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Assuming one could instantly have access to an loan to cover his recent loss/accident at an affordable interest, would it make sense to pay premium for an insurance plan?

I was thinking that it would make more sense to save the premium money and use it to pay part of a loan to cover the losses if it happens. That way one would save the trouble of paying insurance and never using it and make better use of that money.

But I believe in economics. It wouldn't be such a big thing if it didn't make sense. So, how can one evaluate when it makes sense to pay for insurance or just count on a loan?

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    $\begingroup$ Are you assuming that the loss did not impact the ability to work or access to the labor market? $\endgroup$
    – Dave Harris
    Commented Apr 4, 2017 at 4:48
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    $\begingroup$ Isn't a major aspect of insurance the distribution of risk? And doesn't that mean that this question only relates to losses that are a) larger than immediately affordable, but b) smaller than long-term unaffordable? - and isn't that quite a narrow band? I have liability insurances that far exceed anything I can expect to cover myself. I'd never get a loan to cover what my house insurance covers, were I to suffer a rare, large loss. $\endgroup$
    – SusanW
    Commented Apr 4, 2017 at 14:05
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    $\begingroup$ It really depends what you're insuring! Don't forget that motor insurance is mostly there to insure third parties, and that buildings insurance is there to protect your mortgage lender. People generally don't have instant access to large amounts of credit. $\endgroup$
    – pjc50
    Commented Apr 4, 2017 at 14:20
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    $\begingroup$ Insurance makes sense for large incidents, not for small things, like "product protection plans", where you are going to replace the product eventually anyway. Few houses burn down, and trying to pay your existing mortgage plus one to replace the house would probably be impossible. Medical costs can be essentially unlimited, so don't even go there. $\endgroup$
    – user12110
    Commented Apr 4, 2017 at 19:24
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    $\begingroup$ Probably relevant: For very high-net worth individuals, does it make sense to not have insurance? on Personal Finance & Money $\endgroup$
    – user
    Commented Apr 4, 2017 at 19:37

8 Answers 8

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As Dave Harris' points out in his comment, I assume that your question deals with events that do not compromise the individual's ability to work, which would prevent her from taking on debt.

Let's take the example of a fire which happens with probability $\pi \in [0,1]$. The damage of a fire equals $D$ dollars, i.e. it costs $D$ dollars to repair the house. If the individual buys full insurance at a price $p$ she ends up with a final wealth equal to $w-p$ in all contingencies since the insurance company covers the damages.

Suppose now that she does not buy any insurance but contracts a loan to cover the damages. In order to repair the house she has to borrow $D$ dollars. This loan costs her $D'$ dollars over her lifetime, taking into account the interest and her time preferences (discounting). She then ends up with final wealth equal to

  • $w$ with probability $1-\pi$
  • $w-D'$ with probability $\pi$

Thus, the two distributions that she faces are different under these two solutions. If she buys insurance, she completely eliminates the risk of having to pay for the damages. By contrast, if she contracts a loan ex post she still has a large downside risk of $w-D'$ in case of a fire. Under the standard conditions of insurance theory (e.g. risk aversion, actuarially fair insurance prices) it is clear that buying insurance would be strictly preferable.

