This is pretty useful!
I note that it assigns infinite badness to going bankrupt (e.g., if you put the cost of any event as >= your wealth, it always takes the insurance). But in life, going bankrupt is not infinitely bad, and there are definitely some insurances that you don't want to pay for even if the loss would cause you to go bankrupt. It is not immediately obvious to me how to improve the app to take this into account, other than warning the user that they're in that situation. Anyway, still useful but figured I'd flag it.
I think the solution to this is to add something to your wealth to account for inalienable human capital, and count costs only by how much you will actually be forced to pay. This is a good idea in general; else most people with student loans or a mortage are "in the red", and couldnt use this at all.
This is good! But note that many things we call 'insurance' are not only about reducing the risk of excessive drawdowns by moving risk around:
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There can be a collective bargaining component. For example, health insurance generally includes a network of providers who have agreed to lower rates. Even if your bankroll were as large as the insurance company's, this could still make taking insurance worth it for access to their negotiated rates.
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An insurance company is often better suited to learn about how to avoid risks than individuals. My homeowner's insurance company requires various things to reduce their risk: maybe I don't know whether to check for Federal Pacific breaker panels, but my insurance company does. Title insurance companies maintain databases. Specialty insurers develop expertise in rare risks.
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Insurance can surface cases where people don't agree on how high the risk is, and force them to explicitly account for it on balance sheets.
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Insurance can be a scapegoat, allowing people to set limits on otherwise very high expenses. Society (though less LW, which I think is eroding a net-positive arrangement) generally agree that if a parent buys health insurance for their child then if the insurance company says no to some treatment we should perhaps blame the insurance company for being uncaring but not blame the parent for not paying out of pocket. This lets the insurance company put downward pressure on costs without individuals needing to make this kind of painful decision.
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Relatedly, agreeing in advance how to handle a wide range of scenarios is difficult, and you can offload this to insurance. Maybe two people would find it challenging to agree in the moment under which circumstances it's worth spending money on a shared pet's health, but can agree to split the payment for pet health insurance. You can use insurance requirements instead of questioning someone else's judgement, or as a way to turn down a risky proposition.
Annoying anecdote: I interviewed for an entry-level actuarial position recently and, when asked about the purpose of insurance, I responded with essentially the above argument (along the lines of increasing everyone's log expectation, with kelly betting as a motivation). The reply I got was "that's overcomplicated; the purpose of insurance is to let people avoid risk".
By the way, I agree strongly with this post and have been trying to make my insurance decisions based on this philosophy over the past year.
I like this! improvement: a lookup chart for lots of base rates of common disasters as an intuition pump?
Whether or not to get insurance should have nothing to do with what makes one sleep – again, it is a mathematical decision with a correct answer.
I'm not sure how far in your cheek your tongue was, but I claim this is obviously wrong and I can elaborate if you weren't kidding.
I'm confused by the calculator. I enter wealth 10,000; premium 5,000; probability 3; cost 2,500; and deductible 0. I think that means: I should pay $5000 to get insurance. 97% of the time, it doesn't pay out and I'm down $5000. 3% of the time, a bad thing happens, and instead of paying $2500 I instead pay $0, but I'm still down $2500. That's clearly not right. (I should never put more than 3% of my net worth on a bet that pays out 3% of the time, according to Kelly.) Not sure if the calculator is wrong or I misunderstand these numbers.
Kelly is derived under a framework that assumes bets are offered one at a time. With insurance, some of my wealth is tied up for a period of time. That changes which bets I should accept. For small fractions of my net worth and small numbers of bets that's probably not a big deal, but I think it's at least worth acknowledging. (This is the only attempt I'm aware of to add simultaneous bets to the Kelly framework, and I haven't read it closely enough to understand it. But there might be others.)
There's a related practical problem that a significant fraction of my wealth is in pensions that I'm not allowed to access for 30+ years. That's going to affect what bets I can take, and what bets I ought to take.
The reason all this works is that the insurance company has way more money than we do. ...
I hadn't thought of it this way before, but it feels like a useful framing.
