Group Assignment Risk Insurance 2022 Ki  May 24, 2022 1 

 Contents 

1 Introduction 2 Dividend Process Example 3 Assignment tasks 4 What do you have to hand in? 5 How will you be graded? 6 Recommendations 7 Answers to some student questions 3 4 5 7 8 9 10 2 1 Introduction The goal of this assignment is to investigate the implications of extracting profit for the capital process for the simulated ruin probabilities. An insurance company is a for-profit entity. As such, it would want to not only keep growing their capital reserve U(t), but also to extract profit from the reserve for distribu- tion to share holders or to further develop the business. In this assignment, you have to incorporate a dividend extension either from the literature or from your own thinking. Assume that dI(t, u, x) is the dividend gathered at time t and it can depend on the current wealth level as well as the incoming claim1 2 and dU(t) is the change of capital process U with respect to time, the extended capital process would follow the dynamic: U(0) =u dU(t) =c X(t) dI(t, U(t), X(t)). (1) Here X(t) = Xi if the i-th claim comes at time t, for i = 1,2,...,k, and 0 otherwise. The main task for each group is to design the dividend rule dI(t, u, x), conduct simulation for the process and derive numerical solutions for values of interest, plan extra experi- ments as comparison and deliver recommendation. We would give further details to the assignment with an example. 1Here we use the differential operation to define the divident process as it is more general but not necessarily most rigorous mathematically. Think of dI as the change of dividend at each moment. 2The definition here is not the most general, please feel free to use your creativity to define this process. You just have to consult us on your definition before actually working on the assignment. 3 2 Dividend Process Example Consider the continuous-time ruin model and assume that the insurance company takes the naive approach and just take out all the premium it receives as dividend, so dI(t, u, x) = cdt, where c is the premium rate. The total dividend gathered till time t would be t t I(t) = dI(s, U(s), dS (s)) = cds = ct. 00 Therefore, the extended ruin process according to Equation (1) can be simplified to U(t) = u + ct S (t) I(t) = u S (t). We can define the time of the k-th claim arrive as Tk for k = 1,2,... and the time of ruin beforeoratk-thclaimask = min{t | 0 t Tk&U(t) < 0}.Weletk = ifnoruin occurs at or before the k-th claim. We can study the probability of ruin before or at k-th claims with this new model: k(u) = IP(k Tk). More interestingly for us, we can calculated the profit extracted until ruin, which is I(k) = ck or if there is no ruin in the first k claim, I(Tk) = cTk. But this calculation ignores the time value of money and the funding cost of the shareholder. Assume that the funding rate for the shareholder is r, we can calculate the present value of the dividend as k k c D(k)= ersdI(s)=c ersds= r(1erk). 00 Similarly, the present value would be D(Tk) = cr (1 erTk ) without ruin occurring at or before k-th claim. You are tasked to find the sample mean of k and the profit extracted in this assignment with Monte Carlo simulation. 4 3 Assignment tasks There will be 3 versions of this group assignment and you are free to pick which version you prefer. Basic Assignment: You assume dI(t, U(t), X(t)) = dt for any constant and use pro- cess D(t) to measure the profit gathered to finish this assignment. Your grade will not be higher than 7.5. Standard Assignment: You design any form of dividend process other than the constant version in the basic assignment and use process I(t,U,S) to measure the profit gathered to finish this assignment. The claim size distribution is also no longer i.i.d.,butX Gamma(2+i1,2),butremainindependentamongeachotherandof i 100 N(t). Your grade will not be higher than 9. AdvancedAssignment: Youdesignanyformofdividendprocessotherthantheconstant version in the basic assignment and use process D(t,U,S) to measure the profit gathered to finish this assignment. There is an automatic 0.5 bonus to your groups grade. You can finish this assignment with the following steps: Step 0: You have to implement a standard ruin model so that everyone has the same baseline and we can compare your results effectively. The baseline capital model follows Example 4.3.7 from Modern Actuarial Risk Theory, the textbook. You do not have to show this part in your report. For advance and basic version, the model setting is Claims arrives according to the Poisson process N(t) with parameter = 1. Each incoming claim is independently distributed as Gamma(, ) with the shape and rate parameterization and = 2. 5 The premium is set according to safety loading of 0.3. The initial wealth is 7.5 The simulation ends after 400 claims. For standard version, the model setting is Claims arrives according to the Poisson process N(t) with parameter = 1. Each incoming claim is independently distributed as Gamma(2 + i1 , 2) with 100 the shape and rate parameterization. The premium is set to 2.2. The initial wealth is 7.5 The simulation ends after 400 claims. Step 1: Design the rule to gather profit dI(t, u, x) and discuss your economic reasoning. The dividend process has to be non-negative in all cases. There is no capital injec- tion in this assignment. Step2: FormulateyourextendedcapitalmodelandimplementitinR.Reportthefollow- ing value. 