Robust Bayesian Analysis of Loss Reserves Data using Scale Mixture Distributions

Robust Bayesian Analysis of Loss Reserves Data using Scale Mixture Distributions Boris Choy (boris.choy@uts.edu.au) Department of Mathematical Sciences University of Technology Sydney, Australia Collaborators: Udi E. Makov and Jennifer S.K. Chan Robust Bayesian Analysis of Loss Reserve Data

Outline • Loss Reserve Data • Model for Mean Function • Scale Mixtures Error Distributions and Bayesian Inference • Results • Further Work Robust Bayesian Analysis of Loss Reserve Data

Loss Reserve • Is the fund reserved by an insurance company to cover the outstanding claims incurred but not yet settled • Is the liability of the insurance company and must be predicted accurately. Robust Bayesian Analysis of Loss Reserve Data

Our Objectives • To predict the amount of loss reserves for an insurance company more accurately. • To allow for extreme claims and protect inference using robust error distributions. • To cope with the structural changes of the claims. Robust Bayesian Analysis of Loss Reserve Data

The Data • Claims paid to the insureds of an insurance company from 1978 to 1995. • There are N=171 observations.Those in red and blue are outliers. Robust Bayesian Analysis of Loss Reserve Data

The Outliers • Large claims in red will overestimate the loss reserve and hence lower the profitability. • Small claims in blue will underestimate the loss reserve and hence lower the solvency and increase the risk of bankruptcy. • Accurate prediction of the levels of loss reserves is important. Robust Bayesian Analysis of Loss Reserve Data

Loss reserves – Structural Change Achieve a max. in 3-4 yrs * ¤ Achieve a max. in 5-6 yrs * Y=15546 ¤ Y=11920 Robust Bayesian Analysis of Loss Reserve Data

Loss reserves – Structural Change • From 1978 to 1983 • Claims rise up to a maximum in 3-4 years • Then decline gradually until almost zero. • From 1984 to 1994 • Claims rise up to a maximum in 5-6 years • Then decline gently until almost zero. Robust Bayesian Analysis of Loss Reserve Data

Log-normal Model Renshaw & Verrall, 1998 • Log-ANOVA log(Yij) = θij +εij , εij ~N(0, σ2) θij =μ + αi + βj,i,j=1,2,….n Constraint: • Log-ANCOVA log(Yij) = θij +εij ,εij ~N(0, σ2) θij =μ + α×i + βj , i,j=1,2,….n Constraint: Robust Bayesian Analysis of Loss Reserve Data

Threshold Model Hazan and Makov, 2001.To cope with structural changes of the claims Log-ANOVA θij =μ1 + α1i + β1ji ≤ T θij =μ2 + α2i + β2ji > T Log-ANCOVA θij =μ1 + α1 ×i + β1j i ≤ T θij =μ2 + α2×i + β2ji > T Different T are adopted and they are selected according to some goodness-of-fit measures. Robust Bayesian Analysis of Loss Reserve Data

Shapes of the normal, Student-t, Laplace, Cauchy and logistic distributions. The Laplace and logistic curves are adjusted to have mean 0 and variance 1. Heavy-tailed Error Distributions Robust Bayesian Analysis of Loss Reserve Data

Scale Mixtures Distributions • Scale mixtures of normal (SMN) Andrews and Mallows (1974): X = Z λ With location and scale parameters: • Scale mixtures of uniform (SMU) Robust Bayesian Analysis of Loss Reserve Data

Examples of SMN Distributions • Student-t (v) • Symmetric Stable () • Logistic • EP(β) Robust Bayesian Analysis of Loss Reserve Data

Examples of SMU Distributions • Normal • EP • GT Robust Bayesian Analysis of Loss Reserve Data

Bayesian Inference • Use Bayesian analysis for modelling the loss reserves • Use Markov chain Monte Carlo (MCMC) algorithms to simulate the posterior realizations. • Use Bayesian software “WinBUGS” to obtain the posterior functionals. • Use scale mixture form for the well-known distributions to speed up the MCMC algorithms. • Use the mixing parameters from the scale mixture form to identify outliers. Robust Bayesian Analysis of Loss Reserve Data

SM Error Distributions and Prior Specification • The normal error distribution for is replaced by a heavy-tailed scale mixture distribution. • Vague and non-informative prior distributions are adopted to express ignorance of the model parameters. Robust Bayesian Analysis of Loss Reserve Data

Outline • Loss Reserve Data • Scale Mixtures Error Distributions and Bayesian Inference • Model for Mean Function • Results • Further Work Robust Bayesian Analysis of Loss Reserve Data

Result - goodness-of-fit • Mean-Square of Error (MSE) • Model with the smallest MSE is preferred. • Posterior expected utility (U) • Model with the largest U is preferred Robust Bayesian Analysis of Loss Reserve Data

Result MSE is in the scale of 1,000,000 Threshold ANOVA model with t errors is chosen to be the best model Robust Bayesian Analysis of Loss Reserve Data

Prediction of the Lower Triangle Fitted & predicted outstanding claims Robust Bayesian Analysis of Loss Reserve Data

Comparison with Chain Ladder • Chain-ladder (CL) method is a very common model for loss reserve • Plot observed against predicted claims using CL method and best model shows that errors on large claims are lower for the best model Y=15546 Y=11290 Robust Bayesian Analysis of Loss Reserve Data

Detection of Extremely Claims Unusual claims are detected using the scale parameter Robust Bayesian Analysis of Loss Reserve Data

Future Work • More complicated structure for the mean function. • Develop asymmetric scale mixture distributions that can protect statistical inference as well as identify extremely large claims. Robust Bayesian Analysis of Loss Reserve Data

Thank You Robust Bayesian Analysis of Loss Reserve Data

Robust Bayesian Analysis of Loss Reserves Data using Scale Mixture Distributions