Static versus dynamic longevity risk hedging

Transcription

1 ISSN Saic versus dynamic longeviy risk hedging Clemene De Rosa Elisa Luciano Luca Regis No. 403 March by Clemene De Rosa, Elisa Luciano and Luca Regis. Any opinions expressed here are hose of he auhors and no hose of he Collegio Carlo Albero.

2 Saic versus dynamic longeviy-risk hedging Clemene De Rosa, Elisa Luciano, Luca Regis April 1, 2015 Absrac This paper provides he saic, swap-based hedge for an annuiy, and compares i wih he dynamic, dela-based hedge, achieved using longeviy bonds. We assume ha he longeviy inensiy is disribued according o a CIR-ype process and provide closed-form derivaives prices and hedges, also in presence of an analogous CIR process for ineres rae risk. Our calibraion o 65-year old UK males shows ha once ineres rae risk is perfecly hedged he average hedging error of he dynamic hedge is moderae, and boh is variance and he hickness of he ails of is disribuion are decreasing wih he rebalancing frequency. The spread over he basic "swap rae" which makes 99.5% quanile of he disribuion of he dynamic hedging error equal o he cos of he saic hedge lies beween 0.01 and 0.06%. Keywords: longeviy risk, saic vs. dynamic hedging, longeviy swaps, longeviy bonds. JEL classificaion: G22, G32. 1 Inroducion Life insurance companies porfolios are affeced by so-called longeviy risk, which is he risk ha people live longer han expeced when he company priced and reserved heir policies. So, while increasing longeviy is welcome from he social poin of view, i is considered one of he risks ha life insurance companies have o face. The ways in which a presen hey can cope wih longeviy is eiher by reinsuring i, as hey used o do in he pas, or by using more recen hedging approaches. Such sraegies rely on he use of so-called moraliy derivaives, which were firs inroduced by Blake and Burrows (2001). In he las decade he marke for such insrumens slowly developed and, while sill lacking liquidiy, allows insurers and pension funds o pursue hese alernaive de-risking sraegies. A presen, acors seeking coverage agains longeviy risk can choose beween a full, saic hedge hrough a derivaive, such as an s-forward or a longeviy swap, or a parial, dynamic hedge. In he firs case he whole excess of longeviy is The Auhors graefully acknowledge financial suppor from he Global Risk Insiue, Canada. Collegio Carlo Albero, clemene.derosa@carloalbero.org Universiy of Torino and Collegio Carlo Albero, elisa.luciano@unio.i. IMT Insiue for Advanced Sudies Lucca and Collegio Carlo Albero; luca.regis@imlucca.i. 1

