Actuariat de l Assurance Non-Vie # 10 A. Charpentier (Université de Rennes 1)

Transcription

1 Actuariat de l Assurance Non-Vie # 10 A. Charpentier (Université de Rennes 1) ENSAE ParisTech, Octobre / Décembre

2 Généralités sur les Provisions pour Sinistres à Payer Références: de Jong & Heller (2008), section 1.5 et 8.1, and Wüthrich & Merz (2015), chapitres 1 à 3, et Pigeon (2015). Les provisions techniques sont les provisions destinées à permettre le réglement intégral des engagements pris envers les assurés et bénéficaires de contrats. Elles sont liées à la technique même de l assurance, et imposées par la réglementation. It is hoped that more casualty actuaries will involve themselves in this important area. IBNR reserves deserve more than just a clerical or cursory treatment and we believe, as did Mr. Tarbell Chat the problem of incurred but not reported claim reserves is essentially actuarial or statistical. Perhaps in today s environment the quotation would be even more relevant if it stated that the problem...is more actuarial than statistical. Bornhuetter & Ferguson 2

3 La vie des sinistres date t 1 : survenance du sinistre période [t 1, t 2 ] : IBNR Incureed But Not Reported date t 2 : déclaration du sinistre à l assureur période [t 2, t 3 ] : IBNP Incureed But Not Paid période [t 2, t 6 ] : IBNS Incureed But Not Settled dates t 3, t 4, t 5 : paiements date t 6 : clôture du 3

4 La vie des sinistres À la survenance, un montant estimé est indiqué par les gestionnaires de sinistres (charge dossier/dossier). Ensuite deux opérations sont possibles : effectuer un paiement réviser le montant du 4

5 La vie des sinistres: Problématique des PSAP La Provision pour Sinistres à Payer est la différence entre le montant du sinistre et le paiement déjà 5

6 Triangles de Paiements : du micro au macro Analyse micro Sinistres i = 1,, n Montant de i à la date t, C i (t) = j:t i,j t Y i,j Provision (idéale) à la date t, R i (t) = C i ( ) C i (t) survenances, date T i,0 déclaration, date T i,0 + Q i paiements, dates j T i,j = T i,0 + Q i + k=1 et montants (T i,j, Y i,j ) Z 6

7 Triangles de Paiements : du micro au macro Analyse macro On agrége les sinistres par année de survenance : S i = {k : T k,0 [i, i + 1)} On regarde les paiements effectués après j années Y i,j = k S i T k,l [j,j+1) Z k,l Ici Y n 7

8 Triangles de Paiements : du micro au macro Analyse macro On agrége les sinistres par année de survenance : S i = {k : T k,0 [i, i + 1)} On regarde les paiements effectués après j années Y i,j = k S i T k,l [j,j+1) Z k,l Ici Y n 1,1 et Y 8

9 Triangles de Paiements : du micro au macro Analyse macro On agrége les sinistres par année de survenance : S i = {k : T k,0 [i, i + 1)} On regarde les paiements effectués après j années Y i,j = k S i T k,l [j,j+1) Z k,l Ici Y n 1,2, Y n,1 et Y 9

10 Triangles de Paiements : du micro au macro les provisions techniques peuvent représenter 75% du bilan, le ratio de couverture (provision / chiffre d affaire) peut dépasser 2, certaines branches sont à développement long, en montant n n + 1 n + 2 n + 3 n + 4 habitation 55% 90% 94% 95% 96% automobile 55% 80% 85% 88% 90% dont corporels 15% 40% 50% 65% 70% R.C. 10% 25% 35% 40% 10

11 Triangles de Paiements : complément sur les données micro On note Ft n l information accessible à la date t, Ft n = σ {(T i,0, T i,j, X i,j ), i = 1,, n, T i,j t} n et C( ) = C i ( ). i=1 Notons que M t = E [C( ) F ] est une martingale, i.e. E[M t+h F t ] = M t alors que V t = Var [C( ) F ] est une sur-martingale, i.e. E[V t+h F t ] V 11

