Trends and Volatility in Mortality and Longevity Risk: Insights from Econometric and Actuarial Modeling

Transcription

1 Trends and Volatility in Mortality and Longevity Risk: Insights from Econometric and Actuarial Modeling Presentation at Institut Fuer Finanz-Und Aktuarwissenschaften Universitaet Ulm Prof with Carolyn Ndigwako Njenga (UNSW), Severine Gaille (University of Lausanne) and David Odell (Insureware) THE UNIVERSITY OF NEW SOUTH WALES Australian School of Business School of Actuarial Studies September, 9

2 Research Aims Research program: Australian Research Council Linkage Grant (with PwC, APRA, World Bank) Longevity Trends by Country and major Cause of Death Stochastic Trends for Country Level and by Age Principal Components and Number of Factors Cointegration and Common Stochastic Trends Australian data by Age group - Smooth by Age and Cointegration Cause of Death data: Trends by Age, Cohort and Time Actuarial Loss Reserving Models - Trends by Age (Development period), Cohort (Accident period) and Time (Calendar period)

3 Background Australia has a compulsory 9 per cent Superannuation Guarantee contribution (introduced 99) Significant accumulation of assets for retirement Only one life insurer offers life annuities and longevity insurance variable annuity (Lifestream Guaranteed Long Term Income) Government means tested pension Henry Tax Review - lack of longevity insurance market High current interest in longevity risk from government and industry

4 Longevity Trends by Country Figure: Average Death Rates for Developed Countries

5 Longevity Trends by Country - Standardised Rates Females AUSTRALIA FEMALE ENGLAND FEMALE JAPAN FEMALE NORWAY FEMALE USA FEMALE Males AUSTRALIA MALE ENGLAND MALE JAPAN MALE NORWAY MALE USA MALE

6 Longevity Trends for Australia by Age Declining Mortality Age Declining Mortality Age Declining Mortality Age Year Year Year Declining Mortality Age 7 Declining Mortality Age 9 Decline in Mortality Rates Age Year Year Years Figure: Age Specific Death Rates at Different Ages

7 Unit roots Trend-stationary time series with φ < The random walk with drift y t = µ + φy t + u t () y t = µ + y t + u t () If φ = then the series has a unit root and as T the effect of the shocks persist and accumulate as stochastic trends in the series: y t = µ + y + u t () t=

8 Unit Root Tests - by Country Constant Lags Constant, Trend Lags Australia Australia England.998. England Japan.9.97 () Japan Norway () Norway USA () USA Table: ADF Tests on Male Standardised Mortality Rates () Indicates significant trend

9 Principal Components Females Relative Importance of Principal Components Males Relative Importance of Principal Components.78. Variances Variances Comp. Comp. Comp. Comp. Comp. Comp.6 Comp.7 Comp.8 Comp.9Comp. Comp. Comp. Comp. Comp. Comp. Comp.6 Comp.7 Comp.8 Comp.9Comp. Figure: Principal Components for Australia - Mortality Rate Levels

10 Principal Components by Country - Time and Cohort Trends Country Difference Number of Factors Percentage Variation Australia Diagonal Horizontal England Diagonal Horizontal Japan Diagonal Horizontal 98. Norway Diagonal Horizontal USA Diagonal 97. Horizontal 97.7 Table: PCA Factors using for differences in rates for countries in study

11 Principal Components Australia England Japan Comp Comp Comp Comp Comp Comp6 Comp7 Comp Comp Comp Comp Comp Comp6 Comp Comp Comp Comp Comp Comp6 Norway USA Comp Comp Comp Comp Comp Comp6 Comp7 Comp Comp Comp Comp Comp Comp6 Comp7 Figure: Multiple PCA Factors for Mortality Differences

12 VAR and VECM VAR(p) for p lags: m t = A + A m t + A m t + + A p m t p + e t where m t = (m t,..., m kt,..., m Kt ) for k =,..., K time series. Long-run specification of VECM: m t = Γ m t + + Γ p m t p+ + Πm t p + A + e t () where Γ i = (I A A i ), i =,..., p Π = (I A A p )

13 VAR and Cointegration Lee-Carter model special case of VAR model VAR Models developed for standardised mortality rate at country level Autoregression in country mortality rates: Norway AR(); other countries AR(); Cross country lags significant Countries in the study have stochastic trends with drift; no evidence of common stochastic trends (cointegration) Cross country diversification.