3 ransferred o a hird pary, once and for all, and he coverage is no changed over ime. In he second case, coverage is parial, ofen done hrough cusomized derivaives, which we will call longeviy bonds. The parial naure of coverage calls for adjusmen over ime. In his paper longeviy risk is represened hrough he so-called sochasic longeviy, i.e. by an inensiy of moraliy arrival which is iself a sochasic process. In order o keep he model racable and o provide easy o implemen hedges, we work in coninuous ime. In order o ensure posiiviy of he inensiy and o have a longeviy model which nicely couples wih he modeling of ineres raes, we assume ha longeviy iself follows a Feller, or Cox e al. (1985)-ype (CIR) process. Some previous works have focused on saic hedging ools. Ngai and Sherris (2011), in paricular, compared he effeciveness of saic hedging hrough various derivaives. Raher few previous sudies focus insead on dynamic hedging. Among hem, Dahl e al. (2011) analyzed dynamic hedging via longeviy swaps, analyzing he differen performance of a consan and of a rebalanced sraegy. The original conribuion of our paper lies in providing boh saic and dynamic, closed - form hedges for he CIR longeviy process. The saic hedge enails he use of a longeviy swap, while he dynamic hedge is performed applying Dela-Gamma hedging sraegy, as proposed by Luciano e al. (2012b). We couple he heoreical conribuion wih a calibraed example, and we compare he efficiency of he saic versus he dynamic hedge. We deermine he cos of he saic hedge which would equae, in a sense ha we specify below, he hedging error of he parial coverage. We explore sensiiviy wih respec o differen assumpions on he rebalancing frequency of he sraegy, which is expeced o affec he qualiy of hedging. We leave he sudy of he role of ransacion coss and basis risk for furher research. The paper unfolds as follows: in Secion 2 we se up he model for longeviy and financial risk evaluaion, in Secion 3 we describe he liabiliies o be hedged, in Secion 4 we explain he saic and dynamic hedging sraegy, in Secion 5 we compare heir effeciveness on a calibraed model. The las Secion summarizes and oulines furher research. 2 Longeviy and ineres rae risk modelling In order o model longeviy and ineres rae risk, we assume ha moraliy for a specific generaion occurs according o a Poisson process, whose inensiy is sochasic. We consider a sandard filered probabiliy space (Ω, F, Q), which saisfies he usual assumpions, and on which a filraion F is defined. The measure Q is already he so-called risk-neural measure. We will discuss below he relaionship beween his measure and he effecive one. We le he moraliy inensiy of a specific generaion be described by a socalled Cox-Ingersoll and Ross (CIR) process, which is acually a Feller process, of he ype: dλ() = (a + bλ())d + σ λ()dw (), (1) wih a > 0, b > 0, σ > 0, λ(0) = λ 0 R ++. The reason behind he assumpion b > 0 is ha he process is expeced o have no mean reversion. The previous SDE describes he evoluion (for a given generaion) of he inensiy of moraliy arrival over calendar ime. Because he generaion ages over ime, he previous 2

4 drif simply ells ha he expeced change in inensiy is affine and increasing wih he inensiy iself. If he iniial poin λ 0 is sricly posiive and he coefficiens saisfy he following condiion: a σ2 2, (2) hen he moraliy inensiy λ() will be sricly posiive for every, almos surely. Hence, in order o obain a saisfacory calibraed model for he inensiy process, we impose his condiion on he parameers during he calibraion. Consisenly, we assume ha he spo ineres rae - or ineres rae inensiy - follows a CIR process of he ype: dr() = (ā br())d + σ r()dw (), (3) wih ā > 0, b > 0, σ > 0, r(0) = r 0 R ++, where he Wiener process W is independen of W. 1 The las assumpion enails independency beween he whole longeviy and ineres inensiy processes. The negaive sign preceding b and is sric posiiviy guaranee ha he process for he ineres-rae incorporaes mean reversion, which is a usual assumpion in he ineres-rae domain. The coefficien b is called speed of mean reversion and represens he speed a which he he shor rae r() reurns o is long-run value ā whenever r < ā or viceversa. Similarly o he longeviy case, he resricion on he parameers ha, ogeher wih he posiiviy of he iniial poin r 0, guaranees ha he ineres rae r() never urns negaive is given by: ā σ2 2. (4) A each single poin in ime, he condiional disribuions of he moraliy inensiy and he ineres rae are given, up o a scale facor, by a noncenral chi-square disribuion. In deails, given wo ime insans u <, hen he disribuion of λ() condiional on λ(u) is given by: λ() σ2( e b( u) 1 ) X 2 d (ν), (5) 4b where X 2 d (ν) denoes he densiy of a noncenral chi-square random variable wih degrees of freedom d = 4a σ 2, (6) and noncenraliy parameer ν = 4be b( u) λ(u). (7) σ 2 (e b( u) 1) Similarly, he disribuion of r() condiional on r(u) is given by: r() σ2( 1 e ) b( u) X 2 4 b d ( ν), (8) 1 Le he filraion F be he filraion generaed by he wo Brownian moions. 3

