iBankCoin
Joined Oct 26, 2011
153 Blog Posts

2014 Goals: Streamlining The Process Part 1

I find goals are far more likely to be successful when you allow time to let the old goals develop beyond what you intended and see where they take you and then make goal setting more of an active process to manage rather than a single event that could otherwise overwhelm. Goal setting around New Year’s day I feel has become more of a marketing ploy and a distraction if anything from what you really want to accomplishment. So what I like to do is let the old play out for awhile. Then I start with a relatively unspecific long term goal that I observe my past works progressing towards. In this case it is “streamlining the investment/trading process”.

It is certainly not the only goal as constant improvement, education, and improving my ability to carry out the strategies more efficiently and effectively are other important trading goals. But by limiting it to just one general concept, I can apply the focus this year towards it. That is a “big picture” idea as the spreadsheet will help me with just one of those elements which is analyzing and managing risks with more precision in what I can expect. From this idea I can begin work on aspects of it, for awhile. Based upon how much I can get done in a sample period, how much time I can spend per week, and how much I have to do… I can then have a goal that reflects reality.

I find that this method of goal setting allows for expedient results.

Setting financial goals is a bit tricky as it involves chance and uncertainty, but setting financial goals is something that can be done as you can see here.

So from that I have a number of ideas as how different elements may coordinate together to help me streamline the process. First I need to sort out some of the ideas of things I want in a perfect world
1)Market analysis, Sector Analysis and Industry Analysis.
2)Using the above, develop a Ranking or Grading Systems of individual stocks based upon UNIQUE classifications on whether or not the market, sector, industry, and classification of stock and market cap size are currently “in phase” currently, “on deck” or “not in phase”, that has DIFFERENT applications/formulas for ranking them according to the particular TYPE/classification of stock it is AND what stage it and the industry/sector is in.
3)A tool that can quickly look at both stock and option pricing and a manual assessment of expectation, probability and timing to compare risk/rewards and analyze the effects of using any one particular option with the others and with the stock and position sizing.
4)An ability to take the inputs and add it to the trading journal spreadsheet where I can track and manage the various allocations by a number of categorical breakdowns if wanted.
5)A more multifaceted trading simulator which considers multiple, simultaneous and overlapping approaches on different timelines and compares strategies of shifting allocation towards each strategy, adjusting risk and adding/reducing capital from your account over time.
6)Ultimately a more flexible, dynamical approach to allocation that not only has the flexibility to adjust to maintain certain general allocations without having to sell individual positions short of their targets to do so, but also adapt in their allocations according to the behavior and numbers and relates expectations from every asset class or strategy allocation to others as well as future opportunity.
7)Summarize all the data in order to factor in everything (fees, risk tolerance, alternative investments, expectation) and quickly convert the data into a recommendation based upon my own inputs to optimize the portfolio.

This will be a process that will take a lot of time in developing, that will likely be an ongoing project over the next few years, so I have to make the spreadsheet flexible enough to be able to change with my strategies and positions and expectations, so very few elements will be “set in stone” and most will be inputs which I can change and very few assumptions will be made that are fixed.

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Going Against The Great

baberuth
I find myself in an unusual position. For the first time that I know of, I see things from the exact opposite perspective as the Option Addict.. And with conviction. I purchased some LNKD today near the end of the day on a day where option addict was selling his calls. Going forward is my TOP idea right now.

I say this with a tremendous amount of respect. I really do look up to “OA” and 80% of what I do is because of him.  Much of what I do differently than him is still because of “OA”. What I mean by that the approach of slicing up the market in multiple layers and identifying a good risk/reward among other things I got from him. So even if I am using different criteria and specific techniques to analyze are different, a lot of my ability to sort through it all and come up with a highly actionable setup can only be because I learned the majority of the process from him.

Betting against “a great” is not something that I like to do, and betting against the Fly or a tabbed blogger typically will get you nowhere fast. Yet I truly believe I am in a position where I have an edge on this trade. Even if I am wrong, I have learned that sometimes the only way you can learn to develop a strong intuition is going with your gut when your eyes confirm what you feel. I’ve watched great trades pass me by as a result of not trusting my gut but both my gut and eyes have been fine-tuned to being very productive. In this case, BOTH are signalling to me that going forward there is or will be tremendous opportunity in LNKD for at least a setup with profitable expectation.

Here is what I am looking at.

Item 1)The longer term rotation by the big money creates long term trends. Here I am comparing LNKD with another social media giant FB and using the GOOG/BIDU relationship to show how it works. daily and weekly chart. Although there is often an overlap and pretty significant correlation being in the same space, in terms of trends you can tell what’s in favor and how it rotates. You can often anticipate who is the next to lead and which is next to lag.Jeff of course has talked about the risk cycle in a slightly more advanced way for more of a swing trader’s perspective, but I believe in many cases when the stocks are similar enough in size and type of company you can look at a longer term cycle of multiweek/multimonth moves. Eventually based upon this analysis, I believe a multimonth move is coming in LNKD eventually.

The first one to bottom or breakout typically leads with stronger trends upwards. The first one to peak typically begins to consolidate first and the other follows.

leader laggard cycle

To emphasize what I see.

GOOG starts with a stronger trend while bidu consolidates. Bidu then forms a relative low while GOOG is having troubles showing strength and as bidu rips higher IT then begins to catch up aggressively and thus becomes a leader for the time being. THEN the rate of ascent slows as GOOG continues consolidating. Now GOOG makes an initiative move and the rate of ascent favors GOOGLE as a leader confirmed as BIDU is the first to top out which at some point may be good for BIDU as at some point it will consolidate again while GOOG then is late to decline and BIDU will at some point in the future likely take off another leg higher while GOOG continues to consolidate again and the cycle continues.

Now FB/LNKD. LNKD was strong while FB lagged. There are some subtle differences in the relationship as to the nature of how they lead and lag. As FB peaked in feb 2013 suggesting LNKD may follow and eventually it did, but LNKD remained very strong and actually even began the leg up first before FB, actually continuing a second leg of leadership. FB followed late, midway through LNKD’s advance and then continued as LNKD sputtered, then continued again but this time the rate of decline slowed as basically FB consolidated and made it’s next leg up as LNKD continued to tighten much like GOOG did to BIDU as BIDU still advanced, but as soon as GOOG broke out, it’s rate of ascent was much higher.

