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We Know That the Immunized Portfolio Doesn't Exist in Most Cases

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We Know That the Immunized Portfolio Doesn't Exist in Most Cases Alina Kondratiuk-Janyska1, Marek Kaluszka2 BOND PORTFOLIO IMMUNIZATION IN ARBITRAGE FREE MODELS Acta Universitatis Lodziensis. Folia Oeconomica, Łódź University Press, Accepted 2005 Abstract The aim of this paper is to examine arbitrage free models and investigate which of them imply well known and widely applied classical duration strategy. Problem of a noncallable and default-free bond portfolio immunization is studied in a 3-period model of time with a fixed investment strategy either under known optimization criterions such as maxmin, Bayesian, Gamma-maxmin, Markowitz-type and others. In some models there are anomalies since, it is proved that, any strategy is optimal. A notable fact is that the Markowitz approach is free from such anomalies and, moreover, in some cases occurs to be a duration strategy. Keywords: Bond portfolio; Immunization; Duration; Maxmin Procedure; Bayesian Method; Markowitz Criterion; Gamma-maxmin Approach JEL Classification: G11, G19 1. Introduction 1 Assistant in the Center of Mathematics and Physics, Technical University of Lodz ; Institute of Mathematics, Technical University of Lodz. E-mail address:[email protected] 2 Assistant Professor in the Institute of Mathematics, Technical University of Lodz 1 Early work on immunization was based upon the Macaulay definition of duration (1938) and it was shown by Samuelson (1945) and Redington (1952) that if the Macaulay duration of assets and liabilities are equal, the portfolio is protected against a local parallel change in the yield curve. Fisher and Weil (1971) formalized the traditional theory of immunization defining the conditions under which the value of an investment in a bond portfolio is hedged against any parallel shifts in the forward rates. The main result of this theory is that immunization is achieved if the Fisher-Weil duration of the portfolio is equal to the length of the investment horizon. A generalization of the Fisher and Weil results (1971) can be found in Rządkowski and Zaremba (2000). Unfortunately, this traditional approach has serious limitation since it implies arbitrage opportunity inconsistent with the rules of modern finance theory. To overcome it, Fong and Vasicek (1984), Nawalkha and Chambers (1996), Balbás and Ibánez (1998) estimated the lower bound of the change in portfolio value which leads to different risk controlling strategies. See also Kaluszka and Kondratiuk-Janyska (2004ab). The comprehensive treatment of the present state of the art can be found in Nawalkha and Chambers (1999) and Jackowicz (1999). The aim of this paper is to present immunization problem of a noncallable and default-free bond portfolio in a 3- period model of time referring to the Fong and Vasicek (1984), the Nawalkha and Chambers (1996), the Balbás and Ibánez (1998), the Balbás et al. (2002) studies among others. A fixed investment strategy is examined with respect to known optimization criterions: maxmin, Bayesian, Gamma-maxmin or completely new: Markowitz-type and others. It is expected to indicate which of them imply well known and widely applied duration strategy. However, in some models there are anomalies since, it is proved that, any strategy is optimal. The most crucial fact is that the Markowitz approach is free from such anomalies and, moreover, in some cases occurs to be a duration strategy. 2 2. Preliminary notations Denote by []0,3 the time interval with t=0 the present moment. Assume that two types of bonds are available: 1-year and 3-year default-free noncallable strip bonds. An investor is interested in acquiring a bond portfolio in order to discharge a liability in 2 years from today. Therefore his aim is to purchase an immunized portfolio against shocks in interest rates. We write a1 , a3 for the amounts of money spend on 1-year and 3-year bonds bought at time 0. By g(t) we mean the current instantaneous forward rate. 2 Hence the time-2 present value of a and a is A = a exp g s ds and 1 3 1 1 ∫ () 0 2 A = a exp g s ds , respectively. Obviously, 3 3 ∫ () 0 A1 + A3 = L (1) or equivalently 2 a + a = Lexp− g s ds . 