ACTL3162 General Insurance Techniques
ACTL5106 Insurance Risk Models
Assignment, 2021 T3
Due date: 1 November 2021 (Monday of Week 8) at 5pm sharp
1 Learning outcomes
The assignment aims at developing the course learning outcomes in relation to those stated in the course
outline. It also assesses the program learning outcomes �Knowledge�, �Problem solving and critical thinking�,
as well as �Communication�. You are expected to demonstrate your ability to analyse an actuarial problem,
apply appropriate theories and logic to interpret the problem, and develop solutions and conclusions. The
communication of those will also be assessed.
2 Two tasks
Task 1. [40 marks]
Your task is to use Maximum Likelihood Estimation (MLE) to �t an appropriate accident severity distribution
for individual claims. The claims data are stored in Loss.csv. You are required to �t the Log-normal,
Gamma, Burr, and Mixture of Two Exponentials distributions to the claims data, and use appropriate
goodness-of-�t tests to decide and subsequently justify which of the four distributions is the most appropriate
for modelling the claim severity distribution. You may wish to further support your conclusions via graphical
approaches.
You must brie�y describe your methodology in reaching the MLE estimates of the parameters. Although
you may choose to include mathematical formulas if you think it helps your explanations, providing detailed
mathematical formulas is not necessary. Your entire R code (or the code of any other software used) should be
submitted as a separate �le (see Sections 3 and 4 regarding required documents and submission procedure).
Note that the probability density function of the Mixture of Two Exponentials distribution, which has three
parameters 0 < p < 1, α > 0, and β > 0, is given by
f(x) = pαe−αx + (1− p)βe−βx, x > 0.
Task 2. [60 marks]
One of your duties is to ensure that an insurance company satis�es solvency requirements, i.e., the probability
of ruin must be managed well. Based on the recent experiences, you believe that the Pareto distribution with
density function
fS(x) =
α
θ
(
θ
θ + x
)α+1
, x ≥ 0,
where α = 1.7 and θ = 70, describes the individual claims su�ciently well. In addition, you believe that claims
arrive according to a Poisson process with rate λ = 20 per month. Therefore, the surplus of the company at
time t (measured in months) can be described as
Ct = c0 + πt−
Nt∑
i=1
Si, t ≥ 0, (1)
where c0 = 30000 is the initial surplus at time 0, π = 2600 is the constant rate of premium income, Si ∼
Pareto (α = 1.7, θ = 70) is the i-th claim amount, and Nt is the value of Poisson process at time t. (The usual
assumptions for the risk process are made, which include: (i) the claim amounts Si’s are independent and
identically distributed (i.i.d.); and (ii) the Poisson process is independent of the claim amounts.) De�ne
ψ(c0) = P
(
min
t≥0
Ct < 0 ∣∣∣∣C0 = c0 ) , (2) 1 which is the ultimate ruin probability given the initial surplus of c0. In this task, you will explore ψ(c0). You are given the following: 1. The ruin probability (with initial surplus x) can be expressed as ψ(x) = P(L > x), (3)
where
L = max
t≥0
(
Nt∑
i=1
Si − πt
)
is the maximum of the net loss process.
2. For notational convenience, we introduce
d
= to indicate equality in distribution. It can be shown that
L
d
=
M∑
i=1
Zi, (4)
where
• M is a Geometric variable with probability mass function
P (M = k) =
(
1
1 + ρ
)k
ρ
1 + ρ
, k = 0, 1, 2, . . .
• ρ is the pro�t loading given by
ρ =
π
E
[∑N1
i=1 Si
] − 1. (5)
• Zi’s (i = 1, 2, . . .) form a sequence of i.i.d. variables with density function
fZ(z) =
P(S1 > z)
E[S1]
. (6)
Please answer the following questions:
a) Explain why it is di�cult to simulate the ruin probability directly via simulating sample paths with (1).
b) Show that formula (3) is consistent with the original de�nition of the ruin probability in (2).
c) Consider the running maximum of the net loss at time t, i.e.,
Bt = max
0≤v≤t
(
Nv∑
i=1
Si − πv
)
.
(i) Explain why {Bt : t ≥ 0} is an increasing step function of t.
(ii) What is the limit of Bt as t→∞? Explain intuitively why this limit is �nite (with probability 1).
[Hint: relate the jump times and sizes of Bt with the original surplus process.]
(iii) Explain intuitively why the distribution of L can be written in the form of (4). (You do not need to
explain why ρ and fZ(z) are exactly given by (5) and (6).)
d) (i) Based on the above information, explain a procedure (with formulas where appropriate) how you will
simulate the ultimate ruin probability ψ(c0) e�ectively.
(ii) Give an approximate value of ψ(c0) via simulations. Also provide a 99.9% con�dence interval and
explain how such an interval is obtained.
(iii) Comment on you answer in (ii).
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3 Required documents and suggested page limit
You are asked to provide reports for the tasks and R code (or the code of another software you use). There
will be THREE submission boxes in Moodle: (i) Business report for Task 1; (ii) Business report
for Task 2; and (iii) one single word or pdf �le containing the programming code for both tasks.
• Your responses for each of the two tasks should be provided in a word or pdf �le in the
form of a report. The two reports should be submitted as two separate �les. You do not
need to provide a table of contents in your report. You should think of a clear and e�ective structure
for your responses. The following page limits are suggested.
� For Task 1, the main body of the report is expected to be no more than 3 pages.
� For Task 2, the main body of the report is expected to be no more than 4 pages.
