A Risk Minimizing Strategy for Portfolio Immunization

Consider a fixed-income portfolio whose duration is equal to the length of a given investment horizon. It is shown that there is a lower limit on the change in the end-ofhorizon value of the portfolio resulting from any given change in the structure of interest rates. This lower limit is the product of two terms, of which one is a function of the interest rate change only, and the other depends only on the structure of the portfolio. Consequently, this second term provides a measure of immunization risk. If this measure is minimized, the exposure of the portfolio to any interest rate change is the lowest. THE TRADITIONAL THEORY OF immunization as formalized by Fisher and Weil [6] defines the conditions under which the value of an investment in a bond portfolio is protected against changes in the level of interest rates. The specific assumptions of this theory are that the portfolio is valued at a fixed horizon date, that there are no cash inflows or outflows within the horizon, and that interest rates change only by a parallel shift in the forward rates. Under these assumptions, a portfolio is said to be immunized if its value at the end of the horizon does not fall below the target value, where the target value is defined as the portfolio value at the horizon date under the scenario of no change in the forward rates. The main result of this theory is that immunization is achieved if the duration of the portfolio is equal to the length of the horizon. The assumption that interest rates can only change by a parallel shift (that is, by the same amount for all maturities) has been the subject of considerable concern. Bierwag [1, 2], Bierwag and Kaufman [3], Khang [7], and others have postulated alternative models of interest rate behaviors. Each of these specifications implies a different measure of duration, with immunization attained if this duration measure is equal to the horizon length. A limitation of this approach is that the portfolio is protected only against the particular type of interest rate change assumed. In a more recent development, Cox et al. [5], Brennan and Schwartz [4], and others have investigated immunization conditions when interest rates are governed by a continuous process consistent with a market equilibrium. Depending on the specilfication of the interest rate process, there is a duration-like measure (possibly multidimensional, as in Brennan and Schwartz) such that the portfolio is immunized if a proper value of this measure is maintained. This assumes a continuous rebalancing of the portfolio. Again, immunization is achieved only if interest rate changes conform to the specific process assumed. In this paper, we wish to pursue a different approach. If it turned out that the