Longevity/Mortality Risk Modeling and Securities Pricing

Patrick Brockett, Yinglu Deng, Richard MacMinnUniversity of Texas at Austin Illinois State University

Introduction• Background

• Data Description

• Model– Model Framework and Requirement– Model Specification

• Numerical Calculation– Parameter Calibration– Model Comparison– Implied Market Price of Risk– Example: q-forward Pricing

• Conclusion

Longevity Risk

Participants– Pension funds

• Corporate Sponsored

• Government Sponsored

– Annuity Providers• Insurance

companies• Reinsurance

companies

• Definition• Dramatic improvements in longevity

during the 20th century– In developed countries, average life expectancy has

increased by 1.2 months per year– Globally, life expectancy at birth has increased by 4.5

months per year

• The impact of the longevity risk– In U.K., double the aggregate deficit from £46 billion

to £100 billion of FTSE100 corporation pension– In U.S., the new mortality assumptions for pension

contributions, increase pension liabilities by 5-10%– up-to-date mortality tables, pension payments,

increase 8% for a male born in 1950

Mortality Risk

Participants– Life Insurance

Providers• Insurance

companies• Reinsurance

companies

• Definition• Catastrophe mortality events

– 1918 pandemic influenza, more than 675,000 excess deaths from the flu occurred between September 1918 and April 1919 in U.S. alone

– H5N1 avian influenza occurred in Hong Kong in 1997, and H1N1 occurred globally in 2009

• The impact of the mortality risk– The reserves for U.S. life insurance policies

stand at around $1 trillion

Securitization

Participants– Investment

Banks• JP Morgan• Goldman

Sachs– Reinsurance

Companies• Swiss Re• Munich Re

• Insurance linked securities– The interaction and combination of the

insurance industry and the capital market– Load off the non-diversified risk from the

insurer or pension balance sheet– An efficient and low-cost way to allocate and

diversify risk in the capital market– Enhance the risk capacity of the insurance

industry

• Examples:– Catastrophe Mortality Bond– Life settlement securitization

Model• Modeling the mortality rate

– Quantify and measure the longevity risk and mortality risk– Forecast the future mortality rate and life expectancy– Manage longevity risk for pension funds and annuity providers– Manage mortality risk for life insurers– Price mortality rate linked securities

• Catastrophe bonds• Longevity bonds• Life-settlement securities• Annuities

• Criterion for the model– Incorporate underlying reasons (stochastic, cohort effect, jump effect)– Goodness of fit– Mathematical tractability– Easy calibration and implementation– Concise, neat and practical

Contribution• The first model to give a closed-form solution to the expected mortality rate, and q-

forward type products. The closed-form solves the computing time-consuming problem encountered by most of the complicated structured derivatives

• The first model to address the longevity jump and the mortality jump separately in a concise model with only 6 parameters

• The model parameterization is very easy and straightforward, which enables the model implementation very efficient

• The model fits the data better than the classical Lee-Carter model and other previous jump models

Literature Review• Lee-Carter (1992), benchmark, without jump, extended by Brouhns, Denuit

and Vermunt (2002), Renshaw and Haberman (2003), Denuit, Devolder and Goderniaux (2007), Li and Chan (2007)– Our model incorporates the jump diffusion process

• Biffis (2005), Bauer, Borger and Russ (2009), with affine jump-diffusion process, model force of mortality in a continuous-time framework– Our model incorporates the cohort effect

• Chen, Cox and Peterson (2009), with compound Poisson normal jump diffusion process– Our model incorporates the asymmetric jump diffusion process

• Lin, Cox and Peterson (2009), modeling longevity jump and mortality jump– Our model provides a concise and practical approach

Data• HIST290 National Center for Health Statistics, U.S.

• Death rates per 100,000 population for selected causes of death

• Death rates are tabulated for age group (<1), (1-4), (5-14), (15-24), then every 10 years, to (75-84), and (>85)

• Both sex and race categories

• Selected causes for death include major conditions such as heart disease, cancer, and stroke

Data

Figure 1. 1900-2004 Mortality Rate

Data

Figure 2. Comparison of the Age Group Mortality Rates

Model Framework• Lee-Carter Framework

– Mortality improvement– Different improvement rate for age groups– Dynamic improvement trend

• Model Set-up

effect residual : ratemortality they tosensitivit group age :

schedulemortality theof age theacross shape general :

effectshift group age :

timeand agefor ratemortality the:

(2) ) exp( or

(1) )ln(

,

,

,,

,,

tx

x

a

x

tx

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txtxxtx

ebe

a

tx

ekba

ekba

x

Model Framework

• Two-stage procedure Single Value Decomposition (SVD) method– Regression

– Re-estimate

xkaT

a

bk

tx

T

ttxx

xt

group ageeach for separately on )-)(ln( regress

(3) )ln(1 then

1 tosums and 0 tosums set hat conditon t thenormalize

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at time group agein population theis at time deaths of sum actual theis where

(4) ))exp((

satisfiedequation theenable toestimate-rey iterativel

,

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txPtD

kbaPD

tx

t

xtxxtxt

Model Framework

tk series-timemortality The 3. Figure

Model Framework

2004-1900 during and parameters specific agefor valueFitted 1. Table xx ba

Model Requirement• Stochastic Process • Brownian Motion • Transient Jump• Asymmetric Jump

• Non stochastic process• Geometric Brownian Motion• Permanent Jump• Symmetric Jump

V.S.

