代写statistical R 程序、代做留学生R统计

日期：2018-04-26 04:09

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 ? ?kti

, i = 1, 2, . . . N are stored in the vector Residuals_k and the residuals yti ? y?ti

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, etc.

c University of Illinois at Urbana-Champaign, Department of Mathematics