Fixing The Flaws In Fixed Fractional Position Sizing

A few days ago I accidentally ran into a white paper by a certain Christian B. Smart, PhD that had mysteriously made its way into my dropbox. It immediately grabbed my attention for several reasons. For one you simply cannot pass up a paper authored by someone with that name, although it may just be a pen name. More importantly it not only highlights a major flaw in fixed fractional position sizing, a method we use religiously here at Evil Speculator, but also promises to fix the problem. How could I resist? Of course, nothing in life is free and if you suspect that there may be a price to be paid for keeping up with theoretical compounding then I promise that you won’t be disappointed. More on that further below.

Now after familiarizing myself with the math I spent a bit of time running the numbers. The aim of this post is not to regurgitate Dr. Smart’s white paper but to share some rather interesting findings on how his approach affects a variety of trading systems. Before you continue reading I strongly recommend you read his paper first. No worries, it’s pretty light on the math and you’ll get away with basic algebra. But just to set the stage here’s the skinny:

  • Fixed fractional position sizing is a popular and time tested method for money management. In the strategy a fixed percentage of equity (e.g. 1%) is risked per trade. We call that ‘R’ here at Evil Speculator and it refers to a unit of risk per campaign.
  • Fixed-fractional money management is an intuitive method in which bet size increases when equity increases and bet size decreases when equity decreases. This form of money management is conservative in that it dramatically decreases risk of ruin.
  • A concept related to money management is system expectancy. A system’s expectancy is the average, or expected, amount of money an investor expects to make per dollar risked. For example, a trading system with a winning percentage of 40%, whose average win is equal to twice the average loss, has an expectancy approximately equal to 0.40 * 2 – 0.60 = 0.80 – 0.60 = 0.20.
  • Another key concept related to money management is that of compounding. Dr. Smith actually does not make mention of the word which surprised me a bit as the entire aim of his paper revolves around permitting effective compounding. If you’re unfamiliar with the concept of compounding then Google will be your best friend.
  • With fixed fractional position sizing, any system does not achieve its expectancy (and thus its theoretical compounding) in the long run, but an amount less than the system expectancy. With a progressive betting system like fixed fractional sizing, in which returns are reinvested, the total return is the product of a series of numbers.
  • The underperformance of fixed fractional position sizing has a basis in mathematics. The system expectancy is the system’s arithmetic mean. The average amount made per trade with fixed fractional position sizing is the system’s geometric mean. A well known inequality in mathematics states that the geometric mean is always less than or equal to its arithmetic mean. So in the long run, fixed fractional position sizing will never achieve system expectancy but will underperform.

So in a nutshell – in reality trading systems will always lag behind their theoretical expectancy. Over time the difference actually adds to quite a bit of lost profits:


This graph is taken off the white paper and shows a Monte Carlo simulation of an hypothetical system over 5000 trades. Clearly this should not be taken as a realistic projection of a real life system and only serves to demonstrate the concept. The take away message is this: Over the life time of a system there is an exponential delta between its theoretical and geometric compounding results.

Clearly missing out on ill-gotten gain is not a situation we take lightly here at Evil Speculator. So I sat down and actually produced a handy spreadsheet which I invite you all to download and have a go with on your own. I’m only mildly versed in Excel and if you are able to offer an improvement then please share it with the rest of us. Now as you know I have quite a few systems in the running and what I wanted to find out if and how they would be affected. The results quite frankly were a bit surprising.

You may recall that I am currently live testing Scalpius which despite its name is more of an intra day swing trading system. It is based on some of my work on volatility cycles and thus far it’s looking extremely promising. After painstakingly running the stats on 10 forex and futures symbols over the past five years I arrived at the following stats:

  • Win/Loss Percentage: 1:1.11 – which is 47 to 53.
  • Average Win: 1.23R
  • Average Loss: 0.76R
  • Expectancy: 0.18R

The stats slightly vary between the various symbols but not excessively. There is a common theme which Scott often refers to as a ‘forest of good numbers’ – a term I really like as it describes the process of testing for possible system form fitting. I would characterize Scalpius as having a small but consistent edge. It won’t make you rich overnight but if you keep it in the running and if you are able to grab good fills then it’s a very promising system. I am currently forward testing it over at Vankar and we are getting excellent fills – thus I’m cautiously optimistic.


Now if you ran Scalpius via 0.5% fixed fractional position sizing (FFPS) then you would theoretically bank a $147k profit over 1,000 campaigns based on a theoretical expectancy of 0.1815R. Yes, that is just a theoretical model – we need to be clear on that. In any case, when comparing that with the geometric expectancy of 0.1765R we arrive at geometric returns of $144k, the delta in US$ being $3,031. Not exactly chump change but given the overall context it represents only 2% of the theoretical returns, thus I’d submit it to the BFD department and move on.


However look at what happens when we increase the base risk percentage from 0.5% to 1.0%. Suddenly the numbers jump quite a bit. Obviously larger position sizing increased the compounding effect significantly but the delta between the arithmetic and geometric returns now amounts to 5.7%.

