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