Departamento de Economfa de la Empresa Business Economics Series 06

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1 ------_.. Wrking Paper Departament de Ecnmfa de la Empresa Business Ecnmics Series 06 Universidad Carls III de Madrid Nvember 994 Calle Madrid, Getafe (Spain) Fax (34) WHEN CAN YOU IMMUNIZE A BOND PORTFOLIO? Alejandr BalMs de la Crte and Alfred Ibanez Rdrfguez Abstract _ The bject f this paper is t give cnditins under which it is pssible t immunize a bnd prtfli. Maxmin strategies are als studied, as well as their relatins with immunized nes. Sme special shcks n the interest rate are analyzed, and general cnditins abut immunizatin. are btained. When immunizatin is nt pssible, capital lsses are measured. Key Wrds and Phrases Immunized prtfli;,maxmin prtfli; Weak immunizatin cnditin; The set f wrst shcks. Balbas, Departament de Ecnmfa de la Empresa de la Universidad Carls III de Madrid, and IMiiez, Departament de Ecnmfa de la Empresa de la Universidad Carls III de Madrid. Research partially supprted by DGICYT PS94/0050

2 Intrductin The bject f this paper is t present a general framewrk including and hmgenizing the main results in standard literature related t financial immunizatin, as well as t give a new necessary and sufficient cnditin (the s called weak immunizatin cnditin) t guarantee the existence f a immunized prtfli. Paper's utline is as fllws. First sectin establishes a minimal set f hyptheses which are cmmn t mst f mdels, and frm them, the existence f maxmin prtflis and the weak immunizatin cnditin are prved. Once we have fund cndi tins under which immunized prtflis exist, we will devte the secnd sectin t characterize them. As in previus literature, we will prve that immunized prtflis are thus fr which the (s called) wrst shck is the null shck and then, a special kind f differential must be zer. This permi ts t' btain the classical duratin measures and sme new nes, depending f the pssible shcks. We will intrduce the set f wrst shcks in sectin three and will find general expressins which give us hw much mney we culd lse if immunizatin were nt pssible. All the btained results are applied in furth sectin t analyze classical results abut immunizatin. T be precise, we will study the results f Bierwag and Khang (979) Prisman and Shres (988) Fng and Vasicek (984) and thers authrs, and will find new expressins abut financial immunizatin. Finally, we pint up the mst imprtant cnclusins f the paper in sectin five. I The weak cnditin and maxmin prtflis. Let [D,T] be the time interval being t=d the present mment. Let us cnsider n default free and ptin free bnds with maturity less r equal than T, and with prices P,P,...,P respectively. 2 n We will represent by K the set f the admissible shcks ver the interest rate, and therefre, K will be a subset f the vectr space f real valued functins defined n [D,T]. If the elements f K are nly cnstant functins we will be wrking with additive shcks. If these elements are plynmials we will have plynmial shcks like the nes cnsidered by Prisman and Shres (988) r Chambers et al (988) between thers, and if these elements are cntinuusly differentiable functins we are under the hypthesis f Fng and Vasicek (984). Clearly, mre situatins abut the functins in K may be cnsidered. Let.m (D<m<T) be the investr planning perid and cnsider n real valued functinals V :K--~)R i=,2,...,n such that V (k) (where kek is any admissible shck) is the i-th bnd value at time m if the shck k takes place. We will assume the fllwing hyptheses 2

