Pricing Longevity Bonds using Implied Survival
Probabilities
DANIEL B AUER
with Jochen Russ
U LM U NIVERSITY – RTG 1100 AND I NSTITUT
A KTUARWISSENSCHAFTEN
FÜR
F INANZ -
APRIA Annual Meeting, Tokyo, August 2006
UND
Introduction
Extending the Ideas of Lin and Cox (2005)
Forward Mortality Approach
Volatility Estimation
Outlook
Contents
1
Introduction
Longevity Bonds
The Model of Lin and Cox (2005)
2
Extending the Ideas of Lin and Cox (2005)
A General No Arbitrage Model
Implied Survival Probabilities
3
Forward Mortality Approach
The HJM Framework
Pooling the Approaches
4
Volatility Estimation
5
Outlook
APRIA
Tokyo, August 2006
D. Bauer
Pricing Longevity Bonds using Implied Survival Probs
2 / 28
Introduction
Extending the Ideas of Lin and Cox (2005)
Forward Mortality Approach
Volatility Estimation
Outlook
Longevity Bonds
The Model of Lin and Cox (2005)
Contents
1
Introduction
Longevity Bonds
The Model of Lin and Cox (2005)
2
Extending the Ideas of Lin and Cox (2005)
A General No Arbitrage Model
Implied Survival Probabilities
3
Forward Mortality Approach
The HJM Framework
Pooling the Approaches
4
Volatility Estimation
5
Outlook
APRIA
Tokyo, August 2006
D. Bauer
Pricing Longevity Bonds using Implied Survival Probs
3 / 28
Introduction
Extending the Ideas of Lin and Cox (2005)
Forward Mortality Approach
Volatility Estimation
Outlook
Longevity Bonds
The Model of Lin and Cox (2005)
Longevity Risk
Mortality improvements behave in a stochastic fashion, i.e.
mortality improvements are unpredictable over time.
Unsystematic mortality risk, idiosyncratic risk.
→ diversifiable...
Systematic mortality risk – mortality changes in either
direction. Mortality improvements → Longevity Risk.
→ not diversifiable!
⇒ In order to secure the liquidity of companies who face
longevity risk, solutions/means are needed to
bare/manage this risk.
Reinsurance → risk transfer.
Securitization1 : isolation and repackaging of cash flows in
order to be traded in capital markets.
1
cf. Cowley and Cummins (2005)
APRIA
Tokyo, August 2006
D. Bauer
Pricing Longevity Bonds using Implied Survival Probs
4 / 28
Introduction
Extending the Ideas of Lin and Cox (2005)
Forward Mortality Approach
Volatility Estimation
Outlook
Longevity Bonds
The Model of Lin and Cox (2005)
Longevity Risk
Mortality improvements behave in a stochastic fashion, i.e.
mortality improvements are unpredictable over time.
Unsystematic mortality risk, idiosyncratic risk.
→ diversifiable...
Systematic mortality risk – mortality changes in either
direction. Mortality improvements → Longevity Risk.
→ not diversifiable!
⇒ In order to secure the liquidity of companies who face
longevity risk, solutions/means are needed to
bare/manage this risk.
Reinsurance → risk transfer.
Securitization1 : isolation and repackaging of cash flows in
order to be traded in capital markets.
1
cf. Cowley and Cummins (2005)
APRIA
Tokyo, August 2006
D. Bauer
Pricing Longevity Bonds using Implied Survival Probs
4 / 28
Introduction
Extending the Ideas of Lin and Cox (2005)
Forward Mortality Approach
Volatility Estimation
Outlook
Longevity Bonds
The Model of Lin and Cox (2005)
Longevity Risk
Mortality improvements behave in a stochastic fashion, i.e.
mortality improvements are unpredictable over time.
Unsystematic mortality risk, idiosyncratic risk.
→ diversifiable...
Systematic mortality risk – mortality changes in either
direction. Mortality improvements → Longevity Risk.
→ not diversifiable!
⇒ In order to secure the liquidity of companies who face
longevity risk, solutions/means are needed to
bare/manage this risk.
Reinsurance → risk transfer.
Securitization1 : isolation and repackaging of cash flows in
order to be traded in capital markets.
1
cf. Cowley and Cummins (2005)
APRIA
Tokyo, August 2006
D. Bauer
Pricing Longevity Bonds using Implied Survival Probs
4 / 28
Introduction
Extending the Ideas of Lin and Cox (2005)
Forward Mortality Approach
Volatility Estimation
Outlook
Longevity Bonds
The Model of Lin and Cox (2005)
Longevity Risk
Mortality improvements behave in a stochastic fashion, i.e.
mortality improvements are unpredictable over time.
