Class 6 — What insurance is worth

Tuesday, September 15

This recap comes from a recording of class, transcribed with Whisper and summarized with Claude, with a light edit from me. It’s meant as a convenience rather than a careful document, so the phrasing will sometimes be a little off. If something here looks wrong, please tell me and I’ll fix it.

What we covered

Certainty equivalent, risk premium, and willingness to pay, with the $100,000 example worked in class, then what moves the risk premium and why that makes high-risk pools hard to run. A question at the end got us into unraveling a week early. We skipped the in-class problem for time.

Key points

  • The certainty equivalent is a value on the wealth axis, not a payment. The risk premium is the gap from expected wealth down to it, and willingness to pay for insurance is the bigger gap from starting wealth down to it, which is the same as expected cost plus the risk premium. This came up several times, so it’s worth one more read.
  • Expected utility only uses the two endpoints, so it’s a line. The curve comes back when you ask what certain amount gives the same utility. Invert the utility function to get there, and go a few decimals out first. Exponentiating undoes a log and nothing else.
  • The expected-cost part of what you pay is a transfer to providers. The risk premium is where an insurer’s profit can come from: risk-averse buyers, a risk-neutral insurer.
  • Willingness to pay keeps rising with the probability of illness, but the risk premium peaks near a coin flip and shrinks as illness becomes near certain. A high-risk pool is almost all expected cost and very little risk premium, so there’s little room for profit.
  • Raising the price after a bad year loses the healthiest enrollees first, which leaves the price too low again. That’s unraveling, and next Tuesday does it properly.

One clarification

In the worked example the full willingness to pay is $5,426, the $5,000 expected cost plus the $426 risk premium.

Thursday

Short quiz, open note, then the simulation. Calculations on expected value, certainty equivalent, risk premium, and willingness to pay are fair game. On the homework, the New Jersey enrollment history sits under the tabs labeled historical below the recent quarters, and quarterly enrollment is the same people each quarter, so pick one quarter per year rather than adding them.