Conditional Least Squares and Copulae in Claims Reserving for a Single Line of Business Michal Pešta Charles University in Prague Faculty of Mathematics and Physics Ostap Okhrin Dresden University of Technology Institute of Economics and Transport
Introduction 2 30 Overview Motivated by claims reserving in non-life insurance Joint work with Ostap Okhrin (TU Dresden) Triangular data models Published in Insurance: Mathematics and Economics, 56 (May 2014): 28 37.
Data structure 3 30 Triangular data i\j 1 2 n 1 n 1 Y 1,1 Y 1,2 Y 1,n 1 Y 1,n 2 Y 2,1 Y 2,2 Y 2,n 1...... Y i,n+1 i n 1 Y n 1,1 Y n 1,2 n Y n,1 n copies of stochastic process The first realization consists of n observations The last one has only one observation
Main aims 4 30 Terminology and goals Y i,j... cumulative payments in origin year i after j development periods (accounting year i + j) n... current year corresponds to the most recent accident year and development period Our data history consists of right-angled isosceles triangles Y i,j, where i + j n + 1 Predict Y i,n and R i = Y i,n Y i,n+1 i (claims reserve) Estimate distribution of the reserves
CMV model 5 30 Conditional mean and variance (CMV) model CMV model Y i,j = µ(y i,j 1, α, j) + σ(y i,j 1, β, j)ε i,j (α, β) α and β are unknown parameters, which dimensions do not depend on n µ is a continuous function in α σ is a positive and continuous function in β Errors ε i,j (α, β)
CMV model 6 30 CMV model s errors Disturbances {ε i,j (α, β)} n+1 i j=1 are independent sample copies of a stationary first-order Markov process for all i All ε i,j (α, β) have the common true invariant distribution G α,β which is absolutely continuous with respect to Lebesgue measure on the real line Filtration F i,j = σ(y k,l : l j, k i + 1 j) denotes the information set generated by that trapezoid E[ε i,j (α, β) F i,j 1 ] = 0 var[ε i,j (α, β) F i,j 1 ] = s(α, β)
CMV model 7 30 Properties of the CMV model Unknown true values [α, β ] of parameters [α, β ] set (due to identifiability purposes): s(α, β ) = 1 Model s name come from the fact that E[Y i,j F i,j 1 ] = µ(y i,j 1, α, j) var[y i,j F i,j 1 ] = σ 2 (Y i,j 1, β, j)s(α, β) Conditional mean models: types of ARMA models, vector autoregressions, linear and nonlinear regressions,... Conditional variance models: ARCH and any of its numerous parametric extensions (GARCH, EGARCH, GJR-GARCH, etc.), stochastic volatility models,...
CMV model 8 30 Candidates for the mean and variance function From the nature of data: Y i,j C i R + almost surely as j, i (stabilizing property) One may propose, e.g., µ(y i,j 1, α, j) = η(α, j)y i,j 1 σ(y i,j 1, β, j) = ν(β, j) Y i,j 1 η(α, j) should be decreasing in j with limit 1 as j ν(β, j) should be decreasing in j with limit 0 as j
CMV model 9 30 Dependence modeling Since the mean and variance trends are removed by the CMV model, the rest of the relationship among claim amounts Y i,j can be additionally captured by modeling dependent errors {ε i,j (α, β)} n+1 i j=1 are independent sample copies of a stationary first-order Markov process for all i generated from (G α,β ( ), C(, ; γ)) C(, ; γ) is the true parametric copula for [ε i,j 1 (α, β), ε i,j (α, β)], which is given and fixed up to unknown parameter γ
CMV model 10 30 Copula-based model It is believed that there exist a kind of information overlap between the claims from consecutive development periods Joint bivariate distribution of [ε i,j 1 (α, β), ε i,j (α, β)] has distribution function H(e 1, e 2 ) = C(G α,β (e 1 ), G α,β (e 2 ); γ) Conditional copula density can be derived as h(e 2 e 1 ) = g α,β (e 2 )c(g α,β (e 1 ), G α,β (e 2 ); γ) where c is the copula density and g α,β is the marginal density corresponding to G α,β Play an important role in making" the dependent errors conditionally independent
