Introduction The model for basis risk Hedging Strategies Empirical Results Basis risk in static versus dynamic longevity-risk hedging Luca Regis 1 joint work with: Clemente De Rosa 2 and Elisa Luciano 3 1 IMT Lucca, Collegio Carlo Alberto, 2 Collegio Carlo Alberto, 3 University of Torino, Collegio Carlo Alberto Longevity 11 Conference 7-9 September 2015, Lyon Luca Regis Basis risk in static versus dynamic longevity-risk hedging 1/32
Introduction The model for basis risk Hedging Strategies Empirical Results Contents 1 Introduction 2 The model for basis risk 3 Hedging Strategies 4 Empirical Results Luca Regis Basis risk in static versus dynamic longevity-risk hedging 2/32
Introduction The model for basis risk Hedging Strategies Empirical Results Motivation How can Insurance Companies hedge their exposure to Longevity Risk ? 1 Static hedging - using customized, Over-The-Counter, Derivatives written on the actual Portfolio Population. • Pros: Perfect Hedge, no need for readjustments • Cons: Informational asymmetry 2 Dynamic hedging - using Standardized, traded, products written on a Reference Population. • Pros: Less Opacity, easier valuation for both counterparts • Cons: Non-perfect hedge, requires readjustment over time, basis risk may arise. Luca Regis Basis risk in static versus dynamic longevity-risk hedging 3/32
Introduction The model for basis risk Hedging Strategies Empirical Results Aim • How do Static and Dynamic hedging strategies for Longevity risk compare with each other in terms of cost/efficiency? 1 When hedging error is due to the discrete rebalancing frequency only. 2 When also basis risk is present. 1 Provide a parsimonious model for basis risk; 2 Evaluate the cost of a dynamic hedging strategy to derive the “acceptable” cost of a static hedge: what would be the cost of the static hedge equivalent to a certain percentile of the hedging error of the dynamic hedging strategy? Luca Regis Basis risk in static versus dynamic longevity-risk hedging 4/32
Introduction The model for basis risk Hedging Strategies Empirical Results Mortality Intensity/ Reference Population We model the mortality intensity of a specific generation x belonging to the Reference Population , under the risk-neutral measure Q , as a non-mean reverting CIR process: � dλ rp x ( t ) = ( a + bλ rp λ rp x ( t )) dt + σ x ( t ) dW x ( t ) , (1) with a > 0 , b > 0 , σ > 0, λ x (0) = λ 0 ∈ R ++ . Properties: • If a ≥ σ 2 2 , then λ rp ( t ) > 0 for every t ≥ 0. • Denoting τ the time to death, then we have a closed form expression for S rp x ( t, T ) = P ( τ � T | τ > t ). Luca Regis Basis risk in static versus dynamic longevity-risk hedging 5/32
Introduction The model for basis risk Hedging Strategies Empirical Results Survival Probability/ Reference Population The conditional Survival Probability at time t for the horizon T ≥ t is given by: x ( t, T ) = A rp ( t, T ) e − B rp ( t,T ) λ rp S rp x ( t ) , (2) where A rp ( t, T ) and B rp ( t, T ) are solutions of an appropriate system of Riccati equations. � 2 a � 1 2 ( γ − b )( T − t ) σ 2 2 γe A rp ( t, T ) = (3) , � � e γ ( T − t ) − 1 ( γ − b ) + 2 γ � e γ ( T − t ) − 1 � 2 B rp ( t, T ) = + 2 γ , (4) � � e γ ( T − t ) − 1 ( γ − b ) √ b 2 + 2 σ 2 . with γ = Luca Regis Basis risk in static versus dynamic longevity-risk hedging 6/32
Introduction The model for basis risk Hedging Strategies Empirical Results Survival Probability/ Reference Population Definition: We define the Longevity Risk Factor I ( t ) as the difference between the actual intensity at time t and its forecast made at time 0 (the time-0 forward rate): I ( t ) = λ rp x ( t ) − f rp x (0 , t ) (5) It turns out that: x ( t, T ) = e − X rp ( t,T ) I ( t )+ Y rp ( t,T ) , S rp (6) where X rp ( t, T ) B rp ( t, T ) , = Y rp ( t, T ) lnA rp ( t, T ) − B rp ( t, T ) f rp = x (0 , t ) . Luca Regis Basis risk in static versus dynamic longevity-risk hedging 7/32
Introduction The model for basis risk Hedging Strategies Empirical Results Mortality Intensity/ Portfolio Population We assume that the mortality intensity of generation x belonging to the Portfolio population follows a 2-factor CIR process given by: λ pp δ x λ rp ′ x ( t ) = x ( t ) + (1 − δ x ) λ x ( t ) (7) , � �� � � �� � Common Factor Idiosyncratic Factor with x ( t ) = ( a ′ + b ′ λ ′ x ( t )) dt + σ ′ � dλ ′ x ( t ) dW ′ λ ′ x ( t ) . (8) • a ′ > 0, σ ′ > 0, b ′ ∈ R , with a ′ ≥ ( σ ′ ) 2 2 , ′ • W x and W x are independent standard Brownian, • 0 ≤ δ x ≤ 1. Luca Regis Basis risk in static versus dynamic longevity-risk hedging 8/32
