When Is Insurance Worth It?

kqr

172 When Is Insurance Worth It?

19th Dec 2024

5 min read

172

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

This is a philosophical question; my spouse and I differ in views.

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.

Get insurance if you need it to sleep well at night.

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.

If you love your children, you should get disability insurance for them.

You should insure only what you cannot afford to lose.

These are the things I would say in response.

It is not a philosophical question, it is a mathematical one.
Technically, some insurance is worth its price, even when the insurance company makes a profit.
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.

Saving up the premium instead of getting insurance is making the mistake of conflating an ensemble average with a time average.
Love does not make insurance a mathematically appropriate decision. Running the numbers does.

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:

What is our current wealth W?
How much is the premium P?

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

What is the probability p_i that this event happens? and
If it does happen, what would be the uninsured cost c_i?

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:

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:

It ends up needing expensive maintenance due to its age. We expect this to happen once in the next three years, meaning there’s roughly a 33 % probability it’s needed the next year. This costs maybe $2000.
We end up riding irresponsibly and wreck it, or it gets stolen, or somehow needs to be replaced completely. Maybe there’s a 1/40 risk of this any given year, and it would cost $8000.

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:

The Kelly criterion is bad, and
The probability distribution of bad events is unknown.

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 = \sum [p_{i} log \frac{W - P - d_{i}}{W - c_{i}}]

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.

BettingFinancial InvestingKelly CriterionProbability & StatisticsPracticalWorld Modeling

Curated

172

Mentioned in

21Non-Obvious Benefits of Insurance

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[-]lincolnquirk3mo5425

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.

9Bunthut3mo

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.

2kave3mo

This doesn't play very well with fractional kelly though

2CronoDAS3mo

Human capital is worth nothing after you die, though.

[-]jefftk3mo4713

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:

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.
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.
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.
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 f

... (read more)

[-]philh3mo*2620

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 simultan... (read more)

[-]gwillen3mo2512

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 agree with you, and I think the introduction unfortunately does major damage to what is otherwise a very interesting and valuable article about the mathematics of insurance. I can't recommend this article to anybody, because the introduction comes right out and says: "The things you always believed about rationalists were true. We believe that emotions have no value, and to be rational it is mandatory to assign zero utility to them. There is no amount of money you should pay for emotional comfort; to do so would be making an error." This is obviously false.

3kqr3mo

The probability should be given as 0.03 -- that might reduce your confusion! 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. 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 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: * Kelly is not dependent on log utility -- we agree. * 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. * 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.

7tup993mo

Perhaps you should make this more clear in the calculator, to avoid people mistakenly making bad choices? (Or just change it to percent. Most people are more comfortable with percentages, and the % symbol will make it unambiguous.)

5philh3mo

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.)

1kqr3mo

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.

6philh3mo

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.

3kqr3mo

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.

2Martin Randall3mo

The example of flood vs fire insurance is explainable with the Kelly framework. Offering flood insurance is more like making one large bet. Offering fire insurance is more like offering many small bets. Under Kelly the insurance company needs a larger edge to justify a larger bet. Standard disclaimer about simplified models here.

3philh3mo

I think we're disagreeing about terminology here, not anything substantive, so I mostly feel like shrug. But that feels to me like you're noticing the framework is deficient, stepping outside it, figuring out what's going on, making some adjustment, and then stepping back in. I don't think you can explain why you made that adjustment from inside the framework. Like, how do you explain "multiple correlated bets are similar to one bigger bet" in a framework where * Bets are offered one at a time and resolved instantly * The bets you get offered don't depend on previous history ?

1Martin Randall3mo

It feels to me like a natural modeling choice. To apply simple physics to child-on-swing, model the child as a point mass. To apply Kelly to many highly correlated bets, model them as one large bet. To apply Kelly to many weakly correlated bets, model them as many independent bets. And we also need to make some modeling choice to convert insurance policies to a series of independent bets. Maybe you're saying that we need a reason to justify those choices, because bad model choices can give bad results. So we need to "go outside the Kelly framework" to get that justification. If so, I agree.

[-]notfnofn3mo255

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.

[-]Sunishchal Dev3mo1513

This seems to assume that 100% of claims get approved. How can the equation be modified to account for the probability of claims being denied?

