Immunization Bounds, Time Value and Non-Parallel Yield Curve Shifts

Transcription

1 ARICLES ACADÉMIQUES ACADEMIC ARICLES Assurances e gesion des risques, vol. 75(3), ocobre 2007, Insurance and Risk Managemen, vol. 75(3), Ocober 2007, Immunizaion Bounds, ime Value and Non-Parallel Yield Curve Shifs by Geoffrey Poiras absrac Since Redingon (1952) i has been recognized ha classical immunizaion heory fails when shifs in he erm srucure are no parallel. Using parial duraions and convexiies o specify immunizaion bounds for non-parallel shifs in yield curves, Reiano (1991a,b) exended classical immunizaion heory o admi non-parallel yield curve shifs, demonsraing ha hese bounds can be effecively manipulaed by adequae selecion of he securiies being used o immunize he porfolio. By exploiing properies of he mulivariae aylor series expansion of he fund surplus value funcion, his paper exends his analysis o include ime values, permiing a connecion o resuls on he ime value-convexiy radeoff. Measures of parial duraion, parial convexiy and ime value are used o invesigae he generaliy of he duraion puzzle idenified by Bierwag e al. (1993) and Soo (2001). Keywords: Yield curve, immunizaion heory, duraion, convexiy. résumé Nous savons depuis Redingon (1952) que la héorie classique d immunisaion ne foncionne pas lorsque les mouvemens dans la srucure par ermes des aux d inérê ne son pas parallèles. En uilisan des mesures de durée e de convexié parielles pour idenifier les bornes d immunisaion pour des mouvemens non-parallèles, Reiano he auhor: Geoffrey Poiras, Faculy of Business Adminisraion, Simon Fraser Universiy, Burnaby, B.C., CANADA V5A 1S6, «poiras@sfu.ca». his paper benefied considerably from he insighful commens of wo anonymous referees and he suppor and encouragemen of he Edior. his paper was parially wrien while he auhor was a visiing Professor in he Faculy of Commerce and Accounancy, hammasa Universiy, Bangkok, hailand. Suppor from he Social Science and Humaniies Research Council of Canada is graefully acknowledged. 323

2 (1991a, 1991b) a généralisé la héorie classique d immunisaion à des mouvemens non-parallèles, permean ainsi de manipuler les bornes d immunisaion en sélecionnan les ires appropriés dans le porefeuille. Au moyen des propriéés d expansions de aylor mulivariées sur la valeur des fonds, ce aricle analyse le comporemen des bornes d immunisaion en prenan en considéraion la valeur dans le emps de l argen e en lian cee valeur à la convexié de la srucure par ermes. Des mesures de durée e de convexié parielles son alors uilisées pour éudier la généralisaion du puzzle de durée el que présené par Bierwag e alii (1993) e Soo (2001). Mos-clés : Srucure par erme des aux, héorie d immunisaion, durée, convexié. 1. Inroducion In he seminal work on fixed income porfolio immunizaion, Redingon (1952) uses a univariae aylor series expansion o derive wo rules for immunizing a life insurance company porfolio agains a change in he level of ineres raes: mach he duraion of cash inflows and ouflows; and, se he asse cash flows o have more dispersion (convexiy) han he liabiliy cash flows around ha duraion. From ha beginning, a number of improvemens o Redingon s classical immunizaion rules have been proposed, aimed a correcing limiaions in his classical formulaion. Paricular aenion has been given o generalizing he classical model o allow for non-parallel shifs in he yield curve, e.g., Soo (2004, 2001), Nawalka e al. (2003), Navarro and Nave (2001), Crack and Nawalka (2000), Balbas and Ibanez (1998) and Bowden (1997). While mos sudies aim o idenify rules for specifying opimal porfolios ha are immunized agains insananeous non-parallel shifs, Reiano (1992, 1996) explores he properies of he immunizaion bounds applicable o non-parallel shifs. In paricular, parial duraions and convexiies are exploied o idenify bounds on porfolio gains and losses for an insananeous uni shif in he yield curve. he objecive of his paper is o exend he parial duraion framework by incorporaing ime value changes ino he immunizaion bound approach. his exends he resuls of Chrisensen and Sorensen (1994), Chance and Jordan (1996), Barber and Copper (1997) and Poiras (2005) on he ime value-convexiy radeoff. 2. Background Lieraure hough Redingon (1952) recognized ha classical immunizaion heory fails when shifs in he erm srucure are no parallel, Fisher and Weil (1971) were seminal in siuaing he problem in a 324 Assurances e gesion des risques, vol. 75(3), ocobre 2007

3 erm srucure framework. he developmen of echniques o address non-parallel yield curve shifs led o he recogniion of a connecion beween immunizaion sraegy specificaion and he ype of assumed shocks, e.g., Boyle (1978), Fong and Vasicek (1984), Chambers e al. (1988). Sophisicaed risk measures, such as M 2, were developed o selec he bes duraion maching porfolio from he se of poenial porfolios. Being derived using a specific assumpion abou he sochasic process generaing he erm srucure, hese heoreically aracive models encounered difficulies in pracice. For example, minimum M 2 porfolios fail o hedge as effecively as porfolios including a bond mauring on he horizon dae (Bierwag e al. 1993, p. 1165). his line of empirical research led o he recogniion of he duraion puzzle (Ingersoll 1983; Bierwag e al. 1993; Soo 2001), where porfolios conaining a mauriy-maching bond have smaller deviaions from he promised arge reurn han duraion mached porfolios no conaining a mauriy-maching bond. his resul begs he quesion: are hese empirical limiaions due o failings of he sochasic process assumpion underlying he heoreically derived immunizaion measures or is here some deeper propery of he immunizaion process ha is no being accuraely modelled? Insead of assuming a specific sochasic process and deriving he opimal immunizaion condiions, i is possible o leave he process unspecified and work direcly wih he properies of an expansion of he spo rae pricing funcion or some relaed value funcion, e.g., Shiu (1987,1990). Immunizaion can hen proceed by making assumpions based on he empirical behaviour of he yield curve. Soo (2001) divides hese empirical muliple facor duraion models ino hree caegories. Polynomial duraion models fi yield curve movemens using a polynomial funcion of he erms o mauriy, e.g., Crack and Nawalka (2000), Soo (2001), or he disance beween he erms o mauriy and he planning horizon, e.g., Nawalka e al. (2003). Direcional duraion models idenify general risk facors using daa reducion echniques such as principal componens o capure he empirical yield curve behaviour, e.g., Elon e al. (2000), Hill and Vaysman (1998), Navarro and Nave (2001). Parial duraion models, including he key rae duraion models, decompose he yield curve ino a number of linear segmens based on he selecion of key raes, e.g., Ho (1992), Daareya and Fabozzi (1995), Phoa and Shearer (1997). Whereas he dimension of polynomial duraion models is resriced by he degree of he polynomial and he direcional duraion models are resriced by he number of empirical componens or facors ha are idenified, e.g., Soo (2004), he number of key raes used in he parial duraion models is exogenously deermined by he desired fi of immunizaion procedure. Immunizaion Bounds, ime Value and Non-Parallel Yield Curve Shifs 325

