- June 23, 2020

ACTL2102 Foundations of Actuarial Models ACTL5103 Stochastic Modelling for Actuaries Term 2 2020 Assignment 1 Background You are an actuarial analyst at Skippie Insurance. Your company offers a Bonus-Malus System (BMS) car insurance scheme with 10,000 policyholders in the current policy year. 5,000 policyholders are males aged 20 to 25, and the remaining 5,000 are males aged 50 to 55. The base premium charged to policyholders is $300 per annum for males aged 50-55 and $400 per annum for males aged 20-25. For simplicity, you are to assume that the individual claim cost is $1,000 for both groups. Details of the current scheme (which applies to both groups of policyholders) is as follows: There are 5 levels of discount: Table 1: Existing scheme Level -2 -1 0 1 2 Discount (%) -20 -10 0 10 20 New policyholders enter the scheme on discount Level 0. If a policyholder has one year with no claims, he/she moves up to the next higher level of discount in the following year unless he/she is already on the highest level (Level 2). If a policyholder has one claim during a year, he/she moves down to the next lower level of discount, and with more than two claims (≥ 2), he/she moves down 2 levels unless he/she is already on the lowest level (Level -2). Your company has been modeling transitions for all policyholders in the current portfolio using a discrete-time stationary Markov chain. Your manager is currently considering two alternatives to this scheme: Option 1: Cut costs by simplifying policy administration and reducing the number of discount levels to three: Table 2: Simplified scheme: applies to all policyholders Level -1 0 1 Discount (%) -10 0 10 The transition rules are the same as before; new policyholders start at discount level 0. After a year with no claim, policyholders move up a discount level. If they have one claim in a year, they move down one level (or stay at level -1), and if they have more than two claims in a year they move down to the lowest level (-1) in the next year. Option 2: Use a different discount scheme for both levels. Your manager believes that the claims experience for the younger policyholders is much more variable than for older policyholders. Therefore, he proposes a new discount scheme only for the younger group of policyholders: Table 3: More complex scheme: applies only to younger policyholders Level -3 -2 -1 0 1 2 3 Discount (%) -30 -20 -10 0 10 20 30 The same transition rules apply (more than 2 claims = move down 2 levels, 1 claim = move down 1 level, no claims = move up one level). The current discount scheme in Table 1 will still apply to older policyholders. 1 2 Three tasks You have been asked by your manager to perform analyses on the current scheme. In particular, you have been asked to do the following: Task 1: 1. Determine the probability transition matrices for the three different schemes, for each group of policyholder. Your manager wants you to assume that the number of claims per year follows Poisson distribution with parameters λold = 0.15 and λyoung = 0.25. 2. For the current scheme only (see Table 1), simulate the number of policyholders in each of the discount levels for the year of 2021 (i.e. next year), 1000 times. Describe the distribution of next year’s gross premium (provide summary statistics). Task 2: 1. Plot the Loimaranta efficiency of the three different schemes as a function of λ, the claim frequency and explain your results. 2. Compare the long-run profitability. 3. Recommend which system should be used and discuss the pros and cons of your choice (justification does not need to be limited to the above findings). In order to determine the efficiency of the scheme, your manager wants you to calculate the long-run proportions in each discount level, and the Loimaranta effiency for each scheme, defined as η(λ) = dP (λ) P (λ) dλ λ = λ P (λ) P ′(λ) where λ is the claim frequency and P (λ) is the mean premium charged for a policyholder with claim frequency λ. You can search more information regarding Loimaranta efficiency in Google. In this context, the mean premium term P (λ) refers to the average premium that a policyholder would be charged by the scheme, if the distribution of the number of the claims per year for that particular policyholder is Poisson with parameter λ. Therefore, P (λ) = Σni=1pii(λ)ci where ci is the premium charged in discount level i and pii(λ) is the long-run probability of being in discount level i. Note that the Loimaranta efficiency η(λ) can now be approximated by using the forward difference approximation for P ′(λ), P ′(λ) ≈ P (λ+ ∆λ)− P (λ) ∆λ In order to get an accurate plot for η(λ), your manager recommends evaluating P (λ) at many evenly spaced points. 3 Data Your manager is providing you with a data extract of their policyholder data, claimsdata.csv. This file contains the age of the policyholder, the number of claims made in 2019 and BMS levels for each policyholder in 2019 and 2020. It serves as an example of what happens in an NCD scheme but you would just be working with the age and ncdlevel20 columns. 4 Required document Your manager has asked you to perform your analyses in R, which is the standard software used for analyses in your company. You are asked to provide a business report and R code to your manager. Following are the requirements of the report: The report should have an executive summary and provide results for all of the above two tasks. You do not need to provide a table of contents in your report. 2 The main body of the report should be no more than 4 pages (i.e. maximum 4). 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. You must prepare a separate word or pdf document for R codes (not as an R file so that it can be checked by Turnitin). Your codes need to be well presented with sufficient guidelines such that your colleagues from other departments can replicate your results. Your codes must run without further modification by just copying and pasting from your submitted document, otherwise no mark will be awarded for those criteria that relate to R; note that we will check all codes. 1 For example, you should at least include some comments in your codes (by using the # sign in R) to guide your audience. 5 Assignment submission procedure 5.1 Business report and R code: Turnitin submission through Moodle Your assignment must be uploaded as a unique word or pdf 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 – we will check all codes. 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. You must have a cover page with your name and student number. Assignments must be submitted via the Turnitin submission box that is available on the course Moodle website. There are two submission boxes for business report and R code separately. Turnitin reports on any similarities between the student’s 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: https://student.unsw.edu.au/turnitin. 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. 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 ([email protected]). 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 your LIC ([email protected]) 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 at the discretion of the LIC. 5.2 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. 1 There is plenty of documentation on the web about how to do this. This link could get you started: http://www.wikihow. com/Write-Software-Documentation. 3 In particular, this means that any R 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 with other students performing the same task, big patches of identical code (even with different 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 students. 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 different persons marking the assignment does not decrease this probability. 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 before the deadline is already considered as plagiarism? 2 2 Yes, that’s right, just sending it, even if the the third party promises not to copy, is already plagiarism in the UNSW policy! 4