1. Stochastic Excess-of-Loss Pricing within a Financial Framework CAS 2005 Reinsurance Seminar Doris Schirmacher Ernesto Schirmacher Neeza Thandi

2. Agenda Extreme Value Theory Central Limit Theorem Two Extreme Value Theorems Peaks Over Threshold Method Application to Reinsurance Pricing Example Collective Risk Models IRR Model

3. Central Limit Theorem Consider a sequence of random variables X 1,…,X n from an unknown distribution with mean  and finite variance  2. Let S n =  X i be the sequence of partial sums. Then, with a n = n and b n = n  (S n -b n )/ a n approaches a normal distribution

4. Visualizing Central Limit Theorem

5. Distribution of Normalized Maxima M n = max(X 1,X 2,…,X n ) does not converge to normal distributions:

6. Fischer-Tippett Theorem Let X i ’s be a sequence of iid random variables. If there exists constants a n > 0 and b n and some non-degenerate distribution function H such that (M n – b n )/a n  H, then H belongs to one of the three standard extreme value distributions: Frechet:   (x) = 0 x 0 exp( -x -  ) x>0,  >0 Weibull:   (x) = exp(-(-x)  ) x 0 0 x>0,  > 0 Gumbel:  (x) = exp(-e -x ) x real

7. Visualizing Fischer-Tippett Theorem

8. Pickands, Balkema & de Haan Theorem For a large class of underlying distribution functions F, the conditional excess distribution function F u (y) = (F(y+u) – F(u))/(1-F(u)), for u large, is well approximated by the generalized Pareto distribution.

9. Tail Distribution F(x) = Prob (X<= x) = (1-Prob(X<=u)) F u (x-u) + Prob (X<=u)  (1-F(u)) GP  (x-u) + F(u) for some Generalized Pareto distribution GP  as u gets large.  GP  * (x-u*)

10. Peaks Over Threshold Method Mean excess function of a Generalized Pareto: e(u) =  /(1-  ) u +  /(1-  )

11. Agenda Extreme Value Theory Central Limit Theorem Two Extreme Value Theorems Peaks Over Threshold Method Application to Reinsurance Pricing Example Collective Risk Models IRR Model

12. Example Coverage: a small auto liability portfolio Type of treaty: excess-of-loss Coverage year: 2005 Treaty terms: 12 million xs 3 million xs 3 million Data: Past large losses above 500,000 from 1995 to 2004 are provided.

13. Collective Risk Models Look at the aggregate losses S from a portfolio of risks. S n = X 1 +X 2 +…+X n X i ’s are independent and identically distributed random variables n is the number of claims and is independent from X i ’s

14. Loss Severity Distribution Pickands, Balkema & de Haan Theorem  Excess losses above a high threshold follow a Generalized Pareto Distribution. - Develop the losses and adjust to an as-if basis. - Parameter estimation: method of moments, percentile matching, maximum likelihood, least squares, etc.

15. Mean Excess Loss

16. Fitting Generalized Pareto

17. Claim Frequency Distribution Poisson Negative Binomial Binomial Method of Moment Maximum Likelihood Least Squares

18. Combining Frequency and Severity Method of Moments Monte Carlo Simulation Recursive Formula Fast Fourier Transform

19. Aggregate Loss Distribution

20. Risk Measures Standard deviation or Variance Probability of ruin Value at Risk (VaR) Tail Value at Risk (TVaR) Expected Policyholder Deficit (EPD)

21. Capital Requirements Rented Capital = Reduction in capital requirement due to the reinsurance treaty = Gross TVaR  – Net TVaR  Gross Net

22. IRR Model Follows the paper “Financial Pricing Model for P/C Insurance Products: Modeling the Equity Flows” by Feldblum & Thandi Equity Flow = U/W Flow + Investment Income Flow + Tax Flow – Asset Flow + DTA Flow

23. Determinants of Equity Flows Asset Flow DTA Flow U/W Flow Invest Inc Flow Tax Flow Equity Flow = Cash Flow from Operations - Incr in Net Working Capital Increase in Net Working Capital Cash Flow from Operations = U/W Flow + II Flow + Tax Flow - Asset Flow + DTA Flow

24. Equity Flows U/W Cash Flow = WP – Paid Expense – Paid Loss Investment Income Flow = Inv. yield * Year End Income Producing Assets Tax Flow = - Tax on (UW Income Investment Income) Asset Flow =  in Required Assets DTA Flow =  in DTA over a year

25. Overall Pricing Process Inputs Asset flows U/W flows Investment flows Tax flows DTA flows Target Return on Capital Parameters Equity Flows Pricing Model Target Premium