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ICC, Birmingham
The APCI model — a stochasticimplementation.
Stephen Richards
23rd November 2017
Copyright c© Longevitas Ltd. All rights reserved. This presentation may be freely distributed, provided itis unaltered and has this copyright notice intact.
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Overview
1. Contributors
2. Background
3. APCI model
4. Fitting and constraints
5. Parameter estimates
6. Smoothing
7. Value-at-Risk (VaR)
8. Conclusions
9. Constraints (again)
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2 Background
CMI released new projection spreadsheet.
Calibration is done by new APCI model.
See Continuous Mortality Investigation [2017].
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2 Background
CMI intended APCI model for calibratingdeterministic targeting spreadsheet.
Richards et al. [2017] show how to implement it asa fully stochastic model.
Presented at sessional meeting of IFoA on 16thOctober 2017.
Paper and materials at www.longevitas.co.uk/apci
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3 APCI model
logmx,y = αx + βx(y − y) + κy + γy−x (1)
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3 Related models for logmx,y
Age-Period : αx + κy (2)
APC : αx + κy + γy−x (3)
Lee-Carter : αx + βxκy (4)
APCI : αx + βx(y − y) + κy + γy−x (5)
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3 Model relationships
Age-Period:αx + κy
APC:αx + κy + γy−x
Lee-Carter:αx + βxκy
APCI:αx + βx(y − y) + κy + γy−x
Add βxChange nature of κy
Add γy−xChange nature of βxChange nature of κy
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3 Model relationships
Age-Period:αx + κy
APC:αx + κy + γy−x
Lee-Carter:αx + βxκy
APCI:αx + βx(y − y) + κy + γy−x
Add γy−x
Add βxChange nature of κy
Add γy−xChange nature of βxChange nature of κy
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3 Model relationships
Age-Period:αx + κy
APC:αx + κy + γy−x
Lee-Carter:αx + βxκy
APCI:αx + βx(y − y) + κy + γy−x
Add γy−xAdd βx
Add βxChange nature of κy
Add γy−xChange nature of βxChange nature of κy
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3 Model relationships
Age-Period:αx + κy
APC:αx + κy + γy−x
Lee-Carter:αx + βxκy
APCI:αx + βx(y − y) + κy + γy−x
Add γy−xAdd βx
Add βxChange nature of κy
Add γy−xChange nature of βxChange nature of κy
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3 Model relationships
Age-Period:αx + κy
APC:αx + κy + γy−x
Lee-Carter:αx + βxκy
APCI:αx + βx(y − y) + κy + γy−x
Add γy−xAdd βx
Add βxChange nature of κy
Add γy−xChange nature of βxChange nature of κy
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3 APCI model
APCI model can be viewed superficially as either:
An APC model with added Lee-Carter-like βxterm, or
A Lee-Carter-like model with added γy−x cohortterm.
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3 APCI model
. . .but there are important differences:
In the Lee-Carter model the change in mortality isage-dependent: βxκy.
In the APCI model only the expected change isage-dependent: βx(y − y).
κy in the APCI model is very different to κy in theother models.
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3 APCI model
⇒ Although related to the APC or Lee-Carter models,the APCI model is not a generalization of either.
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4 Identifiability
All of these models have an infinite number ofpossible parameterisations.
Pick the Age-Period model as a simple example. . .
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4 Identifiability — problem
If we set:
α′x = αx + v,∀xκ′y = κy − v,∀y
then the model will have the same fitted values for anyreal-valued v.
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4 Identifiability — solution
Use an identifiability constraint to impose desiredbehaviour without changing fit.
Choice of identifiability constraints helpsinterpretation and can make parameters like κyforecastable.
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4 Constraints
Age-Period model:
Imposing∑
y κy = 0 does not change the fit. . .
. . .but it means that αx is (broadly) the average oflog µx,y over the period.
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4 Constraints used
AP :∑y
κy = 0 (6)
LC :∑y
κy = 0,∑x
βx = 1 (7)
APC :∑y
κy = 0,∑x,y
γc = 0,∑x,y
(c− cmin + 1)γc = 0
(8)
where c = y − x.