1 3 ∫ () 0 The terminal value of a portfolio depends on two factors. The first one is connected with the way of reinvesting coupons paid before an investment horizon. The other appears when bonds are priced before their expiry dates in order to sell them. Therefore the portfolio terminal value depends on types of bonds, and investment decisions such as way of reinvesting coupons or the moment of rebalancing it. In the paper consider a strategy such that at t = 1 a bond portfolio is sold and immediately, an investor reinvests money by purchasing 1-year strip bonds. Hence the time-2 present value of a portfolio depends on a character of a single shock just before the moment when first bond expires. By g1()t denote the instantaneous forward rate over the time interval []1,t , that 3 t is, that investing $1 at time 1 in a zero-coupon bond we get exp g s ds at timet . ∫ 1() 1 Then at t = 2 the portfolio is valued at 1 3 3 2 V = a exp g()s ds + a exp g ()s ds − g ()s dsexp g ()s ds = 1 ∫ 3 ∫ ∫ 1 ∫ 1 0 0 1 1 2 3 = A exp g s − g s ds + A exp− g s − g s ds . (2) 1 ∫ ()1 () () 3 ∫ ()1 ()() 1 2 Introduce notation for shocks in the instantaneous forward rate i+1 ki = ∫ ()g1()s − g ()s ds, i = 1,2. i Then (2) simplifies to k 1 −k 2 V = A1e + A3e . (3) In the classic approach it is assumed that the term structure of interest rates is subject to parallel movements g1 ( s) = g( s)+ ε for all s , where ε is an arbitrary real number. Then ε −ε V = A1e + A3e ' and immunization V ≥ L is secured if Vε = 0 which implies that A1 = A3 . Clearly, this is a duration strategy since portfolio duration at t = 0 equals 2 and at t = 1 is 1. However, the above approach has serious limitation. The assumption of parallel shocks yields an arbitrage opportunity that is in contradiction to the rules of modern finance theory as well as empirical research. We present a few different sets of shocks and bring them together with different optimal criterions. What can indicate which of the presented models are proper and reflect real behaviour of interest rates is intensive empirical research. However, such an investigation is beyond of this paper. 4 3. Maxmin approach Viewing immunization as a maxmin strategy is widely considered (see e.g. Bierwag and Khang (1979), Prisman (1986) among others) but the question of determining an appropriate class of shocks on the interest rates is still unanswered. Consider shocks such that k 1 ≤ ε 1 and k 2 ≤ ε 2 . One of the arbitrage free approaches to the immunization problem is a maxmin strategy relying on finding −ε1 −ε2 −ε1 −ε2 −ε 2 max min V = max(A1e + A3e )= max (A1 (e − e )+ Le ). (4) ()Ai k1,k2 ()Ai 0≤A1≤L In particular, • if ε 2 > ε1 , then a maxmin portfolio is A1 = L , A3 = 0 • if ε 2 < ε1 , then a portfolio A1 = 0 , A3 = L is maxmin • if ε 2 = ε1 , then an arbitrary portfolio is maxmin Remark 1. Applying (4) we get at t = 2 under the worst scenario for ε 2 > ε1 that max min V = Le−ε1 < L , ()Ai k1,k2 otherwise max min V = Le−ε2 < L . ()Ai k1,k2 The maxmin strategy does not hedge a bond portfolio perfectly against changes in the 2 interest rates. To assure it, one should invest more than Lexp− g s ds at t = 0 . ∫ () 0 Remark 2. The above approach secures a bond portfolio against the biggest loss in the portfolio’s value. Such a situation happens extremely rare because only for k 1 = −ε 1 , k 2 = ε 2 . 5 Now focus on another class of shocks such that 2 2 2 k1 + k2 ≤ ε . Then (4) yields max min (A ek1 + A e−k2 ) 2 2 2 1 3 A1+ A3 =L k1 + k2 ≤ε and a lower bound on V is estimated applying either the inequality e x ≥ 1+ x or the Jensen inequality. Here, using the latter tool we get A A A k − A k k1 −k2 1 k 1 3 −k 2 1 1 3 2 A1e + A3e = L e + e ≥ Lexp L L L under the condition A1, A3 ≥ 0. Thus k1 −k2 max min ()A1e + A3e ≥ L exp max min ()A1k1 − A3k2 . A + A =L 2 2 2 A + A =L 2 2 2 1 3 k1 + k2 ≤ε 1 3 k1 + k2 ≤ε A1, A 3≥0 A 1, A3 ≥0 Since by the Cauchy-Schwarz inequality 2 2 2 2 A1k1 − A3k2 ≥ − A1 + A3 k1 + k2 , we get 1 max min ()A ek1 + A e−k2 ≥ L exp − min A2 + A2 max k 2 + k 2 2 2 2 1 3 1 3 2 2 2 1 2 A 1+ A 3=L k1 + k2 ≤ε L A1+ A 3=L k1 + k2 ≤ε A , A ≥0 1 3 A 1, A 3≥0 1 2 2 ε L ε = Lexp− min A1 + ()L − A1 ε = Lexp− = Lexp− . L 0≤A 1≤L L 2 2 Summing up, if a bond portfolio is sold at t = 1 and for received money an investor purchases 1-year zero-coupon bonds expired at t = 2 , then ε max min V ≥ Lexp− . (5) A + A =L 2 2 2 1 3 k1 + k2 ≤ε 2 A1, A 3≥0 6 Analysis of equality conditions in the Jensen and the Cauchy-Schwarz inequalities L implies that equality in (5) is attained if and only if A = A = .
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