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Beating A Hypothetical market

How Edge Influences Position Size/Strategy

Volume profiles provide objective support/resistance levels that can measure points of possible future orders, with the psychology of whether or not we are above or below the price often determining who’s in control.

Lets take a look at a hypothetical volume profile and what zones of price mean for the stock.

volume profile

The upside target could also represent a selling point should prices draw near, as a move lower could re-enter the fast zone and enter “price discovery lower” until there’s more substantial orders coming in where support can be found. Similarly, the downside target could represent a buying point as a move higher would re-enter the fast zone and enter “price discovery higher”.

One of the key reasons these levels measure as good locations is there’s a clear risk and reward in which the reward outweighs the risk.

As a market gets closer and closer to a support/resistance level, the risk you take lessens, assuming your stop is a particular point beyond support/resistance and your target remains the next point of control. As such, assuming the same probability of outcome, your edge actually increases.

This is the one type of pattern in which you could actually add to a losing position, provided you still maintain the same stop.

This is supported by mathematics as you will see on a kelly criterion calculator.

Say for example, you have $5 of upside, $25 of downside. We can simplify this to a risk of 1 and a reward of 5 or a payout of 5:1. If we assume 40% chance of a “win”, we can use a kelly criterion calculator to get an expected gain on an equal risk basis.


-According to the Kelly criterion your optimal bet is about 28% of your capital

-Your fortune will grow, on average, by about 15.31% on each bet.

Now let’s say the 5:1 payout changes as the price advances closer to your stop. Offering $3 risk to $27 reward or 9 to 1. If the odds stay the same at 40%,

-According to the Kelly criterion your optimal bet is about 33.33% of your capital.

-Your fortune will grow, on average, by about 31.12% on each bet.

Note: the position size can increase by about 20%, and over an infinite time horizon maintains the measurement of portfolio risk measured in this case as “one full kelly bet”.

Now let’s say the stock moves to the stop. I would suggest using a full ATR as the “risk” if you are placing the trade that day, even if you intend on closing out the trade if it doesn’t close above the stop. Let’s say this is a risk of $1 to $29 reward or 29:1. We have to at least adjust the odds of a win slightly since the order will stop out for a small loss at least a bit more often. But for the sake of demonstration, in this case we’ll say it won’t.

You can now risk 37.93% of your capital or about 35% more than the initial 28% position to reach one full kelly of risk. Assumptions are made incorrectly all the time. The infinite time horizon is an absurd assumption, as is the assumption baked into the kelly criterion of 100% volatility tolerance, as is the assumption of a perfectly, fixed, known edge. But these assumptions can still allow us to draw conclusions.

We can use this as a means for comparison and to draw some objective lessons. If you are increasing your portfolio size substantially (double and triple) because your edge improves, you are probably either under doing the first bet, or over doing the 2nd and 3rd, or both. If you are pyramiding higher position sizes, that certainly may make sense if the probability of the asset going higher outweighs the shifting risk/reward and/or if you have an undefined upside target that expands based upon the strength of the breakout.

Also, if you are adding to a loser, you still need a clearly defined exit that shouldn’t change just because you’ve lowered your cost basis.

Additionally, adding lower and reducing higher is best for entire asset classes where you are either maintaining set allocations or changing the allocations based upon the relative edge against all other alternatives on a risk adjusted basis (including cash and adjusting for personal risk tolerance). If you increase a stock ETF lower, you have very little risk that the entire market will make a huge overnight move, whereas stocks are at greater risk of gapping substantially up or down as well as far greater risk of going to zero. Even though you are setting clear stops, either that overnight risk should be reflected in your position size, or you should only consider this tactic when buying asset classes or broad funds.

So how does edge influence position size? Slightly. The difference from 0% allocation to some allocation is infinite, so there are instances where a slight change in edge from negative, zero, or a very small edge to a more significant one can make a huge difference.. but if you are talking about a single setup developing, that difference is probably going to have less impact than most people would position for.

The more complex question of how one should change overall allocations as an outlook ranges from very bullish to very bearish to everything in between is a more difficult one to answer, particularly when we are talking about multiple trades held simultaneously and at overlapping intervals of time. There are other aspects of game theory that could help answer this question perhaps.

It would depend upon holding period which would depend upon fees and market’s tendency to trend and the edge and magnitude of that edge.  Nevertheless, with all things equal, a portfolio should allocate according to the probability of outperformance in the asset class, with additional capital tied towards “risk off” and cash related allocation as a person’s tolerance for risk decreases.

In other words, if it was simply between cash and stock, 50% stock would represent 50/50 chance of market up or down. If not for additional risks of margin 50% short stock and 50% cash would be acceptable as well. 80% chance would represent 80% chance of a rally, and 20% of decline of equal amount (adjusted for risk which actually would suggest holding more than 20% cash due to costs from volatility). the inflection point of 50/50 would actually shift towards shorts, but assuming shorting isn’t worth the additional risk, 80% cash and 20% stock would represent 20% chance of stock outperformance.

Your holding period makes a huge difference. For example, say there’s a 52% chance of the market going up on a given day… what’s the odds of the market finishing with more days up than down on a given year?

It turns out to be around a 72.5% chance that there are more up days than down. So a strategy that would rebalance yearly would theoretically have to be positioned much more aggressively bullish than one that rebalances daily if you assume a bull market or slight bias higher each day.


One final note: If you are aiming for targeted allocations using options, you have to also understand that a small movement in price can change the dollar amount allocated towards bullish ideas or bearish ideas substantially. So this is where finding a new bet or doubling down may make some sense, but it’s to maintain targets, or slightly increase them with your edge, not to drastically change them.

If you own 10% call options and 10% put options a small move upwards in the market can easily cause the balance to change to say 25% call options and 4% put options. This is where adding to a broad market hedge and/or holding broad market calls can be helpful in allowing you to quickly restore the balance of your intended allocation without selling individual positions short of their target or before your system dictates that you should, without having inconsistencies and over exposure to directional bias, beyond what you intend.

If you want to have 10% calls and 5% puts, mostly short dated options and the market moves, you now might have 20% calls and 2% puts. If you use index options on the dominant direction, you can quickly take off an IWM call and reverse it with an IWM put or find a put of something relatively correlated to the market (higher than .5) and have something like 15% call and 7.5% puts. This isn’t perfect, but it helps you stay a lot closer to your targets.

There is a substantial cost over time in paying additional premium for assets that perhaps have a lower edge just to normalize some of your volatility, ut it can be measured against the benefit of reducing the risk and even if you can break even on this tactic, if it normalizes your exposure you can see similar returns with lower drawdowns.


Keeping somewhat inversely correlated assets or low correlated assets can provide a substantial benefit, even if neither system is better than the other, or even if one system is break even as can be illustrated below.

normalized systems2

normalized systemsHypothetically, even a slightly unprofitable system that can virtually eliminate or substantially mitigate all risk of drawdowns can improve your strategy, as you can simply leverage up and see higher returns with the same or lower drawdown. Long term capital management failed to stress test their portfolio against a long enough history that included major credit risks and bond market collapses and/or “black swan” type of events, and didn’t protect their portfolio against the unlimited loss, and they used assumptions about liquidity that weren’t true but if they were able to adjust for these things, they probably would have been able to accomplish a better risk/reward and eliminate the big blow up. It’s the margin risk and unlimited loss potential and lack of liquidity without adequate time provided to raise additional funds that ruined their strategy, not the soundness of the strategy itself or the math involved. They also tried to keep up with a historically very strong market with enormous leverage, rather than just accepting a very profitable system without trying to benchmark their gains to an underlying market.


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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):


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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