ACTSC 432 Loss Models
II
Introduction
One of the main jobs of an actuary is to set premiums.
Life insurance:
Example: Whole life policy which pays $100,000 at the
moment of death.
When is the payment
Confusion over the full credibility
condition for compound Poisson
We can base our decision on the accuracy
of X or N.
Credibility based on accuracy of X
Pr
X X
X
0
yp
r
r
pn 0
2
and Pr y p
2
X
2
Table of Common Distributions for ACTSC 432/832
1. Discrete Distributions
(a) Poisson with parameter > 0: A random variable X is said to have a Poisson distribution denoted by X P () if X has the foll
NORMAL DISTRIBUTION TABLE
The first table below gives values of the distribution function, (DIX), of the
standard normal distribution for selected values of x. The integer part of x is
given in the to
Project ACTSC 832, Fall 2012
Deadline: 5:30pm, Thursday, December 20. Please email a PDF le of your project
to [email protected] by the deadline.
Project requirement: Projects must be typed using LaTe
ACTSC 432/832 LOSS MODELS 2, FALL 2012
Instructor: Professor Jun Cai, M3 4012, 519-888-4567 ext. 36990, [email protected]
Instructors oce hours: 3:305:30, Wednesday.
Teaching assistants: TAs names, an
Review Notes for Loss Models 2 ACTSC 432/832, Fall 2012
Part 3 Empirical Bayes Parameter Estimation
1. Data structure of the empirical Bayesian parameter estimates: Suppose that
there are r group poli
Review Notes for Loss Models 2 ACTSC 432/832, Fall 2012
Part 2 Credibility Theory
1. The classical credibility or the limited uctuation credibility: Let X1 , ., Xn
be the losses in the past n years or
Review Notes for Loss Models 2 ACTSC 432/832, FALL 2012
Part 1 Statistical Concepts
Conditional distributions: If X and Y are discrete random variables, the joint
probability function (pf) of (X, Y )
Lectures week 5
Example: Auto insurance
Assume that the number of automobile claims per year
for a policyholder in a given portfolio of policyholder
with the same underwriting characteristics averages
Lectures week 4
Partial Credibility
If the full credibility condition is not satisfied then use
partial credibility, where the premium is a weighted
average between the past experience X and the
manua
Example: Normal-normal
Suppose that X| ~N, v and that ~N, a.
1
2a
e
1
2a 2
model distribution
f X| x|
n
i1
1
2v
posterior distribution
e
1
2v x i 2
f X| x|
f |X |x
f X x
f X| x|
n
i1
1
1 e 2vv
Practice Questions 4 ACTSC 432/832, Fall 2015
1. Past claims data on a portfolio of policyholders are given below:
Policyholder Year 1
1
750
2
625
3
900
Year 2
800
600
950
Year 3
650
675
850
(a) Calcu
Practice Questions 3 ACTSC 432/832, Fall 2015
1. Let Xj be the loss in year j. Given , losses X1 , ., Xn , Xn+1 are conditionally independent. Assume
E(Xj |) = rj
and
V ar(Xj |) =
r2j 2
mj
for j = 1,
Solutions to Practice Questions 1 ACTSC 432/832, Fall 2015
1. (a) The expected amount of a claim is E(X) = E(E(X|) = E(/2) = 25.
(b) V ar(X) = E(V ar(X|) + V ar(E(X|) = E(32 /4) + V ar(/2) = 2078.33.
Practice Questions 1 ACTSC 432/832, Fall 2015
1. Let X be the amount of a claim. Given = > 0, the conditional distribution of
X is a Pareto distribution P areto(3, ). The distribution function of is a
Solutions to Practice Questions 5 ACTSC 432/832, Fall 2015
1. We have
x2
x 10 (x10)2
2 2
x
e
(x + 10)x e 2 dx
E(X) =
dx =
2
10
0
2
2
2 x2
2 x
x
2 dx + 10
2 dx =
=
x e
xe
x e 2 dx + 10
2
0
0
2
Practice Questions 5 ACTSC 432/832, Fall 2015
1. A loss random variable X has the following pdf:
x10
2
f (x) =
0,
e
(x10)2
2 2
, x > 10,
x 10,
where the parameter > 0. A random sample of ve losses fr
Solutions to Practice Questions 3 ACTSC 432/832, Fall 2015
1. (a) We have E(Xj ) = E(E(Xj |) = E(rj ) = rj E(),
V ar(Xj ) = E(V ar(Xj |) + V ar(E(Xj |) = E
(
= r2j V ar() +
( r 2j 2 )
mj
+ V ar(rj )
E
Practice Questions 2 ACTSC 432/832, Fall 2015
1. The amounts of claims in the past n exposure units were X1 , X2 , , Xn , which are
independent and have a common distribution as X = N Yj , where N, Y1
Lectures week 7
Example: Urn model
0s
1s
Urn 1 1 60% 40%
Urn 2 2 80% 20%
1 2 1 (dont know which one we have
2
selected)
Select 3 balls with replacement. The sum of the
number on the three balls is 2.
Lectures week 8
Nonparametric Estimation Bhlmann-Straub model
v
EX ij | i i i , VarX ij | i i m iji ,
E i , v Ev i and a Var i
The goal is to estimate the structural parameters , v
and a.
Xi
ni
m
Lectures week 6
Example: Urn model
Two urns with balls marked either 0 or 1
0s
1s
Urn 1 60% 40%
Urn 2 80% 20%
An urn is selected at random 1 2 1 (dont
2
know which one we have selected)
Select 3 balls
Unbiased Estimators
Suppose X 1 , X 2 , , X n are iid with PDF (PF) f X x;
which depends on a scaler parameter . Let
X 1 , X 2 , , X n be an estimator of
Definition: An estimator is unbiased for