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
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Stochastic Excess-of-Loss Pricing
within a Financial Framework
CAS 2005 Reinsurance Seminar
Doris SchirmacherErnesto Schirmacher
Neeza Thandi
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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
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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
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Visualizing Central Limit Theorem
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Distribution of Normalized MaximaMn = max(X1,X2,…,Xn) does not converge to normal distributions:
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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
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Visualizing Fischer-Tippett Theorem
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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.
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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*)
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Peaks Over Threshold MethodMean excess function of a Generalized Pareto:
e(u) = /(1-) u + /(1-)
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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
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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.
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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
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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.
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Mean Excess Loss
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Fitting Generalized Pareto
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Claim Frequency Distribution
• Poisson
• Negative Binomial
• Binomial
• Method of Moment
• Maximum Likelihood
• Least Squares
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Combining Frequency and Severity
• Method of Moments• Monte Carlo Simulation• Recursive Formula• Fast Fourier Transform
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Aggregate Loss Distribution
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Risk Measures
• Standard deviation or Variance• Probability of ruin• Value at Risk (VaR)• Tail Value at Risk (TVaR)• Expected Policyholder Deficit (EPD)
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Capital RequirementsRented Capital = Reduction in capital requirement
due to the reinsurance treaty = Gross TVaR – Net TVaR
Gross
Net
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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
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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
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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
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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