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
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
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
function
1
failure
rate
function )
'k*
Mean
excess
loss
(
me
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
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
Distribution of S for continuous severity
Loss Model I
ACTSC 431/831, Spring 2016
Chapter 8: Frequency Distributions (Part II/II)
Creating New Frequency Distributions and Effect
on Frequency
Textbook Mapping: Chapter 7, Section 8.6
Outline
1
Creating new frequency distributions
Mixed frequency di
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
Overview
Generally speaking, a loss random variable is said
heavytailed if it has a large probability to
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
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 =
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
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 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.
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
Recall a collective risk model for the aggregate claim amount of a
portfolio is of the f
Loss Model I
ACTSC 431/831, Spring 2016
Chapter 10: Introduction to Ruin Theory
Surplus process
In order to quantify the insurers risks, it is necessary to model
the insurers surplus over time.
We consider the simplest model, i.e.,
surplus = initial capit
Loss Model I
ACTSC 431/831, Spring 2016
Chapter 3: Distributional Quantities and Risk
Measures
Textbook Mapping: Section 3.5
Overview
In this chapter, we will introduce some distributional
quantities for a given random variable X . These distributional
qu
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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 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