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

In Redington's conditions for immunisation of a liability stream by an asset portfolio at the current interest rate, which condition on convexity is required, given that present values are equal and the discounted mean terms are equal?

Assets must have greater convexity than liabilities. With equal present values and equal discounted mean terms, a larger asset second derivative makes the surplus function have a local minimum at the current rate, so any small rate move increases surplus.

  1. AThe convexity of assets must be less than the convexity of liabilities
  2. BThe convexity of assets must equal the convexity of liabilities exactly and be zero
  3. CThe convexity of assets must be greater than the convexity of liabilitiesCorrect
  4. DThe convexity of assets must be negative
  5. The convexity of assets must equal the discounted mean term of liabilities

Explanation

Redington requires PV(A)=PV(L), equal first derivatives (equal discounted mean terms), and the second derivative of the asset PV exceeding that of the liabilities. This ensures the surplus has a local minimum, so small interest rate changes in either direction raise surplus. Equal convexity would only give a stationary point of the surplus without guaranteeing a profit.

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