[PPT] - Managing Longevity Risk: Tontines vs. Annuities An Chen, Peter PowerPoint Presentation

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Managing Longevity Risk: Tontines

vs. Annuities

An Chen, Peter Hieber, Jakob Klein | Lyon, September 2015 | University of Ulm

Jakob Klein

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Page 2 Managing Longevity Risk: Tontines vs. Annuities | An Chen, Peter Hieber, Jakob Klein | Content

Introduction

Annuity providers face systematic mortality risk:

◮ Solvency regulations force insurers to set aside capital ◮ Possible consequences: High annuity/reinsurance

premiums, solvency risk when capital requirements are not sufficient, . . . Measures taken:

◮ Risk transfer to other parties (e.g. Swaps) or policyholders

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Page 5 Managing Longevity Risk: Tontines vs. Annuities | An Chen, Peter Hieber, Jakob Klein | Introduction

Objectives

◮ Derive optimal payouts for expected-utility-maximizers ◮ Fairness restrictions ◮ Analyze risks borne by providers ◮ Calculation of risk-adequate loadings (→ Solvency II) ◮ Multiple perspectives

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Page 6 Managing Longevity Risk: Tontines vs. Annuities | An Chen, Peter Hieber, Jakob Klein | Introduction

Tontines: Past and present

◮ Early suggestion by Tonti (17 th century) ◮ Collection of money in the UK ◮ Popular product in the US - now forbidden ◮ Milevsky, Salisbury (2015): Optimal Retirement Tontines

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Page 7 Managing Longevity Risk: Tontines vs. Annuities | An Chen, Peter Hieber, Jakob Klein | Model setup

Relevant quantities

◮ Tontine contract:

◮ Provider pays a fixed amount to a group of policyholders ◮ alive policyholders share the payout

◮ Annuity contract:

◮ Provider pays a fixed amount to each alive individual

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Page 9 Managing Longevity Risk: Tontines vs. Annuities | An Chen, Peter Hieber, Jakob Klein | Model setup

Contract payoffs

At time t > 0

◮ an individual tontine-policyholder receives

b•(t) := 1{ζ>t} nd(t) N(t) , (1)

◮ an annuitant receives

b◦(t) := 1{ζ>t} c(t) . (2) where ζ is the residual lifetime of the individual and N(t) is the number of policyholders at time t.

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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

 

∞

e−rtb•(t)dt

  =

∞

e−rt tp∗

x n−1

k=0

n − 1 k

(tp∗

x)k(1 − tp∗ x)n−1−k nd(t)

k + 1 dt =

∞

e−rt

1 − (1 − tp∗

x)n

d(t) dt . (3)

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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

 

∞

e−rtb◦(t)dt

  =

∞

e−rt tp∗

xc(t) dt .

(4)

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Page 12 Managing Longevity Risk: Tontines vs. Annuities | An Chen, Peter Hieber, Jakob Klein | Model setup

Expected utility - Policyholder perspective

Assume an investor with; Power utility with constant relative risk aversion (CRRA) u(X) = X 1−γ 1 − γ Expected utility of a tontine policyholder: U•

·p∗ x, d(·), n

:= E

 

∞

1{ζ>t}e−rtu

nd(t) N(t)

dt

  =

∞

e−rt

n−1

k=0

n − 1 k

u

nd(t) k + 1

(tp∗

x)k+1 (1 − tp∗ x)n−1−k dt .

(5)

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Page 13 Managing Longevity Risk: Tontines vs. Annuities | An Chen, Peter Hieber, Jakob Klein | Model setup

Optimal tontine payout

d∗(t) := max

d(t) U• ·p∗ x, d(·), n

,

(6) s.t.

∞

e−rtd(t)
1 − (1 − tp∗

x)n

dt ≤ 1 . Solution: d∗(t) =     

n−1

k=0

n−1

k n k+1

1−γ (tp∗

x)k+1 (1 − tp∗ x)n−1−k

λ∗ 1 − (1 − tp∗

x)n

    

1 γ

, (7)

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Page 14 Managing Longevity Risk: Tontines vs. Annuities | An Chen, Peter Hieber, Jakob Klein | Model setup

Optimal annuity payout

c∗(t) := max

c(t) U◦ ·p∗ x, c(·), n

= max

c(t) ∞

e−rt tp∗

x u (c(t)) dt,

(8) s.t.

∞

e−rtc(t) tp∗

x dt ≤ 1 .

