"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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## 41 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

Hi Ernest,

Thank you for the enlightening post. I have made a few observations upon perusing your arguments:

1. There is a key distinction between loss aversion and risk aversion; in particular the former concept relies heavily on the dependency structure of monetary rewards across time. In particular expectations alone do not convey any information associated with the risk level of the investment. If given two scenarios:

A: you get 100 / lose 100 with probability measure 1/2

B: You get 0 for sure

Favouring B over A is perfectly rational from a risk aversion point of view.

2. I suppose your computation using stochastic calculus assumes all capital available are placed into the trade instead of keeping it constant? Will the expected growth rate of capital be negative if the trader always puts $100 in every game?

3. Tying back to my first point, since the "utility" of a trader depends upon previous events, the reference point plays a significant role in determining a "rational" player's action. For example, suppose a trader started at $1000 and reached $500 following t trades. Then at T = t, it is irrational (using the definition proposed by Daniel) to continue the investment if your reference point is at T=0 instead of T=t-1.

Hi Anon,

1) Yes, but Kahneman would argue in your example that both will be unattractive to a layman, and the layman would be rational in finding them unattractive. Only if in A one wins a bit more than one loses, and yet the layman rejects the bet, would he be irrational.

2) Yes in my example. However, you are right to point out that if a fraction of his wealth is risked each time, there is a Kelly optimal leverage when the bet become rational.

3) Indeed, if initial wealth is larger (i.e. leverage is lower), then this bet can be rational, as per 2) above. I do not like utility theory and does not like to think in terms of utility theory. But Kelly formula reaches the same conclusion.

Ernie

Ernie,

I think you're being a bit unfair to Kahneman in this post.

(1) The example he used in his book was $100 loss vs. a $150 gain, and refusing *that* gamble would be irrational. He goes on to note that studies show most laymen would need a $200 gain to accept the gamble, which is seriously irrational.

(2) What you're calling "loss aversion" is actually just standard "risk aversion" as defined in standard utility theory. There's nothing irrational about risk aversion, and the logarithmic utility function U(w)=Log(w) fully explains the expected capital growth vs. capital loss issue you discussed (i.e., the kelly criterion). Kahneman and Tversky's Prospect Theory differentiates loss aversion from risk aversion in a more subtle manner, and they prove that utility functions like Log(w), and therefore the Kelly criterion, can't explain how people actually behave.

Regards,

aagold

Hi aagod,

1) Actually, whether a gamble is rational or not depends not only on the game itself: it depends on the relative size of one's networth to the size of the bet, as per Kelly's formula. In the example above, I showed that if the initial networth is $1,000, then it is not rational to take the gamble. However, if the initial networth were $10,000, then it would have been rational. This is because with a $10,000 networth, the probability is vanishinly small that a series of losses would have ruined the gambler. Kelly formula in fact suggests what the optimal leverage is, or conversely, what the optimal initial net worth should be if we want to maximize compound growth. In our example, it is $2,205.

What I objected to in Kahneman's discussion is not the particular numerical values of whether that specific gamble is rational or not. I am objecting to his criterion for rationality: he based that on the expectation value of a single gamble, which is shown to be an erroneous criterion.

2) Indeed Kelly criterion is equivalent to log utility. That is well-known. I also don't doubt that many people are irrational even according to Kelly criterion. But again, what I, and other practitioners and researchers such as Peters, objected to is the criterion the behavioral economists use for rationality. They invariably use ensemble averages, which is incorrect.

Ernie

Ok, well I guess this is an example where we're all in what they call "violent agreement".

Just for the record, I really liked that Peters paper about ergodicity appreciate you bringing it to my attention in this blog. I also think Kahneman is correct and is not saying what you think he is (i.e., he's not saying people should consider only expected value of a bet and not consider range of possible outcomes / volatility). I also think you and Peters are correct, so it's all good :-)

Hi Ernie,

I watched Peter's presentations, and in the end it boils down to the fact that low-positive-expectation/high-risk gambles can still be attractive even in the timeseries world, just the leverage should be less than 100%.

So basically, if presenting the question very unmathematically, would you agree that EVERY bet with a positive expectation should be taken (i.e. it's rational to do so) provided that the leverage was choosen correctly(not just betting 100% of all the money every time) ?