Unsystematic mortality risk, idiosyncratic risk.
→ diversifiable...
Systematic mortality risk – mortality changes in either
direction. Mortality improvements → Longevity Risk.
→ not diversifiable!
⇒ In order to secure the liquidity of companies who face
longevity risk, solutions/means are needed to
bare/manage this risk.
Reinsurance → risk transfer.
Securitization1 : isolation and repackaging of cash flows in
order to be traded in capital markets.
1
cf. Cowley and Cummins (2005)
APRIA
Tokyo, August 2006
D. Bauer
Pricing Longevity Bonds using Implied Survival Probs
4 / 28
Introduction
Extending the Ideas of Lin and Cox (2005)
Forward Mortality Approach
Volatility Estimation
Outlook
Longevity Bonds
The Model of Lin and Cox (2005)
Longevity Risk
Mortality improvements behave in a stochastic fashion, i.e.
mortality improvements are unpredictable over time.
Unsystematic mortality risk, idiosyncratic risk.
→ diversifiable...
Systematic mortality risk – mortality changes in either
direction. Mortality improvements → Longevity Risk.
→ not diversifiable!
⇒ In order to secure the liquidity of companies who face
longevity risk, solutions/means are needed to
bare/manage this risk.
Reinsurance → risk transfer.
Securitization1 : isolation and repackaging of cash flows in
order to be traded in capital markets.
1
cf. Cowley and Cummins (2005)
APRIA
Tokyo, August 2006
D. Bauer
Pricing Longevity Bonds using Implied Survival Probs
4 / 28
Introduction
Extending the Ideas of Lin and Cox (2005)
Forward Mortality Approach
Volatility Estimation
Outlook
Longevity Bonds
The Model of Lin and Cox (2005)
Longevity Bonds
Basic building block is (T , x0 )-Bond2 (diagonal approach):
pays T px0 at time T , where T px0 is the (realized) proportion
of the population of x0 -year olds at time t = 0 who are still
alive at time T .
If such a bond exists for all maturities and all ages, it is
straight forward how insurers can hedge their longevity
risk.
Questions:
1
How to price such a contract?
2
Model for the evolution of the price? → needed to
determine prices of mortality derivatives.
2
cf. Cairns at al. (2005)
APRIA
Tokyo, August 2006
D. Bauer
Pricing Longevity Bonds using Implied Survival Probs
5 / 28
Introduction
Extending the Ideas of Lin and Cox (2005)
Forward Mortality Approach
Volatility Estimation
Outlook
Longevity Bonds
The Model of Lin and Cox (2005)
Longevity Bonds
Basic building block is (T , x0 )-Bond2 (diagonal approach):
pays T px0 at time T , where T px0 is the (realized) proportion
of the population of x0 -year olds at time t = 0 who are still
alive at time T .
If such a bond exists for all maturities and all ages, it is
straight forward how insurers can hedge their longevity
risk.
Questions:
1
How to price such a contract?
2
Model for the evolution of the price? → needed to
determine prices of mortality derivatives.
2
cf. Cairns at al. (2005)
APRIA
Tokyo, August 2006
D. Bauer
Pricing Longevity Bonds using Implied Survival Probs
5 / 28
Introduction
Extending the Ideas of Lin and Cox (2005)
Forward Mortality Approach
Volatility Estimation
Outlook
Longevity Bonds
The Model of Lin and Cox (2005)
Longevity Bonds
Basic building block is (T , x0 )-Bond2 (diagonal approach):
pays T px0 at time T , where T px0 is the (realized) proportion
of the population of x0 -year olds at time t = 0 who are still
alive at time T .
If such a bond exists for all maturities and all ages, it is
straight forward how insurers can hedge their longevity
risk.
Questions:
1
How to price such a contract?
2
Model for the evolution of the price? → needed to
determine prices of mortality derivatives.
2
cf. Cairns at al. (2005)
APRIA
Tokyo, August 2006
D. Bauer
Pricing Longevity Bonds using Implied Survival Probs
5 / 28
Introduction
Extending the Ideas of Lin and Cox (2005)
Forward Mortality Approach
Volatility Estimation
Outlook
Longevity Bonds
The Model of Lin and Cox (2005)
The Model of Lin and Cox (2005)
Important issues for pricing models of mortality derivatives:
How to calibrate the model?
In particular: what’s the risk premium?
Most existing models don’t give answers...
Lin and Cox (2005):
→ Use annuity data to obtain risk-adjusted (risk-neutral?)
survival probabilities!
Idea: transform best estimate survival probabilities, such that
the sum of expected discounted cash flows of an annuity
equals the one observed on the market.
