# Health insurance is hard

Health insurance has been front-and-center in the news recently, following Congressional Budget Office (CBO) reports that the current House and Senate healthcare bills may cause millions of Americans to lose their healthcare coverage while also removing several taxes imposed by the Affordable Care Act, colloquially known as Obamacare. Insurers haven’t been silent; some have stated that the proposed restructuring of Affordable Care Act exchanges would do wonders for their business, while others have vocally advocated for retaining the individual mandate. Here, I’ll outline a simple model of a health insurance company to demonstrate why the problem of covering as many people as possible while also ensuring low premiums for those who can’t afford insurance is so difficult. I’ll also suggest guidelines for future policy.

### An illustrative model

The model I’ll detail is really quite simple; no Humana or Aetna here. The insurance company I’ll attempt to describe is the sort of company you might set up: premiums aren’t invested in a financial market, and there aren’t any employees. Let’s begin with the health insurance consumer. She has a probability distribution of incurring an illness or injury (or not-illness-or-injury; having nothing wrong with her at all) at time \(t\) given by \(p_i(x, t)\), for events \( x \in \mathbb{X}\). Associated with each event \(x\) at time \(t\) is a cost \(C_i(x, t)\). Thus, her expected discounted cost due to injury and illness over her life is given by

Like anyone, she doesn’t want to have to pay this lifetime cost. More than that, though, she wants *certainty*; she’d rather pay a fair amount of money each month than pay what some event out on the tail of the distribution \(p(x,t)\) might cost her. (I’m making the (quite reasonable) assumption that rare events are positively correlated with higher cost.)
A health insurance company is willing to purchase that risk from her at a constant price per month (or whatever other time period you like), call it \(\pi_i\), but in return for taking on this risk they’ll increase the total cost to the consumer by \(\rho_i\), the risk premium.
To find out what they should charge the consumer per month, they’ll solve for \(\pi_i\) in the following equation:

Summing the geometric series and doing some algebra, we see that she’ll be charged \(\pi_i = (\mathbb{E}[C_i] + \rho_i)( 1 - e^{-rt} ) \). So far, so good. If the insurance company does the same thing for each customer \(i\), they’ll have revenue given by \(R = \sum_i \sum_{t=0}^{\infty} e^{-rt}\pi_i\) and expected cost given by \(C = \sum_i \mathbb{E}[C_i]\) for expected profit of \(\mathcal{P} =\sum_i \rho_i\). (The reader will note that I’m talking about long-run revenue and cost here; we’ve done all of the discounting “up front”, so to speak, so that we can algebraically manipulate quantities when estimating expected profit.) Already we see the profit incentive for insurance companies to seek out consumers \(i\) with cost functions of low magnitude or event probability distributions with thin tails.

Now suppose a policy is implemented such that all consumers \(i’\) must have monthly premiums set to \(\theta_{i’} = \pi_{i’} - \delta_{i’}\), where \(\delta_{i’} > 0\). For now, we’ll assume that this policy appears for no reason at all, and thus that there’s no correlation between consumer \(i’\) and the consumer’s cost function or event probability distribution. The company can respond in one or both of two ways:

- Eat the cost; that is, let expected profits become

so that the company’s risk of ruin is higher and expected profitability is lower; or

- Charge other customers more; that is, given that
*initially*the company will have the same number of consumers before and after the policy implementation, the revenue function becomes

where \( \sum_i \delta_i = \sum_{i’} \delta_{i’} \); that is, some consumers are in effect subsidizing others. The first option is pretty clearly flawed; the company exists to make a profit, and even more important that quarterly earnings is the company’s survival. Anything that increases its risk of ruin is going to be a no-go.

The second option is flawed, too, since it will discourage those consumers \(i \neq i’\) from entering the insurance market because of rising premiums, and likewise encourage consumers \(i’\) to enter the market when they might not have earlier. This is problematic because the assumption that the assignment of a discount \(\delta_{i’}\) and \(i’\) cost function / probability distribution is not at all realistic. In fact, the usual rationale for health insurance subsidies is to provide previously-unobtainable coverage for the poor. Yet there is overwhelming evidence to support the claim that poverty and sickness are positively correlated. Thus, increasing enrollment of \(i’\) and decreasing enrollment of \(i \neq i’\) has the effect of increasing the company’s risk profile and thus its risk of ruin, along with decreasing its profitability. The result? Insurance companies will withdraw from markets that enforce these premiums—which is exactly what we see happening right now.

### Individual mandate?

There isn’t an easy fix to this problem; if there were, I wouldn’t be writing about it. The Affordable Care Act, flawed as it is, does contain one important mechanism that the current Congress would do well to retain in their bill, at least in spirit: the individual mandate. This enforces penalties for not purchasing insurance to encourage the spry, healthy consumers (\(i \neq i’\)) to purchase insurance, thus reducing the risk of ruin to the insurance companies and thereby encouraging them to stay in the market. In fact, my criticism of the individual mandate is simply that its penalties for not purchasing insurance aren’t high enough; many consumers decide they incur less disutility from paying the penalty than from purchasing insurance they feel they don’t need. The Swiss figured this out a long time ago; they mandate that all residents purchase a basic private healthcare plan, while ensuring that insurance companies don’t raise prices of these basic plans to unaffordable levels.