Testing the Optimality of Immunization Strategies With Transaction Costs

Transcription

1 Testig the Optimality of Immuizatio Strategies With Trasactio Costs Eliseo Navarro ad Jua M. Nave Departameto de Ecoomía y Empresa. Uiversidad de Castilla-La Macha (*) Abstract I this paper, a dyamic fixed-icome portfolio selectio model is developed uder differet stochastic ad o stochastic term structure regimes. This model allows the itroductio of trasactio costs ad shows that, i this cotext, the maximi strategy agaist iterest rate risk may ot be the immuizatio strategy but that it may cosist of buildig up a portfolio with a iitial duratio less tha the ivestor plaig period. The optimality of the solutios provided by the model are tested usig simulatio techiques. JEL: G Keywords: Iterest Rates; Fixed-Icome Potfolio, Immuizatio Risk; Dyamic Immuizatio; Trasactio Costs. (*) Correspodig author: Eliseo Navarro. Facultad de Ciecias Ecoómicas y Empresariales. Uiversidad de Castilla-La Macha. Area de Ecoomía Fiaciera. Plaza de la Uiversidad,. E ALBACETE, Spai. Tel Fax: aavarro@idrab.uclm.es

2 . Itroductio Oe of the mai results cocerig the developmet of portfolio strategies agaist iterest rate risk is the Dyamic Global Immuzatio Theorem euciated by Khag (983). Accordig to this theorem, to guaratee a fial portfolio value at the ed of a give period of time, idepedetly of iterest rate chages, the optimal strategy would be to keep alog time portfolio duratio equal to the ivestor s plaig period. Due to the ature of portfolio duratio such a strategy would imply a cotiuous portfolio rebalacig. However, the optimality of this strategy is based o a set of assumptios, icludig: (a) the forward iterest rates structure g(t), t > 0, chage to g * (t,λ) where λ is a stochastic shift parameter. Specifically he assumes that, g * (, t λ) = g() t + λ () (b) absece of trasactio costs. The first assumptio avoids the problem of the risk of misestimatio of the term structure behavior called by Fog ad Vasicek (983) "immuizatio risk". Bierwag (987) called it "stochastic process risk", poitig out that if a ivestor assumes a icorrect hypothesis about term structure chages, the percieved duratios would be differet from the actual oes ad so, the ivestor losses from misestimatio (or misguesstimatio) of the correct process could be substatial". This problem has bee studied i several papers assumig alterative hypothesis about he term structure behavior ad two differet approaches ca be distiguished. The first oe could be called, accordig to De Felice ad Moricoi (99) semidetermiistic ad cosists of assumig that istataeous forward rates may chage by a radom factor. These sort of models, however, may allow arbitrage opportuities (Igersoll, Skelto ad Weil, 978) ad lead to differet defiitios of 2

3 duratio depedig o the iitial hypothesis about iterest rate chages. The secod approach is based o o arbitrage models of the term structure as those suggested by Cox, Igersoll ad Ross (985) or Vasicek (977). The immuizatio strategy uder these term structure models, has bee aalyzed by Boyle (978) ad Cox, Igersoll ad Ross (979) where the cocept of stochastic duratio was first defied. With respect to the hypothesis of the absece of trasactio costs, it becomes crucial i a dyamic cotext; if trasactio costs are take ito accout, the strategy of a cotiuous portfolio rebalacig, may become ot optimal due to the high expeses this strategy would icur. This problem has bee aalyzed by Maloey ad Logue (989). These authors measure the impact of trasactio costs o the immuizatio strategy. Bierwag (987) suggests that cotiuous portfolio rebalacig i order to keep portfolio duratio equal to the remaiig ivestor s plaig period may be iefficiet. Some years later, Lee ad Cho (992) developed a portfolio selectio model usig dyamic programmig techiques, cocludig that if trasactio costs are high eough, the optimal strategy agaist iterest rate risk may cosist of a partial immuizatio i order to avoid trasactio costs. However, they assume as a boudary coditio that the optimal portfolio path must start with a immuized portfolio; this is a arbitrary costrai, which may lead to a suboptimal strategy. I this article, we try to deal with these problems developig a dyamic portfolio selectio model uder differet assumptios about the term structure of iterest rates (TSIR) behavior. Particularly, three differet cases are aalyzed: - the most simplistic oe cosistig of assumig a flat term structure ad parallel TSIR shifts ad - two alterative o arbitrage stochastic term structure models which assume that the istataeous spot iterest rate follows a diffusio process: Vasicek (977) ad Cox, Igersoll ad Ross (985) models. The model is based o a previous static oe that behaves accordig to classical Fisher ad Weil (97) Immuizatio Theorem i the first case ad Boyle s (978) 3

4 Stochastic Immuizatio i the two stochastic cases. The, the model is adapted to a dyamic cotext ad, the the model is elarged i order to icorporate trasactio costs. I this case, it is show that, if trasactio costs are high eough, the the optimal strategy may differ from that proposed by Khag. Fially, the optimality of the solutios provided by the model is tested usig simulatio techiques ad showig their superiority agaist immuizatio strategy. 2. The static model. As Bierwag ad Khag (979) proved, immuizatio ca be described as a maximi strategy i a game agaist Nature where the ivestor target is to guaratee a miimum retur over his plaig period or, equivaletly, a miimum value at the ed of his horizo plaig period (HPP). Thus, accordig to Datzig (97), this maximi solutio ca be worked out by solvig a equivalet liear program. This liear program depeds o the hypothesis about the term structure of iterest rates assumed by the model. So, we describe first a set of commo assumptios ecessary to model this portfolio selectio problem ad the differet hypothesis about the TSIR are itroduced leadig each of them to alterative models which are described i sectios 2. (ostochastic term structure) ad 2.2 (stochastic term structure). The set of iitial assumptios are the followig: - Fiacial markets are competitive: idividual ivestors decisios do't affect iterest rates which are give exogeously. - Perfect divisibility of fiacial assets. - Absece of trasactio costs. - Short sales are ot allowed. This costrai is imposed i the model as a sufficiet coditio i order to guaratee that the et icome geerated by the portfolio is always o egative throughout the plaig period, which is oe of the hypothesis of Khag's theorem. 4

