R语言代写 ACTL2102 Foundations of Actuarial Models ACTL5103 Stochastic Modelling for Actuaries S2 2018 Assignment

ACTL2102 Foundations of Actuarial Models ACTL5103 Stochastic Modelling for Actuaries S2 2018 Assignment

Due dates and times:
􏰈 5 October 2018, 5:00pm (sharp).

1 Learning outcomes

This assignment aims to develop the the Business Undergraduate Program Learning Outcomes: 1 Business knowledge, 2 Problem solving, and 3 Business communication, and the following course learning outcomes:

(1) Describe and explain concepts and principles of actuarial modelling.

6. (6)  Develop an understanding of Markov processes that can be used for insurance, survival, sickness

   and 􏰊nancial modelling

7. (7)  Develop an understanding of the main concepts of Monte Carlo simulation of a stochastic process

   and a capability to carry out simple simulation procedures

8. (8)  Explain the concepts of Monte Carlo simulation of a stochastic process using a series of pseudo-

   random numbers and apply simulation to simple actuarial problems.

11. (11)  Express his/her views on, and understanding of, an aspect of statistical models.
12. (12)  Develop communication skills for the presentation of complex statistical models in written report

    form.

13. (13)  Construct written work which is logically and professionally presented.

It aims at giving students an opportunity to apply the knowledge of stochastic process and simulation skills they learn in the context of predicting insurance losses. It requires development of simulation algorithms, and implementation of the simulation algorithms with the software R.

This assignment is an individual assignment.

2 Background

You are an actuarial analyst at the insurance company AL Insurance. Your team is required to estimate the number of claims in the next four quarters for a home insurance product. This product provides cover for damages to the buildings and contents, including those from natural catastrophes.

Your manager has provided you with two models, and you are required to estimate the claim counts under the assumptions of each model. Furthermore, you are required to report your results along with recommendations on the appropriateness of each model. Your manager has also provided you with the following data to assist with your analysis.

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Data

The numbers of claims and policies in force in the past 12 months are given in the following table.

Month Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Number of poli- 15 20 20 25 30 30 35 40 50 50 50 60 cies in force (in
thousands)

Number of 290 382 270 352 392 307 326 442 732 759 912 1203 claims observed

2.1 Model I – a homogeneous Poisson process

The arrival of insurance claims is assumed to be a homogeneous Poisson process. The intensity is 17.44 claims per day.

2.2 Model II – an inhomogeneous Poisson process

The arrival of insurance claims is assumed to be an inhomogeneous Poisson process. The intensity depends on the month in which the claim occurs. Let us denote the intensity function by λ(t) which is speci􏰊ed such that

11.02 per day, during March, April, and May;
11.68 per day, during June, July, and August;
26.41 per day, during September, October, and November; 20.83 per day, during December, January, February.

3 Assignment

Your manager has required you to perform the three following tasks and provide the results in a report. You are also required to provide the R codes that you used for the analyses in the appendix of the report.

3.1 Task I: Simulating the number of claims

You are required to simulate the number of claims in the next 12 months under the assumptions of each model. You should perform an adequate number of simulations under the assumptions of each model and report:

􏰈 three trajectories of the counting process of claims counts;
􏰈 the distributions of total claims counts in the next 12 months;
􏰈 statistics on the simulated claims counts that could assist with your analysis.

For the simulation of a non-homogeneous Poisson process under the assumption of Model II, you can either create your own algorithm or adopt an existing algorithm. If an existing algorithm is used, reference details need to be provided. One available algorithm for this simulation is provided to you in Section 3.6, however you are not restricted to use this algorithm. You are also provided R codes (see Section 3.7) that can be used to simulate a homogeneous Poisson process, and you are welcome to develop your own algorithm and codes.

λ(t) =

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3.2 Task II: Commenting on the assumptions of both models

You are required to comment on the appropriateness of the two models. You should explain the advantages and disadvantages of each model. Your analysis should consider the background of your work, the data you are given, and the theoretical properties of the models.

3.3 Task III: Providing recommendations on the model assumptions

You should recommend an alternative model that is more appropriate in estimating the total number of claims in the next 12 months. You should justify the assumptions of your model and provide numerical values for the parameters.

3.4 Requirements

You must perform the analysis with the software R, which is used as the standard software for numerical analysis in your company. You are required to write a report to your manager. The following are the requirements of the report:

• 􏰈  You should create a business report to your manager. Your report should include (at least)
  􏰈 an executive summary;
  􏰈 the results for each of the three tasks. You do not need to present the results in the same

  order as the tasks.

