Full Immunization

Study guide: Exam FM-2

Full immunization protects an asset-liability position against any immediate change in the interest rate — small or large, up or down — in contrast to Redington Immunization, which only covers small changes. A single liability L due at time T is fully immunized when all three conditions hold:

1. PV(\text{assets}) = PV(\text{liability}) at the current rate i
2. The Duration of the assets equals the duration of the liability (Duration Matching)
3. There is an asset cash flow before the liability date and one after it

With two zero-coupon assets, A at time T-a and B at time T+b, conditions 1 and 2 are the two equations

A\,v^{T-a} + B\,v^{T+b} = L\,v^{T}

(T-a)\,A\,v^{T-a} + (T+b)\,B\,v^{T+b} = T\,L\,v^{T}

• v = 1/(1+i); condition 3 is a > 0 and b > 0
• Dividing through by v^T reduces them to A(1+i)^{a} + B(1+i)^{-b} = L and a\,A(1+i)^{a} = b\,B(1+i)^{-b}
• Under these conditions the surplus PV_A - PV_L is zero at i and positive at every other rate, whichever way the rate moves
• For several liabilities, repeat the construction for each one: two asset cash flows per liability. As with Redington immunization, the portfolio must be rebalanced periodically to keep the durations equal

Full Immunization Setup A single liability of L is due at time T. Show how to fully immunize with two assets.

Answer Purchase two zero-coupon bonds: one maturing before T (at time T_1 < T) and one after T (at time T_2 > T), with amounts A_1 and A_2 such that A_1 v^{T_1} + A_2 v^{T_2} = L v^T and the durations match, T_1 A_1 v^{T_1} + T_2 A_2 v^{T_2} = T L v^T. Present values and durations matched, with the liability bracketed by the two asset dates, the surplus rises whichever way the rate moves.

Immunizing a 15-Year Liability with 10- and 20-Year Zeros An investor owes 1{,}000 in 15 years. Zero-coupon bonds of every term yield 4\%. Find the maturity values A (10-year) and B (20-year) that fully immunize the liability, and the surplus if the rate immediately moves to 2\% or 6\%.

Answer Here a = b = 5, so the reduced equations are

\begin{align*}
A(1.04)^{5} + B(1.04)^{-5} &= 1{,}000 \\
5A(1.04)^{5} &= 5B(1.04)^{-5}
\end{align*}

Each term is therefore 500:

\begin{align*}
A &= 500(1.04)^{-5} = 410.96 \\
B &= 500(1.04)^{5} = 608.33
\end{align*}

At 4\% both sides are worth 1{,}000(1.04)^{-15} = 555.26. After an immediate move:

\begin{align*}
i = 2\%:\ \ 410.96(1.02)^{-10} + 608.33(1.02)^{-20} &= 746.52 \\
1{,}000(1.02)^{-15} &= 743.01 \\
i = 6\%:\ \ 410.96(1.06)^{-10} + 608.33(1.06)^{-20} &= 419.16 \\
1{,}000(1.06)^{-15} &= 417.27
\end{align*}

The surplus is 3.50 at 2\% and 1.89 at 6\% — positive in both directions.