I would guess lower cost insurance policies tend to come from companies with lower claim approval rates, so it seems appropriate to price into the calculator. I believe there are also softer elements in insurance costs like this that should be considered, such as customer service quality, but that's probably out of scope for this calculator.

2kqr3mo

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.

1Richard1213mo

A refused claim is (legally) an event that was never covered by the insurance, and is therefore irrelevant if the question is "take policy A or not at all". After all, if that event occurred without insurance, it is still not covered. However, this is important to consider when comparing different policies with different amounts of coverage. Eg "comprehensive" car insurance compared with "third party, fire, and theft". "Rates" of unpaid claims only make sense in a situation where the law allows the insurer to breach their contracts. In that situation, the value of insurance plummets, and possibly reaches zero.

1Jake_NB2mo

>A refused claim is (legally) an event that was never covered by the insurance, and is therefore irrelevant if the question is "take policy A or not at all". This implicitly assumes all legal cases are clear-cut and independent of perspective or legally-related capabilities; and also perfect knowledge of the fine print (which is desirable but not always realistic). Usually the insurance company has more relevant experience, better specialized lawyers and generally more resources to spend fighting claims than the consumer. So they can often successfully claim that your particular circumstances are not covered after the fact, even when you believe they were.

[-]romeostevensit3mo113

I like this! improvement: a lookup chart for lots of base rates of common disasters as an intuition pump?

[-]awenonian3mo104

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.

Don't be overly naive consequentialist about this. "Nothing" is an overstatement.

Peace of mind can absolutely be one of the things you are purchasing with an insurance contract. If your Kelly calculation says that motorcycle insurance is worth $899 a month, and costs $900 a month, but you'll spend time worrying about not being insured if you don't buy it, and won't if you do, I fully expect that is worth more than $1 a month.

But do be actual consequentialist about it. If the value of the insurance is more like $10, but the cost is $900, I doubt peace of mind about this one thing is worth $890 a month.

[-]Tom Davidson3mo1014

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

With that assumption the equation compares your utility with and without insurance. Simple!

If you had some other utility function, like utility = $, then you should make insurance decisions differently.

I think the Kelly betting stuff is a big distraction, and that ppl with utility=$ shouldn't bet like that. I think the result that Kelly betting maximizes long term $ bakes in assumptions about utility functions and is easily misunderstood - someone with utility=$ probably goes bankrupt but might become insanely rich AI is happy not to Kelly bet. (I haven't explained this point properly, but recall reading about this and it's just wrong on it's face that someone with utility=$ should follow your formula)

2Matt Goldenberg3mo

Is this true? I'm still a bit confused about this point of the Kelly criterion. I thought that actually this is the way to maximize expected returns if you value money linearly, and the log term comes from compounding gains. That the log utility assumption is actually a separate justification for the Kelly criterion that doesn't take into account expected compounding returns

2philh3mo

I've written about this here. Bottom line is, if you actually value money linearly (you don't) you should not bet according to the Kelly criterion.

1notfnofn3mo

From the original post: Click the link for a more in-depth explanation

1kqr3mo

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.

2philh3mo

No it's not. In the real world, money compounds and I want more of it at lower risk. Also, in the real world, "utility = log($)" is false: I do not have a utility function, and if I did it would not be purely a function of money.

1kqr3mo

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."

3Tom Davidson3mo

Yep I'm saying you're wrong about this. If money compounds but you don't have utility=log($) then you shouldn't Kelly bet

3philh3mo

I still disagree with that.

2jake_mendel3mo

I either think this is wrong or I don’t understand. What do you mean by ‘maximising compounding money?’ Do you mean maximising expected wealth at some specific point in the future? Or median wealth? Are you assuming no time discounting? Or do you mean maximising the expected value of some sort of area under the curve of wealth over time?

[-]tup993mo812

Whether or not to get insurance should have nothing to do with what makes one sleep

This (and much of the rest of your article) seems needlessly disdainful of people’s emotions.

Wealth does not equal happiness!

If it did, then yes, 899 < 900 so don’t buy the insurance. But in the real world, I think you’re doing normal humans a big disservice by pretending that we are all robots.

Even Mr. Spock would take human emotions into consideration when giving advice to a human.