APRIA
Tokyo, August 2006
D. Bauer
Pricing Longevity Bonds using Implied Survival Probs
6 / 28
Introduction
Extending the Ideas of Lin and Cox (2005)
Forward Mortality Approach
Volatility Estimation
Outlook
Longevity Bonds
The Model of Lin and Cox (2005)
The Model of Lin and Cox (2005)
Important issues for pricing models of mortality derivatives:
How to calibrate the model?
In particular: what’s the risk premium?
Most existing models don’t give answers...
Lin and Cox (2005):
→ Use annuity data to obtain risk-adjusted (risk-neutral?)
survival probabilities!
Idea: transform best estimate survival probabilities, such that
the sum of expected discounted cash flows of an annuity
equals the one observed on the market.
APRIA
Tokyo, August 2006
D. Bauer
Pricing Longevity Bonds using Implied Survival Probs
6 / 28
Introduction
Extending the Ideas of Lin and Cox (2005)
Forward Mortality Approach
Volatility Estimation
Outlook
Longevity Bonds
The Model of Lin and Cox (2005)
The Model of Lin and Cox (2005) –
Questions/Considerations
General theoretical framework: why are annuities a
reasonable/good starting point?
Application of the Wang transform adequate? Would
another transform be more adequate?
Cairns et al. (2005): ...(unclear) how different transforms
for different cohorts and terms to maturity relate one to
another and form a coherent whole?
APRIA
Tokyo, August 2006
D. Bauer
Pricing Longevity Bonds using Implied Survival Probs
7 / 28
Introduction
Extending the Ideas of Lin and Cox (2005)
Forward Mortality Approach
Volatility Estimation
Outlook
Longevity Bonds
The Model of Lin and Cox (2005)
The Model of Lin and Cox (2005) –
Questions/Considerations
General theoretical framework: why are annuities a
reasonable/good starting point?
Application of the Wang transform adequate? Would
another transform be more adequate?
Cairns et al. (2005): ...(unclear) how different transforms
for different cohorts and terms to maturity relate one to
another and form a coherent whole?
APRIA
Tokyo, August 2006
D. Bauer
Pricing Longevity Bonds using Implied Survival Probs
7 / 28
Introduction
Extending the Ideas of Lin and Cox (2005)
Forward Mortality Approach
Volatility Estimation
Outlook
Longevity Bonds
The Model of Lin and Cox (2005)
The Model of Lin and Cox (2005) –
Questions/Considerations
General theoretical framework: why are annuities a
reasonable/good starting point?
Application of the Wang transform adequate? Would
another transform be more adequate?
Cairns et al. (2005): ...(unclear) how different transforms
for different cohorts and terms to maturity relate one to
another and form a coherent whole?
APRIA
Tokyo, August 2006
D. Bauer
Pricing Longevity Bonds using Implied Survival Probs
7 / 28
Introduction
Extending the Ideas of Lin and Cox (2005)
Forward Mortality Approach
Volatility Estimation
Outlook
A General No Arbitrage Model
Implied Survival Probabilities
Contents
1
Introduction
Longevity Bonds
The Model of Lin and Cox (2005)
2
Extending the Ideas of Lin and Cox (2005)
A General No Arbitrage Model
Implied Survival Probabilities
3
Forward Mortality Approach
The HJM Framework
Pooling the Approaches
4
Volatility Estimation
5
Outlook
APRIA
Tokyo, August 2006
D. Bauer
Pricing Longevity Bonds using Implied Survival Probs
8 / 28
Introduction
Extending the Ideas of Lin and Cox (2005)
Forward Mortality Approach
Volatility Estimation
Outlook
A General No Arbitrage Model
Implied Survival Probabilities
Mortality Risk Premium
sells endowment policy/annuity
Life Insurer
−→
sells longevity bond
Market/Investors
←−
→ Since the price of the longevity bond and of the life
insurance correspond to each other, there should be an
interdependence such that there are no arbitrage opportunities.
APRIA
Tokyo, August 2006
D. Bauer
Pricing Longevity Bonds using Implied Survival Probs
9 / 28
Introduction
Extending the Ideas of Lin and Cox (2005)
Forward Mortality Approach
Volatility Estimation
Outlook
A General No Arbitrage Model
Implied Survival Probabilities
Mortality Risk Premium (2)
→ Unsystematic mortality risk:
Deversifiable, i.e. risk premium depends on number of
insured.
⇒ In a no arbitrage framework, there cannot be a premium,
since then only the largest insurer would survive.
Insurers’ key competence LOLN: can reduce variance by
sophisticated marketing of different products/product lines.
Large investor: can utilize market imperfections and
generate returns higher than yields of market zero-bonds.
Assumption: No risk premium for unsystematic mortality risk!