5 2. No stochastic term structure models I this case we make the followig hypothesis about TSIR: a) The term structure is flat; b) TSIR chages cosist of parallel movemets of the whole term structure, i.e., short ad log term iterest rates chages are equal. Uder this set of assumptios the immuizatio problem ca be modeled as follows. We assume a ivestor who wats to allocate a amout of I dollars i a market where differet coupo bearig bods 2 are available. Differet portfolio allocatios ca be cosidered as the strategies played by the ivestor. Also, we assume that just after the purchase of the selected portfolio, iterest rates may chage from its curret level (deoted by r c ) to ay of the values r,..., r c,... r m ; where r <... < r c <... < r m. Fially we assume that it does't take place ay additioal uexpected iterest rate chage durig the remaiig HPP 3. These possible ew iterest rate values ca be regarded as the strategies played by Nature. Let p i (i =,...,) be the curret price of oe uit of asset i ad x i be the umber of uits of this asset icluded i the optimal portfolio. The a ivestor strategy cosists of a vector (x, x 2,..., x ) which must verify the followig budget costrai: i= xp i i = I (2) 2 I order to isolate iterest rate risk we assume these are default-free risk bod without ay callable feature. 3 We assume alog this paper that the Pure Expectatios Hypothesis about the TSIR holds; so uder a flat term structure regime ay iterest rate chage is cosidered to be uexpected. 5

6 If just after selectig a strategy, iterest rates move from r c to r j, portfolio value at the ed of the HPP (if o additioal uexpected iterest rate chage takes place) is give by the followig expressio: i= xv i where v ij deotes the value at the ed of the HPP of a ivestmet of p i dollars i asset i ad, as before, iterest rates chage just after the purchase, from r c to r j remaiig uchaged util the ed of the HPP. Note that v ij is calculated assumig that coupo ad pricipal paymets due before the ed of the HPP are reivested at the expected iterest rate immediately after the TSIR shift accordig to the Pure Expectatios Theory. i (3) Deotig by V the miimum fial portfolio value the ivestor wishes to maximize, the portfolio selectio process ca be modeled as follows: Max V xv, S. to: i= xv V j=,..., m i ij xi pi = I i= x, V 0 i (A) 2.2 Stochastic term structure models At this poit we assume the followig hypothesis about the term structure: a) Istataeous spot iterest rate (r(t)) follows a diffusio process so its behavior is described by the followig stochastic differetial equatio: dr() t = f ((),) r t t dt +ρ ((),) r t t dz ~ (4) 6

7 where dz ~ is a Wieer Process with zero mea ad variace dt. b) There are o arbitrage opportuities. The price, P(r(t),t,s) at time t of a pure discout bod which matures at time s (t=s) is a stochastic variable that verifies (by Ito's lemma) the followig differetial equatio where P dp = P P P f dt dz P dt P dz t + r + ρ 2 + ρ 2 ~ = µ + σ ~ (5) 2 2 r r µ ρ 2 P P 2 P = + f + (6) 2 P t r 2 r ad σ = ρ P (7) P r If we assume the Expectatios Hypothesis ad there exists o arbitrage opportuities, the the price of a uit discout bod must satisfy the followig partial differetial equatio [Vasicek (977)]: ρ 2 P P 2 P + f + rp = 0 (8) 2 t r 2 r At maturity, the bod price must be equal to oe so P(r(t),s,s)=; this provides the boudary coditio ecessary to solve Eq. (8). Now we will assume two specific assumptios about the stochastic process followed by spot rate. Case A: Vasicek model [Vasicek (977)]. dr() t = α ( γ r()) t dt + ρ dz ~ (9) where α, γ ad ρ are positive costats. 7

8 I this case the solutio to (8) is give by where: 2 ρ 2 Prt ((),,) ts = exp FT ( )( G rt ()) TG FT ( ) 4α (0) T = s t () FT ( ) = ( exp( α T)) α (2) 2 ρ G = γ 2 2α (3) Uder this term structure hypothesis, the relative basis risk of a discout bod is give by P/ r = FT ( ) (4) P As CIR (979) poit out "if we wat stochastic duratios be a proxy for basis risk 4 of coupo bods with the uits of time, it is atural to defie it as the maturity of a discout bod with the same risk". The portfolio duratio uder Vasicek TSIR model, D V, is give by: D V = F s CsPrt () ((),,) tsft () CsPrt () ((),,) ts s where C(s) is the stream of cash flows geerated by that portfolio ad F - [X] is: F X X = l( [ ] α ) α (5) (6) Case B.- Cox, Igersoll ad Ross model [CIR (979)]. dr() t = κ ( µ r()) t dt + σ r() t dz (7) where κ, µ ad σ are positive costats. 4 Basis risk ca be defied as the possibility that a istitutio margi will rise or fall as a cosequece of market rates movemets. 8

9 where Now the solutio to (8) is give by P( r( t), t, s) = A( T)exp( r( t) B( T)) (8) 2λexp [( κ λ) T / 2] [ T ] 2 AT ( ) = ( λ + κ) exp( λ ) + λexp( λt) 2κµ σ 2 (9) BT ( ) = 2 ( exp( λt)) [ T ] ( λ + κ) exp( λ ) + 2λexp( λt) (20) 2 2 λ = κ + 2 σ (2) ad portfolio duratio uder CIR term structure model, D CIR is: where D CIR = B s CsPrt () ((),,) tsbts (,) CsPrt () ((),,) ts s (22) B 2 ( κ λ) X [ X] = l λ 2 ( κ + λ) X (23) It is importat to poit out that uder these two term structure regimes the whole TSIR depeds o the curret istataeously compouded spot iterest rate r(t). Now the selectio model should be restated i terms of the variable r(t). Thus v ij must be redefied as follows: v ij Ci() s P( rj,,) t s s = Pr (, ts, ) j where C i (s) deotes the paymet stream geerated by oe uit of bod i ad P(r j, t, s) is the price at time t of a pure discout bod with maturity at s if spot istataeous iterest rate becomes r j. 9 (24)