But I do note that, there are theoretical reasons to expect flood insurance to be harder to get than fire insurance. If you get caught in a flood your whole neighborhood probably does too, but if your house catches fire it's likely just you and maybe a handful of others. I think you need to go outside the Kelly framework to explain this.
I have a hobby horse that I think people misunderstand the justifications for Kelly, and my sense is that you do too (though I haven't read your more detailed article about it), but it's not really relevant to this article.
I'm confused by the calculator.
The probability should be given as 0.03 -- that might reduce your confusion!
Kelly is derived under a framework that assumes bets are offered one at a time.
If I understand your point correctly, 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.
Of course, the insurance calculator does not offer you the interface to enter a periodful of simultaneous bets! That takes a dedicated tool. The calculator can only tell you the ROI of insurance; it does not compare this ROI to alternative, more complex portfolios which may well outperform the insurance alone.
If you get caught in a flood your whole neighborhood probably does too
This is where reinsurance and other non-traditional instruments of risk trading enter the picture. Your insurance company can offer flood insurance because they insure their portfolio with reinsurers, or hedge with catastrophy bonds, etc.
The net effect of the current practices of the industry is that fire insurance becomes slightly more expensive to pay for flood insurance.
I have a hobby horse that I think people misunderstand the justifications for Kelly, and my sense is that you do too
I don't think I disagree strongly with much of what you say in that article, although I admit I haven't read it that thoroughly. It seems like you're making three points:
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Kelly is not dependent on log utility -- we agree.
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Simultaneous, independent bets lower the risk and applying the Kelly criterion properly to that situation results in greater allocations than the common, naive application -- we agree.
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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.
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.
1Make sense. I suppose we assume that the insurance pays out the value of the asset, leaving our wealth unchanged. So assuming we buy the insurance, there's no randomness in our log wealth, which is guaranteed to be log(W-P). The difference between that, and our expected log wealth if we don't buy the insurance, is V. That's why log(W-P) is positive in the formula for V, and all the terms weighted by probabilities are negative.
It seems to me that another common and valid reason for insurance is if your utility is a nonlinear function of your wealth, but the insurance company values wealth linearly on the margin. E.g. for life insurance, the marginal value of a dollar for your kids after you die so that they can have food and housing and such is much higher than the marginal value of a dollar paid in premiums while you’re working.
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.)
That is assuming you live sufficiently long. The point in life insurance is to make sure you leave something for your kids/spouse if you die soon.
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.
TL;DR: If you want to know whether getting insurance is worth it, use the Kelly Insurance Calculator. If you want to know why or how, read on.
Note to LW readers: this is almost the entire article, except some additional maths that I couldn't figure out how to get right in the LW editor, and margin notes. If you're very curious, read the original article!
Misunderstandings about insurance
People online sometimes ask if they should get some insurance, and then other people say incorrect things, like
or
or
or
or
or
These are the things I would say in response.
The last quote (“things you cannot afford to lose”) is the closest to being true, but it doesn’t define exactly what it means to afford to lose something, so it ends up recommending a decision based on vibes anyway, which is wrong.
In order to be able to make the insurance decision wisely, we need to know what the purpose of insurance really is. Most people do not know this, even when they think they do.
The purpose of insurance
The purpose if insurance is not to help us pay for things that we literally do not have enough money to pay for. It does help in that situation, but the purpose of insurance is much broader than that. What insurance does is help us avoid large drawndowns on our accumulated wealth, in order for our wealth to gather compound interest faster.
Think about that. Even though insurance is an expected loss, it helps us earn more money in the long run. This comes back to the Kelly criterion, which teaches us that the compounding effects on wealth can make it worth paying a little up front to avoid a potential large loss later.
This is the hidden purpose of insurance. It’s great at protecting us against losses which we literally cannot cover with our own money, but it also protects us against losses which set our wealth back far enough that we lose out on significant compounding effects.
To determine where the threshold for large enough losses is, we need to calculate.
