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Fixing The Flaws In Fixed Fractional Position Sizing

Fixing The Flaws In Fixed Fractional Position Sizing

by The MoleAugust 1, 2015

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.


About The Author
The Mole
Mole created Evil Speculator amidst the chaos of the financial crisis in early August of 2008. His vision for Evil Speculator is a refuge of reason, hands-on trading knowledge, and inspiration for traders of all ages and stripes. You can follow him and his nefarious schemes at various social media waterholes below.
  • molecool

    I spent most of my Saturday putting this together so I’m hoping to see some kick butt discussion by the time I wake up 😉

  • molecool

    Either that or buy a sub. I can be bought.

  • fearful_syymmetry

    Thanks for your work! Into my third read…the first graph is not available. The Scalpius link needs correcting.

  • Scott Phillips

    Superb analysis, this one goes straight to bookmarks. Well done!

  • SirDagonet

    Wasn’t it Paul Tudor Jones that said I’d rather make a dollar on a million trades than a million dollars one trade?

  • phylum

    So it goes, there’s always work, no short cuts, just the grind ….


    Sometimes I cheat and look to the summary. Thanks for sharing all of that hard work.

  • mugabe

    Hey Mole,

    V good stuff. One thing. I don’t think that the difference is between 1% and 1.91% after 200 trades. The difference is between 1%* (actual account value after 200 trades) and 1%*(expected account value after 200 trades).

    Of these figures, the second is fixed (ie $1,910) but the first is unknowable. In any case, in most cases the difference will be far less than you state. And if the system has outperformed, the difference will be negative.

    The *problem* with this approach to position sizing (is it a problem?) is that you are betting a greater % of your account size when the system is doing badly and less when it is doing well.

    I think what I’ve just said is correct …

  • hellbent

    Yeah I think that’s right, mate.The doc ignores actual equity and uses projected/expected equity instead. So if actual equity is outperforming the projected the bet size will be smaller than FF at that point. In a drawdown he is still ignoring actual equity and bet size will be bigger than FF. Might work for some systems but it doesn’t work for me :-]

  • mugabe

    Numbers aside, psychologically it’s a very tough ask when your system isn’t donig well.

  • hellbent

    An old bad habit of mine was go the Martingale way and up the anti after a loss. It works a treat, until it doesn’t. Psychologically, I’ll do whatever… once I understand how and why it works. Which is why this has been a great exercise. Depth of understanding is everything.

    Good bit of team work on this one. Thanks Zero Edge. Thanks Rat brothers :)

  • hellbent

    Nice work, Mole and much appreciated.

  • molecool

    I fixed the first graph – please reload!

  • molecool

    Thank you sir!

  • molecool

    I fixed it – thanks.

  • molecool

    Actual equity cannot outperform the projected, guys. It’s a compensation mechanisms for the loss incurred by your geometric expectancy.

  • hellbent

    This is a curly one… If we kick off with 5 wins in a row, for example, I’d have thought we’d be well above the the expected equity curve and thus a simple FF 1% would have us betting more than the EEFF system proposed. I am still a bit unsure about the actual formula as printed on the paper though. Never a good sign:)

  • mugabe

    That’s the way I see it, too. The idea is to keep position size consistent with expected equity after n trades, regardless of actual equity.

  • mugabe

    ‘I’ll do whatever… once I understand how and why it works’

    I’m sort of like this too ie pretty rational/non-emotional, but I think almost everyone’s tolerance for drawdown is less than they think

    Martingale works great with roulette (or a coin toss) until it doesn’t!

  • supervilin

    In line with what mugabe said, what if you stick with the regular fractional position sizing and just tune up your R just a tad, so it is in line with the expected equity. Let’s say you plan to have your R as 1% but you determine via simulation that 1.2% will actually yield closer to the expected equity. Perhaps this is less risky than going with 1% R via EFFPS (please not another acronym lol). I guess it will depend on a system so one should run the numbers..

