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

An insurer pools n independent, identically distributed policies, each with mean claim Rs 5,000 and standard deviation Rs 20,000. It charges each policyholder a premium equal to the mean claim. For n = 400, what is the standard deviation of the average claim per policy, and the insurer's probability (using normal approximation) of the total claims exceeding total premium by more than one standard deviation of total claims?

The standard deviation of the average claim is Rs 1,000, found as 20,000 divided by the square root of 400. Since premium equals expected claims, exceeding it by one standard deviation of total claims has normal probability 1 minus Phi(1), about 16%.

  1. ARs 1,000; about 16%Correct
  2. BRs 1,000; about 2.3%
  3. CRs 20,000; about 16%
  4. DRs 400,000; about 50%
  5. Rs 50; about 16%

Explanation

Standard deviation of average = 20,000/sqrt(400) = 20,000/20 = Rs 1,000. Total claims have mean equal to total premium, so exceeding it by one standard deviation of total claims has probability 1 - Phi(1) = about 15.9%, i.e. about 16%. The 2.3% option corresponds to two standard deviations, and Rs 20,000 ignores pooling.

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