3mruwnik3mo

Wealth not equaling happiness works both ways. It's the idea of losing wealth that's driving sleep away. In this case, the goal of buying insurance is to minimize the risk of losing wealth. The real thing that's stopping you sleep is not whether you have insurance or not, it's how likely it is that something bad happens, which will cost more than you're comfortable losing. Having insurance is just one of the ways to minimize that - the problem is stress stemming from uncertainty, not whether you've bought an insurance policy. The list of misunderstandings is a bit tongue in cheek (at least that's how I read it). So it's not so much disdainful of people's emotions, as much as it's pointing out that whether you have insurance is not the right thing to worry about - it's much more fruitful to try to work out the probabilities of various bad things then calculate how much you should be willing to pay to lower that risk. It's about viewing the world through the lens of probability and deciding these things on the basis of expected value. Rather than have sleepless nights, just shut up and multiply (this is a quote, not an attack). Even if you're very risk averse, you should be able to just plug that into the equation and come up with some maximum insurance cost above which it's not worth buying it. Then you just buy it (or not) and sleep the sleep of the just. The point is to actually investigate it and put some numbers on it, rather than live in stress. This is why it's a mathematical decision with a correct answer. Though the correct answer, of course, will be subjective and depend on your utility function. It's still a mathematical decision. Spock is an interesting example to use, in how he's very much not rational. Here's a lot more on that topic.

1Michael Cohn3mo

100% agree with the principle that buying peace of mind can be a good deal whether the peace of mind is quantitatively justified or not, and the broader principle that we shouldn't be disdainful of emotions. But while emotions aren't entirely rational, they're not entirely irrational either. I expect that learning and applying the method in this post will help the user feel peace of mind at a level of insurance that's somewhat closer to optimal than they would have otherwise. It's also probably useful to learn the quantiatively optimal strategy so you can be consciously aware of what premium you're paying for peace of mind, and make thoughtful decisions about how much it's worth to you. I said that buying peace of mind can be a good deal, but I'll bet there are people who -- if they could see exactly how big the risk-aversion premium they're paying is -- would decide that they'd rather deal with more anxiety and pocket the cash.

[-]Ruby3mo7-7

Curated. Insurance is a routine part of life, whether it be the car and home insurance we necessarily buy or the Amazon-offered protection one reflexively declines, the insurance we know doctors must have, businesses must have, and so on.

So it's pretty neat when someone comes along along and (compellingly) says "hey guys, you (or are at least most people) are wrong about when insurance makes sense to buy, the reasons you have are wrong, here's the formula".

While assumptions can be questioned, e.g. infiniteness badness of going bankrupt and other fact... (read more)

6Davidmanheim3mo

The math is good, the point is useful, the explanations are fine, but the embracing straw vulcan version of rationality and dismissing any notion of people legitimately wanting things other than money seems really quite bad, which leaves me wishing this wasn't being highlighted for visitors to the site.

[-]philh3mo60

Appendix B: How insurance companies make money

Here's a puzzle about this that took me a while.

When you know the terms of the bet (what probability of winning, and what payoff is offered), the Kelly criterion spits out a fraction of your bankroll to wager. That doesn't support the result "a poor person should want to take one side, while a rich person should want to take the other".

So what's going on here?

Not a correct answer: "you don't get to choose how much to wager. The payoffs on each side are fixed, you either pay in or you don't." True but doesn't... (read more)

4philh3mo

Oh, I think that also means that section is slightly wrong. You want to take insurance if log(Wyou−P)>plog(Wyou−c)+(1−p)log(Wyou) and the insurance company wants to offer it if log(Wthem)<plog(Wthem+P−c)+(1−p)log(Wthem+P). So define V(W)=log(W−P)−(plog(W−c)+(1−p)log(W)) as you did above. Appendix B suggests that you'd take insurance if V(Wyou)>0 and they'd offer it if V(Wthem)<0. But in fact they'd offer it if V(Wthem+P)<0.

[-]CronoDAS3mo50

Does the answer to "should I buy insurance" change if the interest rate that you earn on your wealth is zero or even negative?

1PoignardAzur2mo

Something about the article felt off to me, and "should I buy insurance if interest rates are zero" is a good intuition pump for why. Yes, I think you should still buy insurance. The reason I'd come up with, peace of mind aside, is losing a lot of money at once is worse than losing a little money over time, because when you lose a lot of money your options are more limited. You have less flexibility, less ability to capitalize on opportunities, less ability to withstand other catastrophes, etc.

