportional to each other, i. e. there exist constants f ; ; : : : ; J ; such that C i; f C i;. (5) The constants f are called claims-development factors, factors or age to age factors. Given the information D I it is natural to estimate these unknown constants by Due to (5) ; the ultimate claim of accident year i is predicted by P i C i; P i C i; : (6) C i;i 5 i b f (7)

6 and the C i; in the lower right part of the triangle are estimated by i; C i;i Y k i b f k for i i ; : : : ; I; i ; : : : ; J: () () are called the -predictions. It is also useful and meaningful to de ne i; : C i; for C i; D I : (9) The reserve b R i of accident year i is an estimate of the outstanding liabilities JX R i C i; () i of the ultimate claim and the cumu- and is the di erence between the prediction C b lative claim C i;i known at time I; i.e. Finally br i b C C i;i () br tot is the total reserve over all accident years. b R i () 3. The stochastic -model of Mack The following distribution-free stochastic model underlying the method was introduced in [7] by Mack. Model Assumptions 3. (Mack-model) i) Cumulative claims C i; of di erent accident years are independent. ii) There exist positive parameters f ; : : : ; f J and ; : : : ; J i ; : : : ; I; and all ; : : : ; J such that for all i E[C i; C i; ; C i; ; : : : ; C i; ] f C i; ; (3) Var(C i; C i; ; C i; ; : : : ; C i; ) C i; (4) It is useful to introduce the individual ratios F i; : C i; C i; : (5) Because of model assumptions 3. F i; belonging to di erent accident years are independent and it holds that E[F i; B ] f ; (6) Var(F i; B ) w i; with w i; C i; : (7) 6

7 We have deliberately written (7) in this more complicated way, because conditional on C i; ; the denominator C i; in F i; is no longer a r.v., but a constant playing the role of a weight w i; C i; : The following properties with regard to the estimators f b easily be veri ed: are well known and can Properties 3. i) The estimators (6) can be written as a weighted mean i i X bf w i; X F i; ; where w ; w i; ; () w ; ii) Conditional on B ; it holds that h i E B Var B f ; (9) i X, where w ; w i;; () w ; and (9) has minimum variance among all linear unbiased estimators. iii) can be estimated by and it holds that b i X w i; F i; i i E b B : f b () Most of the following properties with regard to the F i; are well known. For completeness a proof is given in appendix C. Properties 3.3 It holds that i) Conditional expectation: ii) Conditional dependence structure: E [F i;k B ] f k for k J ; () ff i;k B : k ; : : : ; J g are uncorrelated, (3) iii) Conditional coe cient of variation: s CoV (F i;k B ) k E f k C i;k B for k J : (4) 7

8 3.3 Mean square error of prediction, telescope formula and estimation principle As measure for the reserve uncertainty we use the conditional mean square error of prediction (msep). De nition 3.4 The conditional mean square error of prediction (msep) of the prediction C b is de ned by msep CD I : E C D I The conditional msep of other predictors and estimators are analogously de ned. Note that msep Ri D br I i msep CD I ; since b R i b C C i;i and since C i;i is known at time I. To derive estimators of this msep we will make extensive use of the following telescope formula. Lemma 3.5 (Telescope Formula) For any real numbers x and y ; ; ; : : : ; J, it holds that!! Y J x Y J JX i y Y IY i x k (x y ) y m : (5) k mi Proof: This result is well known. We show it for a product with J 3: The extension to any number J is self evident. x x x 3 y y y 3 x x x 3 x x y 3 x x y 3 x y y 3 x y y 3 y y y 3 (x y ) y y 3 x (x y ) y 3 x x (x 3 y 3 ) : In the formulas of the conditional msep there appear the unknown factors f : To nd an estimator for the msep we have to estimate these unknown constants. As a general principle we replace them by the estimators f b : But we can t do it for the quadratic di erence terms f ; because this would give an estimator of zero, which is not meaningful. To nd an estimator we have to study the uctuation of f b around f : For this purpose we take into account as many observations of D I as possible and consider the conditional r.v. f b B with moments h i E B E f B Based on (7) we will use the following estimation principle: f ; (6) P i w i; : (7)

9 Estimation Principle 3.6 a) estimator of f : b E (F f ) B b P i w i; : () b) Other functions of f such as Q J i f by f b : are estimated by replacing the unknown f Remarks: - Note that f P i w i; b P bf i w i; CoV B and hence dcov B : (9) 4 The ultimate run-o prediction uncertainty The ultimate claims development results () are de ned by where i;ultimate b C C for single accident years i ; : : : ; I; tot;ultimate b C X I C for the total over all accident years, b C ; C X I C : In this section we derive the ultimate run-o uncertainties de ned as the msep of the in a slightly di erent way than in Mack (see [7]). a) single accident year i msep CD I C E C E 4(C E [C D I ]) {z } D I A {z i } process variance P V i C D I 3 5 E 6 4 E [C D I ] C d D I {z } B i {z } estimation error EE i 3 7 5:(3) 9

