Insurance Deductibles and Covid-19 Counts

I was thinking about the boring topic of insurance deductible pricing. The deductible is the part of the claim that the insured customer is responsible for. E.g. if you select a $1,000 deductible and you make a claim on $5,000 worth of damage to your house, you cover $1,000 and the insurance company covers $4,000. (Insurers like deductibles because they discourage small, frivolous claims. Insurance customers like them because they make the premiums more affordable. Win-win!)

We actuaries have to be careful about the “unseen claims.” If everyone has a $1,000 deductible, we don’t know how many claims there would have been between 0 and 1,000. Those would-be claims don’t enter our data sets. They are censored from our view, because the customers don’t bother to make them. (The term for these losses that don’t show up at all is “truncated”; “censored” is reserved for a policy limit, in which you know about the claim but don’t know how big it would have been if not for the limit.) But it’s okay, because we have an approach that mathematically “re-inserts” these claims. It’s an application of basic probability theory. We have to make some assumptions about the shape of the severity curve below the deductible. This takes the form of assuming the distribution type: is it gamma, log-normal, Pareto? But you have to do this if you want to offer, say, a $500 deductible or full coverage. You can’t just assume those unseen claims don’t exist. You make various assumptions about those would-be claims that never come to your attention, and mathematically they reenter your data set. Perhaps you test the sensitivity of the result to those assumptions, but you still have to adjust. 

I was very disappointed to realize that epidemiologists either don’t have these methods or don’t take them seriously. I still see people reporting the “case fatality rate”, or CFR, as if it’s a meaningful metric of the disease’s deadliness. It’s not. The CFR divides known deaths by known cases. A case only becomes known if it comes to the attention of healthcare workers. We disproportionately find out about severe cases that lead to an office visit, hospitalization, or death. The mild and (apparently common) asymptomatic cases are censored from our view and don’t enter the CFR’s denominator. (Unless we did extensive random testing, which we’re not yet doing.) I have seen some attempts to adjust the raw CFR downward, but still ending up with a fatality rate that’s way too high. I’m seeing that some epidemiologists are building in the assumption that 1% or 2% of people who get infected eventually die. I think this is way too pessimistic. It is more alarmist still to report the raw CFR.

Time will tell how big a deal this is. My reading of the Princess Diamond cruise ship and the data from Iceland (which has done extensive random testing) is that there is a broad distribution of disease severities, including lots of mild and even asymptomatic cases. The true mortality could be in the range of the seasonal flu, or it might be a few times higher. There is still a lot of uncertainty. Still, I think it’s just needlessly alarmist to exaggerate the true risk by taking raw data, with all it's well understood flaws, at face value. I can't tell if people actually find the high estimates of Covid-19's mortality to be credible, or if they're trying to achieve some kind of "consistent messaging" to get across to a public with a short attention span. If anyone is consciously taking this approach, I want to gently suggest that such deceptions are likely to backfire. The noble lie gets found out. Look at the backlash against the messaging about face masks being ineffective. "Don't buy them, because they're ineffective. Also, medical staff need them more than you." People will start to resent it if their public health bureaucracy is making value judgments and trade-offs for them and "juking the stats" to back into their desired policy recommendations.

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I know that Covid-19 could be worse than the flu even if it has a similar fatality rate. Maybe it's worse because a lot more people get it, or because the lingering damage to the lungs is worse than any lasting effects of a bad case of the flu. None of that justifies exaggerating the mortality risk.

Suppose insurance losses follow a distribution, a "density function", f(x). The density function tells you the relative likelihood of observing a claim at a given dollar value. The cumulative distribution function, called F(x), is the integral of f(x). F(x) is the probability of a claim being x or less.  To find the shape of f(x), you would gather a bunch of observed losses, x1, x2, x3, and so on. The likelihood of observing a loss at x1 is f(x1). Assuming the losses are independent, the likelihood of the observed set of losses is the product f(x1)*f(x2)*...*f(xn). You can use this likelihood function to estimate the parameters of your distribution and find out the shape of f(x).

With no deductible, f(x) just means "I observe a claim of x dollars." If there is a deductible, say $1,000, and the total cost of the damage is x (insurer's share plus the insured's $1,000 share), the observation enters the likelihood function as f(x) / [1 - F($1,000)]. Which is like saying, "I observe a claim of x dollars given that it's greater than $1,000 dollars."

Early-stage epidemiology could benefit from this kind of thinking. Stop thinking of the known case counts as "I observe a case of Covid-19." It is rather "I observe a case of Covid-19, given that it is above a severity threshold that brings a patient to the hospital." Granted there is uncertainty initially about the distribution of severities, some intelligent guesswork can convert "known cases" to a (possibly wide) range of true cases.