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

A health insurer pools policyholders whose claims are positively correlated because an epidemic affects many of them at once. Compared with independent risks, which statement is correct as the pool grows very large?

The variance of the average claim does not fall to zero, because the covariance term remains. With common correlation r, the variance tends to r times the individual variance as n grows. Pooling removes only the independent part of the risk, not correlated, systematic risk.

  1. AThe variance of the average claim does not fall to zero, because the covariance term remainsCorrect
  2. BThe variance of the average claim still falls to zero at rate 1/n
  3. CThe mean claim per policy rises with n
  4. DThe Law of Large Numbers ensures the average claim becomes deterministic regardless of correlation
  5. Diversification removes all systematic risk

Explanation

For n identical risks with variance s² and common correlation r, the variance of the average is s²/n + s²r(n−1)/n, which tends to r s² as n grows. So a non-diversifiable component stays. The claim that the average becomes deterministic requires uncorrelated or weakly correlated risks.

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