Hua Chen and Samuel H. Cox RMI “Brown Bag” Seminar August 31, 2007
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Transcript of Hua Chen and Samuel H. Cox RMI “Brown Bag” Seminar August 31, 2007
August 31, 2007 Modeling Mortality with Jumps: Transitory Effects and Pricing
Implication to Mortality Securitization 1Hua Chen and Samuel H. Cox
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
August 31, 2007 Modeling Mortality with Jumps: Transitory Effects and Pricing
Implication to Mortality Securitization 2Hua Chen and Samuel H. Cox
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)
August 31, 2007 Modeling Mortality with Jumps: Transitory Effects and Pricing
Implication to Mortality Securitization 3Hua Chen and Samuel H. Cox
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 M
øller 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; Rensh
aw and Haberman, 2003; Denuit, Devolder and Goderniaux, 2007; Cairns, Blake and Dowd, 2006b;
August 31, 2007 Modeling Mortality with Jumps: Transitory Effects and Pricing
Implication to Mortality Securitization 4Hua Chen and Samuel H. Cox
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
August 31, 2007 Modeling Mortality with Jumps: Transitory Effects and Pricing
Implication to Mortality Securitization 5Hua Chen and Samuel H. Cox
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;
August 31, 2007 Modeling Mortality with Jumps: Transitory Effects and Pricing
Implication to Mortality Securitization 6Hua Chen and Samuel H. Cox
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
August 31, 2007 Modeling Mortality with Jumps: Transitory Effects and Pricing
Implication to Mortality Securitization 7Hua Chen and Samuel H. Cox
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.
August 31, 2007 Modeling Mortality with Jumps: Transitory Effects and Pricing
Implication to Mortality Securitization 8Hua Chen and Samuel H. Cox
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.
August 31, 2007 Modeling Mortality with Jumps: Transitory Effects and Pricing
Implication to Mortality Securitization 9Hua Chen and Samuel H. Cox
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
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August 31, 2007 Modeling Mortality with Jumps: Transitory Effects and Pricing
Implication to Mortality Securitization 10Hua Chen and Samuel H. Cox
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.
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August 31, 2007 Modeling Mortality with Jumps: Transitory Effects and Pricing
Implication to Mortality Securitization 11Hua Chen and Samuel H. Cox
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 mortalityindex K, from 1900 to 2003
Mortality improvement Mortality jump in 1918
August 31, 2007 Modeling Mortality with Jumps: Transitory Effects and Pricing
Implication to Mortality Securitization 12Hua Chen and Samuel H. Cox
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
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August 31, 2007 Modeling Mortality with Jumps: Transitory Effects and Pricing
Implication to Mortality Securitization 13Hua Chen and Samuel H. Cox
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
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August 31, 2007 Modeling Mortality with Jumps: Transitory Effects and Pricing
Implication to Mortality Securitization 14Hua Chen and Samuel H. Cox
Model K(t)
Table 3: Parameter Estimation via CMLEModel with jumps-transitory effect: Ln(likelihood) = -62.52
-87.917473
-87.917473
-65.471774
Parameter Estimate Parameter Estimate
u -0.2173 0.3733
m 0.8393 s 1.4316
p 0.0436
Model with jumps:-permanent effect
-87.917473
-87.917473
-65.471774
Parameter Estimate Parameter Estimate
u -0.2172 0.3872
m -0.3062 s 2.3133
p 0.0396
Model without jumps: Ln(likelihood) = -94.27
-94.26548
Parameter Estimate Parameter Estimate
u -0.2172 0.6043
Likelihood Ratio Test (LRT) statistics = 63.49
August 31, 2007 Modeling Mortality with Jumps: Transitory Effects and Pricing
Implication to Mortality Securitization 15Hua Chen and Samuel H. Cox
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.
August 31, 2007 Modeling Mortality with Jumps: Transitory Effects and Pricing
Implication to Mortality Securitization 16Hua Chen and Samuel H. Cox
Pricing Mortality Securities
The Swiss Re Mortality Bond (2003)
Payoff schedule:
Loss ratio:
Bond HoldersSwiss Re Vita Capital
Off balance sheet
Principal $400m
Mortality index
Up to $400m upon extreme mortality
events
Up to $400m without extreme mortality events
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August 31, 2007 Modeling Mortality with Jumps: Transitory Effects and Pricing
Implication to Mortality Securitization 17Hua Chen and Samuel H. Cox
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.
August 31, 2007 Modeling Mortality with Jumps: Transitory Effects and Pricing
Implication to Mortality Securitization 18Hua Chen and Samuel H. Cox
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
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August 31, 2007 Modeling Mortality with Jumps: Transitory Effects and Pricing
Implication to Mortality Securitization 19Hua Chen and Samuel H. Cox
Pricing Mortality Securities
where , and under P.
.
where , , and under Q.
Here
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August 31, 2007 Modeling Mortality with Jumps: Transitory Effects and Pricing
Implication to Mortality Securitization 20Hua Chen and Samuel H. Cox
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 f
or 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.
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August 31, 2007 Modeling Mortality with Jumps: Transitory Effects and Pricing
Implication to Mortality Securitization 21Hua Chen and Samuel H. Cox
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.
1
August 31, 2007 Modeling Mortality with Jumps: Transitory Effects and Pricing
Implication to Mortality Securitization 22Hua Chen and Samuel H. Cox
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.
August 31, 2007 Modeling Mortality with Jumps: Transitory Effects and Pricing
Implication to Mortality Securitization 23Hua Chen and Samuel H. Cox
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?