# Rational risk-assessment decision framework: Should I buy health insurance?

Update: I've posted a sub-question on the math stack exchange to help answer part of this primary question. Sub-question: Toy problem: How much should I rationally be willing to pay for this hypothetical and simplified insurance?

Question (roughly): My feeling is that health insurance is insanely over-priced (for me) and not at all worth it for me to buy. But how can i know? How do i answer that question rationally? What variables / factors should be considered?

(If you're curious: cost is approx \$8,400 / yr premium + ~\$14,000 / yr deductible for spouse + me total for "Bronze" level coverage (this is in U.S.)).

Of course the answer to this will be "fuzzy" (there will be some gray area between the "yes, buy" and "no, don't buy" outcomes). That's okay. I just want some basis outside of my head for which to make a decision. I don't want to just say "That looks so ridiculously expensive, there's no way i'm going to pay for that."

Ideally there would be some tool where i could enter a number of personal questions (about age, health, financial assets, income, location (relevant for health risk?), etc), it would have the probabilities of risk for various diseases, etc, and would then output some reasonable cutoff range for insurance cost for me. I could then compare that price to what i would actually need to pay to buy insurance and then make a rational decision about whether or not i should buy the insurance.

Possible variables / factors to consider:

• my health.
• probability of needing medical assistance (how likely am I to break an arm, get hit by car, get cancer, have a heart attack, etc).
• what is cost of coverage (without insurance) for the things I may need medical assistance with?
• my assets.
• my income.
• what is my comfort level with risk (low, high, etc)?

If assessment calculator can tell me "yes, buy" or "no, don't buy", then it should know:

• insurance cost: premiums, deductible
• insurance coverage (ie what is covered)

What methods can I use (probability tree, risk calculator, etc) to take the variables into consideration rationally and yield some reasonable price that I could consider as the cutoff for whether or not I should buy insurance?

other keywords: rational decision making, logic, probability, health care

• Basically, this is the very complex job of an Actuary that insurance companies use to decide what level their premiums should be (before adding a multiplying factor to give them a profit margin). I doubt (but would be happy to be proved wrong) that anything accurate-enough to be of use would be freely available. Commented Feb 11, 2019 at 16:04
• What probability of being forced into bankruptcy because of medical debt is an acceptable risk to you? Commented Feb 13, 2019 at 21:35
• One problem is that it is often very difficult to determine "cost of coverage without insurance" because there is no transparency in medical pricing. This makes it difficult to get started with a probabilistic analysis. Commented Feb 13, 2019 at 21:58
• @TripeHound On the surface it seems similar to actuarial analysis, but an actuary is evaluating risk across large numbers of people, so the law of large numbers applies and they can calculate a reasonable expected value for their actual costs. But i am asking: how should i rationally evaluate risk as an individual? in other words, the actuary (as insurer) asks: "how much should we charge?" whereas i (as an individual) ask: "how much should i (be willing to) pay?" A full solution to the problem may rely on much of the same data, but analyzed with different methods. Commented Feb 14, 2019 at 18:18
• @Kevin I would like the level of acceptable risk to be a variable in the model. That is, i would like to be able to play around with changing its value to see how it affects the outcome (the decision that should rationally be made based on the model). Is it a significant variable in the model or of relatively little importance, etc? Commented Feb 14, 2019 at 18:21

## 6 Answers

A simple model of insurance is that it minimizes risk by lowering both the expected value of your gains (increasing the expected value of your losses) while decreasing the variance of those gains. The most insurance can do is to pay 100% of every covered claim with no limitations, for a presumably rather large premium. The least you can do is to buy no insurance and accept all the risk (and the potential for minimal losses).

Suppose the premium for the covered period is P and the insurance covers a fraction r of any covered claim with no limitations. Furthermore, assume the probability of covered claim e[i] occurring during the period is p[i] and incurs a loss c[i]. To keep things simple, assume for the moment each event can happen at most once per period. Then, the expected value E[loss] of the loss is the sum of all terms (1 - r) x p[i] x c[i] / n, while the variance is the sum of all terms pow((1 - r) x p[i] x c[i] - E[loss], 2) / n, where n is the number of terms. In the foregoing, the premium should be included as one of the covered claims which will occur with probability 1 and cost P / (1 - r). Note that if r = 1, the mean is P and the variance is 0.