You need to provide a reference list if any references are used in your report. Cover pages, appendices
and reference lists are not counted towards the suggested page limits. There is no speci�c formatting
requirement (except it should be portrait); however, you should ensure that the report is professional in
the business context. While the above page limits are not strict requirements, it is important to note
that an excessively long report may be an indication that the student fails to communicate the key ideas
concisely. Grading is based on the quality of the responses, not the length of the responses. You may
be able to write short and excellent reports!
• Intermediate steps for questions involving any form of derivation are required. Your comments and
conclusions should be well justi�ed and charts/graphs can be used to support your conclusions where
applicable.
• You are strongly recommended to use the software R for programming, although the use of
other software will also be accepted. Some sample R codes for �tting are available on the course website
which may be of use. In addition, feel free �nd packages online to perform your computations (but
always check that your answer is sensible!).
• When making a comment or conclusion based on R outputs (or other software outputs), you should
include the relevant outputs (such as graphs, tables etc) in the main body of your report. You should
make sure that the marker can read and understand your arguments and statements without referring
to the separate R code �le.
• Your R code (or code of another software) should be submitted as a separate word or pdf
document. This �le should contain the code for BOTH tasks. Your programming code will
NOT be graded. The marker will choose a portion of the reports and check the code. He/she may
copy and paste the code, run it and check whether it is implementable and consistent with the output
presented in your answer. Students will risk failing the assignment if no code is submitted, or
the submitted code cannot be run, or the output provided in the answer is not consistent
with the output generated by the code.
• You should NOT
� Include a chunk of programming codes in the main body of your report
� Have �gures or tables that are not referred to or analysed in the main body of your report
� Include materials that are not highly relevant in the main body of your report
4 Assignment submission procedure
4.1 Report and R code: Turnitin submission through Moodle
Each of your reports must be uploaded as a word or pdf document, and all parts must be in portrait format.
The R code must be provided as a separate word or pdf �le, in a format that we can check. As long as the
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due date is still future, you can resubmit your work; the previous version of your assignment will be replaced
by the new version.
Assignments must be submitted via the Turnitin submission box that is available on the course Moodle
website. There are THREE submission boxes: two for business reports and one for your code
separately. Turnitin reports on any similarities between their own cohort’s assignments, and also with regard
to other sources (such as the internet or all assignments submitted all around the world via Turnitin). More
information is available at: [click]. Please read this page, as we will assume that you are familiar with its
content.
Please note that when an assessment item had to be submitted by a pre-speci�ed submission
date and time and was submitted late, the School of Risk and Actuarial Studies will apply the fol-
lowing policy. A penalty of 25% of the mark the student would otherwise have obtained, for each full (or part)
day of lateness (e.g., 0 day 1 minute = 25% penalty, 2 days 21 hours = 75% penalty). Students who are late
must submit their assessment item to the Lecturer-in-Charge (�LIC�) via e-mail (eric. .au).
The LIC will then upload documents to the relevant submission boxes. The date and time of reception of the
e-mail determines the submission time for the purposes of calculating the penalty.
You should check your document(s) once they are submitted (check it on-screen) in case you have mistakenly
submitted an earlier version of your work or you have unknowingly faced internet interruption which may
result in a corrupt/incomplete �le. We will not mark assignments that cannot be read on screen.
Students are reminded of the risk that technical issues may delay or even prevent their submission (such as
internet connection and/or computer breakdowns). Students should allow enough time (at least 24 hours
is recommended) between their submission and the due time so that you have time to solve issues
that may arise and �nd alternative internet connection or computer. The Turnitin module will not let you
submit a late report. No paper copy will be either accepted or graded.
In case of a technical problem, the full documents must be submitted to the LIC (eric. .au)
before the due time by e-mail, with explanations about why the student was not able to submit on time. In
principle, this assignment will not be marked. It is only in exceptional circumstances where the assignment was
submitted before the due time by e-mail that it may be marked�and this only if a valid reason is established
(and the LIC has the discretion in deciding whether a given reason is valid).
4.2 Plagiarism awareness
Students are reminded that the work they submit must be their own. While we have no problem with students
discussing assignment problems if they wish, the material students submit for assessment must be their own. In
particular, this means that any code you present are from your own computer, which you yourself developed,
without any reference to any other student’s work.
While some small elements of code are likely to be similar, big patches of identical code (even with di�erent
variable names, layout, or comments�Turnitin picks this up) will be considered as plagiarism. The best
strategy to avoid any problem is not to share bits and pieces of code with other students.
Note however that you are allowed to use any R code that was made available during the course (e.g. from
the lectures or developed in the tutorial exercises). You don’t need to reference them formally, and this will
not be considered as plagiarism.
Students should make sure they understand what plagiarism is�cases of plagiarism have a very high proba-
bility of being discovered. For issues of collective work, having di�erent persons marking the assignment does
not decrease this probability. For more information on plagiarism, see [click].
Students should consult the �Write well; Learn deeply” website and consult the resources provided there. In
particular, all students should do the quiz about plagiarism to make sure they know how to avoid any issue.
For instance, did you know that sharing any part of your work with other students (outside your group for
group project) before the deadline is already considered as plagiarism? 1
1Yes, that’s right, just sending it, even if the third party promises not to copy, is already plagiarism in the UNSW policy!
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https://student.unsw.edu.au/turnitin
https://www.business.unsw.edu.au/degrees-courses/course-outlines/policies
5 Assessment criteria
Please see the �le, �Rubric”.
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Learning outcomes
Two tasks
Required documents and suggested page limit
Assignment submission procedure
Report and R code: Turnitin submission through Moodle
Plagiarism awareness
Assessment criteria