## Thursday, July 12, 2018

### Loss Aversion

A frequent email correspondent asked "I’d love to hear your take on “loss aversion.” I just finished listening to Kahneman’s book." My response seems worth sharing with blog readers.

Expected Utility

Let’s review expected utility first. The utility you get from consumption or wealth is a concave function of consumption or wealth. An extra dollar makes you more happy than it makes Bill Gates. So, compare either getting C for sure, or a 50/50 bet of getting C+Delta or C-Delta, i.e. having C or betting 50/50 on a coin flip. The expected utility of C for sure is just U(C). The expected utility of the bet is

EU = prob(loss) * U(consumption if loss) + prob(gain) * U(consumption if gain)

EU = 1/2 * U(C - Delta) + 1/2 * U(C + Delta).

As the graph shows, this is less than the expected utility of C for sure. So, people should decline fair value bets. They are “risk averse”.

Comments. Behavioral fans (New York times has done this often in its economics coverage) criticize “classical economics” by saying it ignores the fact that people fear losses more than they value gains. That’s absolutely false. Look at the utility function. People fear losses more than they value gains. That’s the whole point of expected utility. (You’ll see the confusion in a second).

A common mistake: EU( C) is not the same as U [ E(C )]. You do not find the utility of expected consumption, you find the expected utility of consumption. In my graph, C is equal to the expected value of C-Delta and C+Delta, and the whole point is that the utility of C is bigger than the expected utility of (C-Delta) or (C+Delta). You can take E inside a linear function, but you cannot take E inside a nonlinear function.

Loss aversion

OK, on to loss aversion. In the usual sort of experiments Kahneman found that people seem reluctant to lose money. They have a “reference point” and work  hard to avoid bets that might put them below that reference point. He models that as expected utility with a kink in it, as in the second drawing.

I was careful to draw the reference point as different than C. People do not necessarily place the reference point at the expected value of the bet. In fact, usually they don’t. If betting on stocks, the expected value of the bet is to gain 7% per year. The “don’t lose money” point would be do not go below 0, not do not go below the mean. Here people are especially afraid only of the very left part of the distribution.

Now, really, how are these models different? Expected utility can be any function, and nobody said it doesn’t have a kink in it. The key distinguishing feature of loss aversion – and its Achilles heel – is that the reference point shifts around. If you make some money, and play again, then your kink shifts up to the new amount of money you made. Expected utility is supposed to stay the same function of consumption or wealth. People might change behavior – most likely the utility curve is flatter at high levels of consumption, so rich people are less risk averse. But the curve itself does not shift. The key assumption that distinguishes loss aversion from expected utility is that the kink point shifts around as you gain and lose money.

That’s also the Achilles heel.  The first problem is how do you handle sequential bets. If I go to the casino, and know I will play twice, how do I think about my strategy? With expected utility this is easy, because the expected utility works backwards. Suppose you win the first bet, then figure out what you do in the second bet. For each of win or loss in the first bet, then, you have an expected utility from taking the second bet. The expected utility of the first bet is then the expected vaule of the expected utilities you would have if you won or lost.

Equations might be better than words. Let Chh, Chl, Clh, and Cll be consumption if you win twice, win and then lose, lose and then win, lose and then lose, and U(Chh), U(Chl), etc be their utilities. So, suppose you won the first bet. You evaluate the second bet by

EUh == EU given you won first bet = Ph*U(Chh) + Pl*U(Chl).

Similarly, if you lost the first bet, then

EUl = Eu given you lost the first bet = Ph*U(Clh) + Pl*U(Lll).

So now, the expected utility of the first bet is just a one-stage bet, with these expected utilities as payoffs.

EU = Ph *Euh + Pl*EUl

I won’t go through more equations, but the same thing holds for decisions. You can work optimal decisions backwards.

Now, this all gets to be a big problem with loss aversion. The key question: Do loss averse people in stage 1 think about the fact that if they win or lose in stage 1 that will change their reference point for stage 2? Again, the reference point must shift, or we just have expected utility. If people ignore that their actions today shift the reference point tomorrow, then they’re unbelievably dumb. If people don’t, then we have a mess on our hands. Since your preferences today disagree with your preferences tomorrow, you might act strategically, deliberately winning or losing today to shift the reference point for tomorrow and influence how you will behave tomorrow. There have been brave papers trying to work this out (Barberis, Huang and Santos) but the result has been, in my view, so complex that it hasn’t come in to common use.

