# Is it rational to buy insurance if I have the same sum in a savings account?

I considered to buy a house-contents insurance. The total value of the contents is quite small, say X. Meanwhile, I have X in a savings account, that I can withdraw at any time. I am not sure whether it is rational to buy such insurance at all. My rationale is as follows.

Suppose the price of the insurance is Y per year. Since the insurer wants to profit, this means that, according to their calculations, the probability of a total loss of my house contents (e.g. by fire or earthquake) is p, which is smaller than Y/X. This means that:

• If I buy insurance, I pay Y per year for sure;
• If I do not buy insurance, I pay X once in 1/p years on average, which means X*p on average per year, which is less than Y.

Is this consideration correct?

• To be clear, this is a question about insuring your property in the house, not insuring the house itself? Oct 7, 2021 at 19:56
• @user662852 Yes. The house itself is not mine (I live on a rent). Oct 7, 2021 at 20:20
• A good way to reduce insurance payments is to do it partways: Increase your copay. Most companies will be more than happy to reduce your monthly cost in trade for a higher copay if there is damage to the house. This is basically saying that you do self assurance for any values lost up to the value of the copay, values exceeding will be insured. Oct 8, 2021 at 6:57
• Is money.stackexchange.com/questions/84411/… a dupe or just very close? Oct 8, 2021 at 7:55
• I think you also make the assumption that you only will make use of the insurance once. What if your house burns down twice? It's probably not likely but you should keep in mind multiple possible events that makes you use your insurance for which the sum of the events will make you go over the amount of X Oct 8, 2021 at 12:14

It will generally be true that the expectation value of insurance is negative (so that the insurer makes a profit). This is true regardless of whether you have X (or any amount) in savings.

If you have an uninsured loss, then whether you pay X out of pocket to replace your belongings, or simply accept the loss of your belongings, the overall loss is about the same.

The relevant point is whether X is an acceptable loss. Typically, a loss becomes unacceptable (and worth insuring) if it either is a significant fraction of (or greater than) your net worth, or requires a cash outlay that is a significant fraction of (or greater than) your liquidity.

So, if you would definitely replace your belongings, even X in savings may not be enough to forgo insurance, if it would leave you with little or no remaining savings to meet other needs. If you could and would do without most of your belongings, and/or would not suffer an unacceptable personal or financial impact, then the insurance may not be necessary.

• @ErelSegal-Halevi It's not rational vs irrational. It's up to you to decide what level of coverage you are comfortable with. If you want to spend less and have less coverage that's perfectly acceptable. Oct 7, 2021 at 22:19
• @HartCO: I think that's a bit unfair. It's perfectly reasonable to frame this question in terms of rational/irrational decision theory. However, a rational purchase of insurance will be based on expected utility value, rather than expected dollar value. I'm a big fan of this approach as a response to people who seek overly personalized advice, since it provides a more formal explanation for why such answers have to come from the asker (i.e., because we don't know the asker's utility function). Oct 8, 2021 at 13:16
• @MichaelW.: As Brian has already explained, insurance is only irrational if you assume that everyone's utility function is linear. But most people have nonlinear utility functions (i.e. if I find \$100 on the ground, I'm going to feel differently about it depending on how much money I already have - if I'm broke, \$100 means I can eat tonight, but if I'm rich, then \$100 may not have a significant impact on my lifestyle), so this assumption is usually invalid. Oct 8, 2021 at 20:08
• @MichaelW.: Utility is only about feelings. It's an inherently subjective measurement of one's happiness. That's literally the whole point of utility theory. Oct 8, 2021 at 20:28
• Also consider that a potential loss might not happen in isolation. A natural disaster might destroy your house and also leave you too injured to work for a while. Your savings not only need to cover the property loss, but also cover the gap between your lost wages and whatever disability benefits you get, plus cover temporary housing until you can find a new house. Insurance helps greatly in cases like this.
– bta
Oct 9, 2021 at 0:38

This is essentially the concept of self-insuring. Near the end of my previous car's time with me, it's book value was pretty low. Low enough that it didn't make sense to me to pay for insurance for replacement value when I could literally pay myself that replacement value should the worst happen. Similar to your calculation there, I figured that so long as I dropped that replacement insurance and didn't wreck for a year, I ended up ahead in savings by not paying that insurance.

The company I work for considers itself self-insured for a few of its assets, i.e. the value of all the parts in a warehouse.

The key to both scenarios is to actually have the money around to be available to replace things if the worst happens. On a personal level, this is usually considered an emergency fund. I feel this is why insurance is mostly still valuable to have -- most people don't have the ability or the discipline to have a large enough fund sitting around to be truly self-insured, hence the passing of the risk of the total loss on to another entity for a price.