4 Reiano (1991a,b) provides a seminal, if no widely recognized, analysis of bond porfolio immunizaion using he parial duraion approach. 1 hough Reiano evaluaes a mulivariae aylor series for he asse and liabiliy price funcions specified using key raes, he approach is more general. Muliple facors derived from he spo rae curves, bond yield curves, cash flow mauriies or key raes can be used. In his paper, a refined spo rae model is used where each cash flow is associaed wih a spo rae. Because his approach can be cumbersome as he number of fuure cash flows increases, in pracical applicaions facor models, key raes and inerpolaion schemes ha exogenously deermine he dimension of he spo rae space are used, as in Ho (1992) and Reiano (1992). he advanage of using a spo rae for each cash flow is precision in calculaing he individual parial duraion, convexiy and ime value measures for he elemenary fixed income porfolios ha are being examined. his permis exploraion of heoreical properies of he immunizaion problem where he spo rae curve can change shape, slope and locaion. While i is possible o reinerpre he refined spo rae curve shif in erms of a smaller number of fixed funcional facors, his requires some mehod of aggregaing he individual cash flows. While such aggregaion is essenial where he number of he possibly random individual cash flows is large, as in pracical applicaions, in his paper analyical precision is enhanced by having a one-o-one correspondence beween spo raes and cash flows. Defining a norm applicable o a uni parallel yield curve shif, Reiano explois Cauchy-Schwarz and quadraic form inequaliy resricions o idenify bounds on he possible deviaions from classical immunizaion condiions. In oher words, even hough classical immunizaion rules are violaed for non-parallel yield curve shifs, i is sill possible o pu heoreical bounds on he deviaions from he classical oucome and o idenify he specific ypes of shifs ha represen he greaes loss or gain. his general approach is no unique o Reiano. Developing he Gaeaux differenial approach inroduced by Bowden (1997), Balbas and Ibanez (1998) rediscover he possibiliy of defining such bounds, albei in an alernaive mahemaical framework. Balbas and Ibanez also inroduce a linear dispersion measure ha, when minimized, permis idenificaion of he bes porfolio wihin he class of immunizing porfolios. More precisely, a sraegy of maching duraion and minimizing he dispersion measure idenifies he porfolio ha will minimize immunizaion risk and, as a consequence, provides an opimal upper bound for possible loss on he porfolio. Considering only he implicaions of insananeous non-parallel yield curve shifs, he parial duraion approach o idenifying immun- 326 Insurance and Risk Managemen, vol. 75(3), Ocober 2007

5 izaion bounds is no subsanively superior o he direcional duraion and polynomial duraion models. However, when he analysis is exended o include he ime-value convexiy radeoff, he parial duraion approach has he desirable feaure of providing a direc heoreical relaionship beween he convexiy and ime value elemens of he immunizaion problem. his follows because convexiy has a ime value cos associaed wih he iniial yield curve shape and he expeced fuure pah of spo raes for reinvesmen of coupons and rollover of shor-daed principal. Despie he essenial characer of he ime value decision in overall fixed income porfolio managemen, available resuls on he ime value-convexiy radeoff have been developed in he classical Fisher-Weil framework involving monoonic erm srucure shifs and zero surplus funds. he parial duraion approach permis ime value o be direcly incorporaed ino he performance measuremen of surplus immunized porfolios for non-parallel yield curve shifs ha are of pracical ineres. 3. he Reiano Parial Duraion Model he Reiano parial duraion model akes he objecive funcion o be fund surplus immunizaion. his is a suble difference from he classical immunizaion approach used in Redingon (1952) where he fund surplus is se equal o zero and he soluions o he opimizaion problem produce duraion maching and higher convexiy of asses condiions for immunizaion. Following Shiu (1987, 1990), Messmore (1990) and Reiano (1991a,b), immunizaion of a nonzero surplus involves explici recogniion of he balance shee relaionship: A = L + S, where A is he asses held by he fund, L is he fund liabiliies and S is he accumulaed surplus. In discree ime, he fund surplus value funcion, S(z), can be specified wih spo ineres raes as: S( z) = C A L = ( 1+ z ) ( 1+ z ) = 1 = 1 where: C is he fund ne cash flow a ime deermined as he difference beween asse (A ) and liabiliy (L ) cash flows a ime ; z is he spo ineres rae (implied zero coupon ineres rae) applicable o cash flows a ime ; = (1, 2,...); z = (z 1, z 2,..., z ) is he x1 vecor of spo ineres raes; and is he erm o mauriy of he fund in years. Recognizing ha S is a funcion of he spo ineres raes Immunizaion Bounds, ime Value and Non-Parallel Yield Curve Shifs 327

