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

Two portfolios have the same present value and the same discounted mean term. Portfolio P has cashflows concentrated near its mean term, and portfolio Q has cashflows spread widely either side of the mean term. Which statement about their convexities is correct?

Q has higher convexity. Convexity equals the squared discounted mean term plus the variance of cashflow times, weighted by present value. Since the mean terms are equal, the portfolio with the more widely dispersed cashflows has the larger variance and therefore the greater convexity.

  1. AP has higher convexity than Q
  2. BQ has higher convexity than PCorrect
  3. CThey have equal convexity because the mean terms are equal
  4. DConvexity is negative for Q
  5. Convexity depends only on the yield, not on cashflow timing

Explanation

Convexity is the PV-weighted mean of t squared, which equals the squared mean term plus the PV-weighted variance of the cashflow times. With equal mean terms, the more widely spread portfolio Q has the larger variance and so greater convexity. Equal mean term does not imply equal convexity.

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