On Discrete-Time Greek Hedging of Longevity Risk Kenneth Q. Zhou - - PowerPoint PPT Presentation

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On Discrete-Time Greek Hedging of Longevity Risk Kenneth Q. Zhou - - PowerPoint PPT Presentation

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Introduction Longevity Greek Hedging Empirical Analysis Conclusion On Discrete-Time Greek Hedging of Longevity Risk Kenneth Q. Zhou Joint-work with Johnny S.-H. Li University of Waterloo Longevity 11, Lyon, France September 9, 2015 Kenneth

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Introduction Longevity Greek Hedging Empirical Analysis Conclusion

On Discrete-Time Greek Hedging of Longevity Risk

Kenneth Q. Zhou Joint-work with Johnny S.-H. Li

University of Waterloo

Longevity 11, Lyon, France September 9, 2015

Kenneth Q. Zhou University of Waterloo On Discrete-Time Greek Hedging of Longevity Risk

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Introduction Longevity Greek Hedging Empirical Analysis Conclusion

Agenda

1 Introduction 2 Longevity Greek Hedging 3 Empirical Analysis 4 Conclusion

Kenneth Q. Zhou University of Waterloo On Discrete-Time Greek Hedging of Longevity Risk

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Introduction Longevity Greek Hedging Empirical Analysis Conclusion

Standardization of Longevity Risk

Standardization could resolve the misalignment of incentives between longevity hedgers and capital market investors. Capital markets could provide sufficient supply for acceptance

  • f longevity risk.

Capital market investors could enjoy diversification benefits and risk premiums. High liquidity and symmetric information could be achieved through standardized longevity-linked securities.

Kenneth Q. Zhou University of Waterloo On Discrete-Time Greek Hedging of Longevity Risk

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Introduction Longevity Greek Hedging Empirical Analysis Conclusion

Greek Hedging for Longevity Risk

Greeks measure the sensitivity of the value of a security to changes in certain underlying parameters on which the value depends. Longevity risk is embedded in mortality liabilities that depend

  • n multiple ages over multiple years.

Greek hedging for longevity requires a mortality model that combines the multiple underlying forces into a handful of period effects. Longevity Greeks are calculated based on the underlying period effects.

Kenneth Q. Zhou University of Waterloo On Discrete-Time Greek Hedging of Longevity Risk

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Introduction Longevity Greek Hedging Empirical Analysis Conclusion

Greek Hedging in Longevity Literature

Cairns (2011) developed a dynamic hedging strategy considering delta-only hedges. Luciano et al. (2012) developed a delta-gamma hedging strategy in the context of continuous-time modelling. Cairns (2013) developed the concept of nuga-hedging to mitigate the recalibration risk of period effects.

Kenneth Q. Zhou University of Waterloo On Discrete-Time Greek Hedging of Longevity Risk

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Introduction Longevity Greek Hedging Empirical Analysis Conclusion

Our Objectives

1 The discrete-time Lee-Carter model with conditional

heteroscedasticity (Chen et al., 2015; Wang et al., 2015).

2 Multiple Greeks hedging with the consideration of ‘vega’. 3 Properties and patterns of longevity Greeks. 4 Suggestions on the selection of hedging instruments. 5 Empirical analysis with the England and Wale mortality data.

Kenneth Q. Zhou University of Waterloo On Discrete-Time Greek Hedging of Longevity Risk

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Introduction Longevity Greek Hedging Empirical Analysis Conclusion

The Lee-Carter Model

ln(mx,t) = ax + bxkt where kt is the period effect following a random-walk process kt = kt−1 + θ + ǫt.

Kenneth Q. Zhou University of Waterloo On Discrete-Time Greek Hedging of Longevity Risk

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Introduction Longevity Greek Hedging Empirical Analysis Conclusion

The Lee-Carter Model

ln(mx,t) = ax + bxkt where kt is the period effect following a random-walk process kt = kt−1 + θ + ǫt.

Kenneth Q. Zhou University of Waterloo On Discrete-Time Greek Hedging of Longevity Risk

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Introduction Longevity Greek Hedging Empirical Analysis Conclusion

The GARCH Model

kt = kt−1 + θ + ǫt where ǫt is the error term following a GARCH process ǫt =

  • htηt

ht = ω + αǫ2

t−1 + βht−1

Kenneth Q. Zhou University of Waterloo On Discrete-Time Greek Hedging of Longevity Risk

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Introduction Longevity Greek Hedging Empirical Analysis Conclusion

The GARCH Model

kt = kt−1 + θ + ǫt where ǫt is the error term following a GARCH process ǫt =

  • htηt

ht = ω + αǫ2

t−1 + βht−1

Kenneth Q. Zhou University of Waterloo On Discrete-Time Greek Hedging of Longevity Risk