Nonlinear generalized semiparametric regression 11 30 Parameter estimation CMV model with copula assume three vector parameters to be estimated Estimation process consists of two stages In the first one, mean and variance parameters α and β are estimated in a distribution-free fashion, since no specific distributional assumptions are proposed nor required for the claims The second stage concerns estimation of the dependence structure, mainly the copula parameter γ, in a likelihood based way
Nonlinear generalized semiparametric regression 12 30 Conditional least squares (CLS) Denote M n (α, β) = 1 n n+1 j 1 n 1 n + 1 j j=2 i=1 [ Yi,j µ(y i,j 1, α, j) ] 2 σ 2 (Y i,j 1, β, j) n n+1 j i=1 V n (α, β) = 1 1 n 1 n + 1 j j=2 { [Yi,j µ(y i,j 1, α, j) ]2 2 σ 2 (Y i,j 1, β, j)}
Nonlinear generalized semiparametric regression 13 30 CLS estimates CLS estimate of the mean parameter α for a fixed value of parameter β Θ 2 is defined as α(β) = arg min α Θ 1 M n (α, β) and CLS estimate of the variance parameter β for a fixed value of parameter α Θ 1 is defined as β(α) = arg min β Θ 2 V n (α, β) Computationally not feasible to find the global minimum of M n and V n with respect to [α, β ] simultaneously
Nonlinear generalized semiparametric regression 14 30 Consistency Under regularity conditions α(β) P n α P (β), β; β(α) n β (α), α Mixingales are to mixing processes as martingale differences are to independent processes
Nonlinear generalized semiparametric regression 15 30 Iterative CLS What is the connection between the true unknown parameter values α and β of the CMV model and true unknown parameter values α (β) and β (α)? [ α(β ) β(α ) ] [ P α n β Iteratively estimate α given the fixed value of β and, consequently, estimate β given the fixed value of α (obtained from previous step) Repeat in turns until almost no change in consecutive estimates of [α, β ] ]
Nonlinear generalized semiparametric regression 16 30 Estimation of dependence structure Estimate the unknown marginal distribution function G α,β of CMV model errors ε i,j (α, β) non-parametrically by the empirical distribution function Ĝ n (e) = 1 n(n 1)/2 + 1 of the fitted residuals n 1 i=1 ε i,j ( α, β) = Y i,j µ(y i,j 1, α, j) σ(y i,j 1, β, j) n+1 i I{ ε i,j ( α, β) e} j=2
Nonlinear generalized semiparametric regression 17 30 Likelihood for copula Full log-likelihood for copula parameter γ L(γ) = + n 2 i=1 n 2 i=1 n+1 i log g α,β (ε i,j (α, β)) j=2 n+1 i log c(g α,β (ε i,j 1 (α, β)), G α,β (ε i,j (α, β)); γ) j=3
Nonlinear generalized semiparametric regression 18 30 Psuedo (quasi) likelihood Ignoring the first term in L(γ) and replacing ε s and G α,β by their estimated counterparts ε s and Ĝ n, parameter γ can be estimated by the so-called canonical maximum likelihood, i.e., maximizing the partial (pseudo) log-likelihood γ = arg max γ L(γ) = n 2 i=1 L(γ) n+1 i log c(ĝ n ( ε i,j 1 ( α, β)), Ĝ n ( ε i,j ( α, β)); γ) j=3
Nonlinear generalized semiparametric regression 19 30 Prediction Predictor for reserve R (n) i R (n) i = Ŷ i,n Y i,n+1 i can be defined as Prediction of unobserved claims may be done in a telescopic way based on the CMV model formulation: start with the diagonal element Y i,n+1 i and predict Y i,j, j > n + 1 i stepwise in each row Ŷ i,j = Y i,j, i + j n + 1 Ŷ i,j = µ(ŷ i,j 1, α, j) + σ(ŷ i,j 1, β, j) ε j, i + j > n + 1
Nonlinear generalized semiparametric regression 20 30 Semiparametric bootstrap Errors ε j are simulated from the fitted residuals Takes advantage of the fact that ε i,j (α, β) = G 1 α,β (X j) for all i (due to the independent rows), where {X j } n j=2 is a stationary first-order Markov process with the copula C(x 1, x 2 ; γ) being the joint distribution of [X j 1, X j ]