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4 Constraints used
APCI model uses five identifiability constraints:∑y
κy = 0 (9)∑y
(y − y1)κy = 0 (10)∑x,y
γc = 0 (11)∑x,y
(c− cmin + 1)γc = 0 (12)∑x,y
(c− cmin + 1)2γc = 0 (13)
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4 Not all cohorts are equal
Continuous Mortality Investigation [2017] uses (forexample)
∑c γc = 0.
⇒ Cohort with one observation gets same weightas cohort with thirty observations?
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4 Not all cohorts are equal
Cairns et al. [2009] weight according to number ofobservations, i.e.
∑x,y γc =
∑cwcγc = 0.
Cairns et al. [2009] approach preferable.
See also Richards et al. [2017, Appendix C].
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4 Fitting
The Age-Period, APC and APCI models:
are linear,
use identifiability constraints, and
have parameters that can be smoothed.
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4 Fitting
Assume Dx,y ∼ Poisson(Ex,yµx,y).
AP, APC and APCI models are penalized,smoothed GLMs.
Lee-Carter model can fitted as pairwise conditionalpenalized, smoothed GLMs.
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4 Fitting
Currie [2013] sets out generalized GLM-fittingalgorithm to:
maximise likelihood,
apply linear identifiability constraints, and
smooth parameters.
Note that the Currie algorithm achieves thesesimultaneously, not in separate stages as in ContinuousMortality Investigation [2017].
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4 Constraints
Identifiability constraints do not always have to belinear; see Girosi and King [2008], Cairns et al.[2009] and Richards and Currie [2009].
However, proving that a constraint is anidentifiability constraint is harder if it is non-linear.
The Currie [2013] algorithm works with linearconstraints only.
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5 αx
Parameter estimates αx for four unsmoothed models.
60 80 100
−4
−2
0
Age
Age-Period αx
60 80 100
−4
−2
0
Age
APC αx
60 80 100
−4
−2
Age
Lee-Carter αx
60 80 100
−4
−2
Age
APCI αx
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5 αx
⇒ αx plays the same role across all four models,i.e. average log mortality by age.
. . .as long as∑y
κy = 0.
⇒ αx could be smoothed to reduce effectivedimension of model.
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5 βx
Parameter estimates βx for Lee-Carter and APCI models (bothunsmoothed).
60 80 100
0
1
2
Age
Lee-Carter βx
60 80 100
−2
−1
0
·10−2
Age
APCI βx
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5 βx
Parameter estimates βx for Lee-Carter and −βx for APCI models(both unsmoothed).
60 80 100
0
1
2
Age
Lee-Carter βx
60 80 100
0
1
2
·10−2
Age
APCI −βx
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5 βx
βx plays an analogous role in the Lee-Carter andAPCI models, namely an age-related modulationof the time index.
βx in APCI model operates on a quite differentscale due to (y − y) term.
βx in APCI model would be better multiplied by(y − y) term...
. . .and have a constraint on βx analogous to theLee-Carter one.
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5 βx
Like αx, βx could be smoothed to reduce effectivedimension of model.
Smoothing βx also improves forecasting properties;see Delwarde et al. [2007].
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5 βx
Note that the APCI model has two time-varyingcomponents:
1. An age-dependent central linear trend, (y− y), and
2. An unmodulated, non-linear term, κy.
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5 Conclusions for αx and βx
αx and βx play similar roles across all models.
What about κy and γy−x?
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5 κy
Parameter estimates κy for four unsmoothed models.
1980 2000
−0.4
−0.2
0
0.2
0.4
Year
Age-Period κy
1980 2000
−0.4
−0.2
0
0.2
0.4
Year
APC κy
1980 2000
−0.2
0
0.2
Year
Lee-Carter κy
1980 2000
−0.1
−5 · 10−2
0
5 · 10−2
Year
APCI κy
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5 κy
κy plays a similar role in the Age-Period, APC andLee-Carter models.
κy plays a very different role in the APCI model.
APCI κy values have less of a clear trend patternfor forecasting.
APCI κy values are strongly influenced bystructural decisions made elsewhere in the model.
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5 γy−x
Parameter estimates γy−x for APC and APCI models (bothunsmoothed).
