It’s All in the Hidden States: A Hedging Method with an Explicit Measure of Population Basis Risk Yanxin Liu and Johnny Siu-Hang Li Presenter: Yanxin Liu September 9, 2015 Presenter: Yanxin Liu It’s All in the Hidden States: A Hedging Method with an Explicit Measure of Population Basis September 9, 2015 1 / 44
Outline Introduction 1 The Applicable Mortality Models 2 The Generalized State Space Hedging Method 3 Analyzing Population Basis Risk 4 A Numerical Illustration 5 Conclusion 6 Presenter: Yanxin Liu It’s All in the Hidden States: A Hedging Method with an Explicit Measure of Population Basis September 9, 2015 2 / 44
Introduction Population Basis Risk ⇒ A risk associated with the difference in mortality experience between the population of the hedging instruments and the population of the liability being hedged Multi-Population Mortality Model ◮ Augmented Common Factor Model (Li and Lee, 2005) ◮ Co-integrated Lee-Carter Model (Li and Hardy, 2011) ◮ Gravity Model (Dowd et al., 2011) ◮ Two-population CBD Model (Cairns et al., 2011) ◮ M-CBD Model (Zhou and Li, 2015) Managing Basis Risk ◮ Coughlan et al.(2011) develop a framework for analysing longevity basis risk; ◮ Cairns et al.(2013) use stochastic simulation to analyse several key risk factors, including the population basis risk, that influences the hedge effectiveness of a longevity hedge. Presenter: Yanxin Liu It’s All in the Hidden States: A Hedging Method with an Explicit Measure of Population Basis September 9, 2015 3 / 44
Why An Explicit Measure of Population Basis Risk is Needed? As the market in longevity and mortality-related risk becomes more liquid, the reference population of a standardized longevity contract can be linked to a larger range of populations, ◮ e.g., the Life and Longevity Markets Association provides longevity indices for four different populations: ⋆ United States; ⋆ England and Wales; ⋆ Netherlands; ⋆ Germany. ◮ Those four populations could further be used as the reference population of longevity contracts such as q-forwards. Without an explicit measure of basis risk, ◮ the hedgers are not able to evaluate the basis risk profile for their hedge portfolios; ◮ no guideline for hedgers to select the most appropriate standardized contracts in a longevity hedge. Presenter: Yanxin Liu It’s All in the Hidden States: A Hedging Method with an Explicit Measure of Population Basis September 9, 2015 4 / 44
Research Objectives 1 To introduce a new hedging method called the generalized state space hedging method; ◮ allows us to decompose the underlying longevity risk into a component arising solely from the hidden states that are shared by all populations and components stemming exclusively from the hidden states that are population-specific. ◮ can be used as long as the mortality model can be written in state space form; 2 To develop a quantity called standardized basis risk profile, which is an efficient measure for hedgers to select the most appropriate population among all candidate populations. Presenter: Yanxin Liu It’s All in the Hidden States: A Hedging Method with an Explicit Measure of Population Basis September 9, 2015 5 / 44
Outline Introduction 1 The Applicable Mortality Models 2 The Generalized State Space Hedging Method 3 Analyzing Population Basis Risk 4 A Numerical Illustration 5 Conclusion 6 Presenter: Yanxin Liu It’s All in the Hidden States: A Hedging Method with an Explicit Measure of Population Basis September 9, 2015 6 / 44
State Space Model: A General State Space Structure Observation Equation y t = � � D + B � α t + � ǫ t Transition Equation α t = � � U + A � α t − 1 + � η t where � y t is the vector of observations at time t ; � α t is the vector of hidden states at time t ; D and � � U are the vectors of constants; B is the design matrix (linear transformation between � y t and � α t ); A is a squared matrix (first-order Markov relation of � α t ); i.i.d. i.i.d. � ǫ t ∼ MVN(0 , R ) and � η t ∼ MVN(0 , Q ). Presenter: Yanxin Liu It’s All in the Hidden States: A Hedging Method with an Explicit Measure of Population Basis September 9, 2015 7 / 44
