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ACTSC 625 CAS and Health Insurance Mathematics

ACTSC 625 CAS and Health Insurance Mathematics. Lecture 2 Loss Severity Distributions 10/1/13. Loss Random Variables. Loss severity RVs. Typically, Y j is + skewed and fat-tailed. Claims and losses have different distributions Deductibles – Left Censoring Limits – Right Truncation

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ACTSC 625 CAS and Health Insurance Mathematics

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  1. ACTSC 625CAS and Health Insurance Mathematics Lecture 2 Loss Severity Distributions 10/1/13 ACTSC 625 L2: Loss Severity Distributions

  2. Loss Random Variables ACTSC 625 L1: Intro P&C

  3. Loss severity RVs • Typically, Yj is + skewed and fat-tailed. • Claims and losses have different distributions • Deductibles – Left Censoring • Limits – Right Truncation • Ground Up Data is loss data • Empirical distribution • Uses historical data • Parametric approach • Find a parametric distribution that fits data ACTSC 625 L1: Intro P&C

  4. Some useful severity distributions • Exponential • Weibull • Gamma ACTSC 625 L1: Intro P&C

  5. Some useful severity distributions • Pareto • Lognormal ACTSC 625 L1: Intro P&C

  6. Tail Behaviour • Compare tails of X and Y by looking at • Look at the failure rate function: • DFR -- h’(x)<0  fat-tailed • Look at Mean Residual Loss: • IMRL (e’(d)>0)  fat-tailed ACTSC 625 L1: Intro P&C

  7. Examples • Derive h(x) for the Pareto distribution. • IFR or DFR? • Derive e(d) for the Pareto distribution • IMRL or DMRL? ACTSC 625 L1: Intro P&C

  8. Notes on tail measures 1. 2. 3. 4. Both h(x) and e(d) characterize the distribution. ACTSC 625 L1: Intro P&C

  9. Transforming Distributions • Suppose X is a cs RV, with usual prob functions, fX(x), FX(x), SX(x). • Let X=u(Y) for RV Y, u() cs,monotonic increasing • so Y=u-1(X) • Consider the distribution of Y: • u() cs, monotonic decreasing? ACTSC 625 L1: Intro P&C

  10. Examples • Y=aX a>0 • X=Y >0 • X=Y-1 ACTSC 625 L1: Intro P&C

  11. Mixing Distributions • Discrete mixing: • Continuous mixing ACTSC 625 L1: Intro P&C

  12. Summary • Yjis the severity RV, usually assumed iid and independent of frequency. • Yjis usually assumed cs, > 0, fat-tailed. • Loss severity and claim severity differ • because of policy modifications. ACTSC 625 L1: Intro P&C

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