14 Dynamic Heligman-Pollard (H-P) Model Younger and older ages stationary Mid range of ages evidence of unit roots - non stationary Large number of parameters in VAR model Significant number of stochastic factors (up to 8) Use Heligman-Pollard to smooth Australian data by age group Heligman-Pollard model with stochastic parameters (8 parameters as VECM) q x = A (x+b)c + D exp[ E(log{ x F }) ] + GHx + GH x ()

15 6 8 Age 6 8 Age 6 8 Age 6 8 Age 6 8 Age 6 8 Age 6 8 Age 6 8 Age Heligman Pollard Model Fit q(x)..... log(q(x) 7 6 q(x)..... log(q(x) q(x)..... log(q(x) q(x).... log(q(x) 8 6 Figure: H-P Fit for Australia - Males

16 Time Time Time Time Time Time Time Time Heligman Pollard Model Parameters A B C D E F G H 6 e e 6e 8e e.9... Figure: H-P Parameters for Australia - Males

17 Heligman Pollard as VECM in Parameters VECM - VAR model allows for cointegration and multiple factors for volatility Cross correlations included Improved fit compared to Lee-Carter Captures long run trends although issues with parsimony and forecasting

18 Cause of Death Data - WHO Cancer Circulatory Disease Respiratory Disease Infectious Diseases External causes Others not included Australia Japan Singapore UK Italy Sweden Norway Switzerland USA Table: Major Causes of Death and Countries

19 Cause of Death Analysis Analysis of Cause of Death data by 9 major countries and major causes: Pattern by age similar for each cause across countries Time trends differ by cause, age and country (impact of standard of living, smoking, obesity, etc) Strong evidence of cohort trends for most causes Cancer and circulatory disease pattern different for males and females (impacts time trends and age pattern)

20 Actuarial Models for Trends by Cause of Death Cohort trends significant but also need to smooth age pattern and capture time trends Actuarial loss reserving techniques can be applied: Age trend - development period Cohort trend - accident period Time trend - calendar period Stochastic loss reserving model with parameters for trends fitted to cause of death male data. Preliminary research.

21 Cohorts for Cause of Death Data Country From To USA Australia Switzerland Japan Singapore Italy Norway Sweden UK

22 Trends by Age by Country for Cause of Death - Cancer Australia:Cancer:PTF[M6-]::Dev. Yr Trends Japan:Cancer:PTF[M6-]::Dev. Yr Trends Sweden:Cancer:PTF[M6-]::Dev. Yr Trends Italy:Cancer:PTF[M6-]::Dev. Yr Trends Singapore:Cancer:PTF[M6-]::Dev. Yr Trends UK:Cancer:PTF[M6-]::Dev. Yr Trends Switzerland:Cancer:PTF[M6-]::Dev. Yr Trends Norway:Cancer:PTF[M6-]::Dev. Yr Trends USA:Cancer:PTF[M6-]::Dev. Yr Trends

23 Trends by Cohort by Country for Cause of Death - Cancer Australia:Cancer:PTF[M6-]::Acc. Yr Trends Japan:Cancer:PTF[M6-]::Acc. Yr Trends Sweden:Cancer:PTF[M6-]::Acc. Yr Trends Italy:Cancer:PTF[M6-]::Acc. Yr Trends Singapore:Cancer:PTF[M6-]::Acc. Yr Trends UK:Cancer:PTF[M6-]::Acc. Yr Trends Switzerland:Cancer:PTF[M6-]::Acc. Yr Trends Norway:Cancer:PTF[M6-]::Acc. Yr Trends USA:Cancer:PTF[M6-]::Acc. Yr Trends

