Solution of Sample Exercises
Chapter 4 - Aggregate loss models - Compound distributions
4.1
Mean, variances and PGFs
1. Let N be the number of physicians in a day, where P (N = 1) = P (N = 2) = P (N =
3) = P (N = 4) = P (N = 5) = 1/5. Moreover, let X be t

Stochastic Risk Management Models
MRM 8320, Fall 2015
Assignment 2 - due on October 6th, 4:30pm.
Earn 5 extra points by working and submitting in a group.
1. An investor is considering investing in project A or project B. Project A generates losses that
f

Introduction to Stochastic Models
AS 4320, Fall 2016
Assignment 2 - due on September 19th, 1:30PM.
1. Random variable X has probability density function (1 + 2x2 )e2x , x 0.
(a) Determine the survival function S(x).
(b) Determine the hazard rate h(x).
(c)

Introduction to Stochastic Models AS 4320
Fall 2016
(CRN 82486)
Instructor Information
Instructor: Dr. Ahmad Peivandi
Office: Room 1137, Robinson College of Business (35 Broad Street)
Office Hours: By appointment, send me an email.
Phone: 404-413-7478

Introduction to Stochastic Models AS 4320, Fall 2016
Assignment 1
Due Wednesday September 7th at 1:30 PM
1. Random variable N is distributed as follows:
P (N = n)
N
0.1
0
0.1
1
0.2
2
0.2
3
0.4
4
(a) Calculate the variance of N.
(b) Calculate the CDF of N.

2
PA R A M E T R I C S E V E R I T Y D I S T R I B U T I O N S
Mathematical models are created so that they can represent the information that is available.
Some of the information can be input in the model through parameters. According to the
type of inf

Introduction to Stochastic Models AS 4320
Fall 2014
(CRN 82992)
Instructor Information
Instructor: Dr. Ahmad Peivandi
Office: Room 1136, Robinson College of Business (35 Broad Street)
Office Hours: By appointment, send me an email.
Phone: 404-413-7478

1
REVIEW OF BASIC PROBABILITY CONCEPTS
Many factors are involved in the study of insurance claims some deterministic and some
random. Random events affect both the claim payments (for instance, their size and time
of payment), and if there is actually a p

3
M O D I F I C AT I O N O F L O S S R A N D O M VA R I A B L E S
We consider now modifications of loss random variables that are motivated by the inclusion
of insurance elements as policy deductibles and policy limits.
3.1
modification due to the presenc

Stochastic Risk Management Models
MRM 8320, Fall 2016
Assignment 1 - due on Sep 13th, 4:30pm.
Email attachments and late submissions are not acceptable.
1. [15pt] Question 34 from the practice problems chapter 1.
2. [10pt] A random sample of observations

A
REVIEW OF PROBABILITY
1. The time to death from when insurance was purchased follows a normal distribution
with mean 40 and variance 10. Assuming that a person keeps his insurance until death,
calculate the probability of keeping the insurance for 20 to

H
DISCRETE DISTRIBUTIONS.
1. [From the solution of the SOA Exam MLC May 2007] The exponential inter-arrival
times with mean time between arrivals is equivalent to arrivals following a Poisson
process with a mean of 1 per unit time. We are given that the a

Solution of Sample Exercises
Chapter 2 - Continuous distributions
2.1
Creating distributions
1. Let X be the losses in year 1, then 1.1X are the losses in year 2. Hence,
+
P (1.1X > 2.2) = P (X > 2) =
2
3
1
dx = 3
4
x
x
+
= 0.125.
2
2. Let X be the losses

Solution of Sample Exercises
Chapter 3 - Discrete distributions
1. [From the solution of the SOA Exam MLC May 2007] The exponential interarrival times with
mean time between arrivals is equivalent to arrivals following a Poisson process with a mean of
1

Solution of Midterm Exam - Stochastic Risk Management Models
MRM 8320, Fall 2013
1. This question provides more information than needed. Let N be the number of losses due to
earthquakes per year, and let X be the loss severity random variable. Then, N Poi

Chapter 4
Aggregate loss models - Compound
distributions
4.1
Mean, variances and PGFs
1. [Cox and Pai, p.116, Ex.10] Physicians volunteer their time on a daily basis to provide care to those who are not eligible to obtain care otherwise. The number of phy

Chapter 3
Discrete distributions.
1. Heart/Lung transplant claims in 2008 have inter-arrival times that are independent
with a common distribution which is exponential with mean one month. As of the
end of January 2008 no transplant claims have arrived. C

Chapter 2
Continuous distributions
2.1
Creating distributions
1. [Cox and Pai, p.159, Ex.4] Losses in year 1 follow the density function f (x) = 3x4 ,
x 1, where x are losses in millions of dollars. Ination of 10% impacts all claims
uniformly from year 1