Introduction The model for basis risk Hedging Strategies Empirical Results Basis Risk The weight of the idiosyncratic component (1 − δ x ) can be interpreted as a measure for Basis Risk: λ rp x ( t ) Reference Population: λ pp δ x λ rp x ( t ) + (1 − δ x ) λ ′ Portfolio Population: x ( t ) = x ( t ) � � � λ rp V ar u x ( t ) � � λ pp x ( t ) , λ rp Corr u x ( t ) = δ x ∈ [0 , 1] , � � λ pp V ar u x ( t ) 1 If δ x = 1 ⇒ no Basis Risk ⇒ Benchmark Case 2 If 0 < δ x < 1 ⇒ Basis Risk ⇒ partial coverage possible 3 If δ x = 0 ⇒ Basis Risk ⇒ no partial coverage possible Luca Regis Basis risk in static versus dynamic longevity-risk hedging 9/32
Introduction The model for basis risk Hedging Strategies Empirical Results Survival Probability/ Portfolio Population As in the previous case, the conditional Survival Probability of the Portfolio Population can be written in terms of the longevity risk factor I ( t ): x ( t, T ) = e − X pp ( t,T ) δ x I ( t ) − X ′ ( t,T )(1 − δ x ) λ ′ x ( t )+ Y pp ( t,T ) S pp (9) • Explicit dependence on δ x . • X pp ( t, T ) , X ′ ( t, T ) , Y pp ( t, T ) deterministic coefficients. Luca Regis Basis risk in static versus dynamic longevity-risk hedging 10/32
Introduction The model for basis risk Hedging Strategies Empirical Results Interest rate risk For the sake of symmetry with the longevity case, we assume that the spot interest rate follows, under the risk-netrual measure, a mean-reverting CIR process of the type: � a − ¯ r ( t ) d ¯ dr ( t ) = (¯ br ( t )) dt + ¯ W ( t ) , (10) σ a > 0 , ¯ σ > 0 , r (0) = r 0 ∈ R ++ , • ¯ b > 0 , ¯ • ¯ W independent of W and W ′ . We assume independence between the Financial and the Longevity markets. Luca Regis Basis risk in static versus dynamic longevity-risk hedging 11/32
Introduction The model for basis risk Hedging Strategies Empirical Results Interest rate risk Definition: We define the Interest Rate Risk Factor J ( t ) as the difference between the short rate at time t and the time t forward rate at time 0: J ( t ) = r ( t ) − f (0 , t ) (11) As in the longevity case, the value of a zero-coupon bond D ( t, T ) can be expressed in terms of J ( t ) as: D ( t, T ) = e − ¯ X ( t,T ) J ( t )+ ¯ Y ( t,T ) . (12) Luca Regis Basis risk in static versus dynamic longevity-risk hedging 12/32
Introduction The model for basis risk Hedging Strategies Empirical Results The Insurance Portfolio • We assume that the liabilities of the insurance company are represented by an Annuity contract, with maturity T and annual instalments R paid at year-end, written on an individual belonging to the Portfolio Population and aged x at time t = 0. • The value of the reserves for such contract at time t is: T − t � N pp ( t, T ) = R D ( t, t + u ) S pp x ( t, t + u ) , u =1 T − t ′ ( t,t + u )(1 − δ x ) λ ′ e − ¯ X ( t,t + u ) J ( t )+ ¯ Y ( t,t + u ) · e − X pp ( t,t + u ) δ x I ( t ) − X x ( t )+ Y pp ( t,t + u ) . � = R u =1 Luca Regis Basis risk in static versus dynamic longevity-risk hedging 13/32
Introduction The model for basis risk Hedging Strategies Empirical Results Static Hedging To hedge the unexpected changes in longevity, an alternative for the insurance company is to buy an S -Swap or Longevity Swap . Definition: A Longevity Swap is a contract in which one party (the Insurer) agrees to pay, at a set of specified dates T i (e.g. once a year), a fixed amount K ( T i ) in exchange for the survivorship of a specific generation x belonging to the Portfolio Population. The contract lasts until the last individual of the generation x is dead (at tine w ). Luca Regis Basis risk in static versus dynamic longevity-risk hedging 14/32
Introduction The model for basis risk Hedging Strategies Empirical Results Static Hedging Under the assumption of no arbitrage and independence between the mortality and interest rate risk, the value at time t = 0 of such a contract, from the point of view of the Insurer, is given by: LS (0) = � T � T w � � � � � � �� � λ pp = E 0 exp − x ( s ) ds − K ( T ) E 0 exp − r ( u ) du , 0 0 T =1 w � � � S pp = x (0 , T ) − K ( T ) D (0 , T ) . T =1 • K ( T ) is called the swap rate for the period ( T − 1 , T ), • to ensure that the contract is fairly valued, i.e. it has zero value at inception, ⇒ K ( T ) = S pp x (0 , T ). Luca Regis Basis risk in static versus dynamic longevity-risk hedging 15/32
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