3 Hl K is a cnvex set which cntains the zer shck (dented by k=o). H2 V is a cnvex functinal fr i=,2,...,n H3 There exists a cnstant R>O such that V (O) = RP i=,2,...,n H4 V (k»o fr i=,2,...,n and fr any kek. Fur assumptins are quite clear and simple, and they almst always hld in classical mdels abut immunizatin. In particular, assumptin H3 means that if n shck ver the interest rate takes place, then the i-th bnd value is prprtinal t its present price. If C>O represents the ttal amunt t invest, and the vectr q=(q,q,... q ) 2 n gives us the number f units (q) f the i-th bnd that the investr has bught, then the cnstraints n Lqt = C q~o i=,2,...,n (,) = must hld, and the functinal V(q,k) =[ qv(k) (,2) = gives us the value at time m f prtfli q if the shck k takes place. Obvfusly, V is a cnvex functinal in the k variable since H2 guarantees that it is a nn negative linear cmbinatin f cnvex functinals. The fllwing result nly means that if there is n shck ver the interest rate, then the value (at m) f prtfli q is prprtinal t the capital C. Prpsitin.. V(q,O)=RC fr any q subject t (,). Prf. Frm the assumptins we have n = [ V(q,O) = L qv(o) q RP = RC = = Because f latter prpsitin we will say that RC is the prmised value and it is cmmn fr all feasible prtflis q (that is, prtflis q such that (,) hlds). Let us intrduce the guaranteed value by prtfli q which will be V(q) = Inf {V(q,k);keK} (,3) that is, the infimum f all pssible values (at m) f prtfli q depending f the shck kek. The fllwing result shws that the prmised amunt is greater than the guaranteed ne. Prpsitin.2. The fllwing inequalities hld fr any feasible prtfli q s V(q) s RC 3

4 Prf. First inequality fllws frm H4 and frm H we have Y(q) = Inf {V(q,K);keK} :5 V(q,O) = RC A prtfli is called maxmin if it guarantees as much as pssible and immunized if it guarantees the prmised amunt RC. T intrduce this cncepts in a frmal way we will cnsider the ptimizatin prgram Max Y(q) ) subject t (,) (PU Definitin. 3. A feasible prtfli q is maxmin if it slves prgram P, and immunized if Y(q) = RC. Therem.4. If q is a immunized prtfli, then it is maxmin. Prf. If q is a immunized prtfli then Y(q) = RC and applying prpsitin.2. t any feasible prtfli q' we have V(q') :5 RC =Y(q) and therefre q slves P. Latter result has been prved in a extrardinarily simple way because f the apparent pwer f the intrduced ntatin. It was established at first time by Bierwag and Khang (979) fr a prblem in which the bnds pay a cntinuus cupn, the shcks are additives and there are tw bnds with duratin greater and less than m respectively. Later, Prisman (986) generalized the result f Bierwag and Khang. We are ging t study cnditins under which a immunized (r maxmin) prtfli exists. First we need the fllwing lemma. Lemma.5. Let ~ ~ O. Then, there is a feasible prtfli q such that Y(q) ~ if and nly if fr every admissible shck kek there is at least a bnd i (which depends f k) such that V (k) ~ ~ P 0 ~ C Prf. Let us assume the existence f prtfli q. Then V(q,k) ~ ~C fr any admissible shck k. Frm (,) and (,2) n n \ q V (k) ~ \ q ~ P t.. t.. 0 = = fr any k. Since the terms in bth sides f last inequality are nn negative, this is nly pssible if at least fr ne i we have V (k) ~ ~ P 0 Cnversely, let us cnsider that the given cnditin hlds and 4,

5 let us prve the existence f q prtfli. The fll~ing set is bviusly cnvex A = {(a: a: 2... a: ); a:j~j.lpj j=,2,... n}, n Cnsider als the set B = {(a:,a:...,a:); 3keK with a: ~ V (k) j=,2...,n} 2 n J J Let us prve that B is a cnvex set. In fact. if (a:, a:...,a: ) 2 n and (~.~... ~ ) are in B, we can find tw shcks k and k' in K 2 n such that a: ~ V (k) ~j ~ J V/k'),j=l,2,... n j sine K is a cnvex set. given T with 0 ~ T ~, Tk+(l-T)k' is in K and being V a cnvex j functinal fr any j, we have that Ta: + O-T)~ ~ TV (k) + O-T)VJ(k') ~ V [Tk + (l-t)k'] J J J j j=l,2,...,n and T(a:,a:,...,a:) + (l-t)(~,~,...,~ ) is in B. 2 n 2 n We will prve nw that there are n pints in A O (interir f A) and B simultaneusly. In fact, if (a:, a:,...,a: ) were in bth A 2 n and B, then a: < J.l P j=,2,..,n and we culd find a shck k such J J that a: ~ V (k) j=l, 2,... n J j Therefre J.l P > a: ~ V (k) j=,2,...,n J J J and it is a cntradictin with the assumptins. The separatin therems (see Luenberger (969) ) shw that we can find n real numbers q',q',...,q' such that q' is nt zer fr 2 n at least ne i and n Lq' a: J=l J J if (a:,a:,...,a: ) is in A and (~' ~,...,~) is in B. In 2 n 2 n particular, taking a: = J.l P and ~ = V (k)+r j=,2,...,n where J OJ J J j k is any admissible shck and r is any nn negative number, J n n J.l Lq' P ~ L q' (V (k) + r ) 0.4) J 0J=l J J J=l J J if k is admissible and r ~ 0 j=,2,...,n J We have q' ~ 0 because if we had q <0 then the right side in last inequal i ty wuld tend t minus infinite if r tends t infinite and this is nt cmpatible with the inequality. Analgusly q'~o, 2...,q ~O. Since at least ne q' is nt zer, n 5

