ACTL 3162 / ACTL 5106 2022 Term 3
Assignment
Due: 23:59 October 30, 2022
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
2.1 Task 1 [40 marks]
As an actuarial analyst for a general insurer, you are requested to prepare an analysis for estimating
the form of the accident severity distribution of a recent liability insurance product to the market.
1,000 claims were over the last year.
Data: The claims amounts are stored in Loss.csv.
Your task is to use Maximum Likelihood Estimation (MLE) to fit an appropriate accident severity
distribution for individual claims. You are required to fit the Log-normal, Gamma, Pareto, and
Sum of Two Exponentials distributions to the claims data and use appropriate goodness-of-fit tests
to decide and subsequently justify which of the four distributions is the most appropriate to use for
modelling the claim severity distribution. You may wish to further support your conclusions via
graphical approaches. You must briefly describe your methodology in reaching your MLE estimates
of your parameters.
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
2.2 Task 2. [60 marks]
One of your duties is to ensure that the company satisfies the capital requirements from the regula-
tor, i.e. the probability of ruin within one year is no more than 0.005 (1 in 200 years event). Based
on the recent experience, you believe that a Gamma distribution with shape α = 3 and rate b = 0.5
describes the individual claims sufficiently well. In addition, you believe that the claim arrival is a
Poisson Process with parameter λ = 1 per month. Therefore, the surplus of the company at time
t (measured in months) can be described as
Ct = c0 + pit?
Nt∑
i=1
Xi, t ≥ 0, (1)
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where c0 is the initial surplus at time 0, Xi ～ Gamma(a = 3, b = 0.5) is the i-th claim amount and
pi is the constant rate of premium income paid continuously, and Nt is the value of Poisson process
at time t.
Let ψ(c0) denote the ultimate probability that ruin occurs within time t with initial surplus c0,
i.e. Pr(mins≤tCs < 0). For the efficient use of capital, you want to determine the minimum
capital required to stay solvent. Specifically, you need to ensure that the 1 year survival probability
is at least 99.5% and the 5 year survival probability is at least 99%, i.e. ψ12(c0) ≤ 0.005 and
ψ60(c0) ≤ 0.01. The insurer’s premium is paid continuously at a constant rate pi and is calculated
so that the relative security loading is 30%.
1. Without reinsurance:
(a) With the initial surplus c0 = 35, simulate the ruin probability within 5 years (that is,
the surplus process level falls below zero within 5 year. You can check the minimum of
the process in the simulated surplus for 5 years.)
(b) Find the adjustment coefficient associated with this surplus process and the upper bound
for the probability of ruin.
2. With reinsurance:
The insurer considers to purchase either
(A) a proportional reinsurance from another reinsurance company which charges a premium
loading factor of 50% and the direct insurer retains α = 0.6 of each claim or
(B) an excess of loss (EoL) reinsurance with a limit d = 6 and the reinsurance company
charges a premium loading factor of 50% for this EoL reinsurance.
For the above reinsurance products (A) and (B), perform the following analysis.
(a) With the initial surplus c0 = 35, find the approximated ruin probabilities within 5 years.
(b) To avoid that ultimate ruin is certain, the insurer’s net of reinsurance premium income
per unit time must be larger than the expected aggregate claims per unit time. Find
the range of α in (A) and d in (B) respectively.
(c) Consider a more conservative risk management policy that the ultimate ruin probability
is bound by 1%. This can be achieved by purchasing reinsurance (either (A) or (B)).
Recommend your option with an actuarial analysis. Would you support this policy?
3 Required document
You are asked to provide a report and R code. There will be THREE submission boxes (two
business reports; one for Task 1 and one for Task 2, R code for Task 1 and Task 2) on Moodle.
The report should provide results for all of the above two tasks in word or pdf format. You
do not need to provide a table of contents in your report. and should think of a clear and
effective structure for your responses.
– For Task 1, the main body of the report should be no more than 3 pages (i.e. maximum
3).
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– For Task 2, the main body of the report should be no more than 3 pages (i.e. maximum
3).
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 page limit. No page limit for the
appendix. There is no specific formatting requirement; however, you should ensure that the
report is professional in the business context.
Intermediate steps for questions involving any form of derivation are required. Your comments
and conclusions should be well justified and charts should 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 fitting are available
on the course website which may be of use. In addition, feel free to find 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 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 file.
Your R codes (or codes of other software) should be included in the separate file. The marker
will choose a portion of the reports to check the code. He/she will need to copy the code, run
it and check whether it is correct, implementable and consistent with the output presented
in your answer. Students will risk failing the assignment if the 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 figures 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
Your assignment must be uploaded as a unique document (either pdf or word document) and all
parts must be in portrait format. The R code must be provided as a separate file, in a format
that we can copy and paste to check it. As long as the 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 for two business report and R
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
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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-specified
submission date and time and was submitted late, the School of Risk and Actuarial
Studies will apply the following policy. The late submission will incur a penalty of 5% per
day or part thereof (including weekends) from the due date and time. The submission will not
be accepted after 5 days (120 hours) of the original deadline unless special consideration has been
approved. (e.g., 0 day 1 minute = 5% penalty, 2 days 21 hours = 15% penalty, 5 days 1 minute
= 100% penalty). The submission time will be based on Moodle’s record for the purposes of
calculating the penalty.
You need to check your document once it is submitted (check it on-screen). 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 then consider either
submitting their assignment from the university computer rooms or allow enough time (at least
24 hours is recommended) between their submission and the due time. No paper copy
(e.g. scanned hand writings) will be either accepted or graded.
In case of a technical problem, the full documentation must be submitted to the Lecturer 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 is only if a
valid reason is established (and the Lecturer 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 as-
sessment must be their own. In particular, this means that any code you present are developed
yourself from your own computer, 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 different 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 student outside your group.
Note however that you are allowed to use any R code that was made available during the course
(either with 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 probability of being discovered. For issues of collective work, having different 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
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to avoid any issue. For instance, did you know that sharing any part of your work with other
students (outside your group) before the deadline is already considered as plagiarism? 1