I recently spent more time doing “reading research” rather than “testing research”. As result, this post resembles a collection of links on ideas seen on the web of how to trade futures with a small account – one of the topics I have been interested in.

The Issue: Diversification with Small Account

A small account size – or starting equity – can make it difficult to achieve diversification (a “free lunch” with a high “cover charge” as described in this post – you can read more on diversification and correlation from this blog here and here).

Diversification can be achieved by trading a large number of components in a portfolio, whether “components” represent:

• Instruments
• Systems
• Timeframes

Instruments Diversification

“Instruments” is usually the first aspect that comes to mind when thinking about diversification.
Including more assets/markets/instruments in a portfolio is often described as the “free lunch” – and this is one of the main reasons why large CTAs often include upwards of 100 markets in their portfolio selection.

A small account most likely cannot trade a portfolio of 100+ instrument. This is an issue that Dean Hoffman tries to address in this article: The Conundrum of Small Managed Futures Accounts.

Noting that most diversified trend follower CTAs have a minimum account size of at least $1M, Hoffman describes the advantages of trading larger accounts (able to trade many instruments including those with high margin requirements, more granular position sizing with contract scaling).

Hoffman then describes Dynamic Portfolio Selection as a potential solution for small accounts to achieve increased results from a “virtual high diversification”. The system monitors a large set of instruments but instead of taking all signals (as a diversified trend follower would most likely do), it evaluates and ranks each instrument relatively (based on each market’s potential on a risk-adjusted basis), resulting in about 90% of trading signals being filtered out. This naturally cuts down the number of positions held at the same time, and consequently the required account size.

As this is mostly a “marketing” article for Hoffman’s CTA offering (implementing this concept), there is not much more information on what sort of filtering is applied to select the “best” signals but the general idea is worth investigating (and you can check for yourself whether their performance seems to hold up against the theory).

The subject of dynamic portfolio selection has also been covered in the inevitable Trading Blox forums in this “Dynamic Portfolio Selection” post started by Dean Hoffman himself.

A couple of posts on this blog also describe potential filtering ideas based on relative market volatility and higher-level trend direction.

This idea of filtering trades is not new: the Turtles used to use the concept decades ago, as mentioned by TB forum user sluggo in this post (which contains a link to Trading Blox code implementing similar “heat limitation” mechanism).

Systems (and Timeframes) Diversification: Swarm Behaviour

Combining several systems is also a possibility to achieve diversification. With the extra advantage that it is possible – to some extent – to design systems and control their correlations to the rest of the suite of systems (as opposed to markets, which can have a furious tendency to correlate to +1 or -1 during crisis times).

And as we all know, correlation is a key element of the “diversification benefits” equation (check this thread from user sluggo on TB forums for a good presentation/discussion on the topic).

Adding a profitable mean reversion/counter-trend system to a trend following system will, in all likelihood, reduce the volatility of the combined portfolio, thanks to the negative correlation that it brings. Adding many uncorrelated systems is likely to increase this positive effect.

However, trading a diversified suite of systems has a similar constraint to trading a large portfolio: it increases the required account equity.

A comment from Pumpernickel on a recent post from Quantum Financier (who is starting a series of posts on “signal aggregation: how we form and use an ensemble of signals isolating different pieces of information to build a profitable strategy” ) pointed to a couple of documents from Fall River Capital.

The (pdf) document (part 1 and part 2 of their white paper) describe how they tackle this issue on a large scale, by trading hundreds to thousands systems simultaneously, using the concept of swarm behaviour (which can be seen throughout the natural world, such as in the mesmerising starling flights in the English Somerset Winter, pictured above).

From the white paper (other Fall River white papers and general website are also interesting to read):

An […] approach is to assign each trading system a vote. Each model is polled for its position (long, short, or out) daily, and the total is aggregated into a tally that may be thought of as a “Vox Populi,” or crowd opinion poll. Research showed that aggregating the systems by this simple tally method was a quite workable approach, allowing us to “cheat” by holding a single position per market rather than hundreds or thousands. Regardless of the number of component models, the master strategy holds a position in accordance with the majority of the crowd.

