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
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
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 of 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
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 on 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
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
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
Introduction Longevity Greek Hedging Empirical Analysis Conclusion The Lee-Carter Model ln( m x , t ) = a x + b x k t where k t is the period effect following a random-walk process k t = k t − 1 + θ + ǫ t . Kenneth Q. Zhou University of Waterloo On Discrete-Time Greek Hedging of Longevity Risk
Introduction Longevity Greek Hedging Empirical Analysis Conclusion The Lee-Carter Model ln( m x , t ) = a x + b x k t where k t is the period effect following a random-walk process k t = k t − 1 + θ + ǫ t . Kenneth Q. Zhou University of Waterloo On Discrete-Time Greek Hedging of Longevity Risk
Introduction Longevity Greek Hedging Empirical Analysis Conclusion The GARCH Model k t = k t − 1 + θ + ǫ t where ǫ t is the error term following a GARCH process � ǫ t = h t η t h t = ω + αǫ 2 t − 1 + β h t − 1 Kenneth Q. Zhou University of Waterloo On Discrete-Time Greek Hedging of Longevity Risk
Introduction Longevity Greek Hedging Empirical Analysis Conclusion The GARCH Model k t = k t − 1 + θ + ǫ t where ǫ t is the error term following a GARCH process � ǫ t = h t η t h t = ω + αǫ 2 t − 1 + β h t − 1 Kenneth Q. Zhou University of Waterloo On Discrete-Time Greek Hedging of Longevity Risk
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
Introduction Longevity Greek Hedging Empirical Analysis Conclusion Survival Rate Survival rate: S x , t ( T ) = e − � T s =1 m x + s − 1 , t + s = e − � T s =1 e ax + s − 1+ bx + s − 1 kt + s s =1 e Yx , t ( s ) = e − W x , t ( T ) = e − � T where s � k t + s = k t + s θ + ǫ t + i i =1 � ǫ t + i = η t + i h t + i i − 1 v i − 1 � � ( αη 2 t + i − n + β ) + ( αǫ 2 � ( αη 2 h t + i = ω + ω t + β h t ) t + i − n + β ) . v =1 n =1 n =1 Kenneth Q. Zhou University of Waterloo On Discrete-Time Greek Hedging of Longevity Risk
Introduction Longevity Greek Hedging Empirical Analysis Conclusion Survival Probability Survival probability: p x , t ( T , k t , h t ) : = E[ S x , t ( T ) | k t , h t ] � e − W x , t ( T ) � � = E � k t , h t � s =1 e Yx , t ( s ) � � e − � T � = E � k t , h t . � Annuity Liability: ∞ � (1 + r ) − s p x , t ( s , k t , h t ) . L = s =1 q-forwards: Q = (1 + r ) − T ∗ ( p x f , t + T ∗ − 1 (1 , k t , h t ) − (1 − q f )) . where r is the constant interest rate and q f is the predetermined forward mortality rate. Kenneth Q. Zhou University of Waterloo On Discrete-Time Greek Hedging of Longevity Risk
Introduction Longevity Greek Hedging Empirical Analysis Conclusion Calculation of Delta Annuity liability: T ∂ e Y x , t ( s ) − W x , t ( T ) � � � � ∆ x , t ( T ) = p x , t ( T , k t , h t ) = − b x + s − 1 E � k t , h t � ∂ k t s =1 ∞ ∞ ∂ � � (1 + r ) − s p x , t ( s , k t , h t ) = (1 + r ) − s ∆ x , t ( s ) . ∆ L = ∂ k t s =1 s =1 q-forwards: ∂ ∆ x f , t + T ∗ − 1 (1) = p x f , t + T ∗ − 1 (1 , k t , h t ) ∂ k t ∂ (1 + r ) − T ∗ p x f , t + T ∗ − 1 (1 , k t , h t ) = (1 + r ) − T ∗ ∆ x f , t + T ∗ − 1 (1) . ∆ Q = ∂ k t Kenneth Q. Zhou University of Waterloo On Discrete-Time Greek Hedging of Longevity Risk
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
Introduction Longevity Greek Hedging Empirical Analysis Conclusion Calculation of Gamma Annuity liability: Γ x , t ( T ) = ∂ 2 p x , t ( T , k t , h t ) ∂ k 2 t � T � 2 � T � � � e − W x , t ( T ) b x + s − 1 e Y x , t ( s ) b 2 x + s − 1 e Y x , t ( s ) � = E − k t , h t � � s =1 s =1 � q-forwards: � � � � Yxf , t + T ∗− 1(1) � e Y xf , t + T ∗− 1 (1) − 1 Γ x f , t + T ∗ − 1 (1) = b 2 e Y xf , t + T ∗− 1 (1) − e � x f E � k t , h t � Kenneth Q. Zhou University of Waterloo On Discrete-Time Greek Hedging of Longevity Risk
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
Introduction Longevity Greek Hedging Empirical Analysis Conclusion Calculation of Vega Annuity liability: V x , t ( T ) = ∂ p x , t ( T , k t , h t ) ∂ h t � s � � T � i − 1 � βǫ t + i � � � � e Y x , t ( s ) − W x , t ( T ) ( αη 2 = − b x + s − 1 E t + i − n + β ) � k t , h t � 2 h t + i � s =1 i =1 n =1 q-forwards: � T ∗ � i − 1 � � � βǫ t + i Yxf , t + T ∗− 1 � � � ( αη 2 e Y xf , t + T ∗− 1 − e V x f , t + T ∗ − 1 (1) = b x f E t + i − n + β ) � k t , h t . � 2 h t + i � i =1 n =1 Kenneth Q. Zhou University of Waterloo On Discrete-Time Greek Hedging of Longevity Risk
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
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: u 1 ∆ Q 1 = ∆ L Delta-Gamma: � ∆ Q 1 � � u 1 � � ∆ L � ∆ Q 2 = Γ Q 1 Γ Q 2 u 2 Γ L Delta-Gamma-Vega: ∆ Q 1 ∆ Q 2 ∆ Q 3 u 1 ∆ L = Γ Q 1 Γ Q 2 Γ Q 3 u 2 Γ L V Q 1 V Q 2 V Q 3 u 3 V L Kenneth Q. Zhou University of Waterloo On Discrete-Time Greek Hedging of Longevity Risk
Introduction Longevity Greek Hedging Empirical Analysis Conclusion Example A 30-year temporary annuity sold to a male individual aged 60 at time 0. ∆ L Γ L V L -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
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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