6 conained in z, i is possible o apply a mulivariae aylor series expansion o his bond price formula, ha leads immediaely o he conceps of parial duraion and parial convexiy: S( z) = S( z ) 0 = 1 S( z ), 0 1 ( z z, ) + 0 z 2! i = 1 j = 1 2 S( ) ( z z )( z z i i, 0 j j, 0 ) z z i j S( z) S( z ) 0 + H. O.. S( z ) 0 = 1 D ( z z ), 0 (1) 1 + CON ( z z )( z z ) i, j i i, 0 j j, 0 2! i = 1 j = 1 where z 0 = (z 1,0, z 2,0,..., z,0 ) is he x1 vecor of iniial spo ineres raes, D is he parial duraion of surplus associaed wih z, he spo ineres rae for ime, and CON i,j is he parial convexiy of surplus associaed wih he spo ineres raes z i and z j for i,j defined over (1, 2,..., ). 2 Observing ha he parial duraions a z 0 can be idenified wih a x1 vecor D = (D 1, D 2,... D ) and he parial convexiies a z 0 wih a x marix Γ wih elemens CON i,j, he parial duraion model proceeds by applying resuls from he heory of normed linear vecor spaces o idenify heoreical bounds on D and Γ. In he case of D, he Cauchy-Schwarz inequaliy is used. 3 For Γ he bounds are based on resricions on he eigenvalues of Γ derived from he heory of quadraic forms. o access hese resuls, he direcion vecor specified by Reiano is inuiively appealing. More precisely, aken as a group, he (z z 0, ) changes in he individual spo ineres raes represen shifs in yield curve shape. hese individual changes can be reexpressed as he produc of a direcion shif vecor N and a magniude i: (z z,0 ) = n i where N = (n 1, n 2,..., n ). I is now possible o express (1) in vecor space form as S( z) S( z ) 0 2 = 0 i[ N D ] + i [ N Γ N]. (2) S( z ) 0 From his, he spo rae curve can be shocked and he immunizaion bounds derived. he dimension of N provides a connecion o alernaive approaches o he immunizaion problem ha reformulae he dimensional refined spo rae curve in erms of a smaller (< ) number of fixed funcional facors. he use of spo raes in he formulaion does differ slighly from shocking he yield curve and hen 328 Assurances e gesion des risques, vol. 75(3), ocobre 2007

7 deriving he associaed change in he spo rae curve. Using he spo rae approach, i follows ha = (1, 1,..., 1) represens a parallel shif in he spo rae curve, wih he size of he shif deermined by i. 4 o derive he classical immunizaion condiions, Redingon (1952) uses a zero surplus fund S(z 0 ) = 0 where he presen value of asses and liabiliies are equal a = 0. In pracice, his specificaion is consisen wih a life insurance fund where he surplus is being considered separaely. his classical immunizaion problem requires he mauriy composiion of an immunized porfolio o be deermined by equaing he duraion of asses and liabiliies. When he fund surplus funcion is generalized o allow non-zero values, he immunizaion condiions change o: 1 ds A S A L S d S D L S D D L = = 0 = = A D. A L A L he classical zero surplus immunizaion resul requires seing he duraion of asses equal o he duraion of liabiliies. his only applies for a zero surplus porfolio. Immunizaion wih a non-zero surplus requires he duraion of asses o be equal o he duraion of liabiliies, muliplied by he raio of he marke value of asses o he marke value of liabiliies, D A = (L / A) D L, where L and A are he marke values of asses and liabiliies. As indicaed, his more general condiion is derived by differeniaing boh sides of S = A L, dividing by S and manipulaing. A similar commen applies o convexiy, i.e., CON A > (L / A) CON L. Allowing for a non-zero surplus changes he inuiion of he classical duraion maching and convexiy condiions. Observing ha he duraion of a porfolio of asses is he value weighed sum of he individual asse duraions, a posiive fund surplus wih a zero coupon liabiliy allows surplus immunizaion using a combinaion of asses ha have a shorer duraion han ha of he liabiliy. As such, surplus immunizaion for a fund wih a single liabiliy having a duraion ha is longer han he duraion of any raded asse can be achieved by appropriae adjusmen of he size of he surplus. In general, a larger posiive fund surplus permis a shorer duraion of asses o immunize a given liabiliy. Because yield curves ypically slope upward, his resul has implicaions for porfolio reurns. In he classical immunizaion framework, such issues do no arise because he force of ineres (Kellison 1991) is a consan and, in any even, he force of ineres for a zero surplus fund is unimporan. 6 However, when he ineres rae risk of he surplus has been immunized, he equiy value associaed wih he surplus will earn a reurn ha depends on he Immunizaion Bounds, ime Value and Non-Parallel Yield Curve Shifs 329

8 force of ineres funcion (see Appendix). his reurn is measured by he ime value funcion, e.g., Chance and Jordan (1996). Nonparallel shifs in he yield curve will aler he ime value. 4. Convexiy and ime Value Following Redingon (1952), classical immunizaion requires he saisfacion of boh duraion and convexiy condiions: duraion maching is required o be accompanied wih higher porfolio convexiy, e.g., Shiu (1990). he convexiy requiremen ensures ha, for an insananeous change in yields, he marke price of asses will ouperform he marke price of liabiliies. Ye, higher convexiy does have a cos. In paricular, when he yield curve is upward sloping, here is a radeoff beween higher convexiy and lower ime value (Chrisensen and Sorensen, 1994; Poiras, 2005, ch. 5). his connecion highlighs a limiaion of he aylor series expansion in (1) and (2): he fund surplus value funcion depends on ime as well as he vecor of spo ineres raes, i.e., S = S(z,). If yields do no change, higher convexiy will likely resul in a lower porfolio reurn due o he impac of ime value. hough some progress has been made in exploring he relaionship beween convexiy and ime value (Chance and Jordan, 1996; Barber and Copper, 1997), he precise connecion o he calculaion of he exreme bounds on yield curve shifs is unclear. he exreme bounds associaed wih changes in convexiy are disinc from hose for duraion. How shifs in exreme bounds for duraion and convexiy are associaed wih changes in he porfolio composiion and, in urn, o he ime value is, a his poin, largely unknown. Assuming for simpliciy ha cash flows are paid annually, evaluaing he firs order erm for he ime value in he surplus value funcion, S(z,), produces: 6 1 S C 1 z 1 S = + ln( ) = Θ (3) = 1 ( 1+ z ) S where Θ = (θ 1, θ 2,... θ ) and θ = (C / S) (ln(1 + z ) / (1 + z ) ). he sign on he ime value can be ignored by adjusing ime o coun backwards, e.g, changing ime from = 20 o = 19 produces = 1. aking he o be posiive permis he negaive sign o be ignored. Using his convenion and rearranging he expansion in (2) o incorporae ime value gives he surplus change condiion: 330 Insurance and Risk Managemen, vol. 75(3), Ocober 2007

