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

Which of the following correctly lists the three conditions for Redington's immunisation of a portfolio of assets against a liability stream at the force of interest or rate i0?

The conditions are equal present values, equal discounted mean terms, and asset convexity greater than liability convexity, i.e. asset cashflows more spread around the mean term. Together these make the surplus a local minimum at the current interest rate, so small changes in rates cannot create a loss.

  1. APV of assets equals PV of liabilities; discounted mean term of assets equals that of liabilities; asset cashflows are more spread around the mean term than liability cashflowsCorrect
  2. BPV of assets exceeds PV of liabilities; discounted mean terms are equal; asset convexity is less than liability convexity
  3. CPV of assets equals PV of liabilities; asset mean term exceeds liability mean term; asset convexity equals liability convexity
  4. DUndiscounted totals of assets and liabilities are equal; discounted mean terms are equal; asset convexity exceeds liability convexity
  5. PV of assets equals PV of liabilities; discounted mean terms are equal; asset convexity is less than liability convexity

Explanation

Redington requires V_A(i0)=V_L(i0), V_A'(i0)=V_L'(i0) (equal discounted mean terms) and V_A''(i0)>V_L''(i0), meaning the assets are more spread out about the mean term. Options with lower or equal convexity, or with undiscounted totals, fail to make the surplus have a local minimum at i0.

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