APRIA
Tokyo, August 2006
D. Bauer
Pricing Longevity Bonds using Implied Survival Probs
10 / 28
Introduction
Extending the Ideas of Lin and Cox (2005)
Forward Mortality Approach
Volatility Estimation
Outlook
A General No Arbitrage Model
Implied Survival Probabilities
Mortality Risk Premium (2)
→ Unsystematic mortality risk:
Deversifiable, i.e. risk premium depends on number of
insured.
⇒ In a no arbitrage framework, there cannot be a premium,
since then only the largest insurer would survive.
Insurers’ key competence LOLN: can reduce variance by
sophisticated marketing of different products/product lines.
Large investor: can utilize market imperfections and
generate returns higher than yields of market zero-bonds.
Assumption: No risk premium for unsystematic mortality risk!
APRIA
Tokyo, August 2006
D. Bauer
Pricing Longevity Bonds using Implied Survival Probs
10 / 28
Introduction
Extending the Ideas of Lin and Cox (2005)
Forward Mortality Approach
Volatility Estimation
Outlook
A General No Arbitrage Model
Implied Survival Probabilities
Mortality Risk Premium (2)
→ Unsystematic mortality risk:
Deversifiable, i.e. risk premium depends on number of
insured.
⇒ In a no arbitrage framework, there cannot be a premium,
since then only the largest insurer would survive.
Insurers’ key competence LOLN: can reduce variance by
sophisticated marketing of different products/product lines.
Large investor: can utilize market imperfections and
generate returns higher than yields of market zero-bonds.
Assumption: No risk premium for unsystematic mortality risk!
APRIA
Tokyo, August 2006
D. Bauer
Pricing Longevity Bonds using Implied Survival Probs
10 / 28
Introduction
Extending the Ideas of Lin and Cox (2005)
Forward Mortality Approach
Volatility Estimation
Outlook
A General No Arbitrage Model
Implied Survival Probabilities
Mortality Risk Premium (2)
→ Unsystematic mortality risk:
Deversifiable, i.e. risk premium depends on number of
insured.
⇒ In a no arbitrage framework, there cannot be a premium,
since then only the largest insurer would survive.
Insurers’ key competence LOLN: can reduce variance by
sophisticated marketing of different products/product lines.
Large investor: can utilize market imperfections and
generate returns higher than yields of market zero-bonds.
Assumption: No risk premium for unsystematic mortality risk!
APRIA
Tokyo, August 2006
D. Bauer
Pricing Longevity Bonds using Implied Survival Probs
10 / 28
Introduction
Extending the Ideas of Lin and Cox (2005)
Forward Mortality Approach
Volatility Estimation
Outlook
A General No Arbitrage Model
Implied Survival Probabilities
No Arbitrage Model (risk free rate set to zero)
t =0
n individuals buy endowment (face $1) over one year and
(together) pay n 1 p̃xins
.
0
Insurer buys n longevity bonds and pays n 1 p̃xinv
.
0
t = 1 Individuals obtain n 1 px0 from insurer, who obtains same
amount from investor.
!
NO ARBITRAGE: 1 p̃xins
= 1 p̃xinv
.
0
0
APRIA
Tokyo, August 2006
D. Bauer
Pricing Longevity Bonds using Implied Survival Probs
11 / 28
Introduction
Extending the Ideas of Lin and Cox (2005)
Forward Mortality Approach
Volatility Estimation
Outlook
A General No Arbitrage Model
Implied Survival Probabilities
Pricing Longevity Bonds
Price of a longevity bond = Price of a T-year pure
endowment policy:
h
i
T-measure
=
P(0, T )EQT [T px0 ] .
Π0 (T , x0 ) = EQ BT−1 T px0
We call EQT [T px0 ] implied survival probability.
It can be estimated by using a continuum of endowment
and annuity data; e.g., for a deferred annuity:
|n ax0
=
∞
X
P(0, t) EQt [t px0 ] .
t=n
→ Derive EQT [T px0 ] , 1 ≤ T ≤ ∞ by least square methods
(parametrical approach).
APRIA
Tokyo, August 2006
D. Bauer
Pricing Longevity Bonds using Implied Survival Probs
12 / 28
Introduction
Extending the Ideas of Lin and Cox (2005)
Forward Mortality Approach
Volatility Estimation
Outlook
A General No Arbitrage Model
Implied Survival Probabilities
Pricing Longevity Bonds
Price of a longevity bond = Price of a T-year pure
endowment policy:
h
i
T-measure
=
P(0, T )EQT [T px0 ] .
Π0 (T , x0 ) = EQ BT−1 T px0
We call EQT [T px0 ] implied survival probability.
