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
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
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
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
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
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
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
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
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 3 3 2 2 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
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
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
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
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
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
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
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 ( ) ( ) { 1 1 1 1 = = exp ( ( η )) + φ ( ( η )) } ( $ ) η with for m.l.e. Inversion by F.F.T. if no standard m.g.f. 16
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
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 3 9 6 x µ γ γ γ σ ( NP) R ( ) ( ) =Φ + + 1+ 2 1 1 1 γ q µ σ q q 6 ( NP) 1 2 1 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
VI.Example incremental claims amounts for some line of Marine business Years 0 1 2 3 4 5 6 7 0 1 381 4 399 4 229 435 465 205 110 67 1 859 6 940 2 619 1 531 517 572 287 2 6 482 6 463 3 995 1 420 547 723 3 2 899 16 428 5 521 2 424 477 4 3 964 15 872 8 178 3 214 5 6 809 24 484 27 928 6 11 155 38 229 chain ladder reserve : 133750 7 10 641 Comparing models by extended quasi-likelihood : 1. Log / Gamma 2/.Log / overdispersed Poisson 19
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 1 80 329% 80 348% 71 331% 2 442 134% 442 142% 427 123% 3 1 631 70% 1 631 74% 1 624 62% 4 2 811 53% 2 811 56% 2 777 46% 5 117 86 32% 11 786 34% 11 706 28% 6 41 864 20% 41 864 21% 41 799 18% 7 75 137 31% 75 137 33% 75 595 29% Total 133 750 19% 133 750 20% 134 000 20% 20
VI.Example Estimates of the percentiles of R directly and using Normal Power approximation Gamma approximations not available ( γ 1 = 0.075) 0.50 0.75 0.80 0.90 0.95 0.99 q$1 ( R) η 133750 140733 142463 147019 150781 157836 ( NP) q$ 1 η ( R) 123396 130441 132205 136885 140789 148205 21
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 1 2 3 4 5 6 7 Total 101 37% 101 49% 101 50% 494 25% 494 33% 494 34% 1 286 22% 1 286 29% 1 286 30% 2 793 22% 2 793 28% 2 793 29% 11 262 23% 11 262 30% 11 262 31% 36 702 27% 36 702 35% 36 702 36% 69 563 36% 69 563 48% 69 563 49% 122 200 22% 122 200 29% 122 200 30% 22
VI.Example Estimates of the percentiles of R by Normal Power and Gamma approx. 1 η 0.50 0.75 0.80 0.90 0.95 0.99 ( NP) q1 η ( R) 112621 127662 131632 142550 152071 171164 ( GT ) q1 η ( R) 132 027 146 885 150 820 161 671 171 188 190 445 23
Conclusion 24