6 and then, taking n 5 = Lq' P > 0 J=l J J qj = Cq~ / 5 j=,2,...,n that (q,q,...,q ) verifies (,) and frm (,2) and (,4) (with 2 n r = 0 fr any j) J fr any shck k j.l C ~ V(q,k) As it has been shwn, the lemma is prved with technics f cnvex analysis, which were applied t immunizatin thery in Prisman (986 ). The first interesting cnsequence f lemma.5. is that under the hypthesis Hl H2 H3 and H4 ne can always find a maxmin prtfli. Therem. 6. Prgram Pl has slutin, that is, there always exists a maxmin prtfli. Prf. Let us cnsider the fllwing real valued functinal ver the admissible shcks U(k) = Max { V (k)/p ' V (k)/p '..., Vn(k)/P } 2 2 n fr kek. Define j.l = Inf {U(k);keK} Then, fr any shck k we have U(k) ~ j.l and then there exists a bnd (which depends f k) such that V (k) i > -P- - j.l i The latter lemma shws that we can find a prtfli q such that fr any kek and then We V(q,k) ~ V(q) = Inf {V(q, k); kek} ~ j.l C will have prved that q is a slutin f Pl if we shw that V(q') s j.l C fr any prtfli q'=(q',q',...,q') subject t 2 n (, ). Clearly, fr any feasible shck k we have j.lc v (q') = C U(k) Therefre V (q') ~ C Inf{U(k); kek} = Cj.l Let us intrduce nw the "weak immunizatin cnditin". n ~U(k)Lq't = 6

7 -: Definitin.7. We will say that the set f the admissible shcks K and the n bnds cnsidered verify the weak immunizatin cnditin if fr any shck kek there exists al least ne bnd i (which depends f k) such that V (k) ~ RP The interpretatin f the latter cncept may be as fllws. Let us cnsider a investr interested in a immunized prtfli, that is, a prtfli which guarantees the prmised amunt RC. Then, if ur investr knew the real future shck k then he (r she) wuld buy that bnd which des nt lse value, that is, the bnd such that V (k) ~ V (0) = RP (see assumptin H3). If the investr can find this bnd fr any feasible shck, then we have the "weak immunizatin cnditin" and this name is because if it hlds and we knw the future shck then we can immunize. Nw we are ging t present a surprising result which shws that immunizatin is pssible under the weak immunizatin cnditin (f curse, withut assuming that we knw the future shck k). This is perhaps the mst imprtant result in the present paper and will be applied in. future sectins t explain why immunizatin is nt pssible in sme classical mdels. We will als present situatins in which immunizatin is viable and will intrduce the cncept f "set f wrst shcks". Therem. 8. The weak immunizatin cnditin is necessary and sufficient t guarantee the existence f a immunized prtfli. Prf. It is a immediate cnsequence f lemma.5. taking ~ =R Latter therem has anther interpretatin. "Immunizatin is nt pssible if and nly if there is an admissible shck fr which all the bnds lse value at m" Let us remark that therems.6. and.8. shw that the cnverse f therem.4. is false in general. In fact, the maxmin prtfli (i.e. a prtfli that makes maximum the guaranteed amunt at time m) always exists but it will be seen that the weak immunizatin cndi tin is nt always satisfied, and then immunized prtfli des nt exist. Mrever, it is well knwn that in classical literature ne can find many mdels in which immunizatin is nt pssible. Anyway, it can be easily prved in ur general cntext that if immunized prtfli des exist then immunized and maxmin prtfli are equivalent cncepts. Lking fr immunized bnd prtflis Once we have characterized the existence f immunized prtflis, we will shw in this sectin a way t find them. Fr it, given a feasible prtfli q we cnsider the fllwing ptimizatin prgram Min V(q, k) ) (Pq) subject t kek 7