How they choose the models/systems to be included in the portfolio is mostly driven by each system’s correlation to other systems:

The portfolio of individual candidate systems consists of between several hundred and a few thou‐ sand members that share both low correlations to one another and robust returns over many years of market history. The result is a “swarm” of trading models, each attacking the market from a different direction. This process of system development, evaluation, and selection does not prioritize superior standalone system performance, but rather seeks to uncover profitable trading rules that complement one another when implemented together.

Their testing results seem to show that this approach tracks fairly well an “equal allocation” approach with hundreds/thousands of systems, which itself benefits greatly from low correlated system diversification (reduced volatility, or increased vol-adjusted returns).

This “systems voting” strategy has also been discussed on the TB forums there (again started by user sluggo…).

Other Alternatives

These are ideas to stimulate research on how to alleviate the “futures trading diversification with a small account” issue. Other ideas can also be found on other threads from the TB forum (examples 1, 2 and 3 – search the forum for more discussions), showing that the topic is a “popular” one.

Another alternative would be to move away from trading actual futures but instead focus on “proxy” instruments such as ETFs (see this post for a quantification of how ETFs can track futures) or spread betting (they usually offer lower minimum trading lots, allowing for lower required trading equity, but can have other disadvantages, such as counterparty risk, less instruments available or cost of funding/leverage). Another trade-off to make in system/strategy design..
 

I was recently reading a blog post discussing “trading the equity curve” of a system. This usually entails adapting trading size based on whether the equity curve of the system is below or above its equity curve.

A similar concept is described in The Way of the Turtle, by ex-Turtle Curtis Faith:

The Turtles were instructed to decrease the size of the notional account by 20 percent each time we went down 10 percent of the original account.

The idea seems to make sense in a way: the further the notional account size is reduced during a drawdown, the lower the maximum drawdown amount figure should be. Similar to a racing driver “hitting the brakes” as and when their vehicle starts going off-course, to avoid going “into the ditch”.

In periods of prolonged losing periods, the result should be a lesser impact on the equity curve, and a lower drawdown figure.

Testing The Concept

Trading Blox has this functionality built-in, so it is easy to test. You can set parameters for this Drawdown Reduction technique: the Threshold dictates at which point the notional account size is reduced (10% drawdown in the example above) and the Amount dictates by how much the size is reduced (20% in the example above). The reduction is cumulative, meaning that every time a new 10% decrease is observed, a further reduction of trading size by 20% is applied.

I ran a 20-day Donchian breakout system with “classic” volatility-adjusted position sizing (risk per trade = 0.75% of equity = 2 ATR) as a starting point. The results obtained were as follows:

 
The next run applied the Drawdown Reduction logic to the exact same system. As expected, the Max Drawdown figure does decrease by a fair amount, however note that the return also decreases by a greater amount, actually hurting the MAR ratio. The Sharpe ratio does not improve either:

 
In hindsight, it simply looks that reducing the overall leverage of the original system might achieve the same results (reduction in both drawdown and return). Here are the results of the first system with reduced leverage (0.64%) so that the Max Drawdown amount matches that of the second system:

 
Sharpe and MAR are closer to the first run results, with the CAGR being higher than in the second run, implying that a simple leverage reduction could be a better option.

Post Update: Of course, this is a single one-off test, from which you can hardly draw any conclusions, so do not go and dismiss it solely based on this post/test. A proper and more complete test with a large number of system variations – as kindly pointed out by Pumpernickel in the comments – would be a good start to evaluate the impact of the Drawdown Reduction technique. Something to play with in your own system development or to follow-up in a later post.

Managing the Unexpected

This comparison test benefits from hindsight. This money management technique might still bring a way to deal with extra-ordinary negative periods (where drawdowns would exceed the expected figures from the back-test). This way, the system could start trading with the “optimal” leverage derived from the back-test, with an extra safeguard (possibly with higher threshold triggers) only for cases when the system starts diverging substantially from the back-tested results (whether the system is “broken”, or experiences its worst period to date).

Another point to note is that the downside of reducing leverage during drawdowns usually increases the time required to get out of the drawdown.

