GAMBOOLLL

[simpledude16]

You mean Risk Management theory at its most basic. I mean quite literally this is the most basic application of risk management theory; one party takes on risk, earning a premium from a risk-averse agent. LDFO

if its so basic then why didnt you post it :P

[Penneywize]

if its so basic then why didnt you post it :P

Well the way this came about initially was a study where respondents were asked to list their preferences among lotteries. The Allais paradox originates, essentially, from the following (the numbers are not correct obv, but the result is the same):

Given choices between lottery A and B:

Lottery A
90% chance of winning 100$
10% chance of winning 0$
(expectation of 90$)

Lottery B
50% chance of winning 500$
50% chance of winning 0$
(expectation of 250$)

Respondents tended to choose lottery B. This is in line with expected utility theory, which says individuals would choose the lottery with the highest expectation.

When the lotteries were structured as follows:

Lottery A
90% chance of winning $10,000
10% chance of winning $0
(expectation of $9,000)

Lottery B
50% chance of winning $50,000
50% chance of winning $0
(expectation of $25,000)

Respondents had a much higher tendency to choose lottery A. This flew in the face of contemporary economic thinking, as Lottery A is strictly dominated in terms of expectation by Lottery B. So that's where the paradox originates. If we multiply the payoffs in the first scenario by 100, we have the same payoffs as in the second. Since this is a linear, monotonic transformation, the consumers are expected to make the exact same choices.

I may not be getting my timelines correct but I believe it was about that time (1970s or so) that theory with regards to risk aversion, certainty equivalence and other related topics began to really fill out in the econ literature. Utility of money was obviously an older idea, but the form of log-utility was more or less in the background back in those days.

Anyhow, to answer your question, the reason I didn't mention it before is because that was outside of the equation i.e. you are not given an option to look for insurance on choosing the risky lottery. Certainty equivalence, by the way, is essentially what you're referring to in your post. It is defined as the sum of money you'd need to be paid not to accept the lottery.

If you remember "Deal or No Deal", this was pretty much the crux of the show; the 'bank' would offer sums that were below the expectation of the money remaining in the cases, essentially trying to guess at the lowest possible amount they can pay the contestant in order for them not to accept the gamble.

[Pascal]

Anyhow, to answer your question, the reason I didn't mention it before is because that was outside of the equation i.e. you are not given an option to look for insurance on choosing the risky lottery. Certainty equivalence, by the way, is essentially what you're referring to in your post. It is defined as the sum of money you'd need to be paid not to accept the lottery.

But at the same time we weren't specifically told that we could not look for insurance, or anything else for that matter. I remember having a big argument a while back with a friend who was convinced that insurance was -EV and anyone who took it was being an idiot - I'm of the opinion that insurance can actually be +EV. If you break your leg 1% of the time and the medical costs are $100k each time and you pay $2k for insurance, obviously it's -EV. But if you getting hold of $100k to pay for the bills every 1% actually costs you $300k in interest etc then it's actually +EV for you and also +EV for the insurance company as the costs are different. IDK if I'm missing anything but access to liquid money could make insurance +EV for both parties right?

[Penneywize]

666th post, Pascal.

Yeah, I figured that searching for insurance defeats the purpose of the question as in that case, clearly, no one would choose red, a great majority would take insurance on the second gamble for a guaranteed payoff greater than 1 mil, and a few risk lovers would choose green. There is really nothing interesting about that. Anyway, we are only presented with two buttons, and not a third saying "I need time to find insurance on this risky gamble".

As for your discussion on insurance, in terms of "EV", we are speaking clearly in terms of expectation (exactly what EV is). In a broad sense, your friend is correct. Let's leave out the potential for insurance companies to invest money for a moment - we can assume that, with a large enough number of recipients, the insurance company is simultaneously paying out and accepting payments from insurees to the point where there exists no frictions, for instance. In this case, any given insurance contract offered has to have a negative expectation for the recipient, as otherwise the insurance company would lose money on the contract and ultimately go out of business.

So broadly speaking, insurance contracts should have a negative expectation for anyone accepting them. Again let's leave out cases of moral hazard (i.e. hey, I'm insured, time to start engaging in risky behaviour -- the insurance company's got my back anyway!) and adverse selection (i.e. hey I'm super sickly and have shitty health, but I can lie to the insurance company and pay a lower premium than I should given the risks they are taking on).