10 2.3. Model results To illustrate umerically the former models we have applied them to a very simplistic case. Let's assume a ivestor with a horizo plaig period of 8 moths ad a fixed icome market where there are four differet default-free coupo bods available which characteristics are described i table. Additioally we assume that the ivestor has a iitial budget of oe millio dollars to allocate amog these assets. The hypothesis about the term structure are the followig: Case.- Flat term structure. I this case we assume a flat term structure beig the curret iterest rate level 0 % (compouded semiaually). Iterest rates may move up ad dow by 00 basis poits to 9% or to %, i.e. r = 9%; r 2 = r c = 0%; r 3 = %. The optimal solutio of this program is show i table 2. As we ca see, this result is cosistet with Fisher ad Weil immuizatio theorem: optimal solutio cosist of a portfolio with a duratio equal to the HPP. Case 2.A. Vasicek model. We assume that the istataeous spot iterest rate follows the diffusio process proposed by Vasicek. I order to give realistic values to the model we used those estimated i Nowma (997) for both US (from Treasury Bill market) ad UK (sterlig oe-moth iterbak rate). (See table 3). Notice that ow, Nature strategies cosists of the differet values the curret istataeous spot rate ca take which we assume ca vary 00 basis poits (upwards 0

11 or dowwards) from its curret level (5.6% for US ad 5.99% for UK). The optimal solutios are show i table 2. Case 2.B. Cox, Igersoll ad Ross model. Now, it is assumed that istataeous spot iterest rate follows CIR process. Model parameter are as before those estimated i Nowma (997) for US ad UK. (See table 3). As before, we assume that istataeous spot iterest rate may chage by 00 basis poits from its curret value. The optimal solutios are preseted i table 2. It is importat to see that "Redigto's basic idea is still valid although there are importat differeces withi the framework of the stochastic model of the term structure" [Boyle (978)], i.e., uder stochastic term structure models portfolio immuizatio cosists of makig portfolio duratio (properly defied) equal to the remaiig HPP. 2.4 Immuizatio risk. I former sectios we have assumed explicitly a specific term structure behavior where the whole term structure is supposed to deped o a uique factor (short term iterest rate). However, as it is well kow, the ature of the dyamics of iterest rates is by far much more complex 5. So, immuizatio strategies may fail if TSIR behavior differs sigificatly. This is kow as immuizatio risk. 6 5 There are some iteratioal evidece that at least 95% of term structure movemets ca be explaied by three factors; parallel shifts, slope chages ad curvature chages. Depedig o the coutry aalyzed ad the period covered by differet studies, parallel shifts ca explai a percetage of the variace o iterest rates chages betwee 72.2 ad See for further details Steeley (990), Stricklad (993), D'Ecclesia ad Zeios (994), Navarro ad Nave (995) ad Sherris (995) for UK, US, Italy, Spai ad Australia respectively. 6 For a review of the effects of o-parallel yield curve shifts o traditioal immuizatio strategy see Reitao (992). I Reitao (99) Kag's result was geeralized to ay directioal yield curve model ad to a geeral multivariate odirectioal model.

12 I order to miimize the immuizatio risk derived from a uexpected behavior of the term structure several proposals have bee suggested but o the whole most of them cosist of selectig amog the set of immuized portfolios those that geerate a paymet stream as cocetrated as possible aroud the ed of the HPP. A trivial example would be a portfolio cosistig of zero coupo bods maturig at the ed of the HPP: it is a immuized portfolio (Duratio equal to the HPP legth) ad simultaeously, due to total cocetratio of paymets at t k, it would be free of immuizatio risk. Although there are several alteratives to measure immuizatio risk oe of the most usually accepted dispersio measures is that proposed by Fog ad Vasicek 7 kow as M 2. As they proved, by miimizig this quadratic dispersio measure, the effect o fial portfolio value of a TSIR movemet differet from that assumed is miimized. Particularly, they aalyze the effect of a TSIR shift cosistig of a liear movemet of the istataeous forward rate aroud the ed of the HPP; i this case it is ot possible to build up a immuized portfolio but there is a lower boud for portfolio fial value that depeds o M 2. This dispersio measure correspodig to bod i,, ca be defied as follows: M 2 i = s 2 ( s HPP) C( s) P( r( t), t, s) s i C() s P((),,) r t t s i i where : C i (s) deotes the paymet stream geerated by bod i; i i=,, (25) 7 There are other alterative dispersio measures, as M-Absolute, derived from differet assumptios about term structure movemets; i a recet paper Chalmers ad Nawalka (996) test the use of this latter measure as a first order coditio to protect a ivestmet agaist iterest rate risk istead of usig it as a secod order coditio to miimize immuizatio risk. Other authors have criticized M 2 measure, suggestig the coveiece of icludig a asset with maturity at the ed of the HPP as the best strategy agaist immuizatio risk (see Bierwag et al. (993)). However, i practical terms, these three alterative measures lead to very similar results. For a geeralizatio of immuizatio risk measures see Balbás e Ibáñez (998) 2

13 HPP is the legth of the ivestors horizo plaig period; P(r(t),t,s) is the price at t of a zero coupo bod with maturity at s if the outstadig iterest rate at t is r(t). This dispersio measure is itroduced i the model by pealizig the objective fuctio which becomes: V - A Mi 2 xi (26) i= where A>0 is a coefficiet that depeds o the ivestor's Immuizatio risk aversio. If this dispersio measure is applied to the model the optimal portfolio path cosist of immuized portfolios of miimum dispersio, idepedetly of the term structure assumptio 8. (See table 4). 3. The dyamic model The portfolio selectio model described i the previous sectio provides a portfolio that is immuized agaist iterest rate risk oly at the begiig of the HPP. However, the dyamic behavior of portfolio duratio makes it impossible to keep that portfolio immuized durig the whole plaig period. Moreover, the immuizatio solutio provided by the model is oly valid for the curret iterest rate, so the portfolio has to be adjusted cotiuously. Oly if portfolio cosists of zero coupo bods with maturity at the ed of the HPP it would be possible to keep it immuized alog the HPP without ay additioal rearragemet. The selectio model we describe ext, tries to obtai a optimal rebalacig path to keep portfolio free of iterest rate risk. First, we make a partitio of the HPP ito k subitervals of equal legth - [t 0,t ], [t,t 2 ],..., [t k-,t k ] - beig t 0 the begiig 8 I fact, ay other decreasig fuctio of M 2 could be added to the objetive fuctio to pealized portfolio dispersio as far as we are oly tryig to obtai the immuized portfolio of miimum dispersio. 3

