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One Birdcage Walk, Westminster, London, SW1H 9JJ
Stochastic mortality models for actuarial work
Stephen Richards22nd March 2011
Copyright c© Longevitas Ltd. All rights reserved. Electronic versions of related papers andpresentations can be found at www.longevitas.co.uk
Overview1. Deterministic model risk2. Stochastic model risk3. Advanced ages4. Conclusions
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1. Deterministic model risk• How do you know your model is correct?• What are the consequences if it is not?
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1. Deterministic model risk — CMI 2010• Deterministic• Defaults to projecting decelerating mortality improvements. . .
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1. Deterministic model risk — CMI 2010
Age
Mor
talit
y im
prov
emen
t rat
e (p
er c
ent)
50 60 70 80 90
1.5
2.0
2.5
3.0
3.5
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● Actual improvements to 2007 (smoothed)
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Source: Improvement rates labelled “actual” in CMI 2010.
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1. Deterministic model risk — CMI 2010
Age
Mor
talit
y im
prov
emen
t rat
e (p
er c
ent)
50 60 70 80 90
1.5
2.0
2.5
3.0
3.5
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● Actual improvements to 2007 (smoothed)CMI 2010 with long−term target of 1%CMI 2010 with long−term target of 2%
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Source: Smoothed actual mortality-improvement rates for males born in 1946, together with pro-jected rates according to CMI 2010 model using default parameters and a long-term target of 1%or 2% improvement per annum.
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2. Stochastic model risk• Model risk most obviously applies to deterministic models. . .• . . .but applies to all models, including stochastic ones
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2. Stochastic model risk
log(
mor
talit
y)
1960 1980 2000 2020 2040
−6.5
−6.0
−5.5
−5.0
−4.5
−4.0
−3.5●
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● Crude ratesTime Series LCSmooth LC
Source: Richards and Currie (2009). CMI assured-lives data, fitted and forecast rates for malesat age 65 under two Lee-Carter variants: DDE model (time series) and Currie-Richards model(smooth).
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2. Stochastic model risk
log(
mor
talit
y)
1960 1980 2000 2020 2040
−8
−7
−6
−5
−4
● ● ●● ● ●
● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
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● Crude ratesTime Series LCSmooth LC
Source: Richards and Currie (2009). ONS population data, fitted and forecast rates for malesat age 65 under two Lee-Carter variants: DDE model (time series) and Currie-Richards model(smooth).
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2. Stochastic model risk
Best-estimate 99.5% CapitalModel reserve reserve required
DDE 13.92 14.42 3.6%CBD5 13.96 15.04 8.0%
Source: Richards (2011). Best-estimate and stressed annuity values for a male aged 65 followingpopulation mortality in England and Wales. Continuous temporary annuities to age 105, valuedat 3% per annum. DDE model is that of Delwarde, Denuit and Eilers (2007), while CBD5 model isthat of Cairns, Blake and Dowd (2006), as modified by Currie (2010) to allow for a non-linear effectby age. The DDE model projects a single constant drift term, while the CBD5 model projects abivariate drift term.
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2. Stochastic model risk• DDE model says a reserve of 14.42 suffices with 99.5% probability• CBD5 model says the same reserve suffices with 87.3% probability
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2. Model risk• How do you know your model is correct?• You don’t• Must use a variety of models to explore model risk
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3. Advanced ages
• ONS data only goes up to age 105• Usable CMI data stops even earlier. . .
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3. Advanced ages
Age
log(
crud
e m
orta
lity
rate
)
50 60 70 80 90 100
−6
−5
−4
−3
−2
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Source: CMI assured-lives data. Logarithm of crude force of mortality for CMI data at ages50–100 aggregated over the years 2001–2005. The data above age 95 are unreliable and cannotbe included in any projection model. However, actuarial calculations for annuities and pensionstypically require mortality rates up age 120, so some form of extrapolation is required from age95 to age 120.
