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Immunisation

Match present value and duration, and make the assets more convex than the liabilities: small rate moves in either direction then leave a surplus.

The formulas

Redington condition 1
PVA=PVLPV_A = PV_L
Redington condition 2
DA=DLD_A = D_L

equivalently P'_A = P'_L

Redington condition 3
CA>CLC_A > C_L

assets strictly more convex

Full immunisation
one asset each side of the liability, PV and duration matched\text{one asset each side of the liability, PV and duration matched}

Where it comes from

  1. Write the surplus S(i)=PVA(i)−PVL(i)S(i) = PV_A(i) - PV_L(i). Condition 1 makes S=0S = 0 at the current rate, condition 2 makes S′=0S' = 0, so the current rate is a stationary point.
  2. Condition 3 makes S′′>0S'' > 0, so that stationary point is a MINIMUM: any small move in either direction produces a surplus.
  3. Full immunisation replaces 'small' with 'any', at the cost of requiring assets that straddle each liability in time.

Worked example

A liability of 100,000 falls due in 4 years. Assets are zero-coupon bonds maturing at years 2 and 7, chosen so that present value and duration match at 6%. Does the portfolio satisfy Redington's conditions?

  1. Matching PV and duration at i=0.06i = 0.06 requires maturity amounts of 53{,}399.79 at time 2 and 47{,}640.64 at time 7.
  2. The present value of each side is 79,209.3779{,}209.37, and both durations are 4 years.
  3. Asset convexity exceeds liability convexity because the asset cash flows are spread either side of year 4 - the Jensen effect.
  4. All three conditions hold, so the portfolio is immunised against small rate moves.

Answer: Yes - all three conditions hold

This answer is recomputed from the site’s own interest-theory and probability functions every time the test suite runs, so the page and the mathematics cannot drift apart.

Memory hooks

  • PV, then duration, then convexity - in that order, and the third is an INEQUALITY.
  • Spreading assets either side of the liability date is what buys the extra convexity.

Traps

  • Matching duration but forgetting to check the convexity inequality - that gives a stationary point which could be a maximum.
  • Assuming immunisation survives a large or non-parallel shift. Redington protects against small parallel moves only.

Related

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