Another possible use of this technique might be to use it to dynamically allocate to several systems in a suite: reduce the allocation of poorly-performing systems and shift the allocation to better performing systems (“starve the dogs and feed the stars”). This way, if a system starts becoming “broken”, its allocation is automatically decreased, which should reduce its (negative) impact on the overall suite performance.

The last study on the Leverage Space Portfolio Model was interesting from a theoretical point of view. Its main value was in illustrating the concepts of the framework with real-life data, but it did not have much practical application: on top of some “unpractical” assumptions, the study was done using hindsight: the optimal f values were calculated using past data, then applied to the same past data.

Works fine as a simple exercise but it is a critical flaw in real-life backtesting.

This post looks at a more realistic approach on how to practically apply the Leverage Space Model to Portfolio Construction.

What sort of Test?

In this test, 6 systems, arbitrarily chosen from the State of Trend Following report, will be considered as potential components of the portfolio:

• Bollinger Bands Breakout – 20 days (BBO-20)
• Bollinger Bands Breakout – 50 days (BBO-50)
• Donchian Channel Breakout – 20 days (Donchian-20)
• Donchian Channel Breakout – 200 days (Donchian-200)
• Moving Average Cross-over – 50/200 days (MA-50-200)
• Triple Moving Average – 20/50/200 days (TMA-20-50-200)

The data used is the set of monthly returns for these 6 systems from 1990 to 2010.

One important constraint to consider for this test is to make sure that the optimal f values are not applied to the data used to calculate them: we’re aiming to prevent hindsight bias.

For the Leverage Space Model, we could simply divide the testing data in 2 sets: the optimization set (1990 to 1999) and the testing set (2000 to 2010) to which the values found from the optimization would get applied.

However, I decided to look into a more flexible approach: a Walk-Forward test (you can check a quick recap on Walk-Forward testing).

Every year, from 2000 to 2010, the f values applied to the portfolio for that year are derived from the optimal f values for the 10 years directly preceding (i.e. optimize f values from 1990 to 1999 and apply to 2000, optimize f values from 1991 to 2000 and apply to 2001, etc.).

Note that this involves running 11 optimizations instead of one. Ideally, the test would be run using “realistic” conditions, meaning an optimization with drawdown constraints, as opposed to pure equity growth optimization.

Because of the difference in running times for each optimization (literally hours vs seconds), I went with the “equity-growth” optimization: the classic optimal f concept, if you like. Optimal f, in this classical form, is often brushed aside as being too risky…
And indeed, in the fourth year of the period under test, a loss greater than 100% appeared. This ruin made it impossible to obtain a meaningful result with these conditions. Instead I decided to arbitrarily apply half of the f value (f/2) to be able to run the test

The main point of this test is to compare the performance of a walk-forward, practical approach versus the theoretical return obtained using hindsight. For both these tests, f/2 will be used instead of the optimal f.
Additionally, since the walk-forward results start from 2000 (1990 to 1999 being only used for optimization), other tests and results in this post will also cover the same period.

Base Reference Portfolio

For the most simplistic portfolio construction, we can just divide the equity equally, without leverage, into each of the six components. With a monthly rebalancing, this gives us the following equity curve for the period 2000-2010:

The Hindsight Result

The hindsight approach is very similar to the last study: take all historical data (from 2000 to 2010), feed it to the optimizer and apply the optimal f values to this same data: this is the best portfolio construction that an investor could have used (when considering equity growth only). Based on the previous note, f/2 is used instead of the Optimal f, which gives us the following allocation/leverage factors:

• BBO-20: 0
• BBO-50: 0.022
• Donchian-20: 0.842
• Donchian-200: 0
• MA-50-200: 0.967
• TMA-20-50-200: 0

The numbers above are the “fraction” of total equity to be traded for each system. If trading these 6 systems with a 10M account size, the notional account size for the “Golden Cross” system (MA-50-200) would be 9.67M and so on.
Note that the optimized allocation excludes some of the systems: these are not required to reach the maximum account equity growth.
The total leverage on the portfolio is therefore 1.83: the 10M equity would be traded as 18.3M across the three systems with a non-zero allocation.

There is notable difference in CAGR despite the drawdown being nearly identical between the two allocations:

 
Let’s now look at the actual Walk-Forward test of the LSPM allocation.