The only rational reason to buy insurance in this model is if you prefer a higher guaranteed minimum loss but a lower variance. If you had an arbitrary amount of money - much higher than the mean cost plus several standard deviations - you'd have no reason to buy insurance; it would be rational to self-insure. We must add to our model a notion of a "cliff" beyond which additional losses are sustained in addition to those of the claims themselves: bankruptcy, death, losing the 401k, etc. To keep the model simple we imagine a single discontinuity at loss L, after which an additional cost C is incurred. At this point, we can work out the expected loss for each insurance plan (P, r) by adding a term C x P(loss >= L) to the expected value from earlier. At this point, the lowest expected loss wins the day.

For example, suppose I can get a cold with probability 50% and incur costs \$100, and I can have a stroke and incur costs of \$320,000 with probability 0.05% chance, over the next year, and that I can pay \$6000 in premiums for 75% coverage. If I have to spend more than this, I am ruined; I am a materialistic hedonist and value my own life immensely, say at a billion dollars. Where I live, failure to pay means you're left to die. I have relatively liquid savings of \$100,000 I can use to self-insure in a pinch. With no insurance my expected loss is \$210; with insurance it is \$6052.50. Without insurance, the probability of dying due to failure to pay is 0.05%, so we add \$500,000 to the expected value of no insurance; with this insurance, the bank cannot be broken, so the expected value remains the same. With these parameters, it makes sense to buy the insurance.

You may have noticed this really hunges on the cost of going over the cliff. This is what people mean by risk tolerance. In the example above, the purchaser was not risk tolerant: a very high cost was associated with the cliff. For instance, a family man in good health with a generous life insurance policy may have no fear of death and assign a small value, say even zero, to going over the cliff. In that case, it would be foolish to buy health insurance for \$6,000 per year; he might rationally choose to die if he gets a stroke.

It's my hope this is the sort of framework within which to reason quantitatively about insurance you were looking for.

• This is a good model (and answer). Complex enough to address the primary issues and provide some insight for rational decision making, but simple enough to be practically useful. Commented Feb 14, 2019 at 18:40
• Do you have any suggestions for further reading? The majority of information available elsewhere seems to be written from the perspective of the insurer, not the insuree. Or it is written to address the risk management of a large entity (eg a company). In other words, there seems to be little written about how an individual should rationally assess risk, like the model in your answer adresses. Commented Feb 14, 2019 at 18:42
• @cpotter I put this together extemporaneously based on thinking it through when I read your thought-provoking question. I suppose I am drawing on decision theory and game theory primarily. It makes sense to me that any literature you find on this will be from the seller's perspective; they're the ones with an interest in making money. You might find something similar to this by looking into the question of investing, where r is hoped to exceed 1, P are fees and C is a bonus for making a goal. How does an investment banker rationally decide whether to invest in something? Maybe very similar. Commented Feb 14, 2019 at 19:00
• @cpotter The reasons for that are that individual risk can't be predicted very well in most cases (we can know how many people like you, out of 100k, experience a random health event, but we can't put a very precise number on your personal actual risk), and your choices for indemnification are limited, arbitrary, and usually uneven. Further, your options for treatment can radically change based on whether or not you are insured. Budget and financial risk tolerance are usually going to be more important than your health risk in producing an estimate like you are seeking. Commented May 8, 2019 at 18:18

Medical insurance in the US is a little crazy because people routinely expect it to cover all sorts of little things, every visit to the doctor's office, every prescription, etc.

Let's ignore that for the moment and think just of the more "normal" part of insurance: protection against an unusual but potentially huge expense.

Suppose you buy fire insurance for your house. Say your house costs, whatever, \$100,000. Then if your house burns down, the insurance company pays to replace your house. But most people go their whole lives without their house ever burning down. So is it a waste of money to buy fire insurance? You will PROBABLY never get anything from it. But if you do have a claim, it can be huge. For most people, if they lost their house and had no insurance, it would take decades to recover.

Last year I was paying \$740 a month for health insurance for 2 people. Wow, that's a lot of money. Is it worth it? But then I had a heart attack. The hospital bill was \$95,000, and there were expenses after that for rehab and various follow-ups. If I hadn't had insurance, that would have been a hard blow. I was 59 years old at the time. It might well have prevented me from ever being able to retire comfortably, or at least delaying my retirement for years.

An acquaintance of mine had a baby who was born with birth defects. At one point he said that the bills had just passed \$1 million. If he didn't have insurance, what would he have done?

So yeah, medical insurance is expensive. But if you don't have medical insurance, one serious illness could destroy your finances for life. It would be complicated to calculate what a reasonable amount to pay is, as you'd have to consider every medical condition you could possibly get and what it would cost to treat it if it happened. I certainly don't consider \$700+ per month a bargain, but I think it's worth it considering the huge risk.

