Daniel Reeves (co-founder of Beeminder) blogged about how collision insurance is a ripoff (8 years ago, but I just found it today). He got some pushback and didn’t have the exact math to win the argument. As a professional gambler who also reads a lot of company financials I can provide the exact math for when the average driver should buy collision insurance.
To pick a random insurance company, Progressive has a 10-year average underwriting margin of 7.6%1 (that's the amount that premiums paid exceed claims). On average a collision policy is claimed against once every 17.2 years2. So on average you’re getting paid out 92.4% of what you put in, 8.6 years after you put it in. You could have instead invested the money at 6% and 1.65xed it in 8.6 years. So effectively this wager returns only 0.924/1.65 = 56% of what you put in. Insurance is a wager with a 44% disadvantage, or equivalently not-buying-insurance is a wager with a 1/0.56-1=78.5% advantage.
Whether you should buy the insurance then boils down to whether you should wager your car on a coin flip with a 78.5% advantage. You lose your car if it comes up tails, and you gain 1.785*2-1=2.57 cars if it comes up heads.
Kelly betting
Define utility as 10^(E(log(wealth))).
Optimal betting aka Kelly betting maximizes that utility function. The effect of a specific wager on that utility function is known as the wager’s certainty equivalent (CE) because getting paid that many dollars as a certainty has the same effect on utility as placing the wager. CE is the expected value (EV) minus the cost of the variance. As the size of the wager grows, the EV increases linearly, but the cost of the variance increases quadratically. That parabola of utility has a maximum somewhere, and the Kelly formula helps you find it easily.
f* = p - (1-p)/b
f* is the fraction of your bankroll you should wager
p=0.5 is the probability of winning
b=2.57 is the proportion of the bet gained on a win
Substituting:
f* = 0.5 - 0.5/2.57 = 0.305.
This is the optimal bet size and the x-coordinate of the maximum of the utility parabola. The parabola also goes through (0,0) so the other zero is at (2f*,0). Any bet size between 0 and 2f* has positive utility. So the “bet” of not-buying collision insurance has positive utility until the car is >=61% of your wealth.
https://investors.progressive.com/files/doc_financials/2022/q2/Progressive-2022-Q2.pdf
https://www.bankrate.com/insurance/car/auto-insurance-statistics/#what-types-of-auto-insurance-claims-are-filed
I think your edge from having knowledge of the drivers on the policy can outweight the house edge, speaking as someone who's "won" that bet :)
Ha! Beautiful! I'm finally vindicated :)