10 Process Variance P V i : A i C i;i i F i; i f where the diagonal element C i;i is a multiplicative constant playing the role of a weight w i;i : By applying the telescope formula (5) we obtain! ; Hence A i w i;i ((F i;i f i ) XJ : : : i C i; (F i; f ) P V i E A i DI E 4 i f : : : Y k i F i;k (F f J ) k f k : F i; (F i; f ) k i ) XJ C i; (F i; f ) Y J i : k f k! lk D I f l 3 5 : (3) The cross terms vanish since E [C i; C i;k (F i; f ) (F i;k f k ) D I ] E [E [C i; C i;k (F i; f ) (F i;k f k ) B k ] D I ] for k > ;(3) where we have used that ff i;k : k > g are conditionally unbiased given B k : Hence XJ Y P V i C i; (F i; f ) J D I i E XJ E i XJ 4E 4 C i; (F i; f ) i E [C i; D I ] XJ E [C D I ] f i k f k! k f k k By applying the estimation principle 3.6 we obtain dp V i BXJ i! 3 3 f k B 5 D I 5 (33) E [C i; D I ] : (34) w i; C A ; (35)

11 where w i; are the weights as de ned in (4). Remarks: (35) is the same formula as in Mack [7]. From (4) follows that CoV (F i; D I ) E f C i; D I : Hence the summands in (35) have the following intuitively accessible interpretation b CoV w d (F i; D I ) : (36) i; Estimation error EE i : The estimation error EE i as de ned in (3) is given by EE i E Bi DI ; (37) where B i E [C D I ] d C w i;i i f i b f! : (3) By use of the telescope formula (5) we obtain B i XJ i E [C i; D I ] f f b bf k : (39) k Conditional on D I ; B i principle 3.6 we obtain is an unknown constant and by applying the estimation dee i b B i X >:J i b X i w i; 9 > >; : (4) Remarks: (4) is the same estimator as in Mack [7].

12 In (4) we have seen hat b X i w i; d CoV B ; (4) which is an intuitively accessible interpretation of the summands in (4) : b) Total over all accident years Process Variance P V tot Since di erent accident years are independent it follows that dp V tot d P V i ii J ii J dp V i B i b w i; C A : By changing the order of summation between i and we obtain dp V tot XJ b : (4) w ii i; Estimation Error EE tot where B tot EE tot E Btot DI ; B i ii J ii J XJ B i i E [C i; D I ] k Changing the order of summation between i and yields XJ B E [C i; D I ] A k f ii k By by applying the estimation principle 3.6 we nd dee tot XJ i b PI ii C b k! : f b : Pii w i; : (43)

13 Remarks: Note that dee tot > dee i i XJ i b P I ii : w i; Pii c) result The results in the following Theorem except formula (46) are a summary of (3) ; (35) ; (4) ; (4) ; (43) : Theorem 4. The msep of the ultimate run-o risk can be estimated by a) single accident year i [msep CD I X >:J i where the weights w i; are de ned in (4) ; b) total over all accident years XJ [msep CD I where Remarks: >: b b w i; ii P I ii b C X I bc PI ii w i; b C :! b! 9 > P i w i; >; (44) b C P i w i; P i 9 > C A >; (45) w i; ; (46) The rst summand in (44) and (45) represent the process variance and the second one the estimation error. (44) is the same formula as the Mack formula (95) in appendix A. However, (45) and (46) look di erent to the Mack formula (96) in appendix A. The covariance terms have disappeared, they are much simpler and have an intuitively understandable interpretation (see the second last bullet point of these remarks). But they give the same results as (96) : Formula (46) was already found by Ancus Röhr in []. But his derivation is less stringent, as he did not calculate the msep of the ultimate claims development result ult ; but the msep of ^ ult, where ^ ult is a rst order Taylor expansion of ult. 3

14 For the process error the uncertainties due to the F i; sum up, whereas for the estimation error several accident years are a ected simultaneously by the uncertainty of f b. This is the reason why dp V tot d P V i ; but d EE tot > d EEi : intuitively accessible interpretation From (36) and (9) we see that (44) and (45) can be written as [msep CD I [msep CD I ( XJ where q i dcov (F i; B ) CoV d ) B ; (47) XJ q dcov B ; (4) P I ii b C is the fraction of b C, which is a ected by the uncertainty of b f : (49) (47) and (4) give a good intuitive understanding of (44) and (46) ; as the coe cients of variation CoV (F i; B ) and CoV B are good intuitive measures for the relative deviation of F i; and f b f : In particular, (4) is a very intuitive formula. from the "true" -factors Instead of distinguishing between the process variance and the estimation error we could apply the telescope formula (5) directly to the ultimate run-o risk Z i;ult C C b w i;i i i F i; ( XJ Y w i;i XJ k i F i;k i b f F i;! ) f b bf m m m C i; (F i; f ) i m {z } A i XJ i C i; f f b bf m m {z } B i :(5) However the summands in (5) are correlated, such that the calculation of E Zi;ult DI is rather complicated. But note that the Ci; in B i play the role of "stochastic weights", which are not yet known. A natural procedure is to 4

