Innovations in the CMI’s approach to
graduation and modelling
Tim Gordon & Jon Palin
15 September 2014
The CMI is wholly owned by the Institute and Faculty of Actuaries
Agenda
• Overview
• Data quality and implications
• Sub-annual deaths
• Co-graduation
15 September 2014
3
Overview
15 September 2014
4
CMI modelling
• Aim for CMI outputs to be commensurate with
– its users’ needs – expectations in the longevity space have increased
– credibility of the data – available data has increased
• The CMI is not aiming to lead research …
… but may well lead in its application
• Favour objective / simple / robust / cross-validated
• As stable as possible, but no more
• CMI models are not the whole answer …
… but they are typically the (a) framework and (b) lingua franca
15 September 2014
5
Areas of development
• Reviewing and updating
– Technical baseline – Graduation and Modelling Working Party
– Use applicable standard methods where appropriate
– Transparency – make relevant data and software easier to access
• New initiatives
– High Age Mortality Working Party
– SAPS Mortality Improvements Sub Committee
– CMI_2016 – possible new projections model
– International comparability / population coherency
15 September 2014
6
Data quality and implications
15 September 2014
7
A mortality improvement puzzle
• ONS England & Wales data
– p-spline model
• CMI pensioner data
– implied change between SAPS
S1PMA and S2PMA tables
• Both for males 2002-2007
• Why are the shapes different?
15 September 2014
8
A mortality improvement puzzle
• ONS England & Wales data
– p-spline model
• CMI pensioner data
– implied change between SAPS
S1PMA and S2PMA tables
• Both for males 2002-2007
• Why are the shapes different?
• Are the SAPS improvements
really higher?
15 September 2014
9
Looking into the ONS data
• ONS England & Wales data
– Males in 2007
– Crude mortality rates
– P-spline fit
• Age 88 looks a little odd
15 September 2014
10
ONS data – deviance residuals by age
• Deviance residuals by age for
the p-spline fit for 2007
• Age 88 looks very odd
– 12+ standard deviation event
15 September 2014
11
ONS data – deviance residuals by cohort
• Plot by cohort, not by age
15 September 2014
12
ONS data – deviance residuals by cohort
• Deviance residuals by cohort
for calendar years 2005-2009
• Persistently odd results for
1919 cohort
15 September 2014
13
Why is the ONS 1919 cohort unusual?
• Caused by the 1918 H1N1
influenza pandemic?
• If so then we would expect to
see it in other data sets
15 September 2014
14
CMI and ONS – deviance residuals
15 September 2014
15
Why is the ONS 1919 cohort unusual?
• Caused by the 1918 H1N1
influenza pandemic?
• A quirk of the ONS data rather
than a genuine mortality effect?
15 September 2014
16
Why is the ONS 1919 cohort unusual?
• Exposure ≈ mid-year population
• But not a good approximation
when birth pattern is irregular
• Baby-boom following WWI
• Knock-on effect on projections
• See Cairns et al for more
details (workshop B1 tomorrow)
15 September 2014
17
Deviance residuals
• Deviance residuals for p-spline
fit to ONS data
1961
100
2012
• Highlighted age/year cells have
absolute deviance ≥3
• Main features
– 1919 cohort
– 1960s
40
15 September 2014
18
Data problems – what to do about it?
• What to do about it?
– Exclude the 1960s
– Exclude or adjust outliers
– Allow for overdispersion
• Results in smoother
improvements
15 September 2014
19
Sub-annual mortality
15 September 2014
20
Sub-annual mortality
Weekly deaths
• Modern best practice is to allow for mortality experience to
date using weekly deaths
15 September 2014
21
Temperature as a driver
Weekly deaths v mean Central England temperature
−5 C
12,000
Weekly deaths
(darker line,
right axis)
11,000
10,500
+5°C
10,000
9,500
+10°C
9,000
8,500
+15°C
8,000
Temperature inverted
(lighter line, left axis)
+20°C
Jan 2010
7,500
Weekly deaths
Temperature inverted
0°C
11,500
• Correlation with
(inverted) temperature
is striking …
• … but not predictive
• (unless you can
predict temperature)
7,000
Jan 2011
15 September 2014
Jan 2012
Jan 2013
Jan 2014
22
Sub-annual mortality – SMRs
• Need to use SMRs to
account for ageing
Sub-annual SMR – average and variation
120%
8%
115%
Standard deviation (right axis)
7%
110%
Average (left axis)
6%
105%
5%
100%
4%
95%
3%
90%
2%
85%
1%
80%
0%
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug Sep
Oct
Nov Dec
• Pronounced pattern to
average and variance
• Annual noise arises
because of
– True annual variation
– Calendar year cut-off
• August / September
cut-off ideal
15 September 2014
23
Sub-annual mortality – individual years
Sub-annual SMRs v average by calendar year
+20%
2005
+15%
2006
2007
+10%
2008
2009
+5%
2010
Nil
2011
2012
−5%
2013
2014
−10%
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
15 September 2014
• 2013 was an
exceptionally heavy
year for mortality
• The CMI model not
built from ground up to
deal with volatility
• Propose to take
account of 2014 data
in CMI_2014
24
The annual cut-off can be misleading
SMR using calendar years
• Improvements have stalled?
