Redington Immunization

Study guides: Exam FM-2, Exam ILA-101

Redington immunization is a strategy for protecting a portfolio's surplus S = V_A - V_L against small parallel shifts in interest rates. It requires three conditions to be satisfied simultaneously at the current yield rate j:

1. PV(A) = PV(L) — asset and liability present values are equal (surplus S = 0)
2. D_{Mac}(A) = D_{Mac}(L) — asset and liability Macaulay durations are equal (equivalently their modified durations, since both are measured at the same rate j)
3. \text{Convexity}(A) > \text{Convexity}(L) — asset Convexity exceeds liability convexity

• When all three hold, a small change \Delta j causes S \geq 0 because the convexity term dominates: \Delta S \approx \tfrac{1}{2}(C_A - C_L) \cdot V \cdot (\Delta j)^2 \geq 0
• Redington immunization only protects against small, parallel rate shifts and must be rebalanced as time passes and rates change
• Less restrictive than Cash Flow Matching since exact cash flow timing is not required

Verifying Redington Immunization A company has a single liability of $10{,}000 due in 4 years. It immunizes using two zero-coupon bonds: Bond A maturing in 2 years and Bond B maturing in 6 years. The current yield rate is 5\%. Find the face values of the two bonds that satisfy the first two Redington conditions, and verify the third.

Answer Let X = face of the 2-year bond, Y = face of the 6-year bond, v = 1/1.05.

Condition 1 (PV_A = PV_L): X v^2 + Y v^6 = 10{,}000 v^4 Condition 2 (D_{Mac}(A) = D_{Mac}(L) = 4), multiplying through by the common present value: 2 X v^2 + 6 Y v^6 = 4(10{,}000 v^4) Subtracting twice Condition 1 from Condition 2 gives 4Yv^6 = 20{,}000v^4, so each bond carries half the liability's present value:

\begin{align*}
X v^2 = Y v^6 &= 5{,}000 v^4 = 4{,}113.51 \\
X = 5{,}000 v^2 &= 4{,}535.15 \\
Y = 5{,}000 v^{-2} &= 5{,}512.50
\end{align*}

Condition 3: a zero-coupon payment at time t has (modified) convexity t(t+1)/(1+j)^2, and a portfolio's convexity is the present-value-weighted average of its pieces'. The two bonds carry equal present values, so

\begin{align*}
C_A &= \frac{\tfrac{1}{2}\left[2(3) + 6(7)\right]}{1.05^2} \\
&= \frac{24}{1.1025} \\
&= 21.77 \\
C_L &= \frac{4(5)}{1.05^2} \\
&= 18.14
\end{align*}

Since 21.77 > 18.14, the asset convexity exceeds the liability convexity and Condition 3 holds: the position is Redington immunized.