A Study on Longevity Risk Hedging in the Presence of Population Basis Risk

Transcription

1 A Sudy on Longeviy Risk Hedging in he Presence of Populaion Basis Risk by Kenneh Qian Zhou A hesis presened o he Universiy of Waerloo in fulfillmen of he hesis requiremen for he degree of Maser of Mahemaics in Acuarial Science Waerloo, Onario, Canada, 215 c Kenneh Qian Zhou 215

2 I hereby declare ha I am he sole auhor of his hesis. This is a rue copy of he hesis, including any required final revisions, as acceped by my examiners. I undersand ha my hesis may be made elecronically available o he public. ii

3 Absrac Longeviy risk refers o uncerainy surrounding he rend in human life expecancy. Sandardized hedging insrumens ha are linked o broad-based moraliy indexes can be used o offload longeviy risk from pension plans and annuiies. However, hedges ha are based on such insrumens are subjec o populaion basis risk, which arises from he difference in moraliy improvemens beween he hedger s populaion and he reference populaion o which he hedging insrumens are linked. This hesis aemps o address some issues ha are relaed o longeviy risk hedging in he presence of populaion basis risk. In he firs chaper, a graphical risk meric is proposed o inuiively measure populaion basis risk, which is believed o be a major obsacle o marke developmen. I allows marke paricipans o no only visually evaluae he exen of populaion basis risk, bu also deermine he mos appropriae reference populaion. Compared o exising populaion basis risk merics which are mosly numerical, he proposed graphical risk meric is more informaive in ha i capures more aspecs of populaion basis risk. Along wih he exising numerical risk merics, he proposed graphical risk meric may help hedgers beer undersand populaion basis risk and hence make heir risk managemen decisions. In he second chaper, he feasibiliy of dynamic longeviy hedging wih sandardized hedging insrumens is sudied. To his end, he dynamic hedging sraegy developed by Cairns (211) is generalized o incorporae he siuaion when he hedger s populaion and he reference populaion are differen. The empirical resuls indicae ha dynamic hedging can effecively reduce he longeviy risk exposures of a ypical pension plan, even if populaion basis risk is aken ino accoun. Furher, by considering daa from a large iii

4 group of naional populaions, i is found ha populaion basis risk and small sample risk can possibly be diversified across differen hedgers. Hedgers may herefore be able o compleely eliminae heir longeviy risk exposures by removing he underlying rend risk wih a dynamic index-based hedge and ransferring he residual risks hrough a reinsurance mechanism. iv

5 Acknowledgemens I would like o express my deepes graiude o my supervisor, Professor Johnny Li, who has inspired, guided and helped me hrough my maser s sudies wih his insighful ideas and unreserved suppor. I would also like o hank Professors Mary Hardy and Jun Cai for heir ime and effor in examining his hesis. Special hanks also o Professors Wai- Sum Chan and Rui Zhou for heir consrucive suggesions. I would like o acknowledge he Deparmen of Saisics and Acuarial Science s funding suppor, and Ms. Mary Lou Dufon s assisance. Finally, I would like o express my appreciaion o Cynhia McLauchlan, who has spen endless amouns of ime encouraging and supporing me. v

6 Dedicaion This hesis is dedicaed o my parens for heir limiless love, suppor and encouragemen. vi

7 Table of Conens Lis of Tables x Lis of Figures xi 1 Towards a Large and Liquid Longeviy Marke: A Graphical Populaion Basis Risk Meric Inroducion Mehodology An Illusraion Conclusion Dynamic Longeviy Hedging in he Presence of Populaion Basis Risk: A Feasibiliy Analysis from Technical and Economic Perspecives Inroducion The Dynamic Longeviy Hedging Sraegy vii

8 2.2.1 The Assumed Model The Se-up The Approximaion Mehods Deriving Hedge Raios Evaluaing he Hedge Analyzing he Dynamic Longeviy Hedge Assumpions Baseline Resuls Robusness Managing he Residual Risks Assumpions An Exploraory Analysis A Cusomized Surplus Swap An Illusraion Discussion and Conclusion Appendices 71 A Evaluaing he Qualiy of he Approximaion Mehods 72 B Deriving he Approximaion Formula for p (i) x,u(t, K, k (i) ) when u > 77 viii

9 References 8 ix

10 Lis of Tables 1.1 The calculaed values of h (R) The name and index of he 25 male populaions The sample correlaion coefficiens of he log cenral deah raes for he 25 male populaions The sample correlaion coefficiens of he log cenral deah raes wihou he common ime rend for he 25 male populaions x

11 Lis of Figures 1.1 The graphical risk meric for one reference populaion The graphical risk meric for four reference populaions A graphical illusraion of he proposed hedging framework The disribuions of he parial derivaives of F L and Q (), and h The disribuions of he liabiliies when populaion basis risk is presen The values of he hedge effeciveness over ime The disribuions of he liabiliies when populaion basis risk is absen The disribuions of he liabiliies when model risk is presen The disribuions of he liabiliies when small sample risk is presen The disribuions of he liabiliies when using q-forwards wih differen reference ages The disribuions of he liabiliies when using q-forwards wih differen imeo-mauriies xi

12 2.1 The disribuions of he liabiliies for he 25 pension plans An illusraion of he cash exchange of a cusomized surplus swap The disribuions of he ne cash flows of he cusomized surplus swap for he 25 pension plans The disribuions of he average ne cash flows The variances of he average ne cash flows and he individual ne cash flows 66 A.1 The degrees of accuracy in esimaing p (H) x,(s, K, k (H) ) A.2 The degrees of accuracy in esimaing p (R) x f, +T 1 (1, K, k (R) )) xii

13 Chaper 1 Towards a Large and Liquid Longeviy Marke: A Graphical Populaion Basis Risk Meric 1.1 Inroducion Rapid, unexpeced increases in human life expecancy have posed wha is known as longeviy risk. On a macroeconomic level, longeviy risk affecs curren accoun (Lee and Mason, 21), GDP (Skirbekk, 24), and produciviy (van Groezen e al., 25). From a microeconomic viewpoin, longeviy risk undermines he profis and growh opporuniies of corporaions offering defined-benefi pension schemes, ulimaely affecing heir share prices. According o he Inernaional Moneary Fund (212), if individuals live hree years 1

