Theory and Methodology
A maxmin policy for bond management

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Abstract

A simple immunization problem is formulated as a maxmin optimal control problem and analytically solved by means of dynamic programming. The optimal control law, namely the immunization policy, turns out to be quite different from any duration-based immunization policy. Moreover, it is seemingly able to discriminate between favourable and unfavourable changes in the yield curve.

Introduction

Since bond markets can be more volatile than stock markets, risk control is a key issue in bond management (see, e.g., Fabozzi and Fong, 1994). Immunization theory provides suitable techniques for dealing with such a problem.

The basic setting considered by immunization theory is as follows. An investor, who cares only about his/her wealth on a specific future date, must select a bond portfolio. Only an unexpected change can occur in the term structure of interest rates instantly after the purchase. The simplest immunization policy is that of buying a portfolio with duration equal to the length of the planning horizon. If the unexpected change in the term structure is either “parallel” (Fisher and Weil, 1971) or “suitable” (see, e.g., Bierwag, 1987; Shiu, 1987; Montrucchio and Peccati, 1991), the final wealth will be at least that promised at purchase. Under the above assumptions, the immunization policy has a maxmin property: it maximizes the final wealth in the worst possible case, namely that of no unexpected change (Bierwag and Khang, 1979; Prisman, 1986).

However, if the unexpected change in the term structure can be arbitrary, as is in practice, the abovementioned results do not hold any longer; for the final wealth to be that promised at purchase, the portfolio must include only pure discount bonds maturing at the end of the planning horizon. The immunization policies therefore aim at establishing a suitable trade-off between risk, i.e., the variability of the final wealth, and return (see, e.g., Fong and Vasicek, 1983Fong and Vasicek, 1984). As remarked by Prisman and Shores (1988), some immunization policies do not require that duration and length of the planning horizon be the same.

Although several unexpected changes are likely to occur in the term structure along the planning horizon, few theoretical works have tackled this question. The results available (e.g., Bierwag, 1979; Khang, 1983) have been obtained under the assumption of “parallel” or “suitable” unexpected changes in the term structure.

Such an assumption is relaxed in this paper, where a novel approach to risk control is put forward. In our setting, the wealth (the portfolio value) is a state variable of a discrete time dynamic system and each point of the yield curve acts as a disturbance confined within a well defined range. This paves the way for stating an immunization problem as a maxmin optimal control problem: the immunization policy is the control law which maximizes the final wealth under the worst possible evolution of the yield curve. Needless to say, if the problem is to be analytically tractable, the yield curve must be modelled in a parsimonious manner. A simple, yet financially meaningful example is considered in this paper, where the portfolio can only include pure discount bonds maturing in a time unit as well as irredeemable bonds. As a consequence, only two points of the yield curve are of interest so that the immunization policy can be analytically derived by means of dynamic programming. We recall that optimal control problems have been considered for a long time in the field of bond management (see, e.g., the pioneering work by Bradley and Crane (1972), and the review by Dahl et al. (1989)). However, such problems are usually large scale and, hence, analytically intractable. To the best of our knowledge, none of them is of the maxmin type.

Although very simple, the model can offer some hints in the asset allocation between short and long term bonds.

The paper is organized as follows. The maxmin optimal control problem is stated in Section 2; the optimal solution is found in Section 3; the basic properties of the optimal solution are disclosed in Section 4; final remarks are given in Section 5.

Section snippets

Problem statement

Consider an investor who has to manage a bond portfolio over the planning horizon [0, T]. Assume for simplicity that at time t the portfolio includes a portion ut of pure discount bonds maturing in a time unit as well as a portion (1−ut) of irredeemable bonds. Pure discount bonds have yield rt. Irredeemable bonds have price Pt and pay a coupon C at the end of each time unit so that lt=C/Pt is their yield to maturity measured at time t. Note that the investor knows at time t that holding a pure

Optimal solution

Remark 3.1. Consider the following dynamic programming algorithm (adapted from Bertsekas (1987); see also Piccardi (1992)):JTxT=xT,Jtxt,rt,lt=suputinfrt+1,lt+1Jt+1(1+rt)ut+1+1lt+1lt1−utxt,γt+1+lt+1,t=T−1,…,1,0,where u, x, r, l meet the constraints stated in Section 2. If there exists an optimal control law, then W∗(x0,r0,l0)=J0(x0,r0,l0) andf∗txt,rt,lt=argsuputinfrt+1,lt+1Jt+11+rtut+1+1lt+1lt1−utxt,γt+1+lt+1,t=T−1,…,1,0.

Lemma 3.1. The optimal control law at time T − 1 isu∗T−1=1if1+rT−1⩾1+1LlT−1,0

Basic properties of the optimal solution

Remark 4.1. As already mentioned, if the optimal control law (11) is implemented, the final wealth xT has W∗(x0,r0,l0) as a lower bound. Also note that xT⩾Jt(xt,rt,lt), because Jt(xt,rt,lt) is the lower bound for the problem with planning horizon [t, T] and initial wealth xt. Therefore, ifxt>max1+r0,1+1l∗1l0max1+rt,1+1l∗t+1lt(1+r)tx0,substituting such an inequality into Eq. (12)yields xT⩾Jt(xt,rt,lt)>W∗(x0,r0,l0)=J0(x0,r0,l0). In other words, it can happen that as time proceeds and new

Conclusions

The paper has dealt with the management of a bond portfolio when the planning horizon is given and the target is to maximize the final wealth under the worst possible evolution of the yield curve. Attention has been confined to the case of a barbell portfolio which can solely include pure discount and irredeemable bonds. The resulting problem has taken the form of a maxmin optimal control problem. The optimal control law, i.e. the immunization policy, has been analytically derived by using

Acknowledgements

This work was supported in part by MURST 40% Teoria dei Sistemi e del Controllo. The author would like to thank L. Buzzacchi, F. Hunter, L. Peccati, C. Piccardi, S. Rinaldi and two anonymous referees for their helpful comments and suggestions.

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