15 replace them by the forecasted weights w i; C b i; and to consider ez i;ult XJ i C i; (F i; f ) Then we immediately get; m XJ bf m w i; f i f b bf m : (5) m be h ez i;ult i D I X >:J i b w i;! 9 > P i w i; >; ; which is the same result as the one found in Theorem 4.: Hence by replacing the not yet known stochastic weights in B i by the forecasted weights w i; and then calculating the msep of the modi ed r.v. e Zi;ult we end up with the "correct" estimator (44) : Proof of Theorem 4.: It only remains to prove that (45) can be expressed by (46) : XJ b B bc PI 9 ii C b > P >: w i A ii i; w i; >;! XJ P b I C b ii A P i >: k ii XJ >: b k b C ba ii P i w i; P I ii P i b C C A : b C b C b C! 9 > >;! 9 > A >; 5 The one-year run-o prediction uncertainty of the next accounting year In the new solvency regulation (SST, Solvency II), a time horizon of one year is considered. Therefore the reserve risk relevant for solvency purposes is the one-year run-o risk of the next accounting year. In the previous section, claims reserving was considered from a static perspective. For solvency purposes, the claims reserving has to be seen as a dynamic process, where the 5

16 predictions are updated based on new information which become available during the run-o process. At the end of the next accounting year I there will be available the data D I. The -factors and the prediction of the ultimate claim will then be made on the basis of D I : development year development year i i i J i i i J.. accident year i accident year i.. I I claims development triangle at time at time I claims development triangle at time at time I As in the previous sections we denote by f b and C b i; the factors and forecasts at time I: But for the future accounting year I we will indicate by a superscript the timepoint of the corresponding estimates, e.g. b f (I) on D I ; b f (I) and b C (I) and b C (I) : Note that, conditional are r.v., whereas f b and C b i; are given constants. In the following we derive the msep of this one-year run-o risk strictly within the classical model of Mack. a) single accounting year The claims development result of accident year i in the P&L statement of the next accounting year I is given by and (I) i Z (I) i (I) i ( w i;i F i;i b C i (I) (I) i i f ) : (5) With the telescope formula (5) we can write (5) as n Y Z (I) J o Y i w i;i F i;i f b i bf w F J Y i;i i;i (I) J ) : i {z } i i {z } A i B i By de nition of best estimate reserves we forecast a of. Hence the msep of the one-year run-o risk of the next accounting year is given by msep (I) () E Z (I) i D I i D I E A i DI E B i DI E [Ai B i D I ] : (53) 6

17 For the rst summand in (53) we get E A i DI w i;i E (F i;i f i ) DI f i Y J w i;i i w i; i f i i be A i DI b i i b f i YJ i b f i Y J i which, by use of the the estimation principle 3.6, is estimated by >: w i;i X i ki w k;i 9 > ; >; : (54) For estimating the second and the third summand in (53) we rst note that P i P i (I) bf ii w i; F i; w i; F i; P i P i w i; w i; a F i ; ; (55) where w i ; a P i : (56) w i; Since the observations of di erent accident years are independent it follows that n o F i;i ; f b(i) ; : : : ; f (I) are independent, (57) n (I) i ; b f (I) J o ; : : : ; f (I) J are independent. (5) From the model assumptions 3. and from (55) and (57) follows that E Fi; DI i f i i ; w i;i h i (I) D I E Var (I) D I E (I) D I b f a a f i w i;i ; and, by applying the estimation principle 3.6, a f f b ; f b a ; w i;i be b Fi; DI i i i (59) w i;i h i be (I) D I f b be (I) D I ; (6)! a b P i w i; w i;i b b ; (6) 7

18 where b w i ; Pi w i; Pi w i; : (6) With the estimation principle 3.6 we immediately obtain that be [A i B i D I ] ; (63) since hy J be i (I) Y J f ) i i DI : From (57) and (59) follows that the second summand in (53) can be estimated by be (! b Bi!) J Y DI w i;i i b b wi; i i be Z (I) i D I i >: w i;i w i;i b i i i b i C B i By plugging (54) ; (63) ; (64) into (53) we get >: w i;i >: w i;i b i C B i i i X i B b ki w k;i 9 > B b i 9 > C C A A >; : (64) >; (65) 9 > C C A A >; : Remarks: From (6) and (6) we see that b b which is an intuitively accessible interpretation. CoV d (I) D I ; (66) Note the di erence between dcov Bi i measure, how much f b i will deviate from f ; if only the data B i are considered as known and non random and dcov (I) D I measure how much b f (I) will deviate from b f all data D I are considered as known and non random. if

19 b) total over all accident years We complement the observed triangle by lling up the lower right not yet observed part with the -forecasts C b and take the total over each column, that is we de ne X I bc tot; : w tot; w i; ; where the weights w i; are de ned in (4) : Analogously we de ne (I) tot; : X I w (I) i; ; where w (I) C i; (I) i; if C D I otherwise : By de nition of b f and b f (I) and of the -forecasts it holds that by and tot; w tot; (I) tot; w tot; Y Y b f ; b f (I) : The of the next accounting year for the total over all accident year is given (I) tot b C (I) Z (I) tot (I) tot w tot; b f (I) b f! : From the independence property (5) and from (6) we immediately get 9 Y be Z (I) tot D I B b >:J > C A >; (67) c) result The following Theorem is a summary of (65) and (67) : 9

