Chapter 4

Question 1

What are three things to examine in relation to mortality

Answer 1

1. Data from a mortality investigation
2. a Probabilistic model for the data generating process - poisson/binomial
3. Standard mortality/ graduated table

Question 2

Explain what published life tables/ standard tables are

Answer 2

Based on large amounts of data from life insurance companies - Tables are usually based on mortality experience over a 3 year period where there is a reliable census and record of deaths. Describes mortality experience of a group

Question 3

Given observed deaths and exposed to risk for each age and our crude estimates what do we want to check with standard tables

Answer 3

1. Is our crude estimates consistent with our own past mortality experiences or is it changing?
   Is it consistent with published life tables and if not how much does it differ?

Question 4

What if you cannot find a suitable standard table to compare your data

Answer 4

You will have to graduate the crude rates yourself.

Question 5

Why is there a need to graduate the crude estimates?

Answer 5

They tend to be erratic as they have been estimated independently and so have independent sampling errors. we want the true rates to be a smooth function in x
How do we know its a smooth function? For reasons learnt :)
Used for forecasting

Question 6

How do we know the true mew x is a smooth function of age x

Answer 6

Observed that number of lives increasing in an investigation then typically the crude rates estimated trend to become smoother
Aging is a smooth process so age related mortality inherits this
Crude rates estimated at age x+1 and x-1 tell us something about true mortality rate at x
We want to forecast future mortality so we need a smooth function for this

Question 7

Explain graduation

Answer 7

Aims to produce a smooth set of rates from the crude rates that are suitable for a particular purpose. It doesn’t make the rates more accurate or remove bias - just makes function smooth

Question 8

Why in life assurance is having a smooth mortality estimate fucntion improtant?

Answer 8

These estimates are used to calculate premiums and reserves for policies. Irregular jumps in mortality are hard to justify to customers

Question 9

What are three desirbale features of a graduation

Answer 9

Smoothness, Adherence to data and graduated rates that are suitable for purpose

smoothness – the graduated rates should progress smoothly, So third differences of
graduated rates small and progress regularly.
- adherence to data – the graduated rates are consistent with the crude estimates of
mortality. They give a close fit.
- the graduated rates are suitability for the purpose for which we wish to use them.

Question 10

What si the criterion for smoothness of graduated quanitties

Answer 10

If the third differences of the graduated quantities are small in magnitude compared to the quantities themselves and progress smoothly and regularly

Question 11

Explain undergraduated

Answer 11

Insufficient smoothing has been carried out

Question 12

Explain overgraduated

Answer 12

Graduation process results in rates that are smooth but show little adherence to the data

Question 13

Name six goodness of fit or adherence to data tests

Answer 13

Chi squared test
Standard deviations test
Sign test
Cumulative deviations test
Stevens sign test
Serial correlations test

Question 14

When comparing the crude estimates of mortality against a standard table what is the null hypothesis

Answer 14

The true underlying mortality rates at each age x are the rates in the standard table

Question 15

When comparing the crude estimates of mortality against a graduation what is the null hypothesis

Answer 15

The true underlying mortality rates at each age x are the graduated rates

Question 16

What minimum amount of data do you need to use the tests to compare crude rates with standard table/ graduation

Answer 16

You need more than 5 deaths in each age group

Question 17

What is the purpose of the whittaker-henderson method of graduation

Answer 17

States to choose a graduated rates that minimise the WH expression. It assigned weights to ensure the result is fit for purpose (allows for more weight on fit or smoothness as the person requires)

Question 18

In life assurance vs pensions how does the suitability of a graduation change

Answer 18

In life assurance losses happen with premature deaths so we do not want to underestimate mortality, In pensions or annuity work losses happen with delayed deaths so we do not want to overestimate mortality

Question 19

What do we also assume under H0 for tests

Answer 19

We assume the distribution of Dx under the null hypothesis

Question 20

What is the deviation at age x

Answer 20

Actual deaths at age z - expected deaths at age x

Question 21

What do we know about the standardised deviations?

Answer 21

Zx is approx standard normal for all x under CLT
The Zx’s at different ages are mutually exclusive

Question 22

What is the purpose of the chi squared test

Answer 22

Tests for fit over age range - tests if the crude mortality rates are consistent with a standard table or the graduated rates overall goodness of fit.