14 of the HPP ad t k the ed of the HPP ad we assume that portfolio rebalacig is oly allowed at the begiig of each subiterval. At these poits Nature ca play a set of strategies (scearios about iterest rates) aalogous to those described i the former static model. But ow, we have to make a previous hypothesis about the expected iterest rate level at the begiig of each subiterval i order to aalyze the impact of iterest rate chages just after these rebalacig dates. It must be poited out that actual iterest rates at each t s will differ from those assumed iitially but what does really matters i this model is ot the level of iterest rates but the effects of iterest rate chages o the fial portfolio value. However, the hypothesis about those future iterest rates will differ depedig upo the term structure regime we had assumed. I case (flat term structure) if we assume the Expectatio Hypothesis about the term structure we have that E to [r(t s )] = r c, t s >t 0, i.e., iterest rates are assumed to remai uchaged. If a stochastic term structure model is assumed (case 2.A, Vasicek's model, ad case 2.B, Cox, Igersoll ad Ross model) we have 9 : Case 2.A: Vasicek model: α ( ts t0 ) E [( r ts)] = γ + (( r t0) γ) e ts > t (27) 0 t 0 Case 2.B: CIR model: κ ( ts t0 ) Et [( r ts)] = µ + (( r t0) µ ) e ts > t (28) 0 0 The, what we aalyze i this dyamic model are the effects of uexpected 9 See Vasicek (977) ad Cox, Igersoll ad Ross (985). 4

15 iterest rate chages at each t s o the fial portfolio value. For the sake of simplicity ad without loss of geerality we assume that there are differet coupo bods available at t 0 each of them maturig at t, t 2,..., t k-, t k,..., t respectively; coupo paymets are also due at the rebalacig poits. Now let x(s,i) be the umber of uits of asset i icluded i the portfolio at t s (s=0,,...,k-). The x(s,i)-x(s-,i) is the umber of uits of asset i bought ad sold at t s (depedig upo x(s,i)-x(s-,i) beig positive or egative). Let b(s,i) ad z(s,i) be the umber of uits of asset i bought ad sold at t s respectively ad p(s,i) the value at t s of oe uit of asset i as if E to [r(t s )] were the spot rate prevailig at t s, i.e., as if o uexpected iterest rate chage had take place durig period [t 0, t s ]. The, the followig set of costrais must be satisfied: () i x(0,i) = b(0,i) i =,..., ( ii ) x(s,i) - x(s -,i) - b(s,i) + z(s,i) = 0 s =,..., k i = s+,..., ( iii ) x(s -,s) = z(s,s) s =,..., k ( iv ) x(k -,i) = z(k,i) i = k +,..., The first set of costrais (i) idicates the umber of assets bought at the begiig of the HPP ad the secod group of costrais (ii) the purchases ad sales at each subsequet t s. The third group of costrais (iii) represets the umber of uits of bod s maturig at t s (s=,..., k-) so it must be equal to the umber of assets s held i the optimal portfolio at t s- i.e. with oe period to maturity; this set of costrais is icluded i the model to distiguish those assets that are sold at t s from those with maturity at t s. Fially, the last group of costrais (iv) idicates that at t k (the ed of the HPP) all assets must be sold. 5

16 The iitial budget costrai must be replaced by: x( 0, i) p( 0, i) = I (29) i= where I is the amout of moey available at the begiig of the HPP. Now the budget costrai must be satisfied ot oly at t 0 but durig the whole plaig period, so we have to add the followig set of budget costrais: b(,) s i p(,) s i z(,) s i p(,) s i z(,) s s p(,) s s cx i ( s,) i = 0 i= s+ i= s+ where C i is the coupo paymet of oe uit of bod i. p(s,s) is the face value of asset s, i.e. maturig at t s. p(s,s)z(s,s) are the pricipal repaymets at t s correspodig to asset s. i= s s =,... k (30) What this set of costrais is tryig to catch up is the fact that the amout of moey ivested i ew purchases at each t s (b(s,i)p(s,i)) must come from coupo paymets ( i x(s,i)), sales ( z(s,i)p(s,i)), ad/or pricipal repaymets (p(s,s)z(s,s)). As i the static model, we assume that ivestor's aim is to maximize at each t s the guarateed portfolio value at the ed of the HPP if a uexpected iterest rate chage takes place immediately after portfolio rebalacig, i.e., just after each t s. The, if we deote by V s the miimum fial portfolio value to guaratee at t s, the followig set of costrais must be satisfied: i= s+ xsiv (,) () s V i, j s s = 0,..., k (3) j =,..., m where v i,j (s) deotes ow the fial value of a ivestmet of p(s,i) dollars i asset i at t s if the istataous spot iterest rate chages, just afterwards, from E to [r(t s )] to r j ad o additioal uexpected iterest rate chage takes place util the ed of the HPP, i.e. 6

17 vi, j() s r s = = + Ci( tr) P( rj, ts, tr ) (32) Pr (, t, t) j s k where C i (t r ) represets the paymets stream geerated by bod i from t s awards ad P(r j,t s,t r ) the value at t s of a uit zero coupo bod with maturity at t r if the prevailig iterest rate at t s is r j. As far as ivestor's aim is to maximize at each t s these miimum portfolio fial values ad simultaeously to miimize immuizatio risk at each t s the objective fuctio would be restated as follows: k k 2 s s= 0 s= 0 i= s+ V A M (,)(,) s i x s i (33) where M 2 (s,i) is the Fog ad Vasicek dispersio measure correspodig to bod i at t s ad is defied as follows: M 2 (,) s i = t i j= s+ 2 ( t t ) C( t ) P( E [ r( t )], t, t ) j k i j t s s j t i j= s+ C( t ) P( E [ r( t )], t, t ) i j t s s j 0 0 s = 0,..., k (34) i = s+,..., where C i (t j ) deotes the paymet stream geerated by bod i, t i is the term to maturity of bod i t k is the ed of the HPP P(E to [r(t s ),t s,t j ) is the price at t s of a zero coupo bod with maturity at t j if the outstadig iterest rate at t s is the expected oe at t 0. The the whole model is: 7