5 where X 2 d ( ν) denoes he densiy of a noncenral chi-square random variable wih degrees of freedom d = 4ā σ 2, (9) and noncenraliy parameer ν = 4 be b( u) r(u). (10) σ 2 (1 e b( u) ) In order o proceed o insurance producs pricing and hedging, he riskneural dynamics of he wo previous processes is needed. However, for calibraion purposes, is effecive or hisorical version may be useful, a leas for he longeviy case. In order o keep he noaion simple, we jus assume ha here is no risk premium in he longeviy marke or, equivalenly, ha equaion (1) holds under boh measures. Therefore, he calibraion of he longeviy inensiy is performed by esimaing is dynamics under he hisorical measure and, hen, using i also under he risk-neural measure. The calibraion of he ineres rae dynamics is, on he oher hand, performed direcly under he risk-neural measure, hus incorporaing he risk premium. If we call τ he ime o deah, he condiional survival probabiliy from o T is S(, T ) = P (τ T τ > ). where P is in he Q measure. In he presence of a sochasic inensiy, i can be represened as [ ( ) ] T S(, T ) = E exp λ x (s)ds F. (11) The expecaion E, here and below, is sill under Q. Under he CIR assumpion, ha probabiliy becomes: S(, T ) = A(, T )e B(,T )λ(), (12) where A(, T ) and B(, T ) are soluions of an appropriae sysem of Riccai equaions. These funcions are A(, T ) = B(, T ) = ( 2γe 1 2 (γ b)(t ) (γ b) ( e γ(t ) 1 ) + 2γ ) 2a σ 2, (13) 2 ( e γ(t ) 1 ) (γ b) ( e γ(t ) 1 ) + 2γ, (14) where γ = b 2 + 2σ 2. As shown in Fung e al. (2014), he above specificaion guaranees also ha he limi of he survival probabiliy, when T diverges, is zero. For any given, i is possible o compue he log derivaive of he survival probabiliy, which is somewha inappropriaely called he " forward" moraliy inensiy for ime T, since i represens is forecas a ime. By definiion f(, T ) = lns(, T ) = lna(, T ) 4 + B(, T ) λ(), (15)

6 where lna(, T ) B(, T ) [ ] = 2a 1 σ 2 2 (γ b) γe γ(t ) e γ(t ) 1 + 2γ, (16) γ b = 4γ 2 γ(t ) e [ ( (γ b) e γ(t ) 1 ) + 2γ ] 2. (17) Using a echnique described in Jarrow and Turnbull (1994) and Luciano e al. (2012a), which explois he definiion of "forward" inensiy, we can wrie he survival as S(, T ) = e X(,T )I()+Y (,T ), (18) where I() = λ() f(0, ), while X(, T ) and Y (, T ) are deerminisic funcions of parameers and ime and T : X(, T ) = B(, T ), Y (, T ) = lna(, T ) B(, T ) [ lna(0, ) + ] B(0, ) λ(0). The erm I is called longeviy risk facor and is he difference beween he acual and forecased inensiy for ime. Using he fac ha λ() = I() + f(0, ), (18) becomes S(, T ) = A(, T )e lna(0,) B(,T )[I() + B(0,) λ(0)]. (19) Hence, we have an expression for he survival equivalen o (12). This expression will play a crucial role in hedging, because i encapsulaes all riskiness in he I facor, which has he inuiively nice inerpreaion of difference beween he forecased and acual inensiy. This is exacly wha we have in mind when we hink of longeviy risk. The discoun facor or bond price for ime, under any sochasic process for he spo rae, is [ ( ) ] T D(, T ) = E exp r(u)du F, (20) which, in he CIR case, becomes D(, T ) = Ā(, T )e B(,T )r(), Ā(, T ) = B(, T ) = ( 2 γe 1 2 ( γ+ b)(t ) ( γ + b) ( e γ(t ) 1 ) + 2 γ ) 2ā σ 2, (21) 2 ( e γ(t ) 1 ) ( γ + b) ( e γ(t ) 1 ) + 2 γ, (22) wih γ = b2 + 2 σ 2. As in he longeviy case, he bond value can be reformulaed as D(, T ) = e X(,T )K()+Ȳ (,T ), (23) 5

