Constructing Out-of-the-Money Longevity Hedges Using Parametric Mortality Indexes. Johnny Li

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1 1 / 43 Consrucing Ou-of-he-Money Longeviy Hedges Using Parameric Moraliy Indexes Johnny Li Join-work wih Jackie Li, Udiha Balasooriya, and Kenneh Zhou Deparmen of Economics, The Universiy of Melbourne Asian Acuarial Conference Sepember 18, 2018

2 2 / 43 Ouline Inroducion Parameric Moraliy Indexes The CBD Moraliy Indexes Objecives of This Research Securiies Wrien on he CBD Moraliy Indexes K-forward, K-call and K-pu Applicaions Pricing he Securiies The Risk-Neural Cairns-Blake-Dowd Model Pricing Formulas and Longeviy Greeks Hedging Se-up Empirical Resuls and Insighs Conclusion

3 3 / 43 Ouline Inroducion Parameric Moraliy Indexes The CBD Moraliy Indexes Objecives of This Research Securiies Wrien on he CBD Moraliy Indexes K-forward, K-call and K-pu Applicaions Pricing he Securiies The Risk-Neural Cairns-Blake-Dowd Model Pricing Formulas and Longeviy Greeks Hedging Se-up Empirical Resuls and Insighs Conclusion

4 4 / 43 Moraliy Indexes Purposes of moraliy indexes: To summarize levels of moraliy a differen ime poins To quanify moraliy and longeviy risks To consruc sandardized moraliy-linked securiies Advanages of index-based longeviy risk ransfers: More ransparen, poenially cheaper and more conducive o liquidiy Opens up an addiional pool of buyers, e.g., hedge funds

5 5 / 43 Exising Moraliy Indexes Credi Suisse s longeviy index: Tracks he life expecancy a birh of he general US populaion QxX Index by Goldman Sachs: Tracks he number of survivors in he underlying reference pool LifeMerics Indexes by JP Morgan/LLMA: Tracks he graduaed deah probabiliies of various naional populaions Xpec Cohor Indexes by Deusche Börse: Tracks he number of survivors of a cerain birh cohor

6 6 / 43 The Need for Model-Based Moraliy Indexes A non-parameric moraliy index conveys only a limied amoun of informaion. To build an effecive longeviy hedge, a large number of non-parameric indexes are needed. Parameric (model-based) moraliy indexes: Moraliy indexes developed from he ime-varying parameers in a sochasic moraliy model Richer in informaion conen

7 7 / 43 The CBD Moraliy Indexes Developed from he Cairns-Blake-Dowd (CBD) model: ( ) qx, ln = κ (1) + κ (2) (x x), 1 q x, where qx, is he probabiliy ha an individual aged x a ime will die beween and + 1; x is he mean age over he sample age range; and κ (1) and κ (2) are ime-varying parameers. The ime- values of he firs and second CBD moraliy indexes are defined as κ (1) and κ (2), respecively.

8 8 / 43 Inerpreing he CBD Moraliy Indexes ( ) qx, ln = κ (1) + κ (2) (x x), 1 q x, The firs CBD index (κ (1) ): Represens he level of he moraliy curve (afer ransformaion) (1) A reducion in κ means an overall moraliy improvemen. The second CBD index (κ (2) ): Represens he slope of he logi-ransformed moraliy curve (2) An increase in κ means ha moraliy a younger ages improves more rapidly han ha a older ages

9 9 / 43 K1 and K2 Risks Chan e al. (2014) defined K1 and K2 risks as he risks surrounding he fuure values of κ (1) and κ (2), respecively: More improvemen a younger ages More improvemen a older ages K2 risk The 2 nd CBD Moraliy Index More risk o pension plan sponsors More risk o life insurers The 1 s CBD Moraliy Index K1 risk Expeced values Overall improvemen is faser han expeced Overall improvemen is slower han expeced

