Risk theory: loss models and risk measures, MATH 4280 3.00 F
Instructor: Edward Furman
Homework 6
1. Demonstrate that the lognormal distribution as parameterized in Klugman,
Panjer and Willmot (2008) (see the additional reading section of the website)
is
Risk theory, MATH 4280 3.00 F
Instructor: Edward Furman
Homework 8
1. Use your knowledge of the permissible ranges for the parameters of the
Poisson, negative binomial, and binomial to determine all possible values
of a and b for these members of the (a,
Risk theory
Elementary dependence concepts
Edward Furman
Department of Mathematics and Statistics
York University
November 2, 2011
Edward Furman
Risk theory 4280
1/1
Denition 0.1 (Positive quadrant dependence (PQD)
The pair (X , Y ) or its c.d.f. is said
Risk theory. Loss models and risk measures.
Loss models. Risk measures for loss models
Edward Furman
Department of Mathematics and Statistics
York University
October 5, 2011
Edward Furman
Risk theory 4280
1 / 27
Size - biased c.d.f.s
Denition 0.1 (Size -
Quiz 1
MATH 4280
Jan 14, 2011
Given name and surname:
Student No:
Signature:
INSTRUCTIONS:
1. Please write everything in ink.
2. This quiz is a closed book test, duration 15 minutes.
3. Only non-programmable calculators are permitted.
4. The text has two
Lemma 0.1. For the r.v.s X, Y, Z all in X , we have that
E[E[X| Y, Z]| Y ] = E[X| Y ].
Proof. In the discrete case, for all y realizations of the r.v. Y , we have that:
E[E[X| Y, Z]| Y = y] =
E[X| Y = y, Z = z]P[Z = z| Y = y]
z
=
xP[X = x| Y = y, Z = z]P[
Risk theory, MATH 4280 3.00 F
Instructor: Edward Furman
Homework 5
1. Let X have cdf FX (x) = 1 (1 + x) , x, > 0. Determine the pdf and cdf
of Y = X.
2. One hundred observed claims in 1995 were arranged as follows: 42 were
between 0 and 300, 3 were betwee
50 CHAPTER 6 SOLUTIONS
For the Poisson, /\ > 0 and so it must be a : O and b > O. For the
binomial, m must be a positive integer and O < q < 1. This requires a < 0
and b > 0 provided b/a is an integer 2 2. For the negative binomial, both 7
and ,8 must be
Risk theory: Loss models and risk measures, MATH 4280 3.00 F
Instructor: Edward Furman
Homework 5
1. A portfolio contains 16 independent risks, each with a gamma distribution
with parameters = 1 and = 250. The pdf is
f (x) =
ex/ (x/)1
, x > 0.
()
Give an
Actuarial models
Proposition 1.5
Let X have a p.d.f. fX | (x|) and cdf FX | (x|) where is a
parameter and it is also a realization of a random variable
with p.d.f. f . Then the mixed p.d.f. of X is
fX (x) = fX | (x|)f ()d =
Edward Furman
Risk theory:loss
Proposition 0.1 (Central limit theorem.)
Let X1 , . . . , Xn be a sequence of i.i.d. r.v.s, and let
Sn := n =1 Xk . We also assume that the m.g.f.s exist, E[Sn ]
k
= E[X1 ] + + E[Xn ] and Var[Sn ] = Var[X1 ] + + Var[Xn ].
Then
Sn E[Sn ]
z = (z).
lim P
n
An introduction and basic quantities of interest
Risk Theorey: Loss Models and Risk
Measures, MATH 4280
Lecture 1. Introduction.
Edward Furman
Department of Mathematics and Statistics
York University
September 12, 2011
Edward Furman
Risk Theorey 1
1 / 12
Actuarial models
Denition 1.10 (Univariate elliptical family)
We shall say that X E(a, b) is a univariate elliptical random
variable if when X has a density, it is given by the form
c
1 x a 2
, x R,
f (x) = g
b
2
b
where g : [0, ) [0, ) such that
Edwa
Actuarial models
Denition 1.9 (Linear exponential family)
A random variable X has a distribution belonging to the linear
exponential family if its pdf can be reformulated in terms of a
parameter as
p(x)er ()x
.
f (x; ) =
q()
Here p(x) does not depend on ,
Basic quantities of interest
Risk Theorey: Loss Models and Risk
Measures, MATH 4280
Lecture 1. Basic quantities of interest.
Edward Furman
Department of Mathematics and Statistics
York University
September 19, 2011
Edward Furman
Risk Theorey 1
1 / 18
Basi
Heavy tailed distributions
Risk theory
Basic quantities of interest
Edward Furman
Department of Mathematics and Statistics
York University
November 6, 2010
Edward Furman
Risk theory 4280
1 / 23
Heavy tailed distributions
Denition 1.1 (Distributions tail)
Risk theory. Loss models and risk measures. MATH 4280 3.00 F
Instructor: Ed Furman
Homework 1
1. Let X
F such that
1 0.01x,
P[X > x] =
1.5 0.02x,
0 x < 50
50 x < 75
.
Determine the cumulative distribution, density, and hazard rate functions
for X. Recal
Risk theory. Loss models and risk measures, MATH 4280 3.00 F
http:/math.yorku.ca/~ efurman/MATH4280_2014
Lecture time and location
11:30-13:00 MW VH 3006
Tutorial time and location
14:30-15:30 F
VH 3006
Instructor name and contacts Ed Furman
efurman@maths
Risk theory: Loss models and risk measures, MATH 4280 3.00 F
Instructor: Edward Furman
Homework 3
1. A random variable has density function f (x) = 1 ex/ , x, > 0. Determine e(), the mean excess loss function evaluated at .
2. Losses have a Pareto distrib
Loss models and risk measures. Math 4280 3.00 F
Instructor: Edward Furman
Homework 1
1. An insurance company determines that N, the number of claims received in a week,
is a random variable with P [N = n] =
1
,
2n+1
where n 0. The company also determines
Final test
MATH 4280
December 11, 2011
Given name and surname:
Student No:
Signature:
INSTRUCTIONS:
1. Please write everything in ink.
2. This exam is a closed book exam, duration 180 minutes.
3. Only non-programmable calculators are permitted.
4. The tex
Risk theory. Loss models and risk measures.
Risk measures
Edward Furman
Department of Mathematics and Statistics
York University
September 28, 2011
Edward Furman
Risk theory 4280
1 / 20
How risky is an r.v.?
Denition 0.1 (Risk measure.)
Let X be a set of
Risk theory. Loss models and risk measures. MATH 4280 3.00 F.
Instructor: Edward Furman
Homework 2
1. Determine the 50th and the 80th percentiles for Examples 0.1 - 0.4 (see the
handouts on loss models on the the web) and the random variable in Problem
1,