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Managing Longevity Risk: Tontinesvs. Annuities
An Chen, Peter Hieber, Jakob Klein | Lyon, September 2015 |University of Ulm
Jakob Klein
Page 2 Managing Longevity Risk: Tontines vs. Annuities | An Chen, Peter Hieber, Jakob Klein | Content
Table of contents
Introduction
Model setupContract specificationsContract valueOptimization problemMortalityNew product
Numerical illustrations
Conclusion
Page 3 Managing Longevity Risk: Tontines vs. Annuities | An Chen, Peter Hieber, Jakob Klein | Introduction
Table of contents
Introduction
Model setupContract specificationsContract valueOptimization problemMortalityNew product
Numerical illustrations
Conclusion
Page 4 Managing Longevity Risk: Tontines vs. Annuities | An Chen, Peter Hieber, Jakob Klein | Introduction
Introduction
Annuity providers face systematic mortality risk:I Solvency regulations force insurers to set aside capitalI Possible consequences: High annuity/reinsurance
premiums, solvency risk when capital requirements are notsufficient, . . .
Measures taken:I Risk transfer to other parties (e.g. Swaps) or policyholders
Page 5 Managing Longevity Risk: Tontines vs. Annuities | An Chen, Peter Hieber, Jakob Klein | Introduction
Objectives
I Derive optimal payouts for expected-utility-maximizersI Fairness restrictionsI Analyze risks borne by providersI Calculation of risk-adequate loadings (→ Solvency II)I Multiple perspectives
Page 6 Managing Longevity Risk: Tontines vs. Annuities | An Chen, Peter Hieber, Jakob Klein | Introduction
Tontines: Past and present
I Early suggestion by Tonti (17 th century)I Collection of money in the UKI Popular product in the US - now forbiddenI Milevsky, Salisbury (2015): Optimal Retirement Tontines
Page 7 Managing Longevity Risk: Tontines vs. Annuities | An Chen, Peter Hieber, Jakob Klein | Model setup
Table of contents
Introduction
Model setupContract specificationsContract valueOptimization problemMortalityNew product
Numerical illustrations
Conclusion
Page 8 Managing Longevity Risk: Tontines vs. Annuities | An Chen, Peter Hieber, Jakob Klein | Model setup
Relevant quantities
I Tontine contract:I Provider pays a fixed amount to a group of policyholdersI alive policyholders share the payout
I Annuity contract:I Provider pays a fixed amount to each alive individual
Page 9 Managing Longevity Risk: Tontines vs. Annuities | An Chen, Peter Hieber, Jakob Klein | Model setup
Contract payoffs
At time t > 0I an individual tontine-policyholder receives
b•(t) := 1{ζ>t}nd(t)N(t)
, (1)
I an annuitant receives
b◦(t) := 1{ζ>t} c(t) . (2)
where ζ is the residual lifetime of the individual and N(t) is thenumber of policyholders at time t .
Page 10 Managing Longevity Risk: Tontines vs. Annuities | An Chen, Peter Hieber, Jakob Klein | Model setup
Value of a tontine
P•(·p∗x ,d(·),n
):=E
∞∫0
e−rtb•(t)dt
=
∞∫0
e−rttp∗x
n−1∑k=0
(n − 1
k
)(tp∗x)
k (1− tp∗x)n−1−k nd(t)
k + 1dt
=
∞∫0
e−rt(1− (1− tp∗x)n)d(t) dt . (3)
Page 11 Managing Longevity Risk: Tontines vs. Annuities | An Chen, Peter Hieber, Jakob Klein | Model setup
Value of an annuity
P◦(·p∗x , c(·),n
):=E
∞∫0
e−rtb◦(t)dt
=
∞∫0
e−rttp∗xc(t) dt . (4)
Page 12 Managing Longevity Risk: Tontines vs. Annuities | An Chen, Peter Hieber, Jakob Klein | Model setup
Expected utility - Policyholder perspectiveAssume an investor with; Power utility with constant relative riskaversion (CRRA)
u(X ) =X 1−γ
1− γExpected utility of a tontine policyholder:
U•(·p∗x ,d(·),n
):= E
∞∫0
1{ζ>t}e−rtu(
nd(t)N(t)
)dt
=
∞∫0
e−rtn−1∑k=0
(n − 1
k
)u(
nd(t)k + 1
)(tp∗x)
k+1 (1− tp∗x)n−1−k dt .
(5)
Page 13 Managing Longevity Risk: Tontines vs. Annuities | An Chen, Peter Hieber, Jakob Klein | Model setup
Optimal tontine payout
d∗(t) := maxd(t)
U•(·p∗x ,d(·),n
), (6)
s.t .
∞∫0
e−rtd(t)(1− (1− tp∗x)
n) dt ≤ 1 .
Solution:
d∗(t) =
n−1∑k=0
(n−1k
) ( nk+1
)1−γ(tp∗x)
k+1 (1− tp∗x)n−1−k
λ∗(1− (1− tp∗x)n
)
1γ
, (7)
Page 14 Managing Longevity Risk: Tontines vs. Annuities | An Chen, Peter Hieber, Jakob Klein | Model setup
Optimal annuity payout
c∗(t) := maxc(t)
U◦(·p∗x , c(·),n
)= max
c(t)
∞∫0
e−rttp∗x u (c(t))dt , (8)
s.t .
∞∫0
e−rtc(t) tp∗x dt ≤ 1 .
