Building blocks for a mortality index in an international context

Transcription

1 1 Building blocks for a mortality index in an international context Tiziana Torri Max Planck Institute for Demographic Research Munich, 7 th September 2009

2 2 Outline Longevity risk: Identification Assessment Management Securitization Mortality Indexes Stochastic forecasts of life expectancy Univariate ARIMA Multivariate VAR Vector Error Correction

3 3 Identification Mortality risk is divided into: Process risk: random fluctuations due to the stochastic nature of mortality Catastrophe risk: unexpected shocks of mortality Uncertainty risk: random deviations due to choice of the projection model (model risk) estimated parameters of the model (param.risk) Longevity Risk refers to old age mortality

4 4 Assessment The impact of the LR is measured via: Discounted cash flows Random annual cash flows Random number of annuitants alive in t Loss function Combined with different indexes: Expected value Variance Quantile

5 5 Management Portfolio strategies of loss control Loss prevention Loss severity Portfolio strategies of loss financing Natural hedging Risk transfer Securitization Risk retention

6 6 Securitization LR is transferred to capital markets via mortality-linked securities: Longevity Bonds (EIB/BNP in 2004) Survivor Swaps (Swiss Re in 2007) Mortality Futures Mortality Forwards (JP Morgan in 2007) Mortality Options

7 7 Securitization LR is transferred to capital markets via mortality-linked securities: Longevity Bonds (EIB/BNP in 2004) Failed due to: Presence of basis risk Lack of transparency Capital strain

8 8 Mortality indexes Recently proposed: Longevity Index by Credit Suisse (2005) LifeMetrics Indices by JP Morgan (2007) Xpect-Indices by Deutsche Börse (2008) Our proposal: Life expectancy at birth Publicly available mortality data (HMD) Multivariate analysis between countries

9 9 Stochastic models to forecast mortality Univariate ARIMA models De Beer and Alders (1999) Keilman, Pham and Hetland (2001) Alders Keilman and Cruijsen (2007) Multivariate VAR models Multivariate VEC models Economics

10 10 ARIMA models Based on the theory of stationary stochastic processes. ARIMA(p, d, q): Box and Jenkins iterative model selection strategies: Model identification Model estimation Diagnostic checking of model adequacy

11 ARIMA models 11

12 Actual + Forecast 95% prediction interval ARIMA models ITALY FRANCE e 0 20 SWEDEN NORWAY years

13 13 VAR models Variables are explained by their own past values and the past values of all the other variables in the system.

14 14 VAR models Variables are explained by their own past values and the past values of all the other variables in the system. VAR(p): Model identification Model estimation Diagnostic checking of model adequacy

15 VAR(2) model 15

16 Actual + Forecast 95% prediction interval VAR models ITALY FRANCE e 0 20 SWEDEN NORWAY years

17 Cointegration 17

18 18 Cointegration If each element of a vector of time series Y t achieves stationarity after differencing, but a linear combination Z t = β ' Y t is already stationary, the time series are said to be cointegrated

19 19 Cointegration If each element of a vector of time series Y t achieves stationarity after differencing, but a linear combination Z is already t = β ' Y t stationary, the time series are said to be cointegrated Grangers' representation theorem: For each cointegrated system exists a VEC representation; if exists a VEC representation and the series are integrated, then they are also cointegrated

20 20 VEC models A VAR in a VEC representation: where, Example of a VAR(1): where

21 21 VEC models Studying the rank of Π we obtain information on the number of cointegrating relationship: 0 r n Π = α' β is decomposed into the loading matrix α and the cointegrating matrix β VEC is a VAR model on the first differences plus a vector of cointegrating residuals

22 22 VEC models Z t =βy t Lag-order: 2 Rank( Π)=1 Z t = β ' Y t stationary Z t years

23 Actual + Forecast 95% prediction interval VEC models ITALY FRANCE e 0 20 SWEDEN NORWAY years

24 Comparison of results:

25 Changing time-window 25 T0: e0 e0 T0: T0: e0 e0 T0: e0 T0:

26 Changing time-window: ITALY predicted values 95% prediction intervals 26 ARIMA VAR VEC e 0 (2050) T 0

27 Changing time-window: FRANCE predicted values 95% prediction intervals ARIMA VAR VEC e 0 (2050) T 0

28 Changing time-window: SWEDEN predicted values 95% prediction intervals ARIMA VAR VEC 95 e 0 (2050) T 0

29 29 Changing time-window: NORWAY predicted values 95% prediction intervals ARIMA VAR VEC 95 e 0 (2050) T 0

30 Changing the in-sample period 30 MAE T0: MAE T0: MAE T0: MAE T0: MAE T0:

31 Changing the in-sample period ITALY FRANCE Mean Absolute Errors 1 0 SWEDEN NORWAY 4 ARIMA VAR VEC T 0

32 32 Conclusions Our models: Multivariate framework VEC more coherent, VAR performs better forecasts Questionable the use of e0: Not strictly representing the annuitants mortality Basis risk present for the insurance company however: A well known and understood measure Easily available and constantly updated Transparency guaranteed to investors

33 33 Thanks for your attention! Comments or questions?

34 VAR models Diagram of fit and residuals for NOR ACF Residuals PACF Residuals Diagram of fit and residuals for NOR ACF Residuals PACF Residuals