12 Triangles de Paiements : les méthodes usuelles Source : 12

13 Triangles de Paiements : les méthodes usuelles Source : 13

14 Triangles de Paiements : les méthodes usuelles Source : 14

15 Triangles de Paiements : les méthodes usuelles Source : 15

16 Exemples de cadence de paiement: multirisques 16

17 Exemples de cadence de paiement: risque incendies 17

18 Exemples de cadence de paiement: automobile 18

19 Exemples de cadence de paiement: automobile 19

20 Exemples de cadence de paiement: automobile 20

21 Exemples de cadence de paiement: responsabilité civile 21

22 Exemples de cadence de paiement: responabilité civile 22

23 Exemples de cadence de paiement: assurance 23

24 Les triangles: incréments de paiements Notés Y i,j, pour l année de survenance i, et l année de développement j,

25 Les triangles: paiements cumulés Notés C i,j = Y i,0 + Y i,1 + + Y i,j, pour l année de survenance i, et l année de développement j,

26 Les triangles: nombres de sinistres Notés N i,j sinistres survenus l année i connus (déclarés) au bout de j années,

27 La prime acquise Notée π i, prime acquise pour l année i année i P i

28 Triangles 1 > rm( list =ls ()) 2 > source (" http :// freakonometrics. free.fr/ bases.r") 3 > ls () 4 [1] " INCURRED " " NUMBER " " PAID " " PREMIUM " 5 > PAID 6 [,1] [,2] [,3] [,4] [,5] [,6] 7 [1,] [2,] NA 9 [3,] NA NA 10 [4,] NA NA NA 11 [5,] NA NA NA NA 12 [6,] 5217 NA NA NA NA 28

29 Triangles? Actually, there might be two different cases in practice, the first one being when initial data are missing In that case it is mainly an index-issue in 29

30 Triangles? Actually, there might be two different cases in practice, the first one being when final data are missing, i.e. some tail factor should be included In that case it is necessary to extrapolate (with past information) the final loss (tail 30

31 The Chain Ladder estimate We assume here that C i,j+1 = λ j C i,j for all i, j = 0, 1,, n. A natural estimator for λ j based on past history is λ j = n j i=0 n j C i,j+1 for all j = 0, 1,, n 1. C i,j i=0 Hence, it becomes possible to estimate future payments using Ĉ i,j = [ λn+1 i λ ] j 1 C i,n+1 31

32 La méthode Chain Ladder, en pratique λ 0 =

33 La méthode Chain Ladder, en pratique λ 0 =

34 La méthode Chain Ladder, en pratique λ 0 =

35 La méthode Chain Ladder, en pratique λ 1 =

36 La méthode Chain Ladder, en pratique λ 1 =

37 La méthode Chain Ladder, en pratique λ 2 =

38 La méthode Chain Ladder, en pratique λ 2 =

39 La méthode Chain Ladder, en pratique λ 3 =

40 La méthode Chain Ladder, en pratique λ 4 =

41 La méthode Chain Ladder, en pratique One the triangle has been completed, we obtain the amount of reserves, with respectively 22, 36, 66, 153 and 2150 per accident year, i.e. the total is 41

42 Computational Issues 1 > library ( ChainLadder ) 2 > MackChainLadder ( PAID ) 3 MackChainLadder ( Triangle = PAID ) 4 5 Latest Dev.To. Date Ultimate IBNR Mack.S.E CV( IBNR ) 6 1 4, , NaN 7 2 4, , , , , , , , , ,367 2, Totals 14 Latest : 32, Dev : Ultimate : 35, IBNR : 42

43 18 Mack.S.E CV( IBNR ): 43

44 Three ways to look at triangles There are basically three kind of approaches to model development developments as percentages of total incured, i.e. consider ϕ = (ϕ 0, ϕ 1,, ϕ n ), with ϕ 0 + ϕ ϕ n = 1, such that E(Y i,j ) = ϕ j E(C i,n ), where j = 0, 1,, n. developments as rates of total incured, i.e. consider γ = (γ 0, γ 1,, γ n ), such that E(C i,j ) = γ j E(C i,n ), where j = 0, 1,, n. developments as factors of previous estimation, i.e. consider λ = (λ 0, λ 1,, λ n ), such that E(C i,j+1 ) = λ j E(C i,j ), where j = 0, 1,, 44

45 Three ways to look at triangles From a mathematical point of view, it is strictly equivalent to study one of those. Hence, γ j = ϕ 0 + ϕ ϕ j = 1 1 1, λ j λ j+1 λ n 1 ϕ j = γ 0 if j = 0 γ j γ j 1 if j 1 = λ j = γ j+1 = ϕ 0 + ϕ ϕ j + ϕ j+1 γ j ϕ 0 + ϕ ϕ j 1 1 λ 0 λ λ j+1 λ j+2 1, if j = 0 λ n λ n 1 λ j λ j+1 1 λ n 1, if j 45