14 longer han expeced, hen he already large pension coss would increase by 5% of he 21 GDP in advanced economies and 25% of he 21 GDP in emerging economies. Recenly, some pension plan sponsors and annuiy providers have chosen o offload longeviy risk from heir balance shees. One way o accomplish his ac is by ransferring he risk o capial markes, hrough sandardized derivaive securiies ha are linked o broad-based moraliy indexes. The firs of such ransacions occurred in 28 when Lucida PLC passed par of is longeviy risk exposure ono J.P. Morgan by means of a moraliy q- forward conrac. The risk was subsequenly ransferred o various insiuional invesors, who acceped he risk exposure for a risk premium (Blake e al., 213). Compared o oher risk ransfer mehods such as reinsurance, capial markes soluions are advanageous in erms of being less cosly and having, in heory, no capaciy consrain (Cummins and Trainar, 29). Neverheless, a his poin he marke for sandardized moraliy-linked securiies is small and lacks liquidiy. The indusry leaders believe ha one major obsacle o marke developmen is an inadequae undersanding of populaion basis risk, he residual risk ha originaes from he difference in moraliy improvemens beween he hedger s populaion and he reference populaion o which he hedging insrumen is linked (Life and Longeviy Markes Associaion (LLMA), 212). This problem has been sudied by several researchers, who quanified he risk by numerical merics including percenage reducion in expeced shorfall (Ngai and Sherris, 211), percenage reducion in variance (Cairns e al., 214; Li and Hardy, 211; Li and Luo, 212), and minimal required buffer (Sevens e al., 211). 1 1 The minimal required buffer refers o he minimum asse value (in excess of he bes esimae value of he liabiliies) such ha he probabiliy ha he insurer or pension fund will be able o pay all fuure liabiliies is sufficienly high. 2

15 However, as he exising mehods canno be easily communicaed o marke paricipans, hey sill canno mee he indusry s need for a simple, inuiive meric for populaion basis risk. To aid in filling his gap, in his chaper we conribue a graphical risk meric for assessing populaion basis risk. The graphical risk meric is consruced from a series of join predicion regions, allowing users o visually evaluae he ranges of possible oucomes a various confidence levels. Our conribuion also enables hedgers o deermine, ou of all available reference populaions, he populaion ha resuls in he minimum amoun of populaion basis risk. We believe ha our conribuion is likely o gain wide accepance among praciioners, who are increasing relying on graphical mehods such as survivor fan chars (Blake e al., 28), longeviy fan chars (Dowd e al., 21), and hea maps of moraliy improvemen raes (Coninuous Moraliy Invesigaion Bureau, 29) in making heir risk managemen decisions. We explain he consrucion of he graphical populaion basis risk meric in he nex secion, followed by a secion ha includes a demonsraion based on a hypoheical example and real moraliy daa. Finally he las secion concludes he chaper wih some suggesions for fuure research. 1.2 Mehodology Le us consider a pension plan whose liabiliy value is proporional o a random survivor index, S (H), where H represens he populaion of individuals associaed wih he plan. To 3

16 hedge is longeviy risk exposure, he plan rades a longeviy-linked derivaive, whose payoff is proporional o anoher random survivor index, S (R), where R denoes he derivaive s reference populaion. We le I (H) = S (H) E(S (H) ) and I (R) = S (R) E(S (R) ) be he exceedances of S (H) and S (R) over heir expeced values, respecively. We base he graphical risk meric on I (H) and I (R) raher han S (H) and S (R), parly because users primary ineres are he possible deviaions from he expeced oucomes, and parly because he use of I (H) and I (R) ensures all resuling risk merics are cenered a he origin, hereby allowing users o compare he risk merics for differen reference populaions readily. The firs sep in consrucing he graphical populaion basis risk meric is o simulae realizaions of I (H) and I (R) from a muli-populaion sochasic moraliy model, examples of which include he augmened common facor model (Li and Lee, 25) and he graviy model (Dowd e al., 211). Such a model incorporaes he correlaion beween he uncerain moraliy improvemens of he populaions being modeled, and exhibis mean-reversion o avoid resuling in ani-inuiive diverging long-erm moraliy forecass. The second sep is o opimize he longeviy hedge. We consider a saic hedge, which seems more feasible han a dynamic hedge in oday s marke for longeviy risk ransfers. 2 Specifically, we aim o find, per dollar amoun of he pension liabiliy, he noional amoun h (R) of he longeviy-linked derivaive ha would lead o a perfec hedge in he ideal siuaion when populaion basis risk is absen, i.e., when I (H) and I (R) are perfecly correlaed. We find h (R) by a linear approximaion, which implies h (R) = I(H) I (R). The value of h (R) is esimaed by he slope of he firs order linear regression of I (H) on I (R), derived from he 2 Saic hedging is more realisic, because dynamic hedging requires liquid longeviy-linked securiies ha are no ye available in he curren marke for longeviy risk ransfers. See Fung e al. (214). 4

17 simulaed values of I (H) and I (R) obained in he firs sep. Our choice of h (R), which can be expressed as h (R) = Cov(I(R), I (H) ), Var(I (H) ) is jusified in he sense ha i minimizes he variance of he hedged porfolio; ha is, h (R) is he value of h ha minimizes he following expression: Var(I (H) hi (R) ) = Var(I (H) ) + h 2 Var(I (R) ) 2hCov(I (H), I (R) ) ( ) = Var(I (R) ) h Cov(I(H), I (R) ) + c, Var(I (R) ) where c is a consan ha is free of h. Of course, when populaion basis risk is acually presen in realiy, I (H) is no necessarily equal o h (R) I (R). If I (H) > h (R) I (R), hen he pension liabiliy is under-hedged, and if I (H) < h (R) I (R), hen he opposie is rue. Populaion basis risk can herefore be undersood as he variabiliy associaed wih he random deviaions beween I (H) and h (R) I (R). The hird sep is o express he uncerainy surrounding I (H) and h (R) I (R) by a series of join predicion regions. Mahemaically, J α is a join predicion region for he duple (I (H), h (R) I (R) ) wih coverage probabiliy < 1 α 1 if Pr((I (H), h (R) I (R) ) J α ) = 1 α. The region J α should encompass 1(1 α)% of he possible combinaions of I (H) and 5

