The probability should be given as 0.03 -- that might reduce your confusion!
Aha! Yes, that explains a lot.
I’m now curious if there’s any meaning to the result I got. Like, “how much should I pay to insure against an event that happens with 300% probability” is a wrong question. But if we take the Kelly formula and plug in 300% for the probability we get some answer, and I’m wondering if that answer has any meaning.
I disagree. Kelly instructs us to choose the course of action that maximises log-wealth in period t+1 assuming a particular joint distribution of outcomes. This course of action can by all means be a complicated portfolio of simultaneous bets.
But when simultaneous bets are possible, the way to maximize expected log wealth won’t generally be “bet the same amounts you would have done if the bets had come one at a time” (that’s not even well specified as written), so you won’t be using the Kelly formula.
(You can argue that this is still, somehow, Kelly. But then I’d ask “what do you mean when you say this is what Kelly instructs? Is this different from simply maximizing expected log wealth? If not, why are we talking about Kelly at all instead of talking about expected log wealth?”)
It’s not just that “the insurance calculator does not offer you the interface” to handle simultaneous bets. You claim that there’s a specific mathematical relationship we can use to determine if insurance is worth it; and then you write down a mathematical formula and say that insurance is worth it if the result is positive. But this is the wrong formula to use when bets are offered simultaneously, which in the case of insurance they are.
This is where reinsurance and other non-traditional instruments of risk trading enter the picture.
I don’t think so? Like, in real world insurance they’re obviously important. (As I understand it, another important factor in some jurisdictions is “governments subsidize flood insurance.”) But the point I was making, that I stand behind, is
Correlated risk is important in insurance, both in theory and practice
If you talk about insurance in a Kelly framework you won’t be able to handle correlated risk.
If one donates one’s winnings then one’s bets no longer compound and the expected profit is a better guide then expected log wealth—we agree.
(This isn’t a point I was trying to make and I tentatively disagree with it, but probably not worth going into.)
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.
Ah, my “what do you mean” may have been unclear. I think you took it as, like, “what is the thing that Kelly instructs?” But what I meant is “why do you mean when you say that Kelly instructs this?” Like, what is this “Kelly” and why do we care what it says?
That said, I do agree this is a broadly reasonable thing to be doing. I just wouldn’t use the word “Kelly”, I’d talk about “maximizing expected log money”.
But it’s not what you’re doing in the post. In the post, you say “this is how to mathematically determine if you should buy insurance”. But the formula you give assumes bets come one at a time, even though that doesn’t describe insurance.
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.
Aha! Yes, that explains a lot.
I’m now curious if there’s any meaning to the result I got. Like, “how much should I pay to insure against an event that happens with 300% probability” is a wrong question. But if we take the Kelly formula and plug in 300% for the probability we get some answer, and I’m wondering if that answer has any meaning.
But when simultaneous bets are possible, the way to maximize expected log wealth won’t generally be “bet the same amounts you would have done if the bets had come one at a time” (that’s not even well specified as written), so you won’t be using the Kelly formula.
(You can argue that this is still, somehow, Kelly. But then I’d ask “what do you mean when you say this is what Kelly instructs? Is this different from simply maximizing expected log wealth? If not, why are we talking about Kelly at all instead of talking about expected log wealth?”)
It’s not just that “the insurance calculator does not offer you the interface” to handle simultaneous bets. You claim that there’s a specific mathematical relationship we can use to determine if insurance is worth it; and then you write down a mathematical formula and say that insurance is worth it if the result is positive. But this is the wrong formula to use when bets are offered simultaneously, which in the case of insurance they are.
I don’t think so? Like, in real world insurance they’re obviously important. (As I understand it, another important factor in some jurisdictions is “governments subsidize flood insurance.”) But the point I was making, that I stand behind, is
Correlated risk is important in insurance, both in theory and practice
If you talk about insurance in a Kelly framework you won’t be able to handle correlated risk.
(This isn’t a point I was trying to make and I tentatively disagree with it, but probably not worth going into.)
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.
Ah, my “what do you mean” may have been unclear. I think you took it as, like, “what is the thing that Kelly instructs?” But what I meant is “why do you mean when you say that Kelly instructs this?” Like, what is this “Kelly” and why do we care what it says?
That said, I do agree this is a broadly reasonable thing to be doing. I just wouldn’t use the word “Kelly”, I’d talk about “maximizing expected log money”.
But it’s not what you’re doing in the post. In the post, you say “this is how to mathematically determine if you should buy insurance”. But the formula you give assumes bets come one at a time, even though that doesn’t describe insurance.
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.
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.