Operational Risk and Insurance Analytics Page 1
In Course Assessment:
STAT3022/G022/M022
This assessment counts for 10% of your nal mark for STAT3022/G022/M022.
You should work in groups of no more than 3 students. You may work on the project
alone if you wish, but note that this is not e cient. It is up to you to form. your
own groups.
I will request the student ID number of each group member on Tuesday 6th March
at the end of the lecture.
You may use R, Matlab, or any other statistical software of your choice in order to
solve some parts of the questions in this assessment.
Non-submission of this assessment may mean that your overall examination mark
is recorded as\non-complete", i.e., you may not obtain a pass mark for the course.
The report should be uploaded to Moodle, and o cially ‘submitted’ by 4pm on
Thursday, 22nd March.
Each student must provide a statement of their contribution to the project signed
with the submission.
One member of every group should upload the nal version of the report, using the
relevant submission link on Moodle (which will appear closer to the deadline), and
press ‘submit’. This will require you to read and accept a plagarism statement.
All remaining members of the group should their press their ‘submit’ button; again
this will require each student to read and accept a plagarism statement.
All groups will be able to see which of their members are yet to press ‘submit’.
The report won’t be considered submitted until every member has o cially
submitted the report, and this needs to be complete by the deadline above.
Please note: Each group should submit a single pdf le and a single R or Matlab
script.
The R or Matlab script. will include your code and comments which may also refer-
ence to which question the section of code relates.
The pdf le will consist of your answers to the non-coding parts of the questions
and the required output plots.
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How will the report be marked?
Your report will be marked out of 100, with 30 percent of the marks allocated for the pre-
sentation of the report (including clarity of expression and presentation of proofs, gures,
plots), and 70 percent of the marks available for the analyses and interpretation.
I will be looking for:
Clarity of writing the expressions, proofs etc;
(Please remember you Do Not have to type all your report.)
Ability to select the most important points from an analysis;
Sensible use and display of plots and tables (if necessary);
Accuracy of interpretation of results etc.
Clarity and readability of computer code.
The mark you will receive is your group mark - everyone in the group will be awarded
the same mark, except in exceptional circumstances (e.g. a member of a group did not
contribute fairly to the project).
Continued
Operational Risk and Insurance Analytics Page 3
1 Compound process models under LDA framework
Question A
(a) Consider a severity model for losses in a given business unit risk type given by
Xi(t) Gamma( ; ) (1)
Write a closed-form. expression for the distribution and density function of the sum
of n-losses, considering all Xi’s are independent and identically distributed random
variable with Gamma distribution as mentioned in equation (1). [4]
(b) Write the closed-form. expression for the distribution and density function of the
compound process LDA model with Poisson frequency with intensity parameter
and severity model characterised by Gamma distribution, as in equation (1). [4]
(c) Create a Matlab or R script. to plot the distribution and density functions of the
compound process given by the Poisson-Gamma model, labelling clearly all compo-
nents of the plots. Consider a range of parameter values for the LDA model that
you may choose, and specify the choices made in the reporting of the result. [8]
(d) Write the steps you would take to simulate the annual loss distribution from this
LDA model (Poisson-Gamma) using a Monte Carlo algorithm. [6]
(e) Implement your Monte Carlo algorithm in R or Matlab for 10,000 simulated years
for the same range of parameter values for the LDA model (Poisson-Gamma) that
you chose previously (specify the choices made in the reporting of the results). Plot
histogram estimates of the annual loss distribution and density and compare them
to the exact results obtained. [8]
Question B
(a) Consider an -Stable severity model for losses in a given business unit risk type
given by
Xi(t) S ( ; ; ; 0) (2)
Write a closed-form. expression for the distribution and density function of the sum
of n-losses when = 0:5 and = 1. [4]
(b) Write a closed form. expression for the distribution and density function of the
compound process LDA model with Poisson frequency with intensity parameter
and severity model characterised by distribution in equation (2). [4]
(c) Create a Matlab or R script. to plot the distribution and density function of the
compound process given by the Poisson-Stable model, labelling clearly all compo-
nents of the plots. Consider a range of parameter values for LDA model that you
may choose, and specify the choices made in the reporting of the result. [9]
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(d) Write the steps you would take to simulate the annual loss distribution from this
LDA model (Poisson-Stable) using a Monte Carlo algorithm. [5]
(e) Implement your Monte Carlo algorithm in R or Matlab for 10,000 simulated years
for the same range of parameter values for the LDA model (Poisson-Stable) that
you chose previously (specify the choices made in the reporting of the results). Plot
histogram estimates of the annual loss distribution and density and compare them
to the exact results obtained. [8]
2 Heavy tailed loss models and quantiles
Question A
(a) Provide a detailed de nition of a generic heavy tailed severity model, with distri-
bution function F that admits a density function f, from the perspective of the
properties of the hazard function and hazard rate. Start by de ning the hazard
function and hazard rate, then explain its properties and link this to the heavy
tailedness of the distribution. Provide a worked example to illustrate this point
using one of the one or two parameter family of severity models. [10]
(b) Consider the annual loss random variable Z given by compound process LDA model
Z =
NX
n=0
Xn (3)
for frequency random variable N G(n) and independent severity random variables
that are independent identically distributed according to Xn F(x) for all losses.
Find an expression for the mean and variance of the annual loss in terms of moments
of N and Xn. In the process show that following bound holds
Var[Z] E[X2] max(E[N];Var[N]) (4)
[13]
3 Risk measure calculations and asymptotic approx-
imations
Question A
(a) De ne the VaR measure in a su ciently general form. to deal with di erent types
of supports for the loss random variable. [2]
(b) Provide a de nition for the Expected shortfall risk measure. [2]
Continued
Operational Risk and Insurance Analytics Page 5
Question B
(a) Consider Poisson-Lognormal LDA model. Provide a rst order asymptotic approx-
imation for the tail function of this model. [3]
(b) Consider the rst order asymptotic to the tail function derived in part (a). Create
an R or Matlab script. to plot this function for a range of parameters , and
that you choose. [10]
End of Paper