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    $\begingroup$ One could also talk about the monthly payment for insurance vs monthly payment for the potential loan. That can bring out actual reasons to be risk averse here, if the loan payments would mean a downgrade in quality of life elsewhere. $\endgroup$
    – JollyJoker
    Commented Apr 4, 2017 at 10:18
  • $\begingroup$ That kind of makes sense, but considering that the premium is a single payment and it is lower than the actual D. But what happens is that you actually pay a monthly premium, that if paid for a long time can and is calculated to cover π plus profits. In the end (w - p) will converge to something similar to (w - D') or higher. I think that the fact that you pay premium and cannot use it anymore, is also a risk. It's money that you can't get back and happens with usually a higher probability (1-π). With the credit case you are just dealing with π and can invest the money until it happens. $\endgroup$
    – Leandro Machado
    Commented Apr 4, 2017 at 18:04
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    $\begingroup$ @LeandroMachado "The fact that you pay a premium and cannot use it anymore is also a risk". It is not a risk: you know that you will pay $p$ every month for sure. This is why insurance is desirable to risk-averse individuals even if $p = \pi D$. What you are saying is that if people are sufficiently ``averse'' to spending $p$ every month for nothing with probability $1-\pi$, they would prefer borrowing money after the loss. But this is possible only if they are risk-loving, in which case they wouldn't even buy insurance in the first place. $\endgroup$
    – Oliv
    Commented Apr 4, 2017 at 18:34
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    $\begingroup$ @LeandroMachado Also, the fact that $p$ is paid every month does not affect the argument above. In fact you can consider this choice problem as repeating itself every month: every month you choose whether you spend $p$ on insurance for the month or incur the risk of a damage $D$. $\endgroup$
    – Oliv
    Commented Apr 4, 2017 at 18:38
  • $\begingroup$ @LeandroMachado Finally, I am not sure about your argument regarding investing $p$ instead of buying insurance. This would be beneficial only if the interest rate is high, but then the cost of credit would be high as well. $\endgroup$
    – Oliv
    Commented Apr 4, 2017 at 18:41
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I think the best way to answer your question is with a simple example showing why insurance is so prevalent in the economy. Hopefully, it should then be clear that it can still make sense to purchase insurance even if you can take out a loan to "cover the recent loss/accident".

Suppose that there is a 1% chance of a £10,000 loss occurring. For a concrete example, suppose that it is your £10,000 car being stolen and you absolutely need to have a car. Therefore, if your car gets stolen, you have to buy a new car. Further, suppose that you can become fully insured by paying £100. There are two states of nature to consider:

  1. The car is stolen so you lose £10,000 plus the premium of £100, but the insurance company pays you £10,000. This leaves you with a total loss of £100.
  2. The car is not stolen so you pay the premium giving you a total loss of £100.

Now suppose that instead of paying the insurance premium, you put the £100 towards a loan. Once again there are two states of nature to consider:

  1. The car is stolen so you lose £10,000. Suppose that you do not have the cash to pay for a new £10,000 car up front so you need to take out a loan, say at a generous interest rate of 0%. After you pay it back, your loss is still £10,000. Notice how the loan is irrelevant to the final loss; the only reason you would take out a loan is if you do not have the cash to pay for a replacement car up front.
  2. The car is not stolen so you lose nothing.

The question then is, do you prefer the scenario where you bought full insurance? This depends on your attitudes to risk. If you dislike risk (technically, if you are risk averse), you dislike the fact that if you do not buy insurance, then there is a great amount of uncertainty over the final amount of your loss: it could be a huge £10,000 or it could be nothing. Therefore, you are probably willing accept a relatively small loss of £100 to be rid of this uncertainty. It is generally accepted that most people in the economy are risk averse, hence the prevalence of insurance.

However, if you like to gamble (technically, if you are risk loving), then you might be willing to go without insurance in the hope that the loss does not occur and you do not lose anything. In other words, you take the chance that if the loss occurs, you will have to take out a loan to pay for a new car.

An additional note: As you might expect, if you increase the chance that your car gets stolen, more people will be willing to pay the £100 premium. Similarly, if you keep the chance that your car gets stolen at 1%, more people will be willing to buy insurance if you lower the premium.