46 Three ways to look at triangles On the previous triangle, n λ j 1, , , , , ,0000 γ j 70,819% 97,796% 98,914% 99,344% 99,529% 100,000% ϕ j 70,819% 26,977% 1,118% 0,430% 0,185% 46

47 d-triangles It is possible to define the d-triangles, with empirical λ s, i.e. λ i,j

48 The Chain-Ladder estimate The Chain-Ladder estimate is probably the most popular technique to estimate claim reserves. Let F t denote the information avalable at time t, or more formally the filtration generated by {C i,j, i + j t} - or equivalently {X i,j, i + j t} Assume that incremental payments are independent by occurence years, i.e.. C i1, and C i2, are independent for any i 1 and i 2 [H 1 ] Further, assume that (C i,j ) j 0 is Markov, and more precisely, there exist λ j s and σj 2 s such that E(C i,j+1 F i+j ) = E(C i,j+1 C i,j ) = λ j C i,j [H 2 ] Var(C i,j+1 F i+j ) = Var(C i,j+1 C i,j ) = σj 2 C i,j [H 3 ] Under those assumption (see Mack (1993)), one gets E(C i,j+k F i+j ) = (C i,j+k C i,j ) = λ j λ j+1 λ j+k 1 C 48

49 Testing assumptions Assumption H 2 can be interpreted as a linear regression model, i.e. Y i = β 0 + X i β 1 + ε i, i = 1,, n, where ε is some error term, such that E(ε) = 0, where β 0 = 0, Y i = C i,j+1 for some j, X i = C i,j, and β 1 = λ j. { n j } Weighted least squares can be considered, i.e. min ω i (Y i β 0 β 1 X i ) 2 where the ω i s are proportional to Var(Y i ) 1. This leads to { n j } 1 min (C i,j+1 λ j C i,j ) 2. C i,j i=1 As in any linear regression model, it is possible to test assumptions H 1 and H 2, the following graphs can be considered, given j plot C i,j+1 s versus C i,j s. Points should be on the straight line with slope λ j. i=1 plot (standardized) residuals ε i,j = C i,j+1 λ j C i,j Ci,j versus C i,j 49

50 Testing assumptions H 1 is the accident year independent assumption. More precisely, we assume there is no calendar effect. Define the diagonal B k = {C k,0, C k 1,1, C k 2,2, C 2,k 2, C 1,k 1, C 0,k }. If there is a calendar effect, it should affect adjacent factor lines, { Ck,1 A k =, C k 1,2, C k 2,3 C 1,k,,, C } 0,k+1 = δ k+1, C k,0 C k 1,1 C k 2,2 C 1,k 1 C 0,k δ k and A k 1 = { Ck 1,1 C k 1,0, C k 2,2 C k 2,1, C k 3,3 C k 3,2,, C 1,k 1 C 1,k 2, C 0,k C 0,k 1 } = δ k δ k 1. For each k, let N + k denote the number of elements exceeding the median, and N k the number of elements lower than the mean. The two years are independent, N + k and N k should be closed, i.e. N k = min ( N + k, N ) k should be closed to ( N + k + N ) k 50

51 Testing assumptions Since N k and N + k are two binomial distributions B ( p = 1/2, n = N k + N + ) k, then E (N k ) = n k 2 n k 1 n [ ] k m k 2 n where n k = N + k k + N k and m nk 1 k = 2 and V (N k ) = n k (n k 1) 2 n k 1 m k n k (n k 1) 2 n k + E (N k ) E (N k ) 2. Under some normality assumption on N, a 95% confidence interval can be derived, i.e. E (Z) ± 1.96 V 51

52 et

53 From Chain-Ladder to Grossing-Up The idea of the Chain-Ladder technique was to estimate the λ j s, so that we can derive estimates for C i,n, since Ĉ i,n = Ĉi,n i n k=n i+1 λ k Based on the Chain-Ladder link ratios, λ, it is possible to define grossing-up coefficients n 1 γ j = λ k k=j and thus, the total loss incured for accident year i is then Ĉ i,n = Ĉi,n i γ n γ n 53

54 Variant of the Chain-Ladder Method (1) Historically (see e.g.), the natural idea was to consider a (standard) average of individual link ratios. Several techniques have been introduces to study individual link-ratios. A first idea is to consider a simple linear model, λ i,j = a j i + b j. Using OLS techniques, it is possible to estimate those coefficients simply. Then, we project those ratios using predicted one, λ i,j = â j i + b 54

