Asset-Liability Matching: Redington and Full Immunization

Redington Immunization Revisited

Redington immunization protects against small parallel shifts in the yield curve. The three conditions are: (1) present value of assets equals present value of liabilities, (2) first moments match (duration equality), (3) the second moment of assets exceeds that of liabilities (convexity condition). Mathematically, if h(i) = PV_A(i) - PV_L(i), we need h(i_0) = 0, h'(i_0) = 0, and h''(i_0) > 0 at the current yield i_0.

This ensures h has a local minimum at i_0 with h(i_0) = 0, so h(i) >= 0 for i near i_0. The key word is "near": Redington immunization provides local protection only. For large rate changes, the surplus could potentially become negative, though in practice the protection extends over a meaningful range of rate movements.

Verifying the Redington Conditions

To check the Redington conditions on Exam FM, compute the present value, duration, and convexity of both the asset portfolio and the liability stream at the current rate. For a set of cash flows A_t at times t_1, t_2, ..., the present value is sum of A_t * v^t, the duration is sum of t * A_t * v^t divided by the present value, and the convexity is sum of t^2 * A_t * v^t divided by the present value (using the second moment form).

Example: Assets consist of $500 at time 1 and $1,000 at time 5. Liabilities consist of $800 at time 3. At i = 5%: PV_A = 500*v + 1000*v^5 = 476.19 + 783.53 = $1,259.72. PV_L = 800*v^3 = $691.07. If these are not equal, condition (1) fails and the portfolio is not immunized. You would need to scale the assets or adjust the investment amounts.

Full Immunization

Full immunization provides a stronger guarantee: protection against any parallel shift, not just small ones. For a single liability of amount L at time T, full immunization requires exactly two asset cash flows at times T - a and T + b (for a, b > 0) such that: (1) PV(assets) = PV(liability) and (2) the asset cash flows straddle the liability in time.

Let A_1 be the asset cash flow at time T - a and A_2 at time T + b. The conditions are: A_1 * v^(T-a) + A_2 * v^(T+b) = L * v^T, and A_1 * (T-a) * v^(T-a) + A_2 * (T+b) * v^(T+b) = T * L * v^T. These two equations in two unknowns (A_1 and A_2) have a unique solution.

The surplus function h(i) = PV_A(i) - PV_L(i) is provably non-negative for all i >= 0, not just near i_0. This stronger result comes from the mathematical structure: h(i) is a convex function that achieves its minimum value of zero at i = i_0, and a convex function with a minimum of zero cannot become negative anywhere.

Comparing Redington and Full Immunization

Redington immunization is more flexible because it does not require specific asset structures; any combination of assets satisfying the three conditions works. Full immunization is more restrictive (exactly two asset cash flows straddling the liability) but provides stronger protection. In practice, Redington is used more often because finding assets that exactly match the full immunization structure is difficult.

On Exam FM, the key distinction is: Redington protects against small shifts (local protection), while full immunization protects against all parallel shifts (global protection). A problem that asks you to construct a portfolio that is "fully immunized" requires the straddling condition. A problem that asks for "Redington immunization" requires checking all three conditions but does not require straddling.

Cash Flow Matching (Dedication)

Cash flow matching (also called dedication or exact matching) exactly matches each liability payment with a corresponding asset cash flow at the same time. If the liabilities are $50,000 at time 3, $80,000 at time 5, and $100,000 at time 10, the dedicated portfolio holds zero-coupon bonds maturing at those exact times with face values matching the liability amounts.

This approach eliminates all interest rate risk because there are no reinvestment decisions and no duration mismatches. The cost is usually higher than Redington immunization because it constrains the asset choice more tightly. You cannot take advantage of the yield curve or use assets with intermediate cash flows.

On Exam FM, problems may ask you to compare the cost of dedication versus immunization, or to construct a dedicated portfolio using zero-coupon bonds of various maturities. The cost of the dedicated portfolio is simply the sum of the present values of the liability cash flows at the current spot rates.

Choosing the Right Strategy

The choice among Redington immunization, full immunization, and cash flow matching depends on the trade-off between cost and risk tolerance. Cash flow matching eliminates interest rate risk entirely but is the most expensive. Full immunization protects against all parallel shifts at a moderate cost. Redington immunization is the least expensive but only protects against small shifts and requires periodic rebalancing.

For Exam FM, understand the theoretical differences and be able to construct portfolios under each strategy. Know that all three strategies start with the same present value condition (PV of assets = PV of liabilities) but differ in the additional constraints they impose. Practice problems that ask you to verify whether a given portfolio satisfies the conditions for each type of protection.

Study Tips and Common Pitfalls

The most common mistake is confusing the three strategies. Remember: Redington requires PV match, duration match, and convexity inequality. Full immunization requires PV match and straddling (which automatically satisfies convexity). Cash flow matching requires exact timing matches (which trivially satisfies everything).

Another frequent error is computing the second moment incorrectly. For Redington's convexity condition, use sum of t^2 * CF_t * v^t, not t*(t+1) (that version is for modified convexity). Check which form the problem requires. Practice at least 3 problems for each strategy type, paying special attention to problems that ask you to determine which strategy a given portfolio satisfies.