2CronoDAS2mo

Straight-up diminishing marginal utility of wealth, then?

1PoignardAzur2mo

Not necessarily. I think any method that calculates the value/utility of your wealth as a timeless function of utility per amount will be pretty disconnected to how people behave in practice. It doesn't account for people making plans and having to scrap them because an accident cost them their savings, for instance. (But then again, I'm not an economist, maybe there are timeless frameworks that account for that.)

[-]isaac.august3mo52

I don't quite understand. Going with the worked out example in the post you link to:

To account for the compounding nature of losses, we can use the geometric expectation of total wealth instead of the arithmetic expectation of gains/losses when evaluating the alternatives. This is the mathematically correct thing to do, although I leave out the proof. If you want to dig into the gory details, see The Kelly Capital Growth Investment Criterion; MacLean, Thorp, & Ziemba; World Scientific Publishing; 2011. Yes, it’s that Thorp.
If we replace the worn out pa

... (read more)

1Paul Lang1mo

Not an expert, but to me this math checks out.

[-]FireStormOOO3mo*41

This seems like a very handy calculator to have bookmarked.

~~I think I did find a bug:~~ At the low end it's making some insane recommendations. E.g. with wealth W and a 50% chance of loss W (50% chance of getting wiped out), the insurance recommendation is any premium up to W.

Wealth $10k, risk 50% on $9999 loss, recommends insure for $9900 premium.

~~That log(W-P) term is shooting off towards -infinity and presumably breaking something?~~

Edit: As papetoast points out, this is a faithful implementation of the Kelly criterion and is not a bug. Rather, Ke... (read more)

1papetoast3mo

The math is correct if you're trying to optimize log(Wealth). log(10000)=4 and log(1)=0 so the mean is log(100)=2. This model assumes going bankrupt is infinitely bad, which is not accurate of an assumption, but it is not a bug.

2FireStormOOO3mo

Hmm, I guess I see why other calculators have at least some additional heuristics and aren't straight Kelly. Going bankrupt is not infinitely bad in the US. If the insured has low wealth, there's likely a loan attached to any large asset that really complicates the math. Making W just be "household wealth" also doesn't model "I can replace the loss next paycheck". I'm not sure what exactly the correct notion of wealth is here, but if wealth is small compared to future earnings, and replacing the loss can be deferred, these assumptions are incorrect. And obviously, paying $10k premium to insure a 50% chance of a $10k loss is always a mistake for all wealth levels. You're choosing to be bankrupt in 100% of possible worlds instead of 50%.

[-]transhumanist_atom_understander3mo30

Make 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.

[-]Luís Campos3mo20

This is great! Is the code hosted in any git repository?

1kqr3mo

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

[-]Anna Vernerová3mo21

As others have pointed out, I've been utterly confused by the linked calculator not telling me whether to use probabilities on 0-1 or 0-100 scale (and accepting meaningless inputs). Also a comment right on the calculator page regarding the question what constitutes wealth should be given, especially regarding the common life situations such as having a mortgage (does it count as negative wealth or not?)

I am also utterly confused by the fact that the article never mentions the strength and timeframe of the compounding effect, and never talks about the base ... (read more)

[-]jmh3mo20

Nice write up and putting some light on something I think I have intuitively been doing but not quite realizing it. Particularly the impact on growth of wealth.

I was thinking that a big challenge for a lot of people is the estimated distribution - which is likely why so many non-technical rationales are given by many people. Trying to assess that is hard and requires a lot of information about a lot of things -- something the insurance companies can do (as suggested by another comment) but probably overwhelms most people who buy insurance.

With that t... (read more)

2Richard1213mo

Indeed. In reality, the vast majority of people do not have sufficient information to make reasonable estimates of the probability of loss - and in many cases, even the size of the loss. Eg a landlord is required to rebuild the place and temporarily rehome the tenants while that's being done in the event of the house being destroyed by fire or flood. They're also liable for compensation and healthcare of affected third parties - and legal cost of determining those figures. They can probably calculate the rebuilding cost and a reasonable upper bound on temporary housing. But third party liability? So, it still comes back to vibes.

[-]Pretentious Penguin3mo21

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.

2kqr3mo

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.)

1Sune3mo

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.

2kqr3mo

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.