7 where X(, T ) = B(, T ), Ȳ (, T ) = lnā(, T ) B(, T ) [ lnā(0, ) + B(0, ] ) r(0), and K is he financial risk facor, measured by he difference beween he shor and forward rae: K() = r() F (0, ). The forward rae F (0, ) is a significan financial quaniy, ha represens he fair price a ime 0 - and in general a ime when i becomes F (, T ) - for a forward conrac on he spo rae a T. I is compued, similarly o he forward moraliy inensiy, as where ln D(, T ) F (, T ) = = lnā(, T ) + B(, T ) r(), (24) lnā(, T ) B(, T ) [ = 2ā 1 σ 2 2 ( γ + b) = ) γe γ(t e γ(t ) γ γ+ b ], (25) 2 ) 4γ e γ(t [ ( ( γ + b) e γ(t ) 1 ) + 2 γ ] 2, (26) So, also in he bond case, he reformulaion in erms of he risk facor allows us o synheize in a unique spread he forecas error ha economic agens can make and ha hey may be willing o hedge. 3 The insurance company porfolio Le us suppose ha he insurance company has sold a number n of annuiies on he generaion x whose moraliy inensiy is λ. In principle, is porfolio is likely o include also erm insurance conracs, pure endowmens or more complex producs, bu, for he purpose of our discussion, i seems sufficien o concenrae on annuiies. The exension o he oher conracs jus lised is quie sraighforward. If liabiliies are evaluaed a fair value, an annuiy - wih annual insallmens R, paid a year-end - issued a ime 0 o an individual belonging o generaion x, lasing up o T and (already) paid hrough a single premium a policy incepion is worh T N(, T ) = R D(, + u)s(, + u). (27) u=1 Assuming a CIR moraliy inensiy (1) and a CIR ineres rae process (3), we have ha T N(, T ) = R e X(,+u)K()+Ȳ (,+u) e X(,+u)I()+Y (,+u). (28) u=1 6

8 This is he so-called fair value of he reserves ha he insurance company should have in order o face he paymens for he generaion under exam. Under he previous assumpions, if here is any unexpeced change in he moraliy inensiy or he ineres rae process, he marginal effec on he reserve is as follows: dn = N I di N 2 I 2 (di)2 + N K dk N 2 K 2 (dk)2, where T N I = R D(, + u) M (, + u), 2 N I 2 u=1 T = R D(, + u)γ M (, + u), u=1 T N K = R F (, + u)s(, + u), u=1 2 T N K 2 = R Γ F (, + u)s(, + u), u=1 and he greeks agains moraliy and ineres rae risk, denoed as M, Γ M, F, Γ F, are defined saring from (19) and (23). Appropriae derivaions lead o S I = M (, T ) = X(, T )S(, T ) 0, (29) 2 S I 2 = Γ M (, T ) = X(, T ) 2 S(, T ) 0. (30) Analogously, he greeks for ineres-rae risk are D I = F (, T ) = X(, T )D(, T ) 0, (31) 2 D I 2 = Γ F (, T ) = X(, T ) 2 D(, T ) 0. (32) 4 Hedging Sraegies: implemenaion We discuss separaely he saic and he dynamic hedge. In order o hedge he unexpeced changes jus formalized, he insurance company can eiher buy a saic hedge, i.e. a derivaive, or se up an approximaed, parial hedge, ha can hen be revised over ime. 4.1 Saic hedge For longeviy, he saic hedge can be provided by a so-called s-swap or longeviy swap. A longeviy swap is a sequence of s-forwards. An s-forward signed a is a conrac in which one pary agrees o pay a fixed amoun in exchange for he number of survivors belonging o a specific generaion x in a given ime period. We normalize he number of individuals in generaion x o one. We hus absrac from idiosyncraic risk and consider 7