It can be estimated by using a continuum of endowment
and annuity data; e.g., for a deferred annuity:
|n ax0
=
∞
X
P(0, t) EQt [t px0 ] .
t=n
→ Derive EQT [T px0 ] , 1 ≤ T ≤ ∞ by least square methods
(parametrical approach).
APRIA
Tokyo, August 2006
D. Bauer
Pricing Longevity Bonds using Implied Survival Probs
12 / 28
Introduction
Extending the Ideas of Lin and Cox (2005)
Forward Mortality Approach
Volatility Estimation
Outlook
A General No Arbitrage Model
Implied Survival Probabilities
Implied survival probabilities from synthetic German
annuity data
APRIA
Tokyo, August 2006
D. Bauer
Pricing Longevity Bonds using Implied Survival Probs
13 / 28
Introduction
Extending the Ideas of Lin and Cox (2005)
Forward Mortality Approach
Volatility Estimation
Outlook
A General No Arbitrage Model
Implied Survival Probabilities
Considerations
Milevsky and Promislow (2001): technical rates used by
companies’ actuaries are in fact forward rates.
Independence assumptions:
Example: Russia – economic development can affect
expected life time.
Market price of mortality may depend on interest
rate/financial market development:
Many authors assuming independence explain current
interest in mortality modeling/longevity risk discussion by
"low interest environment".
Mortality contingent options (GMIB, GAO) are rather
exercised when rates are low.
APRIA
Tokyo, August 2006
D. Bauer
Pricing Longevity Bonds using Implied Survival Probs
14 / 28
Introduction
Extending the Ideas of Lin and Cox (2005)
Forward Mortality Approach
Volatility Estimation
Outlook
A General No Arbitrage Model
Implied Survival Probabilities
Considerations
Milevsky and Promislow (2001): technical rates used by
companies’ actuaries are in fact forward rates.
Independence assumptions:
Example: Russia – economic development can affect
expected life time.
Market price of mortality may depend on interest
rate/financial market development:
Many authors assuming independence explain current
interest in mortality modeling/longevity risk discussion by
"low interest environment".
Mortality contingent options (GMIB, GAO) are rather
exercised when rates are low.
APRIA
Tokyo, August 2006
D. Bauer
Pricing Longevity Bonds using Implied Survival Probs
14 / 28
Introduction
Extending the Ideas of Lin and Cox (2005)
Forward Mortality Approach
Volatility Estimation
Outlook
The HJM Framework
Pooling the Approaches
Contents
1
Introduction
Longevity Bonds
The Model of Lin and Cox (2005)
2
Extending the Ideas of Lin and Cox (2005)
A General No Arbitrage Model
Implied Survival Probabilities
3
Forward Mortality Approach
The HJM Framework
Pooling the Approaches
4
Volatility Estimation
5
Outlook
APRIA
Tokyo, August 2006
D. Bauer
Pricing Longevity Bonds using Implied Survival Probs
15 / 28
Introduction
Extending the Ideas of Lin and Cox (2005)
Forward Mortality Approach
Volatility Estimation
Outlook
The HJM Framework
Pooling the Approaches
HJM Framework for Interest Rate Modeling
Takes the entire forward rate curve as (infinite dimensional)
state variable – assumption:
df (t, T ) = αF (t, T ) dt + σ F (t, T ) dWt .
⇒ Obtains no arbitrage condition on drift term – risk premium
does not have to be specified/estimated.
⇒ In particular, this condition provides
( Z
)
"
( Z
)#
T
T
P(0, T ) = exp −
f (0, s) ds = EQ exp −
rs ds
.
0
0
Miltersen and Persson (2005) were the first to apply these
ideas to mortality modeling.
APRIA
Tokyo, August 2006
D. Bauer
Pricing Longevity Bonds using Implied Survival Probs
16 / 28
Introduction
Extending the Ideas of Lin and Cox (2005)
Forward Mortality Approach
Volatility Estimation
Outlook
The HJM Framework
Pooling the Approaches
HJM Framework for Interest Rate Modeling
Takes the entire forward rate curve as (infinite dimensional)
state variable – assumption:
df (t, T ) = αF (t, T ) dt + σ F (t, T ) dWt .
⇒ Obtains no arbitrage condition on drift term – risk premium
does not have to be specified/estimated.
⇒ In particular, this condition provides
( Z
)
"
( Z
)#
T
T
P(0, T ) = exp −
f (0, s) ds = EQ exp −
rs ds
.
0
0
Miltersen and Persson (2005) were the first to apply these
ideas to mortality modeling.
