### May

#### 11

# The Humbert Quiz, from Rocky Humbert

May 11, 2010 |

**1.** Can anyone on the list state the probability that the Dow Jones would lose (and gain) about 10% in ten minutes, without any external stimuli (news headlines, etc.)? Is this probability comparable to the probability that the Dow Jones will lose (and gain) 90% in ten minutes? And how does this probability compare with the chance that lead unintentionally turns into gold? (more info).

If these odds are similar, is it a defensible investment strategy to buy tons of lead (without leverage), and wait for it to turn to gold? Has anyone pitched this idea to the large state pension funds?

**2.** Brownian motion allows for the possibility that all of the molecules of oxygen in my office move to the other side of the room, and I suffocate at my desk. Is this a more plausible explanation for what happened at trading desks on Thursday?

**3.** What were the "computers" (and people) who were selling the large cap ETF's / index funds at $0.01 thinking? There may have been some intra-day margin calls, but why would anyone or any computer sell the Vanguard Large Cap Index Fund at a penny? I will once again go on record as a willing buyer of the ENTIRE US Stock market at a penny. Just give me a call, or in the words of the Sage, "You have my number. And I can respond quickly."

**4.** Would anyone like to defend a portfolio that runs on the full Kelly Criteria and Optimal F?

**5. **Buy and hold is suddenly looking (comparatively) good again. That is, compared with people who left intraday market stop-orders.5. Lastly, in three months time, who will look smarter, the guys who sold P&G at 50? Or the guys who bought P&G at 50? That's the toughest question of all.

## Phil McDonnell comments:

I will volunteer for the Humbert quiz.

**1.** Can anyone on the list state the probability that the Dow Jones would lose (and gain) about 10% in ten minutes, without any external stimuli (news headlines, etc.)?

I do not think anyone can accurately calculate such probabilities without also taking into account the serial correlation in volatility in the short run. In other words the normal and log-normal models are only a stationery approximation to what is actually going on. In reality the volatility can change and it is positively autocorrelated. Thus the quick swings in the market can happen simply because the parameters of the underlying distribution are changing with positive feedback in the volatility parameter.

**2.** Rocky can stop worrying about suffocation. If all the molecules in his 3d Brownian office were to move to one side it would violate conservation of momentum.

**3.** I cannot defend full Kelly it is too risky IMHO. In my book I explain how to find it but only recommend its use as an upper limit for position sizes. It will lead to sub-optimal Sharpe Ratios among other things.

**4 & 5.** I have to pass on.

# Comments

9 Comments so far

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I like Phil’s answer to #2, but think it’s not quite rigorous It’s true that the sum of the momentum vectors should be conserved (assuming that Rocky’s office stays stationary), but this doesn’t preclude the spatial distribution of all of the molecules ending up on one side of the room. Imagine two molecules, both on one side of the room with equal and opposite momentum. Their average momentum is zero, but if you let them bounce around they can be in any of numerous spatial configurations. Here is someone attempting to calculate the odds Rocky requests http://diaryofnumbers.blogspot.com/2010/01/skeptically-speakingpart-ii.html

I’ll take a stab at #1 and respectfully disagree with Phil. The question (less the restriction on headlines) is fundamentally about how fat are the tales (kurtosis) and how far do they extend (standard deviation).

I used tick data from 5/1/2008 to 12/31/2009 for the most actively traded front month Mini-Dow futures contract. Unfortunately, this all the data I have, but a more rigorous analysis would include a much longer time series to more acurately reflect the underlying statistical properties.

I computed the simple 10 minute price returns overlapping every 5 minutes (8:30-8:40, 8:35-8:45, etc) from 8:30 to 3:00. In reality there are many more 10 minute periods during the trading day if you looked at 1 minute overlapping, for example, and since the market doesn’t wear a watch, any random 10 minute time period is a candidate for inclusion in this analysis.

After collecting this set of return series I computed the four-moments of the return distribution.

mean 0.0038%

stdev 0.4396%

skew 1.0332

kurt 13.4355

Given my small sample size and the time horizon over which this analysis occured I’d suggest that the skew is over-stated and that over a much longer time the skew is likely very close to zero, or since we’re dealing with equities, even slightly negative. Based on this judgement for the next step I set the skew to 0.

I computed 1,000,000 random draws from a Pearson distribution that has the computed statistical properties to create an “implied” empirical distribution. From this data I computed the following table:

hurdle freq implied

-10.00% 0.0000% 0.0000

-7.50% 0.0001% 0.0333

-5.00% 0.0022% 0.7318

-2.50% 0.0287% 9.5468

2.50% 0.0914% 30.4033

5.00% 0.0038% 1.2640

7.50% 0.0006% 0.1996

10.00% 0.0001% 0.0333

The frequency percentages in column 2 are how often we recorded an observation greater (or less) than the hurdle rate in column 1. In the third column is the number of time one would expect to observe these events in the course of a year. This was computed as follows: There are 132 10 minute returns overlapping every 5 minutes in a trading day and ~252 trading days in a year for a total of 33,264 possible 10 minute return observations in one year, so column 2 x 33,264 = column3.

So just about once a year we could expect to see a 5% move over ten minutes. As for 10% moves, well about once every 30 years or so…

However, I would expect that should this analysis be done over a longer period one would find both the kurtosis and standard deviation to be higher (if I had to “guess” stdev = .007%, kurt = 20) which will show up in a similar analysis as being a bit more often than that, something on the order of once every 5-7 years. Also consider that a larger sample size should make the frequencies a bit more symetric. And finally given the speed with which the markets move these days, a more appropriate analysis might include a longer return sample (ie 20 minutes since things happen twice as fast…) as a better forward looking proxy.