9 a single annuiy as equivalen o a well-diversified homogeneous porfolio of annuiies. If he mauriy of he forward is T, and he fixed paymen is K(T ), hen he payoff a mauriy, from he poin of view of who pays fixed, is ( ) exp T λ x (s)ds K(T ), (33) where λ x is he moraliy inensiy of generaion x. An s-forward (uni hedge) helps providers of annuiies o hedge heir exposure: if he provider sold a pure endowmen on generaion x wih mauriy T, and buys an s-forward, he will pay K(T ) for sure insead of being exposed o he randomness of he paymen exp ( T λ x (s)ds ). Under he assumpion of no arbirage, and assuming independence beween moraliy and ineres-rae risk, he fair value a ime of such a conrac is [S(, T ) K(T )] D(, T ) = ( ) = E [exp T λ x (s)ds K(T ) ] E [ exp ( T r(u)du where he index signals ha he expecaion is he F one. Since, in order o ener such a conrac, no price is paid a incepion, he no-arbirage value of K(T ), which equaes he fair value o zero, is S(, T ). A longeviy swap is a sequence of s-forwards. If he exchange of amouns happens once a year, he paymen for he period (T 1, T ) is K(T ) and he conrac lass unil he las individual of he generaion is dead (a age ω), he payoffs are given by (33) for T = 1,.., ω. Under he assumpion of no arbirage, and sill assuming independence beween moraliy and ineres-rae risk, he value a ime of such a conrac is ω T =+1 [S(, T ) K(T )] D(, T ) = = ω T =+1 E [exp ( T λ x (s)ds ) K(T ) ] E [ exp ( T )]. r(u)du which is equal o zero, as a fair pricing would require, if K(T ) is se equal o he survival probabiliy for ime T. We call K(T ) he swap rae for he ime period (T 1, T ). 2 Usually he previous swap is no offered o he insurance company a fair value. I enails a cos, which we ake o be fixed and equal o C 0. I follows 2 An alernaive would be o fix a unique swap rae for all periods, K(T ) = K. In his case fairness would be guaraneed by seing K equal o he following value: ω T =+1 K = [ ( T )] [ ( T )] E exp λ x(s)ds E exp r(u)du )]. ω T =1 E [ ( T exp r(u)du )], 8

10 ha he fees K(T ) are raised o K (T ), where he sequence K (T ) solves C 0 = ω T =+1 [S(, T ) K (T )] D(, T ). For he sake of simpliciy, we assume ha he cos C 0 is evenly disribued along he "life" of he swap, by increasing he swap raes K by he same amoun, i.e. K (T ) = K(T )(1 + m) = S(, T )(1 + m) where m is deermined as follows: which implies ha C 0 = ω T =+1 = m [S(, T ) K(T )(1 + m)] D(, T ) ω T =+1 S(, T )D(, T ), m = = ω T =+1 ω T =+1 C 0 S(, T )D(, T ) C 0 e X(,T )K()+Ȳ (,T ) e X(,T )I()+Y (,T ). In principle, he insurance company can be ineresed in hedging ineres rae risk oo. We neglec his coverage here. However, given he similariy of he wo processes, he formulas for an ineres rae swap would be similar o he survival one. 4.2 Dynamic hedge An alernaive o he previous hedge is he following: cover only he changes in he fair value of he reserve (he liabiliies) approximaed a he firs or second order. This is known as dela, or dela-gamma, hedging (see Luciano e al. (2012a)). Boh he firs and second-order changes in he reserve, dela and gamma, depend on changes of he CIR longeviy inensiy and heir expression has been already given explicily in Secion 3. For consisency wih he saic hedge, we assume ha ineres rae risk is no covered. For he same reason, we also assume ha he dynamic hedge is self-financing, a requiremen formalized below. In order o cover he annuiy agains he wo changes, a possibiliy is ha of seing up a porfolio comprehensive of longeviy bonds. Our longeviy bonds pay a every year-end he survivorship of he reference generaion. 3 Their payoff for year T is hen ( ) exp T λ x (s)ds 3 If here is no longeviy bond for a specific generaion, basis risk arises: see for insance Cairns, Blake, Dowd, and MacMinn (2006).. 9