15 September 2014
SMR using ‘mid years’
• 2012 is a blip?
25
Co-graduation
15 September 2014
26
Mortality by pension amount
• CMI pension scheme data
• Low, medium, high amounts ⇒
heavy, middle, light mortality
• Basis risk: use the right
mortality for the right people
15 September 2014
27
Mortality by pension amount
• Hard to see differences on
previous chart
• Plot “relative mortality”, the
difference between
– log(m) for the amount band;
and
– fitted average log(m)
15 September 2014
28
Scaling mortality tables
• Simplest model
– log 𝑚 𝑥, 𝑖 = 𝑠𝑖 × 𝐵𝑎𝑠𝑒(𝑥)
• A simple scaling isn’t right at
every age
• So produce separate tables
15 September 2014
29
Independent tables
• Independent tables
– log 𝑚 𝑥, 𝑖 = 𝐵𝑖 (𝑥)
• Cubic 𝐵𝑖 (𝑥) for each band 𝑖
• Tables pass standard tests, but:
– Crossover at older ages
– Heavy/middle diverging at 60
15 September 2014
30
Co-graduation
• log 𝑚 𝑥, 𝑖 = 𝐴 𝑥 + 𝐵𝑖 (𝑥)
• 𝐴(𝑥) is a common higher-order
function (eg cubic)
• 𝐵𝑖 (𝑥) is a lower-order
adjustment (eg linear)
15 September 2014
31
Why co-graduate?
• Better relationships between tables.
• Fewer parameters. (Eight versus twelve in our case).
• Better use of limited data. All data affects all graduations.
15 September 2014
32
Mortality improvements
• Co-graduation of different years
– log 𝑚 𝑥, 𝑡 = 𝐴 𝑥 + 𝐵𝑡 (𝑥)
• Illustrative results for SAPS
data with linear 𝐵𝑡 (𝑥)
• Some crossover at high ages
15 September 2014
33
Mortality improvements
• Previously
– log 𝑚 𝑥, 𝑡 = 𝐴 𝑥 + 𝐵𝑡 (𝑥)
• If we suspect a steady
progression by time then
– log 𝑚 𝑥, 𝑡 = 𝐴 𝑥 − 𝑡. 𝐵(𝑥)
• 𝐵(𝑥) is then the mortality
improvement
• No crossover, and fewer
parameters used
15 September 2014
34
Mortality improvements
• Quadratic 𝐵(𝑥) for improvements
15 September 2014
35
Going further
• Co-graduation for mortality improvements
– CMI and ONS data
– Males and females
– Multiple countries
15 September 2014
36
Summary
15 September 2014
37
Questions
Comments
Expressions of individual views by members of the Institute and Faculty
of Actuaries and its staff are encouraged.
The views expressed in this presentation are those of the presenter.
15 September 2014
38
Continuous Mortality Investigation Limited
Registered in England & Wales (Company number: 8373631)
Registered Office: Staple Inn Hall, High Holborn, London, WC1V 7QJ
Correspondence address: Cheapside House, 138 Cheapside, London, EC2V 6BW
Email: [email protected]
Tel: 020 7776 3820
Website: www.cmib.org.uk (redirects to www.actuaries.org.uk)
Continuous Mortality Investigation Limited (‘the CMI’) is wholly owned by the Institute and Faculty of Actuaries.
This Document has been prepared by and/or on behalf of Continuous Mortality Investigation Limited (CMI). Use of this Document is subject to CMI’s current
Terms and Conditions (Terms). The CMI does not accept any responsibility and/or liability whatsoever for the content or use of this Document by any party
that has not agreed to the relevant Terms. Whilst care has been taken during the development of the Document, CMI does not (i) warrant its accuracy; or (ii)
guarantee any outcome or result from the application of this Document or of any of CMI’s work (whether contained in or arising from the application of this
Document or otherwise). You assume sole responsibility for your use of this Document, and for any and all conclusions drawn from its use. CMI hereby
excludes all warranties, representations, conditions and all other terms of any kind whatsoever implied by statute or common law in relation to this Document,
to the fullest extent permitted by applicable law. If you are in any doubt as to using anything produced by CMI, please seek independent advice.
© Continuous Mortality Investigation Limited
15 September 2014
39