the apci model --- a stochastic implementation

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)



1 Contributors



2 Background



2 Background

CMI released new projection spreadsheet.

Calibration is done by new APCI model.

See Continuous Mortality Investigation [2017].



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



3 APCI model



3 APCI model

logmx,y = αx + βx(y − y) + κy + γy−x (1)



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)



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








Add γy−x










Add γy−xAdd βx





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.



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.



3 APCI model

⇒ Although related to the APC or Lee-Carter models,the APCI model is not a generalization of either.



4 Fitting and constraints



4 Identifiability

All of these models have an infinite number ofpossible parameterisations.

Pick the Age-Period model as a simple example. . .



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.



4 Identifiability — solution

Use an identifiability constraint to impose desiredbehaviour without changing fit.

Choice of identifiability constraints helpsinterpretation and can make parameters like κyforecastable.



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.



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.



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)



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?



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].



4 Fitting

The Age-Period, APC and APCI models:

are linear,

use identifiability constraints, and

have parameters that can be smoothed.



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.



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].



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.



5 Parameter estimates



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



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.



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



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



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.



5 βx

Like αx, βx could be smoothed to reduce effectivedimension of model.

Smoothing βx also improves forecasting properties;see Delwarde et al. [2007].



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.



5 Conclusions for αx and βx

αx and βx play similar roles across all models.

What about κy and γy−x?



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



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.



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



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.



6 Smoothing



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?



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?



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


https://www.longevitas.co.uk/site/informationmatrix/signalornoise.html


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?



7 Value-at-Risk (VaR)



7 Trend risk v. one-year view?

“Whereas a catastrophe can occur in aninstant, longevity risk takes decades to unfold”

The Economist [2012]



7 Trend risk v. one-year view

Solution from Richards et al. [2014]:

Simulate next year’s experience.

Refit the model.

Value liabilities.

Repeat. . .



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].



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.



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

https://www.longevitas.co.uk/site/informationmatrix/twinpeaks.html


7 Value-at-risk

Variety of density shapes.

⇒ not all unimodal.

Considerable variability between models.

⇒ need to use multiple models.



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)



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.



8 Conclusions



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.



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.



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.



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.



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




10 Constraints (again)



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




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?




Over-constrained models reduce thegoodness-of-fit. . .

. . .but can be used to impose desirable behaviouron parameters.



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)



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)



10 APC model

κy robust to over-constrained model.

Values for γy−x differ, but shape similar.



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)



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)



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?



the apci model --- a stochastic implementation

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