ACTL3162 General Insurance Techniques

Assignment

Due time: Monday 9th October 2017 5 pm

1 Learning outcomes

The assignment aims particularly at developing the course learning outcomes

associated with Aim A and Aim B. It specifically assesses the program goals

“Knowledge”, “Problem solving and critical thinking”, as well as “Communi-

cation”. 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 Assignment task

Task 1

You are an actuarial analyst for a general insurer who introduced a new

motor insurance product to the market just over one year ago. During this

time, the company has received over 1,000 claims and you now believe the

claims experience is significant enough for you to investigate the form of the

accident severity distribution. Some policy details are given below:

1. Every policy has a standard excess of $700. (A policyholder may have

multiple accidents in the year and may have reported more than one

claim. The excess is applied to each accident (or loss), not the total

loss in the year.)

2. In addition, policyholders can choose to add on an additional excess

in order to lower their premium. There are multiple levels for the

policyholder to choose from.

The claims data are stored in data.csv. This file contains claim amounts paid

by the insurer, along with the amount of any additional deductible elected

by the policyholder.

Your task is to use Maximum Likelihood Estimation (MLE) to fit an appro-

priate accident severity distribution (i.e. the total loss before deducting the

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excess) for individual claims. You are required to fit the Pareto, Log-normal

and Gamma distributions to the claims data and use appropriate goodness-

of-fit tests to decide and subsequently justify which of the three distributions

is the most appropriate to use for modelling the claim severity distribution.

In addition, you must briefly describe your methodology in reaching your

MLE estimates of your parameters. However, providing detailed mathemati-

cal 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 separately).

(Hint: The MLE estimates for parameters may not be analytically tractable

in this case! Consider using a numerical package in R to find the parameters.

Also, try various initial conditions for the optimisation in case you’re getting

errors.)

Task 2

Let Ct be the surplus of an insurer at time t, measured in months. Recall

that Ct is defined as

Ct = c0 +

t∑

i=1

(pii − Si), t = 0, 1, 2, . . . (1)

where Si ∼ Gamma(α, β) is the total loss for month i and pii = pi is the

monthly premium income, equal to the expected loss plus a loading of 20%.

This implies that the premium income is fixed month to month. Let ψt(c0)

denote the probability that ruin occurs within time t given initial surplus c0.

a) Derive an expression for ψ1(c0) in terms of the Gamma distribution func-

tion G.

b) Provide an explanation for the expression

ψ2(c0) =

∫ c0+pi

0

ψ1(c0 + pi − y)g(y;α, β)dy + 1−G(c0 + pi;α, β) (2)

where g and G are the Gamma density and distribution functions, respec-

tively.

In addition, evaluate this expression for α = 20 and β = 0.2 where c0 =

0, 10, 20, 50. (Hint: You may try to use the ‘integrate’ function in R!)

c) Explain the difficulty involved in calculating ψt(c0) for larger values of t.

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d) Suppose Si is replaced by a discrete distribution over the non-negative

integers 0, 1, 2, . . . . We call this new distribution S∗i with probability

mass function g∗ and distribution function G∗. We can now calculate the

surplus at time t as

C∗t = c0 +

t∑

i=1

(pii − S∗i ), t = 0, 1, 2, . . . (3)

Let ψ∗t (c0) denote the probability that ruin occurs within time t given

initial surplus c0 in the above model.

• Give an expression for ψ∗1(c0).

• Provide a recursive expression for ψ∗t (c0).

• Comment on the implementation of this expression in comparison to

the original case when Si was continuous.

e) Your task is now to approximate ψ2(c0) by ψ

∗

2(c0) by replacing S with S

∗

with various methods of discretisation discussed in the lecture slides of

Module 4. Use the same values of c0 as in part b).

(A) Method of rounding

(1) h = 1,m = 150

(2) h = 1,m = 300

(3) h = 5,m = 30

(4) h = 5,m = 60

(B) Method of upper and lower bounds

(1) d = 1

(2) d = 5

Above, d and h refers to the span while m refers to the number of discrete

intervals. For (B), you are required to calculate S∗ based on both the

method of upper and lower bounds.

Compare and comment on differences in your results relative to part b).

(Hint: You may try and find suitable R packages to help you do the

discretisation.)

Task 3

Let Si ∼ Gamma(α, β) where α = 5 and β = 0.8.

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a) • What is the distribution of Y5 =

∑5

i=1 Si?

• Compute the distribution function FY5(x) at x = 10, 20, 30, 40, 50.

b) Now, you will need to discretise Si and perform convolution using Panjer

recursion to approximate the distribution function of Y5.

• Evaluate this at the same values of x as in part a).

• Explain your methodology and justify any assumptions/decisions

you have made in the process of finding an adequate approximation.

(Hint: Try using the R package ‘actuar’.)

c) Compare the above results in a) and b) and comment on any differences.

Additional Instructions

• Answers are to be provided in Word or pdf format.

• 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 pro-

gramming, although the use of other software will also be accepted.

Some sample R codes for fitting are available on the course web site

which may be of use. In addition, feel free find packages online to

perform your computations (but always check that your answer is sen-

sible!).

• 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 appendix.

• Your R codes (or codes of other software) should be included in the

appendix. The marker will choose a substantial 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 assign-

ment if the code cannot be run or the output provided in

the answer is not consistent with the output generated by the

code.

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• There is no page limit. However, you should think of a clear and

effective structure for your responses. Responses provided of excessive

lengths are explicitly penalised in the communications criteria.

• You should not

– Include 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

2.1 Communication skills

We recommend students to seek feedback from the EDU (although you need

to give them time to read your report) - connect to the EDU website on

Moodle “Write well; Learn deeply”. The student enrolment key is “ASB_LTP”.

2.2 Assignment submission procedure

Your assignment must be uploaded as a unique document (either pdf or

Word document) and all parts must be in portrait format. 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. 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.

Please note that the School of Risk and Actuarial Studies will apply the

following policy on late assignments. 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 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.

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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 assign-

ment from the university computer rooms or allow enough time (at least

24 hours is recommended) 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 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 only if a valid reason is established (and the LIC has the discretion in

deciding whether a given reason is valid).

2.3 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 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.

Students should consult the “Write well; Learn deeply” website and con-

sult 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 in-

stance, did you know that sharing any part of your work with other students

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(outside your group) before the deadline is already considered as plagiarism?1

3 Assessment criteria

Please see the file, “Rubric”.

1Yes, 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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