Immunisation and asset-liability matching

2 free worked examples

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1. Exam FMImmunisation and asset-liability matchingCore

   Which set of conditions defines Redington immunisation?

   1. APV matched, duration matched, asset convexity greater
   2. BPV matched, duration matched, convexity matched
   3. CDuration matched only
   4. DPV matched and asset convexity smaller
   5. ECash flows matched exactly at every date

   Solution

   1. Write the surplus $S(i)=P{V}_A(i)-P{V}_L(i)$. Matching present values makes $S=0$ at the current rate.
   2. Matching durations makes $S^{\prime }(i)=0$, so the current rate is a stationary point of the surplus.
   3. Requiring asset convexity STRICTLY greater makes $S^{\prime \prime }(i)>0$, so that stationary point is a minimum.
   4. A minimum at zero means any small rate move in either direction produces a surplus, which is exactly what immunisation promises.

   Trap. Matching convexity as well, which leaves the surplus flat to second order and provides no protection.

2. Exam FMImmunisation and asset-liability matchingCore

   What does Redington immunisation NOT protect against?

   1. AA small parallel rise in rates
   2. BA small parallel fall in rates
   3. CA large or non-parallel shift in the yield curve
   4. DThe passage of one day
   5. EAny change in rates at all

   Solution

   1. The derivation is a second-order Taylor expansion of the surplus around the CURRENT rate, assuming the whole curve moves by the same amount.
   2. That makes it valid for SMALL PARALLEL shifts only.
   3. A large shift takes the expansion outside its useful range, and a twist in the curve breaks the parallel assumption entirely.
   4. Durations also drift as time passes, so an immunised portfolio must be rebalanced rather than left alone.

   Trap. Treating immunisation as permanent protection rather than a local, and perishable, condition.