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

A company must pay Rs 1,000,000 in 6 years. It plans to hold two zero-coupon bonds, maturing in 3 years and 9 years, with present values equal in total to the liability PV, to satisfy the first Redington condition and match duration. The yield is 5%. What proportion of the total present value must be placed in the 3-year bond?

Half of the present value, 0.50, goes into the 3-year bond. Zero-coupon discounted mean terms equal their maturities, so 3w + 9(1−w) = 6 gives w = 0.5. The spread of the 3 and 9 year terms also ensures asset convexity exceeds liability convexity.

  1. A0.25
  2. B0.50Correct
  3. C0.75
  4. D0.33
  5. 0.67

Explanation

Matching mean term: w×3 + (1−w)×9 = 6, so 9 − 6w = 6 and w = 0.5. The yield does not alter this because zero-coupon bonds have discounted mean term equal to maturity. Convexity of assets then exceeds liabilities since 3 and 9 are spread around 6.

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