Question:

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, etc

Answer:

## 1.0. INTRODUCTION

Annuity is a product offered by financial institutions that pays subscribers predetermined fixed amounts of money over a predetermined regular period of time. It is a popular product for retirees and pensioners. Rates of annuity products are determined by a number of factors such as age and income. Identifying such factors is a cog in setting appropriate rates for the annuity product. This report will look at whether mortality rates can affect annuity rates and if so, how mortality should be handle. Mortality rate is a demographic measure that indicates the percentage of death in a population. The association of old age and death thus brings mortality rate and annuity together. If mortality varies over time, then it can be an indicator of annuity rates.

## 2.0. METHODS AND RESULTS

This report considered two models of analysis:

### 2.1. LEE CARTER MODEL

A stochastic model developed in 1992, Lee Carter model is used to predict future mortality rates of a given population.

Time series models are used in the model to make future predictions.

*Figure **i*

The output from the Lee-Carter Modeling in R is represented by the plot in figure i above. The Kappa plot indicates a decrease in the rate of mortality over the years while the Alpha plot indicates that the mortality curve has a universal shape. Improvements in mortality decrease with increase in age of an individual in the given population as shown by the Beta plot. Thus, from the Kappa plot we can conclude that mortality changes with time and a stochastic modeling approach would be sufficient for better setting of annuity rates.

Using and ARMA model with p = 1 and q = 1, generated plot took the form of a Beta plot as shown in the figure below:

*Figure **ii*

The plot shows a decrease in mortality improvements with increase in age.

### 2.2. GENERALIZED LINEAR MODEL

A Poisson based Generalized Linear Model is used to model mortality rate. The model took the form:

D_{x;t }= Poisson (E_{x;t}M_{x;t})

The Poisson model has deviance of 1362873 compared to the models null deviance that is equal to 230294974.This indicates the model’s ability to efficiently predict values of future mortality rates as it points to high accuracy.

For the prediction of the mortality for the year 2016 from the data provided (1933-2015), the plot below is obtained. This is the plot for the universal mortality curve.

*Figure **iii*

DISCUSSION

From the deviance values of the models above it is clear that the Poisson Generalized Linear Model has a higher level of accuracy as compared to the Lee-Carter Model. Cohort effect explains the variations in the characteristics of individuals connected to each other by a shared experience.

The analysis of male and female datasets was done together as the “Total” variable. This helps in obtaining the complete view of the population as opposed to segmented data analysis. analysis. The age was also considered for all ages to provide complete inference about the population.

## 4.0. CONCLUSION

The mortality rate, from the analysis of the U.S.A mortality data, changes with time. This therefore implies that mortality rate should be factored in the computation of the annuity rates. We can thus conclude that the use of the Poisson Generalized Linear Model for the prediction of future mortality rates would produce accurate values which can be factored in the calculations of the annuity rates.