Stochastic programming approaches to pricing in non-life insurance - PowerPoint PPT Presentation

Stochastic programming approaches to pricing in non-life insurance Martin Branda Charles University in Prague Department of Probability and Mathematical Statistics 11th International Conference on COMPUTATIONAL MANAGEMENT SCIENCE 29–31 May, 2014, Lisbon

Table of contents 1 Introduction 2 Pricing of non-life insurance contracts 3 Approach based on generalized linear models 4 Optimization models – expected value approach 5 Optimization models – individual chance constraints 6 Optimization models – a collective risk constraint 7 Numerical comparison

Introduction Table of contents 1 Introduction 2 Pricing of non-life insurance contracts 3 Approach based on generalized linear models 4 Optimization models – expected value approach 5 Optimization models – individual chance constraints 6 Optimization models – a collective risk constraint 7 Numerical comparison

Introduction Multiplicative tariff of rates Motor third party liability (MTPL): Engine volume between 1001 and 1350 ccm, policyholder age 18–30, region over 500 000 inhabitants: 300 · (1 + 0 . 5) · (1 + 0 . 4) . Engine volume between 1351 and 1600 ccm, policyholder age over 50, region between 100 000 and 500 000 inhabitants: 450 · (1 + 0 . 0) · (1 + 0 . 2) .

Introduction Four methodologies The contribution combines four methodologies: Data-mining – data preparation. Mathematical statistics – random distribution estimation using generalized linear models. Insurance mathematics – pricing of non-life insurance contracts. Operations research – mathematical (stochastic) programming approaches to tariff of rates estimation.

Introduction Practical experiences More than 4 years of cooperation with Actuarial Department, Head Office of Vienna Insurance Group Czech Republic (VIG CR). VIG CR – the most profitable part of Vienna Insurance Group. VIG CR – the largest group on the market: 2 universal ’ovna, ˇ insurance companies (Kooperativa pojiˇ st Cesk´ a ’ovna) and 1 life-oriented (ˇ podnikatelsk´ a pojiˇ st Cesk´ a spoˇ ritelna). Kooperativa & ˇ CPP MTPL: 2.5 mil. cars from 7 mil. per year (data for more than 10 years)

Pricing of non-life insurance contracts Table of contents 1 Introduction 2 Pricing of non-life insurance contracts 3 Approach based on generalized linear models 4 Optimization models – expected value approach 5 Optimization models – individual chance constraints 6 Optimization models – a collective risk constraint 7 Numerical comparison

Pricing of non-life insurance contracts Tariff classes/segmentation criteria Tariff of rates based on S + 1 categorical segmentation criteria : i 0 ∈ I 0 , e.g. tariff classes I 0 = { engine volume up to 1000, up to 1350, up to 1850, up to 2500, over 2500 ccm } , i 1 ∈ I 1 , . . . , i S ∈ I S , e.g. age I 1 = { 18–30, 31–65, 66 and more years } We denote I = ( i 0 , i 1 , . . . , i S ), I ∈ I a tariff class , where I = I 0 ⊗ I 1 ⊗ · · · ⊗ I S denotes all combinations of criteria values. Let W I be the number of contracts (exposures) in I .

Pricing of non-life insurance contracts Compound distribution of aggregated losses Aggregated losses over one year for risk cell I N I , w W I � � L T I = L I , w , L I , w = X I , n , w , w =1 n =1 where all r.v. are assumed to be independent ( N I , X I denote independent copies) N I , w is the random number of claims for a contract during one year with the same distribution for all w X I , n , w is the random claims severity with the same distribution for all n and w Well-known formulas for the mean and the variance: µ T E [ L T = I I ] = W I µ I = W I I E [ N I ] I E [ X I ] , I I ) 2 I ) = W I σ 2 E [ X I ]) 2 var ( N I )) . ( σ T var ( L T = I = W I ( I E [ N I ] var ( X I ) + ( I

Pricing of non-life insurance contracts Multiplicative tariff of rates We assume that the risk (office) premium is composed in a multiplicative way from basic premium levels Pr i 0 and nonnegative surcharge coefficients e i 1 , . . . , e i S , i.e. we obtain the decomposition Pr I = Pr i 0 · (1 + e i 1 ) · · · · · (1 + e i S ) . We denote the total premium TP I = W I Pr I for the risk cell I . Example : engine volume between 1001 and 1350 ccm, age 18–30, region over 500 000 inhabitants: 300 · (1 + 0 . 5) · (1 + 0 . 4)

Pricing of non-life insurance contracts Prescribed loss ratio – random constraints Our goal is to find optimal basic premium levels and surcharge coefficients with respect to a prescribed loss ratio ˆ LR , i.e. to fulfill the random constraints L T ≤ ˆ I LR for all I ∈ I , (1) TP I and/or the random constraint I ∈I L T � ≤ ˆ I LR . (2) � I ∈I TP I The prescribed loss ratio ˆ LR is usually based on a management decision. If ˆ LR = 1, we obtain the netto-premium. It is possible to prescribe a different loss ratio for each tariff cell.

