Talking with a friend recently we hit on an interesting point in our discussion. He argued that you need to trade a small account very differently from a larger one, meaning it needed a higher level of risk to make the trading effort worthwhile. Implicitly this shows the conviction, that you have some control over the size of your return, for instance by the amount of time and effort you pour into your trading.

I, on the other hand, was of the opinion that everything depends on your strategy´s or portfolio´s risk-adjusted return and your tolerance for risk. The actual account size doesn’t matter at all and your maximum return is constrained by mathematical principles depending on the return parameters of your strategy – you simply can´t push your returns beyond a certain point.

But of course in real life available capital does matter, if only psychologically. For one your tolerance for risk may vary considerably depending on the account size – a certain level of abstraction is necessary to objectively deal with both large and negligible sums of money at risk.

Let´s look at the problem in more depth.

First I want to approximately define account sizes:

**$/€10.000 or below: Small Account**. Here the main issue is that very high levels of return are necessary to make your efforts worth your while. But is that achievable or largely wishful thinking that is very unlikely to materialize in reality? I would argue that viewing a small account as a “training account“ will usually be much more productive in the long run.**$/€ 100.000: Sustainable Account**. Around this sum realistic, repeatable returns start to make an actual difference to the bottom line of an individual investor. To reach and hold capital above that level over the long term shows considerable discipline, skill and commitment – importantly it is a sign that your financial life is generally sound, especially if you used your savings to build up a large part of your capital.**$/€ 300.000 to 500.000: Semi-professional Account**. Trading becomes a business: with 30.000 to 100.000 sustainable pre-tax returns (above 10% per annum) investing / trading can be a major source of income – comparable to a quality job. The amount also shows a professional level of procuring capital for your trading business – you really need to save your ass off for a long time even with disciplined compounding or gain access to outside money with conviction e.g. from family and friends or through loans, windfalls etc.. But it is still vulnerable to large drawdowns especially if you depend on your trading income to finance your cost of living.**Higher: Professional Money Management**

**Sustainable Returns and the Small Account Fallacy**

Why is the myth, that it is possible to make a living of trading less than say $100.000 of capital, so persistent? Often even much smaller account sizes, e.g. $10.000, are touted as a realistic possibility.

I think, it´s no more than a dream, the amount is arbitrary and simply derived from the fact that most actual trading accounts are of such a size and has nothing to do with realistically achievable returns over a longer period of time. It is nothing but wishful thinking trying to make reality fit one´s expectations – a bit like wanting to make 1 + 1 equal 3. Unfortunately this myth is a major reason most accounts will never rise beyond being a small amount for very long – it directly leads to the often quoted statistic that 95% of active traders never make a sustainable profit. Unrealistic expectations lead to overtrading, which is a major reason so many individual traders blow-up their accounts as the risk level is bound to be unsustainably high – occasional short-term outsized gains notwithstanding.

Conversely, when larger sums are at stake, especially if that happens suddenly, these might impact our emotions and ability to trade rationally.

If we take large risks in dollar terms, observing daily portfolio fluctuations worth month´s of income might trigger emotional responses and overrides to our strategy. It could possibly lead us to take profits ineffectively early or could cause catastrophic losses by not taking our predefined exit because of loss aversion.

Abstraction, not seeing trading money as real money, is a skill that needs to be acquired slowly in both cases.

To actually make the dream of living of a small or medium sized account become reality we would need to generate returns above 50% (or even much higher) year after year. How likely is that and what are realistic expectations?

**Arbitrage**

If we actually managed to design a strategy that can post multiple times what the market returns, sooner or later sophisticated market players, who constantly search for such opportunities, would pile into the strategy and arbitrage those profits away in a zero-sum trading world. An impressive example for this mechanism at work is the erosion of the extraordinary profits from high frequency trading in recent years.

We have to draw the conclusion that:

- Such high returns are temporary due to arbitrage and will shrink considerably over time.
- Or we are trading at an unsustainably high risk level and our risk-adjusted returns are actually much more in line with market returns. This will eventually lead to a blow-up as can be shown mathematically.

A useful method to get an idea of achievable returns is to employ the outside view – to look at the numbers posted by our successful competitors as a benchmark.

