A Practioner`s Guide to Generalized Linear Models
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Transcript A Practioner`s Guide to Generalized Linear Models
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