The Kelly bet for any negative arithmetic EV is 0 when you are fully in control of the bet size. Sometimes you're forced to bet, and then you might want to hedge your bet to avoid large losses which will set you back more than the insurance, when you look at it from a growth perspective.

In other words, when you are choosing between "a loss" and "no loss", then the correct Kelly bet is of course "no loss".

However, when you are choosing between "a loss" and "a different loss", which is the case when it comes to insurance, then you need to whip out your slide rule and do the numbers.

This is an example I used with another group of people in another context:

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Let's say you get the opportunity to try to hover a helicopter close to ground, for whatever reason. There's a real pilot next to you who will take control when you screw up (because hovering a helicopter is hard!)

However, there's a small (2 %) chance you will screw up so bad the other pilot won't be able to recover control and you crash the helicopter. You will be fine, but you will have to pay $10 k to repair the helicopter, if that happens.

You can get insurance before you go, which will cover $6 k of helicopter damage (so even with insurance, you have to pay $4 k in addition to the insurance premium if you crash), but cost you $150 up front.

Do you pay a $150 premium to reduce an unlikely (2 %) loss of $10 k down to a still sizeable $4 k?

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If we do the arithmetic expectations, we'll find that with insurance, the expectation is negative $200 if you don't go for the insurance, and negative $230 with the insurance. So always skip the insurance, right?

Not so fast. That is correct if we had effectively infinite money in the bank. If we have an infinite amount of money in the bank, we can repeat that "no insurance" bet over and over and get the arithmetic expectation.

In practise, we don't always have an effectively infinite amount of money compared to the losses in question, so we need to consider how the growth of what we have is affected by the losses.

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The answer then, according to the Kelly criterion, is "it depends". Specifically, it depends on how much money you have in the bank.

If you have more than $35 k in the bank, then the $10 k loss is small enough to not affect the growth of your money significantly. If you have less than that, the $10 k loss is sizeable enough that it's worth spending $150 to reduce it down to $4 k.

Going the other way around, if you have $20 k in your bank, you should be willing to spend as much as $186 on the insurance, to protect against the risk of halving your available money.

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In terms of how the calculation is done: I find it easiest to do it the way Bernoulli did it back in the 1700's when he invented the mathematical formulation of the Kelly criterion: use the geometric expecation. (This is equivalent to the arithmetic expectation of the log.)

Here's the equation set up to compute what wealth is needed to decline the insurance in the helicopter example: https://www.wolframalpha.com/input?i=solve+w%5Ep+*+%28w+-+L%...

(Hopefully the symbols are obvious, but in case they are not:

- w = current wealth

- L = loss with no insurance

- p = probability of no adverse event

- q = 1 - p = probability of adverse event

- c = premium of insurance

- d = deductible of insurance)