MORTALITY IMPROVEMENT RATE MODELLING AND PROJECTINGMODELLING AND PROJECTING

STEVEN HABERMAN AND ARTHUR RENSHAWCass Business School, City University London

Eighth International Longevity Risk and Capital Markets Solutions ConferenceEighth International Longevity Risk and Capital Markets Solutions ConferenceUniversity of Waterloo7‐8 September 2012

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AGENDAAGENDA

M ti ti• Motivation

Hi t• History

M d l C t ti• Model Construction

C St d M d lli M t lit R t• Case Study: Modelling Mortality RatesModelling Mortality Improvement Rates

• Conclusions

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MODELLING & PROJECTING MORTALITYMODELLING & PROJECTING MORTALITY IMPROVEMENT RATES: HISTORY

MR‐mortality ratesLee & Carter (1992)Brouhns, Denuit, Vermunt (2002)Renshaw & Haberman (2006)

Cairns, Blake, Dowd et al. (2008)Plat (2009)Haberman & Renshaw (2011)

MR‐mortality improvement rates

Renshaw & Haberman (2006) Haberman & Renshaw (2011)

MR mortality improvement ratesWillets (2004)

Richards, Kirkby, Currie (2005)Renshaw & Haberman (2006)

CMI

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DATADATA

, , : age , period xt xt xtd e x t

d,ˆ - empirical mortality ratext

xt x txt

dy me

ˆ ˆ1

, , 1

, , 1

12 - mortality improvement rate

ˆ ˆ1x t x t

xtx t x t

m mz

m m

improving mortality (over time) 0xtz

d i i li ( i ) 0deteriorating mortality (over time) 0xtz

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5

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Fig 3a. England & Wales 1961-2007 male mortality experience.Mortality improvement rates (MIR) and their average over time

(dotted line), plotted against calendar year for various fixed ages.7

Fig 3b. England & Wales 1961-2007 male mortality experience.Mortaility improvement rates (MIR) plotted against year of observation

for various fixed years-of-birth (cohorts). 8

Model & forecast PredictionsModel & forecast Predictions

MR route I indices,ˆ x tm , nx j t jq

MR (Route I ) route II indices

Map Inverse Map

, nx j t jq

Map

(differentiate)(&)

(scale)

Inverse Map(rescale)

(&)(integrate)

(Route II )MIR

Model & forecast

,x tz , nx j t jz

,x tm,x tz

, nx j t jq

‐ Central mortality rate (MR);      ‐mortality improvement rate (MIR)- predicted probability of death MR predictions‐ route I; MIR predictions‐ route II

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MORTALITY RATES (Route I) APPROACH

- central rate of mortalityxtm

TARGET:

xt

PARAMETRIC PREDICTOR STRUCTURES:

LC : xt x x tLC

1 : xt x x t t xH

0 : xt x t t xH

(1) (2)5 : xt x t tM x x xt x t t

(1) (2)6 : xt x t t t xM x x

(1) (2) (3)7 :M x x b x ( ) ( ) ( )7 : xt x t t t t xM x x b x

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MORTALITY RATES (Route I) APPROACH

1 1

2 21 1( ) , k kx x

i x i xb x x x i x x i

k k

MODEL FITTING: ~xt xt xtD P e mLet i.i.d., with constant dispersion xt xt xt

, , xt xtxt xt xt xt

xt xt xt

D mY E Y m Var Ye e

Let , p

xt xt xt

log xt xtm xt xt xtelog link  , weights, parametric predictor  , 

V u uscale parameter  variance function 

To fit: minimise the model deviance.

, 

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MORTALITY IMPROVEMENT RATES (Route II) APPROACH

MOTIVATION:

1log xt xtxt xt

xt

mmm t t

Notational convention: re‐define symbols appropriately

TARGET:

- mortality improvement ratext

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MORTALITY IMPROVEMENT RATES O O S(Route II) APPROACH

:LC

DUAL PARAMETRIC PREDICTOR STRUCTURES:

: xt x tLC

1 : xt x t t xH

0 : xt t t xH

(1) (2)5 : xt t tM x x xt t t

(1) (2)6 : xt t t t xM x x

(1) (2) (3) (1) (2) (3)7 : ( )xt t t t t xM x x b x LINK FUNCTION: IDENTITY

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MORTALITY IMPROVEMENT RATES (Route II) APPROACH

MODEL FITTING (Single stage):

Assume  2~ ,xt xtZ N i.i.d., constant dispersion p

2 1, xt xt xtxt

E Z Var Z

xt xtidentity link, parametric predictor  , weights

2 1Vscale parameter ariance f nction 1V u scale parameter  variance function , .

