University of Waterloo
ACTSC 432
Loss Models 2
Course Notes
Author:
David Shi
Professor:
Dr. Jun Cai
January 7, 2013
Contents
1 Statistical Concepts
2
1.1
Mixture Distribution Functions . . . . . . .
ASSIGNMENT 4, ACTSC 432/832, FALL 2012
Due at the beginning of the class on Friday, November 9.
1. Past claims data on a portfolio of policyholders are given below:
Policyholder Year 1
1
750
2
625
3
9
ACTSC 432/832 - Loss Models II
Lecture 9
June 6, 2014
Lecture 9,
ACTSC 432/832 - Loss Models II
1/7
Empirical distribution
Parametric distribution: a set of distribution functions, each
member of whi
ACTSC 432/832 - Loss Models II
Lecture 10
June 9, 2014
Lecture 10,
ACTSC 432/832 - Loss Models II
1/9
Graphic comparison
F (x): the model distribution function
Fn (x): the empirical distribution
Mode
ACTSC 432/832 - Loss Models II
Lecture 11
June 11, 2014
Lecture 11,
ACTSC 432/832 - Loss Models II
1/7
Chi-square goodness-of-t test
t = c0 < c1 < < ck =
pj
= F (cj ) F (cj1 )
pnj
= Fn (cj ) Fn (cj1
ACTSC 432/832 - Loss Models II
Lecture 12
June 16, 2014
Lecture 12,
ACTSC 432/832 - Loss Models II
1/8
Likelihood ratio test
H0 : The data came from a population with distribution A
H1 : The data came
ACTSC 432/832 - Loss Models II
Lecture 13
June 18, 2014
Lecture 13,
ACTSC 432/832 - Loss Models II
1/9
Credibility
Credibility Theory a set of quantitative tools that allows an
insurer to perform pros
ACTSC 432/832 - Loss Models II
Lecture 14
June 23, 2014
Lecture 14,
ACTSC 432/832 - Loss Models II
1/1
Example
Ex14.1 An insurance company has decided to establish its
full-credibility requirements fo
ACTSC 432/832 - Loss Models II
Lecture 16
July 2, 2014
Lecture 16,
ACTSC 432/832 - Loss Models II
1/1
The Bayesian methodology
Past claim r.v: X = (X1 , . . . , Xn )T
Past observations: x = (x1 , .
ACTSC 432/832 - Loss Models II
Lecture 17
July 16, 2014
Lecture 17,
ACTSC 432/832 - Loss Models II
1/8
The Buhlmann credibility
The credibility premium is:
n
j Xj = Z X + (1 Z)
0 +
j=1
with
Z =
k
Lect
ACTSC 432/832 - Loss Models II
Lecture 17
July 14, 2014
Lecture 17,
ACTSC 432/832 - Loss Models II
1/7
The credibility premium
Pure premium: n+1 = E(Xn+1 )
Individual premium (hypothetical mean): n+
ACTSC 432/832 - Loss Models II
Lecture 5
May 21, 2014
Lecture 5,
ACTSC 432/832 - Loss Models II
1/9
Method of percentile matching
Df of the model: F (x|)
= (1 , . . . , p ): the parameter to be esti
ACTSC 432/832 - Loss Models II
Lecture 8
June 4, 2014
Lecture 8,
ACTSC 432/832 - Loss Models II
1/7
Frequentist estimation for (a, b, 0) class
k frequency
nk number of observations
pk () = Pr(N = k
ASSIGNMENT 3, ACTSC 432/832, FALL 2012
Due at the beginning of the class on Friday, October 26.
1. Let Xj be the loss in year j for j = 1, 2, ., n, n + 1. Assume that
2
E (Xj ) = j , V ar(Xj ) = j , a
ASSIGNMENT 2, ACTSC 432/832, FALL 2012
Due at the beginning of the class on Friday, October 5.
1. The amounts of claims in the past n exposure units were X1 , X2 , , Xn , which are
independent and hav
ASSIGNMENT 1 ACTSC 432/832, FALL 2012
Due at the beginning of the class on Wednesday, September 26.
1. Let X be the number of claims in a portfolio. Given = > 0, the conditional
distribution of X is t
ACTSC432/832
Spring 2014
Homework Set III (Due June 16, 2014)
13. Solve the unselected questions of Midterm 1.
4. An automobile insurance policy provides benets for accidents caused by both underinsur
ACTSC432/832
Spring 2014
Homework Set II (Due June 2, 2014)
1. A random sample of ve claims from a lognormal distribution is given as follows:
500
1, 000
1, 500
2, 500
4, 500
Estimate and by the metho
ACTSC432/832
Spring 2014
Homework Set IV (Due July 2, 2014)
1. You are given:
(a) A sample of claim payments:
29
64
90
135
182
(b) Claim sizes are assumed to follow an exponential distribution.
(c) Th
ACTSC432/832
Spring 2014
Homework Set I (Due May 21, 2014)
1. Let X have the uniform distribution over the range ( 2, + 2). That is,
2 < x < + 2.
fX (x) = 0.25,
Show that the median from a sample of
ACTSC 432/832 - Loss Models II
Lecture 3
May 12, 2014
Lecture 3,
ACTSC 432/832 - Loss Models II
1/9
Bayesian estimation: Denition
Prior distribution () - probability distribution over the
space of po
ACTSC 432/832 - Loss Models II
Lecture 2
May 7, 2014
Lecture 2,
ACTSC 432/832 - Loss Models II
1/12
Examples
Ex2.1 The maximum observation from a uniform distribution on
the interval (0, ),
n = maxcfw
ACTSC 432/832 - Loss Models II
Lecture 1
May 5, 2014
Lecture 1,
ACTSC 432/832 - Loss Models II
1/11
Course Information
Instructor: Dr. Xiaoying Han
Oce: 4101 M3
E-mail: [email protected]
Lecture: MW
ACTSC 432/832 - Loss Models II
Lecture 4
May 14, 2014
Lecture 4,
ACTSC 432/832 - Loss Models II
1/10
Conjugate prior distributions
A prior distribution is said to be a conjugate prior distribution
for
ACTSC 432/832 - Loss Models II
Lecture 7
June 2, 2014
Lecture 7,
ACTSC 432/832 - Loss Models II
1/5
Frequentist estimation for discrete distributions
Poisson
Negative binomial
Binomial
The (a, b,
ACTSC 432/832 - Loss Models II
Lecture 15
June 25, 2014
Lecture 15,
ACTSC 432/832 - Loss Models II
1/9
Greatest accuracy credibility
n exposure units of past claim: X = (X1 , . . . , Xn )T
manual ra
ACTSC 432/832 - Loss Models II
Lecture 20
July 23, 2014
Lecture 20,
ACTSC 432/832 - Loss Models II
1/1
Semi-parametric estimation
Parametric form for the conditional distribution fXij | (xij |i )
Ex20
ACTSC 432/832 - Loss Models II
Lecture 19
July 21, 2014
Lecture 19,
ACTSC 432/832 - Loss Models II
1/10
Exact credibility
credibility premium = Bayesian premium
typically occurs in Buhlmannand Buhlma