I look at the Sharpe ratio as a measure of risk-adjusted return to allow comparison of return numbers relative to the risks taken. The Sharpe ratio is the excess return (above the risk free rate) divided by the volatility (as a proxy for risk). As a benchmark we can look at equity markets: they historically posted returns a bit below 10% with volatility above 15% resulting in a long-term Sharpe ratio of around 0,35. A global mix of all asset classes will boost that to about 0,6 because diversification reduces volatility at a similar level of absolute returns.

Looking at the all-time best investors, one has to be aware that they are outliers and simply define an upper ceiling of returns that are possible at all. This gives a reality check when encountering return claims that are way above that and it also shows returns don´t need to be outlandish for someone to be incredibly successful as long as these returns are very good and consistent over a considerable period of time – the magic of compounding at work.

- Star investors like Warren Buffet, George Soros, Paul Icahn and others have managed to earn between 20% and 30% yearly returns over a span of decades boasting Sharpe ratios between 0,7 and 1. This implies a very high risk tolerance, as drawdowns frequently top -50% for such return distributions.
- The best long term hedge fund returns I came across are James Simons´s Renaissance Technology´s Medallion fund at 35% annual return after all fees (5 and 40!) over decades and Edward Thorp´s: Princeton Newport Partners averaging 20% annually with a minute 6% standard deviation and an incredible Sharpe Ratio of 2,33 (the highest I ever came across spanning decades) – this was achieved using leverage of around 8x in a completely hedged statistical arbitrage portfolio and by being the first to do that.
- A very interesting example of concentrated value investing during very bad times is John Maynard Keynes´ investment management of the Chest Fund, King’s College, Cambridge, during the great depression, 1927-1945. Beating the market considerably, he made 9% per year with a Sharpe ratio of 0,4.

These are investors, not nimble traders is a common argument, but looking at hedge funds that does not hold: they can and will go after any superior profit opportunity at any time frame they can find it and will employ the best brain and computational power in the world to get there.

For my own portfolio the absolute highest return level I can hope to reach is a Sharpe ratio of 1, which seems to be a real-world ceiling for virtually all traders and hedge funds over the long term.

**The Kelly Criterion**

What would be the optimal risk to compound our capital at a maximum growth rate without the possibility of going broke?

If we know the return parameters of our strategy or portfolio, our edge, with some accuracy, through backtests or real-time trading data, we can determine the optimal size of our bets or the degree of leverage for our portfolio by a straightforward mathematical formula – the Kelly Criterion.

Trading at the level the Kelly formula spits out will result in the maximum growth rate for our capital that is achievable without risking ruin given our strategy´s return parameters. As can be seen from the graph above, overbetting optimal Kelly will result in quickly diminishing growth rates and betting beyond double Kelly will lead to a certain race to zero for our capital over time, due to constantly negative average returns. The book „Fortune´s formula“ by William Poundstone goes deep into the subject of bet sizing methods.

As inputs we need our strategy´s average win and loss percentages and win probability, if the strategy takes uncorrelated, consecutive bets. As this would be a fairly one-dimensional strategy, I prefer to use the version of the formula calculating the optimal leverage for a continuously invested portfolio. For that we need the annual return and the standard deviation of returns of our strategy, which are also the inputs for the Sharpe ratio. This gives us the tools to run some scenarios based on sound maths.

In a trading context there are some caveats regarding the Kelly Criterion, which was developed in the context of casino games with dependable edge and odds:

- Performance statistics need to be accurate, but in reality future performance is often different from past performance and it may also change considerably across different market regimes.
- Many trading strategies display serially correlated returns and may take simultaneous bets in assets that are not totally uncorrelated.
- Many strategies have a return distribution that is not normally distributed, e.g. negatively or positively skewed, which means the standard deviation does not accurately reflect reality.

Some margin of error needs to be implemented to account for all that. Also, trading at full Kelly has some characteristics that are highly uncomfortable for many people. For example the chance of suffering a drawdown of – 50% before doubling your capital is a whopping 50%, a drawdown of -90% still has a considerable probability of 10%. For these reasons many people trade at half or a quarter Kelly which greatly reduces the volatility while having a lot lower impact on lowering the growth rate.

**Consecutive Bets**

A simple example would be a strategy that wins twice the amount it loses on average while having a 50/50 chance of winning. If our average loss is 50% of the capital we are betting, Kelly tells us to bet 25% of our capital each time (or half to one quarter that for reduced volatility and margin of error).

