Immunization

A set of liabilities is given along with two or more assets, and you must construct a portfolio that satisfies Redington immunization.

Write three conditions: PV_assets = PV_liabilities, Modified duration of assets = modified duration of liabilities, and Convexity of assets > convexity of liabilities. Solve the first two as a linear system for asset weights, then verify the convexity inequality at the end. The convexity check is what makes the immunization work for small rate moves.

Redington: equal PV, equal duration, asset convexity > liability convexity.

Given a liability stream, you must explain which immunization strategy is being used and assess its protection against interest rate movements.

Cash flow matching schedules an asset cash flow for every liability cash flow and removes rate risk entirely, but is the most expensive. Redington is a duration/convexity match that protects against small parallel shifts. Full immunization (Khang/Reitano) bracket each liability with assets before and after so the portfolio dominates the liability for any parallel shift. Pick the strategy based on cost and the shift assumption.

Cash flow match > full immunization > Redington in protection; cost ordering is the reverse.

You are given an immunized portfolio and a small parallel shift in interest rates and must show the surplus is non-negative.

Compute the change in surplus using the second-order Taylor expansion: ΔS ≈ -(D_A · PV_A - D_L · PV_L) Δy + (1/2)(C_A · PV_A - C_L · PV_L) (Δy)^2. The duration term is zero by construction; the convexity term is positive when C_A > C_L. Show that the surplus is non-decreasing in the magnitude of the parallel shift.

Under Redington, ΔS ≈ (1/2)(C_A - C_L) · PV · (Δy)^2 ≥ 0.