55 Variant of the Chain-Ladder Method (2) A second idea is to assume that λ j is the weighted sum of λ,j s, λ j = j 1 ω i,j λ i,j i=0 j 1 ω i,j i=0 If ω i,j = C i,j we obtain the chain ladder estimate. An alternative is to assume that ω i,j = i + j + 1 (in order to give more weight to recent 55

56 Variant of the Chain-Ladder Method (3) Here, we assume that cumulated run-off triangles have an exponential trend, i.e. C i,j = α j exp(i β j ). In order to estimate the α j s and β j s is to consider a linear model on log C i,j, log C i,j = a j +β j i + ε i,j. }{{} log(α j ) Once the β j s have been estimated, set γ j = exp( β j ), and define Γ i,j = γ n i j j C 56

57 The extended link ratio family of estimators For convenience, link ratios are factors that give relation between cumulative payments of one development year (say j) and the next development year (j + 1). They are simply the ratios y i /x i, where x i s are cumulative payments year j (i.e. x i = C i,j ) and y i s are cumulative payments year j + 1 (i.e. y i = C i,j+1 ). For example, the Chain Ladder estimate is obtained as λ j = n j i=0 y i n j k=0 x k = n j i=0 x i n j k=1 x k yi x i. But several other link ratio techniques can be considered, e.g. λ j = 1 n j + 1 n j i=0 y i x i, i.e. the simple arithmetic mean, λ j = ( n j i=0 y i x i ) n j+1, i.e. the geometric 57

58 λ j = n j i=0 x 2 i n j k=1 x2 k yi x i, i.e. the weighted average by volume squared, Hence, these techniques can be related to weighted least squares, i.e. y i = βx i + ε i, where ε i N (0, σ 2 x δ i ), for some δ > 0. E.g. if δ = 0, we obtain the arithmetic mean, if δ = 1, we obtain the Chain Ladder estimate, and if δ = 2, the weighted average by volume squared. The interest of this regression approach, is that standard error for predictions can be derived, under standard (and testable) assumptions. Hence standardized residuals (σx δ/2 i ) 1 ε i are N (0, 1), i.e. QQ plot E(y i x i ) = βx i, i.e. graph of x i versus y 58

59 Properties of the Chain-Ladder estimate Further λ j = n j 1 i=0 C i,j+1 n j 1 i=0 C i,j is an unbiased estimator for λ j, given F j, and λ j and λ j+h are non-correlated, given F j. Hence, an unbiased estimator for E(C i,j F n ) is Ĉ i,j = λ n i λ n i+1 λ ) j 2 ( λj 1 1 C i,n i. Recall that λ j is the estimator with minimal variance among all linear estimators obtained from λ i,j = C i,j+1 /C i,j s. Finally, recall that σ 2 j = 1 n j 1 n j 1 i=0 ( Ci,j+1 C i,j λ j ) 2 C i,j is an unbiased estimator of σ 2 j, given F j (see Mack (1993) or Denuit & Charpentier 59

60 Prediction error of the Chain-Ladder estimate We stress here that estimating reserves is a prediction process: based on past observations, we predict future amounts. Recall that prediction error can be explained as follows, E[(Y Ŷ )2 ] }{{} prediction variance ( ) 2] = E[ (Y EY ) + (E(Y ) Ŷ ) E[(Y EY ) 2 ] + E[(EY }{{} Ŷ )2 ]. }{{} process variance estimation variance the process variance reflects randomness of the random variable the estimation variance reflects uncertainty of statistical 60

61 Process variance of reserves per occurrence year The amount of reserves for accident year i is simply R i = ( λn i λ n i+1 λ ) n 2 λn 1 1 C i,n i. Note that E( R i F n ) = R i Since Var( R i F n ) = Var(C i,n F n ) = Var(C i,n C i,n i ) n n = λ 2 l σke[c 2 i,k C i,n i ] k=i+1 l=k+1 and a natural estimator for this variance is then Var( R n n i F n ) = λ 2 l σ 2 kĉi,k k=i+1 l=k+1 n = Ĉi,n k=i+1 σ k 2 λ 2 61

62 Note that it is possible to get not only the variance of the ultimate cumulate payments, but also the variance of any increment. Hence Var(Y i,j F n ) = Var(Y i,j C i,n i ) = E[Var(Y i,j C i,j 1 ) C i,n i ] + Var[E(Y i,j C i,j 1 ) C i,n i ] = E[σ 2 i C i,j 1 C i,n i ] + Var[(λ j 1 1)C i,j 1 C i,n i ] and a natural estimator for this variance is then Var(Y i,j F n ) = Var(C i,j F n ) + (1 2 λ j 1 ) Var(C i,j 1 F n ) where, from the expressions given above, Var(C i,j F n ) = C i,n i j 1 k=i+1 σ 2 k λ 2 62

