Full Immunization

Fund a single liability \(L\) due at \(t_L\), at current yield \(i_0\), with two asset cash flows \(A_1\) at \(t_1\) and \(A_2\) at \(t_2\) that bracket it: \(t_1 < t_L < t_2\). Impose two conditions at \(i_0\):

1. PV match: \(A_1 v^{t_1} + A_2 v^{t_2} = L v^{t_L}\).
2. Duration match: \(A_1 v^{t_1} t_1 + A_2 v^{t_2} t_2 = L v^{t_L} t_L\).

Write surplus as a function of the yield, \(S(i) = A_1 v^{t_1} + A_2 v^{t_2} - L v^{t_L}\). Conditions 1 and 2 make \(S(i_0) = 0\) and \(S'(i_0) = 0\) — exactly what Redington needs too. The extra ingredient is the bracket itself: with one cash flow strictly before \(t_L\) and one strictly after, \(S(i)\) has a global minimum of zero at \(i_0\) — \(S(i) \ge 0\) for every \(i\), not just nearby ones. Whichever way the rate moves, one of the two asset payments is helped more than the liability is hurt: a rate rise favors the early payment \(A_1\) (it loses less value to discounting), a rate fall favors the late payment \(A_2\). Full immunization implies the two Redington conditions above, but doesn't need a separate convexity inequality — the bracket delivers the required curvature automatically.

A pension fund owes a single lump-sum benefit years from now. Instead of guessing which way rates will move, it buys one bond maturing before the payment date and one maturing after, sized so their combined PV and duration match the liability's today. No matter how far rates drift by the time the benefit is due, the fund's asset side never falls short — a much stronger promise than Redington's "safe for small moves."