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Difficulty: Very hardLaw of Large Numbers

An actuary is redesigning the ratemaking strategy for a regional insurer's commercial auto portfolio, which currently consists of 500 local delivery trucks. To achieve a high degree of premium solvency and rate stability, the actuary proposes expanding the portfolio by aggressively underwriting an additional 50,000 vehicles.

Based on the strict actuarial principles of the Law of Large Numbers, which of the following identifies the necessary underwriting condition for this expansion and its exact mathematical effect on loss prediction?

  1. The expanded pool must consist of homogeneous exposure units facing independent pure risks; as the pool size increases, the difference between actual aggregate losses and expected aggregate losses decreases, improving overall pricing accuracy without identifying which specific trucks will incur a loss.Answer
  2. B
    The insurer must utilize risk avoidance for any vehicles outside its primary geographic region to maintain a tightly localized pool; the Law of Large Numbers dictates that minimizing exposure units is the most effective method to accurately price aggregate hazard occurrences.
  3. C
    The new vehicles must be underwritten to include a balance of pure risks and speculative financial risks, such as guaranteeing against the depreciation of cargo value; applying large numbers across both risk categories mathematically guarantees an underwriting profit.
  4. D
    The primary insurer must execute a treaty ceding the original 500 vehicles to an assuming reinsurer, transferring direct liability so the primary insurer can use the Law of Large Numbers to precisely predict which specific policyholders among the new 50,000 vehicles will suffer a loss.

Answer

The expanded pool must consist of homogeneous exposure units facing independent pure risks; as the pool size increases, the difference between actual aggregate losses and expected aggregate losses decreases, improving overall pricing accuracy without identifying which specific trucks will incur a loss.
For the Law of Large Numbers to function, the insurer needs a substantial volume of similar (homogeneous) and independent exposure units facing pure risks. As this pool expands, the statistical deviation between expected aggregate losses and actual aggregate losses narrows. This allows actuaries to confidently set accurate premiums for the entire group, even though it remains completely impossible to predict exactly which specific vehicles within the group will experience those losses.

Step-by-Step Solution

1
Analyze the conditions required for the Law of Large Numbers to operate effectively in an insurance context.
The principle requires a large volume of homogeneous (similar) exposure units that are independent of one another and exposed to pure risks.
Without similarity and independence, actuarial data cannot be accurately extrapolated to predict future frequency and severity.
2
Determine the specific mathematical and predictive outcome of expanding the pool of exposure units.
As the number of units increases, the relative variance between expected aggregate losses and actual aggregate losses decreases, resulting in greater rate stability.
This allows the insurer to price premiums more accurately for the entire pool.
3
Identify the fundamental limitation of the Law of Large Numbers.
While aggregate prediction improves, the ability to predict which specific individual or unit will suffer a loss remains impossible.
Insurance pools risk precisely because individual losses are uncertain, even when group losses are highly predictable.
4
Evaluate the options against these established parameters while checking for incorrectly applied related concepts.
Only the correct option captures the need for homogeneous pure risks, the improvement in aggregate prediction, and the inability to predict individual losses.
The other choices incorrectly blend speculative risks, invert the size requirement, or falsely claim individual predictability.

Key Concept

The Law of Large Numbers
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Topics

Law of Large Numbers

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