Hua Chen and Samuel H. Cox RMI “Brown Bag” Seminar August 31, 2007

1. Modeling Mortality with Jumps: Transitory Effects and Pricing Implication to Mortality Securitization Hua Chen and Samuel H. Cox RMI “Brown Bag” Seminar August 31, 2007 Modeling Mortality with Jumps: Transitory Effects and Pricing Implication to Mortality Securitization

2. Introduction • Two kinds of mortality risk: • Longevity risk • Short-term catastrophic risk • How to hedge mortality risk? • Reinsurance • Mortality securitization • Examples of mortality securitization: • EIB longevity bond (Nov. 2004) • The Swiss Re mortality bond (Dec. 2003) Modeling Mortality with Jumps: Transitory Effects and Pricing Implication to Mortality Securitization

3. Introduction Stochastic mortality model (Cairns, Blake and Dowd, 2006a) • Continuous-time model, • help us understand the evolution of mortality rates over time • relatively intractable • examples: Milevsky and Promislow, 2001; Dahl, 2004; Biffs 2005; Dahl and Møller 2005; Miltersen and Persson 2005; Schrager 2006; • Discrete-time model • at most measure once a year • relatively easy to be implemented in practice • examples: Lee and Carter, 1992; Brouhns, Denuit and Vermunt, 2002; Renshaw and Haberman, 2003; Denuit, Devolder and Goderniaux, 2007; Cairns, Blake and Dowd, 2006b; Modeling Mortality with Jumps: Transitory Effects and Pricing Implication to Mortality Securitization

4. Introduction • Ignore mortality jumps • Renshaw, Haberman, and Hatzoupoulos (1996); • Sithole, Haberman, and Verrall (2000); • Milevsky and Promislow (2001); • Olivieri and Pitacco (2002); • Dahl (2003); • Denuit, Devolder and Goderniaux (2007) • Do not model mortality jumps explicitly • Lee and Carter (1992): intervention model • Li and Chan (2007): outlier analysis • Model mortality jumps explicitly • Biffis (2005): affine jump-diffusion model for life insurance contracts • Cox, Lin and Wang (2006) : age-adjusted mortality rate, permanent effects Modeling Mortality with Jumps: Transitory Effects and Pricing Implication to Mortality Securitization

5. Introduction Incomplete market pricing • Arbitrage-free framework • Cairn, Blake and Dowd (2006a): detailed discussion • Cairn, Blake and Dowd (2006b): example of EIB • Distortion operator (Wang transform) • Lin and Cox (2005); • Dowd, Blake, Cairns and Dawson (2006); • Denuit, Devolder and Goderniaux (2007); • Cox, Lin and Wang (2006): • normalized multivariate exponential tilting; • account for the correlation of the mortality index across countries; • ignore the correlation over time; Modeling Mortality with Jumps: Transitory Effects and Pricing Implication to Mortality Securitization

6. Outline • Data descriptions and historical facts • Further motivation • Mortality modeling • The classical Lee-Carter model • Model with a jump-diffusion process • Permanent versus transitory effect? • Evidence from the outlier-adjusted Lee-Carter Model • Do outliers matter? • Example of pricing mortality securities • The Swiss Re mortality bond • Conclusion and discussion Modeling Mortality with Jumps: Transitory Effects and Pricing Implication to Mortality Securitization

7. Historical Facts • Mortality improving: longevity risk • The improving mortality has variant effects across age groups. • A proper mortality model should capture this age-specific effect of mortality improving on all ages. Modeling Mortality with Jumps: Transitory Effects and Pricing Implication to Mortality Securitization

8. Historical Facts • Mortality deterioration: Short-term catastrophic risk • The 1918 influenza pandemic raised the mortality rate by 30% overall. • It affected the age groups 15-24 and 25-34 the most, whereas for individuals aged 55 and over the death rates decreased a little bit. • A proper mortality model should reflect the age-specific effect of short-term catastrophic shocks on mortality. Modeling Mortality with Jumps: Transitory Effects and Pricing Implication to Mortality Securitization

9. The Classic Lee-Carter Model • : time-varying mortality index • : the age pattern of death rates • : age-specific reactions to • : the error terms which capture age-specific effects not reflected in the model Modeling Mortality with Jumps: Transitory Effects and Pricing Implication to Mortality Securitization

10. The Classic Lee-Carter Model • The normalization conditions: • Obtain: • A two-stage procedure: • Apply the singular value decomposition (SVD) method to , solve and • Re-estimate the factors by iteration, s.t. where is the actual total number of deaths at time t, and is the population in age group x at time t. Modeling Mortality with Jumps: Transitory Effects and Pricing Implication to Mortality Securitization

