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IAI Actuarial Core Principles · Economic Modelling · Role of insurance in reducing or removing risk

An insurer pools n independent policies, each with the same mean claim m and the same standard deviation s. Which statement correctly describes what happens to the risk per policy as n increases?

The standard deviation of the average claim per policy falls in proportion to 1/√n while the mean stays at m. Total variance is n s², so the average has variance s²/n. Pooling therefore reduces relative risk per policy without changing expected cost.

  1. AThe standard deviation of the average claim per policy falls in proportion to 1/√n, while its mean stays at mCorrect
  2. BThe standard deviation of the total claim falls in proportion to 1/√n
  3. CThe mean claim per policy falls in proportion to 1/n
  4. DThe standard deviation of the average claim per policy falls in proportion to 1/n
  5. The standard deviation of the total claim stays at s regardless of n

Explanation

For independent identical risks, the total has variance n s², so the average has variance s²/n and standard deviation s/√n. The mean of the average remains m. The total's standard deviation rises as s√n, so the option claiming it falls is wrong.

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