Assuming that interest rate shocks are proportional to their values plus one, existence is proven of and a portfolio Z* is constructed with the highest convexity in the class of portfolios that solve the immunization problem to meet the liability to pay C dollars K years from now. Z* appears to be a barbell strategy with two zero-coupon bonds with the shortest and the longest maturities. This intuitively clear result has been obtained in a rigorous way by means of the K-T conditions. In addition, the result is to be strictly related to the problem of maximization of the unanticipated rate of return on a portfolio solving the above immunization problem. Two more results concerning the unanticipated return after K years are provided with proofs. An example illustrating the role of convexity in maximization of the unanticipated return is included. Despite the fact that there exists a pretty vast literature on bond portfolio strategies, the paper offers a new methodological approach to this area.
