Match each actuarial scenario regarding risk pooling to the correct outcome or principle associated with the Law of Large Numbers.
- Aggregating 250,000 independent, identical worker's compensation exposures within a single occupational risk classification.Maximizes statistical credibility, allowing for highly stable premium calculations due to minimized variance in expected aggregate losses.
- Consolidating 10,000 high-severity aviation liability policies with 90,000 low-severity personal umbrella policies to artificially inflate the total exposure count.Violates the prerequisite of homogeneous exposure units, rendering aggregate loss predictions statistically invalid despite the large total pool size.
- Pricing a novel commercial cyber liability product based exclusively on the historical loss data of 150 early-adopter technology firms.Subjects the insurer's rate-setting to excessive volatility and statistical noise, as the limited exposure base fails to trigger actuarial credibility.
- Achieving a 1% margin of error in projecting total annual mortality costs for a mega-group plan, while remaining unable to identify specific decedents.Validates the core premise that macroscopic pool predictability does not eliminate microscopic uncertainty for individual insureds.
Answer
The correct alignments are: 1. Identical worker's comp exposures maximize statistical credibility. 2. Mixing aviation and umbrella policies violates the prerequisite of homogeneous exposure units. 3. 150 cyber policies subjects the insurer to excessive volatility and statistical noise. 4. Projecting total mortality without identifying decedents validates macroscopic predictability without microscopic certainty.
The correct alignments thoroughly demonstrate the operational boundaries of the Law of Large Numbers. It requires a high volume of independent, homogeneous (similar) exposure units to accurately predict aggregate future losses and minimize variance. Aggregating identical exposures perfectly maximizes this statistical credibility. However, mixing heterogeneous risks like aviation and personal umbrella policies strictly violates the homogeneity requirement, rendering predictions statistically invalid regardless of the large total count. Small sample sizes fail to provide credible data, leading to severe rate volatility. Finally, the principle accurately predicts a group's total losses while mathematically unable to pinpoint which specific individuals will suffer those losses.
Step-by-Step Solution
Key Concept
The Law of Large Numbers requires a substantial volume of homogeneous (similar) and independent exposure units to accurately predict future aggregate losses, while remaining unable to predict individual risk occurrences.