¹ Fogler, H. Russell, Groves, William A., and Richardson, James G., “Bond Portfolio Strategies, Returns, and Skewness: A Note,” Journal of Financial and Quantitative Analysis (03 1977), pp. 127–140CrossRefGoogle Scholar.

² Watson, Ronald D., “Tests of Maturity Structures of Commercial Bank Government Securities Portfolios: A Simulation Approach,” Journal of Bank Research (Spring 1972), pp. 34–46Google Scholar.

³ Wolf, Charles R., “A Model for Selecting Commercial Bank Government Security Portfolios,” Review of Economics and Statistics (02 1969), pp. 40–52Google Scholar; and Bradley, Stephen P. and Crane, Dwight B., Management of Bond Portfolios (New York: Wiley and Sons, 1975)Google Scholar. See also, Yawitz, Yess B., Hempel, George H., and Marshall, William J., “A Risk-Return Approach to the Selection of Optimal Government Bond Portfolios,” Financial Management (Autumn 1976), pp. 36–45Google Scholar.

⁴ As noted by the authors, part of the differences in findings also may be attributed to different measures of realized returns and different objective functions. These are not of concern here.

⁵ Fisher, Lawrence and Weil, Roman, “Coping with the Risk of Interest Rate Fluctuations: Return to Bondholders from Naive and Optimal Strategies,” Journal of Business (10 1971), pp. 408–431CrossRefGoogle Scholar. A single planning period is assumed in the remainder of this paper. The planning period is defined by the need to meet a financial obligation or to satisfy an objective target by the end of the period. The consequences for a single planning period differ only in complexity not in substance if multiple planning periods are assumed. The analysis assumes that investors know their planning periods with certainty. The analysis becomes considerably more complex if there is uncertainty about the length of this period. In practice, the length of the planning period may be expected to be affected by many factors including interest rates, inflation, and the availability of future consumption goods. Promised yields are adjusted for coupon effects.

The objective of immunizing the return on the portfolio over a given planning period may be regarded as a suboptimization routine in which an efficient set of portfolios is derived for more complex objective functions. This sub-optimization feature is demonstrated in Bierwag, G. O. and Khang, Chulsoon, “An Immunization Strategy is a Minimax Strategy” (Center for Capital Market Research, 1977)Google Scholar. The duration strategy can be developed as a dynamic programming problem as shown in Bierwag, G. O., “Dynamic Portfolio Immunization Policies” (Center for Capital Market Research, 1977)Google Scholar.

⁶ Bierwag, G. O. and Kaufman, George G., “Coping with the Risk of Interest Rate Fluctuations: A Note,” Journal of Business (07 1977)CrossRefGoogle Scholar; Bierwag, G. O., “Immunization, Duration, and the Term Structure of Interest Rates,” Journal of Financial and Quantitative Analysis (12 1977)CrossRefGoogle Scholar; G. O. Bierwag, “Measures of Duration,” Economic Inquiry (forthcoming); and Khang, Chulsoon, “Bond Immunization When Short-Term Rates Fluctuate More Than Long-Term Rates” (Center for Capital Market Research, University of Oregon, 1977)Google Scholar.

⁷ For securities with uncertain future payments, such as equities or nondefault-free bonds, immunized portfolios cannot be constructed so as to render a certain return. Nevertheless, immunized portfolios with certainty equivalent returns for any planning period may be constructed on the basis of perceived properties of the probability distribution over the payment streams on the securities. Since the range of certainty equivalent durations for equities may be expected to be more restricted than for bonds, immunized portfolios that include only equities may be more difficult to construct over all planning periods.

⁸ Bierwag and Khang, “An Immunization Strategy.”

⁹ Although all portfolios with the same duration immunize for a planning period of the same length so that at least the promised yield is realized, all may not yield the same expected return. The return differences may be related to the higher moment characteristics of the portfolio.

¹⁰ Although these risk-return profiles have not been fully delineated, preliminary conjectures for a single, coupon bond appear in George G. Kaufman, “Measuring Risk and Return for Bonds: A New Approach,” Journal of Bank Research (forthcoming). For an earlier attempt to map this function, see McCallum, John S., “The Expected Holding Period Return, Uncertainty and the Term Structure of Interest Rates,” Journal of Finance (05 1975), pp. 307–323CrossRefGoogle Scholar.

¹¹ Fisher, and Weil, (“Coping with Risk,” p. 424)Google Scholar emphasize the importance of the planning period by arguing “that, unless an investor has a reasonably certain horizon (or series of horizons) at which he will consume his wealth, he should probably not invest in bonds.”

¹² From the data presented it is difficult to be precise about the ordering of durations. The duration of a portfolio also depends upon the coupon rates on the bonds of the portfolio. FGR do not indicate what these structures are.

¹³ Bradley and Crane, Management of Bond Portfolios; Bradley, and Crane, , “A Dynamic Model for Bond Portfolio Management,” Management Science (10 1972), pp. 139–151CrossRefGoogle Scholar; and Bradley, and Crane, , “Management for Commercial Bank Government Security Portfolios: An Optimization Approach under Uncertainty,” Journal of Bank Research (Spring 1973), pp. 18–30Google Scholar.

¹⁴ Bradley, and Crane, , Management of Bond Portfolios, p. 101Google Scholar.

¹⁵ Although selecting a portfolio whose duration is equal to the planning period will guarantee no lower than the promised yield, for coupon bonds it may result in higher yields. As a result, risk surrogates that incorporate variance would show duration immunized portfolios as risky. A more precise risk measure would be some form of semivariance related to the promised yield. For examples, see Fishburn, Peter C., “Mean-Risk Analysis with Risk Associated with Below-Target Returns,” American Ecnomic Review (03 1977), pp. 116–126Google Scholar.

¹⁶ That risk is not invariant with respect to the planning period is also noted by Roll, Richard, “Investment Diversification and Bond Maturity,” Journal of Finance (03 1971), pp. 51–66CrossRefGoogle Scholar.

¹⁷ For an earlier attempt to derive the term structure in this manner, see McCallum.