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181 views13 pagesUntitled
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/ 13
Subject CS2
Revision Notes
For the 2021 exams
Mortality projection
Booklet 7
Covering
Chapter12 Mortality projection
The Actuarial Education Company PAST EXAM QUESTIONS
This section contains all of the past exam questions from the 2019 Paper A
‘exams relating to the topics covered in this booklet
Solutions are given after the questions. These give enough information for
you to check your answer, including working, and also show you what an
outline examination answer should look like. Further information may be
available in the Examiners’ Report, ASET or Course Notes. (ASET can be
ordered from ActEd.)
We first provide you with a cross-reference grid that indicates the main
subject areas of each exam question. You can use this, if you wish, to
select the questions that relate just to those aspects of the topic that you
may be particularly interested in reviewing.
Alternatively, you can choose to ignore the grid, and attempt each question
without having any clues as to its content.
Cross-reference grid
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Page 22 © IFE: 2021 Examinations Subject CS2 September 2019 Question 2
the next 20 years. Life tables are available by single years of age for each
calendar year for the past 10 years. The country has a new Chief
Statistician who suggests fitting an exponential curve to the time trend in
mortality at each age x and forecasting mortality separately at each age
using the parameters estimated for that age.
(i) Comment on this suggestion. [3]
(ii) Suggest an altermative approach which may be more suitable for
projecting mortality in this country. [1]
[Total 4}
© IFE: 2021 Examinations Page 23 SOLUTIONS TO PAST EXAM QUESTIONS
The solutions presented here are just outline solutions for you to use to
check your answers. See ASET for full solutions.
Subject CS2 September 2019 Question 2
(@) Comment
The use of an exponential curve is attractive as there is evidence that
age-specific mortality rates have declined exponentially in some past
periods.
However, the past mortality of the country should be examined to see if this
general trend is exhibited by its population
The approach is also simple to understand and easy to implement.
However, fitting separate curves at each age risks the projected future
mortality rates in any given year progressing roughly with age (and even
decreasing with age in age ranges where this is implausible).
This problem could be overcome by graduating the projected rates, or by
using an alternative projection model.
The approach assumes that developments in medical technology, lifestyle,
etc in the future will progress steadily as they have in the past 10 years.
(i) Alternative approach
Possible alternative approaches include the following:
* Penalised splines could be used to smooth the projected rates between
ages.
* The government could use the Lee-Carter model, which provides an
estimate of the overall average time trend across alll age groups, as well
as how this effects each individual age.
* An age-period-cohort model could also be used, such as adding a
cohort term to the Lee-Carter model. Models that include a cohort effect
have proved superior to two-factor models in some circumstances.
Page 24 © IFE: 2021 Examinations * Amodel based on explanation instead of extrapolation could be
considered. This approach projects future mortality based on expected
changes to certain causes of death.
© IFE: 2021 Examinations Page 25 FACTSHEET
This factsheet summarises the main methods, formulae and information
required for tackling questions on the topics in this booklet.
Types of mortality projection model
Projections of mortality can be made using two-factor models (age x and
time period t , or age x and cohort c), or three-factor models (age, time
period and cohort). In three-factor models the factors are linked by x =t-c.
Cohort has a generally less significant impact on future mortality than age
and period, but still has a non-negligible effect that age-period-only models
will miss. However, models that include cohort have the following
disadvantages:
‘* in age-period-cohort models each factor is linearly dependent on the
other two
* there are heavy data demands for estimating cohort effects, eg requiring
more years of past data than when only age and period are used
* the mortality of recent (currently young) cohorts will be poorly estimated
as based on very few years of past data.
Projection approaches
The three projection approaches are based on expectation, extrapolation,
and explanation.
Methods based on expectation
‘An expectation approach is based around experts’ opinions about how
mortality is expected to change in future.
Simple deterministic models (eg reduction factors) based on expectations of
target future mortality rates are generally used. These expectations may be
based on expert opinion and/or on recent historical trends.
Page 26 © IFE: 2021 Examinations For example, projected mortality rate at age x in time period f is:
Met = M9 * Rep
where mm, is the (known) mortality rate in base year 0, and R,, is the
reduction factor.
