The difference is that insurance is not about the expected return at all - it's about managing risk. The "payout" from insurance is tied to events that can happen whether or not you have insurance, and which would ruin you financially without it.

The expected payout from insurance is only relevant when the above does not hold, i.e. you have enough liquid funds to cover the loss against which you're insured. A good example are non-health-related travel insurances (e.g. against cancellation expenses or baggage loss). Your argument is correct in these cases - these insurances are a net loss and cover a risk that you could simply bear. So you should not buy them.

Exactly. If risk is defined as financial uncertainty, then lottery tickets and insurance are in fact complete opposites with regard to managing risk. Insurance is designed to minimize financial uncertainty and lottery tickets (gambling) to increase it.

Indeed. Also, most people are risk averse, so they are happy to purchase insurance with an EV (expected value) equal to less than the premium simply to avoid the gamble.

Not indeed. Variance matters. Look at the hedge fund industry.

Expected return is insufficient if you can't recover from death to regress to the mean. So yeah, zombies shouldn't buy insurance

Insurance pays out when you lose your job or crash your car; ie, when you get unlucky. Lotteries pay out when your pick of numbers was a better pick than ten million other people; ie, when you get lucky.

Economics has repeatedly suggested that the natural logarithm of dollars is an approximation of the utility of money.

When you're lucky, you go from 50k to 50m. In logarithms, 10.819 to 17.727. When you're unlucky, you go from 50k to effectively 0. Which is 10.819, to negative infinity - but probably, you'll have enough to live on, so like 1 (0). So you gain almost 7 utility for playing the lottery, but lose more than 10 for not playing the insurance. (The closeness of seven and ten explains why so many uninsured people play the lottery.)

Have any links that expand on this idea?

  But our explanation, that "Lottery tickets are foolish 
  because they have a negative expected return" isn't 
  exactly credible if we think insurance is a good deal.

It's my experience that if you offer homeowners the opportunity to gamble all their properties, double-or-nothing on a gamble biased 60:40 in their favor, no-one will take you up on that. Even though the gamble has a positive expected value.

My theory is the marginal utility [1] of your first home, which you stand to lose, is greater than the marginal utility of a second home, which you stand to gain.

[1] http://en.wikipedia.org/wiki/Marginal_utility

Alternatively, this is just more evidence that most people are risk-averse.

Insurance is about trading money for utility. The assumption is that your utility curve in the lossy region is sublinear, i.e. U(-$1e6) << 1e6 x U(-$1), and that the probability of a large loss is low (e.g., 1e-6).

In that case, if you pay a guaranteed -$1, your expected utility loss is U(-$1). If you have a 1e-6 chance of losing $1e6, your expected utility loss is 1e-6 U(-$1e6) < U($1).

Thus, it makes sense to pay $1 to avoid the risk of losing $1e6.

Insurance which pays for high probability, low cost events (e.g., gas for your car, birth control pills) is indeed foolish.

> The assumption is that your utility curve in the lossy region is sublinear

Which makes lottery tickets all the more a bad idea. Not only is the expected return in dollars less than your investment, but thanks to the diminishing marginal utility of money your ten millionth dollar will be worth less than your ten thousandth. Lottery tickets are actually worse than their already crappy EV.

Well, some people hypothesize that your utility can be superlinear in the positive region. Utility = (gain or loss)^3, for example.

Of course, the stats prof says he buys lottery tickets because they are fun. I do something similar - even though the expected gain from video games is precisely $0, I still play them.

An interesting addition to other replies to your comment is that the word "insurance" has come to be thought of as a good, safe thing by people (I'm not saying it shouldn't have).

In blackjack there's a bet called Insurance which you can make if you have blackjack (21 in two cards) and the dealer has an ace. Essentially it's a bet that pays 2-1 if the dealer's second card is a ten/face, the odds of which are 4/13, making this the bet with the worst odds for a player on a blackjack table. But because it's called "insurance", a huge number of people take this bet.

> the odds of which are 4/13

Actually slightly lower, 15/49. (The difference is like 0.15%.)

(I mainly mention this because I used to think my odds of drawing to a flush in poker were 1/4 per draw. It took me embarassingly long to realise that I needed to remove the cards I could see from the deck.)

While you're right that 4/13 is simplistic, 15/49 complicates it without actually making it more accurate. Even if there are no other players, there will always be more than one deck in the shoe (4-8 depending on casino), plus the three cards will never be the first cards out of the shoe - so unless you're counting cards you can't work out the exact odds.