10 Desirable Properies of he CBD Moraliy Indexes Small in dimension, bu able o represen he age-paern of moraliy improvemen (1) (2) = -4, = 0.1 (1) = -3, (2) = 0.15 (1) = -5, (2) = 0.2 (1) (2) = -4, = q x, Inerpreable Age (x) The new-daa-invarian propery 10 / 43

11 11 / 43 Previous Work on Parameric Moraliy Indexes Chan e al. (2014) inroduced parameric moraliy indexes and K-forwards. Tan e al. (2014) examined how a saic K-forward hedge may be calibraed wih a duraion-maching approach. Hao e al. (2017) and Wei (2017) sudied he counerpary credi risk associaed wih K-forwards. Biffis e al. (2017) inroduced a securiy ha is wrien on he populaion-specific ime-varying parameers in he augmened common facor (ACF) model. Alhough hey name heir securiy slighly differenly ( k-forward insead of K-forward ), he spiri behind is essenially he same.

12 12 / 43 Objecives of This Research 1. To explore securiy srucures oher han a zero coupon swap, and examine how he alernaive securiy srucures may benefi he hedger 2. To sudy risk-neural valuaion of K-forwards and oher securiies wrien on parameric moraliy indexes 3. To develop saic/dynamic hedging sraegies based on K-forwards and opions

13 13 / 43 Ouline Inroducion Parameric Moraliy Indexes The CBD Moraliy Indexes Objecives of This Research Securiies Wrien on he CBD Moraliy Indexes K-forward, K-call and K-pu Applicaions Pricing he Securiies The Risk-Neural Cairns-Blake-Dowd Model Pricing Formulas and Longeviy Greeks Hedging Se-up Empirical Resuls and Insighs Conclusion

14 14 / 43 K-forward Consider a K-forward wrien on he ih CBD moraliy index, for i = 1, 2. Suppose ha he K-forward is issued a ime 0 and maures a ime T (where T > 0 ). From he perspecive of he fixed rae receiver, he payoff of his K-forward a mauriy is F (i) (T, K ) = K κ (i) T per $1 noional, where K represens he fixed leg ha is predeermined when he K-forward is issued.

15 15 / 43 K-call Consider a (European) K-call on he ih CBD moraliy index, i = 1, 2. Suppose ha he K-call is issued a ime 0, maures a ime T, and has a srike value of K. The payoff of his K-call a mauriy is per $1 noional. C (i) (T, K ) = max(κ (i) T K, 0)

16 16 / 43 K-pu Consider a (European) K-pu on he ih CBD moraliy index, i = 1, 2. Suppose ha he K-pu is issued a ime 0, maures a ime T, and has a srike value of K. The payoff of his K-pu a mauriy is per $1 noional. P (i) (T, K ) = max(k κ (i) T, 0)

17 17 / 43 Hedging for Pension Plans Liabiliy is higher when... Corresponding index paern Overall moraliy improvemen Fuure values of κ (1) are is faser han expeced lower han expeced Moraliy improvemen is more Fuure values of κ (2) are concenraed a older ages lower han expeced Pension plans are subjec o downside K1 and K2 risks. To hedge heir longeviy risk exposures, pension plans may wrie K1- and K2-forwards as a fixed-rae receiver, ake a shor posiion in K1- and K2-calls, or ake a long posiion in K1- and K2-pus.

18 18 / 43 Hedging for Life Insurance Porfolios Liabiliy is higher when... Corresponding index paern Overall moraliy improvemen Fuure values of κ (1) are is slower han expeced higher han expeced Moraliy improvemen is more Fuure values of κ (2) are concenraed a older ages lower han expeced Life insurance porfolios are subjec o upside K1 risk and downside K2 risk. To hedge heir moraliy risk exposures, life insurers may wrie a K1-forward as a fixed-rae payer and a K2-forward as a fixed-rae receiver, ake a long posiion in a K1-call and a shor posiion in a K2-call, or ake a shor posiion in a K1-pu and a long posiion in a K2-pu.

