On Univariate Extreme Value Statistics and the Estimation of Reinsurance Premiums B. Vandewalle J. Beirlant Katholieke Universiteit Leuven University Center of Statistics Actuarial Science Team G¨ oteborg 2005 On Univariate Extreme Value Statistics and the Estimation of Reinsurance Premiums – p. 1/29
Or ... ... on univariate extreme value statistics and the estimation of insurance premiums for excess-of-loss reinsurance policies in excess of a high retention level ... with special attention to heavy-tailed distributions and Wang’s premium principle as a generalization to the net premium principle. On Univariate Extreme Value Statistics and the Estimation of Reinsurance Premiums – p. 2/29
Overview (Re)insurance premium calculation Net premium principle Wang’s premium principle � applied to excess-of-loss reinsurance setting Extreme value statistics Motivation Extreme value theory (first order framework) Estimating reinsurance premiums Finite sample behavior Simulated data (Fréchet, Burr) Reinsurance premiums (net premium, dual-power transform) On Univariate Extreme Value Statistics and the Estimation of Reinsurance Premiums – p. 3/29
Overview Asymptotic results Motivation Extreme value theory (second order framework) Premium approximation error Asymptotic normality and bias Finite sample behavior Secura Belgian Re data Simulated data (Fréchet, Burr) Reinsurance premiums (net premium, dual-power transform) On Univariate Extreme Value Statistics and the Estimation of Reinsurance Premiums – p. 4/29
Insurance premium calculation Net premium principle Insurance premium calculation ... at the heart of actuarial science ... X � non-negative random variable denoting the total claim amount resulting from a single insurance policy decumulative distribution function ¯ � � F ( x ) = P ( X > x ) Net premium principle Under the assumption that risk is essentially non-existing if the insurer sells enough identically distributed and independent policies, on average the insurer will not lose any money when using premium � ∞ ¯ Π = E ( X ) = F ( x ) dx 0 On Univariate Extreme Value Statistics and the Estimation of Reinsurance Premiums – p. 5/29
Insurance premium calculation Wang’s premium principle However ... experienced losses hardly ever equal expected losses ... � extra loading for risk is desirable if the insurer on average does � � Π ≥ E ( X ) not want to lose any money The search for sound premium calculation principles has been the subject of numerous actuarial papers and remains debatable with respect to choice ... � � Wang, 1996 Wang’s premium principle With g an increasing, concave function, called the distortion, that maps [0 , 1] onto [0 , 1] � ∞ � ¯ � Π = g F ( x ) dx 0 On Univariate Extreme Value Statistics and the Estimation of Reinsurance Premiums – p. 6/29
Insurance premium calculation Wang’s premium principle Examples: g ( x ) = x � Net premium principle ( Π = EX ) g ( x ) = x 1 /α ( α ≥ 1) � Proportional hazard transform principle g ( x ) = 1 − (1 − x ) α ( α ≥ 1) � Dual-power transform principle g ( x ) = (1 + α ) x − αx 2 (0 ≤ α ≤ 1) � Gini principle √ 1+ αx − 1 g ( x ) = ( α > 0) � Square root function principle √ 1+ α − 1 g ( x ) = 1 − e − αx ( α > 0) � Exponential function principle 1 − e − α g ( x ) = log(1+ αx ) ( α > 0) � Logarithmic function principle log(1+ α ) On Univariate Extreme Value Statistics and the Estimation of Reinsurance Premiums – p. 7/29
Insurance premium calculation Wang’s premium principle 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 x On Univariate Extreme Value Statistics and the Estimation of Reinsurance Premiums – p. 8/29
Reinsurance premium calculation Excess-of-loss reinsurance setting X R = ( X − R ) + = max (0 , X − R ) � total claim amount resulting from a single excess-of-loss reinsurance policy in excess of a high retention level R decumulative distribution function ¯ F R ( x ) = ¯ � � F ( x + R ) ⇓ � ∞ � ¯ � Π ( R ) = g F ( x ) dx R � premium given as a function of the decumulative distribution function ¯ F of the original claim amount for the layer from R to infinity On Univariate Extreme Value Statistics and the Estimation of Reinsurance Premiums – p. 9/29
