Standard Error of Technical Cost Incorporating Parameter Uncertainty

Transcription

1 Standard Error of Technical Cost Incorporating Parameter Uncertainty Christopher Morton Insurance Australia Group This presentation has been prepared for the Actuaries Institute 2012 General Insurance Seminar. The Institute Council wishes it to be understood that opinions put forward herein are not necessarily those of the Institute and the Council is not responsible for those opinions.

2 Technical Cost: the best estimate of all the costs from underwriting a policy Key input into: Pricing decisions Introduction Performance and Monitoring Reports However, The variance is not commonly used 2

3 Motivation 3

4 Motivation Multiple claim types combine to form a technical cost: Technical Cost = ClaimFreq AvgClaimSi What distribution does this quantity have? t t ze t Which of the above policies is more risky? Just some of the questions I wanted answered 4

5 Motivation Review of Actuarial and Statistical literature found some answers, but: othing specific about GLMs Used simulation techniques, not a closed form solution Did not always consider parameter uncertainty So, this paper was written. 5

6 Today s Presentation Considering variance is Significant Useful Parameter uncertainty Is present when using models eeds to be quantified 6

7 Scope Fundamental Equation of Insurance Why Include Parameter Uncertainty? Practical Applications Conclusion 7

8 Fundamental Equation of Insurance Y = X i= 1 i Y = Claims Cost for a specific claim type where, = Random number of claims Xi = Random size of claim i This equation is: Fundamental to the study of Ruin Theory The basis of insurance. and X could belong to any distribution. We will assume: is Poisson distributed X is Gamma distributed 8

9 Why Include Parameter Uncertainty? Rank correlation is ~97%. 9

10 Why Include Parameter Uncertainty? Frequency and sizes of a person s claims are randomly distributed Each person has a unique distribution If we knew their true distribution, then our jobs would be significantly easier! However, Random process is observed only through data we collect 10

11 Why Include Parameter Uncertainty? Generalised Linear Models allow a convenient framework to link data and distributions together: T ( ˆ ) X ˆ β ~ Poisson µ T ˆ µ X X i X ˆ X β X ~ iid. Gamma ˆ φx, ˆ φx Because the parameters, β, are estimated by Maximum Likelihood we know they have an asymptotic distribution: = X T 1 ˆ X T X T ( X T ˆ η β ~ β, φ WX ) X This uncertainty in the estimate, is called Parameter Uncertainty 11

12 Why Include Parameter Uncertainty? ˆ η = ( X T WX ) X X T 1 ˆ X T ~, X T β β φ Parameter Uncertainty is affected by the specification of the GLM, X T Fitting low exposure or insignificant parameters will increase uncertainty Model fit is important 12

13 Further Details First condition on our data, D = X T β: Var ( Y ) = ED[ Var( Y D) ] + VarD [ E( Y D) ] = ED{ E[ Var( Y D, ) D] + Var [ E( Y D, ) D] } + VarD{ E[ E( Y D, ) D] } 2 = E Var( X D) E ( D) + E( X D) Var ( D) + Var [ E X D E D ] D [ ] ( ) ( ) i Var(Y) is now a function of the moments of X D and D Solving for the moments: 2 Var( Y ) = exp( η + σ 2ˆ η 2) exp( η + 2) X σ 2ˆ ηx {( ˆ ˆ 1 φ + φ ) exp( σ 2ˆ ) + exp( η + σ 2ˆ 2) [ exp( σ 2ˆ + σ 2ˆ ) 1] } X η X i η η D η X i 13

14 Further Details Morton C, 2012, Standard Error of Technical Cost Incorporating Parameter Uncertainty 14

15 Practical Applications Variance can be used together with the expected claims cost to: Add an additional dimension to analyses Graphically compare two different pricing scenarios based on Profit and Standard Error of Technical Cost 15

16 Pricing Scenario Heat Map Intensity of colour represents expected number of policies at that price based on elasticity models 16

17 Practical Applications Confidence Intervals Allocate capital down to an individual policy and use as an input into DFA modelling Risk based KPIs 17

18 Conclusion Variance of technical cost Has practical uses, including adding an additional dimension to analyses Variance can only be estimated from data we collect and analyse Which, necessarily, adds parameter uncertainty 18

19 Questions? 19