24 Trends by Time by Country for Cause of Death - Cancer Australia:Cancer:PTF[M6-]::Cal. Yr Trends Japan:Cancer:PTF[M6-]::Cal. Yr Trends Sweden:Cancer:PTF[M6-]::Cal. Yr Trends Italy:Cancer:PTF[M6-]::Cal. Yr Trends Singapore:Cancer:PTF[M6-]::Cal. Yr Trends UK:Cancer:PTF[M6-]::Cal. Yr Trends Switzerland:Cancer:PTF[M6-]::Cal. Yr Trends Norway:Cancer:PTF[M6-]::Cal. Yr Trends USA:Cancer:PTF[M6-]::Cal. Yr Trends

25 Volatility by Age by Country for Cause of Death - Cancer e- 6e- e- Australia:Cancer:PTF[M6-]::MLE Variance vs Dev. Yr e Italy:Cancer:PTF[M6-]::MLE Variance vs Dev. Yr e- 6e- e- e Switzerland:Cancer:PTF[M6-]::MLE Variance vs Dev. Yr Japan:Cancer:PTF[M6-]::MLE Variance vs Dev. Yr e e Singapore:Cancer:PTF[M6-]::MLE Variance vs Dev. Yr Norway:Cancer:PTF[M6-]::MLE Variance vs Dev. Yr Sweden:Cancer:PTF[M6-]::MLE Variance vs Dev. Yr UK:Cancer:PTF[M6-]::MLE Variance vs Dev. Yr e-.78.e- e-.e- e-.e- e-.e- e- e USA:Cancer:PTF[M6-]::MLE Variance vs Dev. Yr...e-.. e-...e-.. e-...e-.. e- e- e- e

26 Modeling Cause of Death with Actuarial Loss Reserve Models Five major causes across nine countries analyzed. Summary of key trend and volatility results: Age patterns differ by cause but similar across countries Time trends differ by country and cohort trend significant for causes Differing time trends and differing relative importance of causes adds complexity to modeling country age patterns, trends and volatility

27 Correlations of Model Residuals by Cause of Death Final Weighted Residual Correlations Between Datasets: Japan Final Weighted Residual Correlations Between Datasets: Italy Cancer Circ Ext Infectious Resp Cancer Circ Ext Infectious Resp Cancer Circ Ext Infectious Resp Cancer Cancer Cancer Circ Circ Circ Ext Ext Ext Infectious Infectious Infectious Resp Resp Resp iterations were executed Final Weighted Residual Correlations Between Datasets: Norway Final Weighted Residual Correlations Between Datasets: Australia 8 iterations were executed Final Weighted Residual Correlations Between Datasets: Singapore iterations were executed Final Weighted Residual Correlations Between Datasets: Sweden Cancer Circ Ext Infectious Resp Cancer Circ Ext Infectious Resp Cancer Circ Ext Infectious Resp Cancer Cancer Cancer Circ Circ Circ Ext Ext Ext Infectious Infectious Infectious Resp Resp Resp iterations were executed 7 iterations were executed Final Weighted Residual Correlations Between Datasets: Switzerland 6 iterations were executed Final Weighted Residual Correlations Between Datasets: USA Cancer Circ Ext Infectious Resp Cancer Circ Ext Infectious Resp Cancer Circ Ext Infectious Resp Cancer Cancer Cancer Circ Circ Circ Ext Ext Ext Infectious Infectious Infectious Resp Resp Resp iterations were executed Final Weighted Residual Correlations Between Datasets: UK 8 iterations were executed 6 iterations were executed

28 Conclusions - Country and Age At country level: stochastic trends although not common stochastic trends Age groups within countries: older and younger ages close to stationary and middle ages stochastic trends Significant number of common factors in mortality rate levels as well as differences; multiple factor models and dimension reduction VAR model at country level; VECM model for age groups (stochastic parameter models that smooth across age)

29 Conclusions - Cause of Death Cause of death trends; differ by country and sex Similar age pattern by cause for different countries Significant cohort trends for most causes Volatilities by cause differ across age groups

30 Conclusions - Future challenges Modeling of longevity/mortality trends and volatility has many challenges Number of parameters to capture trends and volatility: fit to data versus forecasting Hierarchical and Bayesian models Financial applications and volatility