8 Many authrs like fr instance Bierwag (977), Bierwag and Khang (979), etc, prve in different mdels (with cntinuus r discrete capitalizatin, with additive, multiplicative, plynmial, differentiable shcks, etc) that the immunized prtflis have the zer shck as the "wrst shck". The fllwing result shws that in a general framewrk, like the ne we are wrking with, the prperty is als valid. Prpsitin 2.. A feasible prtfli q is immunized if and nly the zer shck k=d slves the prgram Pq. Prf. If k=d slves Pq, then V(q,k) i!: V(q,D) fr all kek and frm prpsitin.. V(q,k) i!: RC Therefre V(q) = Inf{V(q,k);keK} i!: RC and q is immunized since the ppsite inequality fllws frm prpsitin. 2. Cnversely, if q is immunizedv(q) = RC and given any kek we have V(q,k) i!: Inf{V(q,k' );k'ek} =V(q) = RC = V(q,D) and the zer shck slves prgram Pq Latter prpsitin may be useful t btain extensins f therem.8. We are ging t d it applying the lcal-glbal therem f mathematical prgramming (see fr instance Luenberger (969» which shws that fr a cnvex ptimizatin prgram (like prgram (Pq) ) the cncepts f lcal minimum and glbal minimum are equivalent. Frm nw n we need t assume the fllwing additinal assumptin which almst always hlds in classical mdels. HS The set K f admissible shcks is a subset f a nrmed space X whse elements are real valued functins ver the interval [D,T]. The cncept f nrmed space can be fund fr instance in Luenberger (969), and examples f X culd be CP[D,T] (functins with p cntinuus derivatives) r LP[D,T], that is, the space f measurable functins f: [D,T] ~R such that T J \f(t)i P dt < ~ where p is a fixed natural number such that pi!:. pssibilities fr X can be cnsidered. Anther many Prpsitin 2.2. There exists a immunized prtfli q if and nly if there exists V neighbrhd f zer in X such that KnV and the n cnsidered bnds verify the weak immunizatin cnditin. Prf. The given cnditin is bviusly necessary since we can take V as the whle space X and apply therem.8. Cnversely, if the neighbrhd V exists, then we can cnsider that V is cnvex, and therem.8. guarantees that there exists a 8

9 feasible prtfli q such that V(q.k) ~ RC =V(q.O) hlds fr keknv. Then. the zer shck is a lcal slutin f prgram (Pq) and the lcal-glbal therem shws that it is a glbal slutin. Therefre, the result fllws frm prpsitin 2.. Latter result has tw interpretatins. First ne is as fllws "immunizatin is pssible if (and nly if) fr any arbitrary small shck k we can find a bnd (which depends f k) which des nt lse value after the shck k". The secnd ne may be written in the fllwing way "immunizatin is nt pssible if and nly if we can find a shck as small as wanted fr which all the bnds lse value at m" Anther interesting cnsequence f prpsitin 2.. is that we can apply the necessary ptimality cnditins t prgram (Pq) and characterize the immunized prtflis. Let us remark that these necessary ptimality cnditins are als sufficient since this prgram is cnvex. The cncepts and results abut ptimizatin which will appear frm nw till the end f this sectin can be fund in Luenberger (969). Frm nw n let us assume the fllwing hypthesis H6 The functinals V i=,2,...,n are Gateaux differentiable with respect t their variable k in an pen set cntaining the zer shck. Latter hyptheses may be written mre easily if we cnsider shcks k which depend f p parameters (fr instance plynmial shcks with p- degree). If this dependence is linear, then H6 means that V is differentiable with respect t the parameters. Therem 2.3. If the zer shck is interir t the set K f admissible shcks, then q is a immunized prtfli if and nly if BV(q, k) I = 0 Bk k=o where the left side term represents the Gateaux differential f the functinal V with respect t its variable k evaluated in k=o. Prf. It is a immediate cnsequence f prpsitin 2.. Latter expressin may be develped in very general situatins. T shw it we are ging t btain equivalent cnditins in the case f cntinuus capitalizatin. The q prtfli pays a cntinuus cupn c(t)~o fr O~t~T. If g(s) (O<s<T) represents the instantaneus frward interest rate and k(s) is a shck n g(s). then the q prtfli value at time m is given by T V(q,k) =Icltlexp[ r:(gis)+kisllds dt (2.) 9