11 Under no-arbirage, if he bond mauriy is T i, is fair value a ime, M i (), is M i () = S(, T i )D(, T i ), which, using he CIR assumpion, can be wrien as M i () = Ā(, T i)e B(,T i)r() A(, T i )e B(,Ti)λx(), or, in he Jarrow and Turnbull formulaion, as M i () = e X(,T i)k()+ȳ (,Ti) e X(,Ti)I()+Y (,Ti). In order o dela-gamma hedge and keep he hedge self financing, we need a each poin in ime hree bonds, whose mauriy is kep consan along he life of he hedge. The hree bonds have mauriies T i, i = 1, 2, 3 and he number of bonds in he porfolio is n i, i = 1, 2, 3. A each rebalancing poin, he amoun of he bonds used o hedge can be found by solving he following sysem n N() di + I n 2 N() I 2 (di) i=1 3 i=1 M i () n i di = 0, I n i 2 M i () I 2 (di) 2 = 0, (34) nn + 3 n i M i () = 0. The firs equaion nullifies he dela of he porfolio, he second nullifies he gamma, while he hird requires i o be self-financing. Noice ha he erms associaed o he annuiy ener wih negaive signs, as hey represen he liabiliy ha he company is endowed wih. Noe also ha he longeviy bond value is equal o an annuiy wih a unique cash flow, or a pure endowmen. The difference, from he sandpoin of an insurance company, is ha i can sell annuiies and pure endowmens or reduce is exposure hrough reinsurance and buy longeviy bonds, while, a leas in principle, i canno do he converse. 4 We could use a number of oher insrumens o cover he annuiy, saring from life assurances or deah bonds, which pay he benefi in case of deah. We resric he aenion o longeviy bonds for he sake of simpliciy. Le us also recall ha longeviy bonds ogeher wih he life assurance and deah bonds represen he Arrow-Debreu securiies of he insurance marke. Once hedging is provided for hem, i can be exended o every more complicaed insrumen. Immediaely before each rebalancing dae we evaluae he porfolio. Is value is he gain or loss of he hedging sraegy, which we finance hrough he bank accoun. In oher words, a each rebalancing dae we sell he enire porfolio and re-apply he self-financing dela-gamma sraegy using he same insrumens and solving equaions (34). Any gain or loss from he hedging revision is sored or charged in he bank accoun, from which he paymens due because of he annuiy conrac are also aken. The bank accoun accrues or charges he shor ineres rae r(). We refer o he absolue value of he bank accoun as o he hedging error. 4 Reinsurance companies have less consrains in his respec. For insance, hey can swap pure endowmens or issue longeviy bonds: see for insance Cowley and Cummins (2005). i=1 10

12 5 Hedging Sraegies: effeciveness and performance comparison In order o compare he wo sraegies (saic and dynamic) above, here we proceed as follows: we calibrae he models o he observed moraliy raes of 65-year old UK males, we imagine differen revision frequencies of he dynamic sraegy, and hen deermine he cos of he saic hedge which would equae he hedging error of he parial coverage, under differen assumpions on he rebalancing frequency of he second. We focus on longeviy risk hedging, hus assuming ha ineres-rae risk has already been hedged perfecly. 5.1 Calibraion We calibrae he parameers of our moraliy model on he generaion of UK males born in 1946, who were aged 65 on 31/12/2010 (i.e. x = 65). We fi our model minimizing he Rooed Mean Squared Error (RMSE) beween he modelimplied and he observed survival probabiliies as compued from daa provided by he Human Moraliy Daabase. Under he consrain given by condiion (2), we fix 01/01/1991 as he observaion poin (individuals have all reached aged 44) and we fi he observed survival probabiliies S(0, ) wih =1, We collec he parameers and he calibraion error in Table 1. Table 1. Moraliy Inensiy Calibraion resuls. a b σ Calibraion Error Because condiion (2) holds, he simulaed moraliy inensiies λ() will be sricly posiive. In he simulaions, we assume ha he maximum life-span of an individual belonging o generaion x is ω = 115, hence he ime horizon we use for he simulaions of he inensiy process is 50 years. Some simulaed sample pahs of he λ x () process are shown in figure 1. Having absraced from hedging issues concerning ineres rae risk, we se he ineres rae o a consan value r = Rebalancing frequency and dynamic hedging performance In his secion we compue he performance of he dynamic hedging sraegy we described in Secion 4.2 under differen rebalancing frequencies and use he resuls o assess reasonable ranges for he cos of a longeviy swap, as described in Secion 4.1. Le us consider an annuiy provider who has sold a whole-life annuiy wrien on UK males aged 65 a ime 0 (mauriy T A = 50y). We assume moreover he exisence of hree longeviy bonds wih rolling mauriies 10, 15 and 20 years, wrien on he same generaion of 65-year-old UK males. Figure 2 provides simulaed sample pahs for he value of hose bonds. Nex, we need o decide afer how many years we wan o assess he perfomance 11