19 19 / 43 Ouline Inroducion Parameric Moraliy Indexes The CBD Moraliy Indexes Objecives of This Research Securiies Wrien on he CBD Moraliy Indexes K-forward, K-call and K-pu Applicaions Pricing he Securiies The Risk-Neural Cairns-Blake-Dowd Model Pricing Formulas and Longeviy Greeks Hedging Se-up Empirical Resuls and Insighs Conclusion

20 20 / 43 The Real-world and Risk-neural Processes Le κ = (κ (1), κ (2) ). ( ) qx, ln = κ (1) + κ (2) (x x) 1 q x, Under he real-world probabiliy measure, κ = µ + κ 1 + Az, where µ = (µ (1), µ (2) ) is he consan drif vecor, A is a 2-by-2 upper-riangular marix, and z = (z (1), z (2) ) is a vecor of wo uncorrelaed sandard normal random variables under he real-world probabiliy measure.

21 21 / 43 The Real-world and Risk-neural Processes ( ) qx, ln = κ (1) + κ (2) (x x) 1 q x, Under he risk-neural probabiliy measure, or equivalenly κ = µ + κ 1 + A( z λ), κ = µ + κ 1 + A z, where z = ( z (1), z (2) ) is a vecor of wo uncorrelaed sandard normal random variables under he risk-neural measure, λ = (λ (1), λ (2) ) is he vecor of marke prices of risk, and µ = µ Aλ.

22 22 / 43 The Marke Prices of Risk The marke prices of risk are calibraed o marke informaion, e.g., marke prices of individual life annuiies. In our baseline calculaions, we se λ (1) = λ (2) = This collecion of marke prices of risk was obained by Cairns e al. (2006) using he marke price of he longeviy bond joinly announced by BNP Paribas and he European Invesmen Bank in We have sensiiviy esed he hedging resuls using a wide range of marke prices of risk.

23 23 / 43 Two Key Quaniies Given informaion up o and including ime, κ (i) T = κ(i) T + (T ) µ (i) + ɛ (i) +k, i = 1, 2, k=1 for T > under he risk-neural probabiliy measure, where ɛ (1) +k = a1,1 z(1) +k + a1,2 z(2) +k, ɛ(2) +k = a2,2 z(2) +k, and a i,j is he (i, j)h elemen in A. Hence, κ (i) T for T > under he risk-neural measure is normally disribued wih a mean of and a variance of E (Q) T ] = κ(i) + (T ) µ (i), i = 1, 2, Var (Q) T ] = { (T )(a 2 1,1 + a2 1,2 ), i = 1 (T )a 2 2,2, i = 2.

24 24 / 43 Pricing Formula for K-forwards For 0 < T, he ime- value (from he fixed-rae receiver s perspecive) of a K-forward wih a fixed leg K is F (i) (T, K ) = E (Q) [(1 + r f ) (T ) F (i) (T, K )] = (1 + r f ) (T ) (K E (Q) T ]) per $1 noional, where r f is he risk-free ineres rae. In pracice, a K-forward may be consruced in such a way ha no cash flow exchanged hands when i is issued. To achieve his, we may se he fixed leg K o E (Q) 0 T ].

25 Longeviy Greeks for K-forwards Per $1 noional, he ime- value of he longeviy dela of a K-forward is (i,f ) (T, K ) = κ (i) for i = 1, 2 and 0 < T. F (i) (T, K ) = (1 + r) (T ), Per $1 noional, he ime- value of he longeviy gamma of a K-forward is Γ (i,f ) (T, K ) = for i = 1, 2 and 0 < T. κ (i) (i,f ) (T, K ) = 0, 25 / 43

26 26 / 43 Pricing Formula for K-calls Per $1 noional, and he ime- price of a K-call wih a srike K is C (i) (T, K ) = E (Q) [(1 + r f ) (T ) C (i) (T, K )] = (1 + r f ) (T ) for i = 1, 2 and 0 < T. Var (Q) T ]φ K E(Q) Var (Q) (K E (Q) T ]) 1 Φ K E(Q) Var (Q) T ] T ] T ] T ],

