# Chapter 24The Tradeoff between Return and Risk in Immunized Portfolios

**By H. Gifford Fong and Oldrich Vasicek**

*Financial Analysts Journal*, 39 (5) (1983), 73–78.

## Abstract

The target value of an immunized portfolio at the horizon date defines the portfolio's target rate of return. If interest rates change by parallel shifts for all maturities, the portfolio's realized rate of return will not be below the target value. To the extent that nonparallel rate changes occur, however, the realized return may be less than the target value.

The relative change in the end-of-horizon value of an immunized portfolio resulting from such an arbitrary rate change will be proportional to the value of its immunization risk. Immunization risk equals the weighted variance of times to payment around the horizon date, hence depends on portfolio composition. For example, immunization risk will be low if portfolio payments cluster around the end of the horizon and high if payments are widely dispersed in time. One may minimize the extent to which a portfolio's realized return differs from its target return by minimizing the portfolio's immunization risk (while keeping the portfolio's duration equal to the remaining horizon length).

Although risk minimization is the traditional objective of immunization, the immunization risk measure may also be used to optimize the risk-return tradeoff. The standard deviation of an immunized portfolio's rate of return over the investment horizon will be proportional ...