There is also the opportunity cost of not using those funds (possibly into significant gains like buying solid stocks, or other land/buildings that may increase significantly in value) that may also make sense to pay the premium to insure with a different entity while freeing up that savings to be used elsewhere.

• To put some numbers to the opportunity cost, the average contents insurance in the UK is £57 on £35000 worth of stuff (moneyadviceservice.org.uk/blog/…). So you would only have to find an investment paying 0.16% more than your savings account for self-insurance not to be worth it. Oct 8, 2021 at 10:16
• Should be noted, with Car Insurance you're just not insuring the value of your car, you're also insuring yourself against other liabilities if you get into an accident like damage to another person's car and their medical bills. If you're in the US the total liabilities you are insuring yourself again are likely well into the 6-figure range. Even if your car is worth \$0.
– Chuu
Oct 8, 2021 at 15:07
• Car insurance also covers damage to public and private property beyond the vehicles themselves and medical bills. If you run into someone's house, you could easily cause tens of thousands of dollars worth of damage. If you destroy the railing on a bridge or highway, that could cost millions to repair. Oct 8, 2021 at 15:22
• Chuu & computercarguy, I tried my best to denote that in the response that I was mostly talking about the replacement value insurance for the car, but it is a fair point. 'Car insurance' is a broad umbrella that covers several different things. In most states, it is illegal not to have at least liability insurance, no matter how much you may have in liquid savings. Oct 8, 2021 at 18:37
• @R.Hamilton I feel your answer is clear that you mean full coverage on your vehicle specifically, not liability coverage.
– Kat
Oct 9, 2021 at 16:55

One thing your financial analysis overlooks is that insurance companies make a profit on retained premiums by investing the money. Depending on the likelihood of a loss, they could conceivably offer insurance at a discount over your expected loss and make their profit from their investments. I believe this is a significant part of the business model of Gen Re, formerly General Reinsurance. Yes, you could invest your savings and make a profit, too, but probably the insurance company has the expertise and resources to manage their assets better than you can manage yours. So you cannot say without question that insurance is more expensive in the long run.

However, even assuming insurance comes at a premium, one can concoct a scenario where the stability of expenses provided by insurance is objectively worth the premium over expected value of losses that is charged by the insurer. This is the same rationale for the existence of commodity futures contracts. Typically those real-life situations are more about controlling the cost of inputs to a product so that the manufacturer/vendor can maintain a stable price for them, but you can probably make one up for your situation if you want to.

People, however, are not fully rational beings. "Peace of mind" is an emotional thing, but people are emotional creatures, making it rational to put a value on emotional benefit. Insurance provides the "peace of mind" of predictability. This is an especially relevant factor when the probability of loss is so low that you will not likely experience one. If the chance of losing the contents of your house is 1 in 200 years, and you only care about it for 50 years, then you with insurance you have a fixed cost of 1/4 X (plus premium) versus without insurance a 1 in 4 chance of a cost of X. Whether insurance is worthwhile in this scenario is a personal decision based on your tolerance for or aversion to risk. Personally I would prefer to pay the insurance and free up 3/4 X of my savings for other needs, but plenty of people feel that with only a 1/4 chance of an event in their lifetime, they would rather take the risk and spend the insurance premiums on something of more immediate benefit. As long as the premium is small compared to the expected loss, I do not think it is fair to call one decision more rational than the other.

• Very relevant! You buy peace of mind with the delta between value of the insured item and the insurance cost. That's why many conservatively minded people, often elderly, are "fully insured": They don't have to worry about a thing any more, and that's worth a lot to them. My pet peeve is the complement to insurance, a lottery. While playing the lottery is on average a net loss you buy taking part in a game, an excitement. Just don't view either one as an investment strategy. Oct 8, 2021 at 10:01
• The issue with peace-of-mind is that, when you buy insurance and something bad happens, you might have to quarrel with the insurance company about the exact amount. They might send you to collect all the receipts of all the property that you lost, etc. In some cases you might even have to go to court, which is the opposite of peace-of-mind. In contrast, if you self-insure, based on money in a savings account, then there is no hassle - if something bad happens, just take the money from the savings account. Oct 8, 2021 at 10:17
• @ErelSegal-Halevi you can look at it in many ways. You might have to quarrel with the insurance company, true, but you also might not be able to retain your savings indefinitely. Perhaps the building you live in collapses with you in it and you end up in hospital and need all your savings to make up for lost wages, leaving you nothing left to replace the contents of your home. From what you have written, it seems that you personally would prefer to self-insure, and I would say that is a reasonable decision: no more and no less reasonable and rational than buying insurance.
– IRR
Oct 8, 2021 at 10:36
• @IRR "the building you live in collapses with you in it and you end up in hospital and need all your savings to make up for lost wages, leaving you nothing left to replace the contents of your home" - note that property insurance would only cover your belongings, and you won't have money in the savings account which you have spent on insurance premiums, possibly leaving you worse off than having no insurance at all. Oct 8, 2021 at 11:01
• @DmitryGrigoryev But the payout to cover your belongings is almost certainly more than double what you've paid in premiums (annual premiums may be <1% of the coverage value). With insurance, you're left with the value of your belongings minus the premiums, which is very likely more than the premiums alone. Not sure if you're making a comment on the timing of the insurance payout, suggesting that it won't be available immediately after a catastrophe like savings would be. But it would be rare for a person who has a catastrophe to be worse off having insurance. Oct 8, 2021 at 14:13

