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IAI Actuarial Core Principles · Actuarial Mathematics for Modelling · Duration, convexity and immunisation

A company has a liability of Rs 1,00,000 due at time 6. It holds assets of zero-coupon bonds paying X at time 3 and Y at time 9, with PV(A)=PV(L) and equal discounted mean term at an effective annual rate of 10%. Let the asset PVs be A3 and A9. Which statement is correct about the immunisation?

Asset convexity is greater, so the position is immunised. Matching the mean term of 6 requires equal PV weights of half on times 3 and 9. Asset convexity is 0.5 times 9 plus 0.5 times 81, which is 45, exceeding the liability convexity of 36.

  1. AAsset convexity is less than liability convexity, so the position is not immunised
  2. BAsset convexity equals liability convexity, so the position is immunised
  3. CAsset convexity is greater than liability convexity, so the position is immunisedCorrect
  4. DAsset convexity cannot be compared without the value of X
  5. Asset convexity is negative because one bond is shorter than the liability

Explanation

Equal PV and mean term mean the weights w3 and w9 satisfy 3w3+9w9=6 with w3+w9=1, so each is 0.5. Asset convexity is 0.5*9+0.5*81=45, against liability convexity 36. Since 45 exceeds 36, Redington's third condition holds and the portfolio is immunised. The comparison does not need X because only weights matter.

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