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 made? At the moment of death
of the policyholder. This
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
X
X X
X/ n
, where
yp
p,
which we usually look up in
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 following probability function (pf):
Prcfw_X = k =
k e
, k
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 top row, and the first decimal place of x is given in the
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 LaTex or Word with a cover page
that shows your name and st
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, and their contact information and oce hours will be
poste
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 policyholders. For policyholder i, there were ni years clai
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 in the n observed exposure units, where each Xj
means
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 ) is denoted by
fX,Y (x, y) = Prcfw_X = x, Y = y.
If X a
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
0.15 and that 95% of the claims frequencies are
betwee
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
manual premium M, or our prior belief about the mean
of X
P
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 x i 2
2v
1
1 e 2a 2
2a
n
exp 1
2v
x i 2 1 2
2a
exp 1
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) Calculate the unbiased estimates of the structural parameter
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, 2, ., n, n + 1, where r > 0 and m1 , ., mn+1 are posit
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.
(c) For x > 0, F (x) =
100 (
0
1
(
+x
)3 )
1
d
100
= 1
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 uniform
distribution U (0, 100).
(a) Calculate the exp
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 + 10.
=
2
Thus, it follows from E(X) =
15+11.6+13.5+17
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 from the pdf is obtained as
follows:
15, 11.6, 13.5, 17.2
Solutions to Practice Questions 2 ACTSC 432/832, Fall 2015
1. (a) We have = E(Xj ) = E(N )E(Yj ) = 240, 2 = V ar(Xj ) = E(N )V ar(Yj ) +
V ar(N )[E(Yj )]2 = 25, 600, 0 =
(
)
yp 2
r
=
(
)
y0.95 2
0.1
=
(
1.96
0.1
The number of exposure units, n, should sat
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 , Y2 , ., are
j=1
independent, N has a negative binomi
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.
We wish to estimate the expected sum of the next two
d
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 ij X ij
1
mi
j1
EX i
1
mi
ni
m ij EX ij
1
mi
j1
X
1
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 with replacement. The sum of the
number on the three b
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 if
E
Definition: An estimator is asymptotically
unbias
3of8
1. [14 points] Assume that the pdf for iid losses Xi, z' = 1, . . . ,n, is given by
(I: 3ex/6
fx (3;): 694 , a: > 0.
(a) [5pts] Determine the maximum likelihood estimator for 6. -' In!
-;x ID a - ems q)
Ll9)- ll >1? 6 = llx; (2 1(619
1:194
lHLl9l=
List of pmf s, pdf s with expectation and variance
Distribution
X Normal(, 2 )
X Gamma(, )
X Beta(, )
X Poisson()
X Bin(n, p)
X U(a, b)
X Neg.Bin.(r, p)
pmf or pdf
2
2
e(x) /2
2 2
x1 ex/
()
(+) 1
(1
()() x
x e
x!
n x
x p (1
x)1
p)nx
1
ba
r+x1 r
r1 p (1
ACTSC 832 Spring 2016 Information on project
Your project will consist in writing a summary and doing a short presentation on a research paper
related to topics covered in class. What you need to do:
1. Choose a paper from the list below (or suggest anoth
ACTSC 432/832 Midterm 2 Spring 2016
Department of Statistics and Actuarial Science, University of Waterloo
July 8, 2016
Last Name:
First Name:
(Please write your UWID number on the back of this page only.)
Put a check mark here if you are an 832 student:
ACTSC 432/832 Example from July 11, 2016
Example: Normal/Normal model Suppose that
Xij |i = i is normally distributed with mean i and variance 2 (known);
i is a normal r.v. with mean and variance 1.
Then, the likelihood function is given by
x 2
ni
r Z