10 It is easily prved in this case that the cnstant R is given by R = exp ( ~(S)dS ) (2,2) and manipulating in (2,) we have T V(q,kl = RJ C(tlexp[-J:g(SldS + I:k(SldS]dt (2,3) The differential f functinal V with respect t its variable k evaluated in the zer shck and applied ver the shck k (that is, the derivative f functinal V evaluated n the zer shck and in the directin given by the shck k) will be given by (2,4) Therem 2.4. The q prtfli is immunized if and nly if ne f the tw fllwing equivalent cnditins hlds fr any admissible shck k. T J c(tl exp [-J:g(SldS ]~(SldS dt 0 (2,5) = (2,6) fr any admissible shck k. Prf. (2,,5) is a immediate cnsequence f (2, 4) and therem and (2,6) fllws frm (2,5) if we change the integratin rder - Expressin (2,5) allws us the fllwing lecture. The sum (in this case integral) f the current value f each cupn multiplied by the shcks which affect it must be zer. A cupn at time t is affected by the shcks in the interval [t,m] 0

11 _.._---_.._ , Expressin (2,6) allws us anther lecture. The sum (integral) f the shcks till m multiplied by the current value f the cupn stream that the shck affects must be equal t the sum (integral) f the shcks frm m till T multiplied by the current value f the cupn stream that the shck affects. Expressin (2,5) and (2,6) give us lectures n the immunizatin cnditin in mre intuitive terms that therem 2.3. Expressin (2,5) and (2,6) can be applied in very general situatins (that is, all situatins in which assumptins H - H6 hld) t find the immunized prtfli q if it exists, that is, if weak immunizatin cnditin (r lcal weak immunizatin cnditin) hlds. III The set f wrst shcks Once we knw when immunizatin is pssible and hw t find the immunized prtflis if they exists, fllwing the apprach f Prisman and Shres (986) r Fng and Vasicek (984) we are nw interested in measuring hw much mney we culd lse if immunizatin were nt pssible. First step is t intrduce a basic cncept Definitin 3.. We will say that a set k,k,...,k f admissible 2 h shcks is a set f wrst shcks, if given any shck kek there exists h real numbers A,A,...,A and 2 h h \ A k e K L j j j = ( h) (which depend f k) such that V (k) ~ V L A k i=l, 2,...,n j= j j The latter cncept just means that the value at m f the n cnsidered bnds is always minred by their values cnsidering linear cmbinatins f elements in the set f wrst shcks. A immediate cnsequence is the fllwing ne. Its prf is very simple and mitted Prpsitin 3.2. Under the assumptins f the latter definitin, the fllwing inequality hlds fr any feasible prtfli q (that is, fr any q subject t (,)). As it is well knwn, a cnvex functin is always minred by its tangent plane. It is als true fr cnvex functinals in nrmed spaces (see Luenberger (989) ) and we can apply this fact t btain sme prperties f the functinal V.