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Introduction Longevity Greek Hedging Empirical Analysis Conclusion

The Conditional Volatility of the Period Effect

Kenneth Q. Zhou University of Waterloo On Discrete-Time Greek Hedging of Longevity Risk

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Introduction Longevity Greek Hedging Empirical Analysis Conclusion

Survival Rate

Survival rate: Sx,t(T) = e− T

s=1 mx+s−1,t+s = e− T s=1 eax+s−1+bx+s−1kt+s

= e− T

s=1 eYx,t (s) = e−Wx,t(T)

where kt+s = kt + sθ +

s

  • i=1

ǫt+i ǫt+i = ηt+i

  • ht+i

ht+i = ω + ω

i−1

  • v=1

v

  • n=1

(αη2

t+i−n + β) + (αǫ2 t + βht) i−1

  • n=1

(αη2

t+i−n + β).

Kenneth Q. Zhou University of Waterloo On Discrete-Time Greek Hedging of Longevity Risk

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Introduction Longevity Greek Hedging Empirical Analysis Conclusion

Survival Probability

Survival probability: px,t(T, kt, ht) : = E[Sx,t(T) | kt, ht] = E

  • e−Wx,t(T)
  • kt, ht
  • = E
  • e− T

s=1 eYx,t (s)

  • kt, ht
  • .

Annuity Liability: L =

∞

  • s=1

(1 + r)−spx,t(s, kt, ht). q-forwards: Q = (1 + r)−T ∗(pxf ,t+T ∗−1(1, kt, ht) − (1 − qf )). where r is the constant interest rate and qf is the predetermined forward mortality rate.

Kenneth Q. Zhou University of Waterloo On Discrete-Time Greek Hedging of Longevity Risk

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Introduction Longevity Greek Hedging Empirical Analysis Conclusion

Calculation of Delta

Annuity liability:

∆x,t(T) = ∂ ∂kt px,t(T, kt, ht) = −

T

  • s=1

bx+s−1 E

  • eYx,t(s)−Wx,t(T)
  • kt, ht
  • ∆L =

∂ ∂kt

∞

  • s=1

(1 + r)−spx,t(s, kt, ht) =

∞

  • s=1

(1 + r)−s∆x,t(s).

q-forwards:

∆xf ,t+T ∗−1(1) = ∂ ∂kt pxf ,t+T ∗−1(1, kt, ht) ∆Q = ∂ ∂kt (1 + r)−T ∗pxf ,t+T ∗−1(1, kt, ht) = (1 + r)−T ∗∆xf ,t+T ∗−1(1).

Kenneth Q. Zhou University of Waterloo On Discrete-Time Greek Hedging of Longevity Risk

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Introduction Longevity Greek Hedging Empirical Analysis Conclusion

Delta of q-forwards

Kenneth Q. Zhou University of Waterloo On Discrete-Time Greek Hedging of Longevity Risk

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Introduction Longevity Greek Hedging Empirical Analysis Conclusion

Calculation of Gamma

Annuity liability:

Γx,t(T) = ∂2px,t(T, kt, ht) ∂k2

t

= E  e−Wx,t(T)   T

  • s=1

bx+s−1eYx,t(s) 2 −

T

  • s=1

b2

x+s−1eYx,t(s)

 

  • kt, ht

  q-forwards: Γxf ,t+T ∗−1(1) = b2

xf E

  • eYxf ,t+T∗−1(1)−e

Yxf ,t+T∗−1(1)

eYxf ,t+T∗−1(1) − 1

  • kt, ht
  • Kenneth Q. Zhou

University of Waterloo On Discrete-Time Greek Hedging of Longevity Risk

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Introduction Longevity Greek Hedging Empirical Analysis Conclusion

Gamma of q-forwards

Kenneth Q. Zhou University of Waterloo On Discrete-Time Greek Hedging of Longevity Risk

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Introduction Longevity Greek Hedging Empirical Analysis Conclusion

Calculation of Vega

Annuity liability:

Vx,t(T) = ∂px,t(T, kt, ht) ∂ht = −

T

  • s=1

bx+s−1 E

  • eYx,t(s)−Wx,t(T)

s

  • i=1

βǫt+i 2ht+i

i−1

  • n=1

(αη2

t+i−n + β)

  • kt, ht
  • q-forwards:

Vxf ,t+T ∗−1(1) = bxf E

  • eYxf ,t+T∗−1−e

Yxf ,t+T∗−1

T ∗

  • i=1

βǫt+i 2ht+i

i−1

  • n=1

(αη2

t+i−n + β)

  • kt, ht
  • .

