DELTA METHOD and RESERVING

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1 XXXVI th ASTIN COLLOQUIUM Zurich, 4 6 September 2005 DELTA METHOD and RESERVING C.PARTRAT, Lyon 1 university (ISFA) N.PEY, AXA Canada J.SCHILLING, GIE AXA

2 Introduction Presentation of methods based on reserve s moments allowing: Estimating: - predictive distribution, - moments, percentiles (VaR) and their function With estimation risk measure, confidence interval Within the GLM approach (including Log-Poisson) 2

3 I.Notations For a given line of business, claims are assumed to be closed in (n + 1) years. GLM approach is based on incremental payments r.v: assumed to be independent ( 1) 2 X, i, j = 0,..., n n+ Among the X r. v. inside the run-off triangle those located in the upper triangle have been observed 3

4 I.Notations Payments delays O r i g i n y e a r s 0 i n n+k 0.. j. n. n+k X 4

5 I.Notations Reserve for the i th origin year Total reserve R i R n = h= n i+ 1 n = i= 1 X R i ih Remark: To analyze future annual cash flows by integrating new business, cash flows for the accident year ( n+ k) : n X i+ j= n+ k 5

6 II. Parameters 2.1. Interest Parameters Interest parameters are linked to the d.f. : F R Π ( ) F R Indicators depending on moments of R : mean, dispersion (variance, standard deviation), margins such : ER ( ) + γ σ ( R) Other indicators: (tail) VaR (percentiles),tailvar, probability of insufficiency, F R Need : estimation of d.f. directly or by inversion of m.g.f. R ( ) ( R = ) M s E s 6

7 II. Parameters 2.2. Estimation ( ) Π=Π ˆ ˆ Π ( ) For an estimator X of uncertainty related i + j n to this estimation will be measured by: F R asymptotic variance: Vas ( Π ˆ ) standard deviation : se.. ( ˆ ) ( ˆ as Π = Vas Π ) se.. as (ˆ Π) Πˆ In addition, a level 95% asymptotic confidence interval for is { ( ) ( ) ( ) } R 0,95 i+ j n i+ j n P A X F B X Π ( ) F R 7

8 III.GLM models 3.1. Random component independent «responses» : with exponential type dist. (, = 0,..., ) X i j n - θ : natural parameters - φ> 0 : dispersion parameter. - b, c specific functions, b being regular Ex. : Poisson, Normal, Gamma, IG, Tweedie dist { } ( ; θ, φ ) = exp θ ( θ ) φ+ (, φ) f x x b c x Moments: E( X ) b' ( ) ( ) = φ ( θ ) =φ ( µ ) µ = = θ V X b V With V variance function 8

9 III.GLM models 3.1. Random component Third moment: µ φ θ µ µ ( ) 2 '''( ) ( ) ( ) 3 X = b = V V Skewness : µ 3( X ) b'''( θ ) V ( µ ) γ1( X ) = = φ = φ V( X ) ''( ) V ( ) b θ µ m.g.f. and cumulant g.f: 1 1 M X ( s) = exp b( θ ) ( ), ( ) log ( ) ( ) ( ) + sφ b θ CX s = M X s = b θ + sφ b θ φ φ moments of X functions of ( θ, φ ), then of ( µ, φ) : κ φ θ r 1 ( r) r( X) = b ( ) 9

10 III.GLM models 3.2. Systematic Component, link function.. Origin Year 0 i Delay 0 j n Calendar year i+j β ( β = 0) j 0 regression parameters : n µ µ i+ j = ( ) ( ) ξ= µ, αi, β i= 1,, n j j= 1,, n α i ( α = 0) 0 10

11 III.GLM models 3.2. Systematic Component, link function systematic component : (, 0,..., ) η =µ+α i +β j i j = n link function : monotone and derivable real function g : ( ) 1 g ( ) η = g µ µ = η Identity link : η = µ = µ + αi + β j Log link : µ+α +β e i j η = log µ µ = 11

12 IV. Estimation based on: - upper triangle likelihood - and Wedderburn equations L ( x ) ( i ) ( j ) i+ j n δlog L = 0 δξ ; µ, α, β, φ As: m.l.e of $ µ, ( α ), ( ) i βj ( ˆ ) ( ˆ ) i j ξ= µ ˆ, α, β ξ= 1 η ˆ ˆ ˆ ˆ, ˆ ( ˆ =µ+α i +β j µ = g η ) η, µ = E ( X ) ( ) R= X E R = µ i+ jfn i+ jfn ER ( ) = µ ˆ i+ jfn E( R) 12