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    $\begingroup$ ` 1. The car is not stolen so you lose £10,000` should be ` 1. The car is stolen so you lose £10,000` :) Can't edit because I would need to edit at least 6 characters. $\endgroup$
    – Gizmo
    Commented Apr 4, 2017 at 7:20
  • $\begingroup$ @Gizmo Thanks for spotting this. Tom.Bowen89 has edited it. $\endgroup$
    – hk39
    Commented Apr 4, 2017 at 9:42
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    $\begingroup$ But no one would give insurance forever at a fixed upfront price (at least not cheaper than the car itself). Rather it's more like £30/month to insure the car and you pay it every month your car isn't stolen. On the opposite side you'd pay £300/month for the loan after the small chance your car is stolen. True £300 > £30, but the comparison is fairer than £10000 >> £100. $\endgroup$
    – csiz
    Commented Apr 4, 2017 at 13:50
  • $\begingroup$ @csiz Treat it over a fixed period of 1 month. You get to pay \$100 to insure against a 1% chance of having your car stolen. Next month you have another 1% chance of having your car stolen and you might want to pay another \$100 to insure against that chance. $\endgroup$
    – Criticizing Israel not allowed
    Commented Apr 5, 2017 at 0:49
  • $\begingroup$ @Oliv I had enough time to read your answer and indeed they are very similar (+1)! The one thing I didn't consider was time discounting. I suppose this sort of thing can always happen when people post answers almost simultaneously. $\endgroup$
    – hk39
    Commented Apr 5, 2017 at 5:37
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Not covered by the other answers: You have an accident in your car and it ploughs into a bus-stop severely injuring and permanently incapacitating several people. Not having insurance, you are presented with a demand for £10,000,000(*) to cover ongoing medical costs. How long do you think it would take to pay off this loan?

(*) Figure plucked from air, but think BIG.

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    $\begingroup$ This is not really part of the economics of insurance. It's part of the fear (risk) that drives people to buy insurance. But truth is if you had a bill like that, you just would end up with bad credit and would move on. $\endgroup$
    – coteyr
    Commented Apr 4, 2017 at 13:12
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    $\begingroup$ I'd say it is part of the economics. For phones, TVs etc, the potential loss (that factor into the economics) is (mostly) limited to what it costs to replace. However, in certain cases, the potential loss is much higher than just the replacement cost, as in 3rd-party liability. $\endgroup$
    – TripeHound
    Commented Apr 4, 2017 at 13:19
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    $\begingroup$ But if your third party liability is larger then what you can actually pay, you just end up with a non-collectible debt. Sure your rating tanks, but no matter how good your credit rating your not going to be borrowing 10,000,000. $\endgroup$
    – coteyr
    Commented Apr 4, 2017 at 13:25
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    $\begingroup$ In countries where motor insurance is mandatory, this is the reason it's mandatory! You can't do £10m of damage and then just walk away with a non-collectible debt, that leaves other people out of pocket. So you have to pay into a scheme that will cover them. Same applies to public liability insurance. $\endgroup$
    – pjc50
    Commented Apr 4, 2017 at 14:08
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    $\begingroup$ @LeandroMachado Assuming you don't want, or aren't allowed, to just write-off the debt, then no. Having easy access to a £10M loan is completely different to paying off a £10M loan! $\endgroup$
    – TripeHound
    Commented Apr 4, 2017 at 18:26
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Insurance: pooled risk.

Loan: borrowing against future income.

With insurance, you may face higher premiums if you make a claim. But you won't pay anything against the claim itself. With a loan, you are responsible for paying the claim. You pay some of it with past income (savings) and some with future income (borrowing). But in the end you pay the whole thing.

Insurance can pay more than your entire future income. Instead of you paying alone, you and everyone else who bought the insurance pays. Once the event happens, you can stop paying if you don't mind not being covered.

A loan has to be less than your expected future income. Otherwise your creditor wouldn't give you the money. Once the event happens, you start paying.

The question basically comes down to how you want to handle risk. Would you rather pay when you may not need it (insurance)? Or would you rather wait to pay until you know you need it (loan)? On average, you'll pay about the same either way. Each person pays less for insurance, but they pay it more often.