APRIA
Tokyo, August 2006
D. Bauer
Pricing Longevity Bonds using Implied Survival Probs
16 / 28
Introduction
Extending the Ideas of Lin and Cox (2005)
Forward Mortality Approach
Volatility Estimation
Outlook
The HJM Framework
Pooling the Approaches
HJM Framework for Interest Rate Modeling
Takes the entire forward rate curve as (infinite dimensional)
state variable – assumption:
df (t, T ) = αF (t, T ) dt + σ F (t, T ) dWt .
⇒ Obtains no arbitrage condition on drift term – risk premium
does not have to be specified/estimated.
⇒ In particular, this condition provides
( Z
)
"
( Z
)#
T
T
P(0, T ) = exp −
f (0, s) ds = EQ exp −
rs ds
.
0
0
Miltersen and Persson (2005) were the first to apply these
ideas to mortality modeling.
APRIA
Tokyo, August 2006
D. Bauer
Pricing Longevity Bonds using Implied Survival Probs
16 / 28
Introduction
Extending the Ideas of Lin and Cox (2005)
Forward Mortality Approach
Volatility Estimation
Outlook
The HJM Framework
Pooling the Approaches
HJM Framework for Interest Rate Modeling
Takes the entire forward rate curve as (infinite dimensional)
state variable – assumption:
df (t, T ) = αF (t, T ) dt + σ F (t, T ) dWt .
⇒ Obtains no arbitrage condition on drift term – risk premium
does not have to be specified/estimated.
⇒ In particular, this condition provides
( Z
)
"
( Z
)#
T
T
P(0, T ) = exp −
f (0, s) ds = EQ exp −
rs ds
.
0
0
Miltersen and Persson (2005) were the first to apply these
ideas to mortality modeling.
APRIA
Tokyo, August 2006
D. Bauer
Pricing Longevity Bonds using Implied Survival Probs
16 / 28
Introduction
Extending the Ideas of Lin and Cox (2005)
Forward Mortality Approach
Volatility Estimation
Outlook
The HJM Framework
Pooling the Approaches
HJM Framework for Mortality Modeling
Price of (T , x0 )-bond at time t:
h
i
Πt (T , x0 ) = Bt EQ BT−1 T px0 | Ft
h
i
= Bt EQ BT−1 T −t px0 +t t px0 | Ft
"
(Z
)
#
T
r (s) + µ̃s (s, x0 ) ds | Ft
= t px0 EQ exp
t
Define forward force of mortality via
(Z
)
T
Πt (T , x0 )
= exp
f (t, s) + µ̃t (s, x0 ) ds .
t px0
t
APRIA
Tokyo, August 2006
D. Bauer
Pricing Longevity Bonds using Implied Survival Probs
17 / 28
Introduction
Extending the Ideas of Lin and Cox (2005)
Forward Mortality Approach
Volatility Estimation
Outlook
The HJM Framework
Pooling the Approaches
HJM Framework for Mortality Modeling
Price of (T , x0 )-bond at time t:
h
i
Πt (T , x0 ) = Bt EQ BT−1 T px0 | Ft
h
i
= Bt EQ BT−1 T −t px0 +t t px0 | Ft
"
(Z
)
#
T
r (s) + µ̃s (s, x0 ) ds | Ft
= t px0 EQ exp
t
Define forward force of mortality via
(Z
)
T
Πt (T , x0 )
= exp
f (t, s) + µ̃t (s, x0 ) ds .
t px0
t
APRIA
Tokyo, August 2006
D. Bauer
Pricing Longevity Bonds using Implied Survival Probs
17 / 28
Introduction
Extending the Ideas of Lin and Cox (2005)
Forward Mortality Approach
Volatility Estimation
Outlook
The HJM Framework
Pooling the Approaches
HJM Framework for Mortality Modeling (2)
Model:
d µ̃t (T , x0 ) = αL (t, T ) dt + σ L (t, T ) dWt .
Obtain condition on drift. No specification of risk premium
necessary.
Same multidimensional Brownian motion for interest and
mortality development – correlations possible.
? What is the initial term structure?
? What does this have to do with the foregoing?
APRIA
Tokyo, August 2006
D. Bauer
Pricing Longevity Bonds using Implied Survival Probs
18 / 28
Introduction
Extending the Ideas of Lin and Cox (2005)
Forward Mortality Approach
Volatility Estimation
Outlook
The HJM Framework
Pooling the Approaches
HJM Framework for Mortality Modeling (2)
Model:
d µ̃t (T , x0 ) = αL (t, T ) dt + σ L (t, T ) dWt .
Obtain condition on drift. No specification of risk premium
necessary.
Same multidimensional Brownian motion for interest and
mortality development – correlations possible.
? What is the initial term structure?
? What does this have to do with the foregoing?
