BA3202-L9 - BA3202 Actuarial Statistics Lecture 8 Run-off...

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BA3203 L9 BA3202 Actuarial Statistics Lecture 8: - Run-off Triangles
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BA3203 L9 Insurance business Actuarial Science NTU Jade Nie [email protected] 2 Policy holders Insurer Premium for 1 year policy Claim payments during 1 st year Reserves: Estimated amounts for possible future claim payments Claim payments when needed Reserves put aside for each policy year
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BA3203 L9 Triangle method Chain ladder method Data: triangle data with incurred/paid claims Incurred Vs Paid Reserve=ultimate claims projected-paid claims Inflation & discounting Average claims method Data: incurred claims, number of claims BF method Data for chain ladder method + Earn premium and expected loss ratio Actuarial Science NTU Jade Nie [email protected] 3
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BA3203 L9 BA3202 Actuarial Statistics Lecture 9: - Generalized linear models (GLM)
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BA3203 L9 Introduction The essential difference for GLMs from linear models in CT3 is that we now allow the distribution of the data to be non- normal. This extension is important in actuarial work. The Poisson distribution is used in modelling the force of mortality, 𝜇 𝑥 (in life), and claim frequency (in GI). The gamma or lognormal distribution are usually used for the claim severity Data analysis: decide which variables or factors are important predictors for the risk being considered quantify the relationship between the predictors and the risk to assess appropriate premium levels Actuarial Science NTU Jade Nie [email protected] 5
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BA3203 L9 Generalized linear model (GLM) Aim: we want to predict a variable (called the response variable) e.g. Number of claims, average claim sizes of a group of policy holder We have a number of other information called the predictors/ covariates e.g. the age, gender, education level, occupation etc. of this group of policy holder GLM relates the response variable to predictors/covariates We need to know the distribution of the response variable belongs to the exponential family Actuarial Science NTU Jade Nie [email protected] 6
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BA3203 L9 Exponential families A distribution for a random variable Y belongs to an exponential family if its density has the following form 𝑓 𝑌 𝑦 ; 𝜃 , 𝜑 = exp 𝑦𝜃 − 𝑏 𝜃 𝑎 𝜑 + 𝑐 𝑦 , 𝜑 where 𝑎 , 𝑏 , 𝑐 are functions, 𝜃 and 𝜑 are parameters 𝜃 : natural parameter 𝜑 : scale/dispersion parameter 𝐸 𝑌 = 𝑏 𝜃 note that 𝐸 𝑌 only depend on 𝜃 When predicting 𝑌 , only 𝜃 is of importance 𝑣𝑎𝑣 𝑌 = 𝑎 𝜑 𝑏 ′′ 𝜃 𝑏 ′′ 𝜃 : defines the way that variance related to the mean 𝑎 𝜑 : function involving the scale parameter Examples of exponential family distributions: Exponential, Normal, Poisson, Binomial, Gamma Actuarial Science NTU Jade Nie [email protected] 7
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BA3203 L9 Exponential families Log-likelihood function of 𝑌 : 𝑙 𝑦 ; 𝜃 , 𝜑 = ln 𝑓 𝑌 𝑦 ; 𝜃 , 𝜑 = 𝑦𝑦−𝑏 𝑦 𝑎 𝜑 + 𝑐 𝑦 , 𝜑 𝜕𝜕 𝜕𝑦 =
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