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Difficulty: MediumLaw of Large Numbers

An insurance actuary is evaluating two potential books of business for the company to underwrite. Portfolio X consists of 40,000 similar mid-sized sedans used exclusively for personal commuting. Portfolio Y also consists of 40,000 vehicles, but includes a highly diverse mix of personal cars, commercial long-haul trucks, agricultural tractors, and high-performance motorcycles. The actuary recommends prioritizing Portfolio X because it better aligns with the Law of Large Numbers. Which of the following best explains why the actuary's recommendation is correct?

  1. The principle relies on pooling a large number of homogeneous exposure units to accurately predict future aggregate losses and establish stable premiums.Answer
  2. B
    Portfolio X represents only pure risks, whereas the varied vehicle types in Portfolio Y create a speculative risk environment where the insurer could potentially profit from the losses.
  3. C
    The diverse exposures in Portfolio Y would legally require the primary insurer to act as a ceding company and transfer all direct liability to a reinsurer.
  4. D
    Selecting Portfolio X is an example of risk avoidance, which is the most statistically predictable method of handling risk in the insurance industry.

Answer

The actuary recommends Portfolio X because the Law of Large Numbers requires pooling a large number of homogeneous (similar) exposure units to accurately predict future aggregate losses and establish stable premiums.
The Law of Large Numbers is a statistical principle stating that as the number of similar (homogeneous) exposure units increases, the actual losses will more closely approach the expected losses. This predictability allows actuaries to establish appropriate premium rates. Portfolio X provides the necessary homogeneity, while Portfolio Y's extreme diversity undermines the mathematical reliability of aggregate prediction.

Step-by-Step Solution

1
Identify the fundamental components of the Law of Large Numbers as it applies to insurance.
The law requires a sufficiently large sample size of exposure units that are substantially similar (homogeneous).
This allows the statistical probability of a loss occurring to become more predictable in the aggregate, even though individual losses remain unpredictable.
2
Compare Portfolio X and Portfolio Y against these fundamental components.
Both portfolios have a large number of units (40,000), but Portfolio X has similar units (homogeneous sedans) while Portfolio Y has diverse units (heterogeneous vehicles).
To apply statistical probability accurately, the units must share similar risk characteristics.
3
Determine why Portfolio X is the better choice for the insurer.
Because Portfolio X consists of homogeneous exposure units, the insurer can rely on the Law of Large Numbers to predict aggregate losses and set accurate premium rates.
A diverse pool like Portfolio Y lacks the similarity needed for actuarial tables to accurately predict expected losses across the entire group.

Key Concept

The Law of Large Numbers requires a large pool of homogeneous (similar) exposure units to accurately predict expected aggregate losses.
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Topics

Law of Large Numbers

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