12 Let v j~,2,...,h be the value f the Gateaux differential f V j with respect t its variable k, evaluated at k=o, and applied ver k j (see Llienberger (969). See als (2,4». Then we have the fllwing result Therem 3.3 Under the assumptins f prpsitin 3.2. the fllwing inequality hlds V(q,k) - V(q,O) ~ h L j=l v A j j Latter inequality gives us the minimum value f V(q,k) if we cnsider shcks such that A, A,. A (which depend f k) are 2 h bunded. Since V(q,O) = RC (the prmised value) the inequality may als be written as fllws V(q,k) - RC RC (3,) and we are measuring the lsses per prmised dllar. It will be shwn in next sectin that in classical mdels the cefficients v IRC can be interpreted as duratin measures, thugh in the j general cntext we are wrking with, many different situatins culd be cnsidered. IV Sme interesting particular situatins Let us apply the develped thery t sme particular interesting cases. First, we will analyze sme classical results, and later, we will give new prperties. Let us cnsider that the admissible shcks are plynmial shcks with degree nn greater than p. Their general frm becmes K(s) ~ A + A s + A s A sp 2 P A base f the space f p degree plynmials is k (s) ~, k (s) = s,..., k (s) ~ sp (4,) P and any shck may be written as a linear cmbinatin f k,...,k. P It may by easily prved that (2,5) hlds fr any shck k if it hlds fr any shck in the base. Clearly and frm (2,5) 2

13 T -Jt J g(s)ds c(t) e 0 (m + _ t + )dt = 0 (4,2) i=o,,2,... p Manipulating + m = C what may be written as t T -J Jt"'c(t) g(s)ds e 0 dt i=o.l... p (4.3) m + = D i=o,...,p (4,4) + where D is i-th duratin f q prtfli and is given by the + right side member f (4.3). That cnstitutes the duratin measures vectr btained by Chambers et al. (988) and characterizes the immunized prtflis. Actually, expressin (2,5) is quite general and therefre. many ther situatins culd be cnsidered, and anther results wuld be btained. If we nly cnsider additive shcks, we are in the latter case with p=o and (4,3) becmes T Jt J - g(s)ds m = C 0 tc (t) e 0 dt (4,5) which is the standard result n immunizatin against additive shcks. See, fr instance Fisher and Weil (97) r Bierwag (977). Bierwag and Khang (979) prved that a bnd with duratin greater than m increases its value (at m) if we have a negative additive shck. Mrever if the bnd has duratin less than m it increases its value if the shck is additive and psitive. Nw, if we have the tw needed bnds we are under the hyptheses f therem.8. and then we can cnclude (like Bierwag and Khang did) that a immunized prtfli des exist because weak immunizatin cnditin hlds. Prisman and Shres (988) prved that (4,3) has n slutin if p~l, and then immunizatin against plynmial shck is nt pssible. Their prf is based in analytic arguments and we are ging t give anther ne with a very simple interpretatin. An example f admissible plynmial shck is given by k (t) = A(t-m) where A is any psitive number. Since K (t)<o if t<m and k (t»o if t>m we have that the instantaneus frward interest rate is ging t fall frm t=o till t=m and it is ging t increase fr t>m. Then, the cupns we have t capitalize (the 3

14 cupns paid befre m) will lse value at m and s will the nes we have t discunt (the nes paid later than m). In this situatin nly the zer cupn bnd wuld nt lse value at m, and if this bnd is nt in the market, t anticipate the shck k des nt permit t save it. Therefre, weak immunizatin cnditin fails and immunizatin is nt pssible. Latter case will appear very ften (nt nly fr plynmial shcks) because in many situatins we can find a shck k(t) such that k(t)<o if t<m and k(t»o if t>m. This fact shws that weak immunizatin cnditin may be mre apprpriate t study the existence f immunized prtflis than analytic technics. Cming back t the case f plynmial shcks, since immunizatin is nt pssible we culd be interested in the ttal amunt f mney we might lse. In such a case, we can apply the results f sectin three. The set given in (4,) is a set f wrst shcks since it is a base f admissible shcks. The cefficients Vi f (3, ) are given, In 'th'is case b y (see als (2,4)) RC(m + _ D ) and (3,) becmes + V(q,k) - RC ~f ;\ (m + - D )/(i+) RC + (4,6) where ;\ i=o,l,...,p are the cefficients f plynmial k. The expressin was btained by Prisman and Shres (988). If we cnsider shcks which are functins with p (p~l) cntinuus derivatives being p any impair number, then Taylr's frmula applied t any admissible shck k gives us k(t)~k (t) if t<m and k(t)~k (t) if t>m, where and =0 k (m) ;\ --:-.":'""",- (t-m) + -, (t-m)p (4,7). p. O~t~T} (4,8) Therefre the value f prtfli q (fr any q) if shck k takes place is minred by its value if k is the future shck. It means that the set f (4,) is a set f wrst shcks in this case, and (4,6) may be applied t btain the maximal lsses. In particular, if p=l then k (t) = k(m) + ;\t - ;\m ;\ = Max {k' (t); 0 ~ t ~ T} V(q,k ) - RC V(q,k) - RC ~ ~ (m-d )(k(m)-m;\) + (m 2 -D );\/2 RC RC 2 (4,9) and taking q such that m = D (that is, immunized against 4

