"You are offered a gamble on the toss of a [fair] coin.

If the coin shows tails, you lose $100.

If the coin shows heads, you win $110.

Is this gamble attractive? Would you accept it?"

(I have modified the numbers to be more realistic in a financial market setting, but otherwise it is a direct quote.)

Experiments show that most people would not accept this gamble, even though the expected gain is $5. This is the so-called "loss aversion" behavioral bias, and is considered irrational. Kahneman went on to write that "professional risk takers" (read "traders") are more willing to act rationally and accept this gamble.

It turns out that the loss averse "layman" is the one acting rationally here.

It is true that if we have infinite capital, and can play infinitely many rounds of this game simultaneously, we should expect $5 gain per round. But trading isn't like that. We are dealt one coin at a time, and if we suffer a string of losses, our capital will be depleted and we will be in debtor prison if we keep playing. The proper way to evaluate whether this game is attractive is to evaluate the expected compound rate of growth of our capital.

Let's say we are starting with a capital of $1,000. The expected return of playing this game once is initially 0.005. The standard deviation of the return is 0.105. To simplify matters, let's say we are allowed to adjust the payoff of each round so we have the same expected return and standard deviation of return each round. For e.g. if at some point we earned so much that we doubled our capital to $2,000, we are allowed to win $220 or lose $200 per round. What is the expected growth rate of our capital? According to standard stochastic calculus, in the continuous approximation it is -0.0005125 per round - we are losing, not gaining! The layman is right to refuse this gamble.

Loss aversion, in the context of a risky game played repeatedly, is rational, and not a behavioral bias. Our primitive, primate instinct grasped a truth that behavioral economists cannot. It only seems like a behavioral bias if we take an "ensemble view" (i.e. allowed infinite capital to play many rounds of this game simultaneously), instead of a "time series view" (i.e. allowed only finite capital to play many rounds of this game in sequence, provided we don't go broke at some point). The time series view is the one relevant to all traders. In other words, take time average, not ensemble average, when evaluating real-world risks.

The important difference between ensemble average and time average has been raised in this paper by Ole Peters and Murray Gell-Mann (another Nobel laureate like Kahneman.) It deserves to be much more widely read in the behavioral economics community. But beyond academic interest, there is a practical importance in emphasizing that loss aversion is rational. As traders, we should

**not**only focus on average returns: risks can depress compound returns severely.

===

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## 13 comments:

You're confusing different issues. It's true that g = m - s^2/2 can be negative for positive m when s is large enough. But this is risk aversion, not loss aversion. Loss aversion refers to the case when g > 0 but skew is negative enough that a skew sensitive utility function has a negative expectation and causes an investor not to take the (series of) g>0 bets. I.e. recognize that writing deep otm puts can kill you even though you've done nothing but collect premium historically :D

The only confusion comes from the fact that there is no such thing as Noble Prize in Economics, but perhaps this is another behavioral bias example. Good article.

Hi Ernie,

but as I recall, the conclusion in the Kahneman's example was something along these lines: "..of course, taking one such bet isn't attractive, but we all in our lifetimes are

constantly making decisions about countless of such small bets, therefore, if we adapt a personal policy to always accept small bets (that is small enough that no single bet

can ever ruin us) with positive expectation, we're almost guaranteed to be better off over our whole lifetime..".

So it seems to me that his idea of a "personal policy to always accept small bets with positive expectations" isn't a "time-series" case, but more of a "many simultaneous

bets" case., even more, almost by definition, our countless real-life bets will be diversified, i.e. it's very rare that someone offers us to flip a coin with him 3 zillion

times in a row, quite the opposite, each time life presents us with a new opportunity (i.e. with a new bet), it's

likely to be something unrelated to the previous bet, i.e. we will also be collecting diversification benefit from adapting such a policy..

"Time for a Change: Introducing irreversible time in economics - Dr Ole Peters". Great video, Ensemble vs. Time Perspective (~7:20 min). Available on youtube and vimeo.

Zyga

Many of Kahneman's loss aversion and irrational theories are disproved long ago by maurice allias (Sveriges riksbanks prize in economics or economics noble ) prize winner even before they were widely propagated, as he only wrote mostly in french they were not widely spread and understood. Just like Hyman Minsky whose theories are there for a long time but only widely known due to financial crisis.

Ironically, your recommendation to maximize the compounded growth rate is similar to the solution from conventional economic theory.

Economic theory says you should maximize expected *utility* of wealth. In a mean-variance world, this is given by:

expected return - (gamma/2)*variance

where gamma is the coefficient of risk aversion.

This is virtually identical to your formula, except you implicitly have gamma=1. Allowing gamma to vary makes more sense to me, because it can account for different attitudes about risk. After all, not everyone would be willing to trade 0.5 units of expected return for 1 unit of variance.

Hi DM: Kahneman is correct if we place bets of the same size every time. However, this is not how wealth is acquired. In order to grow personal wealth, one must scale the size of the bet with our wealth. Imagine Warren Buffet trading 100 shares of IBM each time like he might have done when he was in his twenties! So the world that Kahneman imagines is fictitious and has no bearing on reality.

Ernie

Anon,

I have never heard of defining loss aversion as g>0 but skewness < 0. Do you have the reference for that?

Ernie

Hi Yasser,

See my blog post Kelly vs. Markowitz Portfolio Optimization (https://epchan.blogspot.com/2014/08/kelly-vs-markowitz-portfolio.html) for the equivalence of mean varaince optimization vs Kelly optimization, in which I show that Kelly is superior in one significant aspect (we don't need to assume a utility function!)

Ernie

Hi Ernie,

May I ask where we can get E-mini S&P 500 futures tick data?

Thanks.

You can buy that from tickdata.com, or rent that from quantGo.com.

Ernie

Hi Ernie,

Thank you for quick response.

Have you heard about kibot? Their tick data is cheaper, but I don't know if their quality is ok or not.

Thanks.

Yes, I have heard of them, but have not ascertained their data quality.

One way to find out is to get samples from several different vendors, and compare to see who is missing ticks.

Ernie

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