  • hellbent

    It’s like when your hot missus get’s into some stupid conversation with some guy hitting on her. You think you can handle it but once your switch flips anything can happen. I blew up the other day because I lost control after losing 25% of a 10% account…

  • mugabe

    One thnig that many people have mentioned (Ivan, Bobby, etc.) is writing down rules, reading them every day to internatlise them, and thus sticking to them.

  • hellbent

    That, in a nutshell, is exactly what I’ve got to do.

  • hellbent

    Not that reviews are scientific or anything…

    “the author [Dr CB Smart] merely seems to echo the mistakes propagated by earlier predecessors. I found the book [something about maths] overall to be tedious, sloppily written, and poorly researched. It seems hastily put together, with some glaring grammatical errors that a thorough editor should have caught and corrected. I only give this book two stars because of the subject [was of interest]. (Amazon)

  • molecool

    If you read my post then you will realize that it is really system specific. 1.2% will most likely keep you in sync for a system like Scalpius but a higher expectancy system would require a higher expected equity for your position sizing. Which is why I mentioned that it is a personal choice – the math may suggest 1.5% for a particular system but you may only be comfortable with an increase up to 1.2%.

  • molecool

    Can you please post a link?

  • hellbent

    Had my wires crossed on that earlier excerpt. All I’ve found is this which doesn’t bode well for EEFF if it’s the right man.

  • molecool

    I would not want to jump to conclusions. Many mathematicians make lousy traders and his professional trading success does not invalidate the math.

  • molecool

    No, I think you completely misunderstand the concept, mate. This is not an equity curve filter that uses an SMA. You are ALWAYS using expected equity in expectation of a geometric growth curve.

  • hellbent

    It’s perhaps just talking about two different things. I think what Mugabe was drawing attention to is that conventional FF actually responds to performance which can be variable as we know. The doctor ignores forward performance and substitutes it with expected performance. The glaring difference is that, as Mugabe states: “you are betting a greater % of your account size when the system is doing badly and less when it is doing well”.

    What you’re perhaps referring to above is that the actual forward equity curve tracks the Expected Equity curve without deviating a great deal.

    For the life of me I cannot see how eliminating negative (in a good way) feedback from the much acclaimed FF system of position sizing can be an advantage in real world trading in terms of risk vs reward.

  • i Bergamot

    Great stuff, Mole!
    Printed and saved.

    Grind it out. Don’t shoot for home-runs (they will just happen sometimes… no?).
    Poker. Baseball. Hotdog stand.
    I just never thought it could be proven with simple math.

  • Ivan K

    It is fascinating (from my side) to see how most people focus on the derivative as opposed to the source … translation … any EqC is derived from the sequence of wins and losses … the clues to sizing and EqC filtering lie in the actual distribution of results … some people have referred to ‘dependency in previous comments, almost as an aside … perhaps that concept is worth pursuing more.

  • molecool

    “you are betting a greater % of your account size when the system is doing badly and less when it is doing well”


  • RacerXX

    Awesome work Mole! And good find.

    and thanks for the laugh. Right on.

    “Clearly missing out on ill-gotten gain is not a situation we take lightly here at Evil Speculator.”

  • Billabong

    Demonstrates the “grinding it out” process. Winning Rs larger than losing Rs. Nice job keeping the -R at 1 or less. This is why most people can’t or don’t participate in the trading business…

  • mugabe

    which is why if you start with ten losing trades, based on this system your R as a % of *actual*equity will be proprotionately higher for the 11th trade. and the converse is true if you start with 10 winning trades all above expectancy.

  • Billabong

    One item never discussed is taking partial profits. Do you ever have personalities that come through your program that require taking partial profits at some positive R point before the position runs to its final conclusion?

  • mugabe

    that looks a bit like a Crazy Ivan profile

  • mugabe

    ‘For the life of me I cannot see how eliminating negative (in a good way) feedback from the much acclaimed FF system of position sizing can be an advantage in real world trading in terms of risk vs reward.’

