Stochastic Excess-of-Loss Pricing
within a Financial Framework
CAS 2005 Reinsurance Seminar
Doris Schirmacher
Ernesto 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 m and finite variance s2.
Let Sn = SXi be the sequence of partial sums.
Then, with an = n and bn = nm,
(Sn-bn)/ an approaches a normal distribution
Visualizing Central Limit Theorem
Distribution of Normalized Maxima
Mn = 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: Fa(x) = 0
exp( -x-a)
Weibull: Ya(x) = exp(-(-x)a)
0
Gumbel: L(x) = exp(-e-x)
x<=0, a > 0
x>0, a >0
x<=0, a > 0
x>0, a > 0
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)) GPx,s(x-u) + F(u)
for some Generalized Pareto distribution
GPx,s as u gets large.
 GPx,s*(x-u*)
Peaks Over Threshold Method
Mean excess function of a Generalized Pareto:
e(u) = x/(1-x) u + s/(1-x)
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
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.
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 Requirements
Rented Capital = Reduction in capital requirement
due to the reinsurance treaty
= Gross TVaRa – Net TVaRa
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
Increase in Net Working Capital
 U/W Flow
 Invest Inc Flow
 Tax Flow
Cash Flow from Operations
Equity Flow = Cash Flow from Operations - Incr in Net Working Capital
= 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 = D in Required Assets
DTA Flow = D in DTA over a year
Overall Pricing Process
Asset flows
Parameters
Pricing
Model
Inputs
Target
Return on
Capital
U/W flows
Investment flows
Tax flows
DTA flows
Equity Flows
Target Premium