ACTL3162 General Insurance Techniques ACTL5106 Insurance Risk Models
Assignment, 2020 T3
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]
You are an actuarial analyst for a general insurer who introduced a new liability insurance product to the market just over one year ago. During this time, the company has received 1,000 claims and you now believe the claims experience is signicant enough for you to investigate the form of the accident severity distribution.
Data: The claims amounts are stored in Loss.csv.
Your task is to use Maximum Likelihood Estimation (MLE) to t an appropriate accident severity distribution for individual claims. You are required to t the Log-normal, Gamma, Pareto, and Weibull 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 to use for modelling the claim severity distribution. You may wish to further support your conclusions via graphical approaches.
You must briey describe your methodology in reaching your MLE estimates of your parameters. Your answer should include the following steps:
• Estimate the model parameters for a given model and present the tted model.
• Evaluate the quality of the given model by using graphical approaches and performing hypothesis tests (Hint: when there is no grouping in the data, the K-S and A-D tests make more sense than the χ2-test because no arbitrary decisions need to be made. So you do not have to perform the χ2-test in this assignment.)
• Determine the model that ts best using the criteria introduced in the lectures.
However, providing detailed mathematical formulas and code snippets is not necessary (but the entire R code
or the code of other software if you are not using R must be provided in a separate pdf or word le).
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Task 2. [60 marks]
One of your duties is to ensure that the company satises the capital requirements from regulator, 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 a = 2 and rate b = 0.5 describes the individual claims suciently 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
Nt
Ct=c0+πt−Si, t≥0 (1)
i=1
where c0 is the initial surplus at time 0, Si ∼ Gamma(a = 2, b = 0.5) is the i-th claim amount and π is the
constant rate of premium income paid continuously, and Nt is the value of Poisson process at time t.
Let ψt(c0) denote the ultimate probability that ruin occurs within time t with initial surplus c0, i.e. Pr(mins≤t Cs <
0).
For ecient use of capital, you want to determine the minimum capital required to stay solvent. Specically, 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 π and is calculated so that the relative security loading is 37.5%.
1. Without reinsurance:
(a) With the initial surplus c0 = 40, nd the approximated 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 coecient 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 = 0.7 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 = 40, nd 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) Find (numerically) the direct insurer's retained proportion α ∈ [0, 1] in (A) and limit d in (B) that will maximize the adjustment coecient for the direct insurer. Then calculate the upper bound for the probability of ruin with this choice of α in (A) and limit d in (B).
3. Provide comments on the decision of purchasing these reinsurance products in regards to stability and protability of the company.
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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) in Moodle.
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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 eective structure for your responses.
For Task 1, the main body of the report should be no more than 3 pages (i.e. maximum 3). 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 specic 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 con- clusions should be well justied 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 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 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 codes (or codes of other software) should be included in the separate le. 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 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
Assignment submission procedure
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 le, 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 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.
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Please note that when an assessment item had to be submitted by a pre-specied submission date and time and was submitted late, the School of Risk and Actuarial Studies will apply the following 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 (j.k.woo@unsw.edu.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 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 recom- mended) between their submission and the due time. 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 document must be submitted to the LIC (j.k.woo@unsw.edu.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 markedand 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 dierent variable names, layout, or commentsTurnitin 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 iscases of plagiarism have a very high proba- bility of being discovered. For issues of collective work, having dierent 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) before the deadline is already considered as plagiarism? 1
5 Assessment criteria Please see the le, Rubric".
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Yes, 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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