63 Parameter variance when estimating reserves per occurrence year So far, we have obtained an estimate for the process error of technical risks (increments or cumulated payments). But since parameters λ j s and σj 2 are estimated from past information, there is an additional potential error, also called parameter error (or estimation error). Hence, we have to quantify E ([R i R ) i ] 2. In order to quantify that error, Murphy (1994) assume the following underlying model, C i,j = λ j 1 C i,j 1 + η i,j with independent variables η i,j. From the structure of the conditional variance, Var(C i,j+1 F i+j ) = Var(C i,j+1 C i,j ) = σ 2 j C 63

64 Parameter variance when estimating reserves per occurrence year it is natural to write the equation above C i,j = λ j 1 C i,j 1 + σ j 1 Ci,j 1 ε i,j, with independent and centered variables with unit variance ε i,j. Then E ([R i R ) i ] 2 F n = R i 2 ( n i 1 k=0 λ 2 i+k σ 2 i+k C,i+k + σ 2 n 1 [ λ n 1 1] 2 C,i+k Based on that estimator, it is possible to derive the following estimator for the Conditional Mean Square Error of reserve prediction for occurrence year i, CMSE i = Var( R i F n ) + E ([R i R ) i ] 2 F n. 64

65 Variance of global reserves (for all occurrence years) The estimate total amount of reserves is Var( R) = Var( R 1 ) + + Var( R n ). In order to derive the conditional mean square error of reserve prediction, define the covariance term, for i < j, as ( n ) CMSE i,j = R σ i+k 2 σ j 2 i Rj +, C,k [ λ j 1 1] λ j 1 C,j+k k=i λ 2 i+k then the conditional mean square error of overall reserves CMSE = n i=1 CMSE i + 2 j>i CMSE 65

66 Application on our triangle 1 > MackChainLadder ( PAID ) 2 3 Latest Dev.To. Date Ultimate IBNR Mack.S.E CV( IBNR ) 4 1 4, , NaN 5 2 4, , , , , , , , , ,367 2, Totals 12 Latest : 32, Dev : Ultimate : 35, IBNR : 2, Mack.S.E CV( IBNR ): 66

67 Application on our triangle 1 > MackChainLadder ( PAID )$f 2 [1] > MackChainLadder ( PAID )$f.se 4 [1] e e e e e > MackChainLadder ( PAID )$ sigma 6 [1] > MackChainLadder ( PAID )$ sigma ^2 8 [1] e e e e e 67

68 A short word on Munich Chain Ladder Munich chain ladder is an extension of Mack s technique based on paid (P ) and incurred (I) losses. Here we adjust the chain-ladder link-ratios λ j s depending if the momentary (P/I) ratio is above or below average. It integrated correlation of residuals between P vs. I/P and I vs. P/I chain-ladder link-ratio to estimate the correction factor. Use standard Chain Ladder technique on the two 68

69 A short word on Munich Chain Ladder The (standard) payment triangle, P

70 Computational Issues 1 > MackChainLadder ( PAID ) 2 3 Latest Dev.To. Date Ultimate IBNR Mack.S.E CV( IBNR ) 4 1 4, , NaN 5 2 4, , , , , , , , , ,367 2, Totals 12 Latest : 32, Dev : Ultimate : 35, IBNR : 2, Mack.S.E CV( IBNR ): 70

71 A short word on Munich Chain Ladder The Incurred Triangle (I) with estimated losses,

72 Computational Issues 1 > MackChainLadder ( INCURRED ) 2 3 Latest Dev.To. Date Ultimate IBNR Mack.S.E CV( IBNR ) 4 1 4, , NaN 5 2 4, , Inf 6 3 5, , , , , , , , Totals 12 Latest : 35, Dev : Ultimate : 34, IBNR : Mack.S.E (keep only here Ĉn = 34, 524, to be compared with the previous 35, 72

73 Computational Issues 1 > MunichChainLadder (PAID, INCURRED ) 2 3 Latest Paid Latest Incurred Latest P/ I Ratio Ult. Paid Ult. Incurred Ult. P/ I Ratio 4 1 4,456 4, ,456 4, ,730 4, ,753 4, ,420 5, ,455 5, ,020 6, ,086 6, ,794 7, ,983 6, ,217 7, ,538 7,

