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Last time I checked, E[ln(X)] = ln(G(X)) where G(X) is the geometric expectation of X
Not an expert, but to me this math checks out.
First of all, thank you very much for this thought provoking post. I'm not sure if I've arrived at the right conclusions here, but it seems that
we have made a $12 profit and the insurer has made a $37 profit
is not technically correct. The reason is that you cannot stick units into a logarithmic function. So what you get out on the left hand site is unitless.
But there is another way to think about this. As stated on Wikipedia, "In probability theory, the Kelly criterion (or Kelly strategy or Kelly bet) is a formula for sizing a sequence of bets by maximizing the long-term expected value of the logarithm of wealth, which is equivalent to maximizing the long-term expected geometric growth rate." So if you have a log utility function, you could say you've gained 12 times the value of a reference util.
So I think the following statements remain true from a certain perspective:
Technically no insurance is ever worth its price, because if it was then no insurance companies would be able to exist in a market economy.
if "worth its price" means "linear price" (or quasi-linear, i.e. sufficiently small price so that the curvature of the log doesn't kick in) and you are not a particularly high-risk customer, of course.
Instead of getting insurance, you should save up the premium you would have paid and get compounding market return on it. The money you end up with is on average going to be more than whatever you’ll end up claiming on the insurance.
is true if you want to maximize expected wealth instead of expected log(wealth).
What is indeed false, however, is the heuristic I used for myself up until now.
You should insure only what you cannot afford to lose.
because for a log(wealth) optimizer there are scenarios where insurance makes sense, as shown in the example of the motorbike or the helicopter.
As optimizing log(wealth) seems reasonable to me, I'm very grateful for your post, cause I finally have a precise formula to maximize log(wealth), so I no longer need to rely on the above heuristic.