18 k k 2 Max Vs A M V, x s= 0 s= 0 i= s+ (,)(,) s i x s i (B) Sto. : i= s+ xsiv (,) () s V i, j s s = 0,..., k ; j =,...,m; i =,..., x( 0, i) = b( 0, i) i =,..., x(,) s i x( s,) i b(,) s i + z(,) s i = 0 s =,..., k ; i = s +,... x( s, s) = z( s, s) s =,..., k x( k, i) = z( k, i) i = k +,..., i= x( 0, i) p( 0, i) = I0 b(,) s i p(,) s i z(,) s i p(,) s i z(,) s s p(,) s s Cx i ( s,) i = 0; s =,..., k i= s+ i= s+ zkipki (, ) (, ) + zkk (, ) pkk (, ) + xk (, i) = V k s= k+ i= k xsi (,); bsi (,);(,); ssi V s 0 s, i i= s We proceed to illustrate this model by applyig it to the former example, assumig that the HPP (8 moths) is subdivided ito 3 subperiods of equal legth (6 moths) ad that portfolio rebalacig is oly allowed at the begiig of each subiterval. As before, we make three differet assumptios about the TSIR behavior. Case. Flat term structure. We assume that at the begiig of the HPP the curret iterest rate is 0 % (compouded semiaually) ad that iterest rates may move upwards ad dowwards by 00 basis poits. Also, the expected iterest rate outstadig at the begiig of each subiterval is the curret iterest rate (we assume the Pure Expectatios Hypothesis). 8

19 The optimal solutio paths are reported i pael of table 5. We ca see that this result is cosistet with Khag's Theorem: optimal portfolio duratio cosist of makig duratio equal to the remaiig HPP at every t s. The small differece betwee these two variables are due to the fact of cosiderig a fiite umber of scearios about iterest rate chages. Case 2. Stochastic term structure models. We assumed that the expected iterest rate at the begiig of each subiterval are give by formulae (6) ad (7) ad that the istataeously compouded spot iterest rate may chage by 00 basis poits at the begiig of each subiterval. The curret spot rate ad the model parameters are those assumed i the former example for US ad UK. As we ca see i pael of tables 6 to 9, Khag's Theorem is still valid uder this stochastic term structure regimes. Also these results are cosistet with those obtaied by Gago ad Johso (994) uder stochastic iterest rate i a discrete time framework Itroductio of trasactio costs The models described i the two previous sectios are based o the assumptio that there are o trasactio costs ad they lead to the solutio suggested by Khag. The ext step is the itroductio of trasactio costs i the model to aalyze its effects o the optimal solutio. We will assume that trasactio costs icurred at each portfolio rearragemet are a percetage, α, of the volume traded (i dollars) at each t s. We also assume that pricipal ad coupo repaymets do't geerate ay trasactio 0 I particular they assume the Black, Derma ad Toy (990) arbitrage-free evolutio model. 9

20 costs although other assumptios could be easily implemeted. So, asset purchase prices are icreased by a percetage α ad sale prices are reduce by the same proportio. This ew purchase (sale) prices ca be uderstood as the bid (ask) prices of the assets plus (mius) ay proportioal fee paid to itermediaries. The, the budget costrais have to be modified as follows: i= poi (, )( + α ) b( 0, i) = I0 psi (,)( + α)(,) bsi psi (,)( α)(,) zsi psszss (,)(,) Cxs i (,) i = 0 i= s+ i= s+ z( k, i) p( k, i)( α) + z( k, k) p( k, k) + i= k + i0k C x( k, i) = V i i= s+ i =,..., k (35) Fially, the model is applied to the former example ad the results are preseted i pael 2 of tables 5 to 9 for differet a values (0.5%, 0.3%, 0.45% ad 0.6%). The first outcome that must be poited out is the fact that the optimal path depeds ow o the level of the trasactio costs, i.e., o the level of α. So for α=0 we get the solutio suggested by Khag: at each rebalacig poit portfolio duratio must be equal to the remaiig HPP. But for values of α bigger tha 0.05%, the optimal path has a iitial portfolio that is ot immuized ay loger. It is importat poit out that this critical α-values are very low, implyig that, i practical terms, the immuizatio strategy caot be optimal. k I ay case, this critical α-values will deped o the assumptio about iterest rates volatility which has bee itroduced i the model through a set of scearios about the iterest rates chages (i our example cosistig of chages up to 00 basis poits from its curret level). The the bigger the variability of iterest rates, the bigger the critical α-value. 20

21 A additioal poit is that the differece betwee the iitial portfolio duratio ad the HPP icreases as the level of trasactio costs rises. I fact, i this simple example four differet solutios are obtaied; the iitial portfolio duratio rage from.5 years (for α=0) to approximately,43 years (for a 0.6%). It is worth poitig out that for high eough trasactio costs the optimal solutio cosist of ivestig the whole iitial budget i a bod with maturity at the ed of the HPP. What this strategies suggest is that due to the behavior of portfolio duratio, it is possible to take advatage of its evolutio alog time. If ivestors build up a portfolio with a duratio less tha the HPP, as time passes, portfolio duratio becomes closer to legth of the HPP. So iterest rate risk disappears without ay additioal portfolio rebalacig. Fially, portfolio duratio would be greater tha the HPP, ad the ivestors should woder if it is worth avoidig the potetial loss derived from a adverse iterest rate chage by immuizig their portfolio. If trasactio costs are too high it could be worth assumig this risk, i order to avoid the loss derived from portfolio rebalacig. If portfolio duratio is log eough, this process may be helped by a optimal coupo reivestmet i those bods with a the shortest duratio. However, if the HPP is short, it will ot be possible to keep duratio equal to the HPP uless we proceed to sell bods with log duratios ad ivest the proceeds i bods with shorter duratio. It is importat to poit out that these fidigs are commo to all cases aalyzed, i.e., they are idepedet of the TSIR model assumed. These models provide a first hit to aswer the questio posed by Maloey ad Logue (989) with respect to the "mismatch duratio that is tolerable, give that allowig a modest mismatch will certaily reduce tradig costs". 2