> Even if there are no other players, there will always be more than one deck in the shoe

I hadn't realised this, thanks.

> the three cards will never be the first cards out of the shoe

Does this matter? I'm using a model of "the dealer's other card is equally likely to be any of the cards except the two I have and his face-up one", and it doesn't matter where those three were originally. The model can be improved by counting cards, but it's still strictly (albiet very slightly) more accurate than the model of "the dealer's other card is equally likely to be any of the cards in the deck/shoe".

But perhaps there's something else about Blackjack that I'm not aware of?

Let's imagine using just one deck. Your odds of 15/49 could easily be 12/45, or 15/20, depending on what cards have come before it.

Well, yes. But if you're not keeping track of that, then always using 15/49 will give you marginally better results, on average, than always using 4/13. Perhaps there will be times where, if you had kept track of the cards, you would give odds of 4/13; but you didn't, so you don't know that's the case, and you should give 15/49.

You're correct about your other objections, and 15/49 is indeed harder to work with - but "without actually making it more accurate" is false under the one-deck assumption. It is not wholly accurate, but it is more accurate.

I think you can classify lottery tickets as investment, wheras insurance is, well, insurance. I don't play the lottery because it's extremely unlikely to make me positive returns. I pay for health insurance because I want to cover the unlikely possibility that I have a serious health problem.

There's also the amortization to consider. Over a lifetime I may lose 50% of what I put in to insurance -- but I don't have 50% of what I'd put into it over my lifetime available to me now if I have a large medical bill to pay.

"Investment" is traditionally reserved for activity wth positive expected returns compared to baseline behavior.

Unless you can sustain a $100,000-$300,000 loss before you have have paid into the insurance pool more than that amount, it makes sense. If you had gobs of money, then it wouldn't make sense for you to have insurance. (preconditioned on insurance being a profitable business, etc.)

By analogy - if a person thinks "spending $5 has no impact on my life, but winning $10M will", is that really any different from our justification of insurance? (I agree with you - thats the same way I view insurance. Yet I view lottery tickets as silly, but I can't clearly articulate why).

The cumulative impact of a lifetime of spending $5 per week is much less than winning a lottery. Most people can sustain a $5 per week hit, especially if they get the mental benefits described by the parent article.

There are better things to spend $5 on, but it's not a huge deal either way - as long as you stay out of the addictive/compulsive zone.

The losses covered by insurance are very impactful. Most people can't sustain the massive financial hit that comes from, say, their house burning down. It can make sense to lessen their day-to-day quality of life by spending money on insurance, to guard against that massive downside risk.

Hm. Most people collect on insurance for very-sustainable costs - drug benefits, dental for checkups, routine visits to the doctor. Where do those fit into this argument?

Personally, I would like insurance that pays only for large expenses. I know its a sucker game to pay insurance; since the insurance companies profit I must be losing. But 'group insurance' is not something I can change.

When you buy catastrophic insurance it usually comes with free preventative care, which benefits the insurer as well as the insured.

You can often get a lower premium for a higher deductible, and everyone who knows how to multiply does this.

...unless as noted, its group insurance, in which case you take what you get.

Insurance significantly reduces the possible (or significantly likely) variance in a persons financial well-being. This comes at a cost in total wealth that is worth while because it brings the chance that one would vary into territory where the money one has is not enough for basic utility. Insurance on the other hand increases the variance (meaning the slight increase in expected value comes at a cost) at the other end of the spread in a way that does not guarantee any kind of minimal utility. Minimal utility is more important than huge potential diminished returns.

You also have to consider loss aversion which is a well known psychological phenomenon[0]. In a sentence, people fee when tend to strongly prefer avoiding losses to acquiring gains.

[0] http://en.wikipedia.org/wiki/Loss_aversion

It comes down to the utility of money, or roughly speaking, how happy the money will make you. $20,000 is a lot more valuable to someone with $10,000 than someone with $50,000. In other words, having $10,000 is worth, say, 10 utility, having $30,000 is worth 20 utility, and having $50,000 is worth 25 utility.

Starting with $30,000, buying insurance which has a small chance of saving you $20,000 gives you a +10 utility gain if it pays off. Buying a lottery ticket which has an equal chance of giving you $20,000 gives you a +5 utility gain if it pays off.

With insurance, you're already subject to the risk. With lottery you choose it.