Update

There's no real formula for what is or is not a good deal. It comes down to a subjective decision of how much risk you are willing to take.

Some people try to use "mathematical expectation" here: take the probability of each possible outcome, multiply by the gain or loss in each case, and add them all up treating gains as positive numbers and losses as negative number. If the total is positive, it's a good deal; if it's negative, it's a bad deal.

Like suppose someone offered to play this game with you: You roll a die. If it comes up even, they pay you \$2. If it comes up odd, you pay them \$4. So 50% x +\$2 + 50% x -\$4 = -\$1. The "expected" outcome, i.e. the average outcome if you played many many times, is that you lose \$1. Bad game.

But while this sounds very concrete and mathematical, it's not a useful guide in real life. Suppose I offered you this game: You roll a die. If it comes up 6, you pay me \$100,000. If it comes up 1 to 5, I pay you \$26,000. I will only play with you once. Would you play? By mathematical expectation, it's 1/6 x -100,000 + 5/6 x +26,000 = 5,000. The mathematical expectation is that you gain \$5,000. Nevertheless, I certainly wouldn't play this game. I am not willing to take the risk of losing \$100,000 on one roll of a die, even if the odds are in my favor.

What is the probability that you will have a medical problem that would wipe out your life savings? I don't know. I'm sure the insurance companies have calculated this to several decimal places: that's how they set rates.

Even if by mathematical expectation, \$7,000 a month would be a good deal, I wouldn't do it because that wouldn't leave me much to live on. I'd take the gamble. \$700 a month is a lot but, in my opinion, worth it considering the risk. I can't give you a formula.

• while you're right, that logic works for \$1400 a month as well as it does for \$700. Heck it works for \$7000 a month. So within the framework of "you need catastrophic coverage", how do you evaluate a particular option to know if it's ok or not? Commented Feb 13, 2019 at 14:50
• @KateGregory It doesn't work for \$7k a month. The argument is that without health insurance, your saving could be wiped out. But unless you're rich, \$7k/month definitely will wipe out your savings. Commented Feb 13, 2019 at 17:37
• 7,000 a month still takes 100 months to use up a million. Sure, it becomes something you can't do, but it will always be cheaper than spending a million dollars because you got cancer. Commented Feb 13, 2019 at 17:39
• @KateGregory See my update.
– Jay
Commented Feb 13, 2019 at 20:24
• That is an excellent point that just because the expected value of a game is in your favor does not imply that you should rationally play the game, since it may take a large number of trials until the actual value approximates the expected value. Commented Feb 14, 2019 at 18:25

what is cost of coverage (without insurance) for the things i may need medical assistance with?

Cancer costs something in the neighborhood of \$1,000,000. A complicated child birth can be \$250,000.

The issue is that about 70% of claim dollars are paid toward treatment of about 10% of the covered people. The remaining 30% or so of claim dollars will go toward the people who need a routine surgery, spend a night in the ER or take maintenance medications.

Yes, the cost of mending a broken arm isn't justified by \$9,000 of annual premium for you and your wife. Whether you realize it or not, that's not what you're insuring.

Separately, the plan you cited probably has a variety of things that are covered prior to meeting your deductible; basic office visits and physicals and probably prescription drug copays. Also, you don't have a \$14,000 deductible, you and your wife each have a \$7,000 deductible (and really, it's probably actually closer to \$6,000).

Part of the 2014 healthcare reform did away with underwriting. In theory, you could go without coverage then upon diagnosis of a heinous disease you could wait until November to sign up for coverage for the following year. But the idea that you could just self insure your cost of care isn't reasonable because most people haven't the slightest clue how big the numbers get.

• That's an interesting point that "upon diagnosis of a heinous disease you could wait until November to sign up for coverage for the following year". I wonder how practical this is? I mean, surely there are many problems that would require immediate medical attention or at least near future medical care. So how much of your risk would really be mitigated by this choice? Commented Feb 14, 2019 at 18:30

All insurance is a waste of money until you need it.

Surprisingly, prior to Obamacare, the #1 cause of bankruptcy in the US was catastrophic medical illness. With a greater number of people subsequently having insurance or Medicaid, it's no longer #1. Quality of life, life expectancy, etc. are worse for people without health insurance thought that tends to be skewed toward lower income people.

