stochastic excess-of-loss pricing within a financial framework cas 2005 reinsurance seminar doris...
DESCRIPTION
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 distributionTRANSCRIPT
Stochastic Excess-of-Loss Pricing
within a Financial Framework
CAS 2005 Reinsurance Seminar
Doris SchirmacherErnesto Schirmacher
Neeza Thandi
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
Central Limit Theorem
Consider a sequence of random variables X1,…,Xn from an unknown distribution with mean and finite variance 2.
Let Sn = Xi be the sequence of partial sums. Then, with an = n and bn = n
(Sn-bn)/ an approaches a normal distribution
Visualizing Central Limit Theorem
Distribution of Normalized MaximaMn = max(X1,X2,…,Xn) does not converge to normal distributions:
Fischer-Tippett Theorem
Let Xi’s be a sequence of iid random variables. If there exists constants an > 0 and bn and some non-degenerate distribution function H such that
(Mn – bn)/an H, then H belongs to one of the three standard extreme value
distributions:
Frechet: (x) = 0 x<=0, > 0 exp( -x-) x>0, >0
Weibull: (x) = exp(-(-x)) x<=0, > 0 0 x>0, > 0
Gumbel: (x) = exp(-e-x) x real
Visualizing Fischer-Tippett Theorem
Pickands, Balkema & de Haan Theorem
For a large class of underlying distribution functions F, the conditional excess distribution function
Fu(y) = (F(y+u) – F(u))/(1-F(u)),for u large, is well approximated by the
generalized Pareto distribution.
Tail Distribution
F(x) = Prob (X<= x) = (1-Prob(X<=u)) Fu(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*)
Peaks Over Threshold MethodMean excess function of a Generalized Pareto:
e(u) = /(1-) u + /(1-)
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
ExampleCoverage: a small auto liability portfolioType of treaty: excess-of-lossCoverage year: 2005Treaty terms:
12 million xs 3 million xs 3 million
Data: Past large losses above 500,000 from 1995 to 2004 are provided.
Collective Risk Models
Look at the aggregate losses S from a portfolio of risks.
Sn = X1+X2+…+Xn
Xi’s are independent and identically distributed random variables
n is the number of claims and is independent from Xi’s
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.
Mean Excess Loss
Fitting Generalized Pareto
Claim Frequency Distribution
• Poisson
• Negative Binomial
• Binomial
• Method of Moment
• Maximum Likelihood
• Least Squares
Combining Frequency and Severity
• Method of Moments• Monte Carlo Simulation• Recursive Formula• Fast Fourier Transform
Aggregate Loss Distribution
Risk Measures
• Standard deviation or Variance• Probability of ruin• Value at Risk (VaR)• Tail Value at Risk (TVaR)• Expected Policyholder Deficit (EPD)
Capital RequirementsRented Capital = Reduction in capital requirement
due to the reinsurance treaty = Gross TVaR – Net TVaR
Gross
Net
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
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
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
Overall Pricing Process
Inputs
Asset flows
U/W flows
Investment flows
Tax flows
DTA flows
Target Return on
CapitalParameters
Equity Flows
Pricing Model
Target Premium