27 27 / 43 Longeviy Greeks for K-calls Per $1 noional, he ime- values of he longeviy dela and gamma for a K-call wih a srike K are and (i,c) (T, K ) = (1 + r f ) (T ) Γ (i,c) (T, K ) = (1 + r f ) (T ) 1 1 Φ K E(Q) Var (Q) for i = 1, 2 and 0 < T, respecively. Var (Q) T ] T ] φ K E(Q) T ] Var (Q) T ] T ],

28 28 / 43 Pricing Formula for K-pus Per $1 noional, and he ime- price of a K-pu wih a srike K is P (i) (T, K ) = E (Q) [(1 + r f ) (T ) P (i) (T, K )] = (1 + r f ) (T ) + (K E (Q) T ]) Φ for i = 1, 2 and 0 < T. Var (Q) T ]φ K E(Q) K E(Q) Var (Q) T ] T ] T ] Var (Q) T ],

29 29 / 43 Longeviy Greeks for K-pus Per $1 noional, he ime- values of he longeviy dela and gamma of a K-pu wih a srike K are given by and (i,p) (T, K ) = (1 + r f ) (T ) Φ K E(Q) Γ (i,p) (T, K ) = (1 + r f ) (T ) 1 Var (Q) for i = 1, 2 and 0 < T, respecively. T ] φ Var (Q) T ] T ] K E(Q) Var (Q) T ] T ],

30 30 / 43 Ouline Inroducion Parameric Moraliy Indexes The CBD Moraliy Indexes Objecives of This Research Securiies Wrien on he CBD Moraliy Indexes K-forward, K-call and K-pu Applicaions Pricing he Securiies The Risk-Neural Cairns-Blake-Dowd Model Pricing Formulas and Longeviy Greeks Hedging Se-up Empirical Resuls and Insighs Conclusion

31 31 / 43 The Hedging Sraegy A any ime, he hedge conains wo insrumens, one wrien on each of he wo CBD moraliy indexes. All longeviy hedges are esablished a ime 0. Simple dela hedging is considered. The noional amoun of he ih insrumen is given by h (i) = (i,h) (i,l) (T (i), K (i) ) where H = F, C, or P depending on wheher he ih insrumen is a K-forward, K-call or K-pu.

32 32 / 43 Types of Longeviy Hedge Cash-flow hedges Focus on he variabiliy of he cash flows arising from he liabiliy being hedged and he hedging insrumens. (i) Saic cash-flow hedges: need only h for = 0 (i) Dynamic cash-flow hedges: h for = 0,... are compued. Value hedges Focus on he variabiliy of he values of he hedged posiion a a cerain fuure ime poin, say τ years from ime 0. In our empirical work, we consider paricularly τ = 1, which is he mos relevan o ypical capial requiremens.

33 33 / 43 The Liabiliy Being Hedged The liabiliy being hedged is a whole life annuiy-immediae of $1 ha is sold o individuals aged x 0 a ime 0. The ime- value of he (unpaid) annuiy liabiliy is L = ω x s=1 (1 + r f ) s E (Q) [S x0 + 0,(s)], for = 0, 0 + 1,..., 0 + ω x 0 1, where S x0 + 0,(s) represens he ex pos probabiliy ha an individual aged x a ime would have survived o ime + s. There is no analyical formula for L.

34 34 / 43 Compuaional Challenges In he following siuaions, nesed simulaions are required o esimae he exac values of L for > 0 : 1. Dynamic hedging 2. Evaluaion of he τ-year ahead Value-a-Risk To avoid he need for nesed simulaions, he approximaion of survival funcions mehod is used. (Q) E [S x0 + 0,(s)] depends on κ (1) and κ (2). A Taylor s approximaion is applied o (he probi ransformed) E (Q) [S x0 + 0,(s)] around he bes esimaes of κ (1) and κ (2) as of = 0. The absolue percenage errors are less han 0.03%.