22 5. Model testig The model we developed i the previous sectios is based o the hypothesis of a give behavior of iterest rates. Particularly, we assumed that short term iterest rates ca move upwards or dowwards from its expected value by 00 basis poits. This rage of iterest rate chages was chose arbitrarily, so i this sectio we test the optimality of the differet strategies if iterest rates are allowed to chage without ay boud accordig to the iterest rate model assumed (Vasicek or C.I.R s models). I this case, extreme iterest rate chages are possible although its probability of occurrece may be egligible. Theoretically if we do t assume ay boud for iterest rate chages the maximi strategy is immuizatio idepedetly of trasactio costs; however very extreme iterest rate chages are so ulikely that i practical terms other strategies may yield better results. Oe poit that has to be highlighted is the existece of a trade-off betwee iterest rate risk ad the level of trasactio costs. If we had assume i the previous model a higher level of iterest rate risk (i.e. if we wide the rage of iterest rate chages) we would have icreased the level of the potetial loss derived from adverse iterest rate movemets ad so it may be worth to immuize although this strategy ca cause higher trasactio costs. As we could see there were four differet optimal strategies, depedig o the level of trasactio costs. These strategies are represeted i figure. I this figure we have represeted the differece betwee portfolio duratio ad the ivestor s plaig period as a fuctio of time. Strategy (immuizatio), cosists of a permaet portfolio rebalacig i order to avoid all iterest rate risk alog the whole plaig period by keepig portfolio duratio equal to the remaiig ivestor s plaig period; this strategy, however, geerates the highest trasactio costs. Strategy 4, cosists of ivestig all the ivestor s budget i the 22

23 bod maturig at the ed of the plaig period; thus, at the begiig, its duratio is less tha the plaig period ad so the ivestor is bearig some iterest rate risk, but this strategy is the cheapest i terms of trasactio costs. Strategies 2 ad 3 are itermediate positios betwee miimizatio of iterest rate risk (Strategy ) ad miimizatio of trasactio costs (Strategy 4). I order to test the optimality of these four strategies we have proceed to simulate the behavior of iterest rates accordig to Vasicek s model. Particularly, we have discretized the model by dividig each period betwee rebalacig poits ito 26 subitervals (of approximately oe week legth 2 ). The r r = t+ t r t t + [ α ( γ )] ρ εt t where t = / 52, ε t i.i.d N(0,), r 0 = ad t=, 2,, 78. For each iterest rate path, we have calculated the fial portfolio value we would have got if we had followed each strategy ad repeated the process 0000 times, for differet trasactio costs levels. I figure 2 we have draw the outcomes of these simulatios uder the hypothesis of the absece of trasactio costs. As we ca see, Strategy 4 presets the widest rage of fial portfolio values meawhile Stratey cocetrates all the outcomes aroud the same fial porfolio value (the fial value we would have got if iterest rates had behaved as expected). To illustrate the effects of trasactio costs o portfolio returs (or equivaletly o fial portfolio values ) we have represeted i figure 3 the fial portfolios values uder differet trasactio cost levels. O the whole, trasactio costs make fial portfolio values to move leftwards ad the higher the level, the deeper the movemet to the left. However, as figure 3 shows trasactio costs do t We have also, used CIR s model, ad te results are aalogous so we explai here Vasicek model for the sake of simplicity. 2 We have assumed weeks of 7.09 days. 23

24 affect all strategies i the same way. This leftwards movemet is more itesive for Strategy, i.e. immuizatio strategy, meawhile the effect of trasactio costs is miimized whe followig Strategy 4. Due to this behavior of trasactio costs it may happe that Strategies differet from immuizatio may provide the maximu miimu retur. This is show i table 0 where we provide the maximu, miimu ad mea fial portfolio values (after 0000 iteratios) we got whe followig the four strategies described earlier for three differet levels of trasactio costs (α=0, 0.5 ad per cet). For istace, we ca see that whe α= % strategy 4 yielded a miimum fial portfolio value that is higher tha the miimum portfolio value guarateed by immuizatio although, theoretically, immuizatio is the maximi strategy whe iterest rates are ot bouded. 5. Coclusios I this paper we have developed a dyamic portfolio selectio model for iterest rate risk maagemet uder differet TSIR regimes. This model leads to a result which is cosistet with Khag's Dyamic Immuizatio Strategy cosistig of a cotiuous rebalacig to keep portfolio duratio equal to the ivestor's HPP. The model is the elarged i order to allow the itroductio of trasactio costs to aalyze its effects o the optimal strategy. The results obtaied through a very simple example suggest that if trasactio costs are take ito accout, the strategy cosistig of makig portfolio duratio equal to the HPP is ot optimal ay loger. Moreover, the optimal path has a iitial solutio with a portfolio duratio less tha the HPP. Furthermore, the bigger the level of trasactio costs, the bigger the differece betwee the iitial portfolio duratio ad the HPP. This result held uder differet TSIR models. 24

25 Fially, we have tested usig simulatio techiques, the optimality of these strategies whe iterest rate movemets are ot bouded cocludig that if trasactio costs are high eough, strategies cosistig of portfolios with a iitial duratio less tha the ivestor s plaig period provide the maximu miimu retur, although theoretically imuizatio is the maximi strategy. 25

