ACTL2102 Foundations of Actuarial Models Term 2 2023 Assignment

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ACTL2102 Foundations of Actuarial Models

ACTL5103 Stochastic Modelling for Actuaries

Term 2 2023 Assignment

1 Background

Nowadays, the roles of actuaries have been increasingly expanding beyond the traditional areas of insurance and finance sectors. With strong analytical and problem solving skills, actuaries can apply their skillset to a

range of areas such as energy, environment, health, education, data analytics and government sector.

You are part of the analytics team of the soccer section of Società Sportiva Lazio (Lazio thereafter). One of the key tasks of the analytics team is to assess the availability of a group of 25 players (first starting 11 and 14 substitutes) to play the 38 matches of the Italian Serie A league. For simplicity we assume that all players can play in every position (Goalkeeper, Defender, Midfielder and Forward) and are equally skilled.  Hence, the coach randomly chooses the first starting 11.

The stingy Lazio Chairman Claudio Lotito is unwilling to spend money to sign new players, who would represent further alternatives to coach Maurizio Sarri.  For this reason, coach Sarri is very concerned about players utilization and would like to figure out how many players he can have available for every match (taking place every Sunday), so he can plan substitutions at his convenience, based on the score during the match.

In the current model the analytics team uses a discrete-time Markov Chain to model the transitions associated with player availability.  The current model assumes that if a player is available, then he can be able to train for utilization next week either in the first 11, or as substitute or just stay on the bench during the match. If a player is utilized (either within the first 11 or as substitute) there is the possibility he will get injured.  If injured, the player is unlikely to be available next week, unless he recovers.  In order to preserve the physical conditions of every player, the coach never employs the same player for two or more consecutive matches.

From a previous analysis, the team found that:

— If a player is available, he plays next match with probability 60% or remains on the bench;

— If a player is utilized, he can get injured with probability 10%, or be available next week for training; — If a player is injured, then he can recover with probability 25%.

The head of the medical team reassured coach Sarri that a player can play two matches in a row, instead of resting a week as in the previous case.  However, the medical team warned the coach that if a player is employed for two matches in a row, during the second match his probability of injury will increase to 30%, with a probability of recovery lowered to 5%.  Otherwise, everything remains unchanged as in the current model. Concerned of these warnings, coach Sarri will utilize the same player for two consecutive matches with probability 20%.

Following a bitter fight between coach Sarri and chairman Lotito, the latter promised the coach to make additional investments in the medical equipment, services and staff in order to increase the probability of recovery of each player by 50%. Therefore, he needs to decide between the following two alternatives.

— Purchase additional medical equipment (one-off investment of e3,000,000) and to hire further clinicians (orthopedic surgeons, physiotherapists and so on) for a weekly cost of e40,000.

— Enter in an agreement with a specialized clinic in orthopedic surgery for the treatment of the injured players. Each treatment costs e80,000 per week each player is injured.

2 Tasks

You have been asked by Maurizio Sarri to compare the current model for one match player utilization with the new model where he can use the same player for two matches in a row. More precisely, you need to perform the following tasks. Use only the information included in this paper. Do not overthink about the soccer rules (e.g. maximum number of substitutions, a player can only be substituted once and so on). You are not tested on these.

— Task 1:

1. Write down the transition matrices for the current model and the two-matches model, clearly speci- fying the appropriate state space in each model (hint: use a transition graph to first define the states and the possible transitions, to be defined in the transition matrix.). [6 points]

Provide the transition matrices under the two models at the end of the first half of the tournament, that is after 19 matches (hint: write down how you would calculate the transition matrix, and compute the transition probabilities therein using R). [2 points]

Assuming that at the beginning of the season all 25 players are fully available for playing, calculate the number of players available after half tournament under the two models. [1 point]

2. Suppose the investment decision is taken at the start of the tournament. Perform 1, 000 simulations in order to estimate the final cost of the two investment decisions under the current model and under the two-matches-in-a-row models. Which investment is more convenient under the two models (assume for simplicity 0% interest rate, hence no time value of money)? [4 points]

— Task 2:

1. Based on the two models (before the additional medical investments) do you think coach Sarri should use a players for two matches in a row (in order to have more available players), or is it better for him to allow a player to rest after a match?  To answer this question, assume that Sarri aims at having as many available players as possible during the tournament (hint: you can use simulations). How about after the investment in medical equipment and staff? [4 points]

2. A model is a simplification of the reality.  How would you propose to enhance the two models so far discussed in order to better reflect other aspects which have not been taken into account, whilst maintaining analytical tractability (e.g. the Markov assumption)? [3 points]

3 Required document

Your team leader has asked you to perform your analysis in R, which is the standard software used for analysis in your company. You are asked to provide a business report and R code to your manager. The requirements of the report are as follows:

— 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.

— The main body of the report should be no more than 4 pages (maximum 4). You need to provide a list of references if any references are used in your report.  Cover pages, appendices and the list of references are not counted towards the page limit. There is no specific formatting requirement; however, you should ensure that the report is professional in the business context.  The standard is 12point font with 1.5 spacing

— You must prepare a separate Word or pdf document for the R code for submission (not as an R file

so that it can be checked by Turnitin).  Your code needs to be well presented with sufficient guidelines such that your colleagues from other departments can easily replicate your results.  For example, you comments should be included in your code (by using the # sign in R) to guide your audience.  Your code must run without further modification by just copying and pasting from your submitted document, otherwise no mark will be awarded for the criteria that relate to R. Note that we will check all codes.

4 Assignment submission procedure

4.1 Business report and R code: Turnitin submission through Moodle

Your assignment must be uploaded as a unique 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; we will check all codes. As long as the due date is still in the 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 separate submission boxes for the business report and the R code . 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 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 anyway submit their assignment through Turnitin.

The date and time of reception of the submission determines the 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. No paper copy will be either accepted or graded. In case of a (fully documented) 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 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 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 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 students.

Students should make sure they understand what plagiarism is—cases 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?