After the collapse, those with the lead will try turning into gold without any chemical reactions whatsoever. New Alchemy of Finance?

And you can’t compute probabilities of processes not governed by statistics. Brownian motion-based unlikely events’ probabilities have been computed because Brownian motion is a a statistical phenomenon. Multiple robots reacting to who-knows-what stimuli like European inter-bank liquidity problems or Taleb’s fund option trades is about as “statistical” as somebody having a brand-new dream on a specific night with specific content: it’s too complex and too rare to be described by any statistics.

I’ve always believed in buy and hold and never believed in trading. I read this site for it’s intellectual content not for trading tips. I think it’s particularly difficult to trade in a low-volume, bank-dominated trading environment where said banks can achieve a whole quarter worth consecutive daily gains and where political and central bank events can move the Dow by 400 points. I’m sure some people can do it, but it’s a different game than it used to be. Now even buy and hold is becoming and iffy proposition: you have to hope that whole continents don’t collapse to the point where they can’t recover for buy and hold to work.

Jonathan: Your analysis is interesting — but if I understand your writing, you calculated the probability of a 10% move — but you didn’t calculate the probability of a 10% down-move AND then a 10% up-move all within 10 minutes. That is, the fat tail was on BOTH the left side (down) and the right side (up) within a ten minute period. The Professor grants you a penalty-free 24 hour extension to complete the problem or clarify your result.

Dave Bacon: Thank you for sharing that link. It’s a “breath of fresh air.”

Gary Rogan: I suggest that you check the “use by” date on your canned goods.

I guess only partial credit for not reading the question correctly…You are correct, I did not calculate a 10% down followed by a 10% up in the same 10 minute period in my first response, merely the probablity of a 10% move in 10 minutes.

To address your original question I used the same data as before, 10 minute returns, overlapping every 5 minutes. This time I computed the “return” by measuring the high/low relationship, so a 10% move from H to L captures the magnitude of the move. Since we are really interested in the implied path of the move (moving from high to low and back to the high). I computed two metrics to assess this behavior.

One metric is the ratio of the open to close move relative to the overall high to low range. Thus a reading of 0 implies that the open equals the close and 100 implies that the open equals the high (or low) AND the close equals the low (or high). To make this useful to the analysis I adjusted the scale to go from -100 to 100 so that a reading of -100 means the open and close are the same price.

The second metric is the ratio of the open to the low relative to the high to low range. Thus, a reading of 0 means we opened at the low of the range and a reading of 100 means we opened at the high of the range.

Combining these three series we can develop a metric to answer our question. Using the last two metrics we can assess the path prices took during that 10 minute period. A -1 implies that during the period the market open and closed at the same price AND that we opened at the high of the range, a “perfect” reversal. Coversely a 1 would imply that we opened at the low and closed at the high for a “perfect” trend. Combining this with the high-low return gives us the magnitude of the reversal/trend such that large negative numbers indicate large reversals and large positive numbers indicated large trends.

In the same manner as before, I calculated the four moments of the scale I computed (again setting the skew to 0) and then took 1,000,000 random draws from a Pearson distribution with those properties to get the following:

hurdle freq implied

-7.50% 0.0000% 0.0000

-5.00% 0.0001% 0.0333

-2.50% 0.0034% 1.1310

2.50% 0.0033% 1.0977

5.00% 0.0003% 0.0998

7.50% 0.0001% 0.0333

Using the same metric as before (33,264 yearly observations) implies that about once a year we would expect a 2.50% reversal or trend. The 10% move you requested is not in the data, but I’ll suggest again that given my short sample size and adjusting for the “speed” of information, we could calculate an implied probablility for your question. And it most certainly would happen before lead turned into gold…

Rocky, five years left.

May I tip my hat to the responders some of the best quantifying and qualitating that I have seen in my not totally uneventful and short span in this field. If in the old days, I would have showered you all with gold coins. Now as a preliminary can I invite you to the spec list itself which you can get on by giving a phone call to linda at 203 8400777, or email lap at mantr.com vic

I must disagree with Vic, while the math is great there is a limit to it at the extremes. I believe Wilmott gives a better answer, with his situation where he ask what is the probability that a man claims he can flip a coin 10 times straight and make it come up heads each time. When Wilmott ask this to an audience most respond 1/ (2^10) but the answer is 100%, because the man stating this is a magician with a 2 headed coin.

So you tell me what the probability is for a "plant" to inform the audience the wrong side of inside info that could move the markets 10% in 5 minutes and then have the "magicians" take the other side the next 5 minutes and I will give you an answer to Rocky's first question.

While the answer may not be knowable, I would suggest that this probability has increased exponentially, with the government ownership in of so much of our financial system and also leading the regulatory charge for "change" of those entities.

Further, I believe this currently is much greater probability that US would lose their sovereignty in 10 minutes and only insiders know this for a 90% drop.

In other words what is the probability that the market could be rigged to produce each scenario, this is your answer.

The molecules and lead would be much harder to manipulate, in my opinion, but perhaps someone with chemistry and nuclear fusion knowledge could produce similar manipulation scenarios.

The human mind is capable of naming abstractions and then believing in them just because they’ve been named. “Probability” is a tool applicable to certain classes of real world events. Their occurrence has to follow a Gaussian or at least some other mathematically describable distribution. To be able to ascertain that, sufficient statistics have to be gathered or certain relevant qualities known with great certainty (like physical laws governing coin motion, the degree of honesty of the flipper, etc.) If you ask the probability of the world ending tomorrow or waking up with five unfamiliar monkeys jumping on your bed the questions clearly make no sense because these events have never happened before in any reasonable sense.