Chapter 5

Question 1

What are the three methods of graduating crude mortality

Answer 1

The parametric formula for the whole age range
By reference to a standard table
Or spline functions/ piecewise formulae

Question 2

How many parameters need to be estimated to fit a GM(m,n) curve

Answer 2

M+n

Question 3

In practice what sort of parametric formula does Mew x usually take

Answer 3

Mew x typically follows an exponential curve quite closely for middle to late ages so modelling human mortality we always try include a gompertz term

Question 4

What are Gompertz and Makeham in GM(m,n) form

Answer 4

GM(0,2) - Gompertz
GM(1,2) - Makeham

Question 5

How did the 00 series tables come about

Answer 5

Data was collected by most life companies during 1999-2002 and were analysed by the CMI - produced tables for males and females and for different classes of insurance and pensions business. Graduating formula was of the form GM(r,s)

Question 6

What way can you find the parameter estimated for a GM(r,s) type of formula for mew

Answer 6

Find likelihood ignoring constants of deaths observed given the central exposed to risk and maximise it to find the MLEs of the r+s parameters

Question 7

What are three methods to fit a parametric formula

Answer 7

MLE
Minimise the chi-squared statistic
Minimise the value of the weighted least squares

Question 8

What are three other considerations to take into account when fitting a model

Answer 8

Using additional information form other investigations
Financial risks
Changes in Mortality with time

Question 9

Explain what we should look out for when considering : Using additional information form other investigations when we are fitting a model

Answer 9

Graduated estimates should be compared with other experiences we know of.
Could check: mortality of males is higher than females, mortality of life insurance policy holder is lower than the population as a whole, the mortality of a person recently taking out a policy is lower than a person who took out policy ages ago. etc

Question 10

Explain what we should look out for when considering : financial risks when we are fitting a model

Answer 10

Be aware depending on if its life isnurer or pensions provider where losses will be made.
Life insurance - dont underestimate mortality
Pensions - Dont overestimate mortality!

Question 11

Explain what we should look out for when considering : changes in mortality over time when we are fitting a model

Answer 11

Some companies use graduated mortality tables to estimate future mortality - but mortality investigations must be based on past mortality
Trend is important and past mortality is likely to be on the conservative side for insurance but not adequate for pensions/annuities
For pensions - will have m=to make a projection of future improvement in mrotality

Question 12

What are the steps in graduation by mathematical formula

Answer 12

1. select graduation formula - never choose more than 5 parameters to estimate
2. determine best-fitting parameters
3. calculated the graduated rates at each age once the formula is fitted
4. test for goodness of fit and sense checks. - if tests fail retrun to step 1

Question 13

Why dont we check for smoothness of graduation when using parametric formula

Answer 13

Formula chosen is usually already smooth

Question 14

What are the pro of graduation by mathematical formula

Answer 14

Using a small number of parameters the graduation will be acceptably smooth
When comparing several experiences its useful to fit the same parametric formula to all - difference in parameter show differences in experience
Well suited method to production of standard tables

Question 15

What are the cons of graduation by parametric formula?

Answer 15

Can be difficult to find a curve that fits experience at all ages well
Care is required when extrapolating - may struggle to fit curve at ages with sparse data so these results may need adjustment
If you need more information to fill in for sparse data - try to reference a standard tables if there is one suitable

Question 16

What is graduation by reference to a standard table base don?

Answer 16

The assumption that if the lives involved in the graduation is sufficiently similar to the lives whose experience is the basis of a standard table then the true mortality of our class of lives should be similar to the table.

Question 17

When do we use this method of graduation with reference to a standard table

Answer 17

When there is not a large amount of data
When there is a standard table we think is appropriate to the population - must have a similar shape and a simple relationship evident

Question 18

What are the steps to graduating with reference to a standard table?