LNKD initially peaked first, suggesting it would consolidate but on FB’s next major high LNKD actually took out that high on October 21st while FB failed to do so suggesting that a subtle rotation may be beginning. Even though FB defiantly continued another leg higher this time while LNKD continued it’s long consolidation, LNKD continued to set up with a tighter consolidation readying for a multiweek or multimonth advance to take the baton from FB. While FB is still technically leading, it is on it’s 2nd leg higher. The last time LNKD did that it spent a lot of time consolidating.

 

So right now I see LNKD as having been largely ignored by the big money which create the trends and in my mind, that creates opportunity of greater value for lnkd than it does for FB at SOME point. Based upon technical analysis I think that will cause LNKD to lead SOON, the question is only WHEN.

Now let’s isolate LNKD as a stock and do some more specific technical work with today’s closing chart.

Item 2)Chart 1:

lnkd1

LNKD has about 80% of it’s outstanding shares floating. That means it is heavily traded but not necessarily owned heavily by management types. That means I am more likely to interpret the volume as simply temporary action as opposed to permanent “value” investor positions. Therefore, I think those who bought 150-180 probably took some profits above $220. The aggressive buying everytime you think the stock is breaking down to a lower low tells me they want their shares back badly at lower prices. The smart money to me has been slapping the dumb money around who comes in late and chases the rips higher and panics late lower. Even if that interpretation is incorrect, I think the 150-180 volume will be able to “shoulder” the selling, and possibly even create that one rip higher where the dumb money learns there lesson so they stop chasing higher the next time or two around and suddenly they find themselves missing out and the stock at new highs. At SOME point I think that will happen. Regardless, EVERYTIME you had a new low, you saw an eventual rejection of that low and a very sharp up move from that new low that carried on into the following week.

Item 3)”Bullish Divergence” The RSI and Slow Stochastic have a large degree of overlap which means they essentially are measuring many of the same things (momentum). So I won’t call it a “quadruple divergence. However, on MULTIPLE timeframes (using OA’s triple oversold indicator) the RSI on the daily chart is showing a clear picture of HIGHER relative strength lows even as the stock goes lower. That signals strengthening momentum even as stock is available at lower prices which tends to be bullish.

lnkdtwo

 

I believe we are near levels of support however you look at it. If we get a fast down, I think there will be a quick flush , capitulation and a chase higher and the support as we near 180 will be very aggressively ready to buy. Even if we do not, we may still be at the levels where the consolidation pattern is in a state of support and we could still rip higher once more. If we do not flush lower and reverse we still could see that eventually happen and I may be early, but I like my odds here on multiple timeframes for multiple reasons.

 

HIGH conviction name. I am not saying it might not flush first and reject. Short it if you wish, but to me looking forward there is tremendous potential. I may be far too early in the trade, but nevertheless I like it here even though many will see it as “crazy”.

Finally: For future consideration and what will eventually be described in more details in my 2014 outlook I would like you all to consider that there are generational trends driven by the relationship between bond prices, interest rates and pension funds having to reach certain thresholds of earnings to stay solvent. That these trends are only just starting to potentially become relevant with regard to stock prices. As a consequence these trends along with how the smart money anticipates and capital reacts, create a “liquidity cycle” that potentially change whether or not we are in a stock market or market of stocks and which in turn drives the “risk cycle”.

Disclaimer: I grabbed some LNKD near the end of the day today (2/12)

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How A Portfolio “Kill Switch” Can Change Everything About Long Term Expectations

We all most likely have some sort of psychological breaking point in which we would give up on the system, or in which we should probably re-examine our strategies, and in some cases, in which people go crazy and “break” to the point where they start revenge trading and no longer are actually trading their system. The prior risk management modeling I’ve done using a spreadsheet has not taken into account fees or how one handles draw downs. I have modified the spreadsheet such that there WILL be a fixed fee involved with trading (thus if you have a significant drawdown fees play a much bigger role as a percentage of your account), and I have added the ability to add a “kill switch”. Since I added fees at minimum, it’s necessary to have at least a kill switch of 100% (loss) so you don’t go on calculating $12 fees when your account is less than $12 and continue to go negative. In the meantime I set it up for 1000 hypothetical trades since we now have a more practical model I can simulate it for much longer time periods if need be.

So my hypothesis is people will give up after a drawdown of varying amounts. This drawdown is from the account HIGHS, not from the starting amount. Thus, for some people the end result after 1000 trading periods (whether they participate in all 1000 using the given strategy or not) will actually sometimes be better if they risk less to avoid this breaking point.

PLEASE understand that this simulator only uses one trade at a time as if you only play a single OTM option trade at a time and hold no trades simultaneously. In reality having more relatively uncorrelated trades at a smaller risk % per trade can often improve results at less risk, especially if you have multiple systems such as profitable investing and trading stocks and other asset classes in combination with your options. I will get around to testing partially correlated trades simultaneously and possibly entirely different systems in the future. For now the expectancy is based upon

p1 17%
p2 20%
p3 16%
p4 23%
p5 24%
w1 294%
w2 52%
w3 0%
w4 -64%
w5 -100%

My 2013 OTM option trading results at various risk levels and various kill switches to see a distribution of simulated results. For all accounts I will use a $10,000 starting portfolio and $12 for completed transaction (buy fee plus sell fee) just to use round numbers and so that fees is somewhat significant to the equation.

First let’s start with someone who is very risk averse and cannot handle even a 5% drawdown without feeling incredibly nervous and emotional and plans to quit at a 10%.

I had to think about the results and double check them at first, because I was surprisingly getting a significant skew right with only 1% risk.The mean is greater than the median. (i.e. the “average” is skewed by a smaller number of outliers such that the majority of the people are actually below the “mean”)

The reason was that at 1% with a 10% drawdown, you not only have too little capital at risk to the point where fees will eat away and a large percentage won’t get off the ground, but also that it is extremely common that at some point within the first 200 trades of those 1000 that you will hit the kill switch and very few will make it through all 1000 to produce significant gains which on average will be pretty high even with 1% risk (given that you can last that long). The overall average was still above the starting amount of $10,000, but in most cases, the fear of drawing down killed the traders hope of making money and taking risk. Increasing beyond 1% actually reduced average results because more risk = more volatility and greater probability that you will drawdown before giving your system a chance to compound it’s gains. 10% in theory is around “optimal” in terms of maximizing geometric return, but to someone who is risk adverse, it is probably less than 1%.