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Pricing Longevity Bonds using Implied Survival Probs
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Introduction
Extending the Ideas of Lin and Cox (2005)
Forward Mortality Approach
Volatility Estimation
Outlook
The HJM Framework
Pooling the Approaches
HJM Framework for Mortality Modeling (2)
Model:
d µ̃t (T , x0 ) = αL (t, T ) dt + σ L (t, T ) dWt .
Obtain condition on drift. No specification of risk premium
necessary.
Same multidimensional Brownian motion for interest and
mortality development – correlations possible.
? What is the initial term structure?
? What does this have to do with the foregoing?
APRIA
Tokyo, August 2006
D. Bauer
Pricing Longevity Bonds using Implied Survival Probs
18 / 28
Introduction
Extending the Ideas of Lin and Cox (2005)
Forward Mortality Approach
Volatility Estimation
Outlook
The HJM Framework
Pooling the Approaches
Link to implied survival probabilities
Price of a longevity bond at t = 0
(Z
Π0 (T , x0 ) = P(0, T ) exp
)
T
µ̃0 (s, x0 ) ds
0
= P(0, T ) EQT [T px0 ]
∂
⇒ µ̃0 (T , x0 ) =
log EQT [T px0 ]
∂T
Obtain starting point by implied survival probabilities
Parametric function for mortality term structure proposed
(similar to Nelson/Siegel for term structure of interest
rates).
APRIA
Tokyo, August 2006
D. Bauer
Pricing Longevity Bonds using Implied Survival Probs
19 / 28
Introduction
Extending the Ideas of Lin and Cox (2005)
Forward Mortality Approach
Volatility Estimation
Outlook
The HJM Framework
Pooling the Approaches
Link to implied survival probabilities
Price of a longevity bond at t = 0
(Z
Π0 (T , x0 ) = P(0, T ) exp
)
T
µ̃0 (s, x0 ) ds
0
= P(0, T ) EQT [T px0 ]
∂
⇒ µ̃0 (T , x0 ) =
log EQT [T px0 ]
∂T
Obtain starting point by implied survival probabilities
Parametric function for mortality term structure proposed
(similar to Nelson/Siegel for term structure of interest
rates).
APRIA
Tokyo, August 2006
D. Bauer
Pricing Longevity Bonds using Implied Survival Probs
19 / 28
Introduction
Extending the Ideas of Lin and Cox (2005)
Forward Mortality Approach
Volatility Estimation
Outlook
The HJM Framework
Pooling the Approaches
Link to implied survival probabilities
Price of a longevity bond at t = 0
(Z
Π0 (T , x0 ) = P(0, T ) exp
)
T
µ̃0 (s, x0 ) ds
0
= P(0, T ) EQT [T px0 ]
∂
⇒ µ̃0 (T , x0 ) =
log EQT [T px0 ]
∂T
Obtain starting point by implied survival probabilities
Parametric function for mortality term structure proposed
(similar to Nelson/Siegel for term structure of interest
rates).
APRIA
Tokyo, August 2006
D. Bauer
Pricing Longevity Bonds using Implied Survival Probs
19 / 28
Introduction
Extending the Ideas of Lin and Cox (2005)
Forward Mortality Approach
Volatility Estimation
Outlook
The HJM Framework
Pooling the Approaches
Forward mortality intensities
APRIA
Tokyo, August 2006
D. Bauer
Pricing Longevity Bonds using Implied Survival Probs
20 / 28
Introduction
Extending the Ideas of Lin and Cox (2005)
Forward Mortality Approach
Volatility Estimation
Outlook
Contents
1
Introduction
Longevity Bonds
The Model of Lin and Cox (2005)
2
Extending the Ideas of Lin and Cox (2005)
A General No Arbitrage Model
Implied Survival Probabilities
3
Forward Mortality Approach
The HJM Framework
Pooling the Approaches
4
Volatility Estimation
5
Outlook
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Pricing Longevity Bonds using Implied Survival Probs
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Introduction
Extending the Ideas of Lin and Cox (2005)
Forward Mortality Approach
Volatility Estimation
Outlook
Estimation of the Volatility
Use market data of GMIB in Variable Annuities to obtain implied
mortality volatilities (valuation approach similar to Ballotta and
Haberman (2006)):
value of an annuity starting at future date T and paying $ 1
annually:
" ∞
#
X
ax0 +T = EQ
B(T , k ) k −T px0 +T | FT
k =T
=
∞
X
( Z
exp −
k =T
APRIA
Tokyo, August 2006
)
k
f (T , s) + µ(T , s, x0 + s) ds
T
D. Bauer
Pricing Longevity Bonds using Implied Survival Probs
22 / 28
Introduction
Extending the Ideas of Lin and Cox (2005)
Forward Mortality Approach
Volatility Estimation
Outlook
Estimation of the Volatility (2)
Thus, the value of a Variable Annuity including a roll-up GMIB
and no GMDB:
"Z
#
T
V0 = EQ
e−
Rt
0 rs
+ µ̃s (s,x0 ) ds
At µ̃t (t, x0 ) dt
0
h RT
i
+EQ e− t rs ds T px0 max GA ax0 +T , AT ,
where
GA is the guaranteed annuity within the GMIB, and
A is the insured’s account.