Chapter 1
Review of probability
1.1
Key functions and moments
1. The time to death from when insurance was purchased follows a normal distribution
with mean 40 and variance 10. Assuming that a person keeps his insurance until
death, calculate the probabil

C
DISCRETE DISTRIBUTIONS.
1. Heart/Lung transplant claims this year have inter-arrival times that are independent
with a common distribution which is exponential with mean one month. As of the
end of January no transplant claims have arrived. Calculate th

F
REVIEW OF PROBABILITY
1. Let T be the random variable that represents the time to death from when insurance
was purchased. Then, T Normal(40, 10). Thus, if we define Z = T40 , then Z
10
Normal(0, 1) (Z follows a standard normal distribution). Therefore

Stochastic Risk Management Models
MRM 8320, Fall 2016
Assignment 2 - due on October 4th, 4:30pm.
1. [35pt] The future life time of individuals with blood pressure follows a distribution with
hazard rate . The parameter is distributed uniformly in the inte

Stochastic Risk Management Models
MRM 8320, Fall 2015
Midterm Solution
1.
X P areto(, )
Y W eibull(, )
SX (x)
fX (x)
= lim
= +
x SY (x)
x fY (x)
lim
This is because, exponential term converges to zero faster. Therefore, Pareto has a heavier tail.
2.
E[X]

1
REVIEW OF BASIC PROBABILITY CONCEPTS
Many factors are involved in the study of insurance claims some deterministic and some
random. Random events affect both the claim payments (for instance, their size and time
of payment), and if there is actually a p

HEAVY-TAILED DISTRIBUTIONS AND RATING
BY
J. BEIRLANT, G. MATTHYS
AND
G. DIERCKX
University Center of Statistics, Katholieke Universiteit Leuven
ABSTRACT
In this paper we consider the problem raised in the Astin Bulletin (1999) by
Prof. Benktander at the o

Midterm Exam - Stochastic Risk Management Models
MRM 8320, Fall 2013
1. [20pt] The number of losses due to earthquakes follows a Poisson distribution. In a year, the
average number of losses of this type is 4. Moreover, the severity of earthquake losses h

Stochastic Risk Management Models
MRM 8320, Fall 2014
Assignment 2 - due on October 7th, 7:15pm.
Please note that you can only use the results that I gave you in the lecture notes (excluding the
examples) and also pdf , cdf , and raw moments from Appendix

Stochastic Risk Management Models
MRM 8320, Fall 2014
Assignment 1 - due on September 16th, 7:15pm.
1. [15pt] Losses, denoted by X, have the probability density function
x
x2 e 2
f (x) =
,
16
x 0.
(a) Calculate the coefcient of variation of X.
(b) Calcula

MRM 8320
Assignment 5, due tuesday December 9th at 7:15 pm
1
Please write all steps of your solution to get full credit.
You can submit this assignment by emailing it to the TA jlzhou.001@gmail.com, or leaving it in my
mailbox which is located in the Risk

Stochastic Risk Management Models
MRM 8320, Fall 2014
Homework 4 Solution
1. For a non-homogeneous Poisson process, the intensity function is given by:
5 if t is in (1, 2], (3, 4],
3 if t is in (0, 1], (2, 3],
(t) =
Find the expected time of the second

Model
Random Variables
Tails of Distributions
Risk Measures
Stochastic Risk Management Models
Department of Risk Management and Insurance
Georgia State University
January 18, 2016
Stochastic Risk Management Models
Model
Random Variables
Tails of Distribut

Discrete time multi-state models
Cash flows and actuarial present value
MRM 8320 - Stochastic Risk Management Models
Discrete time models: Markov chains
Department of Risk Management and Insurance
Georgia State University
Atlanta, November 18th, 2013
Mark

Solutions to Final Examination Practice Problems - Stochastic Risk Management Models
MRM 8320, Spring 2016
1.
(a) Losses denoted by X is distributed Exp(10000) also the payment in case of loss is Y =
(X 500)+ (X 2000)+ + (X 3000)+ . Therefore,
E[Y ] = E[X

Poisson processes
Poisson process
Inter-arrival times
Non-homogeneous Poisson processes
A continuous-time model
MRM 8320 - Stochastic Risk Management Models
Continuous time models - Poisson process
Department of Risk Management and Insurance
Georgia State

Final Examination Practice Problems - Stochastic Risk Management Models
MRM 8320, Spring 2016
1. Losses follow and exponential distribution with mean 10000. An insurance policy has a deductible
of 500 and does not cover for the portion of the loss between