Consider the following circumstance. You're twenty-five and expect to live to age seventy-five. When you are thirty-five, you or one of a thousand other people might have to pay \$100,000. An insurance company offers to let all thousand of you buy insurance for \$11 a year. So on one side, you pay \$110 over ten years. On the other side, you pay \$100,000 over forty years. Would you rather a guaranteed \$110 or a possible \$100,000? Assuming an interest/inflation rate of zero.

Of course, real insurance decisions aren't that well bounded. But the principle is the same. You pay a small amount ahead of time to be assured that you won't spend a large amount later.

I was thinking that it would make more sense to save the premium money and use it to pay part of a loan to cover the losses if it happens. That way one would save the trouble of paying insurance and never using it and make better use of that money.

That would work great if the event was guaranteed to happen to you. So you have to pay \$100 in ten years, guaranteed. Or you could pay insurance for ten years. Say \$11 a year. Then obviously, you'd be better off saving \$10 a year than buying insurance. At that level of certainty, there is no risk to offset by pooling.

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Brythan comes close with his answer. The reason that you don't take out a loan is because the lender would have to take the risk on your future income. When the reason you are taking the loan is for health reasons, the risk of not having enough future income to repay the loan is high. Most people can't get a loan for a car even when the car is used for collateral. When you have nothing but your health to secure the loan, the risk to the lender is unrealistic and not likely to be approved. Insurance works because it is expected that some of the people buying it will never use it or use so little of it to not matter. The risk to insurance companies decreases for ever person that falls into the low use group. The best money use to avoid insurance would be to put your premium money into a high yield savings account or other investment and hope for the best.

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I think insurance and loan cover very different problems.

Insurance covers potential losses for a fee. You take insurance to mitigate risk. By accepting a relatively small loss, you do not need to plan or worry about big failures or losses.

On the other hand, loans do not protect you from any losses. Loans provide cash when you do not have enough right now. Remember that you still have to pay your loan back (or go bankrupt with all its demerits).

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Thanks for the contributions. Based on them, I'll give my own interpretation of why I think both have similar outcomes. Considering, of course, that one can still take a debt after the incident.

Lets say you are protecting against damage D that has a probability p of happening and your current wealth is W. If the insurer charges you P for protecting from D, your wealth will be:

  • W - P

In this case p is irrelevant since D is fully covered. If instead one considers to get a loan D' to cover D, then there are two cases:

  • W(1 - p) (no damage happens)
  • (W - D')p (damage happens)

In the case the expected value is W(1 - p) + (W - D')p = W - D'p. If one eliminates the interest from D' and insurance profits from P, both loan and insurance will have the same expected value (D'p = P = Dp). However, since the premium is a upfront payment it also adds a opportunity cost in the equation.

In the end, both are similar and it is just a matter of pricing (interest vs premium). If both have the same price, it would make sense to give preference to a loan instead of the insurance because of opportunity costs. However, depending on the size of the loan interest might grow larger than the premium, mostly because the chance of default will be higher than the chance of accident.

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    $\begingroup$ The expected values are indeed similar but it is widely documented that people do not care only about the expected value of uncertain lotteries: they also dislike facing uncertainty about the final outcome. This is why the sure outcome of $W-P$ would be preferred by a large majority of people to the uncertain outcome $[W(1-p);(W-D)p]$. This behavior is known as "risk aversion" and I suggest that you spend some time reading about it since it is how we usually explain a positive demand for insurance. $\endgroup$
    – Oliv
    Commented Apr 6, 2017 at 6:59
  • $\begingroup$ No sane financial institution will give you such free-ride loan without insurance coverage to mitigate their risk. $\endgroup$
    – mootmoot
    Commented Apr 6, 2017 at 9:31
  • $\begingroup$ Yes, I agree that risk aversion can explain the positive demand for insurance. I think that the fact people tend to overestimate the probability of rare events might add something to that to. But I was more interested if it would be rational to have insurance and your contributions helped shed some light on it. $\endgroup$
    – Leandro Machado
    Commented Apr 6, 2017 at 16:03
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It makes sense to pay the premium if that is how much you value certainty over uncertainty.

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