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Pricing Longevity Bonds using Implied Survival Probs
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Introduction
Extending the Ideas of Lin and Cox (2005)
Forward Mortality Approach
Volatility Estimation
Outlook
Estimation of the Volatility (3)
Estimate volatility by:
1
specify volatility structure;
2
determine value of the GMIB given values for the volatility
using Monte Carlo simulation;
3
compare the value to market prices3 (fees continuously
!
deducted from A), assume A0 = V0 ;
4
adjust volatility values;
5
repeat procedure until value matches market prices.
3
Data: E.g., Equitable Accumulator
APRIA
Tokyo, August 2006
D. Bauer
Pricing Longevity Bonds using Implied Survival Probs
24 / 28
Introduction
Extending the Ideas of Lin and Cox (2005)
Forward Mortality Approach
Volatility Estimation
Outlook
Contents
1
Introduction
Longevity Bonds
The Model of Lin and Cox (2005)
2
Extending the Ideas of Lin and Cox (2005)
A General No Arbitrage Model
Implied Survival Probabilities
3
Forward Mortality Approach
The HJM Framework
Pooling the Approaches
4
Volatility Estimation
5
Outlook
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Pricing Longevity Bonds using Implied Survival Probs
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Introduction
Extending the Ideas of Lin and Cox (2005)
Forward Mortality Approach
Volatility Estimation
Outlook
Remarks and Future Research
! Model can be analogously applied to value other mortality
derivatives, e.g. survivor swaps (cf. Dowd et al. (2006)).
! No restriction to diagonal approach – can handle whole
mortality structure (ages and time) and, therefore, model
bonds for different maturities and ages simultaneously.
→ ? Number of factors in driving Wiener process? Age
dependence?
? Specifications of Volatility.
? Empirical assessment.
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Tokyo, August 2006
D. Bauer
Pricing Longevity Bonds using Implied Survival Probs
26 / 28
Introduction
Extending the Ideas of Lin and Cox (2005)
Forward Mortality Approach
Volatility Estimation
Outlook
Remarks and Future Research
! Model can be analogously applied to value other mortality
derivatives, e.g. survivor swaps (cf. Dowd et al. (2006)).
! No restriction to diagonal approach – can handle whole
mortality structure (ages and time) and, therefore, model
bonds for different maturities and ages simultaneously.
→ ? Number of factors in driving Wiener process? Age
dependence?
? Specifications of Volatility.
? Empirical assessment.
APRIA
Tokyo, August 2006
D. Bauer
Pricing Longevity Bonds using Implied Survival Probs
26 / 28
Introduction
Extending the Ideas of Lin and Cox (2005)
Forward Mortality Approach
Volatility Estimation
Outlook
References
Ballotta, L., Haberman, S., 2006. The fair valuation problem of
guaranteed annuity options: The stochastic mortality environment
case. Insurance: Mathematics and Economics, 38: 195 – 214.
Cairns, A.J., Blake, D., Dowd, K., 2005. Pricing Death: Frameworks
for the Valuation and Securitization of Mortality Risk. To appear in
ASTIN Bulletin.
Cowley, A., Cummins, J.D., 2005. Securitization of Life Insurance
Assets and Liabilities. The Journal of Risk and Insurance, 72: 193 –
226.
Dowd, K., Blake, D., Cairns, A.J., Dawson, P., 2006. Survivor
Swaps. The Journal of Risk and Insurance, 73: 1 – 17.
Lin, Y., Cox, S., 2005. Securitization of Mortality Risks in Life
Annuities. The Journal of Risk and Insurance, 72: 227 – 252.
APRIA
Tokyo, August 2006
D. Bauer
Pricing Longevity Bonds using Implied Survival Probs
27 / 28
Introduction
Extending the Ideas of Lin and Cox (2005)
Forward Mortality Approach
Volatility Estimation
Outlook
References (2)
Milevsky, M.A., Promislow, S.D., 2001. Mortality Derivatives and the
Option to Annuitize. Insurance: Mathematics and Economics, 29:
299–318.
Miltersen, K.R., Persson, S.A., 2005. Is mortality dead? Stochastic
force of mortality determined by no arbitrage. Working Paper,
University of Bergen.
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Pricing Longevity Bonds using Implied Survival Probs
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