26 Refereces Bierwag, G.O., 987, Duratio Aalysis. Maagig Iterest Rate Risk (Ed. Belliger Cambridge Mass). Bierwag, G.O.; Fooladi, I. ad Roberts, G.S., 993, Desigig a immuized portfolio: Is M-squared the key?, Joural of Bakig ad Fiace, Bierwag, G.O. ad Khag, C., 979, A immuizatio Strategy is a Miimax Strategy, Joural of Fiace, May. Black, F.; Derma, E. ad Toy, W., 990, A Oe Factor Model of Iterest Rates ad its Applicatio to Treasure Bod Optios, Fiacial Aalysts Joural, Jauary/February. Boyle, P.P., 978, Immuizatio Uder Stochastic Models of the Term Structure, Trasactios of the Faculty of Actuaries, Cox, J.C.; Igersoll, J.E. ad Ross, S.A., 979, Duratio ad the Measuremet of Basis Risk. Joural of Busiess, 5-6. Cox, J.C.; Igersoll, J.E. ad Ross, S.A., 985, A Theory of the Term Structure of Iterest Rates, Ecoometrica, Datzig, G.D., 97, A Proof of the Equivalece of Programmig Problem ad the Game Problem, i: T.C. Koopmas, ed., Activity Aalysis of Productio ad Allocatio (Yale Uiversity Press). D'Ecclesia, L. ad Zeios, S.A., 994, Risk Factor Aalysis ad Portfolio Immuizatio i the Italia Bod Market, Joural of Fixed Icome, September. De Felice, M. ad Moricoi, F., 99, La teoria dell immuizzazioe fiaziaria. Modelli e strategie (Il Mulio, Bologa). Fisher, L. ad Weil, R.L., 97, Copig with the risk of iterest rates fluctuatios: Returs to bodholders from aive ad optimal strategies, Joural of Busiess. October. Fog, H.G.ad Vasicek, O., 983, The Tradeoff Betwee Retur ad Risk i Immuized Portfolios, Fiacial Aalysts Joural, September-October. Gao, L. ad Lewis, D.J., 994, Dyamic Immuizatio uder Stochastic Iterest 26

27 Rates, The Joural of Portfolio Maagemet, Sprig. Igersoll, J.E.; Skelto, J. ad Weil, R.L., 978, Duratio Forty Years Later, Joural of Fiacial ad Quatitative Aalysis, 3,. 4. Khag, C., 983, A Dyamic Global Portfolio Immuizatio Strategy i the world of Multiple Iterest Rates Chages: A Dyamic Immuizatio ad Miimax Theorem., Joural of Fiacial ad Quatitative Aalysis, September. Lee, S. B. ad Cho, H. Y., 992, A Rebalacig Disciplie for a Immuizatio Strategy, The Joural of Portfolio Maagemet, Summer, Maloey, K.J. ad Logue, D.E., 989, Neglected complexities i Structured Bod Portfolio, The Joural of Portfolio Maagemet, Witer. Navarro, E. ad Nave, J.M., 995, Aálisis de los factores de riesgo e el mercado español de Deuda Pública, Cuaderos Aragoeses de Ecoomía, Vol 5, Nawalkha, S.K. ad Chambers, D.R., 996, A Improved Immuizatio Strategy: M-Absolute, Fiacial Aalysts Joural, September/October. Nowma, K.B., 997, Gaussia Estimatio of Sigle-factor Cotiuous Time Models of the Term Structure of Iterest Rates, Joural of Fiace, 52, Reitao, R.R., 99, Multivariate Immuizatio Theory, Trasactios of the Society of Actuaries, XLIII, Reitao, R.R., 992, No-Parallel Yield Curve Shifts ad Immuizatio, The Joural of Portfolio Maagemet, Sprig. Sherris, M., 995, Iterest Rate Risk Factors i the Australia Bod Market, AFIR Iteratioal Colloquium, Vol 2, Steeley, J.M., 990, Modelig the Dyamics of the Term Structure of Iterest Rates., The Ecoomic ad Social Review, Vol 2, Stricklad, C.R., 993, Iterest Rate Volatility ad the Term Structure of Iterest Rates, FORC Preprit., Vol 93/37. Vasicek, O., 977, A Equilibrium Characterizatio of the Term Structure, Joural of Fiacial Ecoomics, Vol 5,

28 TABLE. ASSET CHARACTERISTICS. Maturity Coupo 2 Duratio 3 Asset 0.5 0% 0.5 Asset 2 0% Asset 3.5 0%.4297 Asset 4 2 0%.866 () Years. (2) Paid half-yearly. (3) Macaulay duratio. 28

29 TABLE 2. OPTIMAL STRATEGIES IN A STATIC FRAMEWORK. Pael NON STOCHASTIC MODEL x x2 x3 x4 DURATION Pael 2 VASICEK MODEL x x2 x3 x4 DURATION US UK Pael 3 CIR MODEL x x2 x3 x4 DURATION US UK () Portfolio duratio are calculated accordig to Macaulay, ad formulae [9] ad [2] for the o stochastic model, Vasicek model ad CIR model respectively. 29

30 TABLE 3. PARAMETER VALUES OF VASICEK AND CIR MODELS. These parameters were estimated by Nowma (997) usig a discrete time model which reduces some of the temporal aggregatio bias. The data used are US Treasury Bill oe moth yields from Jue 964 to December 989 ad oe moth sterlig iterbak rate from March 975 to March 995. Pael Vasicek model a g r 2 Curret r(t) US UK Pael 2 CIR model k m s 2 Curret r(t) US UK () April

31 TABLE 4. OPTIMAL STRATEGIES OF MINIMUM DISPERSION IN A STATIC FRAMEWORK a. Pael NON STOCHASTIC MODEL x x2 x3 x4 DURATION b Pael 2 VASICEK MODEL x x2 x3 x4 DURATION b US UK Pael 3 CIR MODEL x x2 x3 x4 DURATION b US UK (a) Dispersio measure is calculated accordig to Fog ad Vasicek M 2 formula (4). (b) Portfolio duratio are calculated accordig to Macaulay, ad formulae (9) ad (2) for the o stochastic model, Vasicek model ad CIR model respectively. 3