11. The Classical Lee-Carter Model How to model the mortality index K ? • Cox, Lin and Wang (2006) combine a geometric Brownian motion and a compound Poisson process to model the age adjusted mortality rates for US and UK • Cannot model it with a geometric Brownian motion • Cannot model it with permanent jump effect • We model it with a standard Brownian motion and a Markov chain with jumps which only have transitory effects. Figure 1: The dynamic of the mortality index K, from 1900 to 2003 • Mortality improvement • Mortality jump in 1918 Modeling Mortality with Jumps: Transitory Effects and Pricing Implication to Mortality Securitization

12. Model K(t) • Assumptions: • , , . • The Brownian motion W, the jump severity Y, and the jump frequency N are independent with each other • Transitory effect model • Permanent effect model Modeling Mortality with Jumps: Transitory Effects and Pricing Implication to Mortality Securitization

13. Model K(t) • Transitory Effect Model: • Let • If , then is independent on . • If , then is correlated with because of the . • Solution: Conditional Maximal Likelihood Estimation Modeling Mortality with Jumps: Transitory Effects and Pricing Implication to Mortality Securitization

14. Model K(t) Table 3: Parameter Estimation via CMLE Modeling Mortality with Jumps: Transitory Effects and Pricing Implication to Mortality Securitization

15. The Outlier Adjusted Lee-Carter Model • Li and Chan (2005, 2007) • Mortality series are often contaminated with discrepant observations • Outliers may result from recording or typographical errors, or from non-repetitive exogenous interventions. • 7 outliers from 1900 to 2000, most of which resulted from influenza epidemics. Modeling Mortality with Jumps: Transitory Effects and Pricing Implication to Mortality Securitization

16. Pricing Mortality Securities Off balance sheet Swiss Re Vita Capital Bond Holders Principal $400m Up to $400m upon extreme mortality events Up to $400m without extreme mortality events Mortality index The Swiss Re Mortality Bond (2003) • Payoff schedule: • Loss ratio: Modeling Mortality with Jumps: Transitory Effects and Pricing Implication to Mortality Securitization

17. Pricing Mortality Securities • Pricing difficulties: • The mortality index is a weighted average across five countries. the correlation of mortality risks across countries • Cox, Lin and Wang (2006): normalized multivariate exponential tilting. • The principal repayment is based on the experience of the mortality index in three consecutive years. the correlation of the mortality index over time • Cox, Lin and Wang (2006): take the maximum of the mortality index in three years and link the principal repayment to this maximum value. • I will take into account correlations of the mortality index over time. Modeling Mortality with Jumps: Transitory Effects and Pricing Implication to Mortality Securitization

18. Pricing Mortality Securities • The Wang transform: • Transform from physical measure P to risk-adjusted measure Q where is the standard normal cumulative distribution and is the market price of risk. • Calculate , discount back to time zero using the risk-free interest rate, we can get the fair value of the asset X. • preserve the normal and lognormal distribution Modeling Mortality with Jumps: Transitory Effects and Pricing Implication to Mortality Securitization

19. Pricing Mortality Securities • where , and under P. • . • where , , and under Q. Here Modeling Mortality with Jumps: Transitory Effects and Pricing Implication to Mortality Securitization

20. Pricing Mortality Securities • Pricing procedures • Simulate 10,000 paths of K(t) ( t =2004, 2005, and 2006) • Calculate K*(t) (t = 2004, 2005, and 2006) on each path, given initial values of the market prices of risk , , and . • Calculate and the weighted average mortality index for each year, using the year 2000 standard population and corresponding weights. • Calculate and the expected principal repayment at time T • Calculate the discounted expected payoff under Q and let it equal to $400m, we can obtain ‘s via the numerical iteration such as the Quasi-Newton method. Modeling Mortality with Jumps: Transitory Effects and Pricing Implication to Mortality Securitization

21. Pricing Mortality Securities • ’s are smaller under the model with permanent jumps than those under the model with transitory jumps • the large difference in the jump size volatility and the difference in the intrinsic model setup. • under the model without jumps is much lower than under models with jumps. • The former overestimates the variation of the mortality index while underestimating the probability of catastrophic events. The effect of overestimating the variation predominates the effect of underestimating the catastrophic probability. Modeling Mortality with Jumps: Transitory Effects and Pricing Implication to Mortality Securitization

22. Conclusion • What we do in this paper? • Incorporate a jump-diffusion process into the Lee-Carter model. • Explore alternative models with permanent v.s. transitory jump effects • Estimate the parameters via Conditional Maximum Likelihood Estimation. • Examine the outlier-adjusted Lee-Carter model to provide further evidence of mortality jumps • Develop a pricing strategy to account for the correlation of the mortality index over time. Modeling Mortality with Jumps: Transitory Effects and Pricing Implication to Mortality Securitization

23. Conclusion • Future Research • How to develop an “optimal” transform in an incomplete market? • How to price mortality-linked securities under parameter uncertainty? • How to combine mortality risk with credit risk? • Is the regime shifting model suitable here? Modeling Mortality with Jumps: Transitory Effects and Pricing Implication to Mortality Securitization