Nonlinear generalized semiparametric regression 21 30 Resampling algorithm Generate n 1 independent Un(0, 1) rvs {X j } n j=2 Repeat b = 1,..., B (b) U 2 X 2 (b) ε 2 Ĝn ( (b) U 2 ) (b) U j C 1 2 1 (X j (b) U j 1 ; γ), j = 3,..., n (b) ε j Ĝn ( (b) U j ), j = 3,..., n Center bootstrap residuals (b) ε j (b) ε j 1 n 1 n l=2 (b) ε l (b) Ŷ i,n+1 j Y i,n+1 i (b) Ŷ i,j µ( (b) Ŷ i,j 1, α, j) + σ( (b) Ŷ i,j 1, β, j) (b) ε j, j = n + 2 i,..., n (b) R (n) i (b) Ŷ i,n Y i,n+1 i
Real data example 22 30 1 1 1 1 1 1 1 1 1 1 1 2 4 6 8 10 200000 400000 600000 800000 1000000 Development period j Y ij 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 6 6 6 6 6 6 7 7 7 7 7 8 8 8 8 9 9 9 0 0 a
Real data example 23 30 Real data Data set from Zehnwirth and Barnett (2000) ( µ(y i,j 1, α, j) = 1 + α 1 α 2 j 1 α 2 exp { α 1 j α }) 2 Y i,j 1 σ(y i,j 1, β, j) = β 1 exp{ β 2 j} Y i,j 1 CLS estimates: α 1 = 2.033, α 2 = 1.106, β 1 = 109.8, β 2 = 0.4053
Real data example 24 30 6 Residuals 2 1 0 1 2 9 0 8 ε^i,j 2 7 13 45 6 9 8 7 8 7 6 5 7 46 1 1 1 1 1 23 23 23 5 45 4 45 6 5 4 23 2 3 4 3 3 2 2 12 1 1 1 2 4 6 8 10 Development period j Still some slight pattern (trend) not captured by mean and variance parametric part
Real data example 25 30 Copula godness-of-fit Kendall τ for the pairs of consecutive residuals {[ ε i,j 1 ( α, β), n 2,n+1 i ε i,j ( α, β)]} i=1,j=3 equals 0.43, which indicates at least mild dependence Three Archimedean copulae (Clayton, Frank, and Gumbel) together with Gaussian and Student t 5 -copula considered S (C) n goodness-of-fit test proposed by Genest et al. (2009) Gumbel copula ( γ = 1.776) was chosen Exhibits strong right tail dependence and relatively weak left tail dependence Transformed residuals (by the residuals marginal edf Ĝ n ; having uniform margins) seem to be strongly correlated at high values but less correlated at low values
Real data example 26 30 1.5 0.5 0.0 0.5 1.0 1.5 1.0 0.0 0.5 1.0 1.5 2.0 2.5 Residuals ε^i,j 1 ε^i,j 1 0 1 2 1 0 1 2 Simulated Residuals ε^j 1 ε^j 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Transformed Residuals G^ n(ε^i,j 1) G^ n(ε^i,j) Gumbel(γ^) Copula Density Un[0,1] Un[0,1] 0.25 0.25 0.5 0.5 0.75 0.75 1 1 1.25 1.25 1.5 1.5 1.75 1.75 2 2 2.25 2.5 2.5 3 4 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
Real data example 27 30 Results Benchmark: traditional bootstrapped chain ladder (BCL) Disadvantages, which can be overcome by our approach: Number of parameters depending on the sample size Some parameters estimated by just ratio of two numbers (yielding zero sample variance) Questionable consistency of the estimates Non-realistic assumption of independence of the residuals Our approach: Slightly smaller predictions of reserves But even more important is that the estimates of the reserves distribution are less volatile (interesting finding in terms of CoV: GEE AR(1) 1.95; CLSC 1.99)
Reserves 28 30 BCL+CLSC i=2 BCL+CLSC BCL+CLSC i=3 i=4 0k 25k Reserves 25k 50k 75k Reserves 50k 75k 100k BCL+CLSC i=5 BCL+CLSC i=6 75k 100k 125k 150k Reserves 100k 150k 200k Reserves 150k 200k 250k BCL+CLSC i=8 BCL+CLSC i=7 BCL+CLSC i=9 300k 350k 400k 450k 500k Reserves 600k 675k 750k 825k 900k Reserves 1100k 1250k 1400k 1550k BCL+CLSC i=11 BCL+CLSC i=10 4750k 5000k 5250k 5500k 5750k 6000k Total Reserves 1775k 2000k 2225k 2450k 2675k Reserves Reserves BCL+CLSC Reserves Reserves
Conclusions 29 30 Summary Conditional mean and variance (CMV) time series model for triangular data with innovations being a stationary first-order Markov process Framework is demonstrated to be suitable for stochastic claims reserving in general insurance Very flexible modeling approach, relatively smaller number of model parameters not depending on the number of development periods, and time series innovations not considered as independent Increase in precision of the claims reserves prediction Theoretical justification of the proposed approached shown
Questions? 30 30 Thank you! Michal.Pesta@mff.cuni.cz