20 Theorem 5. The msep of the one year run-o risk in the next accounting year I can be estimated by a) single accident year [msep (I) () i D I >: w i;i b i i >: w i;i b i C B i i X i ki w k;i B b 9 > >; 9 > C C A A >; ; (6) where b w i ; Pi w i; Pi w i; ; (69) b) total over all accident years [msep (I) () tot D I where b is given by (69) : Remarks: w i; as de ned in (4) : Y >:J b b 9 > C A >; ; (7) The rst summand in (6) re ects the risk that the observation on the next diagonal will deviate from the forecast at time I: The second summand in (6) re ects the risk of updating the forecasts of later development years due to an update of the estimated -factors from time I to time I. The estimator (7) is surprisingly simple and even simpler than the estimator (7) for a single accident year. intuitively accessible interpretation From (9), (36) and (66) we see that b i CoV w d (F i; B ) ; i;i i b i X i CoV d i w k;i ki b b B ; CoV d D I ;

21 where the right hand side are easily understandable and intuitively accessible interpretations of the items on the left-hand side. The following Theorem is obtained by taking a rst order Taylor expansion of (6) and (7) : It is a good approximation, if b practical situations. b ( b f ) << ;which is the case in most Theorem 5. The msep of the one year run-o risk in the next accounting year I can alternatively (Taylor approximation) be estimated by a) single accident year [msep (I) () i D I b i X >: w i i;i i >: XJ i b ki w k; b 9 > >; 9 > C A >; (7) where b w i ; Pi w i; Pi w i; ; (7) b) total over all accident years Remarks [msep (I) () tot D I w i; as de ned in (4) : X b >:J b 9 > >; : (73) (7) is the Merz-Wüthrich formula (see (97) in appendix A). Hence the Merz- Wüthrich formula is obtained by a rst order Taylor approximation from (6) : In practical applications the numerical results of the two estimators are very often so close to each other that the di erence is negligible for practical purposes. (73) can directly be compared with the corresponding Merz-Wüthrich formula (99) in appendix A. It looks much nicer, since all the covariance terms occurring in (99) have disappeared.

22 intuitively accessible interpretation Analogously as in Theorem 5. we can look behind the formulas and we can write (7) and (73) as [msep (I) () i D I [msep (I) () tot D I dcov (F i;i B i ) CoV d Bi i ) dcov (I) D I ( XJ i ( XJ ; (74) ) CoV d (I) D I ; (75) which are intuitively accessible interpretations of (7) and (73) : 6 The one-year run-o prediction uncertainty in future accounting years In the SST and in solvency II, the market value margin corresponding to the run-o risk equals the discounted cost of capital, which is needed at the end of each accounting year for the one year run-o risk in the next accounting year. For this purpose we need estimates of the one year run-o risk for the accounting years I k; k ; : : : ; J ; evaluated at time I: 6. The Compatibility Condition Note that the one-year in the accounting years I k; k ; : : : ; J are given by (Ik) (Ik) i b C tot b C (Ik ) (Ik ) (Ik) ; (Ik) : Note also that accident year i is already fully developed in the accounting years fi k : k J and that therefore (Ik) C for k J i : The sum of the one year claims development results over all future development years is equal to the ultimate claims development result. Hence the one-year run-o risks Z (Ik) i (Ik) i and hence X J E X J E k Z(Ik) i k Z(Ik) tot satisfy X J D I D I X J i k Z(Ik) i C C b ; (76) k Z(Ik) tot C C b E E C D I C D I ; (77) msep CD I ; (7) msep CD I :(79) i g

23 By de nition of best estimate reserves the forecast of the claims development result in future periods is always n zero. For this reason o it is often argued that the process of best estimate forecasts bc BE(Ik) : k ; : : : ; J is a martingale and that therefore the one-year, which are the increments of this process, are independent. Based on this martingale argument it is then required that the estimators of the one-year run-o risk should satisfy the following "splitting" property: De nition 6. (splitting property) Estimators of the msep of the one year run-o risk ful l the "splitting property" if X J [msep k (Ik) i D I () [msep CD I for i i ; : : : ; I, () X J [msep k (Ik) tot D I () [msep CD I ; () where the right hand side of () and () are given by Theorem 4.. Remarks: The "splitting property" means that the ultimate run-o prediction uncertainty is split over all future accounting years until nal development. However best estimate forecasts are usually not a martingale, which is also the case for the -forecasts. The forecasts ful l h be (Ik) C b i (Ik) C b(ik) ; but they do not satisfy the martingale condition h E (Ik) C b i (Ik) b C (Ik) ; because the unknown factors f are replaced in the -forecasts by its estimates f b and f b(ik) respectively. Assume, that the required capital (risk margin) for the run-o risk is calculated by a xed percentage of the msep (variance risk measure) and that we consider only nominal cash- ows (no interest income). The requirement, to have enough capital for the ultimate run-o risk is a weaker condition than the requirement, that this capital is also su cient to meet, from a today s perspective, the capital requirement for the one-year run-o risk at the beginning of each future accounting year until nal claims development. Therefore the msep for the ultimate run-o risk is a lower bound for the sum of the msep of the one-year run-o risk. For this reason the estimators for the one-year run-o risk should satisfy the following "compatibility" condition: Condition 6. (compatibility) Estimators of the msep of the one year run-o risk ful l the "compatibility condition" if for i i ; : : : ; I and for the total over all accident years it holds that X J i D I () > [msep CD I for i i ; : : : ; I; () X J k [msep (Ik) [msep k (Ik) tot D I () > [msep CD I where the right hand side of () and (3) are given by Theorem ; (3)