74 11 Totals 12 Paid Incurred P/ I Ratio 13 Latest : 32,637 35, Ultimate : 35,271 35, It is possible to get a model mixing the two approaches 74

75 Bornhuetter Ferguson One of the difficulties with using the chain ladder method is that reserve forecasts can be quite unstable. The Bornhuetter & Ferguson (1972) method provides a procedure for stabilizing such estimates. Recall that in the standard chain ladder model, Ĉ i,n = F n C i,n i, where F n = n 1 k=n i λ k If R i denotes the estimated outstanding reserves, R i = Ĉi,n C i,n i = Ĉi,n F n 1 F 75

76 Bornhuetter Ferguson For a bayesian interpretation of the Bornhutter-Ferguson model, England & Verrall (2002) considered the case where incremental paiments Y i,j are i.i.d. overdispersed Poisson variables. Here E(Y i,j ) = a i b j and Var(Y i,j ) = ϕ a i b j, where we assume that b b n = 1. Parameter a i is assumed to be a drawing of a random variable A i G(α i, β i ), so that E(A i ) = α i /β i, so that E(C i,n ) = α i β i = C i, which is simply a prior expectation of the final loss 76

77 Bornhuetter Ferguson The posterior distribution of X i,j+1 is then E(X i,j+1 F i+j ) = ( Z i,j+1 C i,j + [1 Z i,j+1 ] C i F j ) (λ j 1) where Z i,j+1 = F 1 j βϕ + F j, where F j = λ j+1 λ n. Hence, Bornhutter-Ferguson technique can be interpreted as a Bayesian method, and a a credibility estimator (since bayesian with conjugated distributed leads to 77

78 Bornhuetter Ferguson The underlying assumptions are here assume that accident years are independent assume that there exist parameters µ = (µ 0,, µ n ) and a pattern β = (β 0, β 1,, β n ) with β n = 1 such that E(C i,0 ) = β 0 µ i Hence, one gets that E(C i,j ) = β j µ i. E(C i,j+k F i+j ) = C i,j + [β j+k β j ] µ i The sequence (β j ) denotes the claims development pattern. The Bornhuetter-Ferguson estimator for E(C i,n Ci, 1,, C i,j ) is Ĉ i,n = C i,j + [1 β j i ] µ 78

79 where µ i is an estimator for E(C i,n ). If we want to relate that model to the classical Chain Ladder one, β j is n k=j+1 1 λ k Consider the classical triangle. Assume that the estimator µ i is a plan value (obtain from some business plan). For instance, consider a 105% loss ratio per accident 79

80 Bornhuetter Ferguson i premium µ i λ i 1, 380 1, 011 1, 004 1, 002 1, 005 β i 0, 708 0, 978 0, 989 0, 993 0, 995 Ĉ i,n R i

81 Boni-Mali As point out earlier, the (conditional) mean square error of prediction (MSE) is i.e X is mse t ( X) = E ([X X] ) 2 F t ( = Var(X F t ) + E E (X F t ) }{{} X ) 2 }{{} process variance parameter estimation error a predictor for X an estimator for E(X F t ). But this is only a a long-term view, since we focus on the uncertainty over the whole runoff period. It is not a one-year solvency view, where we focus on changes over the next accounting 81

82 Boni-Mali From time t = n and time t = n + 1, n j 1 n j (n) λ j = i=0 C i,j+1 n j 1 and λ (n+1) j = i=0 C i,j+1 n j i=0 C i,j i=0 C i,j and the ultimate loss predictions are then Ĉ (n) i = C i,n i n j=n i λ j (n) and Ĉ (n+1) i = C i,n i+1 n j=n i+1 λ j 82

83 Boni-Mali and the 83

84 Boni-Mali and the 84

85 Boni-Mali and the 85

86 Boni-Mali In order to study the one-year claims development, we have to focus on R (n) i and Y i,n i+1 + R (n+1) i The boni-mali for accident year i, from year n to year n + 1 is then BM (n,n+1) i = R (n) i [ Y i,n i+1 + ] (n+1) R i = Ĉ(n) i Ĉ(n+1) i. Thus, the conditional one-year runoff uncertainty is mse( ( [Ĉ(n) (n,n+1) BM i ) = E i Ĉ(n+1) i ] 2 Fn 86