Kenneth Q. Zhou University of Waterloo On Discrete-Time Greek Hedging of Longevity Risk

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Introduction Longevity Greek Hedging Empirical Analysis Conclusion

Vega of q-forwards

Kenneth Q. Zhou University of Waterloo On Discrete-Time Greek Hedging of Longevity Risk

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Introduction Longevity Greek Hedging Empirical Analysis Conclusion

Hedge Ratios

Hedge ratios (u) are determined by matching the Greeks of the liabilities and q-forwards. Delta-only: u1∆Q1 = ∆L Delta-Gamma: ∆Q1 ∆Q2 ΓQ1 ΓQ2 u1 u2

  • =

∆L ΓL

  • Delta-Gamma-Vega:

  ∆Q1 ∆Q2 ∆Q3 ΓQ1 ΓQ2 ΓQ3 VQ1 VQ2 VQ3     u1 u2 u3   =   ∆L ΓL VL  

Kenneth Q. Zhou University of Waterloo On Discrete-Time Greek Hedging of Longevity Risk

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Introduction Longevity Greek Hedging Empirical Analysis Conclusion

Example

A 30-year temporary annuity sold to a male individual aged 60 at time 0. ∆L ΓL VL

  • 0.067485
  • 0.0016493
  • 0.0078816

q-forwards with reference age from age 60 to 89 and time-to-maturity from 1 to 30 years. Delta-only, vega-only, delta-gamma and delta-vega hedges are considered. Hedges are evaluated at time 30, where the hedge effectiveness is HE = 1 − var(PH − PL) var(PL)

Kenneth Q. Zhou University of Waterloo On Discrete-Time Greek Hedging of Longevity Risk

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Introduction Longevity Greek Hedging Empirical Analysis Conclusion

Single Greek Hedging

Heat maps for the hedge effectiveness of delta-only (left) and vega-only (right) hedges.

Kenneth Q. Zhou University of Waterloo On Discrete-Time Greek Hedging of Longevity Risk

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Introduction Longevity Greek Hedging Empirical Analysis Conclusion

Multiple Greeks Hedging

Heat maps for the hedge effectiveness of delta-gamma (left) and delta-vega (right) hedges.

Kenneth Q. Zhou University of Waterloo On Discrete-Time Greek Hedging of Longevity Risk

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Introduction Longevity Greek Hedging Empirical Analysis Conclusion

Multiple Greeks Hedge Ratios

Kenneth Q. Zhou University of Waterloo On Discrete-Time Greek Hedging of Longevity Risk

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Introduction Longevity Greek Hedging Empirical Analysis Conclusion

Multiple Greeks Hedge Ratios

For multiple Greek hedging with q-forwards, hedge ratios need to be all positive. Recall that the hedge ratios are calculated as ∆Q1 ∆Q2 ΓQ1 ΓQ2 u1 u2

  • =

∆L ΓL

  • ⇒

u1 = ∆LΓQ2 − ΓL∆Q2 ∆Q1ΓQ2 − ∆Q2ΓQ1 u2 = ΓL∆Q1 − ∆LΓQ1 ∆Q1ΓQ2 − ∆Q2ΓQ1 . To have positive u1 and u2, we want the following inequality to hold: ∆Q1 ΓQ1 > ∆L ΓL > ∆Q2 ΓQ2 .

Kenneth Q. Zhou University of Waterloo On Discrete-Time Greek Hedging of Longevity Risk

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Introduction Longevity Greek Hedging Empirical Analysis Conclusion

Ratios of Delta, Gamma and Vega

Ratios of delta over gamma (left) and delta over vega (right) for q-forwards with different reference ages. The horizontal Bold line is the ratio for the liability.

Kenneth Q. Zhou University of Waterloo On Discrete-Time Greek Hedging of Longevity Risk

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Introduction Longevity Greek Hedging Empirical Analysis Conclusion

Conclusion

Greek hedging can be an useful hedging tool in mitigating longevity risk. The choice of reference ages and time-to-maturities of the hedging instrument used is important. Single Greek hedging is robust to the choice of reference ages, but not to the choice of time-to-maturities. Multiple Greeks hedging can further improve the hedge effectiveness, but careful calibrations are required. With limited choices of reference ages, delta-vega hedging is more robust than delta-gamma hedging.

Kenneth Q. Zhou University of Waterloo On Discrete-Time Greek Hedging of Longevity Risk

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Introduction Longevity Greek Hedging Empirical Analysis Conclusion

Further Remarks

The empirical results heavily depend on the assumptions underlying the Lee-Carter model. Other mortality models and their model estimates, especially non-parametric models, should be considered to verify the results. The robustness of Greek hedging relative to other risks such as population basis risk deserves further studies. Other benefits of using multiple Greeks such as reduction of left-tail risk should be further investigated.

Kenneth Q. Zhou University of Waterloo On Discrete-Time Greek Hedging of Longevity Risk