I then decided someone who is mildly risk adverse and doesn’t like 10% drawdowns, really starts to go crazy after 15% and ultimately feels compelled to shut down the account after a 20% drawdown. This was interesting because the average ending amount really increased from 1% to 2% but then decreased to from 2% to 5% to the point where you are better off risking 1% than 5% if you can’t handle a 20% drawdown.

I didn’t gather the histograms yet but here are the data points for 1000 monte carlo simulations of 1000 trades under each given condition.

drawdown switch 10p and 20p

30% and 40% drawdown was up next.

drawdown switch 30p and 40p

 

As you increase risk and become more lax on the “kill switch”, the worst case scenario gets worse, but so does the best case scenario. The skew actually becomes less noticable with a larger drawdown switch AND less risk per trade. Over the long run if you LET the trading system work, the results begin to normalize and cluster around the average. However increased risk increases the skew and depending on the drawdown kill switch may not improve even the MEAN average results. With these trades the approximate kelly criterion is 15% of capital but with even the risk aversion to withstand a 30% drawdown from high you are still best risking less than 1/3rd of the kelly.

 

So that brings us back to the kelly criterion graph which is entirely misleading if you are not aware of the other variables including personal aversion to risk, sample size, time period, etc. This graph is with the assumption of infinite time on your side because it was mathematically convenient. However, based upon some understanding of real data in a finite number of time given certain psychological barriers one has to cross among other things you know that for practical application even risking 1/3rd of this amount is incredibly aggressive when using an option based market strategy assuming one single uncorrelated bet at a time. We are also assuming KNOWN information and a fixed edge as opposed to a more uncertain one, both of which favor more caution. On the other hand, I believe I left a lot of money on the table and can improve my system by executing it and managing it more efficiently, and that eventually I will get around to testing how multiple trading systems work on the account. My theory is you can reduce risk and increase return by using complimentary systems (a stable consistent risk adverse system combined with an aggressive one both of which are relatively uncorrelated with the right mixture will compliment each other and both improve return and reduce risk while “normalizing the returns” over a finite number of time)

 

Hopefully this post is enlightening and helps you really analyze and understand risk, and I look forward to advancing my spreadsheet in the future to help me more thoroughly analyze a more multi-dimentional (synergistic) approach to risk.

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Modeling Your Past Trading Results At Different Risk Levels

So In the last few post I have really focused in on objectively modeling risk within a portfolio given a particular system. Before I go and amend the heck out of the spreadsheet to upgrade it to include the possible input of fees, of monthly addition of new capital to combat some of the decay, adding a “kill switch” input/function where the results are automatically capped at a particular loss if you draw down below a certain amount of your initial risk, and possible other features at some point I may add, I wanted to actually use it as is using objective numbers rather than an arbitrary 20% probability of a particular result with a set expectation.

So I have set up all my trading results not including open positions over 2013 using options while setting stock trades aside separately. I chose to include hedges in the calculation. I did not update a few trades including my 1200% gain in twitter calls. Here are how I looked at the results. I had 281 closed trades since I started tracking. I have room for 5 inputs of theoretical “results” so how I break this down may create a slight difference from reality and theory here, but this is just a model. I want to keep all 100% losses together separately. I have 71 trades that expired worthless. That is 71/281=~25.267%. That is better than I thought I would get because of the aggressiveness of the options.

Now, I want to take a few slightly better than break even and slightly worse that average zero%. I basically took any option trade that made between -18 to 18% and got 33 trades that effectively equal a scratch 33/281=~11.744%

Then I want to average all of the remaining losses not included in the “scratch” area. These most likely will be premium that I salvaged to avoid the 100% loss and those nearing expiration that had failed to move enough.. The average loss here is ~63.8143% and there were 74 trades in this category 74/281=~26.3345% The actual expectation was positive but less than 1% but I will round down to 0%

Now the WINS. Any win over 100% deserves its own category. There are 43 of these 43/281=~15.3025% of all trades with average ROI of ~293.0585%.

And the remaining WINS. These were most likely trades that either I managed poorly and took off before it reached my target, or ones in which I sold and/or rolled out as expiration forced the issue. There are 60 of these 60/281=~21.3523% for an average ROI of ~53.87795%.

So… Now we can define our system. I like to list the GAINS from highest to lowest for easy and consistent interpretation when I look at other systems or modify the expectations.
This is one way to show what the system looks like.
P1=.153025
P2=.213523
P3=.117438
P4=.263345
P5=.252669

W1=2.930585
W2=.5387795
W3=0.000000
W4=-.638143
W5=-1.00000

We confirm that the probabilities are correct because these all add up to one. 37% of my trades produce a win, 48% win or approximately break even. But the largest gains clearly outweigh the losses.

The old way I used to do things would plug this in a kelly criterion calculator, find out that a full kelly I could risk 10% per trade if traded a single trade at a time, and then use my own calculator that factored in fees and correlation and multiple bets at the same time and fees and based upon a $10,000 account would conclude that the “optimal” number of trades at a 60% combined correlation would be 14 trades at 2.6% risk per trade for total of 36.4% capital at risk. Then I would curb that to aim for maybe 7 trades at 2% each.

But now I have learned that 10% resembles more of a “lotto ticket” even after 300 trades. But, I have my baseline of 10% as the max and can make an entirely new distribution at 1% 2% 5% and 10% to show you the difference between this system and one with a 20% probability of either 150% 50% 0% -50% or -100%.
First let us redisplay the results from the arbitrarily determined system
results

Now that we have real numbers, I want to be a bit more thorough than just using 1000 simulations. I will bump it up to 10,000 simulations per risk level. I will leave it at 1,000 just so the numbers remain the same. Keep in mind that the kelly criterion for my trades in reality is 10% vs the theoretical system is 14% so 1% risk actually is MORE aggressive with my system than the theoretical one. As a result you should expect a higher standard deviation and a higher average. With a larger simulation alone you will get a larger minimum and a larger maximum as well anyways.

And here is what the histograms and data looks like

resultme

I think more telling than the distribution since it is so difficult to see at what level the large decline really starts when you deal with such large numbers is the sample equity curves. So I will run a few of those. Click here to look at the equity curve of the theoretical model. and here are a few sample equity curves modeled after my trading abilities.