Answer 18

1. select the standard table
2. Find simple relationship to the standard table (from crude rates)
3. Determine best fitting parameter values
4. Calculate the graduated rates given assumed relationship
5. Test result graduation for goodness of fit and sense checks

Question 19

To find the simple relationship between tables and crude rates what should you do?

Answer 19

Aim is to find a function with standard rates as a parameter to get the graduated rates - unless youre dealing wiht a short age range plot the crude data on a logarithmic scale against the standard rates. Usually you’ll fidn a lienar relationship between crude rates (now graduated) and standard table rates

Question 20

What are the two ways you can find the best fitting parameters when graduating with reference to a standard table

Answer 20

MLE by maximising the likelihoods or could find least squares estimates

Question 21

What are the pros of graduation with reference to a standard table

Answer 21

Can be used to fit small data sets where suitable table exists
If a simple function is chosen and the table rates are smooth a smooth graduation will result
Information obtained from the standard table cna be useful in deciding the shape of graduation at extreme ages

Question 22

What are the cons of graduation with reference to a standard table?

Answer 22

Method is not suitable for preparation of standard tables
Its not easy to choose a standard table always - features of experience we observe must be very similar to the tables experience

Question 23

Define a spline function

Answer 23

Polynomials of a specified degree defined on a piecewise basis across an age range. Piece join at knots where certain continuity conditions are required for smoothness

Question 24

What type of spline is usually used for mortality rates and what does more knots mean?

Answer 24

Commonly cubic splines are used (polynomial degree 3 ). The more knots the more closely the graduated rates adhere to the crude rates but the less smooth the graduation will be.

Question 25

Describe in words why we fit spline functions

Answer 25

We are assuming a simple probabilistic model is fitted by a polynomial over a certain age range with the different polynomial (different age range functions) meeting at knots - If mew x changes very rapidly then there will be more frequent knots

Question 26

Give an exmaple of a piecwise formula that has been used to fit a mortality curve

Answer 26

Heligman pollard formula

Question 27

Is there any prominent trend in mortality curves acorss different scenarios

Answer 27

Mortality form age 40 plus tends to be quite linear but earlier ages have different shapes

Question 28

What are the two types of statistical tests for graduations we can carry out

Answer 28

Comparing two experiences or testing a graduation

Question 29

How much do we reduce the degrees of freedom of the chi squared statistic if testing the adherence to a graduation

Answer 29

If we used a parametric formula for graduation : reduce d.o.f by 1 for every parameter fitted
If we used standard table : reduce d.o.f by 1 for every parameter fitted and some further number (depending on constraint imposed by choice of table)
If we used graphical graduation its tough to say

Question 30

Why do we often not use the cumulative deviations test to test a graduation

Answer 30

Some methods of graduation may force the cumulative distribution to be close to or equal 0 over the range of ages being graduated- so test cannot be sued.

Question 31

Explain how CMI in UK records data for mortality investigations in practice

Answer 31

Usually instead of recording lives it’s based on policies. Instead of recording the number of person-years observed and a number of deaths. The CMIB observes the Number of policy years observed and the number of policies becoming claims by deaths.

Question 32

What problem does recording data based on policies vs lives have?

Answer 32

Known as duplicate policies problem: we’re no longer observing a collection of independent claims as policyholder could have multiple policies. - Increases the variance of the observed “Deaths”

Question 33

In what way could you quickly graduated rates if you had access to a previous graduation

Answer 33

Adjust graduated rates by multiplying by factor less than/more than 1 - keeps smoothness and goodness of fit which is important if our data passed chi squared test.
Reduces bias present.
Ratio to use migth be actual/expected deaths and mutliply this by the original graduated rates for a new graduation

Question 33

In what way could you quickly graduated rates if you had access to a previous graduation

Answer 33

Adjust graduated rates by multiplying by factor less than/more than 1 - keeps smoothness and goodness of fit which is important if our data passed chi squared test.
Reduces bias present.
Ratio to use migth be actual/expected deaths and mutliply this by the original graduated rates for a new graduation