9 S( z) S( z ) 0 2 i[ N D ] + i [ N Γ N] + [ N Θ]. 0 (4) S( z ) 0 In his formulaion, he ime value componen is evaluaed using (3) wih he spo raes observed a he new locaion. Wih a non-zero surplus, saisfying he surplus immunizaion condiion in (2) means ha he value of he porfolio surplus will increase by he ime value. One final poin arising from he implemenaion of Reiano s parial duraion model concerns he associaed convexiy calculaion. Consider he direc calculaion of he parial convexiy of surplus, CON i,j, where i j: C S z i C ( ) S( z) = = i ( 1 + z ) z ( 1 + z ) = 1 i S( z) = 0 for i z z i j j where, as previously, C is he cash flow a ime for = (1, 2,..., ). From (1) and (2), i follows ha he quadraic form N N reduces o: 2 N N = n CON = n S z ( ) 0 Γ., S ( z 2 ) z = 1 = 1 0 In erms of he exreme bounds on convexiy (see Appendix), his is a significan simplificaion. Because he x convexiy marix is diagonal, he exreme bounds are now given by he maximum and minimum diagonal (CON i,i ) elemens. If he ih elemen is a maximal elemen, he associaed opimal N vecor for he convexiy bounds is a x1 wih a one in he ih posiion and zeroes elsewhere. Similar o he duraion adjusmen, o compare N 1 N 1 wih eiher he classical convexiy or requires muliplicaion by. 5. Key Raes, Cash Flow Daes and he Norm Reiano (1991a, 1992) moivaes he analysis of surplus immunizaion wih a sylized example involving a porfolio conaining a 5 year zero coupon liabiliy (= $63.97) ogeher wih a barbell combinaion of wo asses, a 12% coupon, en year bond (= $43.02) and 6 monh commercial paper (= $25.65; surplus = $9.28). he iniial yield curve is upward sloping wih he vecor of yields being y = (.075,.09,.10) Immunizaion Bounds, ime Value and Non-Parallel Yield Curve Shifs 331

10 for he 0.5, 5 and 10 year mauriies. Consisen wih he key rae approach: Yields a oher mauriies are assumed o be inerpolaed (Reiano 1992, p. 37). Reiano derives he vecor of parial duraions of surplus, D, for he hree relevan mauriy ranges as (4.55, 35.43, 30.88). I follows, for he parallel shif case, = (1, 1, 1), ha D = 0 corresponds o he classical surplus immunizaion condiion: when he duraion of asses equals he appropriaely weighed duraion of liabiliies, he duraion of surplus is zero. As a consequence, for a parallel yield curve shif, he change in he porfolio surplus equals zero. o derive he bounds for cases involving non-parallel shifs, Reiano selecs he parallel yield curve shif case as he norming vecor, ha involves imposing a sandard shif lengh of: N = N N = n , = 3. 3 = 1 From his Reiano is able o idenify he exreme bounds on he change in he parial duraion of surplus as (N * D =) N D 81.78, ha correspond o an esimae derived from he parial duraions for he max % S from he se of all shifs of lengh 3. he exreme negaive yield curve shif is idenified as N * = (0.167, 1.3, 1.133) and he exreme posiive shif as N *. Similar analysis for he convexiy of surplus produces exreme posiive and negaive bounds of N N wih associaed shifs of (0.049, 0.376, 1.69) and (.306, 1.662, 0.379). he specificaion of he hree elemen norming vecor in he key rae example of Reiano is moivaed by making reference o marke realiy where i is no pracical o mach he dimension of he yield vecor wih he large number of cash flow daes, e.g., Ho (1992) and Phoa and Shearer (1997). Key raes are used o reduce he dimension of he opimizaion problem. Reiano selecs an example wih 3 relevan key rae mauriies, one mauriy applicable o a 5 year zero coupon liabiliy and wo mauriies applicable o a 6 monh zero coupon asse and a 10 year coupon bond. Even hough here are parial duraions and convexiies associaed wih he regular coupon paymen daes, only hree mauriy daes are incorporaed ino he analysis. Because he parial duraions, convexiies and ime values depend on he cash flow over a paricular paymen period, he aggregaion of cash flows o key rae mauriies permis comparable marke value o be used across yield curve segmens. In addiion o empirical simpliciy, anoher advanage of using key raes in Reiano s model is ha he elemens of he exreme shif vecor, as well as he individual parial duraions and convexiies, have a more realisic appearance. Where 332 Assurances e gesion des risques, vol. 75(3), ocobre 2007

