Sure! https://git.sr.ht/~kqr/insurance-calculator

Depends significantly on where you live! I don't worry about hurricanes, floods, earthquakes, etc.

Among the things that remain are fire, and my government says the fire services get called to 6000 domestic fires every year. Divided by a population of, say, 5 million households that's a risk of 0.12 % per year. Maybe not all fires get fire services involvement, so we'll bump it up to 0.2 %.

You won't find actuarial tables, but they can often be constructed from official sources and/or press releases with some ingenuity. We'd do this for other risks too, like burglary, water damage, etc.

Of course, we could also gut feel our way there. Maybe we consider the past 20 years, and that we'd be told if any one in a circle of 5 friends would tell us about a serious event in their household, and we have been told twice in that time. That's twice in 100 person-years, i.e. a 1/50 all-cause risk.

I agree -- sorry about the sloppy wording.

What I tried to say wad that "if you act like someone who maximises compounding money you also act like someone with utility that is log-money."

Your formula is only valid if utility = log($).

This is a synonym for "if money compounds and you want more of it at lower risk". So in a sense, yes, but it seems confusing to phrase it in terms of utility as if the choice was arbitrary and not determined by other constraints.

The insurance company does not have logarithmic discounting on wealth, it will not be using Kelly to allocate bets. From the perspective of the company, it is purely dependent on the direct profitability of the bet - premium minus expected payout and overheads.

Not true. Risk management is a huge part of many types of insurance, and that is about finding the appropriate exposure to a risk -- and this exposure is found through the Kelly criterion.

This matters less in some types of insurance (e.g. life, which has stable long-term rates and rare catastrophic events) but significantly in other types (liability, natural disaster-linked.)

This is only about maximising profit for a given level of risk, it has nothing to do with specific shapes of utility functions.

Fundamentally we are taking the probability-weighted expectation of log-wealth under all possible outcomes from a single set of actions, and comparing this to all other sets of actions.

The way to work in uncompensated claims is to add another term for that outcome, with the probability that the claim is unpaid and the log of wealth corresponding to both paying that cost out of pocket and fighting the insurance company about it.

It is under no such assumption! If you have sufficient wealth you will leave something even if you die early, by virtue of already having the wealth.

If it's easier, think of it as the child guarding the parent's money and deciding whether to place a hedging bet on their parent's death or not -- using said parent's money. Using the same Kelly formula we'll find there is some parental wealth at which it pays more to let it compound instead of using it to pay for premia.

Even so, at some level of wealth you'll leave more behind by saving up the premium and having your children inherit the compound interest instead. That point is found through the Kelly criterion.

(The Kelly criterion is indeed equal to concave utility, but the insurance company is so wealthy that individual life insurance payouts sit on the nearly linear early part of the utility curve, whereas for most individuals it does not.)

I just wouldn't use the word "Kelly", I'd talk about "maximizing expected log money".

Ah, sure. Dear child has many names. Another common name for it is "the E log X strategy" but that tends to not be as recogniseable to people.

you say "this is how to mathematically determine if you should buy insurance".

Ah, I see your point. That is true. I'd argue this isolated E log X approach is still better than vibes, but I'll think about ways to rephrase to not make such a strong claim.

what do you mean when you say this is what Kelly instructs?

Kelly allocations only require taking actions that maximise the expectation of the joint distribution of log-wealth. It doesn't matter how many bets are used to construct that joint distribution, nor when during the period they were entered.

If you don't know at the start of the period which bets you will enter during the period, you have to make a forecast, as with anything unknown about the future. But this is not a problem within the Kelly optimisation, which assumes the joint distribution of outcomes already exists.

This is also how correlated risk is worked into a Kelly-based decision.

Simultaneous (correlated or independent) bets are only a problem in so far as we fail to construct a joint distribution of outcomes for those simultaneous bets. Which, yeah, sure, dimensionality makes itself known, but there's no fundamental problem there that isn't solved the same way as in the unidimensional case.

Edit: In more laymanny terms, Kelly requires that, for each potential combination of simultaneous bets you are going to enter during the period, you estimate the probability distribution of wealth outcomes (and this probability distribution should account for any correlations) after the period has passed. Given that, Kelly tells you to choose the set of bets (and sizes in each) that maximise the expected log of wealth outcomes.

Kelly is a function of actions and their associated probability distributions of outcomes. The actions can be complex compound actions such as entering simultaneous bets -- Kelly does not care, as long as it gets its outcome probability distribution for each action.