Walk-Forward LSPM Results

As explained above, each year’s leverage is calculated by running the LSPM optimization on the previous 10 years. This is done year after year, going forward.

The results are actually surprisingly good: they even top the hindsight “f/2″ results. When comparing to the “full f” hindsight results, the CAGR is only slightly lower (CAGR = 126% for “full f” hindsight results).

The allocation and leverage in the portfolio vary year after year. One could argue that this is an adaptive approach, which should perform better than a static approach and this single result seems to confirm that idea.
For reference, below are charted the different leverages in the portfolio and how they evolved over time:

A good share of the portfolio is always allocated to the Donchian-20 system, the rest being mostly divided between the TMA-20-50-200 at the start of the decade and the MA-50-200 at the end. Note also how the overall leverage is constantly dropping.

The equity curve comparison also seems to indicate an early over-performance, which does not seem to carry forward. This is confirmed when looking at the returns year by year:

A Note on Drawdowns

The drawdowns in most of these tests are very high and no money manager would really consider trading at these levels. This is a classical reception to the Optimal f concept. The Leverage Space Model can cater for this by including constraints on probability of drawdowns – to keep them at reasonable levels.

The reason this type LSPM optimization (with drawdown contstraints) was not used is simply because of the very high computing times for the optimization but I would expect similar relative comparisons between the hindsight and walk-forward results – something to verify: material for a later post when my machine has a spare dozens of hours to run.

More To Come

This is obviously a single test, which does not provide much significance, nevertheless, the results are quite exciting. But there are a few caveats. The test only uses equity-growth optimization, and we can only assume (or take a whole weekend of computing time to verify) that adding drawdown constraints to the optimizer would have given us a similar relative comparison (between the hindsight results and the walk-forward/adaptive results).

But this definitely warrants further investigation. One of the aspects that I like is how the process weeds out unnecessary systems. You could imagine running a stable of systems and let the optimization pick the systems to trade for each coming year and with which leverage.

Note also that there are other assumptions in the tests that make them less realistic (no account for slippage, funding costs, scalability, margin constraints, etc.) but these would apply to all the systems under comparison here.

On a final note, I have heard that a major index firm has decided to launch LSP indexes based on Vince’s implementation. These would be licensed to ETF providers, with the first LSP-style ETF probably being launched mid-2011. The concept seems to be catching on.

I’ll probably also share in a next post the code that was used to run the Walk-Forward test as it nicely automates the running of it. Stay tuned… (Update: the code can be found on this post: http://www.automated-trading-system.com/r-code-walk-forward-lspm/)

Psychologically, it is harder to trade a system that produces more losing trades. Despite this, Trend Following is a profitable strategy. Could there be a sort of “psychological premium” received by traders willing to use low winning percentage system?

When recently looking at Ralph Vince’s Leverage Space Model, I could see a potential reason why low winning percentage systems might be more robust.

Robust Systems are Volatile

An aspect of robust systems is the volatility in the system results and its equity curve. Similarly to low winning percentage, volatility is also a system feature that traders/investors nearly always want to avoid – at least from a psychological point of view.

Here is a quote from David Druz – a recent addition to the Trend Following Wizards report – which explains the link between robustness and volatility:

The robustness of a trading system is proportional to its volatility. This is the no-free-lunch part. A robust system is one which works and is stable over many types of market conditions and over many timeframes. It works in German Bund futures and it works in Wheat. It works when tested over 1950-1960 or over 1990-2000. Robust systems tend to be designed around successful trading tactics, classical money management techniques, and universal principles of market behavior. These systems are not designed around specific types of markets or market action. And here is the amazing thing about robust systems: The more robust a system, the more volatile it tends to be! This is because robust systems are not optimized to particular markets or market conditions. The converse is also true. You can design systems with excellent returns and low volatility on historical testing, but which work only for given periods in given markets. These systems tend to be curve-fit or market-fit and are not robust. For a system to have the highest odds of profitability over time and markets, the inescapable tradeoff is volatility. Diversification can be used of course, but it will only dampen the volatility so much.

A typical illustration of low-volatility, high-performance equity curve is this one below:

LTCM: smooth equity curve… for a while. No robustness there.