11 he yield curve or spo rae vecors are specified wih he acual number of cash flow paymens, he opical appearance of he elemens of he exreme shif vecor ofen has a sawooh paern ha is unrealisic. 7 While he key rae approach may generae shifs ha have a more realisic appearance, one fundamenal limiaion of he parial duraion model, including he key rae varian, is he use of only mahemaical resricions on he se of admissible shifs. A leas since Cox e al. (1979), i has been recognized ha sochasic models of he erm srucure ha saisfy absence of arbirage can be used o resric admissible shifs. By employing a norm ha is defined mahemaically relaive o a uni elemen shif vecor, he example provided by Reiano is only able o idenify duraion bounds associaed wih exreme shifs ha may, or may no, admi arbirage opporuniies. From an iniial yield curve of y = (.075,.09,.10) for he 0.5, 5 and 10 year mauriies, he exreme negaive shif for i =.01 is o y * = (.0767,.077,.1133). In his relaively simple one-shif-only example using key raes, he dynamics of he erm srucure a he duraion bound are empirically unrealisic. However, his is only a disadvanage if i is he (unlikely) exreme bounds ha are of ineres. If bounds arising from specific shif vecors ha are empirically deermined or mos likely are of ineres, hen assigning spo raes o cash flow daes along he yield curve is revealing. he approach in his paper assigns a spo rae o each cash flow dae, wihou regard o he size of he paymen on ha dae. While his is cumbersome in pracical applicaions where here are a large number of cash flow daes, here are disinc heoreical advanages for measuring he impac of a paricular shif scenario on surplus value. In his approach, shif vecor scenarios are exogenously specified, eiher from empirical or ex ane esimaes. Insead of seeking he se porfolios ha solve he immunizaion problem, he objecive is o measure changes in he duraion, convexiy and ime value properies of specific porfolios for a given shif in he yield curve. Economic resricions imposed by no arbirage can be assessed prior o measuring he impac of a given shif on a specific porfolio. In his case, Reiano s mahemaically deermined exreme duraion and convexiy bounds areuseful o benchmark changes in surplus value. hough oher benchmark exreme bounds are possible, such as he empirical benchmark examined in Chance and Jordan (1996), he approach developed in secion VII below has reained he Reiano consrucion. Immunizaion Bounds, ime Value and Non-Parallel Yield Curve Shifs 333

12 6. he Duraion Bounds ables 1-3 provide resuls for he duraion componen of he aylor series expansion: he individual parial duraions; he elemens, n *, of he exreme shif vecor N * ; and he calculaed exreme duraion bounds. Convexiy and ime value componens are no considered in ables 1-3. Solving for he parial duraions, exreme shif vecor and duraion bounds requires he specificaion of S, A and L. Because here are a heoreically infinie number of poenial combinaions of A and S ha can immunize a given L, some form of sandardizaion is required o generae plausible and readily analysed scenarios. o his end, ables 1-3 have been sandardized o have an equal marke value for he liabiliy. Similar o Reiano, ables 1 and 2 involve a 5 year zero coupon liabiliy while able 3 uses a 10 year annuiy wih he same marke value as he 5 year zero. able 1 immunizes he liabiliy wih he wo asses from he Reiano example, a six monh zero and a 10 year, 12% semi-annual coupon bond. o illusrae he impac of surplus level on asse porfolio composiion, resuls for a high surplus and a low surplus immunizing asse porfolio are provided. All ables use he yield curve and spo raes from Fabozzi (1993) as he iniial baseline. 8 his curve is upward sloping wih a 558 basis poin difference beween he 6 monh (.08) and en year (.1358) spo ineres raes. able 1 repors he parial duraions, he n * and exreme duraion bounds calculaed from he Cauchy-Schwarz inequaliy (see Appendix). Comparison of he bounds beween he low and high surplus cases depends crucially on he observaion ha he bounds relae o he percenage change in he surplus, e.g., an exreme bound of ± 8.86 means he exreme change in surplus is 8.86%. Due o he smaller posiion in he 6 monh asse, he larger bounds for he low surplus case also ranslae o a slighly larger exreme marke value change when compared o he high surplus case. his resul is calculaed by muliplying he repored bound by he size of he surplus. 9 As expeced, because all cash flow daes are used he exreme shif vecor for duraion, N *, exhibis a sawooh change, wih abou 80% of he wors shif concenraed on a fall in he 5 year yield and 17-20% on an increase in he 10 year rae. 10 his is an immediae implicaion of he limied exposure o cash flows in oher ime periods. However, even in his relaively simple porfolio managemen problem, he n * provide useful informaion abou he wors case shif. Wih he proviso ha he precise connecion o uni shifs is obscured, here is no much loss of conen o fill in he sawooh paern as in he key rae approach, due o he small parial duraions in he inervening periods. 334 Insurance and Risk Managemen, vol. 75(3), Ocober 2007

13 able 1 Parial duraions, {n } and exreme bounds for he 5 Year Zero Coupon Liabiliy Immunized wih High and Low Surplus Examples* High Surplus Low Surplus Dae (Yrs.) D n * D n * Cauchy = D Cauchy = D Exreme Duraion Bounds ± 8.860% ± % Surplus * he marke value of he High Surplus Porfolio is composed of ($ ) 1/2 year zero coupon and ($ ) 10 year semi-annual coupon bonds. he marke value of he Low Surplus Porfolio is composed of ($ ) 1/2 year and ($74.343) 10 year bonds. he liabiliy for boh he High Surplus and Low Surplus Porfolios is a 5 year zero coupon bond wih $150 par value and marke value of $ he exreme Cauchy bounds are derived using N = 1. Immunizaion Bounds, ime Value and Non-Parallel Yield Curve Shifs 335