 

1%

1p risk

2%

2p risk

2p risk2

5%

5p risk

AND half a dozen examples of 10% risk and the vicious account volatility

10p6 10p5 10p4 10p 3 10p2 10p risk

It’s important to understand that 10 simultaneous trades at 1% functions much differently than 1 trade at 10% or 10 trades over 10 trading periods at 1%. Unfortunately it isn’t so easy to model this and the results depend greatly upon how correlated the trades end up being (the lower, the better provided you can do so while still having the system be as profitable). For informational purposes, assuming no fees, you will see 10 simultaneous trades at 1% each function as a cross between 1 trade over 10 trading periods and 10% as you get some of the low risk volatility benefits of small position size and some of the high return benefits of 10% risk. The result is usually a better return per risk.

Going forward, I am working on improving this simulator so that it can allow for additional inputs that will help test how fees, adding capital, multiple simultaneously partially correlated bets and having “complimentary systems” can potentially positively influence return while also reducing risk.

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Equity Curve Of Risk – How Risk Influences Expecations

In the last post I discussed how I used a system and position sizing simulator to look at the ENDING equity of thousands of traders trading a theoretical system. I mentioned I would be showing sample equity curves at a given amount of risk by pulling up a random trader. It’s a lot easier on the spreadsheet to get a better sense as you can just press the F9 key to recalculate the random iterations and thus instantly bring up an entirely new random equity curve with all the same settings. You can go through several examples in a short amount of time. It is a bit more time consuming to create new JPEG images of each of them and then post them here so I will only be showing a few.

To further illustrate the type of “risk” you are taking by a particular strategy I provide just one random “trader’s” equity curve of each of them. Understand that results may not be entirely typical but pay attention to the % drawdowns to get a broad sense of the type of risk you may look at and endure.

Please note: The actual expectations of the system you use will drastically impact the type of volatility you see with every 1% change in risk. These sample equity curves are only made with the trading system with an expectation of a 20% chance of each of a 50% loss, 50% gain, no change, 100% loss and 150% gain.

1% risk

1p risk

2% risk

2p risk

5% risk

5p risk

As you increase risk, the results become more polarized and more extreme, so I will provide a few examples for those at the supposed “optimal” risk percentage of 14% risk

14d2 14d 14.2 14

The phenomenal results of a few skew the results of the rest. The drawdowns are insane as you see 70% and 80% drawdowns.

Can you stand 80 trades of being down steadily as your account drives lower to HALF of what it started with? Most people can not and would capitulate so even putting 5% of your capital into this “system” becomes problematic. Granted multiple bets with a lower correlation that adds up to 5% or even more may be actually “lower risk” than 5%. Granted, you can potentially use strategies that actually profit from market overall volatility such as allocation models and rebalancing and modern portfolio theory and hedging and pairs trades and such, you can put in some income and weight a lot of your portfolio with stock that have more of a slow and steady drift upwards that 70% of the time actually provides more stability and increased liquidity that can comba the negative effects of account volatility. Granted, a MORE profitable system can allow you to risk quite a bit more without the same drawdown expectations…. But even so, we are talking about a winning system where even at 1/3rd of what some quants would suggest to be “optimal” over a finite amount of time the returns are very likely to be terrible over a significant period of time.

Can you see why long term capital management went bust now as they did not test their assumptions while taking only a small sliver of time in the past by which to evaluate their “expected risk”?

I could get into how uncertain the world is and how your estimated “edge” within a system is also not a certainty which is still an assumption that this model must make to provide results, but at least can be recalculated with different sets of expectations. But I hope that this post has been educational enough for you to make at a minimum slight, productive adjustments to your way of thinking, if nothing else.

Don’t blow up like LTCM… Test all of even your most basic assumptions… Evaluate your risk in as many ways as you can. Understand risk and how to manage it. Control your destiny rather than being a victim of your own emotional compulsions to sell at the worst point of time and capitulate just before your system takes off because your system is too volatile. Understand the dynamic nature of reality and how increasingly large leverage and risk may be increasingly more volatile while also being more vulnerable to small changes in the conditions by which you based your assumptions. Understand the need to be well capitalized and that fees aren’t factored in and more negatively impact the volatile systems that have an increased probability of drawing down significantly from the starting point. Constantly seek to let the facts guide your conclusions, and seek productive improvement on the way you look at things. Then risk can serve you, rather than you “getting Serrrrrrved” by risk.

 

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How I Let Data Guide My Conclusions and Results Of Thousands Of Monte Carlo Simulated Trades!

I stand to you today to announce that I have used data and simulation to prove myself wrong. Call me a flip flopper if you like, but I view this as a constructive thing as I have chosen to take the most profitable and beneficial path rather than the most comfortable. While some remain attached to certain ideas, I let the data guide my conclusion whenever possible. The human mind is full of biases and often too rigid on our ideas. Be open to examining the assumptions that you take for granted on a daily basis just as testing the assumption that “the world was flat” was a productive one, it is possible you may make significantly greater progress than all of those around you who resist the change in your ideas.

My previous paradigm was guided by this understanding of the relationship of risk:
kellycriterion
Unfortunately, every model has certain “assumptions” it must make to construct any particular generalized “model”. It is usually not the model itself you should test, but the assumptions within the model, as well as your own personal assumptions which can only be done by data first. After adjusting and testing these assumptions and thinking more dynamically I can see that this is simply not practical as you will also see in a bit.

At first I had a simulator created to calculate all possible permutations of theoretical trades, but realized the simulator could be improved. Rather than continuing down the direction I was headed, I “flip-flopped” again, instead opting for constant improvement. I instead came up with a spreadsheet that uses the random number generator and a “Monte Carlo simulator” plugin that I view as much more efficient and flexible in terms of the duration of trades in which I want to test. Although it lacks the same degree of precision, it is a productive tradeoff as you can still increase precision in exchange for a more timely monte carlo simulation (with more random iterations).

I used the spreadsheet to look at returns dynamically over a finite amount of time such as 300 trades. Out of a thousand traders for example, some percentage may gain 20% while another percentage gains 100% and another percentage loses 50%. Using this data, A histogram plotting all simulated results of each of the thousand random iterations of 300 trades was made for various levels of risk given a particular system. The simulation allows for a change of any one of these inputs (probability of 5 different “results” of the trade, the ROI given each of these 5 “results” the number of traders randomly selected and the number of trades they make). You can even look through random equity curves across all 300 trades at a given risk factor and refresh it with a push of a button to pull up another random trader to get a better sense of drawdown within different points of the system over the course of those 300 trades.

Without further ado, here are some results!

