代写ACTL 3162、代写R语音程序

日期：2022-10-31 09:10

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