18 h (R) I (R). For a given value of α, a larger J α reflecs a higher amoun of populaion basis risk. We consruc nine join predicion regions, wih α =.1,.2,...,.9. Finally, he graphical risk meric is creaed by ploing on a Caresian coordinae plane he 1% join predicion region wih he darkes shading, surrounded by he 2%, 3%,..., 9% join predicion regions wih progressively ligher shadings. From he areas of he predicion regions and he degrees of shading, one can visualize ranges of possible hedging oucomes and heir associaed probabiliies of occurrence. The proposed risk meric is somewha similar o he well-known Bank of England inflaion fan char, which simulaneously depics inerval forecass of fuure inflaion raes a differen confidence levels by using differen shades of colour (Wallis, 23). I also has a close resemblance o he exising survivor/longeviy fan chars (Blake e al., 28; Dowd e al., 21). 1.3 An Illusraion We now illusrae he graphical populaion basis risk meric wih a hypoheical example. Le us suppose ha H, he populaion associaed wih he pension plan (he hedger), is Canadian males. Suppose furher ha a he ime when he hedge is esablished, here is no longeviy-linked derivaive linked o Canadian males. However, he plan may use a longeviy-linked derivaive ha is linked o an alernaive reference populaion (R), which can be eiher U.S. males, German males, Duch males, or English and Welsh males. 3 The survivor index used is he ex pos probabiliy ha an individual currenly aged 65 3 As a maer of fac, he LLMA provides moraliy indexes for hese four naional populaions. Derivaive securiies can be wrien on LLMA s moraliy indexes. 6

19 will survive o age 9: S (i) = 24 = (1 q (i) 65+,), i = H, R, where q (i) x, is he probabiliy ha an individual from populaion i dies in year, given ha he individual is alive and aged x a he beginning of year. This survivor index is very similar o he one ha is associaed wih he 25-year longeviy bond ha was announced by BNP Paribas and he European Invesmen Bank in 24 (Blake e al., 213). We use he augmened common facor model proposed by Li and Lee (25) o concurrenly model he fuure moraliy of all five populaions. The model can be expressed as ln(m (i) x,) = a (i) x + B x K + b (i) x k (i) + ɛ (i) x,, i = H, R, where m (i) x, denoes populaion i s cenral deah rae a age x and in year, a (i) x is a parameer measuring populaion i s average level of moraliy a age x, K is a ime-varying index ha is shared by all populaions being modeled, k (i) is specific o populaion i, parameers B x and b (i) x and k (i) a age x, and ɛ (i) x, is he error erm. Following Li and Lee, we esimae a (i) x is he ime-varying index ha respecively reflec he sensiiviy o K by seing i o he average of ln(m (i) x,) over he daa sample period. To esimae B x and K, we apply a firs order singular value decomposiion (SVD) o he marix of i w(i) x,(ln(m (i) x,) â (i) x ), where w (i) x, represens populaion i s number of exposures a age x and year and he ˆ sign denoes an esimae. Anoher firs order SVD is applied o he marix of ln(m (i) x,) â (i) x ˆB x ˆK o obain esimaes of parameers b (i) x and k (i). 7

20 The evoluion of K over ime is modeled by a random walk wih drif: K = C + K 1 + ξ, where C is he drif erm and {ξ } is a sequence of i.i.d. normal random variables wih zero mean and consan variance, whereas he evoluion of k (i) over ime is modeled by a firs order auoregressive process: k (i) = φ (i) + φ (i) 1 k (i) 1 + ζ (i), where φ (i) is a consan, φ (i) 1 is anoher consan whose absolue value is sricly less han one, and {ζ (i) } is a sequence of i.i.d. normal random variables wih zero mean and consan variance. The process for k (i) differen populaions do no diverge indefiniely over ime. is mean-revering, so ha he projeced moraliy raes for The model is fied o hisorical daa covering he age range of 6 o 89 and he sample period of 196 o 29. Mos of he required daa are obained from he Human Moraliy Daabase (214). The only excepion is he daa for German males prior o 1991 (when he Berlin Wall fell), which are obained from he LLMA. Under he augmened common facor model, he probabiliy disribuion of S (i) canno be wrien in closed-form; hus, he join predicion regions canno be derived analyically as was done by Chan e al. (214). Insead, we obain he join predicion regions wih he following numerical procedure: 1. Simulae 5, fuure values of m (i) x, from he esimaed augmened common facor 8

21 Populaion h (R) The US Germany The Neherlands England and Wales Table 1.1: The calculaed values of h (R) for he four reference populaions under consideraion. model. Using hese simulaed values and he approximaion q (i) x, = 1 exp( m (i) x,), 4 calculae realizaions of I (H) and I (R). 2. Using he realized values of I (H) and I (R), calculae he value of h (R) using he previously described linear regression mehodology. The calculaed values of h (R) for he four reference populaions under consideraion are displayed in Table Le Y = (I (H), h (R) I (R) ). For each simulaed realizaion of Y, calculae is Mahalanobis disance o he bes esimae as Y Ŝ 1 Y, where Ŝ is he sample covariance marix of Y. Noe ha he bes esimae of Y is E(Y ) = (, ). Geomerically speaking, he Mahalanobis disance may be viewed as he physical disance beween he realizaion of Y and he origin, weighed by he sandard deviaions and covariance of I (H) and h (R) I (R) Sor he 5, simulaed realizaions by heir Mahalanobis disances o he bes esimae. Choose he 5, (1 α) realizaions wih he shores Mahalanobis disances. 5. Draw a convex hull o enclose he 5, (1 α) chosen realizaions. In geomerical 4 The approximaion is exac if he force of moraliy beween wo ineger ages is consan. 5 See Gnanadesikan and Keenring (1972) for furher informaion abou Mahalanobis disances. 9

22 erms, he convex hull is he smalles convex se ha conains he seleced 5, (1 α) pairs of I (H) and h (R) I (R). The convex hull drawn is a 1(1 α)% join predicion region for I (H) and h (R) I (R), because by consrucion i conains a randomly seleced pair of I (H) and h (R) I (R) in he simulaed sample wih a probabiliy of 1 α. The use of a convex hull (he smalles convex se) prevens he join predicion region from oversaing he underlying uncerainy. Figure 1.1 shows he graphical populaion basis risk meric when he reference populaion is English and Welsh males. The wo doed lines divide he diagram ino four quadrans. The upper-righ (lower-lef) quadran conains he oucomes when fuure moraliy of boh populaions improves faser (slower) han expeced, while he upper-lef (lowerrigh) quadran encompasses he oucomes when he moraliy of Canadian males improves slower (faser) han expeced and he moraliy of English and Welsh males improves faser (slower) han expeced. The dos in he diagram represen he 5, simulaed pairs of I (H) and h (R) I (R). These dos should align perfecly on he 45-degree line in he ideal case when here is no populaion basis risk. The region below he 45-degree line conains he under-hedging oucomes, while he region above conains he over-hedging oucomes. The verical (or equivalenly, horizonal) disance from an oucome o he 45-degree line indicaes he exen of over- or under-hedging associaed wih ha oucome. The likelihood of an oucome is visible from he colour shade of he region in which he oucome is locaed. Essenially, he darker he shading, he more likely he oucome. The area spanned by he risk meric indicaes he overall level of populaion basis risk. Therefore, one may deermine he reference populaion ha leads o he minimum amoun 1

