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

A fund is Redington-immunised at interest rate i0, with all three conditions satisfied. Immediately afterwards the interest rate changes by a small amount, up or down. Which statement is correct?

The surplus rises for a small change in either direction. At the immunised rate the surplus is zero, its first derivative is zero and its second derivative is positive, so the surplus has a local minimum there and any small immediate rate move produces a profit.

  1. AThe surplus (assets minus liabilities) falls whichever direction rates move
  2. BThe surplus rises for a small rise in rates but falls for a small fall
  3. CThe surplus is unchanged to all orders for any size of change
  4. DThe surplus rises for a small change in either direction, because its first derivative is zero and its second derivative is positive at i0Correct
  5. The surplus falls for a small rise in rates and rises for a small fall

Explanation

Surplus S(i)=V_A(i)-V_L(i) has S(i0)=0 and S'(i0)=0, and S''(i0)>0. Taylor expansion gives S(i0+h) ≈ ½S''(i0)h² > 0 for either sign of h. Option 'unchanged to all orders' is wrong since convexity is strictly greater for assets.

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