[-]Paul Lang1mo10

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 maximizin... (read more)

[-]recursivebuilder3mo10

Does anyone have good data sources for how often houses suffer significant damage? (I assume this must be in actuarial tables somewhere, but I'm not seeing any obvious percentage values when searching).

2kqr3mo

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.

[-]JacekLach3mo10

I don't think your case for how insurance companies make money (Appendix B) makes sense. 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.

Separately, the claim that there is no alternative to Kelly is very weak. I guess you mean there is no formalized, mathematical alternative? Otherwise, I propose a very simple one: buy insurance if the cost... (read more)

1kqr3mo

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.

[-]Rationalist112353mo10

The problem with your explanation lies in the way companies calculate the cost of insurance. They do not base it solely on the nominal value of potential losses; instead, they account for the real value. For instance, if there’s a potential loss of $1,000, insurance companies calculate the cost by considering its real value, including factors like the money they’ll collect and invest over time, leveraging the compounding effect. As a result, compounding does not make insurance profitable for the insured in the long run.

When evaluating insurance, the key fa... (read more)

[-]Milli | Martin3mo10

Nice article! I'm practice you can get positive expected value insurance though, since you know more about your risk (in some cases) than the insurance does. The other way around in some cases the insurance is better at estimating your risk and might "rip you off" if they (correctly) assume you overestimate your risk.

4JacekLach3mo

Another common case where you get positive EV insurance is when the cost paid by you and the cost paid by the insurerer, when the bad event happens, are significantly different. For example, if you get extended phone / device insurance from a manufacturer, when the device fails you would have to pay the retail price for a new device. The manufacturer however only needs to pay the production price, which given margins can be a small fraction of the retail price. Thus the manufacturer can set a premium that is (in expectation) somewhere in between those two prices, and you both benefit.

1Richard1213mo

In the majority of cases the insurer has more knowledge, as they can afford the time to do the research. This is of course why any insurance market must have a great many suppliers. If there are only one or two insurers, they can easily set the premiums to make supernormal profits because they are 'competing' against consumers with very little knowledge, instead of other insurers who have similar knowledge levels.

3Milli | Martin3mo

Their research is done on the population level, and they consider adverse selection effects somewhat (people with increased risk are more likely to insure against that risk), but it's far from perfect. I've worked in insurance and profited from insurance financially in cases where I expected the insurance to be positive EV beforehand. Professional experience (Germany): * Some insurance products run at an (accepted) deficit, as they acquire customers who then buy more (money making) insurance products. Examples: Personal Liability Insurance, Motor vehicle liability * There is significant friction between the different departments within an insurance and with the insurance agents, causing mispricing from a risk-perspective (some parties care more about raw sales volume). * Nobody really knows what they are doing. Personal experience: * Personal Liability insurance is dirt cheap in Germany and makes money off people who don't report their damages (e.g. will pay for a friend's phone or camera they dropped themselves). * Insurance for bikes in Berlin. 3 of my bikes were stolen within 3 years and the insurance costing <€100/yr covered all of them.

-2Richard1213mo

If you already know that an adverse event is highly likely for your specific circumstances, then it is likely that the insurer will refuse to pay out for not disclosing "material information" - a breach of contract. Loss leaders are common, but rely on a reasonably accurate prediction of what that loss will be. and simple errors are of course common. I think "nobody knows what they're doing" is an overstatement. Insurance companies make significant profit, and very rarely fail. The usual complaint I see from investors is that some years an insurer has made less profit than others - which is exactly what one should expect, so clearly these investors don't understand what they've invested in. Your bike example sounds like bad luck. An event with a probability of 1% per year can still happen three years running. It would be interesting to consider the inverse Kelly - given an insurance premium X, what are the reasonable bounds of assumed probability of an adverse event costing Y? I'm not sure how the wealth of the underwriter could be estimated though.

4Davidmanheim3mo

"If you already know that an adverse event is highly likely for your specific circumstances, then it is likely that the insurer will refuse to pay out for not disclosing "material information" - a breach of contract." Having worked in insurance, that's not what the companies usually do. Denying explicitly for clear but legally hard to defend reasons, especially those which a jury would likely rule against, isn't a good way to reduce costs and losses. (They usually will just say no and wait to see if you bother following up. Anyone determined enough to push to get a reasonable claim is gonna be cheaper to pay out for than to fight.)

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