**Portfolio Leverage**

Betting at half Kelly (left column) the highest sustainable return we can hope for with a portfolio Sharpe ratio of 1 is 50% per year (which shows just how good a Sharpe ratio of 1 actually is), but at a whopping 50% volatility – almost triple the stock market volatility, which would make it untradable for most people.

More benign is the result on the right, when trading at a quarter Kelly – 25% volatility for a 25% annual return on average. It is still higher than stock market volatility at 19%, but that is the level I aim for in my portfolio.

In general, when we plug in a strategy´s return and standard deviation, we get the leverage we can use on the portfolio as a whole and its Sharpe ratio.

**Practical Conclusions**

If you have mastered the skill of compounding money at a level above 10% per year in a sustainable way, your actual account size shouldn’t be a limiting factor for very long. If you can reach levels above 20%, I´m sure ways to procure the necessary capital can be found. I would argue that a small account size is usually tied to a certain skill level: If you had the necessary skill for sustainable, high returns you wouldn’t have a small account anymore – at least not for long as your capital compounds and the influence of luck averages out more and more.

Therefore to me the most prudent approach would be to figure out your strategy´s performance parameters and keep risk within the constraints of the Kelly Criterion, while slowly building up your skill and capital, adding money through savings, windfalls like bonuses, an inheritance etc..

It´s all about gaining proficiency.

For a small account it simply doesn’t make sense to trade actively for the financial reward in comparison to the time spent working on it. Even if you return 100% per year you would be financially better of as a day laborer than a day trader.

But it does make sense to regard it as a training account to build knowledge, experience and practical skill – the account will gain in line with your level of proficiency. On the way severe setbacks are very likely to happen.

Especially for a beginning trader that would be a smart course of action as it is easy to underestimate the level of skill necessary to become profitable at all. It is a very competitive field with many very smart players. I would venture the guess – judging from my own experience and other trader´s histories – that you need 3 to 5 years of full-time, hands-on education to reach a sustainably profitable level through knowledge and practical experience. This is just about the amount of time and effort necessary to build a new business or to get an education in an area of some complexity. Why should it be any different when trying to learn the craft of investing at a professional level? Keeping an active account small during that time will help to keep the cost of tuition low.

In all honesty any other approach has always lost me money in the end. For all I know I might still be in the state of naive belief that is a long way from sustainable profitability – that only ever becomes clear in hindsight when observed from a higher level of knowledge.

**Ways to trade up a small account as quickly as possible**

I realize that these constraints are not very satisfactory when you are looking at a small account and still have the desire to have a go at trading as a business. Are there ways to hack the slow growth method above?

I think so, but they must embrace risk of failure. I would argue that, if you have less than $10.000 to start your professional trading career, blowing up your account won´t really matter all that much – you´ll be quickly able to begin anew. It would be much easier to embrace a 90% drawdown risk at full Kelly leverage then with a sizable account that would take years to rebuild.

Warren Buffet made an interesting point: he argued, that with a small account (which for him would still be millions) he would be able to compound at 50% annually. That´s an advantage the individual investor has: he can access assets and strategies with low liquidity or capacity and is free from the career risk professional managers face when they look very different from the market.

What are some of the possibilities?

- Count on luck, take a huge risk and either blow-up your account or achieve large growth. This approach has always ended in at least a massive drawdown for me in the past – therefore I think the chances are very small to do it successfully.
- A better way would be to define “luck” and try to find an environment when being lucky is more probable. Consciously running your strategy over-leveraged will work only when this coincides with a phase of a temporarily high Sharpe ratio. I believe most strategies have some performance persistence. If you could define a beginning and end to a period of high leverage that might prove lucrative. US equities, for example, currently show a low volatility of around 7% (instead of the average of 19%). It would be possible to ramp up leveraged equity exposure using Kelly´s formula and scale down as soon as higher volatility materializes.
- Another possibility would be to concentrate on special situations for which you have high conviction paired with a very high reward to risk ratio e.g. 10 : 1. These are usually non-repeatable, somewhat unsystematic and therefore unsustainable one-offs that might make high rewards possible. For many players this is a major incentive and it is a very competitive area that is hard to succeed in.
- Combining several uncorrelated strategies to smooth returns and elevate Sharpe ratio while sticking to Kelly leverage levels is my main approach to progress towards a long-term Sharpe ratio of 1. Specifically combining strategies with opposite return distributions, for example negatively skewed strategies (e.g. short volatility) and those that are positively skewed (e.g. trend following). This has a much higher chance of success than running a single strategy.

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