1900 1950
−0.4
−0.2
0
Year of birth
APC γy−x
1900 1950
0
0.2
0.4
Year of birth
APCI γy−x
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5 γy−x
The γy−x values appear to play analogous roles inthe APC and APCI models. . .
. . .yet the values taken and the shapes displayedare very different.
If values and shapes are so different, what do γy−xvalues represent?
γy−x don’t have an interpretation independent ofthe other parameters in the same model. . .
. . . γy−x don’t describe cohort effects in anymeaningful way.
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6 To smooth or not to smooth?
Continuous Mortality Investigation [2017] smoothsall parameters.
However, only αx and βx exhibit regular behaviour.
Does it make sense to smooth κy and γy−x?
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6 To smooth or not to smooth?
CMI’s smoothing parameter for κy is Sκ.
Smoothing penalty for κy is
10Sκny∑y=3
(κy − 2κy−1 + κy−2)2.
Value for Sκ is set subjectively.
What is the impact of smoothing κy?
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6 Impact of smoothing APCI κy
life expectancies are [...] very sensitive to thechoice made for Sκ, with the impact varyingacross the age range. At ages above 45,changing Sκ by 1 has a greater impact thanchanging the long-term rate by 0.5%.”
Continuous Mortality Investigation [2016, page 42]
See also https://www.longevitas.co.uk/site/informationmatrix/signalornoise.html
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6 Impact of smoothing APCI κy
Sκ has a large impact because κy collects featuresleft over from other parts of the model structure.
Indeed, κy collects every remaining period effectand applies it without any age modulation.
If κy is a “left-over”, should one smooth it at all?
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7 Trend risk v. one-year view?
“Whereas a catastrophe can occur in aninstant, longevity risk takes decades to unfold”
The Economist [2012]
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7 Trend risk v. one-year view
Solution from Richards et al. [2014]:
Simulate next year’s experience.
Refit the model.
Value liabilities.
Repeat. . .
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7 Sensitivity of forecast
Year
Log(
mor
talit
y)
1960 1980 2000 2020 2040
−6.0
−5.5
−5.0
−4.5
−4.0
−3.5
−3.0●●●
●●●●●●●
●●●●●●●●●
●●●●●
●●●●●
●●●●
●●●●●
●●
●●
●●●●
●●●●
● Observed male mortality at age 70 in E&WCentral projections based on simulated 2011 experience
Source: Lee-Carter example from Richards et al. [2014].
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7 Forecasting
Approach from Kleinow and Richards [2016] forparameter uncertainty:
γy−x: use ARIMA model without mean.
κy under AP, APC and LC models: use ARIMAmodel with mean.
κy under APCI model: use ARIMA model withoutmean.
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7 Liability densities
Value-at-risk capital requirements for annuities payable to male70-year-olds. Source: Richards et al. [2017, Table 4].
N = 4986 Bandwidth = 0.05074
11 12 13 14
0.0
0.2
0.4
0.6
0.8
1.0
1.2 AP(S)
N = 5000 Bandwidth = 0.02201
Den
sity
13.5 14.0 14.5 15.0 15.5
0
1
2
3
4APC(S)
11.5 12.0 12.5 13.0 13.5 14.0
0.0
0.5
1.0
1.5
2.0LC(S)
Den
sity
12.5 13.0 13.5 14.0 14.5
0.0
0.5
1.0
1.5APCI(S)
See also https://www.longevitas.co.uk/site/informationmatrix/twinpeaks.htmlwww.longevitas.co.uk 55/74
7 Value-at-risk
Variety of density shapes.
⇒ not all unimodal.
Considerable variability between models.
⇒ need to use multiple models.
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7 Value-at-risk
VaR99.5% capital-requirement percentages by age for four models.Source: Richards et al. [2017].
60 70 80 90
2
4
6
Age (years)
Age-Period(S)
APC(S)
Lee-Carter(S)
APCI(S)
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7 Value-at-risk
Q. Why do capital requirements reduce with agefor Lee-Carter, but not with APCI?
A. κy is unmodulated by age in APCI model.
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8 Conclusions
APCI model is implementable as a fully stochasticmodel.
APCI model shares features and drawbacks withAge-Period, APC and Lee-Carter models.
Smoothing APCI αx and βx seems sensible.