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3. Advanced ages
• How to get rates and projections at advanced ages?• 2D P-spline models can extrapolate by age as well as project in time
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3. Advanced ages — 2D P-spline models
Age
log(
forc
e of
mor
talit
y)
50 60 70 80 90 100 110 120
−5
−4
−3
−2
−1
0
1
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● Observed crude mortality rate2DAP fitted and extrapolated mortality rate2DAP confidence interval
Source: CMI assured-lives data fitted to 2D age-period P-spline model. Logarithm of crude force ofmortality for ONS data for males in 2007 at ages 50–104, together with the fitted and extrapolatedrates to age 120 under the 2D age-period model (Currie, Durban and Eilers, 2004). As with aprojection in time, the uncertainty over the extrapolated rates increases the greater the distancefrom the actual data, i.e. there is an expanding “funnel of doubt”. In this case the funnel is rathernarrow because of the strength of the age signal.
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3. Advanced ages — 2D P-spline models
5060
7080
90100
110120
1980
2000
2020
2040
−6
−4
−2
0
Age
Year
Source: Longevitas Ltd. Fitted and projected logarithm of force of mortality for ONS data formales. Mortality rates are simultaneously extrapolated to age 120 and projected to 2050 using thetwo-dimensional P-spline model of Currie, Durban and Eilers (2004).
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3. Advanced ages — model risk
• Model risk demands we use more than one model• Need other models capable of extrapolation by age• Cairns-Blake-Dowd (2006) model can also extrapolate by age
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3. Advanced ages — CBD 5
logµx,y = κ0,y + κ1,yS (x)
• κ0 and κ1 form a bivariate time series for projecting future rates• S(x) is a smooth function of age
Source: Cairns, Blake and Dowd (2006) as modified by Currie (2010).
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3. Advanced ages — CBD 5
Year
Kap
pa_0
1960 1970 1980 1990 2000
−3.0
−2.8
−2.6
−2.4● ●
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Year
Kap
pa_1
1960 1970 1980 1990 2000
0.085
0.090
0.095
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Age
S(x
)
50 60 70 80 90 110
−30−20−10
0102030
Source: Longevitas Ltd. Parameters for the Cairns-Blake-Dowd model number 5, as modified byCurrie (2010). The function S(x) is smoothed by splines with a five-year knot spacing, whichenables age extrapolation as in the 2D P-spline models.
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3. Advanced ages — CBD 5
5060
7080
90100
110120
1980
2000
2020
2040
−6
−4
−2
0
Age
Year
Source: Logarithm of force of mortality for ONS data for males. Mortality rates are extrapolatedto age 120 using the S(x) function, and projected to 2050 using the bivariate time series for κ0and κ1. The fitted rates have ridges due to period effects.
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4. Conclusions• CMI 2010 model defaults to decelerating mortality improvements• Model risk is very substantial for mortality projections• Don’t rely on a single projection model or methodology• Stochastic projection models can project by age as well as time
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References ICairns, A. J. G., Blake, D. and Dowd, K. 2006 A two-factormodel for stochastic mortality with parameter uncertainty: theory andcalibration, Journal of Risk and Insurance, 73, 687–718Currie, I. D., Durban, M. and Eilers, P. H. C. 2004 Smoothingand forecasting mortality rates, Statistical Modelling 4 279–298Currie, I. D. 2010 On a model of Cairns, Blake and Dowd, Workingpaper available on requestCurrie, I. D. 2011 Forecasting mortality at high ages, Longevitas blogDelwarde, A., Denuit, M. and Eilers, P.H.C. 2007 Smoothing theLee-Carter and Poisson log-bilinear models for mortality forecasting: apenalized likelihood approach, Statistical Modelling, 7, 29–48
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References IIRichards, S. J., Kirkby, J. G. and Currie, I. D. 2006 The im-portance of year of birth in two-dimensional mortality data, British Ac-tuarial Journal, 12 (I), 5–61Richards, S. J. and Currie, I. D. 2009 Longevity risk and annuitypricing with the Lee-Carter model, British Actuarial Journal (to appear)Richards, S. J. and Currie, I. D. 2011 Extrapolating Mortality Pro-jections by Age, submitted articleRichards, S. J. 2011 Applying the brakes, Longevitas blogRichards, S. J. 2011 Model risk, Longevitas blog
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