But really to be useful in economics or finance, we have to have a model that does not apply just to wake up, make a bet, eat, die, but that describes how people make decisions over a lifetime, and with knowledge that they will bet many times.

Some more problems with loss aversion: We certainly see lots of individual behavior that suggests reference points. People get really reluctant to sell houses for less than they bought them for, for example. The problem here is that expected utility refers to your overall wealth. You don’t mind losing on your house, say, if you gained in your stocks, or if the new house you want to buy also went down in price, as your overall wealth or consumption does not change. Now, to apply loss aversion, it seems we need to define loss aversion as a function of each individual purchase, not overall consumption. You need “mental accounting” as well, of which items go into which pot, and a rule not to lose on individual pots, as well as loss aversion.

A deeper skepticism about "irrationality"

I have a deeper skepticism about the rush to label people irrational (and us researchers oh so much smarter). What’s so irrational about following a rule or heuristic that says, avoid losing money on your trades? Another widely adduced “irrationality” is that people tend to trade too much. Given that prices often do bounce back, the heuristic “don’t lose money on your trades” might be a useful guide. Most buying and selling often involves a negotiation. Don’t sell at a loss helps you to avoid sharp negotiators. It’s also a good precommitment device. Real estate agents keep telling us “well, they won’t take that offer because they’d sell at a loss” even when they bought at the peak in 2007.

Likewise, “when people in lab coats come and tell you what the probabilities of things are and offer you bets on them, don’t believe them because they’re trying to trick you” is a pretty good heuristic in life. You are going to run in to a lot more people playing 3 card monte or offering Bernie Madoff investments, and people in lab coats trying to prove you’re a moron,  than you are going to run in to honest researchers explaining how the game really works.

Apparently irrational rules are in fact often good rules for actual life. “Don’t pay attention to sunk costs” say economists, but if you run away from every half finished project because it’s a sunk cost, you don’t get anything done in life. Paying attention to sunk costs is a good way to make sure you actually finish writing, revising, and publishing papers! We live in a deeply information overload environment, and we have limited processing capacity. A lot of apparently irrational behavior seems to me decent rules of thumb for daily life, that appear irrational when extrapolated out of context to cleverly constructed environments that people have never seen before and have little experience with.

Mental accounting is similar. The accounting department of the University of Chicago Booth school, home of efficient markets, subdivides the school’s money into thousands of separate budget items with a rule that  you can’t lose money on any individual one. If you buy and extra bottle of wine on the seminar budget, you can’t offset that even if at the end of the year there is money left over in the Pizzas for MBAs budget. Such budgeting is a practically universal method of controlling costs in all large organizations. It sure feels stupid, but is it really given the information problems?

Moreover, if people do behave this way, there is a way to make a huge amount of money off of them. Asset prices are smooth functions of gains or losses. Suppose your reference point is, you don’t want to lose more than \$1000 on stocks. By buying stocks and options, I can create a portfolio that will never lose more than \$1000. The price of the downside insurance is a smooth function of that loss point, so it will be cheaper for me to buy that downside insurance than it will be for me to sell it to you. Or you can buy it for yourself. There is a lot of behavioral study of people’s portfolios and the nutty investments they make, but we do not see pervasive buying of such downside insurance options, nor do we see the large prices such options would have if people were all loss averse.

In sum, I prefer when seeing unusual behavior to ask a bit harder "just what question is this the right answer to?" before rushing to say people are dumb and ask the government to nudge them my way.

Habits and others

Since I worked on it, I have to mention one halfway alternative – habits. See the  graph. Suppose people have a lower level of consumption that they really don’t want to go below. The obvious lower bound is starvation, but we all develop habits of consumption that would be super-painful to change.  A hedge fund manager’s wife once said at a cocktail party “I’d sooner die than fly commercial again,” and that can be represented by the habit level as well.

