13 Jun MATH 490: STATISTICS FOR RISK MODELING
Math 490: Statistics for Risk Modeling
Name:
April 23, 2018 Final Exam Deadline: May 10, 2018 Teacher
1. This is a take-home final exam.
2. Answers to Questions 1,2,3,4 should be written using a Latex editor.
Question 5 can be answered using your preferred program. All answers should be merged in a single PDF and submitted in Compass. 3. You are asked to write a report in Question 1. Note that your solution to this question should not be limited to just answering the questions I listed. Instead, your submitted result for this question should meet the requirements of a good scientific report, meaning it should be self-contained, have a well-structured organization of the sections, is well-written and uses references in an appropriate way. If your solution to question 1 does not meet the standards of a good scientific report, it will not be marked. 4. Coding should be done using the statistical software R. 5. Clearly indicate the different questions. Start each question on a new page. 6. Clearly explain your solutions. Any R code you use should be added to the report as an appendix. 7. Upload a separate R script, with the R code used in your report. Make sure we can run the R code line by line to create the numbers, tables and figures in your report. 8. It is allowed to discuss the questions with your colleagues, but the report and the R code should be an individual work. The report should contain sufficient elements which make the report unique. Note that plain copying R code, text or ideas without adding your own interpretation is considered as cheating and will be reported to the department and will lead to a grade of zero. 9. GOOD LUCK! c University of Illinois at Urbana-Champaign, Department of Mathematics Question Points Max Question 1 30 Question 2 10 Question 3 10 Question 4 10 Question 5 20 Layout of the report 10 Writing style 10 Total 100 c University of Illinois at Urbana-Champaign, Department of Mathematics Introduction You are working for the consultancy company Linders Consultancy. This company is asked by a annuity provider to provide an in-depth analysis about Stochastic Mortality Modeling. The annuity provider is currently using a single life table for the pricing and risk management of its annuity portfolio. This implies that survival and mortality probabilities are not changing over time. However, this annuity provider may want to change their simple mortality model to a more complicated model taking into account future mortality developments. Therefore, Linders Consultancy is asked to provide a scientific answer to the following questions: • Is there statistical evidence that mortality is changing randomly over time and does it have a significant impact on an annuity portfolio? • Is the Lee-Carter model a good choice if we want to switch to a stochastic mortality model and how can it be used to better estimate annuity values? • How can one use Generalized Linear Models with explanatory variables the Time and the Age, a Poisson distribution and a log-link, to model future mortality rates? How does the GLM approach compare to the Lee-Carter model? • Are there other explanatory variables that one can use in a GLM to improve its fit and prediction power? The annuity provider expects the consultancy company to provide the theoretical foundations used to build these models, but also to explain how to implement these models in R. You can use the data about US mortality, which is stored in the files: dataUSExposures.txt and dataUSDeaths.txt, for your illustrations.
Question 1 Write a report that answers the questions of the annuity provider. Your report should provide sufficient theoretical details about the models which are presented, such that someone with limited knowledge about this subject can understand the report. The report should also provide details about the methodology used to implement the different models and comment on the output of the implementation. Finally, your report should also contain a section discussing various recommendations to improve the model. Below you find some suggestions to include in your report: • Provide empirical evidence that mortality is changing over time. You may also investigate how mortality improvements impact survival probabilities and annuity values. The main message you want to give in this part, is that it is necessary to use advanced stochastic mortality models. • Describe and implement the Lee-Carter model. Investigate the model fit (for example by studying the residuals). Convince your readers in this section that Lee-Carter is a model that very well balances accuracy and complexity. • In order to forecast future mortality, the time coefficient kt has to be modeled by a time series. Fit an ARMA(p, q) model to the data and determine the expected future mortality rates. c University of Illinois at Urbana-Champaign, Department of Mathematics • Forecast future mortality rates for a given age x. Construct 95% confidence bounds using a simulation study. Note that the computation time of the simulations can be reduced by doing the simulation of the innovations outside the for loops. • Assume that Dx,t denotes the number of deaths in year t for age x. The corresponding exposure-at-risk is denoted by Ex,t. Fit the GLM model: Dx,t = Poisson (Ex,tMx,t), where log Mx,t = ax + kt . Investigate the model fit of this model and forecast future mortality. • Compare the GLM model with the Lee-Carter model (use fitted and forecasted values, age and time effects, etc.). Which model is the best? Are there possibilities to improve the GLM model? What is a cohort effect? • Be clear on what data you use. For example, do you use all ages? Do you fit men and women together or separately? Justify your choices. • A good report has a title, an introduction, a well-organized main part and a conclusion. References should be added in the correct way when they are used in the report.