 

results

pX=probability of event X.

wX=win % (ROI) given event X.

System: p1-5=20% W1=150% W2=50% w3=0% w4=-50% w5=-100% A winning system is presented.

Risk defined as capital at risk since this is an option strategy and you can lose the entire premium.

“optimal F % / full kelly = 14% risk”

Note the severe skew right. This means as you increase risk extreme outliers begin to skew the average higher than what is “typical”. Skew right means the mean (average) is way higher than the median (average). The “worst case scenario” grows with risk. The probability that you end with a lower than average result (that is not a typo) increases as risk increases risk given a finite amount of time. Eventually the probability of a poor result is so great that as you increase risk the long term geometric return suffers. If you are a true cowboy looking to become an “outlier” and willing to put in the risk, then perhaps that is okay with you, but just know that going beyond the “optimal” amount is destructive as you approach “an infinite number of trades”. Just know the type of CRAZY account volatility you will have to endure, and a large probability that you actually will end down even after 300 trades. That’s almost 6 years at 1 trade a week!

Since I took the time to create this spreadsheet, I can simulate thousands, or if I like, tens of thousands of traders trading anywhere from 1 to 300 trades (or more if I take 5 minutes to set up more) with a given system with a push of the button. I can instantly adjust the expectations of the trading system and see how the results change.

In the next post Titled “Equity Curve of Risk – How Risk Influences Expectations” I show some example equity curve of a particular risk percentages.

 

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Game Theory and Investing

If you have an edge in the market, or are using an “exploitative” strategy, position sizing is supremely valuable. If not, you should position defensively (use an “equilibrium” strategy) to avoid being “exploited” yourself. As you increase position size and overall risk (with an edge) your geometric return functions like this (over an unlimited time period):

kellycriterion

If we were to continue beyond double the amount that maximizes your geometric return, it would begin to look like this:

This is based upon a formula called the Kelly criterion. There are Kelly criterion spreadsheets available like this one. The exact percentage in which you reach the apex of returns depends upon your edge and odds. 50% of the “optimal” risk produces 75% of the results with 1/2 the volatility. Because of uncertainty over your edge, it statistically is better to err on the side of caution, not to mention better for reasons of psychologically, emotionally, and due to a “less than infinite” duration of one’s trading career, and “utility theory” among other things. Let me be clear… It is far more important to protect what you have while producing a positive return than maximizing the return. As such the exact application of this is perhaps not so important as recognizing that increasing the risk does not proportionally increase the geometric return.

Without fees involved, one single independently correlated position size at the apex of those charts produces a lower return than two independently correlated position sizes at HALF the apex.

We can pretty accurately conclude that uncertainty, randomness (or more accurately “chaos”) and portfolio volatility are usually the enemy of those who look to place wagers and handicap markets with a particular edge.

Illustrated numerically, it occurs because of this concept of the result of a loss not being equal to a gain:

loss gain | needed to get back to even
1.00% | 1.01%
2.00% | 2.04%
3.00% | 3.09%
5.00% | 5.26%
10.00% | 11.11%
20.00% | 25.00%
30.00% | 42.86%
40.00% | 66.67%
50.00% | 100.00%
60.00% | 150.00%
70.00% | 233.33%
80.00% | 400.00%
90.00% | 900.00%
95.00% | 1900.00%
99.00% | 9900.00%

The largest obstacle to overcome with this style of trading is that volatility will more often become your enemy. A conservative approach can still handle most expected volatility, but factor in your “black swan” and “10-sigma events”, and even a conservative strategy can be hit hard and fast by volatility to the point that it can destroy many otherwise profitable strategies. A robust approach that may do only a little better than break even in all environments, yet protect itself from the deadliest rare “black swans” in the long run may prove much better than the aggressive highly profitable approach. Fortunately their is a happy medium that may provide the best of both worlds.

If you have an edge, I believe a portfolio should still have a significant portion designed around mitigating the risks of volatility. The solution to volatility can be to try to adjust strategy based upon the “market type” you are in, but that can often be either retroactive (backward looking), and late to the party, or anticipatory and potentially inaccurate a high percentage of the time. As a result their are still substantial risks of not being able to significantly mitigate the risks of volatility ruining or greatly reducing a strategy’s profitability. In technical terms you might have a “balanced exploitative strategy”, that only slightly deviates from an “equilibrium strategy”.

Quick Primer on Game Theory – “Nash Equilibrium Theory” and “Exploitative Theory”
1)Imagine an alternate universe where there were only 2 investments “stock market” and “dollars” and no leverage where the “stock market” never goes to zero (although individual pieces might). If market were random or if the actual “edge” or expected direction were completely unknown, the “equilibrium solution” to play the market and produce gain based upon others mistake (who transfer capital from stock and cash unevenly causing changes in prices) without directly exploiting them would be 50% cash and 50% stock.
Example: Stocks are worth $1000 and you have $1000 in cash. Stocks double to $2000. You sell 1/4th of your stock position so that you now have $1500 cash, $1500 stock (50% each). Stocks decline by half. You now have $750 stock, $1500 cash $2250 total vs $2000 you started with. $250 more value than you started with or a 12.5% gain due to volatility after stocks “normalized” or “reverted to the mean”. The “equilibrium” strategy seeks not to handicap the market, but let others try to do so while over a very long period of time profit from volatility.
An “exploitative” strategy under the same “universe” is when you decide you have an edge and that the market has neglected “stocks” so you might buy somewhere between 50% and 100% stock (100% would be maximally exploitative, (and an “unbalanced” strategy) where as 50% would not be exploitative). The more “exploitative” you are, the more vulnerable to long term portfolio volatility, uncertainty, and “chaos” (I feel this is a much more accurate term than “randomness”). It’s important to realize that the farther the market is from “equilibrium”, the more an equilibrium strategy will gain. However, the proper exploitative strategy under the same circumstances will gain even more than equilibrium strategies if assumptions are correct in such situations. I will prove this with a brief example: If you had to choose heads or tails with a double sided coin (2 heads) with 3:1 payout (payout is far from equilibrium), calling 50% heads and 50% tails might still gain more than a 2:1 or 1.5:1 payout, but calling heads every time would be MORE profitable. However, if someone was able to switch to a double tailed coin or a regular coin at one point without you knowing, the more aggressively you bet your assumption and position for those assumptions, the more vulnerable to decline and even risk of ruin you are if your assumptions are proven wrong.