Now, as it is, that’s just another expected utility function. But here too, we make the habit point move around depending on experience. The hedge fund manager’s wife was once middle class like us and thought the peanuts on Southwest were a grand treat. As you consume more, or as you observe your neighbors and friends consume more, the minimum acceptable level of consumption rises. It can also model debts and fixed costs -- don't lose so much money that you have to sell the house and move into an apartment.

One reason this is different than loss aversion is that it turns out to be much more tractable for the dynamic question – how do you treat multiple bets. Not having a kink wandering around helps a lot. The strategic element is there. If the habit depends on your consumption, not your neighbor’s, then when you consume more you reset your habits. You might know this ahead of time, and we can work out easily how you behave. Habits make consumption an addictive good, so when thinking about eating more today, you recognize that it will influence how you feel tomorrow. When you drink another cappuccino today, you know it will make you want another one tomorrow, and are less likely to drink today as a result. We can work those things out easily.

That doesn’t address the mental accounting issues, the tendency to treat bets in isolation rather than as parts of a portfolio, and so forth. But it does capture the tendency to really avoid large losses, and how what a loss is shifts over time in response to gains and losses in wealth.

Two other popular approaches model a lot of challenging behavior as well. Recursive or Epstein-Zin utility is a non-expected utility model that captures lab experiments and is also analytically tractable, so it's becoming very common in macroeconomics and asset pricing. Lars Hansen is also been on a long research agenda of even more general utility functions that are  "ambiguity averse," capturing that people shrink away from uncertainty that they really don't know the probabilities of. Alas, once there is a clear and tractable mathematical model, people like this seem not welcome in the behavioral finance club, which says more about academic politics than about substantive matters.

I hope that helps!

PS to all:  Grumpy is back! I was traveling a lot making frequent blog posts difficult.

Update: An email correspondent writes
If a home was purchased at \$500,000 with a mortgage of \$450,000, and it is now sold for less than even \$475,000, then after commissions and closing costs, the seller has less than \$450,000 in hand and will be required to “bring cash to the closing” in order to fully pay off the bank.  Generally, families with homes selling for below mortgage value have trouble raising any cash at all.  All this is further complicated by the problem of where and how can the seller move to next if all his cash is put into getting out of the current mortgage?  No money for new down payment, or first month’s rent and deposit. The current owner is truly stuck at an above market offering price.
This is a good point. Fixed costs and transactions costs can give rise to the extra costs of a loss, and produce a kink. People who develop rules of thumb in that circumstance may not notice the lab experiment is cleverly designed not to have the actual costs of many real-world losses.

1. Excellent discussion. I think it is also true that a lot of "irrationality" washes out in the aggregate (for a variety of reasons) so is of questionable significance. For example, even if people do behave irrationally with respect to investment decisions, there is pretty good evidence that these irrationalities don't affect stock prices-- otherwise there would be more obvious arbitrage opportunities.

1. Agreed, and I'd go further and say economics isn't primarily a theory of individual behavior, it's about how markets function. Assumptions about individuals can be wrong, but the predictions about markets still accurate. This is not to say that they are all accurate, but proving that individuals behave "irrationally" doesn't show anything, and are pretty tangential. You need a lot (all?) individuals behaving irrationally in much the same way, with no one smart enough to take advantage.

2. “A lot of apparently irrational behavior seems to me decent rules of thumb for daily life, that appear irrational when extrapolated out of context to cleverly constructed environments that people have never seen before and have little experience with.”

That’s exactly it! People are actually quite clever about adopting rules that are robust, but seem “irrational”. People play tennis, golf, ride a bike – even glide – and yet have little understanding of the physics involved (not you John!). The mental constructs that people have in their mind about how a ball flies in sports like tennis is completely at odds with the actual physics. If you asked Federer to play tennis on the moon, I suspect he wouldn’t do much better than an amateur – but people never experience that strange situation. Given that people can build excellent heuristics of the physical world, there is little reason to think they wouldn’t do the same when it comes to other decision making aspects of their lives, such as consumption smoothing and risk.

1. Excellent point. I suspect many people who fly do not understand Bernoulli's principle, but board planes anyway, with little trepidation.