Pricing of non-life insurance contracts Sources of risk Two sources of risk for an insurer: 1. Expectation risk : different expected losses for tariff cells. 2. Distributional risk : different shape of the probability distribution of losses, e.g. standard deviation.

Pricing of non-life insurance contracts Prescribed loss ratio – expected value constraints Usually, the expected value of the loss ratio is bounded E [ L T I ] E [ L I ] I = I ≤ ˆ LR for all I ∈ I . (3) TP I Pr I The distributional risk is not taken into account.

Pricing of non-life insurance contracts Prescribed loss ratio – chance constraints A natural requirement: the inequalities are fulfilled with a prescribed probability leading to individual chance (probabilistic) constraints � L T � ≤ ˆ I P LR ≥ 1 − ε, for all I ∈ I , (4) TP I where ε ∈ (0 , 1), usually ε ∈ { 0 . 1 , 0 . 05 , 0 . 01 } , or a constraint for the whole line of business: � � � I ∈I L T ≤ ˆ I P LR ≥ 1 − ε. � I ∈I TP I Distributional risk allocation to tariff cells will be discussed later.

Approach based on generalized linear models Table of contents 1 Introduction 2 Pricing of non-life insurance contracts 3 Approach based on generalized linear models 4 Optimization models – expected value approach 5 Optimization models – individual chance constraints 6 Optimization models – a collective risk constraint 7 Numerical comparison

Approach based on generalized linear models Generalized linear models A standard approach based on GLM with the logarithmic link function g ( µ ) = ln µ without the intercept: Poisson (overdispersed) or Negative-binomial regression – the expected number of claims: I E [ N I ] = exp { λ i 0 + λ i 1 + · · · + λ i S } , Gamma or Inverse Gaussian regression – the expected claim severity: I E [ X I ] = exp { γ i 0 + γ i 1 + · · · + γ i S } , where λ i , γ i are the regression coefficients for each I = ( i 0 , i 1 , . . . , i S ). For the expected loss we obtain E [ L I ] = exp { λ i 0 + γ i 0 + λ i 1 + γ i 1 + · · · + λ i S + γ i S } . I

Approach based on generalized linear models Generalized linear models The basic premium levels and the surcharge coefficients can be estimated as a product of normalized coefficients S S exp { λ i 0 + γ i 0 } � � Pr i 0 = · min i ∈I s exp( λ i ) · min i ∈I s exp( γ i ) , ˆ LR s =1 s =1 exp( λ i s ) exp( γ i s ) = min i s ∈ I s exp( λ i s ) · min i s ∈ I s exp( γ i s ) − 1 , e i s Under this choice, the constraints on loss ratios are fulfilled with respect to the expectations.

Approach based on generalized linear models Generalized linear models The GLM approach is highly dependent on using GLM with the logarithmic link function . It can be hardly used if other link functions are used, interaction or other regressors than the segmentation criteria are considered. For the total losses modelling, we can employ generalized linear models with the logarithmic link and a Tweedie distribution for 1 < p < 2, which corresponds to the compound Poisson–gamma distributions.

Optimization models – expected value approach Table of contents 1 Introduction 2 Pricing of non-life insurance contracts 3 Approach based on generalized linear models 4 Optimization models – expected value approach 5 Optimization models – individual chance constraints 6 Optimization models – a collective risk constraint 7 Numerical comparison

Optimization models – expected value approach Advantages of the optimization approach GLM with other than logarithmic link functions can be used, business requirements on surcharge coefficients can be ensured, total losses can be decomposed and modeled using different models, e.g. for bodily injury and property damage, other modelling techniques than GLM can be used to estimate the distribution of total losses over one year, e.g. generalized additive models, classification and regression trees, not only the expectation of total losses can be taken into account but also the shape of the distribution , costs and loadings (commissions, tax, office expenses, unanticipated losses, cost of reinsurance) can be incorporated when our goal is to optimize the combined ratio instead of the loss ratio, we obtain final office premium as the output,