2)In the real world capital can migrate to and from multiple asset classes, credit can contract and expand, domestic capital flows can be very different than global ones and potentially move against the grain of global capital flow. Asset classes have components and sub components and so on all of which can receive capital inflows (rally) in the face of a broader capital outflow (gold going up or sideways as commodities go down. Rather than concentrate on a specific equilibrium solution, it’s more practical to just look at a simple one as part of a “core” strategy. One such as this:
simple equilibrium

If you want the technical jargon you can read about “modern portfolio theory” which basically attempts to maximize expected return for a given level of risk using the “efficient frontier” and using “beta” to define risk. To me “beta” is a backwards (retroactive) looking view on price volatility relative to the backwards looking view on the market’s price volatility. The assumption of an efficient market is also a “suboptimal” conclusion in my view. I am not interested in an overly technical portfolio that requires frequent management. Instead I am only looking to exploit the inefficiencies of the market one one hand, manage the risks to do so effectively, while mitigating risks via defending against chaos via a portion of capital that is in some pseudo-equilibrium solution that basically looks to establish a “baseline”

 

My portfolio composition will actually function more like this: (click on the image to see full size photo)

dynamic allocation strategy

This is how the same concept could look and function with mostly leverage:

leveraged dynamic strategy

Depending on how low the correlation of each trades, you may be able to add a lot more individual short and long term trades as the natural “baseline” of risk, but it still must be monitored and have certain rules and “limits” in order to keep the exposure and risk in check at all times.

You might reduce cash and add in “long term income” based risk in your capital that could be anything from buying a 12 month cash CD every single month (so that every month you have a CD from a year ago pay out and become liquid), individual stocks that pay dividends, ETFs in any number of many categories that pay dividends and so on. The income will drip feed into the cash account, and because of a growing cash reserve that can be used to rebalance, their is less of a priority to have as much excess cash, provided the income is fairly robust. (A broad S&P ETF probably isn’t at risk to see its dividend get cut).

A basket of various income ETFs such as PFF,CORP,TLT,SPY,HYG,EWSS and SHY as a substitute for a large portion of that cash along with a 12 month CD purchased every month is an effective way to create liquidity without having so much capital tied up in 0 interest yielding cash on the side, yet without having an overly large amount of risk associated with it.

The philosophy is still about having an allocation based portfolio that can manage large swings in volatile capital flows, manage portfolio through large periods of expansion and contraction (inflation/deflation) and any number of unforeseen variables. This management of volatility in combination with individual positions and hedges provides a more stable portfolio overall, but still allows you to do plenty of “core” trading” in the short and long term.

The actual allocation should be dependent upon: your skill in each of the areas you allocate towards, your overall skill, your current market outlook/bias, risk tolerance and a few other variables. I’m not going to go any further in depth than that for now, but I hope this post is able to allow one to rethink their strategy and whether or not their strategy is properly prepared for different periods of volatility while still providing yourself with the flexibility to adapt.

You do NOT have to be a “game theorist” to invest and trade, but it is important that you understand that there is a basic “baseline”. Also, it is useful to know how one might adjust when approaching a market that is not at equilibrium and how to avoid significant risks that are detrimental to your long term returns. We will cover more on making adjustments to a basic balanced philosophy that I have presented.

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The Story Of Success: Do You Have The Hunger To Succeed?

I gave to you a great tool that can help you succeed in my last post.  “Jimmy Bob” asked a good question. The question was along the lines of “how much work do I do to reach success?”

This brings me to a story I have heard, but I will retell it anyways.

There once was a wise guru who had achieved enlightenment sitting on a mountain top

A young boy seeking enlightenment approached him and said, “I want to be successful like you, please teach me your secrets”.

 

“How badly do you want it?” The guru asked.

“I will do anything it takes!” the kid responded.

“Anything? ” The guru questioned

“I would kill for the chance at success, I would die for it!”

The guru having been through the same routine of those who expected him to impart the wisdom over a brief 30 minute chat or less expecting their to be just those few “secrets” had grown tired of people coming to him asking for the quick solution and so often only claiming they would do anything. He knew the path to self improvement and personal development was a long road. Yet nevertheless he had an idea and said “meet me in Long Lake at 5a.m. tomorrow” and I will teach you what you need to know.

To the kid’s surprise, the next day when he came out to meet the guru, the guru was out in the lake already. The guru called to him, “come join me”

The kid said ” I am not prepared”

The guru said, “When you want success, you may not always be prepared, but you must continue on anyways”

The kid said, “But what I mean is, I didn’t bring my swimsuit.”

The guru responded “you may not always have the tools for the job at a particular time in your journey.

The kid said, “but the water is cold”

The guru replied “Life can be cold, and the journey to success can be even colder.”

The kid said “What does this have to do with success, I thought you were going to tell me the secrets? Or tell me a magic success formula? Or give me the magic pill to success?”

 

The guru replied “sometimes in life you have to make “the leap of faith”. Sometimes you have to trust that you’re in the right place and things don’t make sense. Sometimes you may have to rely on your training and experience to know the truth. Other times, to really know the truth and make the right decision, you sometimes have to go against everything you ever learned to find the truth even when it conflicts with your beliefs, and you must have the hunger to carry on anyways. Right now, entering cold water does not seem to have anything to do with success, and yet here we are. But a part of you inside knows that you are here for a reason. A part of you had enough wisdom to seek out someone who was successful and the courage to follow him. A part of you intuitively and instinctively knows that you must come to me now. I will not give you another chance, if you want to have success, join me immediately, no more questions”

morpheus matrix

As the kid finally decided to “take the journey” the guru said to him, “What I have heard from you is nothing but the typical excuses that is not all that much unlike the normal excuses that everyone makes that prevents them from success, “But I don’t have the money, but I don’t have the time, But I don’t know how, but this other guru who’s program I bought told me something different, but It’s too hard, it’s too cold, I don’t want to”. Inevitably some continue the march on, but of those people that try, many more will fail. I assure you, if you take the lessons I am here to teach you, you will succeed, but you have to trust me… Do you trust me?”

The kid approached the guru in water that went up to the kid’s neck.

He was a little nervous, but then he realized he did in fact trust the guru, so he said yes.

“Then allow me to baptize you in this lake.” Said the Guru

The kid nearly was confused, and caught himself from saying that he wasn’t religious and wasn’t interested. He thought perhaps it was a test. So he thought and realized that there were to be no more questions and that if he trusted the guru, he had to prove it. So he said okay.

