The saint mortality model theory and application
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The SAINT mortality model: theory and application. Quant Congress USA New York, 9 July 2008. Søren Fiig Jarner Chief Analyst [email protected] Tryk Alt+F8 og Afspil auto_open for at vise værktøjslinien til opdatering af automatisk indsat tekst (forfatterinfo og præs.overskrift 2 på masterdias).

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The SAINT mortality model: theory and application

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The saint mortality model theory and application

The SAINT mortality model: theory and application

Quant Congress USA

New York, 9 July 2008

Søren Fiig Jarner

Chief Analyst

[email protected]

Tryk Alt+F8 og Afspil auto_open for at vise værktøjslinien til opdatering af automatisk indsat tekst (forfatterinfo og præs.overskrift 2 på masterdias)


Agenda

The SAINT mortality model

Agenda

  • Motivating example: Danish mortality

    • data highly volatile, but with underlying structure

    • Danish vs. international mortality

  • The SAINT framework

    • short-term deviations from long-term trend

  • Population dynamics and frailty

  • The model

    • illustrative example

  • Forecasts and uncertainty


Evolution of danish female mortality

The SAINT mortality model

Evolution of Danish female mortality

Life expectancy 40 yrs (1835)

Life expectancy 80 yrs (2006)

Δlife expectancy = 13 yrs

Δlife expectancy = 21 yrs

Δlife expectancy = 6 yrs

See Jarner et. al (2008) for the life expectancy decomposition


A more detailed look at the recent development

The SAINT mortality model

A more detailed look at the recent development

Danish female mortality

Age

Very little improvement

at the highest ages

100

High annual rates

of improvement

90

80

70

60

50

40

Stagnation/increase

from 1980 to 1995

Sharp decline in

young age mortality

30

20


Simple projections very sensitive to estimation period

The SAINT mortality model

Simple projections very sensitive to estimation period!

Danish female mortality

Reasonable short-term projections

Age

100

90

80

70

60

50

40

30

20

1990

Implausible long-term projections lacking (biological) structure


Data characteristics and modelling challenge

The SAINT mortality model

Data characteristics and modelling challenge

  • General pattern

    • age-specific mortality rates declining over time

    • rates of improvement decreasing with age (rectangularization)

  • Substantial deviations from the general pattern

    • even periods with increasing mortality for some age groups

  • Challenge: Produce plausible, long-term forecasts reflecting both the underlying trend and the ”wildness” seen in data

  • Idea: Estimate the underlying trend from less volatile reference data


Data and terminology

age

x+1

x

t+1

time

t

The SAINT mortality model

Data and terminology

  • Human Mortality Database (www.mortality.org)

  • Danish and international female mortality from 1935 to 2004

    • 18 countries in the international dataset: USA, Japan, Germany, UK, France, Italy, Spain, Australia, Canada, Holland, Portugal, Austria, Belgium, Switzerland, Sweden, Norway, Finland & Iceland

  • Death counts and exposures for each year and each age group

D(t,x) = number of deaths

E(t,x) = exposure (”years lived”)

Death rate, D(t,x)/E(t,x), is an estimate of (the average of) underlying intensity, μ(t,x)

Death probability, q(t,x) = 1-e-∫μ(t,x) ≈∫μ(t,x)


Danish fluctuations around stable international trend

The SAINT mortality model

Danish fluctuations around stable international trend

Danish and international female mortality

Age

Danish life expectancy

among the highest in

the world

Similar development

at the highest ages

100

90

80

70

60

Is this the beginning

of a catch up period?