23 of basis risk by comparing he areas of he risk merics for all available reference populaions. Figure 1.2 displays he graphical risk merics for all four reference populaions under consideraion. I is clear ha he risk meric for U.S. males is smaller han he risk merics for he oher hree available reference populaions. Hence, for his hypoheical example, he hedger should choose o rade a derivaive ha is linked o U.S. male moraliy. 1.4 Conclusion In his chaper, we have proposed a graphical meric o inuiively communicae informaion abou he level of populaion basis risk ha an index-based longeviy hedge is exposed o. The graphical risk meric is composed of a series of join predicion regions of possible hedging oucomes, which are simulaed from an assumed muli-populaion sochasic moraliy model. Various aspecs of populaion basis risk are refleced in he graphical risk meric. Firs, he area of a predicion region indicaes he overall level of he populaion basis risk. Second, he shade of a predicion region reflecs he likelihood of he hedging oucomes enclosed by he region. Third, he shape of he predicion region reveals how he hedger s liabiliy is correlaed wih he survivor index o which he sandardized hedging insrumen is linked. Compared o exising populaion basis risk merics which are mosly numerical and only measure he overall risk level, he proposed meric is more informaive in ha i capures more aspecs of populaion basis risk. Along wih he exising numerical merics, he proposed graphical meric may help poenial hedgers beer undersand populaion basis risk and hence make heir risk managemen decisions. 11

24 We have also illusraed he graphical populaion basis risk meric by a hypoheical example, in which he hedger s liabiliy is associaed wih Canadian moraliy while he available hedging insrumens are linked respecively o he populaions of he U.S., Germany, he Neherlands and England and Wales. Given he resuling join predicion regions, one can easily ell ha among he four reference populaions, he U.S. is he mos appropriae for he hypoheical hedger. We believe ha as he marke grows and sandardized insrumens linked o differen reference populaions become available, he proposed echnique can assis hedgers wih heir choices of hedging insrumens. The graphical populaion basis risk meric depends on he assumed muli-populaion sochasic moraliy model. Admiedly, he conclusions derived from he graphical meric may urn ou o be differen if anoher sochasic moraliy model is assumed. I is warraned o explore in fuure work he robusness of he graphical risk meric relaive o model choices. From a pracical viewpoin, i would be useful o incorporae he proposed echnique ino exising sochasic moraliy modeling sofware such as he LLMA s LifeMerics. Such a developmen would allow poenial hedgers o cusomize he graphical populaion basis risk meric on he basis of heir own choices of moraliy models and daa ses. 12

25 Figure 1.1: The graphical populaion basis risk meric for he siuaion when he hedger s populaion (H) is Canadian males and he derivaive s reference populaion (R) is English and Welsh males. The dos represen he 5, simulaed pairs of I (H) and h (R) I (R). 13

26 U.S. males German males Duch males English and Welsh males.2 h (R) I (R) I (H) Figure 1.2: The graphical populaion basis risk merics for he siuaions when he hedger s populaion (H) is Canadian males and he derivaive s reference populaions (R) are U.S. males, German males, Duch males and English and Welsh males, respecively. 14

27 Chaper 2 Dynamic Longeviy Hedging in he Presence of Populaion Basis Risk: A Feasibiliy Analysis from Technical and Economic Perspecives 2.1 Inroducion The marke for longeviy risk ransfers sared in abou 1 years ago when he European Invesmen Bank and BNP Paribas experimened a 25-year longeviy bond. Since hen, he marke has seen some significan developmens, mos noably in erms of he number and size of deals (Blake e al., 214). However, relaive o he size of he global longeviy 15

28 risk exposure, he presen longeviy risk ransfer marke is sill very small. A small marke no only impedes longeviy risk managemen, bu also poses sysemic concerns, because when longeviy risk is shifed from he corporae secor o a limied number of (re)insurers, wih global inerconnecions, here may be sysemic consequences in he case of a failure of a key player (Basel Commiee of Banking Supervision, 213). The underdevelopmen of he longeviy risk ransfer marke may be aribued o he marked imbalance beween demand and supply. To dae, mos of he longeviy risk ransfers execued are insurance-based, ypically in he form of pension buy-ins, pension buyous or bespoke longeviy swaps. While he insurance indusry has he scope and financial sabiliy o assume longeviy risk, i does no generae sufficien supply for accepance of he risk because of is capaciy consrains. Using he asses for pension plans, in excess of 31 rillion USD, as a proxy for demand and he asses of 2.6 rillion USD held by he global insurance indusry o cover non-life risks as a proxy for supply, Graziani (214) concluded ha he demand for accepance of longeviy risk exceeds supply by a muliple of 1. Michealson and Mulholland (214) also reached a similar conclusion by comparing he poenial increase in pension liabiliies due o unforeseen longeviy improvemen wih he aggregae capial of he global insurance indusry. The demand and supply imbalance will only become worse if he reliance on he insurance indusry o assume longeviy risk coninues. On one hand, he demand is expeced o rise when pension plans in Norh America, where longeviy risk was no widely recognized, begin o realize he maerialiy of he risk as hey replace older moraliy assumpions wih he recenly launched indusry sandards (he MP-214 Scale for he US and he CPM-B Scale for Canada), which reflec he acceleraion of moraliy improvemen happened over 16