14 Consisen wih Reiano s example, he wors ype of shif has a sizeable fall in miderm raes combined wih smaller, bu sill significan, rise in long erm raes. In using only he 6 monh zero and 10 year bond as asses, able 1 is consruced o be roughly comparable o he example in Reiano (1991a, 1992, 1996). o address he duraion puzzle in a conex where porfolios being compared are surplus immunized, he number of asses is increased o include a mauriy maching bond. his example porfolio can hen be compared o a porfolio ha has a similar surplus bu does no conain a mauriy maching bond. able 2 provides resuls for wo cases wih similar surplus levels bu wih hese differen asse composiions. One case increases he number of asses by including a par bond wih a mauriy ha maches ha of he zero coupon liabiliy ( = 5). his is referred o as he mauriy bond porfolio. he oher case does no include he mauriy maching bond bu, insead, increases he number of asses by including 3 and 7 year par bonds. his is referred o as he spli mauriy porfolio. For boh porfolios he 1/2 year and 10 year bonds of able 1 are included, wih he posiion in he 10 year bond being he same in boh of he able 2 asse porfolios. In order o achieve surplus immunizaion, he 1/2 year bond posiion is permied o vary, wih he mauriy maching porfolio holding a slighly higher marke value of he 1/2 year asse. A priori, he spli mauriy porfolio would seem o have an advanage as four asses are being used o immunize insead of he hree bonds in he mauriy maching porfolio. Given his, he resuls in able 2 reveal ha he porfolio wih he mauriy maching bond has a smaller surplus and much smaller exreme bounds even hough more bonds are being seleced in he spli mauriy porfolio and, from able 1, i is expeced ha a smaller surplus will have wider exreme bounds. he parial duraions reveal ha, as expeced, he presence of a mauriy maching bond reduces he parial duraion a = 5 compared o he spli mauriy case. he parial duraions a = 3 and = 7 are proporionaely higher in he spli mauriy case o accoun for he difference a = 5. he small difference in he parial duraion a = 10 is due solely o he small difference in he size of he surplus. Examining he n * reveals ha here is no a subsanial difference in he sensiiviy o changes in five year raes, as migh be expeced. Raher, he spli mauriy porfolio redisribues he ineres rae sensiiviy along he yield curve. In conras, he mauriy bond porfolio is relaively more exposed o changes in 10 year raes even hough he marke value of he 10 year bond is he same in boh asse porfolios. his greaer exposure along he yield curve by he spli mauriy porfolio resuls in wider exreme duraion bounds because he norming resricion dampens he allow- 336 Assurances e gesion des risques, vol. 75(3), ocobre 2007

15 able 2 Parial duraions, {n } and exreme bounds for he 5 Year Zero Coupon Liabiliy Immunized wih he Mauriy Bond and Spli Mauriy Examples* Mauriy Bond Spli Mauriy Dae (Yrs.) D n* D n* Cauchy = D Cauchy = D Exreme Duraion Bounds ± 26.07% ± % Surplus * he marke value of he Mauriy Bond Porfolio is composed of ($5.13) 1/2 year, ($55.54) 5 year and ($37.172) 10 year bonds. he marke value of he Spli Mauriy Porfolio is composed of ($0.4105) 1/2 year, ($ ) 3 year, (27.0) 7 year and ($37.172) 10 year bonds. he liabiliy is a 5 year zero coupon bond wih $150 par value and marke value of $ he exreme Cauchy bounds are derived using N = 1. Immunizaion Bounds, ime Value and Non-Parallel Yield Curve Shifs 337

16 able movemen in any individual ineres rae. In oher words, spreading ineres rae exposure along he yield curve by picking asses across a greaer number of mauriies acs o increase he exposure o spo rae curve shifs of uni lengh. able 3 considers he implicaions of immunizing a liabiliy wih a decidedly differen cash flow paern. In paricular, he liabiliy being immunized is an annuiy over = 10 wih he same marke value as he zero coupon liabiliy in ables 1-2. he immunizing asse porfolios are a mauriy maching porfolio similar o ha in able 2 combining he 6 monh zero coupon wih 5 year and 10 year bonds. he marke value of he 10 year bond is he same as in he able 2 asse porfolios. he oher case considered is a low surplus porfolio, similar o ha of able 1, conaining he 6 monh zero and 10 year bond as asses. able 3 reveals a significan relaive difference beween he exreme bounds for he wo porfolios compared wih he similar porfolios in ables 1 and 2. he exreme bound for he low surplus porfolio has been reduced o abou one hird he value of he bound in able 1 wih he N * vecor being dominaed by he n * value for = 10. he exreme bound for he mauriy bond porfolio has been reduced by jus over one half compared o he opimal bound for he able 2 porfolio wih he N * vecor being dominaed by he n * values a = 5 and = 10. When he exreme bounds for he wo porfolios in able 2 are muliplied by he size of he surplus, here is no much difference in he poenial exreme change in he value of he surpluses beween he wo porfolios in able 3. his happens because, unlike he zero coupon 5 year liabiliy of able 2, he liabiliy cash flow of he annuiy is spread across he erm srucure and he addiion of he five year asse provides greaer coverage of he cash flow paern. In he annuiy liabiliy case, he dramaic exposure o he = 10 rae indicaed by he n * of he low surplus porfolio is a disadvanage compared o he mauriy bond porfolio which disribues he rae exposure beween he = 5 and = 10 year mauriies. 7. Convexiy Bounds and ime Value able 4 provides incremenally more informaion on he porfolios examined in ables 1-3. Cerain pieces of relevan informaion are repeaed from ables 1-3: he surplus and he exreme bounds for duraion. In addiion, able 4 provides he ime value, he sum of he parial convexiies ( ), he maximum and minimum parial convexiies and he quadraic form defined by he duraion-opimal- 338 Insurance and Risk Managemen, vol. 75(3), Ocober 2007

17 able 3 Parial duraions, {n } and exreme bounds for he 10 Year Annuiy Liabiliy Immunized wih he Mauriy Bond and Low Surplus Examples* Mauriy Bond Low Surplus Dae (Yrs.) D n * D n * Cauchy = D Cauchy = D Exreme Duraion Bounds: ± 12.29% ± 32.87% Surplus: * he marke value of he Mauriy Bond Porfolio is composed of ($25.97) 1/2 year, ($34.956) 5 year and ($37.172) 10 year bonds. he marke value of he Low Surplus Porfolio is composed of ($31.388) 1/2 year and ($60.791) 10 year bonds. he liabiliy has marke value of $87.51 wih annual coupon, paid semi-annually, of $ he exreme bounds are derived using N = 1. Immunizaion Bounds, ime Value and Non-Parallel Yield Curve Shifs 339