An example of a model for Ry + is:
Ryp = ay + (1a (1-fy
where a, is the ultimate amount of reduction, and f,,, is the proportion of
the total decline expected to occur in n years.
Expert opinion is used to set the targets a, and fy...
Advantages and disadvantages of an expectations approach:
+ Itis straightforward and easy to implement.
— The method has consistently underestimated rates of improvement
(especially for UK males mortality) between 1990 and 2010.
- Future developments in the prevention and treatment of the main
causes of death, and the effects of changing lifestyles, mean that the
prediction of future mortality by experts is very problematic.
- Using (expert-based) targets tends to cause the true level of uncertainty
in the forecast to be underestimated.
Methods based on extrapolation
The extrapolation approach involves using past data to construct
mathematical models that are then used to predict future mortality.
An important example of these is the Lee-Carter model.
© IFE: 2021 Examinations Page 27 Lee-Carter model
This is a two-factor (age-period) stochastic model:
Ings = ay + Dy ke + Eee
where:
a, describes the general shape of mortality at age x
k, measures how mortality evolves over time (with Jk; = 0 over the
T
historic period used to fit the model)
b,, is the extent to which mortality is affected by the time trend at age x
(with 1, =1)
x
b, ky is the effect of time t on mortality at age x
£y1 is the error term
The £,, values are IID normal random variables with zero mean.
‘The parameters are estimated as follows:
a
AS.in( 7,1) is used to estimate a,
Ft
In(i,.r)-a is used to estimate b, and k,
Modelling k;
Time series methods are usually used, eg the random walk model:
Ky — yy = Aky = + &
where 4 is the mean increase in k, and the « are IID random variables
with variance o?
Page 28 © IFE: 2021 Examinations If f is the latest year of the historical data, the future value of k, in r years’
time can be projected as:
= r
for =k, FD oie)
ix
where ji is the value of 4 estimated from the historical data.
There are two sources of error, or randomness, in this forecast.
* random variation in the error term «; with variance o”, and
* error in the estimation of yz
The standard error of the estimator of Ky, , is given by:
where 1 is the number of past years’ data used to estimate.
Age-period-cohort version of the Lee-Carter model:
Wry = ay + BE Ky + OF Mgt + Ext
where:
a, can be fitted (and therefore smoothed) using p-splines
fh, is the effect of cohort year ¢ on mortality
be is the extent to which mortality is influenced by the cohort effect at
age x
© IFE: 2021 Examinations Page 29 Multi-factor extension of the Lee-Carter model:
s
Wang g = ay +X Ky j BH) Ee
it
where:
k,; 18 the effect of the time trend on mortality at time t for group j
(x) is the extent to which mortality is influenced by time for group j
atage x
Advantages and disadvantages of the Lee-Carter model:
+ Once the parameters have been estimated, forecasting is
straightforward and can proceed using standard time-series methods,
the statistical properties of which are well known.
+ The degree of uncertainly in parameter estimates, and hence the extent
of random error in mortality forecasts, can be assessed.
+ The model can be extended and adapted to suit particular contexts,
eg by smoothing the age pattems of mortality using penalised
regression splines
- Future estimates of mortality at different ages are heavily dependent on
the original estimates of the parameters a, and b, .
- The forecasting assumes that a, and 6, remain constant into the
future.
- The parameters a, and b, will incorporate any roughness in the past
data, including the effect of past period events which could affect
different ages to different extents. This can cause unrealistic
inconsistencies in the progression of projected mortality rates from age
‘to age and from duration to duration.
- There is a tendency for Lee-Carter forecasts to become increasingly
rough over time.
— When the random walk is used to forecast k, , the model assumes that
the ratio of the rates of mortality change at different ages remains
constant over time, when there is empirical evidence that this is not so.
- The two-factor version of the model may be inadequate if there is a
significant cohort effect on mortality.
Page 30 © IFE: 2021 Examinations — Unless observed rates are used for the forecasting, ‘jump-off effects
can occur (ie an implausible jump between the most recent observed
mortality rates and the forecast for the first future period).
Splines
Spline functions can be used for modelling mortality rates by age using:
s s
InfE(D,)]=InEE +38) B(x) => In(m,)= 2% By(x)
f= i
For a cubic spline with s knots at ages x,, x2,...
a 0 x