13 Figure 1. Simulaed sample pahs of he moraliy inensiy process λ x(). of he hedge. In principle, since he value of he annuiy is compued aking ino accoun he maximum life-span ω = 115 of an individual belonging o generaion x, hen, he implemenaion of he dynamic dela-gamma hedging sraegy is of ineres up o 50 years. By so doing, i does no disregard he ails of he disribuion of deahs among policyholders. This procedure is jusified since, when an insurance company decides o implemen a longeviy risk hedging sraegy, she wans o proec herself agains he risk ha he realizaions of deah arrivals among her porfolio of annuians migh belong o he righ-ail of he deah disribuion. However, given he iniial age and he behaviour of survival probabiliies, i is reasonable o assume ha 30 years afer he incepion of he Annuiy conrac, he bulk of her iniial obligaions would be gone. This is why, in order o evaluae he effeciveness of he self-financing dynamic Dela- Gamma hedging sraegy, we fix a ime horizon of 30 years. We consider hree differen rebalancing frequencies of 3 monhs, 6 monhs and 1 year, respecively. Sample pahs of he simulaed evoluion of he annuiy value, using hese hree differen sep sizes, are represened in Figure 3. Simulaed samples of he evoluion of he Bank Accoun, for each rebalancing frequency, are given in Figure 4. The variabiliy of each pah is larger a he beginning of he horizon, when he value of he annuiy is higher, and decreases wih ime. Figure 5 repors he disribuion of he value of he bank accoun a our horizon of ineres, =30 years, for he hree differen rebalancing frequencies. The picure shows ha he average cos of he hedging sraegy is higher he longer he ime inerval beween wo revisions of he sraegy. Also, increasing he rebalancing frequency reduces remarkably he dispersion of he value around is mean. The sraegy rebalanced a 1-year frequency (solid line) presens he faes ails. Table 2 conains, for each case, he mean and sandard deviaion of he hedging error afer 30 years and allows o appreciae he effecs of differen rebalancing frequencies. Less frequen rebalancing leads o higher average hedging errors and higher variabiliy, as expeced. However, we remark ha his resul is obained in he absence of ransacion coss, which 12

14 Figure 2. Simulaed sample pahs of he Longeviy Bonds M i() wrien on generaion x. 13

15 Figure 3. Simulaed sample pahs of a whole-life Annuiy under differen assumpions on he simulaion sep size. 14

16 Figure 4. Simulaed sample pahs of he Bank Accoun under differen assumpions on he heding rebalancing frequency. 15

17 Figure 5 we neglec here and will be higher he higher he frequency. Given he resuls of our implemenaion of he dynamic hedging sraegy, we compue he cos of he swap based on a value-a-risk loading principle. Table 3 repors he cos C 0 and loading m of he swap. The premium charged o he buyer of he swap C 0 is compued as he presen value of he 99.5% value-a-risk of he bank accoun value a =30 years obained applying our hedging sraegy wih differen rebalancing inervals. The resuling loading m, which represens he percenage increase in each observed survival, ranges from 0.01% o 0.06%. This value migh seem low, bu i is worh noicing again ha i is obained in he absence of ransacion coss and basis risk, which migh conribue o increase he coss of he dynamic hedging sraegies. Table 2. Hedging Error s momens under differen rebalancing frequencies. 3 monhs 6 monhs 1 year Mean Sandard Deviaion Table 3. Longeviy Swap premiums and loadings equivalen o he 99.5% Value-a- Risk of he Dela Gamma Hedging sraegy a = 30 years. 3 monhs 6 monhs 1 year C m 0.01% 0.03% 0.06% 6 Summary and furher research This paper compued he saic, swap-based hedge for an annuiy, and compared i wih he dynamic, dela-based hedge, achieved using a longeviy bond. All hroughou, we assumed ha he longeviy inensiy was disribued according 16