    I can see it being an advantage if the system doesn’t suffer from high dependency — but can you ever be sure that it won’t in the future?

  • Ivan K

    Looks can be very deceiving!

  • Ivan K

    As everyone has a different DNA the ‘feel good factor’ is alive and well … until the belief structure (based on stats and RBT) is solidified.

  • Ivan K

    B’bong – there are scores of reasons why ‘most people’ do or do not do things … the ‘real’ reason is seldom cited … and does it actually matter ?

  • Billabong

    In the end no and then there are those who stay … “everyone gets what they want out of the markets”.

  • BobbyLow

    I certainly appreciate all the work that goes into writing a piece like this that even advanced mathematically challenged people like myself might be able to work their way through.

    Regarding systems and position sizing, I’ve come to the conclusion of the following:

    Creating a System is not difficult.
    Learning how SD, Average Positive Expectency, and SQN affect results over time is easy.
    Learning new and creative ways of better position sizing is doable.
    Putting all of the above in one package is not difficult.

    What can be the most difficult part of all comes after the above steps are completed. And this is working and maintaining a system consistently through both good and bad times.

  • Scott Phillips

    In the real world it doesn’t matter as much as we think.

    Unless you are living off thin air and not your trading account, and in a tax free jurisdiction, you will have to take out somewhere between 25-50% of profits anyway to eat and pay taxes.

    Simple truth is that if you can string together 10 in a row years of decent wins (and I’m only talking 25% per annum) Money becomes irrelevant.

  • Grant

    Bingo! Or, one could run 2 businesses as I do. I trade for accumulation and do not touch my trading account. This requires that I get up at 2:30 am during the spring-summer-fall and 1:30 am fall-winter-spring as dictated by the central time zone in the US. I trade from the open of the Euro mkts to lunch in NY (11 AM cdt). I operate another business til 6pm my time which pays my bills, etc…. It doesn’t make for the best social life but I have a wife and 4 kids so the flexibility is great and I do what I love.

  • Grant

    Looks like this student really has it together. Congrats to his/her teacher and him/her.

  • Scott Phillips

    Not even a little bit. Crazy Ivan is a stinking turd compared to this.

  • Scott Phillips

    The decision to take or not take partial profits is entirely system dependent. In a good system it should slightly reduce overall R while dramatically reducing standard deviation of results.

    Reducing standard deviation reduces drawdowns, smooths things out.

    It comes down to the old chestnut. You can’t have it all, do you want to optimise for returns or minimise drawdown.

    It is disingenuous to suggest you can do it any other way.

  • mugabe


  • hellbent

    Lets say our system is very dependent on long term market direction and we failed to pick that up in back testing. Nevertheless we start executing trades with what we believe to be a 0.2 expectancy system. If the market tops coincidentally and changes direction, what happens?

    The performance will not be equal to the initial expectancy and the expectancy will start to change as a result; positive or negative depending on the relationship. Lets assume its negative in this example and from day one our expectancy is actually negative.

    Wont it then mean that we “are betting a greater % of your account size when the system is doing badly” relative to bet size using FF?

  • Ivan K

    B’bong – people say many things … to use a ‘famous’ quote … Frankly my dear, I don’t give a (fill in the blank) !

  • Ivan K

    Looks can be deceptive.

  • Billabong

    Add #5, the psychological battle. Patience, discipline, and avoiding mistakes.

  • Billabong

    Love it! “do you want to optimise for returns or minimise drawdown.” That sums it up.

  • Crofx

    “Rats arse??”

  • Skynard

    Nice piece and good read. Seeing volume pick up on the 5 min /ES. Went long and good to be back.

  • Skynard

    Major implications on a weak dollar at this juncture. We have significant volume across many platfoms now. Time will tell.

  • hellbent

    I’ve come to the conclusion that the formula as printed on the journal article is only good for making projections. What I think it suggests though is that we can calculate an expected equity as we go.