87 Boni-Mali Hence, mse( BM (n,n+1) i ) = [Ĉ(n) + i ] 2 [ σ 2 n i /[ λ (n) n i ]2 C i,n i n 1 j=n i+1 C n j,j n j k=0 C k,j (n) /[ λ n i ]2 i 1 k=0 C k,n i + σ2 n i σ j 2 (n) /[ λ j ] 2 n j 1 k=0 C k,j Further, it is possible to derive the MSEP for aggregated accident years (see Merz & Wüthrich 87

88 Boni-Mali, Computational Issues 1 > CDR ( MackChainLadder ( PAID )) 2 IBNR CDR (1) S.E. Mack.S.E Total

89 Ultimate Loss and Tail Factor The idea - introduced by Mack (1999) - is to compute λ = λ k k n and then to compute C i, = C i,n λ. Assume here that λ i are exponentially decaying to 1, i.e. log(λ k 1) s are linearly decaying 1 > Lambda = MackChainLadder ( PAID )$f [1:( ncol ( PAID ) -1)] 2 > logl <- log ( Lambda -1) 3 > tps <- 1:( ncol ( PAID ) -1) 4 > modele <- lm( logl ~ tps ) 5 > logp <- predict ( modele, newdata = data. frame ( tps = seq (6,1000) )) 6 > ( facteur <- prod ( exp ( logp ) +1) ) 7 [1] 89

90 Ultimate Loss and Tail Factor 1 > DIAG <- diag ( PAID [,6:1]) 2 > PRODUIT <- c(1, rev ( Lambda )) 3 > sum (( cumprod ( PRODUIT ) -1)* DIAG ) 4 [1] > sum (( cumprod ( PRODUIT )* facteur -1) * DIAG ) 6 [1] The ultimate loss is here 0.07% larger, and the reserves are 1% 90

91 Ultimate Loss and Tail Factor 1 > M a ck Ch ai n La dd er ( Triangle = PAID, tail = TRUE ) 2 3 Latest Dev.To. Date Ultimate IBNR Mack.S.E CV( IBNR ) 4 0 4, , , , , , , , , , , ,372 2, Totals 12 Latest : 32, Dev : Ultimate : 35, IBNR : 2, Mack.S.E CV( IBNR ): 91

92 From Chain Ladder to London Chain and London Pivot La méthode dite London Chain a été introduite par Benjamin & Eagles (1986). On suppose ici que la dynamique des (C ij ) j=1,..,n est donnée par un modèle de type AR (1) avec constante, de la forme C i,k+1 = λ k C ik + α k pour tout i, k = 1,.., n De façon pratique, on peut noter que la méthode standard de Chain Ladder, reposant sur un modèle de la forme C i,k+1 = λ k C ik, ne pouvait être appliqué que lorsque les points (C i,k, C i,k+1 ) sont sensiblement alignés (à k fixé) sur une droite passant par l origine. La méthode London Chain suppose elle aussi que les points soient alignés sur une même droite, mais on ne suppose plus qu elle passe par 0. Example:On obtient la droite passant au mieux par le nuage de points et par 0, et la droite passant au mieux par le nuage de 92

93 From Chain Ladder to London Chain and London Pivot Dans ce modèle, on a alors 2n paramètres à identifier : λ k et α k pour k = 1,..., n. La méthode la plus naturelle consiste à estimer ces paramètres à l aide des moindres carrés, c est à dire que l on cherche, pour tout k, { ) n k } ( λk, α k = arg min (C i,k+1 α k λ k C i,k ) 2 ce qui donne, finallement i=1 λ k = 1 n k n k i=1 C i,kc i,k+1 C (k) k n k i=1 C2 i,k C(k)2 k 1 n k C (k) k+1 où C (k) k = 1 n k n k i=1 C i,k et C (k) k+1 = 1 n k et où la constante est donnée par α k = C (k) k+1 λ k C (k) k. n k i=1 C 93

94 From Chain Ladder to London Chain and London Pivot Dans le cas particulier du triangle que nous étudions, on obtient k λ k α k

95 From Chain Ladder to London Chain and London Pivot The completed (cumulated) triangle is then

96 From Chain Ladder to London Chain and London Pivot One the triangle has been completed, we obtain the amount of reserves, with respectively 22, 43, 78, 222 and 2266 per accident year, i.e. the total is 2631 (to be compared with 2427, obtained with the Chain Ladder 96

97 From Chain Ladder to London Chain and London Pivot La méthode dite London Pivot a été introduite par Straub, dans Nonlife Insurance Mathematics (1989). On suppose ici que C i,k+1 et C i,k sont liés par une relation de la forme C i,k+1 + α = λ k. (C i,k + α) (de façon pratique, les points (C i,k, C i,k+1 ) doivent être sensiblement alignés (à k fixé) sur une droite passant par le point dit pivot ( α, α)). Dans ce modèle, (n + 1) paramètres sont alors a estimer, et une estimation par moindres carrés ordinaires donne des estimateurs de façon 97