15 additive shcks) V(q,k) - RC RC which is the Fng and Vasicek frmula. Fng and Vasicek frmula has been extended in Mntrucchi and Peccati (99) (see als Shiu (987) ) t the case f nn 2 differentiable shcks. The authrs shw hw (m - D ) gives us the 2 maximal lsses if m - D = 0 and we change A by the Dini' s derivative f the shck k. Fng-Vasicek and Mntrucchi-Peccati frmulas can be applied when the shck derivative (r its Dini's derivative) can be bunded. If these derivatives are bunded, then the shck variatins <I k (t )-k(t ) ; 0 ~ \ ' t ~ T} 2 2 are als bunded but the cnverse is false in general ( small shcks culd have "very big" derivative). Nw we will apply (3,) t btain a new expressin fr maximal lsses. This expressin gives us the lsses depending f the variatins f shck k, and in practical situatins it culd be easier t determine the shck variatins than its derivative. Let us assume that the set K f admissible shcks is the set f bunded and integrable functins in [O,T]. Let us define [ if t~m if t~m k (t) = k (t) = [ 0 if t>m 2 if t>m (4,0) Then {k,k} is a set f wrst shcks. In fact, given any 2 admissible shck k, cnsider A = Inf {k(t) ; 0 ~ t ~ m} A = Sup {k(t); m ~ t ~ i} 2 ( 4, ) and clearly k(t) ~ A k (t) + A k (t) if t<m and the ppsite 2 2 inequality hlds fr t>m. Therefre V(q,k) ~ V(q,A k + A k ) and 2 2 definitin 3.. hlds. (2,4) shws that t -Jg(S)dS V' (q, 0, k ') = R c(t)e (m-t)dt [ (4,2) and 5

16 -Jtg(S)dS T v' (q, 0, k ) = R c(t)e 0 (m-t )dt 2 I m (4,3) Let us cnsider the present values f the cupns paid befre m and after m respectively, which are given by C' = c(t)e 0 dt t -J g(s)ds T -Jt g(s)ds [ C" =Ic(tle 0 dt m (4,4) and if these numbers are nt zer let us define the duratin f cupns paid befre m and after m respectively by.[ -J t g(s)ds -Jt g(s)ds + T D' = tc(t)e 0 dt D" = +ItcCtle 0 dt m (4,5) Then (4,2) and (4,3) can be written as v' (q, 0, k ) = RC' (m-d' ) V' (q, 0, k ) = RC', (m-d', ) 2 and therefre, frm (3,) we have V(q,k) - RC RC (4,6 ) where k is any admissible shck and A (i=,2) is given by (4,). Latter expressin can be rewritten since it is clear that C=C'+C" and CD = C'D' + C' 'D" and therefre ' and C' C', - (m-d') = (m-d) +--(D' '- m)- (4,7) C C V(q,k) - RC C',?:; A (m-d) + - (D' '- m) (A - A ) RC C 2 (4,8) what gives us anther bundlessness fr capital lsses which may be applied fr very general shcks. Let us remark that if we immunize against additive shcks (that is, if we take m = D ), then the lsses are measured by C', -C- (D" - m) (A - A ) (4,9) 2 and a pssible strategy (alternative t the Fng and Vasicek r Prisman and Shres nes) culd be t minimize C" (D" - m). This strategy is t minimize a dispersin measure (like the Fng and 6