18 o a CIR process. A similar assumpion was done for he ineres rae, in he heoreical par, while he empirical applicaion focused on longeviy risk. We showed ha, once he model is calibraed o a UK individual aged 65, in our case he average hedging error of he dynamic hedge is moderae, and boh is variance and he hickness of he ails of is disribuion are decreasing wih he rebalancing frequency, which we brough from hree monhs o one year. We also compued he spread over he basic "swap rae" which makes 99.5% quanile of he disribuion of he dynamic hedging error equal o he cos of he saic hedge. This spread sayed beween 0.01 and 0.06%. In doing ha, we were more ineresed in providing a mehod o assess which cos of he saic hedge makes i comparable o a given, oleraed error for he dynamic hedge, han o he magniude of he resul iself. The model developed above, indeed, is novel in ha i incorporaes dynamic hedging in a CIR framework and is comparison wih a saic hedge. To fully appreciae he magniudes of he errors one could include basis risk, i.e. he fac ha saic hedges are usually OTC, and herefore he reference populaion is he annuiy one, while dynamic hedges are mos likely based on indices, and herefore have a reference populaion which is no he annuiy one. As a second refinemen, one could include a number of annuians differen from one, and disinguish he idiosyncraic from he common moraliy risk in he group of annuians. Las, bu mos simply, we could enrich our comparison aking ino consideraion he fac ha no only OTC-swaps are usually provided a a cos, which can be a spread on he fair rae or a fixed, iniial amoun, bu also dynamic hedges could involve ransacion coss, and he rebalancing frequency could be chosen so as o opimize in a sense o be defined he rade-off beween he effeciveness of he hedge and is coss. We leave hese hree exensions o furher research. References Blake, D. and W. Burrows (2001). Survivor bonds: Helping o hedge moraliy risk. Journal of Risk and Insurance, Cairns, A., D. Blake, K. Dowd, and R. MacMinn (2006). Longeviy bonds: financial engineering, valuaion, and hedging. The Journal of Risk and Insurance 73 (4), Cowley, A. and J. Cummins (2005). Securiizaion of life insurance asses and liabiliies. The Journal of Risk and Insurance 72 (2), Cox, J. C., J. E. Ingersoll Jr, and S. A. Ross (1985). A heory of he erm srucure of ineres raes. Economerica: Journal of he Economeric Sociey, Dahl, M., S. Glar, and T. Møller (2011). Mixed dynamic and saic riskminimizaion wih an applicaion o survivor swaps. European Acuarial Journal 1 (2),

19 Fung, M. C., K. Ignaieva, and M. Sherris (2014). Sysemaic moraliy risk: An analysis of guaraneed lifeime wihdrawal benefis in variable annuiies. Insurance: Mahemaics and Economics 58 (0), Jarrow, R. and S. Turnbull (1994). Dela, gamma and bucke hedging of ineres rae derivaives. Applied Mahemaical Finance 1, Luciano, E., L. Regis, and E. Vigna (2012a). Dela and gamma hedging of moraliy and ineres-rae risk. Insurance: Mahemaics and Economics (50), Luciano, E., L. Regis, and E. Vigna (2012b). Single and cross-generaion naural hedging of longeviy and financial risk. ICER wp (04). Ngai, A. and M. Sherris (2011). Longeviy risk managemen for life and variable annuiies: The effeciveness of saic hedging using longeviy bonds and derivaives. Insurance: Mahemaics and Economics 49 (1),