50

40

30

Denmark falling behind

the international trend

20


Saint spread adjusted international trend framework

The SAINT mortality model

SAINT (Spread Adjusted InterNational Trend) framework

Parsimonious parametric model for long-term trend

: Family of intensity surfaces (gender specific)

: MLE based on Poisson-model;

Time-series model for short-term deviations (spread)

: Age-dependent vector of regressors (fixed)

: Time-dependent spread parameters (estimated);

Fit multivariate time-series model for


Trend modelling concepts

The SAINT mortality model

Trend modelling concepts

  • Population dynamics

    • Ensure consistent intensity surfaces over time and ages by aggregating individual intensities to population level

    • Individuals living in the same period of time are influenced by common as well as individual factors

    • Some factors have a cumulative effect on mortality

  • Frailty

    • People are genetically different. Only the more robust individuals will attain very high ages

    • Lack of historic improvements among the very old may be due to selection effects. In the future the frailty composition at old ages will change


From individual to population intensity

The SAINT mortality model

From individual to population intensity

  • Mortality intensity for an individual with frailty

  • Individual survival function

  • Survival function for population with frailty distribution

  • Population intensity


Selection effects within a cohort

The SAINT mortality model

Selection effects within a cohort

Individual:

Cohort:

Intensity (μ)

(x)


Selection when mortality is time varying

The SAINT mortality model

Selection when mortality is time-varying

Individual:

Average frailty in population


Trend model

The SAINT mortality model

Trend model

  • Underlying individual intensities

  • Population intensity (mean 1 and variance σ2Γ-distributed frailties)

”treatment” level

”wear-out” rate

”accident” rate

Previous values of κ

are ”remembered” by

the population


Trend fit and forecast

The SAINT mortality model

Trend – fit and forecast

International female mortality

Increasing old age rate

of improvement

Age

100

90

80

70

60

50

40

Early, young are rate

of improvement = 9.1%

General, long-term rate

of improvement = 1.8%

30

20


Spread model

The SAINT mortality model

Spread model

  • Model of Danish mortality

  • The spread is assumed to fluctuate around zero

    • that is, no mean term included in the model

  • The spread controls the length and magnitude of deviations

    • and provides information about projection uncertainty

Mean zero, orthogonal regressors

normalized to (about) 1 at age 20 and 100


Illustration of spread adjustment

The SAINT mortality model

Illustration of spread adjustment

Female mortality in 2004

International trend

Danish data

Danish fit

Estimatesa2004= 21%b2004= 5%c2004=-19%


Long recovery period

The SAINT mortality model

Long recovery period

Estimated and forecasted spread

Fitted at

Fitted bt

Fitted ct

Forecast


Danish mortality fit and forecast

The SAINT mortality model

Danish mortality – fit and forecast

Danish female mortality and international trend

Similar development in

old age mortality

Age

100

90

80

70

60

50

40

Denmark falling behind

… and catching up again

30

20


Forecast uncertainty

The SAINT mortality model

Forecast uncertainty

  • Analytical methods

    • only feasible for very few quantities of interest, e.g. the spread itself

  • Monte Carlo

    • simulate N spread series and calculate mortality forecasts for each

    • calculate quantity of interest, e.g. life expectancy, for each forecast

    • compute uncertainty measures, e.g. 95%-confidence intervals

Females aged 60 in 2005


Summing up

The SAINT mortality model

Summing up

  • Model for small population mortalities showing irregular patterns of improvement

  • Parsimonious trend model

    • estimated from reference population

    • biologically plausible mortality projections

    • future improvements in high age mortality as frailty composition changes

  • Time series model for deviations from trend

    • spread controls length and size of excursions from trend

  • Projection uncertainty calculated by Monte Carlo methods


Selected readings

The SAINT mortality model

Selected readings

  • Lee & Carter (1992). Modelling and forecasting U.S. mortality. JASA, 659-675.

  • De Jong & Tickle (2006). Extending Lee-Carter mortality forecasting. Mathematical Population Studies, 1-18.

  • Cairns et al. (2007). A quantitative comparison of stochastic mortality models using data from England & Wales and the United States.

  • Vaupel et al. (1979) . The impact of Heterogeneity in Individual Frailty on the Dynamics of Mortality. Demography, 439-454.

  • Thatcher (1999). The Long-Term Pattern of Adult Mortality and the Highest Attained Age. JRSS A, 5-43.

  • Jarner, Kryger & Dengsøe (2008). The evolution of death rates and life expectancy in Denmark. To appear in Scandinavian Actuarial Journal.


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