Compound Poisson-Double Exponential Jump Diffusion• Positive Jump

• Small frequency• Large scale

• Negative Jump• Large frequency• Small scale

Asymmetric JumpPhenomenon• Mortality Jump

• Short-term intensified effect• Pandemic influenza, like flu 1918

• Longevity Jump• Long-term gentle effect• Pharmaceutical or medical

innovation

Model Requirement• The descriptive

statistics of shows asymmetric

leptokurtic features.

• The skewness of equals to -0.451

• distribution is skewed to the left

• distribution has a higher peak and two heavier tails

ttt kkk 1

tk

tk

tk

Figure 4. Comparison of actual distribution and normal distributiontk

Model Specification

jumps negative of scale : jumps positive of scale :

jumps negative of proportion : jumps positive of proportion :

jumps theoffrequency : rate with processpoisson : )(

MotionBrownian standard :

.1 ,0, ,0, where

(6) 11)(

and )log(

(5) ))1((

2

1

21

}1{2}0{1

)(

1

21

qp

tNW

qpqp

eqepyf

VY

VddWdtdk

t

yy

yy

Y

tN

iitt

Features

• Differentiating positive jumps and negative jumps

• Mathematical tractability

• Closed-form formula

• Concise

• Widely implemented

Specification

Model Specification

(12) ))111

(21)

21(exp(

)][exp()exp(][

is ratemortality future expected for the expression form-closed theSo,

(11) )111

(21)

21()(

wherefunction sticCharacteri (10) ])(exp[][

nIntegratio (9) )21(

(8) 111

1][

change Measure (7) ))1(()(

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1

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iit

Numerical Calculation• Parameter calibration

– Disentangling jumps from diffusion

– Maximum Likelihood Estimation method

– The form of the DEJD process satisfies the requirement of the transition density for using MLE

– Calibrate parameters

– Results indicates

– Maximum likelihood value

}31.0,20.0;75.0,71.0;45.0,064.0{},;,;,{ 21 p

},;,;,{ 21 p

95.49L

Model Comparison

Figure 5. Comparison of Actual Distribution and DEJD Distributiontk Figure 4. Comparison of Actual Distribution and Normal Distributiontk

57.0:ison Distributi Normal of Deviation Standard

20.0:ison Distributi Normal ofMean

BM

BM

31.0:ison Distributi DEJD of Deviation Standard

20.0:ison Distributi DEJD ofMean

DEJD

DEJD

Model Comparison• Compare fitness of DEJD model with Lee-Carter Brownian Motion model and

Normal Jump Diffusion model (Chen and Cox, 2009)

• Bayesian Information Criterion (BIC)– Allow comparison of more than two models– Do not require alternative to be nested– Conservative, heavily penalize over parameterization– The smaller BIC, the better fitness

(21) )ln(),'|(ln2 mnMCfBIC kkkk

Table 2. Comparison of model fitness

Implied Market Price of Risk• Swiss Re Mortality Catastrophe Bond is issued by the Swiss Reinsurance company ,

as the first mortality risk contingent securitization in Dec. 2003

• The bond is issued through a special purpose vehicle (SPV), triggered by a catastrophe evolution of death rates of a certain population

• The bond has a maturity of three years, a principal of $400m, the coupon rate of 135 basis points plus the LIBOR

• The precise payment schedules are given by the following function:

TtLTt

ft t

t }-max{0,100%spreadLIBOR1,...,1 spreadLIBOR

)(

t

t

t

tt

MMtMMM

MMMMML

0

00

0

00

5.1 allfor 5.13.1

3.1if

%100%100)]2.0/()3.1[(

%0

Risk-Neutral Pricing• Risk-neutral method by Milevsky and Promislow (2001) and Cairns, Blake, and

Dowd (2006a)

• The method is derived from the financial economic theory that posits even in an incomplete market

• No arbitrage At least one risk-neutral measure

• Linear transform instead of the distorted transform function

• Market prices of risk set

32*221

*11

*321 ; ; }.,,{

Risk-Neutral Pricing

expected.discount theequals pricefair theuntil 3,-1 steprepeat and ,set risk of pricemarket adjust they Iterativel 4. Step

payment. principal theof valueexpected theCalculate 3. Step . ofpath

simulated theand (2) formula by the ratemortality theCalculate 2. Step }.,,{

set initial theand 0.31} 0.20,- 0.75; 0.71, 0.45; {0.064,},;,;,{ set parameter on the based series-timemortality future Simulate 1. Step

,

321

21

t

tx

t

k

pk

Table 4. Implied Market Prices of Risk by Risk-Neutral Transform

q-Forward

• Pension funds – hedge against increasing life

expectancy of plan members,– the longevity risk

• Life insurers – hedge against the increase in

the mortality of policyholders,

– the mortality risk

• Basic building blocks– Standardized contracts for a

liquid market

• Exchange – realized mortality of a

population at some future date,– a fixed mortality rate agreed at

inception

q-Forward Pricing

(12) ))}111

(21)

21({exp(

)]}[exp(){exp(][

*2

*2

*1

*1*22**2*

0

*,

*

qpttbtbkbaW

kbEaWE

xxxxx

x

txxx

xtx

• The fixed rate can be calculated with the closed-form formula directly.

Conclusion• Model

– Quantify and measure the longevity risk and mortality risk– Forecast the future mortality rate and life expectancy

• Impact– Manage longevity risk for pension funds and annuity providers– Manage mortality risk for life insurers– Price mortality rate linked securities

• Catastrophe bonds• Longevity bonds• Life-settlement securities• Annuities

• Contribution– Incorporate underlying reasons (stochastic, cohort effect, jump effect)– Goodness of fit– Mathematical tractability– Easy calibration and implementation– Concise, neat and practical

Thank you