• 􏰈  The main body of the report should be no more than 3 pages. Cover page, appendix and

  reference list are not counted towards the page limit. There is no speci􏰊c formatting requirement;

  however, you should ensure that the report is professional in a business context.

• 􏰈  You must include an appendix of R codes such that your colleagues from other departments can

  replicate your results. Please see Section 3.5 for more details.

3.5 R code

You will also have to submit your full R code (which must be included as text in an appendix, not as an R 􏰊le, so that it is also checked by Turnitin). Your audience are your colleagues who may need to replicate your results. Your codes must run without further modi􏰊cation (this does not need to include the installation of packages) 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. You will also have to appropriately document your code so that a third party can readily use it.1 For example, you should at least include some comments in your codes (by using the # sign in R) to guide your audience. See the webpage of R markdown 2 for example.

3.6 One algorithm to simulate a non-homogeneous Poisson process The following algorithm is adapted from Algorithm 1 of Lewis and Shedler (1979) 3

Suppose that you would like to generate a non-homogeneous Poisson process {N(t),t ≥ 0} with an intensity function λ(t) (0 ≤ t ≤ T). Denote by λ∗ the maximum of λ(t) over 0 ≤ t ≤ T, that is, maxt∈[0,T ] λ(t) = λ∗.

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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.

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􏰆Simulation of nonhomogeneous Poisson processes by thinning􏰇 by Lewis, PA W and Shedler, Gerald S: Naval research logistics quarterly 26(3): 403􏰋413, Wiley Online Library

https://rmarkdown.rstudio.com

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Step 1 Simulate a homogeneous Poisson process, {N∗,0 ≤ t ≤ T}, with a constant intensity of λ∗ over the time period [0,T]. If the number of points is 0, then there are no points in the process {N (t), t ≥ 0}; otherwise, proceed to the next step.

Step 2 Denote the ordered statistics of the simulated points of {N∗,0 ≤ t ≤ T} by X1∗,…,Xn∗∗ where n∗ isthenumberofpointsof{N∗,0≤t≤T}andXi∗ isthearrivaltimeoftheith point(1≤i≤n∗).

Step 3 Initialise a variable k = 0 and i = 1
Step 4 Generate a continuous uniform random variable, U, between 0 and 1 Step5 IfU≤λ(Xk∗)/λ∗,thensetkequaltok+1andXk =Xi
Step 6 Set i equal to i+1
Step7 Ifi+1≤n∗,thenstartfromStart4again
Step 8 Return X1,X2,…,Xn where n = k.

This algorithm is based on the thinning property of a Poisson process. One can refer to Lewis and Shedler (1979) for details if interested.

3.7 Sample R codes to simulate a homogeneous Poisson process

  # input:
  # you need to adjust the input values
  # lambda - the intensity of the homogeneous Poisson process
  # T_length - the length of time horizon for the simulation

  # output:
  # arrival_time - the collection of all arrival times
  # last_arrival - the last arrival time

  # the intensity of the homogeneous Poisson process
  lambda <- 0.5

  # the length of time horizon for the simulation
  T_length <- 5

  # initial the most recent arrival time
  last_arrival <- 0

  # initial the collection of all arrival times
  arrival_time <- c()

  # simulate the first inter-arrival time
  inter_arrival <- rexp(1, rate = lambda)

  # simulate the inter-arrival times
  # of the homogeneous Poisson process
  while (inter_arrival + last_arrival < T_length) {
  last_arrival <- inter_arrival + last_arrival
  arrival_time <- c(arrival_time,last_arrival)
  inter_arrival <- rexp(1, rate = lambda)
  }

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3.8 Additional support

We recommend students to seek feedback from the EDU 4(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􏰇.

4 Assignment submission procedure

4.1 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 an appendix, 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. 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-speci􏰊ed sub- mission 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 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 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 marked􏰈and this only if a valid reason is established, and at the discretion of the LIC.

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

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https://www.business.unsw.edu.au/students/resources/learning-support/resources/ resources-on-moodle

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must be their own. 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 di􏰉erent 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 students.

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 di􏰉erent 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?5

5 Assessment criteria Please refer to the following table.

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Yes, that’s right, just sending it, even if the the third party promises not to copy, is already plagiarism in the UNSW policy!

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