17 Vasicek ne) because it is clear that if m = D then frm (4,7) C" (D"- m) = C' (m - D') Anther pssibilities culd be analyzed, since in all cnsidered situatins abut the shcks (plynmial, cntinuusly differentia,ble, bunded and integrable) and in thers that we culd have studied, weak immunizatin cnditin shws that immunizatin against shcks in many cnvex sets is pssible (that is, immunizatin against shcks in a cnvex set which des nt cntain a shck negative fr t<m and psitive fr t>m can be pssible if there are apprpriate bnds in the market). Then, the investr can immunize against shcks in these sets if it is cnsidered that they describe the reasnable changes in the interest rate. We always have that (3,) gives the maximal capital lsses if the real shck des nt belng t the cnsidered set. V Cnclusins I. Under a simple set f assumptins a maxmin prtfli exists and the weak immunizatin cnditin guarantees the existence f a immunized prtfli.. This cnditin shws quite well why immunizatin is nt pssible in many mdels. The reasn is that ne can find a shck k such that all the bnds lse value at the investr planning perid m. This shck k (if it exists) may be fund as small as wanted". Ill. T find this k (in many mdels) ne Just have t lk fr a shck such' that the interest rate falls befre m and increases after m. IV. If the latter shck k des exist, the pssible capital lsses can be measured by mean f the set f wrts shcks. This set is given fr a finite number f shcks (K i =,2,...,h) such that their linear cmbinatins can verify h \ A k (t) ~ k(t) if t ~ m and \ A k (t) L L = = h ~ k(t) if t ~ m Then (3,) gives the maximal lsses in a general cntext. V. Expressin (4.8) is a particular case f (3,) if we are wrking with bunded and integrable shcks. It can be applied fr any prtfli and therefre, a pssible strategy fr the investr culd be t immunize against additive shcks and t minimize C" (D' '-m). This strategy is t minimize a dispersin measure, and it is nt incmpatible with including a bnd with m maturity (empirical result appeared in Bierwag et al. (993)) due t this bnd, which pays the biggest amunt at the instant m, seems t help in minimizing any dispersin measure. Actually,. t minimize the dispersin measure C" (D' '-m) r t minimize the Fng-Vasicek ne D - m 2 (r bth if it is pssible) 2 depends f the investr pinin abut the changes in the interest rate. First ne gives capital lsses depending f the shck variatins, and secnd ne can be applied if we anticipate the 7

18 shck derivative. VI. The investr can als wrk with reasnable assumptins abut the pssible shcks n the interest rates. Then, maxmin prtfli (which always exits) can be determined, and later, (3,) (r its cnsequences (4,6), (4,9), (4,8), etc) can be applied t measure the pssible capital lsses in different cntexts abut the shcks. References Bierwag. G.O. (977) Immunizatin, Duratin, and the Term Structure f Interest Rates. Jurnal f Financial and Quantitative Analysis. December, Bierwag G.O. and Khang C.(979). An Immunizatin Strategy is a Maxmin Strategy. The Jurnal f Finance XXXVII May Bierwag G.O. Fladi I. and Rberts G.S. (993). Designing an Immunized Prtfli is M-squared the Key? Jurnal f Banking and Finance Chambers D.R. Carletn W.T. and McEnaly (988) Immunizing Defaul t-free Bnd Prtflis with a Duratin Vectr. Jurnal f Financial and Quantitative Analysis 23,, Fisher. and Weil R. (97) Cping with Risk f Interest Rate Fluctuatins Returns t Bndhlders frm Naive and Optimal Strategies. Jurnal f Business 52, 5-6. Fng W.J. and Vasicek G.A. (984) A Risk Minimizing Strategy fr Prtfli Immunizatin. Jurnal f Finance Luenberger D.G. (969) Optimizatin by Vectr Space Methds. Jhn Wiley & Sns Inc. Mntrucchi,. and Peccati. (99). A Nte n Shiu-Fisher-Weil Immunizatin Therem. Insurance Mathematics and Ecnmics 0, Prisman E.Z. (986) Immunizatin as a Maxmin Strategy: a New Lk. Jurnal f Banking and Finance 0, Prisman E.2. and Shres M.R. (988). Duratin Measures fr Specific term Structure Estimatins and Applicatins t Bnd Prtfli Immunizatin. Jurnal f Banking and Finance, 2, Shiu S.W. (987) On the Fisher-Weil Immunizatin Therem. Insurance Mathematics and Ecnmics 6,