Something nobody mentioned is that your risk may be higher than the average risk of the insured person, on which the insurer's premiums are based.

As a simple example, insuring your own car may make sense if you drive much more than the average person, or if you know that you are a bad driver. (OK, nobody thinks that. But if they did, they should buy insurance.)

With respect to the house contents it's a matter of which kind of damage or loss is insured and whether that damage is likelier to occur than on average, without the premiums reflecting that. E.g., if theft is insured and all your neighbors have had break-ins last year, by all means insure your belongings. It's economically sensible because the premiums don't reflect your specific high risk.

• Thanks, this is an interesting consideration. On the other hand, thieves usually do not take all property - they won't take the heavy furniture, the many books, etc. So even if the risk of theft is higher, it might not justify the premium for all the contents. Oct 8, 2021 at 10:10
• @ErelSegal-Halevi: It's very common for damage by a thief to far exceed the value of the goods stolen. It's not enough to only insure based on the value that will be stolen. Oct 8, 2021 at 13:22

This is an argument against insurance inspired by a comment by Dmitry under IRR's answer. I didn't want to conflate it with my other answer advocating insurance when there is an above-average risk, hence a second answer.

The argument is that if you are self-insured it's up to you how much of an assumed damage you would actually replace, or in what way. As an example, assume you have an "own damage policy" car insurance which covers damage caused by storms etc. If a tree branch falls on your car in a storm and causes a minor dent and paint damage you go to a garage, tell them it's covered by insurance, and they'll exploit all covered repairs to the maximum. The insurance premiums are based on this assumed behavior of policy holders.

If, by contrast, you are not insured you may decide to do nothing at all! The dent is barely visible. Or you go to a cheap non-brand garage and tell them to just fix the paint in order to prevent rust.

The decision what to replace in which fashion may also be influenced by your specific situation at this point. If you want to spend the money for something else than a full repair you are free to do so.

The economic view is:

• An insurance is a one-fits all: It's not (perfectly) custom-tailored to your needs. It covers some risks that are not important to you, and offers remedies that may exceed your normal choices. You basically overpay for services you don't really need.
• An insurance changes the behavior of the insured, creating higher risks in the insured population than in the uninsured. You pay for that behavior change in other people over which you have no control.
• Money paid as insurance premiums is tied to that specific purpose and cannot be redirected. Self-insuring keeps you in control over your funds.

The general idea of insurance, in my opinion, is to cover disasters. Most people can't afford to keep enough money in reserve for major, unexpected, and unavoidable expenses. If you have \$X in your savings account, it's presumably because you need that money to cover your regular living expenses. If you have a catastophe that requires you to spend \$X to fix the house, what will you use to buy food once you depleted your savings? Insurance allows you to continue living on your savings and earnings.

And if you actually do have enough money to cover such an event, you're wasting it by keeping it in liquid reserves. You can do much better by investing it. If the investments are easy to sell (stocks, mutual funds) you can dip into it to cover emergency expenses, but this would mean you have to sell at inopportune times, maybe without having time to research the best options, and you'll likely incur capital gains taxes as a result. Self-insuring like this is not really a good option for most people.

A good example is medical insurance. In typical years, your medical expenses might be a few hundred dollars for routine doctor visits and medicine. For a family or a person with some chronic conditions this might reach into the thousands. These may be affordable to pay out of pocket, so you might wonder why you're paying several times more than that for insurance.