Solution:
c(t) =
∞∫0
e−rttp∗x dt
−1
. (9)
Page 15 Managing Longevity Risk: Tontines vs. Annuities | An Chen, Peter Hieber, Jakob Klein | Model setup
Mortality assumptions
I Gompertz lawI Binomial distribution for number of survivors up to time tI life tables with mortality shock: tpnew
x = (tpx)1−ε, where ε is
the (random) magnitude of a longevity shock
Page 16 Managing Longevity Risk: Tontines vs. Annuities | An Chen, Peter Hieber, Jakob Klein | Model setup
Risk Margin
Calculation of the risk margin (see e.g. Börger (2010))I Solvency II: Technical Provisions = Best Estimate
Liabilities + Risk MarginI In numerical illustrations: Fair Premium = Technical
ProvisionI Risk Margin = CoC
∑t≥0
SCRt(1+r)t
I Simplifications allowed, e.g. SCR(t) = BELtBEL0
SCR0
I CoC = 6%I SCR = argminx
{P(
BEL1−CF11+r − BEL0 > x
)≤ 0.005
}
Page 17 Managing Longevity Risk: Tontines vs. Annuities | An Chen, Peter Hieber, Jakob Klein | Model setup
Future losses
At time t > 0 the losses generated by a longevity shock can becalculated as
L◦ε(t , ·p∗x ,d
∗(·)):=
∫ ∞t
e−rs((sp∗x)
1−ε − sp∗x)
c∗(s) ds , (10)
L•ε(t , ·p∗x , c
∗(·)):=
∫ ∞t
e−rs((
1− (sp∗x)1−ε)n −
(1− (sp∗x)
)n)
d∗(s) ds ,
(11)
Page 18 Managing Longevity Risk: Tontines vs. Annuities | An Chen, Peter Hieber, Jakob Klein | Model setup
Switching from tontine to annuity
Fix a switching time t∗, at time t a policyholder receives:
1{0≤ζ<t∗}nd(t)N(t)
+ 1{ζ≥t∗}c, (12)
Fair value:
t∗∫0
e−rt(1− (1− tp∗x)n)d(t)dt + e−rt∗
t∗p∗x
∞∫t∗
e−r(t−t∗)t−t∗p∗x+t∗c dt = 1
(13)
Page 19 Managing Longevity Risk: Tontines vs. Annuities | An Chen, Peter Hieber, Jakob Klein | Numerical illustrations
Table of contents
Introduction
Model setupContract specificationsContract valueOptimization problemMortalityNew product
Numerical illustrations
Conclusion
Page 20 Managing Longevity Risk: Tontines vs. Annuities | An Chen, Peter Hieber, Jakob Klein | Numerical illustrations
Tontine vs Annuity: Loss distribution
Figure: loss distribution: age 65, r=4%
Page 21 Managing Longevity Risk: Tontines vs. Annuities | An Chen, Peter Hieber, Jakob Klein | Numerical illustrations
Tontine vs Annuity:Risk margin
Figure: Risk margins: age 65, different longevity shock magnitudes
Page 22 Managing Longevity Risk: Tontines vs. Annuities | An Chen, Peter Hieber, Jakob Klein | Numerical illustrations
Annuity: Risk margin
Figure: Risk margin: age 65, various portfolio sizes at inception
Page 23 Managing Longevity Risk: Tontines vs. Annuities | An Chen, Peter Hieber, Jakob Klein | Numerical illustrations
Tontine: risk margin
Figure: Risk margin: age 65, various portfolio sizes at inception
Page 24 Managing Longevity Risk: Tontines vs. Annuities | An Chen, Peter Hieber, Jakob Klein | Numerical illustrations
Expected utility - risk loading
Figure: Expected utility with risk-based loading: varying interest andshock magnitude
Page 25 Managing Longevity Risk: Tontines vs. Annuities | An Chen, Peter Hieber, Jakob Klein | Numerical illustrations
Switching times - Solvency Capital Requirement
Figure: SCR for deferred payout: age 65
Page 26 Managing Longevity Risk: Tontines vs. Annuities | An Chen, Peter Hieber, Jakob Klein | Conclusion
Table of contents
Introduction
Model setupContract specificationsContract valueOptimization problemMortalityNew product
Numerical illustrations
Conclusion
Page 27 Managing Longevity Risk: Tontines vs. Annuities | An Chen, Peter Hieber, Jakob Klein | Conclusion
Conclusions
I Fairness restrictionsI Products with different risk structures: take into account
compensation for risk transferI New products: multiple perspectives have to be analyzed
Page 28 Managing Longevity Risk: Tontines vs. Annuities | An Chen, Peter Hieber, Jakob Klein | Conclusion
Outlook/Paper
I Mortality modelsI Detailed proofsI Sensitivity analysesI ...
Page 29 Managing Longevity Risk: Tontines vs. Annuities | An Chen, Peter Hieber, Jakob Klein | Conclusion
Thank you for your attention!
Jakob KleinInstitute of Insurance ScienceUlm UniversityGermany
Page 30 Managing Longevity Risk: Tontines vs. Annuities | An Chen, Peter Hieber, Jakob Klein | References
Selected references
I M. Börger. Deterministic shock vs. stochasticValue-at-Risk an analysis of the Solvency II standardmodel approach to longevity risk. Blätter der DGVFM,2010.
I Y. Lin and S. Cox. Securitization of mortality risks in lifeannuities. Journal of Risk and Insurance, 2005.
I M. Milevsky and T. Salisbury. Optimal Retirement Tontinesfor the 21st Century: With Reference to MortalityDerivatives in 1693. Insurance: Mathematics andEconomics, 2015