29 he pas wo decades. 1 On he oher hand, as Solvency II and is equivalence come ino full effec, he insurance indusry will be subjec o more sringen capial requiremens, which furher compress he indusry s abiliy o accep longeviy risk exposures from pension plans. The growh of he longeviy risk ransfer marke herefore depends highly on he creaion of supply, mos likely by inviing paricipaion from capial markes, which are capable of assuming a larger porion of he longeviy risk exposures from pension plans around he world. 2 The longeviy asse class offers capial marke invesors a risk premium, plus poenial diversificaion benefis due o is very low correlaion wih lierally every oher asse class, including inflaion, foreign exchange, commodiies and equiies (Ribeiro and di Piero, 29). However, drawing ineres from such invesors requires he longeviy risk ransfer marke o package he risk as sandardized producs ha are srucured like ypical capial marke derivaives and linked o broad-based moraliy indexes. The ac of sandardizaion is imporan in par because i fosers he developmen of liquidiy, and in par because i removes he informaion asymmery arising from he fac ha hedgers (pension plans) have beer knowledge abou he moraliy experience of heir own porfolios. Towards he goal of sandardizaion, he marke for longeviy risk ransfers has o overcome wo echnical challenges which discourage hedgers from using sandardized hedging insrumens. The firs challenge is o find ou how sandardized insrumens can be used o form a hedge ha can eliminae a meaningful porion of he hedger s longeviy risk exposure. Hedging sraegies have o be developed so ha hedgers know he ype and 1 See he Sociey of Acuaries (214) and he Canadian Insiue of Acuaries (214). 2 According o Roxburgh (211), he oal value of he world s financial sock, comprising equiy marke capializaion and ousanding bonds and loans, is 212 rillion USD a he end of

30 noional amouns of hedging insrumens hey need o acquire. The second challenge is o undersand and more imporanly miigae he residual risks ha are lef behind by a sandardized, index-based longeviy hedge. Of he residual risks he mos significan consiuen is populaion basis risk, which arises from he difference in fuure moraliy improvemens beween he populaion associaed wih he hedger s own porfolio and he populaion(s) o which he sandardized insrumens are linked. However, as explained below, he research quesions on longeviy hedging sraegies and populaion basis risk are sill open. A significan porion of he exising lieraure on longeviy hedging sraegies focuses on saic hedging (Cairns, 213; Cairns e al., 26b, 214; Coughlan e al., 211; Dowd e al., 211; Li and Hardy, 211; Li and Luo, 212). Broadly speaking, he saic hedging sraegies were derived by maching he sensiiviies of he liabiliy being hedged and porfolio of hedging insrumens wih respec o changes in he underlying moraliy raes. Saic hedging sraegies are generally subjec o he shorcoming of he need for long-daed hedging insrumens. For example, in an illusraive saic hedge for a 3-year pension liabiliy, Li and Luo (212) used five securiies, of which he longes ime-o-mauriy is 25 years. Such long-daed securiies do no seem appealing o capial marke invesors. A few researchers including Cairns (211), Dahl (24), Dahl and Møller (26), Dahl e al. (28) and Luciano e al. (212) proposed dynamic longeviy hedging sraegies. Excep he work of Cairns (211), he exising dynamic longeviy hedging sraegies were developed from coninuous-ime models, which provide mahemaical racabiliy bu are no sraighforward o implemen in pracice. Furher, alhough some exising saic hedging sraegies include an adjusmen for populaion basis risk (Dowd e al., 211; Li and Hardy, 18

31 211; Li and Luo, 212), none of he aforemenioned dynamic longeviy hedging sraegies akes populaion basis risk ino accoun. For he problem of populaion basis risk, researchers have recenly conribued significanly o he developmen of muli-populaion sochasic moraliy models (Ahmadi and Li, 214; Cairns e al., 211; Dowd e al., 211; Hazopoulos and Haberman, 213; Jarner and Kryger, 211; Li and Hardy, 211; Li and Lee, 25; Yang and Wang, 213; Zhou e al., 213, 214). Such models can be regarded as a pre-requisie for undersanding populaion basis risk, because hey allow users o gauge he range of possible moraliy differenials beween wo relaed populaions, wih biological reasonableness aken ino consideraion. Researchers have also inroduced merics for quanifying populaion basis risk, for example, reducion in expeced shorfall (Ngai and Sherris, 211), reducion in porfolio variance (Coughlan e al., 211; Li and Hardy, 211) and minimal required buffer (Sevens e al., 211). However, o our knowledge, lile aenion has been paid o how populaion basis risk can be miigaed. In his chaper, we aemp o address he limiaions of he curren lieraure by invesigaing how a dynamic, index-based longeviy hedge can be performed when populaion basis risk is presen and how he residual risks lef behind by he hedge can be miigaed. Figure 2.1 provides a graphical illusraion of he general framework on which his chaper is based. One par of he framework is a dynamic hedging sraegy wih which a pension plan can ransfer he rend risk (i.e., he risk surrounding he rend in longeviy improvemen) o capial markes, even if he securiies available are linked o a broad-based moraliy index. Anoher par of he framework is a specially designed reinsurance reay, called a cusomized surplus swap, which ransfers he residual risks o a reinsurer who 19

32 collecively manages he residual risks from he index-based longeviy hedges of various pension plans. 3 The dynamic hedging sraegy we propose is obained by generalizing he dynamic dela hedging sraegy of Cairns (211) o incorporae he siuaion when he populaions associaed wih he hedger s porfolio and he hedging insrumens are no he same. The generalizaion is derived on he basis of a muli-populaion sochasic moraliy model, under which he moraliy dynamics of differen populaions are non-rivially correlaed. When implemening he proposed hedging sraegy, he hedger needs o hold one only hedging insrumen a a ime and he hedging insrumen can be shorer-daed. The former propery helps he marke o concenrae liquidiy, while he laer propery beer mees he appeie of capial marke invesors. Adding furher o he conribuion of Cairns (211) is a sudy of he robusness of he dynamic hedging sraegy relaive o differen facors including model risk, small sample risk and he properies of he hedging insrumens used. The cusomized surplus swap we design eliminaes all residual risks ha are lef behind by he dynamic longeviy hedge. Therefore, he combinaion of a dynamic longeviy hedge and cusomized surplus swap should produce he same hedge effeciveness as a ypical bespoke longeviy swap. Using real moraliy daa from 25 differen populaions, we demonsrae ha he residual risks can poenially be diversified away when a reinsurer wrie cusomized surplus swaps wih a range of hedgers. A reinsurer should hus have a 3 A similar concep was menioned by Cairns e al. (28). In heir se-up, hedgers ransfer all heir longeviy risk exposures by wriing bespoke longeviy swaps wih a special purposed vehicle (SPV), and he SPV in urn issues a sandardized longeviy bond which ransfers he rend risk o he bondholders. The residual risks are borne by he SPV manager. 2

