When does the law of large numbers stop protecting an insurer's aggregate profit?

## The argument being tested An insurer, a casino or any operator of many small repeated bets makes the same pitch: each individual policy or wager is risky, but the expected margin is positive, and with enough of them the law of large numbers turns a per-unit edge into a near-certain aggregate result. The average outcome per unit converges to the expected margin, so total profit is roughly the margin times the volume. That argument is correct — under its assumptions. A senior answer is a tour of the assumptions and what each failure does to the book. ## Failure 1: dependence collapses your effective sample size The law assumes independent draws. Insurance risks are frequently correlated by construction: - Geographic concentration: one hurricane triggers claims on every policy in the county at once. - Shared cause: a pandemic, a recession, a supply-chain failure, a single legal precedent that reopens a whole class of claims. - Common process: the same faulty underwriting rule applied to every policy written last year. When outcomes share a driver, the number of *independent* pieces of information is far smaller than the policy count. Writing 200,000 policies in one flood plain is closer to one large bet than to 200,000 small ones. The counting of exposures is the easy part; the honest question is always "how many independent bets is this really?" ## Failure 2: heavy tails slow convergence to a crawl Even with genuine independence, convergence speed depends on how variable the individual outcomes are. Claim severity is typically strongly right-skewed: most claims are modest, a few are catastrophic, and the average is dominated by the rare huge ones. With such a distribution the observed average can sit well below the true mean for a long time — you appear profitable while under-reserved — and then be re-set by a single event. In the extreme, a distribution with no finite mean has no value for the average to converge to at all, and averaging never stabilises. The operational consequence: an observed loss ratio over a short history is a weak estimate of the true one when severity is heavy-tailed, and management confidence should be scaled to that weakness, not to the policy count. ## Failure 3: the distribution is not stable "Identically distributed" quietly assumes the risk you are writing today is the risk you priced. It usually is not: - **Adverse selection**: if your price is low relative to competitors for a given risk class, the applicants who accept are disproportionately the bad risks, shifting the mean. - **Moral hazard**: being insured changes behaviour, and therefore changes the loss distribution. - **Drift**: climate, construction costs, medical inflation and litigation norms all move the mean over the years a book is held. Converging beautifully to last decade's mean is not a defence. ## Failure 4: the long run is longer than your solvency The law of large numbers is a statement about a limit. It says nothing about the path taken to reach it, and an operator with thin capital can be wiped out by an early adverse run even when the expected margin is genuinely positive. This is the ruin problem, and it is why capital adequacy is a separate discipline from pricing: pricing establishes that the mean is favourable, capital establishes that you survive the variance long enough for the mean to matter. "We are positive expected value" is an answer about the destination, not about whether you arrive. ## Failure 5: the mean itself is a model output The law converges to the true expected value, not to the number in your pricing model. If the model is wrong — a mis-specified frequency assumption, an omitted correlation, a coverage clause interpreted more broadly by a court than by an actuary — volume delivers convergence to a mean you never intended to accept. Scale multiplies model error rather than diluting it. ## What restores the conditions - **Diversify across genuinely uncorrelated exposures** — different perils, geographies and lines of business — so that effective sample size grows with the policy count. - **Cap individual exposure** with policy limits, deductibles, and reinsurance for the tail, which truncates severity and makes the average behave much better. - **Hold capital sized to the tail**, not to the mean, using scenario and stress analysis for the correlated events the average conceals. - **Re-underwrite continuously**, monitoring for drift and selection effects rather than assuming the priced distribution still holds. ## Answering this well Do not simply say "the law of large numbers protects insurers". Name the four assumptions, give one concrete way each fails in a real book, and finish on the point interviewers are actually probing: a positive expected margin plus large volume is a claim about the limit, and the business also has to survive the path.