Risk Premium and Willingness to Pay

Ian McCarthy | Emory University

Motivation

In 2018, Humana exited the ACA exchanges due to an “unbalanced risk pool based on the results of the 2017 open enrollment period”. CNN Money Article

Fun fact: Humana and Aetna’s merger deal had just been blocked. NPR Article

Motivation

spend

visits

Today’s Roadmap

  1. Certainty Equivalent
  2. Risk Premium
  3. Willingness to Pay for Health Insurance
  4. Examples
  5. Determinants of Risk Premium and WTP

Certainty Equivalent

  • The guaranteed wealth level that yields the same utility as a risky prospect
  • Defined by \(U(CE) = E[U(w)]\)
  • For risk-averse individuals: \(CE < E[w]\)

Certainty Equivalent as a Welfare Measure

  • Expected utility is not directly comparable across individuals or policies
    • “Utility” has no dollar units
    • Hard to report welfare changes in $
  • The certainty equivalent (CE) translates expected utility into money

Certainty Equivalent as a Welfare Measure

  • How CE enters formal analyses (intuition):

    utility in plan choice: \[U_{ij} = V_{ij} - \alpha_i\,\text{prem}_j + \varepsilon_{ij}.\] A $1 increase in any premium changes utility by \(-\alpha_i\) utils \(\Rightarrow\) 1$ \(\equiv \alpha_i\) utils.

    money-metric welfare change: for policy \(\mathbf p \to \mathbf p'\) \[ \Delta CE_i \;=\; \frac{1}{\alpha_i}\Big[\Lambda_i(\mathbf p') - \Lambda_i(\mathbf p)\Big], \quad \Lambda_i(\mathbf p) \equiv \log\!\sum_{j}\exp\!\big(V_{ij}-\alpha_i\,\text{prem}_j\big). \] Divide by \(\alpha_i\) to convert “utils” to dollars.

Certainty Equivalent in Practice

  • Applications in counterfactuals (reported in $):
    • IO: consumer surplus under alternative market structures
    • Health: welfare from premium subsidies/benefit changes across plans
    • Labor: risky jobs, wage contracts, or policy reforms

Risk Premium

Risk premium definition

The risk premium is the difference between expected wealth and the certainty equivalent (CE). It measures the cost of risk for a risk-averse individual: \[ \pi \;=\; \mathbb{E}[w]\;-\;CE. \]

This concept is distinct from willingness to pay for insurance; we use it as one component of WTP.

Depiction of Risk Premium

Willingness to Pay

WTP definition

The willingness to pay (WTP) for health insurance equals the actuarially fair premium (expected cost of illness) plus the risk premium: \[ \text{WTP} = \mathbb{E}[\text{cost}] + \pi. \]

Interpretation: people pay their expected costs plus an extra amount to remove uncertainty.

Example

Consider the utility function, \(u(w)=\ln(w)\). An individual starts with a wealth of $100,000. With probability 0.25, this person will get sick and incur a cost of $20,000.

  1. Calculate expected wealth, \(E[w]\).
  2. Calculate expected utility, \(E[U(w)]\).
  3. Calculate value of wealth that gives you \(u=E[U(w)]\) (based on the utility function).
  4. Calculate the risk premium as the difference between (1) and (3).
  5. Calculate maximum willingness to pay by adding the risk premium and the expected cost.

Answer

We’re asked to find some wealth level, \(y\), such that the person is indifferent between \(y\) with certainty versus the risky wealth levels, \(w_{h}=\) $100,000 with probability 0.75 or \(w_{s}=\) $80,000 with probability 0.25.

  • Expected wealth is: \(E[u]=0.75\times 100,000 + 0.25 \times 80,000 =\) 95,000

  • The person’s expected utility is: \(E[u]=0.75\times \ln (100000) + 0.25 \times \ln (80000) =\) 11.4571396.

  • We need \(y\) such that \(u(y)=\) 11.4571396. Given our utility function, this is satisfied for \(y=\) $94,574. This is the certainty equivalent.

  • The risk premium is then the expected wealth less the CE, or $ 425.84.

  • Since the person starts with $100,000, the full WTP for health insurance is $100,000 \(-\) $94,574 \(=\) $5,425.8, which is also the sum of the actuarily fair premium of $5,000 and the risk premium of $425.84.

In-class Problem: Demand for insurance

Assume that utility takes the log form, \(u(x)=ln(x)\). If someone is healthy, they maintain their current wealth of $100, and if they become ill, they must incur a cost of $50. Answer the following questions based on this setup.

  1. Calculate the risk premium and WTP based on a probability of illness of 0.1.
  2. Repeat part (1) using a probability of illness of 0.2.
  3. Repeat part (1) using a probability of illness of 0.5.
  4. Explain how these values differ and why. What might this say about the profitability of insurance in a market with many sick people?

Determinants of Risk Premium and WTP

What affects the risk premium?

Based on the graph, what do you think are some things that might affect the risk premium and WTP?

  1. Curvature of the utility function
  2. Probability of illness
  3. Cost of illness

Other issues

High risk pools

  • A “high-risk pool” is a way to put people that are more likely to incur high medical costs all in one plan.
  • Recalling the curvature of demand function, probability of illness, and cost of illness…do you think a high-risk pool is sustainable (think about the profit to the insurer)?

Let’s look at this in practice, KFF High-risk Pools

Other reasons to buy health insurance

  1. Increase bargaining power with providers

  2. Manage where care is delivered (due to information problems in health care decisions)

Main takeaways

  1. Calculate risk premium and maximum willingness to pay for health insurance
  2. Explain and show graphically how changes in the utility function, probability of illness, and the cost of illness affect the risk premium
  3. Discuss how these factors (preferences, probability of illness, and cost of illness) help us in understanding the effects of real-world health insurance policy.