State Space Model and Multi-Population Mortality Model Suppose that we consider n p populations. α (1) α ( n p ) α c t ) ′ , ( � t ) ′ , . . . , ( � ) ′ ) ′ , where We partition � α t into � α t = (( � t ◮ � α c t represents the states that are shared by all populations being modeled; α ( p ) ◮ � represents the states that are exclusive to population p . t η (1) η ( n p ) ◮ accordingly, � t ) ′ , ( � t ) ′ , . . . , ( � ) ′ ) ′ ; η c η t = (( � t ◮ Q would be a block diagonal matrix with blocks Q c , Q (1) , . . . , Q ( n p ) η (1) η ( n p ) η c being the covariance matrix of � , respectively. t , � t , . . . , � t Presenter: Yanxin Liu It’s All in the Hidden States: A Hedging Method with an Explicit Measure of Population Basis September 9, 2015 8 / 44
Reformulating the current mortality models into state space form: Illustration I: ACF Model (Li and Lee, 2005) Illustration II: M-CBD Model (Zhou and Li, 2015) Presenter: Yanxin Liu It’s All in the Hidden States: A Hedging Method with an Explicit Measure of Population Basis September 9, 2015 9 / 44
Augmented Common Factor Model (ACF Model, Li and Lee, 2005) The ACF model assumes that ln( m ( p ) a ( p ) t + b ( p ) x k ( p ) + ǫ ( p ) + b c x k c x , t ) = x , t , x t µ c + k c k c t − 1 + η c = t , t µ ( p ) + φ ( p ) k ( p ) k ( p ) + η ( p ) = , p = 1 , . . . , n p , t t t where ◮ m ( p ) x , t is population p ’s central death rate at age x and in year t ; ◮ a ( p ) x , b ( p ) x , µ c , µ ( p ) and φ ( p ) are constants; x , b c x b ( p ) x b c = 1 and | φ ( p ) | < 1; ◮ � x = 1, � x i.i.d. i.i.d. i.i.d. ◮ ǫ ( p ) ∼ N(0 , Q c ) and η ( p ) ∼ N(0 , σ 2 ∼ N(0 , Q ( p ) ). ǫ ), η c x , t t t Presenter: Yanxin Liu It’s All in the Hidden States: A Hedging Method with an Explicit Measure of Population Basis September 9, 2015 10 / 44
ACF Model in State Space Form In ACF model, the observation equation and transition equation can be specified as y t = � � D + B � α t + � ǫ t and α t = � � U + A � κ t − 1 + � η t , where � ′ � ln( m (1) x a , t ) , . . . , ln( m (1) x b , t ) , . . . , ln( m ( n p ) x a , t ) , . . . , ln( m ( n p ) � y t = x b , t ) , � ′ � a (1) x a , . . . , a (1) x b , . . . , a ( n p ) x a , . . . , a ( n p ) � D = , x b � ′ � t , k (1) , . . . , k ( n p ) k c α t � = , t t � ′ � ǫ (1) x a , t , . . . , ǫ (1) x b , t , . . . , ǫ ( n p ) x a , t , . . . , ǫ ( n p ) � ǫ t = , x b , t µ c , µ (1) , . . . , µ ( n p ) � ′ , � � U = Presenter: Yanxin Liu It’s All in the Hidden States: A Hedging Method with an Explicit Measure of Population Basis September 9, 2015 11 / 44
ACF Model in State Space Form and 1 0 0 · · · 0 � b (1) � b c 0 · · · 0 φ (1) 0 0 · · · 0 � � b (2) b c 0 · · · 0 φ (2) 0 0 · · · 0 A = , B = . . . . ... . . . . . . . . ... . . . . . . . . . . . . � � b c b ( n p ) 0 0 · · · φ ( n p ) 0 0 0 · · · with b c = ( b c � x a , . . . , b c x b ) ′ and b ( p ) = ( b ( p ) x a , . . . , b ( p ) � x b ) ′ , p = 1 , . . . , n p . α c In ACF model, the common states vector � t reduces to a scalar and α c t = k c we have � t . Similarly for the population-specific states vector, α ( p ) = k ( p ) we have � , for p = 1 , . . . , n p . t t Presenter: Yanxin Liu It’s All in the Hidden States: A Hedging Method with an Explicit Measure of Population Basis September 9, 2015 12 / 44
M-CBD Model (Zhou and Li, 2015) The M-CBD model assumes that logit ( q ( p ) x ) + κ ( p ) 1 , t + κ ( p ) x ) + ǫ ( p ) κ c 1 , t + κ c x , t ) = 2 , t ( x − ¯ 2 , t ( x − ¯ x , t , κ c µ c i + κ c i , t − 1 + η c = i , t , i , t κ ( p ) µ ( p ) + φ ( p ) κ ( p ) i , t − 1 + η ( p ) = i , t , i , t i i for i = 1 , 2 and p = 1 , . . . , n p , where ◮ q ( p ) x , t is population p ’s death probability at age x and in year t ;; i , µ ( p ) and φ ( p ) ◮ µ c are constants; i i ◮ ¯ x is the average of the sample age range; ◮ | φ ( p ) | < 1; i 2 ( t )) ′ and � η ( p ) = ( η ( p ) 1 ( t ) , η ( p ) ◮ let � 2 ( t )) ′ ; η c t = ( η c 1 ( t ) , η c t ◮ ǫ ( p ) i.i.d. i.i.d. η ( p ) i.i.d. ∼ N(0 , σ 2 η c ∼ MVN(0 , Q c ) and � ∼ MVN(0 , Q ( p ) ). ǫ ), � x , t t t Presenter: Yanxin Liu It’s All in the Hidden States: A Hedging Method with an Explicit Measure of Population Basis September 9, 2015 13 / 44
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