32 TABLE 5. OPTIMAL PORTFOLIO PATH UNDER A FLAT TERM STRUCTURE REGIME. The α value represets the level of trasactio costs as a percetage of the volume traded; α=0 meas the absece of trasactio costs. I this case the optimal strategy is cosistet with Khag's theorem, i.e., at each rebalacig poit the portfolio has to be restructured i order to keep its duratio equal to the remaiig HPP. Pael α = 0.00% s x(s,) x(s,2) x(s,3) x(s,4) DURATION a Pael 2 α = 0.5% s x(s,) x(s,2) x(s,3) x(s,4) DURATION a α = 0.30% s x(s,) x(s,2) x(s,3) x(s,4) DURATION a α = 0.45% s x(s,) x(s,2) x(s,3) x(s,4) DURATION a α = 0.60% s x(s,) x(s,2) x(s,3) x(s,4) DURATION a (a) Portfolio duratio are calculated accordig to Macaulay. 32

33 TABLE 6. OPTIMAL PORTFOLIO PATH UNDER VASICEK TSIR MODEL USING US DATA. The α value represets the level of trasactio costs as a percetage of the volume traded; α=0 meas the absece of trasactio costs. I this case the optimal strategy is cosistet with Khag's theorem, i.e., at each rebalacig poit the portfolio has to be restructured i order to keep its duratio equal to the remaiig HPP. Pael α = 0.00% s x(s,) x(s,2) x(s,3) x(s,4) DURATION a Pael 2 α= 0.5% s x(s,) x(s,2) x(s,3) x(s,4) DURATION a α = 0.30% s x(s,) x(s,2) x(s,3) x(s,4) DURATION a α= 0.45% s x(s,) x(s,2) x(s,3) x(s,4) DURATION a α = 0.60% s x(s,) x(s,2) x(s,3) x(s,4) DURATION a (a) Portfolio duratio are calculated accordig to formula (9). 33

34 TABLE 7. OPTIMAL PORTFOLIO PATH UNDER VASICEK TSIR MODEL USING UK DATA. The α value represets the level of trasactio costs as a percetage of the volume traded; α=0 meas the absece of trasactio costs. I this case the optimal strategy is cosistet with Khag's theorem, i.e., at each rebalacig poit the portfolio has to be restructured i order to keep its duratio equal to the remaiig HPP. Pael α = 0.00% s x(s,) x(s,2) x(s,3) x(s,4) DURATION a Pael 2 α = 0.5% s x(s,) x(s,2) x(s,3) x(s,4) DURATION a a = 0.30% s x(s,) x(s,2) x(s,3) x(s,4) DURATION a α = 0.45% s x(s,) x(s,2) x(s,3) x(s,4) DURATION a α = 0.60% s x(s,) x(s,2) x(s,3) x(s,4) DURATION a (a) Portfolio duratio are calculated accordig to formula (9). 34

35 TABLE 8. OPTIMAL PORTFOLIO PATH UNDER CIR TSIR MODEL USING US DATA. The α value represets the level of trasactio costs as a percetage of the volume traded; α=0 meas the absece of trasactio costs. I this case the optimal strategy is cosistet with Khag's theorem, i.e., at each rebalacig poit the portfolio has to be restructured i order to keep its duratio equal to the remaiig HPP. Pael α = 0.00% s x(s,) x(s,2) x(s,3) x(s,4) DURATION a Pael 2 α = 0.5% s x(s,) x(s,2) x(s,3) x(s,4) DURATION a α = 0.30% s x(s,) x(s,2) x(s,3) x(s,4) DURATION a α = 0.45% s x(s,) x(s,2) x(s,3) x(s,4) DURATION a α = 0.60% s x(s,) x(s,2) x(s,3) x(s,4) DURATION a (a) Portfolio duratio are calculated accordig to formula (2). 35

36 TABLE 9. OPTIMAL PORTFOLIO PATH UNDER CIR TSIR MODEL USING UK DATA. The α value represets the level of trasactio costs as a percetage of the volume traded; α=0 meas the absece of trasactio costs. I this case the optimal strategy is cosistet with Khag's theorem, i.e., at each rebalacig poit the portfolio has to be restructured i order to keep its duratio equal to the remaiig HPP. Error!Marcador o defiido.pael α = 0.00% s x(s,) x(s,2) x(s,3) x(s,4) DURATION a Pael 2 α = 0.5% s x(s,) x(s,2) x(s,3) x(s,4) DURATION a α = 0.30% s x(s,) x(s,2) x(s,3) x(s,4) DURATION a α = 0.45% s x(s,) x(s,2) x(s,3) x(s,4) DURATION a α = 0.60% s x(s,) x(s,2) x(s,3) x(s,4) DURATION a (a) Portfolio duratio are calculated accordig to formula (2) 36

37 TABLE 0.- DESCRIPTION OF THE DATA GENERATED BY THE SIMULATION OF THE OPTIMAL STRATEGIES WITH DIFFERENT LEVELS OF TRANSACTION COSTS TRANSACTION COSTS = 0 % Fial Values MINIMUM MAXIMUM MEAN STRATEGY STRATEGY STRATEGY STRATEGY TRANSACTION COSTS = 0 5 % Fial Values MINIMUM MAXIMUM MEAN STRATEGY STRATEGY STRATEGY STRATEGY TRANSACTION COSTS = % Fial Values MINIMUM MAXIMUM MEAN STRATEGY STRATEGY STRATEGY STRATEGY

38 FIGURE.-INTEREST RATE RISK ASSUMED BY OPTIMAL STRATEGIES 0, Strategy 4 HPP-DURATION 0,05 0-0,05 Strategy 3 Strategy 2 Strategy (immuizatio) -0, 0 0,25 0,5 TIME 0,75,25,5 38

39 FIGURE 2.- OUTCOMES OF OPTIMAL STRATEGIES TRANSACTION COSTS 0 % Strategy 700 Frecuecy Strategy 4 Strategy Strategy Fial portfolio Value

40 FIGURE 3.- EFFECT OF TRANSACTION COSTS ON OPTIMAL STRATEGIES 000 EFFECT OF TRANSACTION COSTS ON OPTIMAL STRATEGIES TRANSACTION COSTS % TRANSACTIION COSTS 0'5 %TRANSACTIION COSTS 0 % Strategy Strategy FRECUENCY Strategy 2 Strategy 3 Strategy Strategy Strategy 4 Strategy FINAL PORTFOLIO VALUE