18 shif convexiies, N * N *, where N * = (n 1*,..., n * ) is he vecor conaining he opimal n * s from ables 1-3 and Γ is a diagonal marix wih he CON, elemens along he diagonal. 11 he quadraic form calculaed using he N * for he duraion bound is of ineres because i provides informaion abou wheher he convexiy impac will be improving or deerioraing he change in surplus when he exreme duraion shif occurs. Using hese measures, able 4 illusraes he imporance of examining he convexiy and ime value informaion, in conjuncion wih he duraion resuls. Of paricular ineres is he comparison beween he mauriy maching and he spli mauriy porfolios of able 2. he primary resul in able 2 was ha he spli mauriy porfolio had greaer poenial exposure o spo rae (yield) curve shifs, as refleced in he wider exreme bounds associaed wih a spo rae curve shif of lengh one. Wheher his was a posiive or negaive siuaion was unclear, as he exreme bounds permied boh larger poenial gains, as well larger poenial losses, for he spli mauriy porfolio. Ye, by idenifying lower poenial variabiliy of he surplus of he mauriy bond porfolio, his provides some insigh ino he duraion puzzle. In his vein, able 4 also reveals ha, despie having a smaller surplus, he mauriy bond porfolio has a marginally higher ime value. his happens because, despie having a higher surplus and a smaller holding of he 1/2 year bond, he spli mauriy porfolio has o hold a disproporionaely larger amoun of he hree year bond relaive o he higher yielding seven year bond. Wih an upward sloping yield curve, his lower ime value is combined wih a higher convexiy, as measured by. his is consisen wih he resuls in Chrisensen and Sorensen (1994) where a radeoff beween convexiy and ime value is proposed, albei for a classical one facor model using a single ineres rae process o capure he evoluion of he yield curve. 12 As such, here is a connecion in he mauriy bond porfolio beween higher ime value, lower convexiy and smaller exreme bounds ha is direcly relevan o resolving he duraion puzzle. able 4 also provides a number of oher useful resuls. For example, comparison of he high and low surplus porfolios from able 1 adds o he conclusions derived from ha able. I is apparen ha, all oher hings equal, he ime value will depend on he size of he surplus. However, as illusraed in able 4, he relaionship is far from linear: he surpluses of he wo porfolios from able 1 differ by a facor of 10.9 and he ime values differ by a facor of 2.5. High surplus porfolios permi a proporionaely smaller amoun of he longer erm securiy o be held wih corresponding impac on all he various measures for duraion, convexiy and ime value. In addiion, unlike 340 Assurances e gesion des risques, vol. 75(3), ocobre 2007

19 able 4 ime Values, Convexiy and Oher Measures for he Immunizing Porfolios* ABLE 1 High Surplus Low Surplus (5 Year Zero Liabiliy) Surplus ime Value = 2 Θ N * N * Max CON Min CON Cauchy Duraion Bound ± 8.86% ± 97.13% D ABLE 2 Mauriy Bond Spli Mauriy (5 Year Zero Liabiliy) Surplus ime Value = 2 Θ N * N * Max CON Min CON Cauchy Duraion Bound ± 26.07% ± 39.90% D ABLE 3 Mauriy Bond Low Surplus (10 Year Annuiy Liabiliy) Surplus ime Value = 2 Θ N * N * Max CON Min CON Cauchy Duraion Bound ± 12.29% ± 32.86% D * See Noes o ables 1-3. Muliplying by 2 o make appropriae adjusmen o conver semiannual o annual raes, he ime value 1 is defined in (3). he sum of he parial convexiies is Θ he quadraic form, N * N *, is he sum of squares for he relevan N * from ables 1-3 muliplied erm-by-erm wih he appropriae parial convexiies. Max CON and Min CON are he maximum and minimum individual parial convexiies. Immunizaion Bounds, ime Value and Non-Parallel Yield Curve Shifs 341

20 he classical inerpreaion of convexiy which is ofen associaed wih he single bond case where all cash flows are posiive, convexiy of he surplus can, in general, ake negaive values and, in he exreme cases, hese negaive values can be larger han he exreme posiive values. However, his is no always he case, as evidenced in he able 3 porfolios where he liabiliy is an annuiy. he absence of a fuure liabiliy cash flow concenraed in a paricular period produces a decided asymmery in he Max CON and Min CON measures for individual C, convexiies, wih he Max values being much larger han he absolue value of he Min values. his is a consequence of he large marke value of he 10 year bond relaive o he individual annuiy paymens for he liabiliy. 8. Immunizing Agains Specific Scenario Shifs he appropriae procedure for immunizing a porfolio agains arbirary yield curve shifs is difficul o idenify, e.g., Reiano (1996). Some previous effors ha have approached his problem, e.g,, Fong and Vasicek (1984), have developed duraion measures wih weighs on fuure cash flows depending on a specific sochasic process assumed o drive erm srucure movemens. his inroduces sochasic process risk ino he immunizaion problem. If he assumed sochasic process is incorrec he immunizaion sraegy may no perform as anicipaed and can even underperform porfolios consruced using classical immunizaion condiions. In general, shor of cash flow maching, i may no be possible o heoreically solve he problem of designing a pracical immunizaion sraegy ha can provide opimal proecion agains arbirary yield curve shifs. In he spiri of Hill and Vaysman (1998), a less ambiious approach is o evaluae a specific porfolio s sensiiviy o predeermined yield curve shif scenarios. In pracical applicaions, his will be sufficien for many purposes. For example, faced wih a seep yield curve, a porfolio manager is likely o be more concerned abou he impac of he yield curve flaening han wih a furher seepening. If here is some prior informaion abou he expeced change in locaion and shape of he yield curve, i is possible o explore he properies of porfolios ha saisfy a surplus immunizing condiion a he iniial yield curve locaion. he basic procedure for evaluaing he impac of specific yield curve shifs requires a spo rae shif vecor ( i) N i = {(z 1,1 z 1,0 ), (z 2,1 z 2,0 ),... (z,1 z,0 )} o be specified ha reflecs he anicipaed 342 Insurance and Risk Managemen, vol. 75(3), Ocober 2007