400, the sample ruin probability after 400 claims. E[I(min{k,Tk},U,S)]orE[D(min{k,Tk},U,S)]withthefundingratebeing 0.5% depends on your assignment version and E denotes the sample mean. The profit score 150c400, where is your answer to the previous bullet point. 150c is the penalty score and it depends on the premium rate c. If you adjust the premium rate for the next step then you would also have to adjust this score. The target of the insurance company is to maximize the profit score. 6 Puttheset.seed(G)commandatthetopofyourRcode,whereGisyourgroup number. Step 3: Repeat Step 2 but for models with different parameter values, claim distributions and/or dividend process. Compare your model with these new simulations. This step to to test your skills on research design and gathering information to draw general conclusion. So you have to come up with the range of study you would like and the above is just a suggestion. Step 4: Base on step 2 and 3, make one recommendation to the board of this fictional insurance company on how to extract profits. 4 What do you have to hand in? You are expected to hand in the following components: A report pdf made in LATEX, explaining your work on the steps above. Make sure the cover page of your pdf contains your names, group num- ber, student numbers and assignment version. You get a group number by enrolling to a group on Nestor. Without a group number you cannot do the assignment. The maximum page length is 5 pages, in a 12 point font with a 1.0 line spacing and normal page margins. Cover page, tables, reference list, graphs and appendix with R code do not count for the page limit. We reserve the right to ignore all work beyond the page limit in your report. The report pdf contains an appendix with the documented/commented R code of your work. Your R code is only in this appendix, not in the main text. 7 Aseparate.Rfile,containingyourcodeaswritteninthereportandallthenecessary supplementary files that allow us to run it. We should be able to run your code on a freshly install R environment without downloading extra files or making extra settings, except installing package as required. Deviations from these instructions will be penalized and may result in a grade equal to 1. 5 How will you be graded? The grading will be done using a rubric, which is also provided on Nestor. The rubric form will give you an idea of what is expected from you. The rubrics indicate what aspects have to be present in the report, but they do not necessarily dictate the structure of your report. You can use the rubric as a checklist on your report, to verify that all components are included. The rubric assesses whether You provide proper economic reasoning throughout the whole report, includ- ing introduction, dividend design, comparison study, and final recommenda- tion. You obtain a satisfactory profit score. You correctly explain and implement your ruin extension. You design an appropriate comparison study to support your conclusion and recommendation. Your writing is clear, complete, concise, and well-organized. You R code is correct and well-documented. 8 Weemphasizethatthewayyouwritethingsdown(assumptions,intermediatesteps, explanation, motivation, notation) is very important in this assignment. We reserve the right to adjust the final grade if you deviate from our instruction. After the groups are final, every member within a group submitted the report will receive a common grade with no exception. 6 Recommendations In case you encounter a problem, ask for help or a hint via the Discussion Board (24/7!) or via our online office hours. In any case, dont waste too much time and dont stay stuck for too long time, but quickly ask for our help! We are there to help you and are happy to do this. Try to explain your steps in a concise way, but stay to the point. Remember that the maximum number of pages is 5. Simulation takes time and programming may prove more challenging than you expect. It is recommended to start this assignment early. It is not necessary to derive the implications of the extension for the Lundberg upper bound analytically, as the main focus is on simulation study and the economic interpretation. If you would like to attempt the standard or advanced version, the design of the dividend process would be crucial and can make or break your assignment. You are highly recommended to contact us for advice and comment before any program- ming takes place. AlthoughSection5.2fromyourtextbookisalsorelatedtodividendandruinmodel, it is not connected to this assignment and may not provide you with too many 9 7 Answers to some student questions inspirations. In the report introduction, you should briefly discuss why an insure company would like to maximize the profit score. In step 2, the accuracy of the simulated values should be investigated. Provideyourcodewithasufficientamountofexplanationtomakeitunderstandable to others. The goal is any programmer should be able to pick up your work even if they are not an actuary. Assignments cannot be handed in after the deadline. Assignments sent by email will not be accepted, all assignments have to be handed in via Nestor and the plagiarism scanner. So make sure to hand in on time, via Nestor! Thereisnoreinvestmentinthisassignment.Oncethedividendisextracted,itleaves the system. 10