13 IV.Estimation 4.1. Estimation risk (Delta method) m.l.e.: - $ξ AN ξ, Σ ( $ ξ ) 1 as Σas ( ξ ) = I ( ξ) $ - $ η = ( $ η ) AN η, Σ ( $ as η) Σ ( $ η) = J Σ ( $ ξ ) J as η as η J η jacobian matrix of η : ξ η( ξ) = ( η ) η η 1 k = i η 1 l = j = 1, = if, = if µ αk 0 k i βl 0 l j - µ = ( µ ) AN µ, Σas ( µ ) Σ ( µ ) = DΣ ( $ η) D as as D Jacobian matrix (diagonal) of = 13 1 ( η ) g ( η ) µ

14 IV.Estimation 4.1. Estimation risk (Delta method) Then AN [ ER ( ) ], { ( ) } i Σ ER i= 1,..., n as i ER ( i ) i= 1,..., n { } With - { ( ) } ( ) Σ as ERi = Jµ Σas µ J µ J µ ( µ ) [ ER ( )] - Jacobian matrix of i And { } 2 ER ( ) AN E( R), σ as E( R) With: - - (1,1,...,1) Jacobian matrix of { } 2 σ as ER ( ) = JR Σ as ER ( i ) J R J R = [ ERi ] i 1,..., = n ( ) ER ( ) Asymptotic s.e. and confidence interval for ER ( ) using only products of matrix (spreadsheet) 14

15 IV.Estimation 4.2. Extensions Same approach could be applied to: - variance: n V( R) = V( X ) = φ V( µ ) i= 1 jfn i i= 1 jfn i n - more generally, to cumulants: n κ( R) κ( X ) = i= 1 jfn i - Then to any regular function of moments of R Giving only a variance function and dispersion parameter by quasi-likelihood V ( µ ) f 0, φ Ex. : over-dispersed Poisson 15

16 V.Predictive distribution 5.1. Inversion of the m.g.f m.g.f: n n n 1 MR s MX s b b g s b b g i 1 i j n φ = + f i= 1 j= n + i 1 ( ) ( ) { = = exp ( ( η )) + φ ( ( η )) } ( $ ) η with for m.l.e. Inversion by F.F.T. if no standard m.g.f. 16

17 V.Predictive distribution 5.2. Approximated distributions using moments n n = i= 1 j= n i+ 1 µ n n From R X and independence of X, moments of R are functions of ( ): µ ER ( ) µ = = i= 1 j= n i+ 1 n n n n ( ) ( ) ( ) 2 σ = V R = V X = φ V µ i= 1 j= n + i 1 i= 1 j= n + i 1 n n n n 2 3 = 3( R) = 3( X ) = V ( ) V( ) i= 1 j= n i+ 1 i= 1 j= n i+ 1 µ µ µ φ µ µ µ 3 γ1 = γ1( R) = 3 σ µ (m.l.e. with ) 17

18 V.Predictive distribution 5.2. Approximated distributions using moments Using : - NP-approximation, - Gamma approximations (Translated, Bowers), based on m.l.e. of ( µ, σγ, 1) For instance F x F x x µ γ γ γ σ ( NP) R ( ) ( ) =Φ γ q µ σ q q 6 ( NP) η = + 1 η + 1 η 1 We obtain m.l.e. of approximations of d.f., VaR, enhanced by their asymptotic s.e. and confidence interval 18

19 VI.Example incremental claims amounts for some line of Marine business Years chain ladder reserve : Comparing models by extended quasi-likelihood : 1. Log / Gamma 2/.Log / overdispersed Poisson 19

20 VI.Example Overdispersed Poisson model Estimation of Φ : Deviance Estimation of Φ : Pearson residuals Bootstrap (1000 samples) Estimates Estimates of Estimates of of E(R i ) se(r i )/R i E(R i ) se(r i )/R i E(R i ) se(r i )/R i % % % % % % % % % % % % % % % % % % % % % Total % % % 20

21 VI.Example Estimates of the percentiles of R directly and using Normal Power approximation Gamma approximations not available ( γ 1 = 0.075) q$1 ( R) η ( NP) q$ 1 η ( R)

22 VI.Example Estimation of Φ : maximisation of the likelihood Estimates Gamma Model Estimation of Φ : Deviance Estimates Estimation of Φ : Pearson Estimates of E(R i ) se(r i )/R i of E(R i ) se(r i )/R i of E(R i ) se(r i )/R i Total % % % % % % % % % % % % % % % % % % % % % % % % 22

23 VI.Example Estimates of the percentiles of R by Normal Power and Gamma approx. 1 η ( NP) q1 η ( R) ( GT ) q1 η ( R)

24 Conclusion 24

Bootstrapping the triangles

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