Question 2 A colleague of you has checked your code and does not agree with your simulation study. Especially the use of the normal distribution is questioned. As an alternative, he proposes to use the following approach when simulating future mortality rates. Assume today is time T and you have fitted the Lee-Carter model with a random walk1 for the kt process using past mortality rates for the years t1, t2, . . . , tN = T. You want to simulate the log mortality rates yT +1, yT +2, . . .. The residuals kti − ˆkti , i = 1, 2, . . . N are stored in the vector Residuals_k and the residuals yti − yˆti are stored in the vector Residuals_y. • Simulate a single path with N_sim steps. The future values kT +1, kT +2, . . . kT +NSim and the future log death rates yT +1, yT +2, . . . are determined as follows using the R function sample: x=25 AA_x=Alpha_hat[AA==x] BB_x=beta_hat[AA==x] kappa_sim=c() y_sim=c() kappa_sim[1]=kappa_hat[n_Years] y_sim[1]=DeathRatesUS[n_Years, AA==x] for(i in 2:N_sim){ kappa_sim[i]=kappa_sim[i-1]+d+sample(Residuals_k,1,replace=TRUE) #d is the estimated drift. y_sim[j]=AA_x+BB_x*kappa_sim[j]+sample(Residuals_y,1,replace=TRUE) } 1You should use your preferred time series model here, instead of the random walk. c University of Illinois at Urbana-Champaign, Department of Mathematics Do you think this methodology makes sense? What is the difference with the methodology that uses the function rnorm? What method do you prefer? Provide statistical evidence of your conclusion.
Question 3 Linders Consultancy has an excellent restaurant for their employees and during lunch, some of your colleagues suggest that you may want to take into account that an annuity provider is not very likely to have a lot of young people in their portfolio. Therefore, it could be an idea to calibrate the models on ages 40+, 50+ or even 65+, where the last scenario would coincide with the portfolio of a pension provider. The main question you want to answer is: does the choice of the age groups in the data set have a major impact on the calibration results and the projected mortality rates?
Question 4 A topic you (probably) did not investigate so far is the prediction power of your model (Lee-Carter or GLM). The standard way to investigate if your model is able to predict future mortality rates, is by dividing the data set in 2 parts, a test and a training data set. For example, you can take a training data set which includes mortality data for the years 1933, 1934, . . . , 1990. Then, you fit the model on this training data set and forecast the mortality rates for the test data set, which are the years 1991, 1992,. . . , 2015. Since you have observed mortality rates for this test data set, you can compare how well the model predicts the realized values. Of course, you can change the test and training data set.
Question 5 The results of your study should be presented during a board meeting of the annuity provider. Directors, managers and actuaries working for the annuity provider will attend the meeting. You are asked by the CEO of Linders Consultancy to prepare 10-15 slides for this meeting. The slides should be self-contained, balance intuition and technical details and show that you are capable to communicate your in-depth knowledge about this subject in an easy, creative and understandable way. You can make the slides in Latex (use the beamer class), Powerpoint, Keynote, Prezi, e
Our website has a team of professional writers who can help you write any of your homework. They will write your papers from scratch. We also have a team of editors just to make sure all papers are of HIGH QUALITY & PLAGIARISM FREE. To make an Order you only need to click Ask A Question and we will direct you to our Order Page at WriteDemy. Then fill Our Order Form with all your assignment instructions. Select your deadline and pay for your paper. You will get it few hours before your set deadline.
Fill in all the assignment paper details that are required in the order form with the standard information being the page count, deadline, academic level and type of paper. It is advisable to have this information at hand so that you can quickly fill in the necessary information needed in the form for the essay writer to be immediately assigned to your writing project. Make payment for the custom essay order to enable us to assign a suitable writer to your order. Payments are made through Paypal on a secured billing page. Finally, sit back and relax.
About Writedemy
We are a professional paper writing website. If you have searched a question and bumped into our website just know you are in the right place to get help in your coursework. We offer HIGH QUALITY & PLAGIARISM FREE Papers.
How It Works
To make an Order you only need to click on “Order Now” and we will direct you to our Order Page. Fill Our Order Form with all your assignment instructions. Select your deadline and pay for your paper. You will get it few hours before your set deadline.
Are there Discounts?
All new clients are eligible for 20% off in their first Order. Our payment method is safe and secure.