Smoothing APCI κy and γy−x is not sensible.
Currie [2013] algorithm makes fitting penalized,smoothed GLMs straightforward.
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References I
A. J. G. Cairns, D. Blake, K. Dowd, G. D. Coughlan,D. Epstein, A. Ong, and I. Balevich. A quantitativecomparison of stochastic mortality models using datafrom England and Wales and the United States.North American Actuarial Journal, 13(1):1–35, 2009.
Continuous Mortality Investigation. CMI MortalityProjections Model consultation — technical paper.Working Paper 91, 2016.
Continuous Mortality Investigation. CMI MortalityProjections Model: Methods. Working Paper 98,2017.
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References II
I. D. Currie. Smoothing constrained generalized linearmodels with an application to the Lee-Carter model.Statistical Modelling, 13(1):69–93, 2013.
A. Delwarde, M. Denuit, and P. H. C. Eilers.Smoothing the Lee-Carter and Poisson log-bilinearmodels for mortality forecasting: a penalizedlikelihood approach. Statistical Modelling, 7:29–48,2007.
F. Girosi and G. King. Demographic Forecasting.Princeton University Press, 2008. ISBN978-0-691-13095-8.
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References III
T. Kleinow and S. J. Richards. Parameter risk intime-series mortality forecasts. ScandinavianActuarial Journal, 2016(10):1–25, 2016.
S. J. Richards and I. D. Currie. Longevity risk andannuity pricing with the Lee-Carter model. BritishActuarial Journal, 15(II) No. 65:317–365 (withdiscussion), 2009.
S. J. Richards, I. D. Currie, and G. P. Ritchie. Avalue-at-risk framework for longevity trend risk.British Actuarial Journal, 19 (1):116–167, 2014.
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References IV
S. J. Richards, I. D. Currie, T. Kleinow, and G. P.Ritchie. A stochastic implementation of the APCImodel for mortality projections, 2017.
The Economist. The ferment of finance. Special reporton financial innovation, February 25th 2012:8, 2012.
More on longevity risk at www.longevitas.co.uk
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10 Corner cohorts
Number of observations for each cohort in the data region.
ymin ymaxxmin
xmax1
2
2
3
3
3
4
4
4
4
12
2
3
3
3
4
4
4
4Data region Forecast region
yforecast
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10 Constraints (again)
Both Continuous Mortality Investigation [2017]and Richards et al. [2017] avoid estimating “cornercohorts”.
This means not all constraints are required foridentifiability.
Continuous Mortality Investigation [2017] andRichards et al. [2017] both fit over-constrainedAPCI models.
What impact does this have?
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10 Constraints (again)
Over-constrained models reduce thegoodness-of-fit. . .
. . .but can be used to impose desirable behaviouron parameters.
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10 APC model — κy
Parameter estimates κy APC(S) model
1980 2000
−0.4
−0.2
0
0.2
0.4
Year
κy (over-constrained)
1980 2000−0.4
−0.2
0
0.2
0.4
Year
κy (minimal constraints)
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10 APC model — γy−x
Parameter estimates γy−x APC(S) model
1900 1950
−0.4
−0.2
0
Year of birth
γy−x (over-constrained)
1900 1950
0
0.2
0.4
Year of birth
γy−x (minimal constraints)
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10 APC model
κy robust to over-constrained model.
Values for γy−x differ, but shape similar.
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10 APCI model — κy
Parameter estimates κy APCI(S) model
1980 2000
−0.1
−0.05
0
0.05
Year
κy (over-constrained)
1980 2000
−0.05
0
0.05
0.1
0.15
Year
κy (minimal constraints)
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10 APCI model — γy−x
Parameter estimates γy−x APCI(S) model
1900 1950
0
0.2
0.4
Year of birth
γy−x (over-constrained)
1900 1950
0
0.2
0.4
0.6
0.8
Year of birth
γy−x (minimal constraints)
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10 APCI model
Neither κy nor γy−x robust to over-constrainedmodel.
κy in APCI model is a term which picks upleft-over aspects of fit.
γy−x changes radically depending on constraintchoices.
⇒ What are the implications for the CMIspreadsheet of using γy−x from APCI model?
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