To fit: minimise the model deviance.

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MORTALITY IMPROVEMENT RATES (Route II) APPROACH

MODEL FITTING (Joint Modelling with 2 stages):

Stage 1:

Assume  i.i.d., variable dispersion

Stage 1:

2~ ,xt xt xtZ N

identity link parametric predictor weights

2, xtxt xt xt

xt

E Z Var Z

xtidentity link, parametric predictor  , weights

2 1V u scale parameter  variance function ,

xt xt

Yields: residuals ˆxt xt xtr z to form Stage 2 responses.

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MORTALITY IMPROVEMENT RATES (Route II) APPROACH

2xtR

Stage 2:Model squared residuals  as follows

2

2 2, xtxt xt xt

xt

E R Var R

log xt x x xtlog link  , predictor  weights ,

2V u uscale parameter variance function V u uscale parameter  , variance function 

Yields: fitted values  x̂t which are then used as Stage 1 weights

To fit: minimise the model deviances.

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FITTING THE MODELSFITTING THE MODELS

• Use England and Wales  male mortality experience for 1961‐2007 ages 20‐89

•Fit Route I, Route II models (single stage and joint stages)

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MODEL DYNAMICSMODEL DYNAMICS

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MAPPING MIR PREDICTIONS TO MRPREDICTIONS

Compute MIR predictions , nx t jz and

convert to MR predictions using ‐

,21 2 3nx t jz

j

,, , 1

,

; 1,2,3,...2

n

n n

n

jx t j x t j

x t j

m m jz

Need starter values mNeed starter values  , nx tmOnce converted, compute

, ,1 expn nx t j x t jq m

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TOPPING OUT BY AGETOPPING OUT BY AGEExtrapolate model projectionsp p j

, : 1, 2,..., ; ( )nx j t j k kq j x x x x

using

log 1x j t j kq a b j x x

,g

1 ;nx j t j k

k k

q j

c j x x j x x

1 1j x x x x x x x 1, , 1,k k kj x x x x x x x

requiring

1, 1 , ,, , k n k k n k n kx t x x x t x x t x xq q q

to determine a b c

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to determine a, b, c.

Fig 5. E&W 1961-2007 males, ages 20-89. AR(1) period index

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processeswith forecasts (RH panels), residuals (LH panels): MIR- models (rows).

INDICES OF INTERESTINDICES OF INTEREST

f dLife expectancy predictions‐

11x j n x j t jl t j q

,1 2 nx j n x j t j

jx n

x n

j qe t

l t

Fixed rate annuity value predictions‐

jl t j v

0

x j nj

x nx n

l t j va t

l t

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ESTIMATING PREDICTION INTERVALSESTIMATING PREDICTION INTERVALS BY SIMULATION: MIR

Generate predictions and prediction intervals.

Period index modelled as an AR(1) processPeriod index modelled as an AR(1) process.

(VAR(1) process if multivariate).

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ESTIMATING PREDICTION INTERVALS BYESTIMATING PREDICTION INTERVALS BY SIMULATION: MIR

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FORECASTINGFORECASTING

U E l d d W l l i f 1961• Use England and Wales male experience for 1961‐1982, ages 55‐89 to fit models and then calculate life expectancies (and annuity values) for age in 1982xexpectancies (and annuity values) for age    in 1982by cohort method where      65,66,...,80x

x

• Compare values for same indices calculated using raw mortality rates for 1983‐2007, by depicting ( di t d t l) l i t(predicted‐actual) values against age.

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Fig 15c EW females Retrospective relative error in 1982 predicted log death rates

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Fig 15c. EW females. Retrospective relative error in 1982 predicted log death rates, averaged over ages, years, cohorts respective. (Route II LH panels; Route I RH panels) based on region bounded by ages 60-89, years 1983-2007.

CONCLUSIONSCONCLUSIONS

• Differencing the mortality rate series can lead• Differencing the mortality rate series can lead to stationarity.

• Mortality improvement rate models have• Mortality improvement rate models have some advantages ‐ age span of data is less restricted (for M5, 

)g p (

M6, M7)‐ age index,    , plays no roleF d di ti h l d

x• Forward predictions show regular and desirable properties.

• Cohort based version of MIR models is being• Cohort‐based version of MIR models is being developed.  

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