3. Great post, John. By the way, I am really looking forward to your comments on the presentations of the Nobel Symposium!

4. This is exactly what I've always taught my students. The differences between expected utility and loss aversion are largely trivial, and any additional features of loss aversion are not necessarily helpful.

5. Hedge fund risk managers, of which I was one, wrestle with problems of reference point shift, habit shift and bets in isolation; all compounded by power-law tail distributions. Fulfilling our fiduciary responsibility, we established hedges that allowed losses of no more than 5% for some, 3% for others and risk free rates for others. There was something of a clientele effect across individuals and firms taking risk with us.

6. There is an asset structure essentially designed to avoid downside loss in common use - bonds. If I wanted not to loose more than \$1000, I could buy \$1000 stocks and put the rest in bonds ...

1. When investing for retirement, your time frame should typically be thirty to forty years (or more). One of the great retirement risks is inflation and bonds leave you exposed to inflation. \$0.02

7. Nice material for my behavioral finance class. You forgot one huge advantage of loss aversion. It motivates students to learn expected utility. They need to understand utility theory to criticize it with prospect theory. Loss aversion is risk-seeking in losses versus expected utility’s asymmetric responses. They really learn more about utility theory when they have to debate it. This is true of most behavioral finance. If it’s taught as a debate they learn both sides better.

8. Thank you for your thoughts on this subject! I've been thinking something similar to the habits idea and wanted to ask a related question. Could you think of habits as investing in behaviors? then you could have safe and risky behaviors to choose from given the budget constraints of environment, capital, human capital, and time. the food i eat, the hours of sleep i get, my morning routine, how i get to work, all these things are investments in a productive behavior...hopefully.

9. I certainly don't want to argue across settings, but I have seen hundreds of poker players who change reference points and style of play in ways that EU does not explain. Nearly all players know not to put their entire bankroll on the table, and few are playing at levels where the differences in likely outcomes change where they are on the EU curve but once they are up money they think it is theirs and behave as such while after they are down they still think the money they lost is theirs and behave as such. All the while neglecting the fact that after a large enough number of hands your earnings are informative about your relative ability compared to the table and even playing opposite of what the information tells you. One could try and argue these people are being rational but their changes in play once up or down are a large part of what allow grinders to make a living. If in cash game you significantly alter your play because you are up or down a bit you have given away a great deal of information and automatically lowered your expected return.

10. This post and many of the comments miss the point of loss aversion. Matthew Rabin and other behavioral economist agree that over large stakes there is diminishing utility of wealth (and thus, things like insurance are useful). However, expected utility theory suggests that utility is virtually linear over small stakes. Loss aversion can explain small-scale risk aversion where expected utility theory can't. That is the biggest empirical fact that loss aversion can explain that expected utility theory cannot.

11. The distinction between expected utility and what you've described as loss aversion is that EU is almost always framed in terms of final wealth vs. initial wealth. While that's a convenient basis, the theoretical development for the EU model doesn't make any reference to the person's wealth. Academics subsequently modified the model to add decreasing, flat, and increasing risk aversion as a function of wealth. My contention is that this led everyone down the wrong path.

Consider what happens if you drop wealth as a consideration in your EU model. In that situation, an individual exchange can be viewed in terms of its effect on income rather than wealth. From this perspective, the kink arises naturally, because (we can assume) a loss is viewed differently from a gain.

12. Vernon Smith's book on constructivist and ecological rationality is relevant here. Basically, he argues that just because people don't follow the standard model is no argument that they are "irrational." If loss aversion is a tolerably good strategy most of the time it is rational in the sense that it works over the long run.

13. John: I thought loss aversion meant being risk-averting with respect to gains but risk-loving wrt losses. Staring from wealth of 100, suppose someone prefers a gain of 50 to the gamble with equal chance of gaining 0 or 100; but the same person, possessed of 200, prefers the gamble in which they lose 100 or 0 with equal probability to the sure loss of 50. Graphically, the first response makes the utility function between 0 and 100 concave to the origin; while the second makes it convex over that same range. This case cannot, in particular, be captures with a kinked, but concave, utility function.

1. Yes. I have the same view about loss aversion.

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