As he was submerged under water, he realized the guru wasn’t letting up and was HOLDING him under water. The child began to become fearful, short of breath. He eventually realized he may have been wrong to trust the guru, he briefly calmed down and knew he could hold his breath longer so he still lay longer. But then he started to panic a bit and tried to stand up to which he was met with even more pressure.

“He’s trying to drown me, I am definitely wrong to have trusted the guru!” he thought, “I knew it”. He pushed and struggled even harder to come up for air but he couldn’t. full on panic set in. He finally started to leap out of the water when the guru covered his mouth and pushed him down one last time.

Just when he began to feel the intense struggle like he couldn’t take it anymore, and just when he reached maximum discouragement, his life started to flash before his eyes as he began to accept his fate and stopped struggling.

Just then the guru ripped him up out of the water and asked “Are you okay”.

The kid responded, “what the hell was that?! no I’m not okay, you almost killed me! I thought you told me to trust you, yet you could have killed me!”

The guru said, “Okay, now we shall do that 3 more times”, “Hell no!” the kid says.

The guru said to him, “ahhh… but I had had the impression you were willing to do anything… even die for a chance to achieve the success that allows you to truly live.”

“I didn’t mean I want you to kill me… How does that help anything anyways? Are you going to tell me what I need to know or just waste my time and practically kill me?!”

Then the guru imparted the most important lesson:

“I’ve already taught you everything you need to know, now only you can realize the lesson here. You see… No one can “tell you” how to succeed, only you can choose that path. It doesn’t matter what anyone says to you. It doesn’t matter who your guru is, without substantial desire and hunger for success, you will not succeed.

Yet when you want success like you want air, success is inevitable.

 

buddha

The guru continued, “The ‘old you’ is now metaphorically dead, and you are symbolically reborn and baptized into the mindset needed for success. You have endured what you have needed to in order to truly understand the hunger one must have. Having nearly accepted death, you probably had a few regrets, you probably thought of what was truly important to you. But if you still want success, it shall be yours… but only if you proceed with the same desire and passion of which you experienced when you wanted air.  So remember this feeling, and draw upon it when you aren’t content with where you are and truely know what it feels like to want something bad enough and know success is only a matter of time as long as you persist with this desire”

 

The kid grew up to be tremendously successful in all walks of life and continues to live a wonderful life.

—————————————————————————————————————–

So there is only one question I have for you…

“How bad do you want it”?

 

 

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How To Study Volume Profiles: TOS “On Demand”

The one problem about using something such as “volume profiles” is that it can be tough to look back on a chart and illustrate what the volume profile looked like “BEFORE” the break to compare it. In other words the volume profile will have changed dramatically since it “filled” the volume pocket. For example, you might notice a similar pattern shaping up and have a small handful of stocks that you think it may have looked like, but you can’t quite remember or even begin to “compare”. You may wish to do a “case study” on volume profiles, or big movers through such profiles. You also may want to look at whether the big moves were an isolated event, or if many chart patterns had similar volume profiles and chart patterns. You also may want to study the daily moves or 30m chart of a setup that happened months ago

 

Without a time machine, you can’t really study the past. Enter “think or swim on demand”… a “time machine” of sorts.

It is excellent for those who wish to develop more trading experience and revisit periods of time in which perhaps emotions got in the way. It should be much easier to see things objectively, yet relive the trade down tot he minute if you’d like. Built in experience It is excellent for studying stocks that have moved very well in the past to see what the volume profile looked like.

For example, we might look at stocks up over 100% in the last year and come across some names we may want to check out. As of 10/21/2013 this is what CZR looks like.

We can see some big moves and an overall major move from a low of under $5 to a high over 25.

CZR

But what did the chart look like before the moves? How much of the volume profile was filled in by the recent movements? What did the volume profile look like before hand?

1)Determine the date when you want to “go back” to.

2)Click on the “On Demand Button”.

2

3)This message will come up. Click “OK”

3

4) In the top left corner you will see this.

4

Click the “Jump to…” button to select a date for the “time machine”

5)The calendar will pop up… search through to find the date you want.

5

6)In this case we will choose January 29, 2013.

I am not looking for a minute by minute chart to see how the breakout developed today so the “time” doesn’t matter to me.

volume pocket

Now what if we want to see what it looked like as it was breaking through this first pocket? We might go a few days forward to 2/7… when the breakout was really getting started.

volume pocket breakout

You can see the stock performed extremely well in a short amount of time as it broke out right into the volume pocket. Now what do you do midway through a volume profile? Fast forward 2 days

volume profile target

BOOM!. Extraordinary gains as the stock actually carried the stock with plenty of momentum and rocketed right through the reference point. If you sold short of the profile target, you could have missed a monster gain. Of course looking at one volume profile may not dictate what type of behavior you might expect. You also would want to look at what the other casinos were doing at the time. i don’t do this now, but you can if you wish.

In hindsight, we also know the move wasn’t over. The stock consolidated for awhile and then broke even higher to the ultimate high above 25.  So we can go forward some time again.

balance area

We can see that the stock consolidated and ultimately formed a “balance region”. Where there was plenty of supply and demand at current prices, but gaps both above and below. We see a flag or pennant pattern of sorts that usually acts as a continuation pattern so even with volume profile below, the bias should be higher if we can get the move to resume above the balance region. If not, it becomes a potential failed move and potential short.

So fast forward

CZR break 2

This volume profile looks much different than the one before it as much of the thin overhead supply got filled in with more volume. The stock then begins potentially flagging in a short term wedge pattern.

 

I could go on and continue to study the run to above $25, but I will stop here, to encourage you to do it for yourselves. I would start with all the stocks that have gone up over 100% in the past year, starting with the biggest gainers over 50k volume. Just use Finviz to run a screen as I have done and linked to in the last sentence.

URL:http://finviz.com/screener.ashx?v=211&f=sh_avgvol_o50,ta_perf_52w100o&ft=4&ta=0&p=w&o=-perf52w

Then use TOS On Demand and get to work. To get out of on demand, you must click on the “OnDemand” button again.