Question 23

What are the degrees of freedom in the chi squared test

Answer 23

X can be assumed to have a chi squared distribution with m degrees of freedom where m is the number of age groups - each age group has to have at least 5 death recorded. If we are testing adherence to a graduation rather than a standard table the degrees of freedom will reduce and they depend on the method of graduation

Question 24

What are the problems with the chi squared test

Answer 24

Could be satisfied where data do not satisfy the distributional assumptions underlying it ex: few large deviations offset by a lot of small ones)
Chi squared test may fail to spot is significant groups of consecutive ages that are biased up or down
Test may fail to detect a consistent small bias in the graduation

Question 25

What is the standardised deviations test used for

Answer 25

Used to detect a few large deviations offset by a lot of very small deviations - overall goodness of fit test. Sees is the distribution of standardised deviations is normally distributed

Question 26

What are the steps to calculate the standardised deviations statistic

Answer 26

Divide real number line into any convenient number of intervals
Count number of Zxs falling into each range
Calculate the expected number of Zx that should fall into that range with Zx distributed as standard normal RV
Compare observed Zxs with expected number of Zxs using Chi squared test

Question 27

Explain the purpose of the signs test

Answer 27

Tests for overall bias - checks if there is an imbalance between positive and negative deviations

Question 28

xplain the steps in the sign test

Answer 28

1. Determine how many graduated rates lie above and below the crude rates - signs of standardised deviations
2. Define P = number of Zx that are positive
3. P is binomial (m, 1/2)
4. Find smallest value k s.t the prob (P<k) is less than 2.5%
   Test is satisfied at the 5% level is k<=P<=m-k where m is number of age groups
5. find p value - if p value is greater than 5 % there is insufficient evidence to reject H0

Question 29

Explain purpose of the cumulative deviations test

Answer 29

Check for overall bias or long runs of deviations of the same sign. Many methods of graduation result in this test being zero

Question 30

How do we assess the test statistic in the cumulative deviations test

Answer 30

If the magnitude of the test statistic is high it indicates graduated rates are biased.
Too low the test statistic is positive and too high the test statistic is negative

Question 31

What is the aim of the grouping of signs test

Answer 31

Aims to detect clumping of deviations of the same sign - stevens test is another name for it. G is the test statistic which is the number of groups of positive deviations. We asusme of m deviations n1 are positive and n2 are negative. You cna look upt he cirtical value of G int he actuarial tables

Question 32

When does goruping of signs test fail

Answer 32

G<k it fails at 5% significance level

Question 33

Purpose of the serial correlation test

Answer 33

Detects grouping of signs or deviations testing for overgraduation. Under H0 two sequence sof deviations of length m-1 should be uncorrelated. basically lagged sequences should be uncorrelated

Question 34

What should we want in terms of correlation between graduated rates

Answer 34

If graduated rates are neither over graduated nor undergraduated we would expect the individual standardised deviations at consecutive ages to behave as if they were independent and uncorrelated.

Question 35

What values cna the serial correlation coefficient (formula in tables) take? and what does the sign of r mean?

Answer 35

Can take values between -1 and 1
Psotive Rj means nearby Zx values tend to be similar
Negative rj means nearby zx values tend to be opposite

Question 36

How do we conduct the test or reach a conclusion for the serial correlations test

Answer 36

1. calculate Zx for each age group
2. Calculate the serial correlation coefficients
3. Find the T ratio and compare the to standard normal distribution
   If t ratio is too high this indicates a tendancy for deviations of same sign to cluster together

Question 37

Why do people tend to use sign tests or grouping of signs test over the serial correlation test

Answer 37

More powerful. In serial correlation text its possible for correlations in one part of the age range to be canclled out by the opposite correlations.

Question 38

What is the purpose of the standardised deviations test

Answer 38

Chi-squared test for residuals. Tests for large deviations offset by a lot of smaller deviations - overall goodness of fit

Question 39

What is the empirical rule

Answer 39

68-95-99.7

Question 40

When should graduation by parametric formula be employed?

Answer 40

Suited to the production of standard tables from
large amounts of data. [1]
Graduation by parametric formula is employed to produce extremely precise results, and good smoothness. [1]
Comparing several experiences to fit the same parametric formula to all of them. Differences between the fitted parameters give an insight into the differences
between the experiences. [1]
Maybe when extrapolating, if good rationale for using the formula.