Solution: c(t) =  

∞

e−rt tp∗

x dt

 

−1

. (9)

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Page 15 Managing Longevity Risk: Tontines vs. Annuities | An Chen, Peter Hieber, Jakob Klein | Model setup

Mortality assumptions

◮ Gompertz law ◮ Binomial distribution for number of survivors up to time t ◮ life tables with mortality shock: tpnew x

= (tpx)1−ǫ, where ǫ is the (random) magnitude of a longevity shock

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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))

◮ Solvency II: Technical Provisions = Best Estimate

Liabilities + Risk Margin

◮ In numerical illustrations: Fair Premium = Technical

Provision

◮ Risk Margin = CoC t≥0 SCRt (1+r)t ◮ Simplifications allowed, e.g. SCR(t) = BELt BEL0 SCR0 ◮ CoC = 6% ◮ SCR = argminx

P
BEL1−CF1

1+r

− BEL0 > x

≤ 0.005

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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 be calculated as L◦

ǫ

t, ·p∗

x, d∗(·)

:=

∞

t

e−rs (sp∗

x)1−ǫ − sp∗ x

c∗(s) ds ,

(10) L•

ǫ

t, ·p∗

x, c∗(·)

:=

∞

t

e−rs 1 − (sp∗

x)1−ǫn −

1 − (sp∗

x)

n d∗(s) ds , (11)

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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∗

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)

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Page 19 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%

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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

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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

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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

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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 and shock magnitude

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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

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Page 26 Managing Longevity Risk: Tontines vs. Annuities | An Chen, Peter Hieber, Jakob Klein | Conclusion

Conclusions

◮ Fairness restrictions ◮ Products with different risk structures: take into account

compensation for risk transfer

◮ New products: multiple perspectives have to be analyzed

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Page 28 Managing Longevity Risk: Tontines vs. Annuities | An Chen, Peter Hieber, Jakob Klein | Conclusion

Outlook/Paper

◮ Mortality models ◮ Detailed proofs ◮ Sensitivity analyses ◮ ...

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Page 29 Managing Longevity Risk: Tontines vs. Annuities | An Chen, Peter Hieber, Jakob Klein | Conclusion

Thank you for your attention!

Jakob Klein Institute of Insurance Science Ulm University Germany jakob.klein@uni-ulm.de

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Page 30 Managing Longevity Risk: Tontines vs. Annuities | An Chen, Peter Hieber, Jakob Klein | References

Selected references

◮ M. Börger. Deterministic shock vs. stochastic

Value-at-Risk an analysis of the Solvency II standard model approach to longevity risk. Blätter der DGVFM, 2010.

◮ Y. Lin and S. Cox. Securitization of mortality risks in life

annuities. Journal of Risk and Insurance, 2005.

Managing Longevity Risk: Tontines

Jakob Klein

Table of contents

Introduction Model setup Contract specifications Contract value Optimization problem Mortality New product Numerical illustrations Conclusion

Table of contents

Introduction Model setup Contract specifications Contract value Optimization problem Mortality New product Numerical illustrations Conclusion

Introduction

Annuity providers face systematic mortality risk:

◮ Solvency regulations force insurers to set aside capital ◮ Possible consequences: High annuity/reinsurance

premiums, solvency risk when capital requirements are not sufficient, . . . Measures taken:

◮ Risk transfer to other parties (e.g. Swaps) or policyholders

Objectives

◮ Derive optimal payouts for expected-utility-maximizers ◮ Fairness restrictions ◮ Analyze risks borne by providers ◮ Calculation of risk-adequate loadings (→ Solvency II) ◮ Multiple perspectives

Tontines: Past and present

◮ Early suggestion by Tonti (17 th century) ◮ Collection of money in the UK ◮ Popular product in the US - now forbidden ◮ Milevsky, Salisbury (2015): Optimal Retirement Tontines

Table of contents

Introduction Model setup Contract specifications Contract value Optimization problem Mortality New product Numerical illustrations Conclusion

Relevant quantities

◮ Tontine contract:

◮ Annuity contract:

Contract payoffs

At time t > 0

◮ an individual tontine-policyholder receives

b•(t) := 1{ζ>t} nd(t) N(t) , (1)

◮ an annuitant receives

b◦(t) := 1{ζ>t} c(t) . (2) where ζ is the residual lifetime of the individual and N(t) is the number of policyholders at time t.

Value of a tontine

P•

·p∗ x, d(·), n

 

∞

  =

∞

x n−1

n − 1 k

x)k(1 − tp∗ x)n−1−k nd(t)

k + 1 dt =

∞

1 − (1 − tp∗

x)n

d(t) dt . (3)

Value of an annuity

P◦

·p∗ x, c(·), n

 

∞

  =

∞

xc(t) dt .

(4)

Expected utility - Policyholder perspective

Assume an investor with; Power utility with constant relative risk aversion (CRRA) u(X) = X 1−γ 1 − γ Expected utility of a tontine policyholder: U•

·p∗ x, d(·), n

 

∞

nd(t) N(t)

  =

∞

n−1

n − 1 k

nd(t) k + 1

x)k+1 (1 − tp∗ x)n−1−k dt .

(5)

Optimal tontine payout

d∗(t) := max

d(t) U• ·p∗ x, d(·), n

(6) s.t.

∞

x)n

dt ≤ 1 . Solution: d∗(t) =     

n−1

n−1

k n k+1

1−γ (tp∗

x)k+1 (1 − tp∗ x)n−1−k

λ∗ 1 − (1 − tp∗

x)n

    

, (7)

Optimal annuity payout