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

Under Redington's theory of immunisation, which of the following sets correctly states the conditions that must hold at the current valuation rate of interest for a fund to be immunised against small changes in interest rates?

The conditions are: equal present values of assets and liabilities, equal discounted mean terms, and asset convexity greater than liability convexity. The first two make the surplus stationary at the current rate, and the third makes it a minimum, so small interest rate changes cannot create a loss.

  1. APresent value of assets equals present value of liabilities; discounted mean term of assets equals that of liabilities; convexity of assets is greater than convexity of liabilitiesCorrect
  2. BPresent value of assets exceeds present value of liabilities; discounted mean terms are equal; convexity of assets is less than convexity of liabilities
  3. CPresent value of assets equals present value of liabilities; discounted mean term of assets exceeds that of liabilities; convexities are equal
  4. DNominal cashflows of assets equal nominal cashflows of liabilities; discounted mean terms are equal; convexity of assets is less than that of liabilities
  5. Present value of assets equals present value of liabilities; discounted mean term of assets is less than that of liabilities; convexity of assets is greater than that of liabilities

Explanation

Redington requires V_A(i)=V_L(i), equal discounted mean terms so the first derivatives of the surplus vanish, and greater asset convexity so the second derivative of the surplus is positive. This makes the surplus a local minimum at the current rate. The option with unequal mean terms fails the first-order condition, and less asset convexity would give a local maximum.

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