Hi DM,

Yes, for every gamble with positive expectation, there is a low enough leverage which will make it rational to take the bet.

Ernie

I see. Thanks!

(All that is of course provided that the transaction costs are low enough and the risk-free rate is at least lower than the bet's expectation..)

Hi Ernie,

Would you please recommend some references for HFT strategies, especially for FX.

Many Thanks.

Jim

Hi Jim,

The best overall book on HFT is Algorithmic and High Frequency Trading by Cartea et al. It is the last item on my Recommended Books list on the right sidebar of this blog. But of course, my own book Machine Trading also has a chapter on that, which touches on FX specifically.

Ernie

Hi Ernie,

May I ask what is the FX mean reverting strategy you trade in your fund?

Many thanks.

Hi,

It is based on the mean reversion of pairs of cointegrated currencies.

Ernie

Hi Ernie,

Thank you for quick response!

What cointegrated currencies do you trade? Do you trade directional strategies in FX markets too? Many thanks.

Hi,

An example of cointegrated currency pairs would be AUD vs CAD.

We don't trade directionally.

Ernie

Trading is speculation. You have to avoid making financial speculation.

Dear Ernie,

May I ask why you don't trade directionally in FX markets?

Thanks.

Naturally, it is because I haven't found reliable signals for that.

Ernie

Hi Ernie,

Could we use Kalman filter to trade currency pairs?

How do we deal with capital allocation?

Many Thanks.

Hi,

Kalman filter is a general signal processing (denoising, weighted moving average, weighted linear regression) technique that can be used to trade anything. So yes, we can use it to trade FX pairs. Capital allocation is determined by the hedge ratio as determined by the filter every period.

Ernie

Dear Ernie,

In www.epchan.com/book2, I cannot find some data set, such as inputData_USDCAD_20120426 and

inputData_AUDUSD_20120426. Thanks.

Hi,

Please email me for those files.

Ernie

Hi Ernie,

Would you please recommend references for Kalman filter proof?

Many thanks.

Hi,

I am not a keen reader of proofs - but you can find many references in this excellent Wiki intro: https://en.wikipedia.org/wiki/Kalman_filter

Ernie

Hi Ernie,

I was hoping you could point me in the right direction. I have a trading system that can place trades daily at close, but holds them for one day if up or two days if down on the first day. The Kelly ratio is ~ 50% for all trades. New signal is generated if one day change is negative. Which means that if I place a trade today for 50% of my account and tomorrow the position is down, I have to keep the position open for an additional day. I should also open a new position tomorrow because I have a new signal on the same stock. Meaning the day after tomorrow I have two positions on the same stock, which are not completely independent as there is one day overlap. This situation can in fact continue for many days as long as there are consecutive down days on the same instrument. What is the right bet size for the second and subsequent positions? I feel like Kelly ratio is only good for new positions, the subsequent should be different. Any suggestion would be greatly appreciated.

Zyga

Hi Zyga,

As I have written elsewhere, Kelly "ratio" is unsuitable for trading, because it assumes discrete bets where you can lose the entire amount. No trader should make bets that can lose 100% (unless you are buying options)!

Much more applicable to trading is the continuous Kelly formula (m/s^2) that I described in my books. If you use that formula, the leverage it suggests would be applicable to the maximum position size you should hold. So if you "pyramid" as you suggested, and have e.g. 5 overlapping positions, the Kelly leverage should be applied to 5*$position.

Hope this helps.

Ernie

Thank you, that's very helpful. I checked the example on page 174 in your book Algorithmic Trading, however, it uses annualized mean and SD. Assuming I hold any position only for a single day and there are no overlapping trades, if my strategy's average return from the last e.g. 250 transactions is 1.1% and SD is 3.4%, utilizing m/s^2 tells me that I could trade with a leverage of 9.5 if I wanted to utilize the full Kelly. Is my interpretation correct?

Zyga

Zyga,

You shouldn't average the last 250 transactions. You should take the average annualized return of the strategy using calendar year. You should set some day's returns to 0 if there are no positions, when you are averaging.

But otherwise, yes, you are correct.

Ernie

Thank you for the clarification, much appreciated.

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