98 Introduction to factorial models: Taylor (1977) This approach was studied in a paper entitled Separation of inflation and other effects from the distribution of non-life insurance claim delays We assume the incremental payments are functions of two factors, one related to accident year i, and one to calendar year i + j. Hence, assume that Y ij = r j µ i+j 1 for all i, j r 1 µ 1 r 2 µ 2 r n 1 µ n 1 r n µ n r 1 µ 2 r 2 µ 3 r n 1 µ n.. et r 1 µ 1 r 2 µ 2 r n 1 µ n 1 r n µ n r 1 µ 2 r 2 µ 3 r n 1 µ n.. r 1 µ n 1 r 2 µ n r 1 µ n 1 r 2 µ n r 1 µ n r 1 µ 98

99 Hence, incremental payments are functions of development factors, r j, and a calendar factor, µ i+j 1, that might be related to some inflation 99

100 Introduction to factorial models: Taylor (1977) In order to identify factors r 1, r 2,.., r n and µ 1, µ 2,..., µ n, i.e. 2n coefficients, an additional constraint is necessary, e.g. on the r j s, r 1 + r r n = 1 (this will be called arithmetic separation method). Hence, the sum on the latest diagonal is d n = Y 1,n + Y 2,n Y n,1 = µ n (r 1 + r r k ) = µ n On the first sur-diagonal d n 1 = Y 1,n 1 +Y 2,n Y n 1,1 = µ n 1 (r 1 + r r n 1 ) = µ n 1 (1 r n ) and using the nth column, we get γ n = Y 1,n = r n µ n, so that r n = γ n µ n and µ n 1 = d n 1 1 r n More generally, it is possible to iterate this idea, and on the ith sur-diagonal, d n i = Y 1,n i + Y 2,n i Y n i,1 = µ n i (r 1 + r r n i ) = µ n i (1 [r n + r n r n i+1 100

101 Introduction to factorial models: Taylor (1977) and finally, based on the n i + 1th column, γ n i+1 = Y 1,n i+1 + Y 2,n i Y i 1,n i+1 = r n i+1 µ n i r n i+1 µ n 1 + r n i+1 µ n r n i+1 = γ n i+1 µ n + µ n µ n i+1 and µ k i = d n i 1 [r n + r n r n i+1 ] k µ k r k in %

102 Introduction to factorial models: Taylor (1977) The challenge here is to forecast forecast values for the µ k s. Either a linear model or an exponential model can be considered

103 Lemaire (1982) and autoregressive models Instead of a simple Markov process, it is possible to assume that the C i,j s can be modeled with an autorgressive model in two directions, rows and columns, C i,j = αc i 1,j + βc i,j

104 Zehnwirth (1977) Here, we consider the following model for the C i,j s C i,j = exp (α i + γ i j) (1 + j) β i, which can be seen as an extended Gamma model. α i is a scale parameter, while β i and γ i are shape parameters. Note that log C i,j = α i + β i log (1 + j) + γ i j. For convenience, we usually assume that β i = β et γ i = γ. Note that if log C i,j is assume to be Gaussian, then C i,j will be lognormal. But then, estimators one the C i,j s while be 104

105 Zehnwirth (1977) Assume that log C i,j N ( X i,j β, σ 2), then, if parameters were obtained using maximum likelihood techniques E (Ĉi,j ) ( σ = E exp (X i,j β 2 )) + 2 ( = C i,j exp n 1 σ 2 ) (1 σ2 n 2 n ) n 1 2 > C i,j, Further, the homoscedastic assumption might not be relevant. Thus Zehnwirth suggested σ 2 i,j = V ar (log C i,j ) = σ 2 (1 + j) 105

106 Regression and reserving De Vylder (1978) proposed a least squares factor method, i.e. we need to find α = (α 0,, α n ) and β = (β 0,, β n ) such or equivalently, assume that ( α, β) = argmin n i,j=0 (X i,j α i β j ) 2, X i,j N (α i β j, σ 2 ), independent. A more general model proposed by De Vylder is the following ( α, β, γ) = argmin n i,j=0 (X i,j α i β j γ i+j 1 ) 2. In order to have an identifiable model, De Vylder suggested to assume γ k = γ k (so that this coefficient will be related to some inflation 106