    Stick with: R = Expected equity * fixed-fraction

    Except that going forward: Expected equity = Equity + (Equity * (1 + Expectancy) * fixed-fraction)

    Expectancy we all know and love but to use it effectively going forward I think we need to use a rolling average so that we can transition form backtested Expectancy to proven Expectancy. If it stays the same – GREAT.

    How say ye?

  • TheRooster

    Hi Ivan

    We have discussed dependency before – how do you measure it other than by eye? I know that my eyes will see patterns that aren’t there so does one needs to test for randomness within the sequence rather than ‘looking’ to see if it exists?

    Would be interested in your thoughts.



  • Billabong

    “…but to use it effectively going forward I think we need to use a rolling average so that we can transition from backtested Expectancy to proven Expectancy.” Your point is well taken. I update my primary equity expectancy every 6-12 months. I find some equity positions flatten out for a period of time and other are in a reversal situation. It helps with limiting drawdowns (mistake trades based on reduced expectancy).

  • Billabong

    DX has been channeling between 25 EMA and upper BB since 24 June

  • Nemo

    No mention of the Kelly Criterion?

  • Ivan K

    Rooster – Why not start with a simple assessment of the number of times a single win is followed by a win on the next campaign … and work your way out from there … building a simple matrix of probabilities … the same idea can be applied regarding a losing campaign … what comes next (on probabilities) a win or another loss.

    Once the matrix is in place the fun stuff can begin.

    PS. Even eyesight / imagination challenged people can create the simple matrix above.

  • Ivan K

    Nor of the ‘”Optimal f” concept of Ralph Vince (Portfolio Management Formulas) from over 20 + years ago

  • saltwaterdog

    Step 1 would appear to be determining how large of a data set of outcomes is statistically valid, no?

  • hellbent

    23ish. Less than 1/10…

  • Ivan K

    H’bent – let the fun games begin then.

  • Ivan K

    S’dog – the more the merrier … the more likely (on probabilities) to be aligned with the unknown outcome of the future.

  • hellbent

    I’m up for the treasure hunt but my brain’s packing it in for the day. Thanks for checking in, Ivan. I’ll be working on it.

  • Ivan K

    H’bent – the hunt itself has mega returns … enjoy the Journey.

  • molecool

    Completely unrelated – the Kelly criterion only offers an alternate approach to arriving at R. You realize that geometric returns via the Kelly criterion would still limp behind the theoretical?

  • captainboom

    A quick search and I find that Mr. Vince has a web site with a link to a white paper he authored recently on Optimal f. Looks a bit dense for the mathematically challenged. Link is a PDF.

  • molecool

    ( ) ( (
    ( ) ) ( ( /( ) ) ( ) ) * )
    ( ) (()/( ( ) )())(()/( ) (()/(` ) /(
    )((_) /(_))) ((((_)( |((_) /(_))((((_)( /(_))( )(_))
    ((_)_ (_)) ((_) ) _ ) |_ ((_)(_))_| ) _ ) (_)) (_(_())
    | _ )| _ | __|(_)_(_)| |/ / | |_ (_)_(_)/ __||_ _|
    | _ | /| _| / _ ‘ < | __| / _ __ | |
    |___/|_|_|___|/_/ _ _|_ |_| /_/ _ |___/ |_|

  • molecool

    Why would I talk about portfolio management when covering compounding inefficiencies? Some of the folks who commented already seem to have a hard time grasping the topic at hand – so why add more complexity?

  • molecool

    Yeah, sure I’m going to use this 😉

  • TheRooster

    Hi Ivan

    My concern would be that the weight of scientific evidence shows that we all see things that are not there in random data ‘when we use our eyes’.

    A probability matrix is a good idea. I have been looking at stochastic/ statistical tests but haven’t found anything yet that fits my 2 criteria: 1) it would achieve the task and 2) I can understand the test sufficiently to understand its flaws!!

    I’ll report back if i find anything useful (if anyone is interested!!)

    Kind Regards


  • Ivan K

    Thanks Roy