Insurance Payouts

A conceptual actuarial algorithm would start with a probabilistic model of the insurable incidents and then try to balance the policy owner's potential payouts against the reserves built up from the incoming premiums. As an objective, an insurance company wants to always keep its head above water, by keeping the balance positive:

Balance = Reserve - Payouts

For the earthquake model, a simple expected pay-out scheme would multiply the strength of the earthquake (S) against the probability of it occurring p(S). In other words, the larger the earthquake, the more the damages and the more the payout.

dPayout/dt = UnitPayout * RateOfEarthquakes * Integral (p(S)*S ds)

The entroplet for the earthquake:

p(S) = c/(c+S)^2

leads to an indefinite integral will diverge as a logarithm if S extends to infinity:

c * [ ln(c+S) + c/(c+S) ]

An infinite payout won't happen due to physical constraints but it does demonstrate precisely why the fat-tail of low-probability events have such an impact on an actuarial algorithm. They essentially drive the time-averaged payout to a surprisingly large number over a policy-owner's lifetime. (In contrast, a thin-tail probability distribution such as a damped exponential will pay-out only on the average earthquake size and not the maximum)

Should the insurer's balance ever go negative, a reinsurance company picks up the rest of the payout. The whole artifice sits precariously close to a Ponzi scheme, and the insurers optimally hope to push the payout as far in the future in possible, praying that the big one won't happen. In California, earthquake insurance does not come automatically with a homeowner's policy. This also explains why insurance companies typically don't offer flood insurance. The 100-year floods occur too often. In other words, the "fat-tail" probability distributions for floods and insurance generate too high an occurrence for supposedly rare events and the companies would end up losing if they offered only affordable premiums to make up their monetary reserve. The alternative of higher-priced premiums would only attract a fraction of the population.

I don't know if I can yet model all the extraneous game theoretic aspects, but if you had an insurance policy against earthquakes, the insurance actuarian would consider this model quite useful. They couldn't tell you when the earthquake would happen, but they could reasonably predict the size in terms of a probability (essentially related the slope of the above curve). The problem comes in when you consider the risk of the insurance company (or the reinsurance company) not covering the spread in the face of a calamity. It happened to AIG and the financial collapse of 2008 forced the government to bail them out.

PostEdit:

This essentially explains why fat-tail distributions can wreak havoc on our predictions. If we know that a given data set follows a thin distribution such as a Normal or Exponential, a large outlier will not affect the results. But for a fat-tail distribution, when a "gray swan" outlier occurs, it will act to sway the expected value considerably, especially if we did not previously account for its possibility. This happens because if we want to keep the mean to a finite value, we have to place limits on our integration range. Yet once we find a data point outside this range, we have to update our average with this knowledge, and this will push the average up. This does not happen with the thin-tail functions because any new data always stays within range and it will only update the mean if you treat it as a Bayesian update of the entire data set. So the trade-off lies between keeping a large enough range to accommodate gray swans versus keeping the range small so as not spook people with large premiums.

Again, this always reminds me that with catastrophic accidents, we don't want to see the fat-tails and gray swans. Yet with a goal of finding more oil, a gray swan becomes a desired outcome. Finding another super-giant that will force us to update our fat-tail statistics keeps the cornucopians fueled with optimism. The finite hope does exist, and that may forever prevent us from facing reality.