24 6. Estimators of the msep of the one-year run-o risk in future accounting years It holds that msep (Ik) () E Z (Ik) i D I i D I E E Z (Ik) i D Ik D I ; (4) msep (Ik) () E Z (Ik) tot D I tot D I E E Z (Ik) D Ik D I : (5) The following estimator of the inner expected value of (4) is immediately obtained with Theorem 5.. where [msep (Ik) () i D Ik B (Ik) i k >: B (Ik) (Ik) b i k i k b i k i k C i k; tot >: C i;i k C B i k X i li C l;i k B (Ik) 9 > >; (6) b 9 > C C A A >; ; Pi k C i; Pi k C i; ; (7) i k X (Ik) C i;i k i k C i; X F i; ; where C ; w i; ; () C ; i k (Ik) : (9) Note that the C i; appearing in (6) play the role of a "weight". The only di erence to (6) is that some of these "weights" are "stochastic weights" and not exactly known at time I. A natural procedure is to replace them with the weights w i; de ned in (4) ; which are either the already known weights C i; D I or the forecasts C b i; at time I: As mentioned in the remarks to Theorem 4. on page 3 in the last bullet point, this procedure has given there the correct Mack-result. We do the same here and check then whether the resulting estimators ful l the compatibility condition 6.. 4

25 Theorem 6.3 The msep of the one-year run-o risk in future accounting years can be estimated by i) single accident year i, accounting years I k ; k ; : : : ; J i where [msep (Ik) () i D I >: w i;i k b (Ik) b i k ) >: i k b i k i k C B w i;i k i k w i k; X i li w l;i k b (Ik) 9 > >; (9) b 9 > C C A A >; Pi k w i; Pi k w i; ; (9) weights w i; as de ned in (4) : ii) total over all accident years, accounting years I k ; k ; : : : ; J 9 J Y [msep (Ik) () B tot D b (Ik) b > C A >: i k >; : (9) iii) The estimators (9) and (9) ful l the compatibility condition: Remarks: - For k (9) and (9) are of course the same as (6) and (7) in Theorem The msep for accounting years with k > can also be calculated by lling up the next k diagonals with the chain ladder forecasts to get an ari cial new data set e D Ik and by applying Theorem (7) on the remaining triangle in e D Ik ; but by keeping the estimators b from the original triangle D I : - (9) is again a surprisingly simple formula. - intuitively accessible interpretation Analogously as in Theorem 5. the expressions appearing in Theorem 6.3 can be interpreted in the following easily understandable and intuitively accessible way: X i w i;i k li w l;i k b (Ik) b i k CoV d (F i;i k D Ik ) ; ) i k b i k CoV d ) i k b i k B k ; CoV d (I) D Ik : 5

26 Proof of Theoerm 6.3: The derivation of the estimators in i) was already explained. ii) is obtained analogously. iii) follows from the next Theorem 6.4 and the fact, that the estimators (9) and (9) are greater than the estimators (93) and (94) : As in section 5 the estimators in Theorem 6.3 can again be approximated by a rst Taylor approximation. Theorem 6.4 The msep of the one-year run-o risk in future accounting years can alternatively (Taylor approximation) be estimated by i) single accident year i, accounting years I k ; k ; : : : ; J i 9 b [msep (Ik) () ik B i D > C X >: w i A i;i k >; ik >: XJ i k b (Ik) li w l;i k b 9 > >; (93) where b (Ik) are given by (9). ii) total over all accident years, accounting years I k ; k ; : : : ; J 9 > [msep (Ik) () tot D I >: XJ k b (Ik) b >; (94) where b (Ik) is given by (9). iii) the estimators (93) and (94) ful l the splitting property: Remarks: - The rst bullet point in the remarks after Theorem 6.3 is analogously valid here. - (93) and (94) are di erent to the Merz-Wüthrich formulas () and () : However they give the same numerical results. This cannot be a pure coincidence. Thus they must be same ust written di erently. - The rst summand in (93) re ects again the risk that the observation on the next diagonal will deviate from the forecast at time I, whereas the second summand entails the risk of updating the forecasts of the later development years: The rst summand looks similar to the one in Merz-Wüthrich, but note, that there is a di erence even in this rst summand. - (94) is again a surprisingly simple formula compared to the rather complicated expression with all the covariance terms in () : 6