21 shif from he iniial locaion a z 0 = (z 1,0, z 2,0,..., z,0 ) o he arge locaion z 1 = (z 1,1, z 2,1,..., z,1 ). his sep begs an obvious quesion: wha is he correc mehod for adequaely specifying N i? I is well known ha, in order o avoid arbirage opporuniies, shifs in he erm srucure canno be se arbirarily, e.g., Boyle (1978). If a sochasic model is used o generae shifs, i is required ha N i be consisen wih absence of arbirage resricions on he assumed sochasic model. hese resricions, which apply o he se of all possible pahs generaed from he sochasic model, are no required when he se of assumed fuure yield curve shif scenarios is resriced o specific shif scenarios based on hisorical experience or ex ane expecaions exogenously checked for consisency wih absence of arbirage. Where such shif scenarios are noional, relevan resricions for mainaining consisency beween individual spo raes are required. In erms of implied forward raes in one facor erm srucure models, necessary resricions for absence of arbirage ake he form: 13 (1 + z j ) j = (1 + z i ) i (1 + f i,j ) j-i = (1 + z 1 ) (1 + f 1,2 ) (1 + f 2,3 ),..., (1 + f j-1,j ) where he implied forward raes are defined as (1 + f 1,2 ) = (1 + z 2 ) 2 / (1 + z 1 ) and f j-1, j = (1 + z j ) j / (1 + z j-1 ) j-1 wih oher forward raes defined appropriaely. his imposes a smoohness requiremen on spo raes resricing he admissible deviaion of adjacen spo raes. In addiion o smoohness resricions on adjacen spo raes, he use of he parial duraion approach requires ha admissible N i shifs saisfy he norming condiion N = 1. In he associaed se of uni lengh spo rae curve shifs, here are numerous shifs which do no saisfy he spo rae smoohness requiremen. Because smoohing will allocae a subsanial porion of he uni shif o spo raes ha have small parial duraions, resricing he possible shifs by using smoohness resricions ighens he convexiy and duraion bounds compared o he exreme bounds repored in ables 1-4. o invesigae his issue, hree scenarios for shifing he iniial yield curve are considered: flaening wih an upward move in level, holding he = 10 spo rae consan (YC1); flaening wih a downward move in level, holding he = 6 monh rae consan (YC2); and, flaening wih a pivoing around he = 5 rae, where he > 5 year raes fall and he < 5 year raes rise (YC3). In empirical erms, hese hree scenarios represen plausible yield curve slope shifs ha have he larges possible move in he shor rae (YC1) and he long rae (YC2), and no change in he rae on he liabiliy (YC3). While oher empirically plausible scenarios are possible, such as a shif in yield curve shape moving he rae on he liabiliy down, he long rae up and holding Immunizaion Bounds, ime Value and Non-Parallel Yield Curve Shifs 343

22 he shor rae consan as in he exreme negaive shifs in ables 1-2, such cases are no examined due o he necessiy of keeping he number of scenarios o a manageable level. Given he hree shif scenarios being considered, wha remains is o specify he elemens of N i for shifs of uni lengh. he smoohness resricions require ha changes in yield curve shape will disribue he shif proporionaely along he yield curve. For example, when flaening wih an upward move in level, he change in he = 6 monh rae would be larges, wih he size of he shif geing proporionaely smaller as increases, reaching zero a = 10. Solving for a facor of proporionaliy in he geomeric progression, subjec o saisfacion of he norming condiion, produces a number of possible soluions, depending on he size of he spo rae increase a he firs sep. he following hree uni lengh N i shif vecors were idenified. ime (Yrs.) Flaen Up (YC1) Flaen Down (YC2) Pivo (YC3) As in (1) and (2), he acual change in a specific spo rae requires he magniude of he shif o be given. Observing ha each of hese hree scenario N i vecors is consruced o saisfy he norming condiion 344 Assurances e gesion des risques, vol. 75(3), ocobre 2007

23 N = 1, he empirical implicaions of his resricion are apparen. More precisely, uni lengh shifs do no make disincion beween he considerably higher volailiy for changes in shor erm raes compared o long erm raes. Imposing boh uni lengh shif and smoohness resricions on spo raes is no enough o resric he se of heoreically admissible shifs o capure all aspecs of empirical consisency. While i is possible impose furher empirically-based resricions on he se of admissible shifs, for presen purposes i is sufficien o work wih hese hree empirically plausible spo rae curve shif scenarios. Given hese hree uni lengh spo rae curve shifs, able 5 provides he calculaed values associaed wih (2) and (3) for he six porfolios of ables 1-3. Because he iniial duraion of surplus is approximaely zero and he porfolio convexiy ( ) is posiive in all cases, classical immunizaion heory predics ha he porfolio surplus will no be reduced by ineres rae changes. 14 he informaion in able 5 illusraes how he parial duraion approach generalizes his classical immunizaion resul o assumed non-parallel spo rae curve shifs. Comparison of he size of N i D wih he Cauchy bound reveals he dramaic reducion ha smoohness imposes on he poenial change in surplus value. In paricular, from (2) i follows ha a negaive value for he parial duraion measure N i D is associaed wih an increase in he value of he fund surplus projeced by he duraion componen. All such values in able 5 are negaive, consisen wih N i D indicaing all hree curve shifs produce an increase in he value of surplus. As in ables 1-3, he change in he surplus from he duraion componen can be calculaed by muliplying he surplus by he N i D value and he assumed shif magniude i. For every porfolio, he YC3 shif produced a larger surplus increase from he duraion componen han YC1 and YC2. Given ha he YC3 shif decreases he ineres rae for he high duraion 10 year asse and increases he ineres rae for he low duraion 6 monh asse wihou an offseing change in he liabiliy value, his resul is no surprising. In conras, while he YC2 shif increased surplus more han YC1 in mos cases, he reverse resul for he mauriy bond porfolio wih he annuiy liabiliy indicaes ha porfolio composiion can maer when he spo rae curve shif is non-parallel. Following Chance and Jordan (1996) and Poiras (2005, p.275), inerpreing he conribuion from convexiy depends on he assumed shif magniude i 2. A posiive value for he convexiy componen (N i N i ) indicaes an improvemen in he surplus change in addiion o he increase from N i D. For all porfolios, here was small negaive conribuion from convexiy for he spo rae curve flaening up (YC1). When muliplied by empirically plausible values for i 2 Immunizaion Bounds, ime Value and Non-Parallel Yield Curve Shifs 345