Loss Model I
ACTSC 431/831, Spring 2016
Chapter 10: Introduction to Ruin Theory

It
will not be tested
the Find Exam !
for
Surplus process
=
In order to quantify the insurers risks, it is necessary to model
the insurers surplus over time.
0
We consider
Actsc 431/831 (Fall 2014)
Tutorial 1  solution
Question 1.
(a) We prove the result by induction principle. First note that
x
et dt = et
x
0
= ex =
k=0
0! k x
x e .
k!
Thus the equation holds for For n = 0.
Next, we assume the equation holds for n = m, i.
Actsc 431/831 (Fall 2014)
Tutorial 2  solution
Question 1. By LHospital rule, investigating the ratio of their survival functions is equivalent to
investigating the ratio of their corresponding pdfs. Dene the limiting ratio of the two random
variables X
Actsc 431/831 (Fall 2014)
Tutorial 5 part I
This is Part I of Tutorial 5. The second part will be posted after the second
midterm exam.
1. Let N have a Poisson distribution with mean . Let have a uniform distribution on
the interval (0,5). Determine the u
Formula Sheet For Midterm Two of ACTSC 431/831
The (a, b, 0) class: let the corresponding random variable be N .
1. Poisson distribution POI()
k e
k!
E[N ] = , Var[N ] = , GN (t) = e(t1)
p0 = e , a = 0, b = , pk =
2. Binomial distribution BIN(q, m)
p0 =
Chapter 1. Introduction and Overview
Course objective: As the name of the course suggests, it is to introduce various mathematical
models which can be used by insurers to forecast and predict future insurance losses (or risks),
primarily in a nonlife ins
Midterm Exam TWO ACTSC 431/831, Fall 2014
Last Name:
First Name:
LD. No:
Date: Nevernber 12, 2014 (Wednesday)
Time: 1:00pm2220pm
Aids: SOA or UW approved calculators
Total Pages: 10 (including the cover page and the formula sheet on page 10)
MARKING SC
Actsc 431/831 (Fall 2014)
Tutorial 1
1. In this course, we often encounter the computation of integrals such as
2 t
x t e dt. The following formula can be used to expedite the computation:
n
tn et dt =
x
k=0
x 2 t
0 t e dt
n! k x
x e , n = 0, 1, . . . .
Actsc 431/831 (Fall 2014)
Tutorial 2
1. [Example 4.4 in lecture notes] Suppose X has pdf fX (x) =
2
,
(1+x2 )
x > 0 and Y has pdf
1
,
(1+y)2
fY (y) =
y > 0. Compare the tail weights of these two distributions by investigating
the ratio of their survival f
Actsc 431/831 (Fall 2014)
Tutorial 3
1. For models involving general liability insurance, actuaries at the Insurance Services Oce
once considered a mixture of two Pareto distributions. They decided that ve parameters
were not necessary. The distribution t
Loss Model I
ACTSC 431/831, Spring 2016
Chapter 7: Frequency Distributions (Part I/II)
Basic Frequency Distributions
=
Textbook Mapping: Chapter 6
=
=
Collective risk model
Fix
.
portfolio
,
and
period
tie
<
Recall a collective risk model for the aggregat
Formula Sheet
A useful integral
Z
1
n
t
t e dt =
x
n
X
n!
k=0
k!
xk e
x
; n 2 N:
Pareto distribution with parameters ; > 0. For x > 0,
k
f (x) =
(x + )
+1
, F (x) =
x+
, E Xk =
(
1)
k!
(
k)
.
Gamma distribution with parameters ; > 0. For x > 0 and k = 1;
Loss Model I
ACTSC 431/831, Spring 2016
Chapter 4: Severity distribution (Part I/III)
Creating Severity Distributions
=
distributions
Textbook Mapping: Chapter 4, Sections 5.2.15.2.4
IKI
Tail
.
of
.
It
KI
:
Policy
adjustment
.
Va Rp
TVARP
=
.
it cfw_
x c
Loss Model I
ACTSC 431/831, Spring 2016
Chapter 1: Introduction and Overview
Course objective
etep1
2
Loss
.
Course objective: Construct various mathematical models which
can be used by insurers to forecast and help predict future
insurance losses, primar
Loss Model I
ACTSC 431/831, Spring 2016
Chapter 6: Severity Distributions (Part III/III)
Policy Adjustments