In practice, insurance companies will lose money on some contracts, but make money on aggregate. People accept insurance contracts for the same reason they buy locks for their front doors: peace of mind. There is an innate value in this, and this is, essentially, what insurance companies are selling.

In more exact technical terms, insurance companies take on risk in return for a premium. Individuals are risk averse agents, and are willing to pay a premium to lessen the risk they are exposed to (in this case, risk due to tje uncertain prospects of health and injury).

Given the scenario you described, I can't agree that an insurance company would offer a contract to an individual with a positive expectation. If the individual has a given chance to break their leg, it's the insurance company's job to evaluate this probability and demand a premium that sufficiently compensates them for taking on the "risk" of having to pay you a large sum in the event you do break your leg.

Now, if we add in the ability to invest and so on, this situation is not likely to change by much. The options for investing money aren't necessarily risk-free either, though you can come up with ways to build an essentially non-risky portfolio that I won't get into here.

I'd imagine that the best you can expect is if there were 'perfect competition' in the insurance market -- something that does not really exist in practice -- the best you could hope for is a neutral EV situation for both parties.

This results, theoretically, from insurance companies continually entering the market so long as a profit can be made; they compete with each other based on their pricing of contracts. Once the marginal firm entering the market prices the contract at cost i.e. for zero expected profits, no further firms enter, all firms in the market make no profits, and contracts are at the lowest possible price for consumers.

[Pascal]

Given the scenario you described, I can't agree that an insurance company would offer a contract to an individual with a positive expectation. If the individual has a given chance to break their leg, it's the insurance company's job to evaluate this probability and demand a premium that sufficiently compensates them for taking on the "risk" of having to pay you a large sum in the event you do break your leg.

Think you might have missed what I was saying - I'll try and break it down.

Person buys insurance from insurance company for $1,000
Risk of accident per holiday assessed at 1%, average cost of accident (as in hospital bills, new travel plans, medicine, etc) = $50k
Therefore, insurance company is making a $1,000 profit in the long run
If person doesn't take insurance, average cost = $500 a holiday
However, what if person doesn't have access to $50k unlike the insurance company which has plenty of money in reserve? $50k of bills would need say $40k borrowed at a high interest rate, possible remortgage, etc) - total cost of borrowing $50k to the person could be say $150k
Real average cost for person = $1.5k per holiday
Therefore insurance is +EV for insurance company and +EV for the person because the insurance company has easy access to pay the money for hospital bills whereas getting that amount of money would cost the person lots in interest etc

[Penneywize]

Ah, I see what you're driving at. Well, the question to ask would be this: do insurance companies offer the service of liquidity (implicitly) free of charge?

I would say no. Whatever costs are associated with the insurance company having to carry a certain balance readily available for payout to insurees have to be capitulated elsewhere; consider that having this liquid money balance carries a non-trivial opportunity cost, versus say being invested in bonds or on the stock market.

I would assume this cost is factored into the price of contracts. It simply doesn't make economic sense to do otherwise; if insurance companies simply provide this service out of the kindness of their heart, they are implicitly losing money due to opportunity cost and inflation. So while the consumer would benefit, the insurance company would lose money, and this is clearly not a +EV situation for both parties. Further, as I alluded to in my last post, if insurance companies lost money on contract, they would go out of business.

It's just not really possible for a "+EV" situation to come up for two parties in which they unequivocally mutually benefit. You can analyze things from insurance contracts to credit default swaps and so on, and you'll find that these instruments exist for the purposes of hedging against risk. One party is always taking on risk, and the other party is always paying a premium for it.

[Penneywize]

re-reading your post, I'd have to think that the total costs of borrowing for a given individual would have to be irrationally high for them to be in a positive expectation position when taking on a contract. While it could happen, it doesn't make sense from a theory perspective, and it certainly wouldn't apply to a great number of individuals.

Strictly speaking, if there existed perfect information, insurance companies would take into account the costs of the individual's potential borrowing and use that to price their contracts. That is, to say, if borrowing costs are excessively high, it'd be irrational for an insurance company not to consider this as part of their opportunity cost of lending out their liquid money. Without market frictions, we would see that the opportunity cost of holding liquid assets would then be exactly the cost of borrowing (i.e. the real interest rate) faced by consumers.