Computing when insurance is worth it
The Kelly criterion is not just a general idea, but a specific mathematical relationship. We can use this to determine when insurance is worth it. We need to know some numbers:
Then we need to estimate the probability distribution of the bad events that could occur. In other words, for each bad event ii we can think of, we estimate
We’re going to ignore the deductible for now because it makes the equation more complicated, but we’ll get back to it. We plug these numbers into the equation for the value V of the insurance to someone in our situation:
V=log(W−P)−(1−∑pi)logW−∑[pilog(W−ci)]If this number is positive, then the insurance is worth it. If it is negative, we would do better to pay the costs out of our own pockets.
Motorcycle insurance
In a concrete example, let’s say that our household wealth is $25,000, and we’ve just gotten a motorcycle with some miles on it already. Insuring this motorcycle against all repairs would cost $900 per year. We might think of two bad events:
Assuming no deductible, would this be worth it? Yes! Solving the equation – or entering the parameters into the Kelly insurance calculator – we see that we should be willing to pay a premium of up to $912 in this situation. If our wealth had been $32,000 instead, the insurance would no longer have been worth it – in that situation, we should not spend more than $899 on it, but the premium offered is $900.
The effect of the deductible
In the same example as above, now set a fixed deductible of $500 for both events, and watch the value of the insurance plummet! Under those terms, we should only accept the insurance if our wealth is less than $10,000.
Helicopter hovering exercise
To test your knowledge, we’ll run with one more example.
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 $10,000 kr to repair the helicopter, if that happens.
You can get insurance before you go, which will cover $6,000 of helicopter damage (so even with insurance, you have to pay $4,000 in addition to the insurance premium if you crash), but cost you $150 up front. Do you take it?
You probably know by now: it depends on your wealth! There’s a specific number of dollars in the bank you need to have to skip the insurance. Whipping out the Kelly insurance calculator, we figure it out to be $34,700. Wealthier than that? Okay, skip the insurance. Have less than that? It’s wise to take the offer up.
It’s not that hard
I am surprised not more people are talking about this. Everyone goes around making insurance decisions on vibes, even as these decisions can be quite consequential and involve a lot of money. There’s just a general assumption that insurance decisions are incalculable – but the industry has calculated with them for at least seventy years! Are people not a little curious how they do it?
More specifically: until now, there has been no insurance calculator that actually uses the Kelly criterion. All others use loose heuristics. Who thinks that leads to better decisions?
Appendix A: Anticipated and actual criticism
I think there are two major points of disagreement possible in the description above:
Both of these points are technically true, but not as meaningful as their proponents seem to think.
Yes, the Kelly criterion is too aggressive for most people, who do not value maximum growth over all else. Most people want to trade off some growth against security. The correct response here is not to throw the baby out with the bathwater and ignore Kelly entirely – the correct response is to use a fractional Kelly allocation. This can be done quite easily by entering a lower wealth in the Kelly insurance calculator. See the Kelly article for more discussion on this.
The probability distribution of anything is unknown, but this is not a problem. Good forecasters estimate accurate probabilities all the time, and nearly anyone can learn to do it.
But, perhaps most fatally, the people who oppose the method suggested in this article have not yet proposed a better alternative. They tend to base their insurance decisions on one of the incorrect superstitions that opened this article.
Appendix B: How insurance companies make money
The reason all this works is that the insurance company has way more money than we do. If we enter the motorcycle example with no deductible into the Kelly insurance calculator again, and increase our wealth by a factor of ten, we see the break-even point moves down to $863. This is the point where the insurance starts being worth offering for someone with 10× our wealth!
In other words, when someone with 10× our wealth meets us, and we agree on motorcycle insurance for $900, we have made a $12 profit and the insurer has made a $37 profit.
It sounds crazy, but that’s the effect of the asymmetric nature of differential capital under compounding. This is the beauty of insurance: deals are struck at premiums that profit both parties of the deal.
Appendix C: The relativity of costs
The clever reader will also see that if we set the deductible to be event-dependent, and create a virtual event for when nothing bad happens (this event has a deductible and cost of zero), a lot of the terms are similar and can be combined. Indeed, the equation can then be given as
V=∑[pilogW−P−diW−ci]This, perhaps, makes it clear that it is not the absolute size of the wealth that matters, but its size in proportion to the premium, deductible, and cost of events.