27 - intuitively accessible interpretation Analogously as in Theorem 5. (93) and (94) can be written as [msep (Ik) () i D I [msep (Ik) () tot D I i k B i k ) dcov (Ik) D Ik ; dcov (Fi;ik B ik) CoV d ( XJ i k ( XJ i k ) dcov (Ik) D Ik ; which are intuitively accessible and understandable formulas. Proof of Theorem 6.4: (93) and (94) are obtained by a straightforward rst order Taylor approximation from (9) and (9) :The proof of the splitting property iii) is given in appendix D 7 Numerical example Table shows in the upper left part the data of a claims development triangle of medical costs in accident insurance and in the lower right part the chain-ladder forecasts. Column contains the -forecasts of the ultimate claims (in red) and the corresponding chainladder reserves. The total reserves at the end of for this line of business amounts to CHF mio. Table : triangel of cumulative payments in CHF ' and forecasts Reserves 94 '34 '3 '373 '465 '55 '559 '593 '63 '755 '49 '96 '999 3'96 3'333 3'4 3'53 3'67 3'74 3'797 3'75 3' '43 '796 3'4 3'36 3'3 3'4 3'56 3'5 3'7 3'95 3'966 4'64 4'3 4'56 4'94 5'394 5'495 5'597 5'79 5'4 6'3 96 '6 '97 3'44 3'35 3'4 3'469 3'56 3'6 3'659 3'6 3'69 3'79 3'75 3'77 3'793 3'799 3'5 3'4 3'37 3'5 3'5 97 '75 3'45 3'74 3' 3'64 3'93 3'974 3'997 4'7 4'4 4'59 4' 4' 4'7 4'44 4'57 4' 4' 4'33 4'46 4'53 9 '673 4'457 4'66 4'96 5'55 5'7 5'35 5'7 5'5 5'47 5'43 5'45 5'465 5'45 5'497 5'57 5'59 5'54 5'56 5'5 5'64 99 '9 4'73 5'3 5'3 5'359 5'437 5'5 5'55 5'569 5'6 5'65 5'6 5'6 5'653 5'67 5'73 5'74 5'77 5'776 5'794 5'4 99 3'5 4'9 5'37 5'6 5'734 5' 5'65 5'93 5'9 6'5 6' 6'4 6'7 6'64 6'39 6'354 6'39 6'4 6'49 6'447 6'5 99 '649 4'34 4'5 4'995 5'97 5'63 5'35 5'375 5'5 5'56 5'635 5'697 5'73 5'75 5'4 5'34 5'59 5' 5'93 5'9 5' '779 4'34 5'3 5'49 5'69 5'67 5'73 5' 5'35 5'935 6' 6'59 6'9 6' 6'43 6'69 6'6 6' 6'6 6'56 6' '49 4'344 4'7 4'96 5'33 5'7 5'9 5'3 5'66 5'349 5'3 5'399 5'43 5'459 5'466 5'47 5'475 5'477 5'5 5'54 5' '6 5'43 5' 6'63 6'37 6'97 6'54 6'5 6'6 6'75 6'5 6'9 6'977 7'59 7'56 7'7 7'63 7'3 7'34 7'396 7' '54 7'574 '4 '36 9'99 9'76 9'44 9'59 9'5 '3 '7 '36 '479 '564 '647 '776 '65 '93 '993 '65 ' '49 6'66 7'5 7'375 7'55 7'646 7'746 7'9 7'6 7'94 7'9 7'947 7'976 '46 ' ' '9 '334 '37 '44 ' '557 7'9 7' '65 '33 '445 '54 '66 '697 '744 '6 '74 '946 9' 9' 9'33 9'39 9'359 9'49 9'4 9' '74 '9 9' '355 '66 ' ' '59 '65 '77 '9 '93 '7 '5 '36 '53 '633 '7 '73 '66 ' '5 '6 3'99 3'9 4'77 4'574 4'55 5'4 5'33 5'469 5'6 5'75 5'93 6' 6'9 6'54 6'65 6'74 6'47 6'956 7'3 '374 6'7 3'3 5'55 5'797 6'3 6'39 6'65 6'65 7'4 7'5 7'3 7'46 7'65 7'53 '5 '99 '45 '549 '66 '79 '93 '66 6'3 '93 3'55 4' 4'65 4'956 5' 5' 5'364 5'474 5'64 5'75 5'9 6'4 6'3 6'56 6'64 6'73 6'39 6'94 7'3 '649 6'97 ' 4'66 4'3 5'36 5'56 5'73 5'973 6'6 6'53 6'4 6'543 6'74 6'95 7'3 7'337 7'4 7'574 7'67 7' 7'96 '94 3 6'369 '594 4'45 5'9 5'56 5'9 6'5 6'6 6'36 6'5 6'74 6'7 7'6 7'57 7'44 7'67 7'33 7'99 '44 '6 '349 '63 4 7'735 5'339 7'74 '77 '73 9'47 9'54 9'5 '96 '335 '53 '69 '95 '63 '39 '69 '7 '9 '9 '7 '53 '9 5 9' 7'45 9'96 '4 '7 '5 '74 3'4 3'47 3'75 3'9 4'75 4'439 4'7 4'99 5'335 5'544 5'6 5'46 6'4 6'3 3'76 6 '3 '5 4'53 6'3 7' 7'736 '7 '55 '95 9'5 9'533 9'77 3'9 3'44 3'77 3' 3'45 3'67 3'9 3'36 3'36 5'6 7 '945 '346 3'646 4'65 5'34 5'4 6'45 6'6 6'93 7'5 7'55 7'73 '4 '36 '675 9'69 9'3 9'466 9'654 9'47 3'56 5'55 '73 '74 5'7 6'39 7'9 7'664 '96 '477 '3 9'73 9'456 9'694 3'9 3'36 3'69 3'9 3'376 3'544 3'746 3'953 3'3 7'3 9 '667 '95 3'57 5'3 5'74 6' 6'63 6'99 7'36 7'65 7'9 '45 '453 '77 9'97 9'496 9'739 9'99 3'9 3'6 3'599 9'34 '35 3'476 6'99 7'574 '346 '95 9'357 9'755 3'3 3'43 3'77 3'7 3'366 3'74 3'76 3'56 3'74 3'96 33'7 33'37 33'73 '347 Total 66'697 f