33 Longeviy risk of a pension plan Trend risks from various pension plans Trend risk Sandardized securiies Capial marke invesors Residual risks Cusomized surplus swaps A reinsurer Residual risks from various pension plans Figure 2.1: A graphical illusraion of he general framework on which his chaper is based. 21

34 much larger capaciy o wrie cusomized surplus swaps han conracs such as pension buy-ous which involve significan sysemaic risk. Overall, our proposed risk managemen framework is likely o be more economical han radiional longeviy risk ransfers ha are enirely insurance-based, because in heory i is less cosly o ransfer he sysemaic rend risk hrough liquidly raded sandardized securiies han ailor-made (re)insurance conracs. The res of his chaper is organized as follows. Secion 2.2 presens he echnical deails of he proposed dynamic hedging sraegy. Secion 2.3 illusraes he proposed dynamic hedging sraegy and evaluaes is robusness relaive o various facors. Secion 2.4 defines he proposed cusomized surplus swap and demonsraes he diversifiabiliy of he residual risks. Finally, Secion 2.5 concludes he chaper and discusses in more deail why he proposed risk managemen framework is likely o be more economical. 2.2 The Dynamic Longeviy Hedging Sraegy The Assumed Model The dynamic hedging sraegy requires an assumed sochasic moraliy model, from which quaniies such as hedge raios can be derived. In he single-populaion se-up of Cairns (211), he original Cairns-Blake-Dowd model (a.k.a. Model M5) was assumed. In our muli-populaion generalizaion, we assume he augmened common facor (ACF) model proposed by Li and Lee (25). The ACF model concurrenly models he moraliy dy- 22

35 namics of muliple, say P, populaions as follows: ln(m (i) x,) = a (i) x + B x K + b x (i) k (i) + ɛ (i) x,, i = 1,..., P, where m (i) x, represens populaion i s cenral rae of deah a age x and in year, a (i) x parameer indicaing populaion i s average level of moraliy a age x, K is a ime-varying index ha is shared by all P populaions, k (i) populaion i, parameers B x and b (i) x is a is a ime-varying index ha is specific o respecively reflec he sensiiviy of ln(m (i) x,) o K and k (i), and ɛ (i) x, is he error erm ha capures all remaining variaions. Following Li and Lee (25), we esimae he ACF model by he mehod of singular value decomposiion. The rend in K deermines he evoluion of moraliy over ime for all populaions being modeled. As in he original Lee-Carer (Lee and Carer, 1992) model, K is assumed o follow a random walk wih drif: K = C + K 1 + ξ, where C is he drif erm and {ξ } is a sequence of i.i.d. normal random variables wih zero mean and consan variance σ 2 K. Deparures from he common ime rend are capured by he populaion-specific index k (i), which is assumed o follow a firs order auoregressive process: k (i) = φ (i) + φ (i) 1 k (i) 1 + ζ (i), where φ (i) and φ (i) 1 are consans, and {ζ (i) } is a sequence of i.i.d. normal random variables 23

36 wih zero mean and consan variance σk,i 2. We require φ(i) 1 < 1 so ha he process for k (i) is mean-revering. This propery ensures ha he resuling forecass are coheren, which means he projeced moraliy raes for differen populaions do no diverge indefiniely over ime. To incorporae any correlaion ha is no capured by he common rend K, we furher assume ha ζ (i) and ζ (j) for i j are consanly correlaed, despie such correlaions are no aken ino accoun in he original ACF model The Se-up We le S (i) x,(t ) = T (1 q (i) s=1 x+s 1,+s) (2.1) be he ex pos probabiliy ha an individual who is from populaion i and aged x a ime (he end of year ) would have survived o ime + T, where q (i) x, denoes he probabiliy ha an individual from populaion i dies beween ime 1 and (during year ), provided ha he/she has survived o age x a ime 1. When compuing q (i) x, from m (i) x, (on which he ACF model is based), we use he approximaion q (i) x, 1 exp( m (i) x,). 4 I is clear from he definiions ha S (i) x,(t ) is no known prior o ime + T, while q (i) x, is no known prior o ime. Define by F he informaion abou he evoluion of moraliy up o and including ime. Due o he Markov propery of he assumed sochasic processes, he value of E(S x,u(t (i) ) F ) for u depends only on he values of K and k (i) bu no he values of K v 4 The approximaion is exac if he force of moraliy beween wo consecuive ineger ages is consan. 24

37 and k v (i) for v <. Hence, we have p (i) x,u(t, K, k (i) ) := E(S x,u(t (i) ) K, k (i) ) = E(S x,u(t (i) ) F ). We call p (i) x,u(t, K, k (i) ) a spo survival probabiliy when u = and a forward survival probabiliy when u >. Le us suppose ha he hedger inends o hedge he longeviy risk associaed wih a pension plan for a single cohor of individuals, who are all from populaion H and aged x a ime. The plan pays each pensioner $1 a he end of each year unil deah. I follows ha he ime- value of he pension plan s fuure liabiliies (per surviving pensioner a ime ) can be expressed in erms of spo survival probabiliies as F L = s=1 (1 + r) s p (H) x +,(s, K, k (H) ), where r is he ineres rae for discouning purposes. The hedging insrumens are q-forwards ha are associaed wih populaion R. A q-forward is a zero-coupon swap wih is floaing leg proporional o he realized deah probabiliy a a cerain reference age during he year immediaely prior o mauriy and is fixed leg proporional o he corresponding pre-deermined forward moraliy rae. In his applicaion, he hedger should paricipae in he q-forwards as he fixed-rae receiver, so ha he/she will receive a ne paymen from he counerpary when moraliy urns ou o be lower han expeced. Consider a q-forward ha is linked o reference populaion R and age x f. Suppose 25

