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.
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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.
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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.
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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 ().
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 ()
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.
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.
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?