And in order to compensate for the loss in profits our position sizing has increased accordingly. If you look at the table in the center of the graph then you will find that the difference in position sizing has to increase alongside the base percentage. Whereas compensating expectancy loss at 0.5% only requires a small increase to 0.545% at a base of 1.0% you are now required to trade at 1.18% position sizes relative to your actual equity, as that percentage represents 1% of your ‘expected equity’.


But there’s more to this story yet. If you read the white paper then you remember that Dr. Smart used the standard hypothetical 40/60 – 2:1 system for his Monte Carlo simulation. I have used those stats in the graph above but have taken the liberty to reduce the risk percentage to 1.0%. Quickly apparent is that this system has a better expectancy of 0.2R which in turn changes the dynamics of the delta between arithmetic and geometric returns. We’re are banking a bit more here obviously but if you look at the table in the center you’ll see that we are also using larger adjusted position sizing. Now instead of $100k we are calculating at an expected equity of $120k. That’s a 20% increase and those numbers grow even more quickly as we are increasing the base percentage.


If you were to use 2% position sizing instead as per the white paper then you would have to calculate your positions via an expected equity of $140k. Also the delta between arithmetic and geometric returns has jumped to a whopping 35.3%! So clearly time is not the only factor that affects compounding. The higher the SQL of your system the larger the loss in expectancy. Which to most of us would be counter intuitive, but the numbers do not lie. Systems with a higher win/loss rate require increased position sizing in order to keep up with their theoretical expectancy.

Take Away Points

Given the above dynamics there are several considerations for system developers. The first one is whether or not compensating for geometric returns via an increase in position sizing is worth the added risk. The white paper makes it clear that the increase in position sizing does also increase maximum drawdowns.


I have not run the numbers on that end myself but judging by the paper’s graph it seems that it is roughly in sync with the increase in risk percentage. So if you’re trading at an 1.2% adjusted position size instead of 1% then it’s realistic to expect at minimum 24% maximum drawdowns instead of 20% at fixed fractional position sizing.

And given that the dynamics between theoretical and geometric returns are highly system specific answering the question of whether it is worth it depends, as always, on your personal risk profile. The real question then changes from whether or not to use expected fixed fractional position sizing (EFFPS) to how much compensation you are willing to allow for. And if nothing else we also need to embrace the fact that theoretical compounding becomes less efficient with increasing SQN. Systems with smaller but consistent edges (as for instance Scalpius) actually benefit quite bit more in comparison with high expectancy but low opportunity systems.


Meaning, if you could choose between a holy grail system which trades 200 times per year and produces 100R and one that takes 1000 entries to produce the same 100R then most of us would probably instinctively choose the former over the latter. The graph above shows such a hypothetical system – I have fiddled with the stats to produce almost exactly the theoretical returns of what Scalpius is promising at 1% position sizes. The delta between the theoretical and geometric returns is smaller, but look at the position sizing required to keep up! At a base of 1% we would have to use 1.91% to account for a 3.9% difference. Hardly worth the additional risk I would say and I’m pretty sure most of you would agree. Of course it

Finally the more risk you are willing to take trading any system the higher will be the additional risk incurred in order to keep up with theoretical compounding. Increasingly larger position sizing means that draw downs will be deeper and generally your system will exhibit higher standard deviation. That is never a benefit to any system but especially systems that thrive via large outliers will be the most negatively affected. Drawing from some of the lessons we’ve learned about equity curve filters I believe that low dependency low standard deviation systems would benefit the most (e.g. Scalpius) and high dependency high standard deviation systems benefit the least from EFFPS. Reason being that a high number of consecutive losers will quickly do you in if you’re trading 3%+ position sizes.

The final take away is that we as traders need to be careful what we wish for. At some point in our career all of us have been on a hunt for that holy grail system which prints money fast with a small amount of trades. Given the inherent power of compounding it remains that elusive path to quick riches which many of us hope for but very few ever achieve (and the one’s who do aren’t talking).

Over the years I however have slowly shifted away from that and am now more focused on creating systems which produce a small but reliable edge over time. And apparently when it comes to compounding it is these types of systems which require the least amount of risk when it comes to compensating for exponential lag due to the delta between theoretical expectancy and geometric expectancy. On paper lofty outlier systems may seem what you want but given enough opportunity (i.e. number of trades) ‘more realistic systems’ with a consistent edge may actually rival hypothetical ones over time courtesy of compounding. Quite some food for thought.

The future is now – so don’t bring a knife to a raygun fight. If you are interested in becoming a Zero subscriber then don’t waste time and sign up here. A Zero subscription comes with full access to all Gold posts, so you actually get double the bang for your buck.


Filling The Gaps

Last night’s dramatic gaps at the open across the board have been followed by systematic reversals. Before I went to bed my wife had actually joked that I should run to the ATM and clean it out. Well, when I got up this morning I wished that I had followed her advice:


Because that sweet EUR/USD exchange rate of 1.095 is no more. The entire gap has been filled and I can only assume that Greece suddenly doesn’t matter anymore or that a new batch of hopium has hit the streets in expectation of the July 5th referendum.