38 ha he q-forward is issued a ime and maures a ime + T. The payoff from he q-forward depends on he realized value of q (R) x f, +T. The corresponding forward moraliy rae q f is chosen so ha no paymen exchanges hands a incepion (ime ). I is assumed ha q f = E(q (R) x f, +T F ), which is equivalen o saying ha no risk premium is given o he counerpary acceping he risk. 5 A =,..., + T 1, he value of he hedger s posiion of he q-forward (per $1 noional) can be expressed as Q ( ) = (1 + r) ( +T ) (q f E(q (R) x f, +T F )) = (1 + r) ( +T ) (q f (1 E(S (R) x f, +T 1 (1) F ))) = (1 + r) ( +T ) (q f (1 p (R) x f, +T 1 (1, K, k (R) )). Under our pricing assumpion, we have Q ( ) =. Noe ha boh F L and Q ( ) are relaed linearly o values of p (i) x,u(t, K, k (i) ), where i = H, R and u. The main idea behind he dynamic hedging sraegy is ha a each discree ime poin, he q-forward porfolio is adjused so ha F L and he adjused q-forward porfolio have similar sensiiviies o changes in he underlying common moraliy index K. Hence, a each discree ime poin, we need o compue F L and Q ( ) and heir parial derivaives wih respec o K. However, because of he way in which S x,(t (i) ) depends on K u and k u (i) for u = + 1,..., T, he values of p (i) x,u(t, K, k (i) ) for u (and hus F L and Q ( )) canno be compued analyically. I follows ha nesed simulaions are required, making 5 Because he counerpary acceping longeviy risk from he hedger deserves a risk premium, in pracice q f should be smaller han E(q (R) x f, F +T ), so ha payoff o he counerpary is posiive in expecaion erms. However, because our focus for now is he echnical aspecs raher han he associaed coss, we assume q f = E(q (R) x f, F +T ) for simpliciy. 26

39 he dynamic hedging framework sraegy compuaionally challenging. In more deail, le us assume ha he hedging horizon is Y years and ha he q-forward porfolio is adjused annually. Suppose ha N sample pahs of fuure moraliy (i.e., values of K, k (H) and k (R) for = 1,..., Y ) are used o evaluae he hedge s performance. For each of hese N sample pahs, we need o evaluae, a each ime poin for = 1,..., Y, F L and Q ( ) on he basis of he realized values of K, k (H) and k (R) in ha paricular sample pah. If we calculae each F L and Q ( ) wih M sample pahs of moraliy beyond ime, hen in oal we need o generae N M Y sample pahs. Because N and M are ypically very large, say 1,, he compuaional burden is huge. To reduce compuaion burden, in he nex subsecion we derive formulas o approximae p (i) x,u(t, K, k (i) ) for u so ha he need for some of he simulaions can be avoided. The accuracy of he approximaion formulas is evaluaed in Appendix A The Approximaion Mehods The approximaion formula for p (i) x,u(t, K, k (i) ) depends on wheher u = or u >. Approximaing p (i) x,u(t, K, k (i) ) when u = Following Cairns (211), we approximae p x,(t, (i) K, k (i) ) by applying a Taylor expansion o is probi ransform, f x,(t, (i) K, k (i) ) := Φ 1 (p (i) x,(t, K, k (i) )), where Φ denoes he sandard normal disribuion funcion. The Taylor expansion is made around ˆK = E(K K ) and 27

40 ˆk (i) = E(k (i) k (i) ). We consider a second-order approximaion, which means f (i) x,(t, K, k (i) ) D x,,(t (i) ) + D x,,1(t (i) )(K ˆK ) + D (i) x,,2(t )(k (i) ˆk (i) ) D(i) x,,3(t )(K ˆK ) D(i) x,,4(t )(k (i) (i) ˆk ) 2 + D x,,5(t (i) )(K ˆK )(k (i) (i) ˆk ), where D (i) x,,(t ) = f (i) x,(t, ˆK, D (i) x,,2(t ) = ˆk (i) (i) f x, (T, ˆK,k (i) ) k (i) D x,,4(t (i) ) = 2 f (i) x, (T, ˆK,k (i) ) k 2,(i) (i) (i) f x, (T,K,ˆk ) ), D x,,1(t (i) ) = K, K= ˆK, D x,,3(t (i) ) = 2 f (i) (i) x, (T,K,ˆk ) (i) k =ˆk (i) K 2, K= ˆK, D x,,5(t (i) ) = 2 f (i) x, (T,K,k(i) ) k (i) =ˆk (i) K k (i) K= ˆK,k (i) The values of D (i) x,,j (T ) for j = 1,..., 5 are compued numerically as follows: =ˆk (i). D (i) x,,1(t ) (f (i) x,(t, ˆK + h, (i) ˆk ) f x,(t, (i) ˆK, D x,,2(t (i) ) (f x,(t, (i) ˆK (i), ˆk + h) f x,(t, (i) ˆK, D (i) x,,3(t ) (f (i) x,(t, ˆK + h, (i) ˆk ) + f x,(t, (i) ˆK h, (i) ˆk ))/h, (i) ˆk ))/h, (i) ˆk ) 2f x,(t, (i) ˆK (i), ˆk ))/h 2, D x,,4(t (i) ) (f x,(t, (i) ˆK (i), ˆk + h) + f x,(t, (i) ˆK (i), ˆk h) 2f x,(t, (i) ˆK (i), ˆk ))/h 2, D x,,5(t (i) ) (f x,(t, (i) ˆK (i) + h, ˆk + h) + f x,(t, (i) ˆK (i) h, ˆk h) f x,(t, (i) ˆK (i) + h, ˆk h) f x,(t, (i) ˆK (i) h, ˆk + h))/4h 2, where h is an arbirarily small posiive value. 28

41 To calculae he above parial derivaives for a fixed, we require nine ses of M sample moraliy pahs, which are respecively based on nine differen ses of saring values, including (K = ˆK, k (i) = (i) ˆk ), (K = ˆK + h, k (i) = (i) ˆk ), (K = ˆK, k (i) (i) = ˆk + h) and so on. For a hedging horizon of Y ime seps, he number of sample pahs required o generae he parial derivaives is 9 M Y. Suppose again ha N moraliy scenarios are used o evaluae he hedge s performance. Because he parial derivaives are independen of hese N moraliy scenarios, he oal number of sample pahs we need o generae is N + 9 M Y, which is significanly smaller han N M Y when N and M are large. Approximaing p (i) x,u(t, K, k (i) ) when u > Using a firs-order approximaion, i can be shown ha x,u(t, K, k (i) ) Φ p (i) E(V (i) u F ) Var(V (i) u F ), where E(V u (i) F ) = D x,u,(t (i) ) D x,u,1(t (i) )(E(K u F ) ˆK u ) D x,u,2(t (i) )(E(k u (i) (i) F ) ˆk u ), Var(V u (i) F ) = 1 + (D x,u,1(t (i) )) 2 Var(K u F ) + (D x,u,2(t (i) )) 2 Var(k u (i) F ), E(K u F ) ˆK u = K K C, E(k u (i) F ) ˆk u = (φ (i) 1 ) u ((φ (i) 1 ) k (i) k (i) ) + (φ(i) 1 ) u (1 (φ (i) 1 φ (i) 1 1 ) ) φ (i), 29

