An Immunization Strategy is a Minimax Strategy

IN A SERIES OF PAPERS-Fisher and Weil [7], Bierwag [2, 3, 4], Bierwag and Kaufman [5], and Khang [14] show that an investor can immunize a portfolio of default free coupon bonds against unexpected interest rate fluctuations by selecting a portfolio that has some appropriate measure of duration which is equal to the length of the investor's planning period. ' Immunization and the implied duration strategy is a powerful result that is of considerable importance in bond portfolio management. It has never been shown, however, that immunization is the consequence of optimizing a multi-period objective function. In this paper we show that an immunization strategy produces a diversified bond portfolio for which the minimal rate of return over a multi-period planning horizon is maximized. Consequently the rate of return on an immunized portfolio can never be less than that which is implied in an initially observed equilibrium term structure of interest rates. Immunization is a maximin strategy. As a maximin strategy, we show that an immunized portfolio of coupon bonds stochastically dominates a portfolio of pure discount bonds (zero coupon bonds) the return on which has a zero variance. Moreover, even an efficient bond portfolio has a random return with a positive variance. Diversification cannot eliminate this risk as it is generally understood. The maximin theorem implies that the immunized portfolio eliminates "downside" risk; the probability of a rate of return over a multiple period horizon less than that implied by the term structure of interest rates is zero. If Fishburn's [8] measure of "downside" risk is an appropriate measure of risk for investors who hold bond portfolios in multiperiod planning models and if the yield implied by the initially observed term structure is the target, risk-free, or promised yield, * University of Oregon 'The relationship between immunization and a concept of duration is well established in the economic literature. Macaulay [15] is generally credited with inventing the concept of duration as a measure of the average life of an income stream, although Hicks [11] independently perceived it and called it the "average period" of an investment. Samuelson [18] also independently discovered it and he showed that a weighted equality of the duration of income and outgo streams will immunize the banking system against interest rate fluctuations. Redington [17] and Wallas [19] demonstrated the usefulness of the concept in actuarial theories. Grove [9, 10] showed that duration, as a property of the maturity profile of the balance sheet, indicates an attitude toward risk. Weil [20] published a survey of much of this work. Kaufman and Hopewell [12] have recently shown that the volatility of a bond price is related to the duration of a bond. Boquist, Racette, and Schlarbaum [15] show that the ,8 of a bond portfolio in a capital asset pricing model is related to duration. Kaufman and Bierwag [6] and Kaufman [13] have suggested that duration, the length of the planning period, and the differencing period (the time interval over which rates of change are calculated) are inter-related so that the interpretation of f8 in the capital asset pricing model (whether over equities or bonds) must