Being a proponent of game theory I personally believe that the Greeks will vote to stick with the Euro – both sides simply have too much to lose in any other scenario. I also imagine all those Greeks standing in line at the bank today during the smoldering summer heat and that may dampen their resolve to ‘stick it to the EC’ and go it on their own. Don’t get me wrong – as I suggested yesterday my notion is that Greece would actually be better off over the long term but I just don’t see it happen. Fortunately we only have a few days to wait and I’m very much looking forward for this dreaded saga to finally meet some level of resolution.


Gold also just filled its gap and we are back from whence we came (Friday). Fortunately Scalpius was able to snag a few R here (see below) – that’s certainly a good start.


Equities also on the rebound but thus far it’s been rather modest. The level we should keep our eye on is 2053.5 which marks the weekly Net-Line Sell Level (NLSL) and we traded merely two ticks above it last night. So definitely a technical level to watch and probably the last bastion of support as the 25-week SMA is currently at 2072.

Skynard grabbed a few longs at the open and that took some balls – big props for that mate. I refrained from getting involved after having spent over 25 hours last weekend coding and refining Scalpius as well as refactoring CrazyIvan into a leaner and meaner 2.0 edition (see previous post). Suffice to say I’m exhausted but clearly this will not be a week to rest or relax as volatility is upon us.

Honorable Mention:


Scalpius grabbed a  nice long position on Gold (and crude) near the lows – not a bad start! Unfortunately it was in a demo/test account – no real action just yet as it’s in early alpha.

Lesson learned:

Let Mrs. Evil handle the forex action moving forward.

Obviously I am not eager to touch anything right now – too much gaping action and it’s too late now to touch any of those moves. Let’s see what happens over the next few hours – I’ll be sure to put up a post later after the open once we see more participation.


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Greece Saga Precautions

It seems that finally Greece has gone past the point of no return and events are quickly escalating. If you truly care about all the gory details then look no further than ZeroEdge which is literally bursting out of the seams with extra sarcastic coverage of the gloom & doom that is bound to descend on us starting Sunday night. Your guess of what happens next is good as mine but I’ve hired my own consultants and per Dr. Venkman the outlook is pretty grim:

So before you all resort to cannibalism (first dips on Skynard) let’s take some basic pre-cautions in anticipation of some massive volatility across Forex which will most likely affect bonds, equities, and various futures contracts as well.  First up, don’t be surprised if your broker may only accept close orders tonight and tomorrow – clearly the industry has learned from the Swiss Franc debacle a few months ago.

Down here at the lair I have turned on CrazyIvan but strongly caution you subs from taking any entries until things normalize, whenever that may be. On the Thor front I will not put up any new setups tonight but revisit the situation on Monday before the roll over. Perhaps this will blow over quickly after some last minute deal but I won’t risk any exposure until we know for sure.

Message to Greece: Good riddance! I have nothing against the country or its people – quite on the contrary (although I’m partial to Moussaka). One of my relatives is actually renovating a villa in Santorini and I plan to visit soon (with my American passport – ahem). But with all due respect – I was sick of hearing about this dreaded Greek saga five years ago and since then have long ceased to care. We’ve teased this out beyond anyone’s pain threshold and we all could do well without hearing about Greece at least a week or two.

Message To All Greeks: Good luck – you’ll need it. But heck – you still have a gorgeous country at your disposal and things could be worse (just ask any Greek over 75). This may be an opportunity to turn the corner and rebuild your nation without outside interference. As the saying (kind of) goes: Be careful what you wish you for because you may have just gotten it.

CrazyIvan News: A few weeks ago I posted about some long overdue statistics I had run on my CrazyIvan results. Turns out that inside periods are a complete waste of time over the long term (as I had long suspected) and I also managed to identify a list of the highest scoring entry patterns. I have now removed those dreaded IPs and today deployed a new version which features a filter that will only permit the 12 highest scoring entry patterns. That in turn however will have implications on how we trade CrazyIvan going forward.

  • Each respective symbol will be taking a lot less entries on a daily basis. I can’t give you exact numbers but I expect about 1/3 of the entries in comparison with the old system.
  • Given the that I felt that we needed to add a bit more opportunity. So I have made the following changes:
    • Added the following futures contracts: Corn, the 10-Year Bond futures, Natural Gas, and the S&P E-Mini.
    • Added the following forex symbols: USD/CAD, EUR/JPY, NZD/USD
    • Removed AUD/JPY. It’s a minor one and I don’t see any advantage over the EUR/JPY for example.

For the coming weeks I recommend that you trade the new futures symbols with only 0.5R position sizes until we are sure that they flow as expected. I don’t anticipate any problems but always prefer to err on the cautious side. The new Forex symbols shouldn’t give us any trouble and those can be traded full steam ahead.

That’s it – I’m off to my training class and then I’ll have plenty more work on my plate before Monday. See you guys on the other side.

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