42 Var(K u F ) = σk 2 (u ) and Var(k(i) u F ) = 1 (φ(i). A proof of he above approximaion formula is provided in Appendix B. 1 )2(u ) σ 2 1 (φ (i) 1 )2 k,i Deriving Hedge Raios Our goal is o ensure ha a each discree ime poin, he q-forward porfolio and he pension plan s fuure liabiliies have similar sensiiviies o changes in he underlying common moraliy index K. To achieve his goal, he hedge raio h (i.e., he noional amoun of he q-forward) a ime is chosen in such a way ha F L K = h Q ( ) K. Because we mach he firs derivaives only, only one q-forward conrac is needed a each. For he same reason, our hedge may be considered as a dela hedge. In principle, one may creae, for example, a gamma hedge by maching also he second order derivaives. The nex chaper explores dela and gamma hedges in a saic se-up. The parial derivaive of F L wih respec o K is compued as follows: F L K = = (1 + r) s=1 (1 + r) s=1 s=1 s p(h) x +,(s, K, k (H) s Φ(f (H) K ) x +,(s, K, k (H) )) K (1 + r) s D (H) x +,,1(s) φ(f (H) x +,(s, K, k (H) )), 3

43 where φ represens he probabiliy densiy funcion for a sandard normal random variable. The parial derivaive of Q ( ) wih respec o K depends on he value of relaive o he q-forward s mauriy dae + T. If = + T 1, Q ( ) K = (1 + r) (R) p 1 x f,(1, K, k (R) K ) (1 + r) 1 D (R) x f,,1(1) φ(f (R) x f,(1, K, k (R) )). If =,..., + T 2, Q ( ) K =(1 + r) p (R) ( +T ) x f, +T 1 (1, K, k (R) ) K (1 + r) ( +T ) Φ K =(1 + r) ( +T ) φ E(V (R) +T 1 F ) Var(V (R) E(V (R) +T 1 F ) Var(V (R) +T 1 F ) +T 1 F ) D(R) x f, +T 1,1 (1). Var(V (R) +T 1 F ) Evaluaing he Hedge As previously menioned, N moraliy scenarios are simulaed o evaluae he effeciveness of he dynamic hedge. Define by P L he ime- value of all pension liabiliies, given he informaion up o 31

44 and including ime ; ha is, ( ) P L = E (1 + r) s S (H) x,(s) F s=1 F L, = = s=1 (1 + r) s S (H) x,(s) + (1 + r) S (H) x,()f L, >. (2.2) The value of P L is non-random, as i is simply a funcion of K and k (H) whose values are fixed. For >, he values of P L are differen under differen simulaed moraliy scenarios. In paricular, he values of S (H) x,(s) for s = 1,..., depend on he realized values of K s and k (H) s K and k (H). for s = 1,...,, whereas he value of F L depends on he realized values of I is assumed ha a each ime poin, he hedger wries a new q-forward conrac (i.e., a q-forward wih incepion dae = ) wih a noional amoun of h. The value of his posiion is h Q () = a ime and becomes h Q +1 () = h (1 + r) (T 1) (q f E(q (R) x f,+t F +1)) = h (1 + r) (T 1) (E(q (R) x f,+t F ) E(q (R) x f,+t F +1)) = h (1 + r) (T 1) (p (R) x f,+t 1 (1, K +1, k (R) +1) p (R) x f,+t 1 (1, K, k (R) )) a ime A ime + 1, he posiion wrien a ime is closed ou, and anoher new q-forward conrac is wrien. The process repeas unil he end of he hedging horizon Y is reached. For simpliciy, we assume ha all q-forwards used over he hedging horizon 6 The second sep is due o our pricing assumpion. 32

45 have he same mauriy T and reference age x f. Le P A be he ime- value of he asses backing he pension plan a ime. We assume ha P A = P L. For = 1,..., Y, we have P A = P A 1 + (1 + r) h 1 Q ( 1). If P A is very close o P L for = 1,..., Y, hen he dynamic hedge can be said as successful. The poenial deviaion beween P A and P L is he residual risk ha is no miigaed by he hedge. Using his reasoning, we measure hedge effeciveness by he following meric: HE = 1 Var(P A P L F ). Var(P L F ) A value of HE ha is close o one indicaes he hedge is effecive. Similar merics have been used by Cairns (211, 213), Cairns e al. (214), Coughlan e al. (211) and Li and Hardy (211). 2.3 Analyzing he Dynamic Longeviy Hedge Assumpions The following assumpions are used in he baseline calculaions. 1. The hedger wishes o hedge he pension liabiliies ha are payable o a single cohor of individuals, who are all aged x = 6 a ime. The moraliy experience of hese 33

46 individuals is idenical o ha of he UK male insured lives. 2. The pension plan pays each individual $1 a he end of each year unil deah or age 9, whichever he earlies The hedging horizon is Y = 3 years (i.e., he hedge sops when he liabiliies have compleely run off). 4. The q-forwards used are linked o England and Wales (EW) male populaion. They all have a ime-o-mauriy (from incepion) of T = 1 years and a reference age of x f = The marke for he q-forwards is liquid and no ransacion cos is required. 6. The ineres rae for all duraions is r = 4% per annum. The ineres rae remains consan over ime. 7. The hedger can inves or borrow a an ineres rae of r = 4% per annum. 8. The values of D (i) x,,j (T ) for i = H, R and j =,..., 5 are compued from an ACF model ha is esimaed o he daa from he populaions of EW males and UK male insured lives over he period of 1966 o 25 and he age range of 6 o To mach he end poin of he daa sample period, ime is se o he end of year We assume ha no pension is payable beyond age 9, because he upper limi of he age range o which he ACF model is fied is 89. This assumpion may be relaxed if one assumes a parameric curve o exrapolae deah probabiliies beyond age The daa for EW males are provided by he Human Moraliy Daabase (214), while he daa for UK male insured lives are obained from he Insiue and Faculy of Acuaries by a wrien reques. 34