Extreme value statistics Motivation In reinsurance applications, emphasis often lies on the modelling of extreme events, i.e. mostly events with low frequency high and often disastrous impact A very practical tool in the analysis of such extreme events can be found in extreme value statistics, where the tail behavior of a distribution is characterized mainly by the extreme value index γ theoretical framework some estimators (extreme value index, small exceedance probabilities ... reinsurance premiums) On Univariate Extreme Value Statistics and the Estimation of Reinsurance Premiums – p. 10/29
Extreme value statistics Extreme value theory (first order framework) Consider X 1 ,n ≤ . . . ≤ X n,n independent and identically distributed random variables with common distribution function F Maximum domain of attraction condition � X n,n − b n � ≤ x n →∞ P lim = H ( x ) a n Limit necessarily of generalized extreme value type � � − (1 + γx ) − 1 exp , 1 + γx > 0 , γ � = 0 γ H γ ( x ) = � � − exp ( − x ) x ∈ R , γ = 0 exp , Then F is said to belong to the maximum domain of attraction of the extreme value distribution H γ , denoted as F ∈ D ( H γ ) On Univariate Extreme Value Statistics and the Estimation of Reinsurance Premiums – p. 11/29
Extreme value statistics Extreme value theory (first order framework) γ < 0 � F belongs to the Weibull class (e.g. uniform, beta and reversed Burr distribution) γ = 0 � F belongs to the Gumbel class (e.g. exponential, normal and gamma distribution) γ > 0 � F belongs to the Fréchet class (e.g. Fréchet, Pareto and Burr distribution) On Univariate Extreme Value Statistics and the Estimation of Reinsurance Premiums – p. 12/29
Extreme value statistics Extreme value theory (first order framework) γ < 0 � F belongs to the Weibull class (e.g. uniform, beta and reversed Burr distribution) γ = 0 � F belongs to the Gumbel class (e.g. exponential, normal and gamma distribution) γ > 0 � F belongs to the Fréchet class (e.g. Fréchet, Pareto and Burr distribution) � ¯ F ( x ) = x − 1 /γ l F ( x ) � Pareto-type tail t →∞ ( l F slowly varying at infinity, i.e. l F ( tx ) − → 1 for all x > 0 ) l F ( t ) U ( x ) = Q (1 − 1 /x ) = x γ l U ( x ) ( l U slowly varying at infinity) On Univariate Extreme Value Statistics and the Estimation of Reinsurance Premiums – p. 12/29
Extreme value statistics Estimating reinsurance premiums (Karamata theorem) Assume G ( t ) = g (1 /t ) to be regularly varying at infinity with index of regular variation β , i.e. G ( t ) = t β l G ( t ) ( l G slowly varying at infinity) ⇓ ¯ F ( x ) = x − 1 /γ l F ( x ) � ∞ x β/γ l GF ( x ) dx Π ( R ) = R ( l GF slowly varying at infinity) Karamata ⇓ γ < − β � ¯ 1 Π ( R ) ∼ ˙ � Π ( R ) = − β/γ − 1 Rg F ( R ) when the retention level R tends to infinity On Univariate Extreme Value Statistics and the Estimation of Reinsurance Premiums – p. 13/29
Extreme value statistics Estimating reinsurance premiums 1 − β/γ − 1 Rg ( ¯ Π ( R ) ∼ ˙ Π ( R ) = F ( R )) ⇓ ˆ 1 Π ( R ) = γ − 1 Rg (ˆ p R ) Premium estimator: − β/ ˆ � large retention level � regularly varying distortion On Univariate Extreme Value Statistics and the Estimation of Reinsurance Premiums – p. 14/29
Extreme value statistics Estimating reinsurance premiums 1 − β/γ − 1 Rg ( ¯ Π ( R ) ∼ ˙ Π ( R ) = F ( R )) ⇓ ˆ 1 Π ( R ) = γ − 1 Rg (ˆ p R ) Premium estimator: − β/ ˆ � large retention level � regularly varying distortion Net premium principle G ( x ) = x − 1 , g ( x ) = x � β = − 1 , l G ( x ) = 1 . On Univariate Extreme Value Statistics and the Estimation of Reinsurance Premiums – p. 14/29
Extreme value statistics Estimating reinsurance premiums 1 − β/γ − 1 Rg ( ¯ Π ( R ) ∼ ˙ Π ( R ) = F ( R )) ⇓ ˆ 1 Π ( R ) = γ − 1 Rg (ˆ p R ) Premium estimator: − β/ ˆ � large retention level � regularly varying distortion Dual-power transform principle ( α > 1 ) x − 1 + o G ( x ) = x − 1 { α − α ( α − 1) x − 1 � � } 2 g ( x ) = 1 − (1 − x ) α β = − 1 � x − 1 + o l G ( x ) = α − α ( α − 1) x − 1 � � 2 On Univariate Extreme Value Statistics and the Estimation of Reinsurance Premiums – p. 14/29
Extreme value statistics Estimating reinsurance premiums 1 − β/γ − 1 Rg ( ¯ Π ( R ) ∼ ˙ Π ( R ) = F ( R )) ⇓ ˆ 1 Π ( R ) = γ − 1 Rg (ˆ p R ) Premium estimator: − β/ ˆ � large retention level � regularly varying distortion Needed: p R of small exceedance probability p R = ¯ estimator ˆ F ( R ) estimator ˆ γ of the tail index γ On Univariate Extreme Value Statistics and the Estimation of Reinsurance Premiums – p. 14/29
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