35 35 / 43 Moneyness For K-opions wih a srike value of K and mauriy dae of T, he following moneyness meric is defined: Moneyness = K E(Q) Var (Q) For any 0 < T, we have he following: T ] T ]. If K = E (Q) T ], he K-opions are called a-he-money. If K < E (Q) T ], hen he K-call is called in-he-money while he K-pu is called ou-of-he-money. If K > E (Q) T ], hen he K-pu is called in-he-money while he K-call is called ou-of-he-money.

36 36 / 43 Baseline Assumpions The annuiy liabiliy is sold o individuals aged x 0 = 65 a ime 0. The limiing age is assumed o be ω = 100. The composiion of he hedge porfolio is eiher one K-forward (wrien as a fixed-rae receiver) on each of he wo CBD moraliy indexes, or one K-pu on each of he wo CBD moraliy indexes. The Moneyness of he wo insrumens used are idenical. T = 15, λ = (0.175, 0.175), and r f = Boh he CBD moraliy indexes and he annuiy liabiliy are linked o he moraliy experience of EW males.

37 L 2013 Baseline Resuls Saic cash flow hedge Dynamic cash flow hedge Value hedge VaR: K-forward hedge VaR: K-pu hedge VaR: unhedged Value 17 Value 17 Value VaR: K-forward hedge VaR: K-pu hedge VaR: unhedged L VaR: K-forward hedge VaR: K-pu hedge VaR: unhedged L Moneyness Moneyness Moneyness 0.8 Saic cash flow hedge 0.8 Dynamic cash flow hedge 0.8 Value hedge VaR: K-forward hedge VaR: K-pu hedge Hedging cos 0.4 Hedging cos 0.4 Hedging cos VaR: K-forward hedge VaR: K-forward hedge VaR: K-pu hedge VaR: K-pu hedge / 43

38 K-pu vs. K-forward Facor (1): Cos of hedging A K-pu hedge is less cosly Reason: only he downside risk has o be paid 0.8 Saic cash flow hedge 0.8 Dynamic cash flow hedge 0.8 Value hedge K-forward hedge Expeced cos of hedging K-forward hedge K-pu hedge Expeced cos of hedging K-forward hedge K-pu hedge Expeced cos of hedging K-pu hedge Moneyness Moneyness Moneyness 38 / 43

39 39 / 43 K-pu vs. K-forward Facor (2): Effeciveness as a hedging insrumen The dela of a K-pu is always smaller in magniude han ha of he corresponding K-forward. As such, he hedge raio (he noional amoun) of a K-pu hedge is always larger in magniude han ha of he corresponding K-forward hedge. Compared o a K-forward hedge, a K-pu hedge is more effecive in he sense ha i will pay a larger payoff o he hedger in an adverse scenario.

40 K-pu vs. K-forward 40 / 43

41 41 / 43 Oher Ineresing Findings For cash flow hedges, a K-pu hedge ends o yield a lower VaR compared o he corresponding K-forward hedge when he marke prices of risk are high. However, if he marke prices of risk are oo high, boh K-pu and K-forward hedges may no longer be economically jusifiable. For saic cash flow hedges, a K-pu hedge ends o resul in a smaller VaR compared o he corresponding K-forward hedge when he imes-o-mauriy of he hedging insrumens are long. Value hedges are highly effecive, even when he marke prices of risk are high.

42 42 / 43 Ouline Inroducion Parameric Moraliy Indexes The CBD Moraliy Indexes Objecives of This Research Securiies Wrien on he CBD Moraliy Indexes K-forward, K-call and K-pu Applicaions Pricing he Securiies The Risk-Neural Cairns-Blake-Dowd Model Pricing Formulas and Longeviy Greeks Hedging Se-up Empirical Resuls and Insighs Conclusion

43 43 / 43 Conclusion Securiy srucures: Inroduced K-opions wrien on he CBD moraliy indexes Explained how K-opions may be uilized by hedgers Pricing: Derived exac analyical pricing formulas for K-forwards and opions Hedging: Developed hree ypes of longeviy hedges Derived analyical expressions for he longeviy Greeks of K-forwards and opions Examined he relaive performance beween K-opion hedges and K-forward hedges in differen circumsances