28 Table shows the square root of the estimated msep for the ultimate run-o and for the one year run-o in the next accounting year where the latter is calculated with the "exact" estimator according to Theorem 5. as well as with the Taylor approximation according to Theorem 5.. We see, that in this example the numerical results of the two estimators di er not more than in the second digit after the decimal point and that the results with the "exact" estimator are greater, but only very minor in the second digit after the decimal point. accident year Reserves Table ultimate run off one year run off msep / in % reserves next accounting year msep / in % reserves exact Taylor appr % % % % % % % % %.4.4 % % % % % % % 999 ' % % ' % % ' %.5.5 7% ' % % 3 ' % % 4 '9 63 % % 5 3' % % 6 5'6 4 6% % 7 5' % % 7'3 6 % % 9 9'34 93 % % '347 '795 % '57.37 ' % Total 66'697 5'33 % '435. ' % Table 3 shows the results obtained for the square-root of the one-year msep in future accounting years until nal development together with the ingoing reserves. The numerical results obtained with the "exact" formula and with the Taylor approximation are again practically the same with di erences only in the second digit after the decimal point. From the table we can also the splitting property of the Taylor-approximation. In the current formula for calculating the risk margin in solvency II it assumed that the required capital for the remaining one-year run-o risk in future accounting years decreases proportionally to the remaining reserves. This was due to the lack of formulas to calculate the prediction uncertainty for future accounting years. But now the formulas have been developed and are here. Comparing the development pattern of the reserves with the development pattern of the msep we see, that the latter decreases much slower. This is not a surprise. Complex and complicated claims such as severe bodily inury claims stay open for a long time, whereas "normal" claims can be settled much quicker. Hence the proportion of the reserves stemming from complex claims is bigger in later development years. But the prediction uncertainty of this kind of claims is bigger as for the "normal" claims. But it also means that one will need more capital in solvency II with this new formulas, since the risk margin will become bigger.

29 Table 3 total over all accident years development pattern reserves and msep accounting year ingoing Reserves in ' dev. pattern "exact" one year run off msep / Taylor appr. dev.pattern 66'697 % '435. '435.6 % ' 4'53 73% '.67 '.67 74% '3 4'99 6% '66.6 '66.5 6% '4 35'76 54% '564. ' % '5 3'64 47% '46.5 ' % '6 7'96 4% '5.7 '5.7 5% '7 4'694 37% '63.4 '63.4 4% ' '66 3% '99. '99. 45% '9 '79 % '7.3 '7.3 4% ' 6'67 4% % ' 3'53 % % ' '64 7% % '3 '933 3% % '4 6'93 % % '5 5'64 7.7% % '6 3'64 5.5% % '7 ' % % ' '6.4% % '9 74.3% % ' % % sum year msep 5'36'94 ultimate msep 5'36' dev. Pattern ingoing reservs dev. pattern msep Acknowledgement I am grateful to Ancus Röhr. Discussions with him on earlier drafts of his paper [] have motivated me to work further on this topic. The result is the present paper. I am also grateful to Hans Bühlmann. He told me that Lemma 3.5 is called the telescope-formula and how it can be proved best. Finally I would like to thank Patrick Helbling, who had done most of the calculations of the numerical example. 9

30 References [] Buchwalder M., Bühlmann, H., Merz M., Wüthrich M. (6). The Mean Square Error of Prediction in the Chain Ladder Reserving Method-Final Remark 36/, [] Buchwalder M., Bühlmann, H., Merz M., Wüthrich M. (6). The Mean Square Error of Prediction in the Chain Ladder Reserving Method (Mack and Murphy Revisited).ASTIN Bulletin 36/, 553. [3] Bühlmann, H., De Felice M., Gisler, A., Moriconi F., Wüthrich, M.V. (9). Recursive Credibility Formula for Chain Ladder Factors and the Claims Development Result. ASTIN Bulletin 39/, [4] Gisler, A. (6). The estimation error in the chain-ladder reserving method: a Bayesian approach. ASTIN Bulletin 36/, [5] Gisler, A., Wüthrich M.V. [6] (). Credibility for the chain-ladder reserving method.3/, [7] Mack, T. (993). Distribution-free calculation of the standard error of chain ladder reserve estimates. ASTIN Bulletin 3/, 3 5. [] Mack, T., Quarg B., Braun C. (6).The Mean Square Error of Prediction in the Chain Ladder Reserving Method- A Comment. ASTIN Bulletin 36/, [9] Merz M., Wüthrich M.V. (). Stochastic Claims Reserving Methods in Insurance, Wiley. [] Merz M., Wüthrich M.V. (). Modelling the claims development result for solvency purposes. CAS Forum, Fall, [] Merz M., Wüthrich M.V. (4). Claims run-o uncertainty: the full picture, SSRN manuscript [] Roehr A. (6). Chain ladder and error propagation, to appear in the next ASTIN Bulletin. 3