The short answer is:

"You've got to ask yourself one question: 'Do I feel lucky?' " ~Dirty Harry

• The bankruptcy data is generally quoted to support this idea includes all bankruptcy petitions with any medical bills which hardly proves causality.
– quid
Commented Feb 11, 2019 at 0:16
• Does your data suggest that having health insurance reduces the number of bankruptcies due to catastrophic medical illness? Commented Feb 13, 2019 at 14:00
• Without a doubt, with someone else responsible for the bill(s) there will be fewer occurrences of that type of bill on bankruptcy petitions. It's very rare that a bill causes a bankruptcy. Individual bankruptcy is caused by drastic and/or prolonged reductions of income which can certainly correlate to health condition. The implication in that stat though is that the cost of care was pushing people to bankruptcy when that's not the case. Health status will always be a major (maybe most important) factor to a person's ability to earn, so if it's not the number 1 cause now, what is?
– quid
Commented Feb 13, 2019 at 22:04

Insurance is a thing for the poor. Why? Because if you have sufficient money on hand to cover your costs (whatever they may be) yourself, it is almost always better not to take an insurance (an exception is, of course, if you have a bad medical history and are likely to suffer various health complications or if for whatever reason you feel it is particularly likely for you to suffer an expensive to treat condition or series of conditions).

You need to remember that the way an insurance company profits is by extracting more money from people than they provide, meaning the average person will pay more in insurance fees than he will get the insurance company to pay for him throughout his life. In terms of probability, you are more likely to end up with a deficit than with a profit. If you are poor, it makes sense for you to get insured - profitable or not, it's better than dying because you had bad luck and no money. But if you do have the money, you do not run such a risk, and can thus enjoy the better probability of net profit (or rather a lesser chance of expenses). Just set up an emergency account where you'll keep some money on the side in case something happens - not only will you still have that money should that "something" never happen (allowing for a wider range of emergency cases than a regular insurance offers and leaving it as an inheritance to your relatives should you die), but you will also get to collect the interest.

Finally, I would advise seeking medical care outside of USA if possible due to often astronomical prices of US healthcare. I hear a lot of Americans go seek medical care to India in private clinics, as the plane ticket, accommodation, and medical fees still end up considerably cheaper than seeking treatment in the US, but if that prospect seems dubious to you, you will still end up saving money by seeking help anywhere in the EU, where you will get at least as good a service as you would get in the US. Such an approach would further lead to lowering the necessary amount in the emergency account, but it's of course just a suggestion.

In the end, my answer to your question is: Don't get an insurance if you can afford not having one.

• "if you have sufficient money on hand to cover your costs (whatever they may be) yourself"—as pointed out in other answers, a heart attack can be ~100k, birth complications ~250k, and cancer over a million, and there are plenty of other random events that could bankrupt the bottom 50+% of Americans, and put financial strain on most of the rest. The risk may be low—though I suspect much higher than you seem to think—but the consequences if it happens are catastrophic. As I asked OP above, what probability of being forced into bankruptcy because of medical debt is an acceptable risk to you? Commented Feb 13, 2019 at 23:22
• That's why I mentioned he should make sure he can afford it. I don't know OP's financial situation, so I gave advice in general. I also mentioned the possibility of seeking treatment somewhere besides US, as the prices you list are, frankly, absolutely ridiculous and are double or even quadruple the price of the same service in other countries. If OP has half a million lying about (a sum that certainly isn't too impossible to accumulate), I'd say he doesn't need to fear the risk of medical debt provided he is willing to seek treatment abroad. Commented Feb 14, 2019 at 8:27
• This answer has some interesting points (so I don't think it should be down-voted), but seems to over-simplify the problem and doesn't fully address the issue. For example: How do I decide if I'm poor? How do I decide if I can cover the cost of services without insurance? What about my risk tolerance? Maybe I'm wealthy but I don't want some unexpected medical costs to wipe out half my assets. What is the largest medical expense I could incur and with what probability? etc. Commented Feb 14, 2019 at 18:35

In some metro's all health insurance does is keep the patient out of public hospitals.

The public hospitals are not free unless the patient doesn't have anything. If the patient has financial assets or significant earnings then the public hospital will demand a payment plan.

Also, a lower-cost narrow-network health insurance essentially just puts the patient back into the public hospital system but with less financial risk to the patient.

So an expensive health insurance puts the patient into private hospitals.

But when some health insurance is estimated to pay just some percentage of the health care costs, the percentage not paid could possibly bankrupt the patient. Also, if the patient is too sick to work they might not be able to continue the health insurance.

• that doesn't seem to answer the question Commented Feb 11, 2019 at 0:06
• If the patient wants to be in a private hospital system then they should buy premium health insurance. If the patient wants no financial risk then they should buy premium health insurance. Commented Feb 11, 2019 at 0:10