But medical costs can be very unpredictable. You could have a heart attack or get cancer, and the costs of treating these are likely to be in the hundreds of thousands. Unless you're rich, this can easily bankrupt you. Maybe if you're lucky, your savings from not paying all those insurance premiums over the years would cover these expenses, but that's hard to say. Can you be so disciplined that you don't spend those savings on frivolous things like vacations and more "stuff"? So in this respect, buying insurance is kind of like putting money in a retirement account -- you're sure the money will be there if you need it. Although unlike a retirement account, if you're lucky enough to never be severely ill, you may come out behind financially.

Of course, the above doesn't apply to places with socialized medicine, and is the major reason for socialized medicine. But the general idea applies to any insurance, although the amounts tend to be lower. This will affect the necessity of paying for insurance. Most people advise against buying extended warranties for small electronics and appliances, because the replacement cost is relatively affordable. But if your home is detroyed in a fire, that would be a disaster comparable to a heart attack (if you have a mortgage, they may even require you to have insurance, because they don't want you to abandon the home and leave them with the worthless husk).

If you have decent savings, you can use it to make a compromise. Most insurance plans have a "deductible" -- insurance doesn't start paying in full until after you've spent that amount out of pocket in a particular year. You can get lower premiums by agreeing to a higher deductible.

• " In typical years, your medical expenses might be a few hundred dollars for routine doctor visits and medicine. " - shows a distinct USA bias :) Oct 8, 2021 at 14:53
• @AakashM True, this obviously doesn't apply to socialized medicine. Oct 8, 2021 at 14:54
• "Most people can't afford to keep enough money in reserve for major, unexpected, and unavoidable expenses" <-- not only can't they - they shouldn't be able to. If you attempt to to do so, you're amassing at least a couple orders of magnitude of wealth more than you actually need, just to deal with unlikely worst-case events. This leads to all kinds of destructive/antisocial behavior, and there is fundamentally no way more than a small portion of people can do it at once. The value offered by insurance (or non-insurance social safety nets) is not having to do that. Oct 8, 2021 at 22:51
• @R..GitHubSTOPHELPINGICE Good point. I may have been thinking also of less liquid funds like investments. Oct 8, 2021 at 22:56
• @AakashM Maybe a first world bias, but not a specifically US bias. If you live in a country with socialized medicine, that doesn't change what your expenses are, it just changes who pays them. Oct 9, 2021 at 2:48

Not exactly. The math is different on the two sides of the deal.

You're comparison makes sense in terms of deciding whether you are willing to pay for it. When the loss of X happens (if it ever actually happens), would you rather have a few Y (or potentially many times over if the loss is in the distant future) and have to pay X to recoup your loss, or be out Y such that you never have to pay X (even if there is never such a loss where you would anyway)? Assuming that Y is calculated based accurately on the risk that you specifically will suffer a loss of X, it really comes down to how you feel about carrying the risk of X yourself vs paying a fair price for someone else to carry the risk.

In actuality, that's not *really *how Y is calculated. The insurance company isn't just taking on your risk; they're covering X (and X-like values) for a whole pool of customers, with the assumption that they will pay X in at least some cases (losing money on that particular deal). They calculate Y such that the premiums collected across the entire risk pool more than covers any payouts. Insurance companies generally have a lot of data and expert math factored into their calculations of risk and price, and expect that they will make a profit even if they pay out X to you without receiving anywhere near that amount from you in premiums.

Essentially, using Y/X to estimate likelihood of a loss won't be accurate, because that's not how Y was calculated. Really what you want to consider is "can I tolerate a loss of X if it happens?" and "how much would I be willing to pay to not take that risk?".

• I disagree with part of this. The point of the "data and expert math" is for the insurer to estimate OP's risk as well as possible. This is how Y is calculated. This will be combined with many other customers' risks to give the insurer a predictable profit (total profit expectation is the sum of profit expectation for each customer, whereas relative variance is much less than for individual customers). But it's optimal to price each customer based on the best estimate of their individual risk (plus profit margin). Oct 7, 2021 at 21:59
• I don't see that this answer is providing a coherent point. I can steelman it to say "Maybe you have a factual basis for believing that you level of risk that is significantly different from the risk class that the insurance company has assigned you to", but if that's what you're saying, why don't you come out and say it? Oct 9, 2021 at 2:45
• @Acccumulation yeah, it's not my best answer (rewrote it several times and kinda lost it's focus). It still kinda makes its point though, especially with help from the comments Oct 9, 2021 at 2:58

If you actually do a valuation of the replacement cost of your property (on the same basis as the insurance would cover, e.g. new-for-old), you might be surprised at the cover you need. The little stuff adds up even if you wear cheap clothes (for example)

You should also consider whether the insurance you would buy is needed to cover anything else - e.g. you may need cover against liability to your landlord, and this could be a condition of your tenancy. The administrative costs come from the insurers take over the expected loss (which also includes their profit), but the admin costs for a more restrictive policy aren't much less, so you might save less than you think by dropping to any required minimum.