31 Appendices A Formulae of Mack and Merz-Wüthrich A. Formula of Mack Written in the notation of this paper Mack [7] has found the following formulas for estimating the total run-o uncertainties for the reserves. Theorem A. (Mack) The msep can be estimated by i) single accident year i [msep CD I XJ i b w i;! P i w i; (95) where b C i; are the -forecasts and where w i; are as de ned in (4) : ii) total over all accident years [msep CD I ii J ii J ki [msep CD I k;j X J i b P i mi w m; C A : (96) A. Formulae of Merz-Wüthrich In [] Merz and Wüthrich found the following formula for the msep of the one year run-o risk in the next accounting year (see formulas (.) and (.3) in []). Theorem A. The msep of the one-year run-o risk in the next accounting year can be estimated by i) single accident year i where [msep i;id I () (I) b i (I) i XJ (I) i (I) w i ;! P w i i;i ki w k;i b (I) (I) P i ki w k;! ; (97) P i ki w k; : (9) 3

32 ii) total over all accident years [msep tot;id I () ii J mi (I) ii J (I) m;j >: [msep i;id I () (99) b i (I) i P i li w l;i XJ i b (I) (I) P i ki w k; In [] Merz and Wüthrich also derived the following formulas for estimating the msep of the one year run-o risk in future accounting years (see formulas (.4) and (.4) in []). Theorem A.3 The msep of the one-year run-o risk of future accounting years I k can be estimated by i) single accident year i (for k ; : : : ; J i ) [msep i;ikd I () b i k i k XJ i k i; i k b ky m ii) total over all accident years (for k ; : : : ; J ) [msep tot;ikd I () ii J k mi ii J k mi b C m;j b C m;j ii J k b i k m i k XJ i k! (I) i m P i k mi w m;i k ky (I) k (I) m m [msep i;ikd I () ky (I) i m b P i mi w m; P i k mi w m;i k ky (I) k (I) m m! () : P i li w l; ()! : 9 > >; : 3

33 B Discussion on the Estimation Error in 6: comparison between BBMW estimator and Mack estimator As mentioned in the introduction, there was was quite some discussion how to estimate the estimation error in the classical Mack model (see [], [], [4]). In [4] Gisler introduced the Bayesian -model. He then considered two speci c Bayesian models, where one of them was essentially the same as the model, which is considered in []. By taking a non-informative prior he obtained the BBMW estimator for the estimation error. Thus the Bayesian approach seemed to speak in favour of the BBMW estimator. The introduction of the Bayesian -model turned out to be very fruitful (see [5],[3]). It is a useful model for many situations in practice. However, the Bayesian -model is a di erent model. It is di erent to the Mack model, and in the mean-time I have come to the conclusion that the BBMW estimator is the appropriate estimator in the Bayesian model, but not in the classical model of Mack. In (37) we have seen that the Mack estimator is given by dee Mack i XJ i b X i w i; : () In [] the authors suggested an alternative estimator dee BBMW J i wi; BY B X i i w i; C A i C A ; (3) what they called "conditional resampling". More discussions on the two estimators at that time are found in [], [], [4]. We now reconsider the two estimators. From the telescope formula we have obtained that the estimation error is given by where B i XJ i EE i B i ; C b i; f {z } i k f k {z } d i : (4) The graphics below visualises (4) for accident year i in J 3 : For simplicity we ohave dropped there the index i: Note that both, the traectory : J 3; : : : ; J as 33

34 well as the traectory : E [C i; D I ] : J 3; : : : ; J ; are xed and non stochastic. To estimate B i we should have in mind that the i are realisations of r.v. i b C i; (F f ) (5) where F i X i w i; X F i; ; w ; w i; : w ; and that i E i D I : Since C b i; is a known and given multiplicative factor it is obvious that one should consider the conditional distribution of i given B for estimating i: This is exactly what Mack does, and he obtains E i B i; X i w i; ; b (Mack) i E b i B i; b X i w i; : (6) Since f i; ; : : : i; g are conditional on B known constants, it follows that the r.v. f i B : i ; : : : ; J g are uncorrelated. Hence fd i : i ; : : : ; J g can be considered as realisations of uncorrelated r.v. which then leads immediately to the Mack estimator () : In terms of the telescope formula, the BBMW estimator is obtained if one considers the r.v. eb i XJ i e Ci; ef f k f k : (7) 34

35 where Y ec i; w i; Fk e and where k i the F e h i are independent r.v. with E ef f and Var (F B ) X i w i; : Hence the known multiplicative constants C b i; in (4) are replaced by r.v. Ci; e in (7) ;which hardly makes sense in the classical Mack model. However, in the Bayesian -model the factors f are realisations of r.v. F B (the superscript B indicates that we are in the Bayesian model). The estimation error is then given by where B i EE i B i ; E [C D I ] d C w i;i i F B With the telescope formula we also obtain that where B i X J i e C i; F B ec i; w i; i b f! Y J bf k Y k i F B k : : () k ; (9) which should be compared with (4) : In the following we always refer to the Bayesian -model considered in []. In this model the r.v. F B are conditional on D I independent. By taking a non informative prior and in the cases where the two rst moments of the posterior distribution exist, these moments are given by E F B DI f b ; () Var F B DI X i : () w i; From () ; () ; () and the conditional independence of the F B we immediately see that h i J E Bi D I wi; BY B C Y X i A C A : i w i; Hence, the BBMW estimator is the appropriate estimator in the Bayesian -model, but not in the classical model of Mack. 35 i