Textbook Mapping: Sections 8.18.5
Important
,
Difficult
,
Confusion
Hazard
h
rate
'k*
Mean
excess
function
loss
(
1
failure
mean
rate
function
Loss Model I
ACTSC 431/831, Spring 2016
Chapter 9: Aggregate Risk Models
Textbook Mapping: Sections 9.19.7
Outline
'
Individual risk model
Collective risk model
Distribution of S for discrete severity distribution
EFE*n
Distribution of S for continuous s
Actsc 431: Term Test 2 (Version 1)  Spring 2013
Department of Statistics and Actuarial Science, University of Waterloo
June 25, 2013
Last name:
First name:
I.D.#:
Notes:
Show all work.
Aid: Calculator (Financial or Scientific)
Unless otherwise specifi
Loss Model I
ACTSC 431/831, Spring 2016
Chapter 2: Random Variables
Stat
24
,
330,333
Overview
In this chapter, we review some basic probability concepts of
random variable. These concepts are central to the
characterization of any mathematical model cons
ACTSC 431/831  Loss Models 1
Problem set 2
The following exercises are from the textbook Loss Models, 4th edition.
Exercises 4.34.5, 4.74.9, 4.11
Exercises 5.1, 5.2, 5.55.7, 5.9, 5.11, 5.18, 5.19
1
ACTSC 431/831  Loss Models 1
Problem set 3 (lecture notes 5, 6)
The following exercises are from the textbook Loss Models, 4th edition.
Exercises 3.25, 3.27, 3.29
Exercises 8.4, 8.5, 8.88.13, 8.15, 8.168.28
In addition to the exercises in the textbook, s
Loss Model I
ACTSC 431/831, Spring 2016
Chapter 5: Severity Distributions (Part II/III)
Tail of Distributions

Textbook Mapping: Sections 3.4.13.4.4

whither
has
a
a
distribution
heavy tail
or
/ loss
light
N
tail ?
severity
Create
distribution (
new
I/
ACTSC 431/831  Loss Models 1
Problem set 1
The following exercises are from the textbook Loss Models, 4th edition.
Exercises 2.4, 2.5
Exercises 3.53.8, 3.12, 3.15, 3.183.20, 3.29 (a,b,d,e), 3.32, 3.33, 3.36, 3.37
In addition to the exercises in the textb
Loss Model I
ACTSC 431/831, Spring 2016
Chapter 3: Distributional Quantities and Risk
Measures
=
Textbook Mapping: Section 3.5
Overview
II
In this chapter, we will introduce some distributional
quantities for a given random variable X . These distribution
Loss Model I
ACTSC 431/831, Spring 2016
Chapter 8: Frequency Distributions (Part II/II)
Creating New Frequency Distributions and Eect
on Frequency

Textbook Mapping: Chapter 7, Section 8.6
Outline
1
nee
Creating new frequency distributions
Mixed frequenc
ACTSC 431/831  Loss Models 1
Problem set 4 (note 7, note 8 (page 18)
practice the fast IBP
1. Suppose that N j! = " follows a Poisson distribution with parameter ", and ! follows a uniform
distribution on (0; 5). Determine the probability that N " 2.
So
ACTSC 431/831  Loss Models 1
Problem set 5 (note 8 (page 933), note 9)
1. Assume that the number of losses NL have a zerotruncated Poisson rv with parameter ".
(a) Identify the pmf and the pgf of NL .
k
k
(b) Assume that a loss results in a positive pa
Formula Sheet
Gamma distribution with parameters
x
f (x) =
and ( ) =
R1
0
x
1
e
x
> 0 and
1
e
( )
> 0. For x > 0,
x=
, E [X] =
2
, Var(X) =
,
dx is the gamma function such that ( + 1) =
Exponential distribution with parameter
f (x) =
1
e
x=
> 0. For x > 0
Actsc 431: Term Test 3 (Version 1)  Spring 2013
Department of Statistics and Actuarial Science, University of Waterloo
July 15, 2013
Last name:
First name:
I.D.#:
Notes:
Show all work.
Aid: Calculator (Financial or Scientific)
Unless otherwise specifi