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



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)



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;



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



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;



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



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.



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.



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

xa

txtxxtx ekbam ,, )ln(

tk

xb

txe ,

tk



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.

0 and 1 t

tx

x kb

)ln( ,t

txx ma

xtx am )ln( ,

xb

tk

tk x

txxtxt kbaPopD )exp(,

tD txPop ,



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



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

],[],[)(~

)(

)(~

thttht

t

NYtktk

dWudttkd

1 if ,)(

0 if ,)()(

],[],[

],[

dtttdtttt

dtttt

NYdWdtpmu

NdWdtpmutdk

pprob

pprobN htt 1 ,0

,1],[ ),(~ 2

],[ smNY tht ),0(~ tNWt



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

][)(~

)(~

tht WWuhtkhtk ],[],[)(

~)( htthtt NYhtkhtk

],[],[][)(~

htthtttht NYWWuhtk

],[],[],[],[ ][)( htthttthtthttht NYWWuhNYtk

)()( tkhtkzt

],[],[],[],[][ thtththtthttthtt NYNYWWuhz

],[],[]2,[]2,[2 ][ htthtthththththththt NYNYWWuhz

0],[ httNtz htz

1],[ httN tz htz ],[ httY

],[],[)(~

)(

)(~

thttht

t

NYtktk

dWudttkd



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


u -0.2172 0.3872

m -0.3062 s 2.3133

p 0.0396

Model without jumps: Ln(likelihood) = -94.27

-94.26548


u -0.2172 0.6043

Likelihood Ratio Test (LRT) statistics = 63.49



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.



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

T t,0,1max

1-1,2,...T t ,

)( 2006

2004ttlossspreadLIBOR

spreadLIBOR

tf

5.1 ,1

5.13.1 ,2.0

3.1 3.1 ,0

0

000

0

0

qq

qqqq

qqqq

loss

t

tt

t

t




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.




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

][* XE

),(~),(~ * NXNX

),(~)ln(),(~)ln( * NXNX

]))(([)( 1* xFxF




where , and under P.

.

where , , and under Q.

Here

],[],[)(~

)(

)(~

thttht

t

NYtktk

dWudttkd

),0(~ tNWt),(~ 2

],[ smNY tht

pprob

pprobN tkt ,1

1,0],[

*

],[*

],[**

**

)(~

)(

)(~

thttht

t

NYtktk

dWudttkd

),(~ 1* ttNWt ),(~ 2

2*

],[ ssmNY tht

*

**

],[,1

1,0

pprob

pprobN tht

])1([1 31* pp




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.

1 2 3

)exp( **, xtxtx bkam *

tq

*tloss

000,10

1

*,

* )]0,1[max(000,10

1000,000,400][

i ttiT lossrepaymentE




’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



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.



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?

hua chen and samuel h. cox rmi “brown bag” seminar august 31, 2007

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