Although you could abuse this and use it to develop overconfidence from hindsight bias, I use it to understand price action through volume profiles, and the volume profiles of some of the largest gainers. This way I avoid having to go through years of waiting and watching to develop that same understanding/experience. Sure, the past isn’t always going to be the same as the future… but if you wait for 2 years worth of watching volume profiles until you feel comfortable trading them, it may be too late. You can use TOS on demand to paper trade a 100k account and perhaps selecting from a list of say 20-50 stocks try to run that 100k up to 200k in a year and repeat until you are comfortable doing so and understand the swings in your account and the action in stocks… Try to pick stocks that you don’t know well enough to know what’s going to happen if possible and don’t study the chart before you “go back”. This will more adequately simulate the real experience and allow you to “immerse yourself” in trading.

 

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Seasonal Gameplan: Technology

rotation equityclock relative outperformance

If the seasonal tendencies are any indication, you should get a real nice opportunity to buy Technology in October.

The sector data is very relevant because when an entire sector shows out-performance during seasonal data, you have an average of hundreds, or thousands of individual stocks that would have performed well enough over that period each year that the study is done.

That sample size is “statistically significant”… geek-speak for “relevant”. If you can show if the entire tech sector as a whole outperforms and offers an excellent entry early in October, then you can be very confident that this is likely not a case of being fooled by randomness. 6 out of 10 coins could end up heads or even 7 and fool you as to whether or not you have an edge, but if 650/1000 coins come up heads then you can with a confidence say that something the coin flipper is doing has given him an edge. That’s not to say that it isn’t possible that there is nothing to this particular set of data and it is just “luck”, but just that it’s much less likely with such a large sample.

That’s also not to say that I might not have a greater edge finding the specific seasonals that perform the best. Tech may outperform by 2% while GPS seasonals may outperform by say 5%, but I will have much more confidence in the data being relevant for the average technology company. And I want to solidify my edge before I look to enhance it.

Thus, I am not inclined to care as much about individual seasonals unless it is first supported by the sector since this could just be a case of a coinflip that came up heads 4 times out of 5. Not statistically unusual when you have so many individual stocks or “coin flippers”. One of them is bound to get heads 5 times in a row. So where is the proof of a mathematical edge? There is only a small level of confidence of an edge in individual names except for maybe a few companies trading for a handful of decades.

However, sample size quality rather than quantity is what determines relevance and usefulness of the data. Since I can be reasonably confident tech will outperform during the month of October and beyond, NOW I want data that I can use. In other words, since I know this coin flipper flips 650 out of 1000 coins, now I can compare when he flipped the coin with his left hand vs his right hand.

That is a metaphor for being able to use data with a smaller sample size, as long as it is in this case contained within the technology sector. Since I am starting with an edge, and then selecting individual seasonal data, I already have the confidence of outperformance, and now it’s just an attempt at improving upon that data.

Maybe because of the lower sample size at this point, you only end up with a 60% chance of improving your odds over the average 650/1000 or 65% chance. But that’s more confidence than if you randomly selected just any tech name, (50% chance of improving beyond the average chance) and you still have a much higher degree of confidence that this will outperform the S&P. Plus you have to narrow down the list of the 900 tech companies to select one of them anyways so using it as a filter after the sector/industry/stock is determined to have an edge to confirm the bias is fine.

Side note: I like materials too because the seasonals are beginning a bottoming process (and individual materials may lead before the sector bottoms), and the sector rotation outlook also supports a rotation into materials soon as well.

All sectors in October: http://charts.equityclock.com/2012/10/page/24

Technology starting with those relevant in October: http://charts.equityclock.com/category/technology/page/6

Materials starting with those relevant in October: http://charts.equityclock.com/category/materials/page/6

(note: Dates are in reverse order)

Maybe you don’t care as much about individual seasonals as a filter. You can filter in other ways. There are 867 stocks in the technology sector via finviz. Put a filter of over 50k volume and it’s down to 675. Filter price over $10, Volume over 100k, optionable/shortable gives you 383 tech stocks. If you only look at those with weekly options, you get down to 49 weekly option tech companies.

Now maybe when you are looking for “short squeeze” candidates you add “float short” over 10%, and only look at those with a total float under 50M, as it won’t take as much buying pressure for these to have their supply of shares consumed. The idea behind this is since you expect tech out-performance, if the sector is being bought up, the shorts will get under water, and the thin supply could mean very explosive price moves accompanied by a major short squeeze.

Low FLOAT:
UNXL,QIHU,SCTY,YELP,VHC,

I also like to include those with a float/market cap under 2% so I may add a couple. In other words, less than 2% of the market cap floats. This is also a recipe for a short squeeze. It is a measure of scarcity of float relative to market cap. A mega cap 100B company having only 60M shares that float is more likely to see a short squeeze than a 80M company with 60M shares that float. That includes the following

QIHU,VMW,YELP,SCTY,DDD,FSLR

Combine the two and that gives you the trading list:UNXL,QIHU,SCTY,YELP,VHC,VMW,DDD,FSLR

So then you can just look for the technical setups you like the best or whatever ultimate filter from that list you want to use.

If you are not looking for low float, short squeezes, but instead laggards, you can come up with another screen for those. I think knowing what the market appetite for risk is, really can improve the trading as you narrow down to individual names as we have all seen Option Addict do it. The point is, you start with an edge, and now you are just trying to improve that while narrowing it down and simplifying the names you look for to trade.

There are plenty of other ways to develop the list and you may want to start from the initial 867 stocks in the tech sector and apply your own filters or manually look for the best setups. I personally really like the names of the results I got above. Yet If you are looking for some specific kind of trigger such as a bull flag or a specific candlestick pattern and stick to trading those, or an oversold RSI you may want a larger list than that and filter it down on your own, perhaps including a manual visual look at all the thumbnails.

I am not suggesting you should just use the statistical edge as the only reason to trade tech. Instead I am adding a “priority” on the sector. If I get any specific setup I like at all, I will make sure to trade them. If I am between two names, I will go with the tech company. If given a close choice, I will be overweight in tech and have more tech. Just a slight bias or extra focus towards scanning through the sector is all you need. How you find an entry point and determine whether the trade qualifies is up to the individual.

Some might want fundamental screens, or go through all the charts to find technical set ups, or copy and paste all those with seasonal data that provides a relevant opportunity in the month of October into the tickers to only go through those. Some may also want to run through fundamental analysis. I find the site wikiwealth.com useful for fundamental analysis, and of course finviz screens.

Putting up relevant filters is about increasing your odds, and trading with multiple tailwinds at your back (as we outlined with ALTR and others). It is about finding the stocks that will move enough, that you have an edge on, that you have confidence will work.

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