Longevity/Mortality Risk Modeling and Securities Pricing Patrick Brockett, Yinglu Deng, Richard MacMinn University of Texas at Austin Illinois State University
Feb 22, 2016
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
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effectshift group age :
timeand agefor ratemortality the:
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Model Framework
• Two-stage procedure Single Value Decomposition (SVD) method– Regression
– Re-estimate
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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
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and )log(
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Features
• Differentiating positive jumps and negative jumps
• Mathematical tractability
• Closed-form formula
• Concise
• Widely implemented
Specification
Model Specification
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(21)
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)][exp()exp(][
is ratemortality future expected for the expression form-closed theSo,
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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
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0
00
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00
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%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
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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