Similarly there may be useful but not legally or contractually required cover included in your policy. There certainly is in my combined buildings and contents (but I own the house with a mortgage so I'm in a different position to you)

Insurance companies have lawyers and adjusters, they'll likely pay out less of their money than you would in the same circumstance.

In simple terms you are right. The insurance company has to make a profit out of insuring you, so they have to charge more than they expect to payout. What you're missing is they also have far more ways to recoup losses than just dipping into their reserves.

One big argument in favor of buying insurance is they can spread the cost of litigation and negotiation over thousands or millions of people. Meaning they're much more likely to be able to pay you out of someone else's money.

In the case of home contents insurance, they may be able to go after the building manager if the building wasn't maintained and it caused a flood or a fire. Also, be sure you have Home Owners or Renters Insurance.

EDIT

One reason "Self Insured" comes up so often with old cars is you'd get very little payout for full coverage. From an insurer's perspective, they'd pay more in investigation and lawyer fees than the payout.

Especially for home, the insurance is likely going to be low. Can you buy a new bed, all new clothes, a new TV, computer and gaming console, and couch all at once?

• So you say that my premium Y may be actually smaller than my expected loss X*p? Oct 11, 2021 at 2:03
• @ErelSegal-Halevi Don't count on it, but an insurance company is much better at negotiating payouts than you. Oct 11, 2021 at 2:40

"Rational" can be a little tricky to define. One reasonable approach used by economists is the VNM axioms, which say that preferences are consistent, preferences are transitive, no outcome is infinitely good or bad, and a preference for A over B holds regardless of some third outcome. Under these assumptions, you can prove the VNM utility theorem, which says that it's possible to assign a utility function U to various outcomes, such that a rational actor will always act so as to maximize the expected value of the utility function.

However, it is not in general true that your utility function equals the amount of money you have. For example, if you lose X dollars and it causes you to go bankrupt and become homeless, that's really bad, whereas gaining X dollars might not even produce much of a noticeable improvement in your life. In these situations, the utility function might be a highly nonlinear function of how much money you have.

People are usually risk-averse. Part of this risk-averseness is rational by the VNM definition, as described in the example above. Part of it is actually irrational, as described in work by people like Daniel Kahneman. For example, a person who is given a coffee mug as a thank you gift for participating in a psychology experiment will often refuse to sell it for a given price X, whereas if you had offered earlier to let them buy it for X, they would have refused.

Anyway, whether the reasons are rational or irrational by the VNM definition, people are generally willing to pay a significant amount of money for the privilege of avoiding some risk. This is why, for example, people are still willing to put their money in a bank account rather than investing every penny they have in the stock market. So if you're insuring the contents of your house, you are certainly losing in the sense of not maximizing your expected wealth, but that may still be a natural thing to do because it's reducing your risk.

In real-life terms, consider that making an insurance claim is an insane hassle, takes a long time, and will sometimes result in your claim being denied. Insurance companies make it that way because they want to discourage claims. This is really the reason I would not self-insure the contents of my house just in order to reduce my risk.

• "consider that making an insurance claim is an insane hassle, takes a long time, and will sometimes result in your claim being denied. This is really the reason I would not self-insure the contents of my house" --- I do not understand the implication: if making an insurance claim is so hard, then this should be a good reason to self-insure! Oct 11, 2021 at 2:06

Insurance is all about risk management. The question should never be can you afford to pay, but rather do you want to take the risk. Insurance is paying to eliminate or reduce the consequences of a risk.

Let’s take the two most obvious insurance types, life and home. If it was about affording, for life insurance, you would never get it, because you are dead and afford no longer applies. For home insurance, it’s mainly mandated by lenders who could in fact easily afford to pay for the loss (i.e just write it off), but don’t want to take that “risk” (in quotes because at scale it’s not a risk it’s a certainty, if you insure a million homes some of them WILL be destroyed by disasters).

How much are you willing to pay so that you won’t have to use your savings to replace the contents of your house? This is a tricky calculation, because it of course depends upon how likely you are to have to do so. If it was 100% guaranteed to happen, then the logical amount would be cost of contents -1. If it was 100% guaranteed to not happen, then of course the logical amount would be zero.

But it’s somewhere in between, so you have to judge both the risk and the consequences to determine how much you are willing to pay.

Insurance is not a gamble, it is purchasing a time limited guarantee. How much that guarantee is worth is up to you.