4 Risk Premium and WTP

4.1 Why do we care?

Let’s motivate some of this with a simple story of Humana and the ACA exchanges. In 2018, Humana withdrew from the ACA exchanges citing an “unbalanced risk pool” due to the results of the 2017 open enrollment period. The risk pool refers to the collection of policyholders and their corresponding risk levels. In this context, Humana essentially claimed that their enrollees were too sick and too expensive relative to the plan premiums. Humana had also just had their proposed merger with Aetna blocked by the Department of Justice (details here), which may have also been related to the decision to drop out of the ACA exchanges.

Managing the risk pool is critical for insurers (not just health insurers). One reason that this is so important is because of something called “community rating” which sets restrictions on how much insurers can change premiums for different enrollees in the same market. While there is some opportunity to charge some enrollees different premiums, it’s useful just to think of community rating as requiring all enrollees in a given market to be charged with the same premium. So, since insurers can’t ask some people to pay more than others, they have to set their premiums so that the revenue collected from the very healthy enrollees (those that don’t use much health care) offsets any losses paid out to providers for very unhealthy enrollees (those that use lots of health care, or those that are just very unlucky in a given year). If an insurer underestimates the share of high-need patients enrolling in a plan, then they will have an “unbalanced risk pool” and incur more expenses than expected, potentially to the point of eroding any profits from the low-need enrollees.

Figure 4.1 highlights the relationship between health care utilization and patients’ underlying health status, where we see that those self-reported to be of low health also have the highest health care expenditures and most frequent encounters with the health care system. As is clear from these figures, insurers can expect to incur more costs as less healthy patients enroll in their plans.

So insurers really need to know how to price their products appropriately. To do that, they need to understand what determines someone’s willingness to pay for health insurance. That’s where we’re headed in this chapter.

4.2 Risk Premium

The risk premium reflects how much money a risk-averse person would pay to avoid risk. In other words, it’s the amount of money that makes a person indifferent between a risky and riskless scenario. The risk premium is reflected graphically in Figure 4.2.

As an example, let’s consider someone with the utility function, $u(w)=\ln(w)$. They start with $100,000. With probability 0.25, this person will get sick and incur a cost of $20,000. Their wealth in the sick state is therefore $80,000. To find the risk premium, we need to calculate how much money they are willing to give up to avoid the risky scenario ($\pi$ in Figure 4.2). In the risky scenario, their expected utility is: $E[u]=0.75\times \ln (100000) + 0.25 \times \ln (80000) =$ 11.457. We therefore need to work backward from this level of utility (the y-axis) to find the value on the x-axis that would provide the same level of utility if instead there were no risk. To do this, we take the inverse of the original utility function at the value of $E[u] =$ 11.457. In this case, the inverse utility is the exponential — this is the operation we need to apply to the original utility function in order to recover the wealth value (i.e., $\exp(u(w))=w$). So we need to take the exponential of 11.457, which yields $w=$ 94,574. This value reflects the amount of wealth in a risk-free situation that would provide the same level of utility as they expect to have in the risky situation. The risk premium is then just the difference between these two wealth values, $\pi =$ 425.84.

4.3 Willingness to Pay

The willingness to pay (WTP) for health insurance reflects the risk premium plus the expected costs of health care. Intuitively, the cost of care is a transfer from the enrollee to the insurer. We have to pay that no matter what. For this reason, the expected cost of care is sometimes referred to as the “actuarily fair” premium. In our example, the actuarily fair premium is $5,000, calculated as the probability of illness times the cost of illness. Adding this to our risk premium yields the maximum WTP for health insurance, $5,425.8.

4.4 Determinants of WTP

From Figure 4.2, we can easily identify three factors that determine someone’s risk premium and WTP. The easiest way to think of how these factors affect the risk premium and WTP is to think about the gap between the expected utility line and the utility curve in Figure 4.2.

Curvature of the utility function. As preferences become more risk averse, utility becomes “more curved” and the distance between the expected utility line and the utility curve increases. Therefore, risk premium and WTP are increasing with risk aversion.
Probability of illness. This one is a little tricky. It helps to look at the extreme values with probability of illness set to 1 or 0 and note that the expected utility and utility curve intersect at those points. Since the risk premium reflects the gap between those curves, the risk premium is 0 when no such gap exists. So the risk premium is 0 when the probability of illness is 1 or 0. Hopefully this makes sense conceptually — there is no risk when the probability of illness is 0 or 1. We know with certainty that we will be either healthy or sick. We may not want to be sick, but we’re not uncertain about the outcome. And since the risk premium is what we would pay to avoid uncertainty, the risk premium is naturally 0 at those points. Extending that logic, the risk premium is largest for probability of illness of 0.5, as this is the point at which we face the most uncertainty about being sick or healthy.
Cost of illness. The cost of illness is reflected by the gap between $x_{1}$ and $x_{2}$ in Figure 4.2. All else equal, the gap between the expected utility line and the utility curve grows as the distance between $x_{1}$ and $x_{2}$ increases.

4.5 Other considerations

While Figure 4.2 is useful to understand how and why someone would pay in excess of the actuarily fair premium for health insurance, there are other benefits of health insurance not easily captured in the figure. One is the role of insurers in price negotiations. An individual patient has little bargaining power relative to a hospital, and therefore little opportunity to negotiate a lower price; however, an insurance plan, in pooling the bargaining power of all of its individual members, is in a better position to negotiate lower prices. Insurance products therefore also serve to reduce the cost of illness relative to remaining uninsured. Another factor involves the information problems in health care decision-making. It can be extremely challenging to identify the “best” health care providers, and insurance plans can help with this process (although whether they actually do serve to funnel patients to higher quality providers is an empirical question).