The Impact of Money Management

Money Management is a pivotal part of a trading system. It can make or break any system, however good it is. Over-trading a really good system will still lose you money.

Indeed, Vince affirms, in the Leverage Space Trading Model intro, that Money Management represents 100% of a trading system (leaving 0% for signal generation, etc.). This is probably a hyperbole, but the impact of money management should not be understated.

By looking at the impact of the the system’s winning rate on the money management part of it, there might be a reason as to why a system might be more robust when it produces more losers than winners.

Optimal f, boundaries and the Tiger Cage

Depending on preferences, there exists an optimal f which maximizes growth and which can take risk factors into account. The optimum approach is obviously to be as close as possible to the optimal f. However, this is not simple: markets change, systems go through different phases of performance and as a result, optimal f moves as well. It is not a fixed value.

From the start, we know that optimal f is bounded between 0 and 1 and that the further we are from it, the less optimal performance will be. Vince had an amusing analogy, comparing the optimal f to a roaming tiger on a football field. The tiger could be anywhere on the field; the hunting trader needs to find its location to place the tiger cage. Any error in locating the tiger would result in sub-optimal performance: the further the tiger is from the cage, the worse the system would perform.

However, there is a corollary from the Optimal f calculations: the optimal f value is bounded by the winning rate of the trading system. F can take values between 0 and 1 but if the system has a win rate of 30%, optimal f will always be between 0 and 0.3.

Going back to the tiger-on-the-football-field analogy, it greatly reduces the task of the “hunting” trader if we know that the tiger never ventures beyond the 30-yard line. And as a result we’ll never be too far off the target. From a trading point of view, this means that there is less room for error in the location of the optimal leverage and therefore less impact of a sub-optimal leverage.

In terms of robustness, this means that as markets change, so will the optimal f. A trading system with a low winning percentage will reduce the possible variation in the optimal f (e.g. [0,0.3] instead of [0,1]) and therefore the error between the actual f value used by a trader and the optimal f. Reducing the error should also reduce its negative impact on the system performance and make the system less sensitive to underlying market changes. In a word: more robust.

This is just a thought. I do not have any hard evidence or theory for it . As a young kid I always dreamt of having a tiger and riding it to school. I’d now like to think that it was an early subconscious call to be a good trader by focusing on “catching the tiger” of good position sizing/money management…

Following the Risk-Opportunity Analysis conference I attended earlier this month, I decided to test the model and the software used to implement it (Vince‘s java app and Joshua Ulrich’s R implementation).

Most of the mathematical formulas supporting the model are in the book Leverage Space Trading Model. I will not paraphrase the book and reproduce all the formulas here – but I will refer to some of them. Getting the book is probably a good idea for a better understanding of the concepts.

Test Case

The trading application of the Leverage Space Model is presented as a generalisation of the Kelly formula, which is well illustrated by the coin-toss betting example (as per Vince’s paper).

In a practical “trading” example, I have decided to look at the four Trend Following Wizards with the longest track records (that I have): Campbell, Dunn, John Henry and Millburn. All track records go back to 1985 (up to end 2009). The question this test tries to answer is this:

As an investor from 1985, what would have been the best allocation between the four managers?

Note: Here “best” simply means the highest CAGR (we’ll see later that “best” can be defined in different ways based on your preferences / utility function).

I will describe the concepts using the simple two-coin toss example and draw a parallel with the “real-world” application on the four Trend Following Wizards.

Optimal f

For a single stream of returns or betting outcomes/probabilities, there is a specific level of leverage, or fraction of capital, to risk on each event, which maximizes the geometric growth of the equity. This is what Optimal f relates to.

In the example of the coin-toss with the following parameters:

• Risk 1 unit for each bet
• Tails: lose 1
• Heads: win 2

The fraction of capital staked on each bet will alter the expected growth rate as per this curve:

This has been covered extensively elsewhere (as a simple application of the Kelly formula) so I will not repeat the details here. The main point is that .25 is the Optimal f (meaning, in that case, that staking 25% of the largest loss on each bet will maximize the growth of the trading stake over time – any other value would be sub-optimal). Note that here the largest loss is equal to 100% of the bet size and therefore the Optimal f and the fraction of capital to stake are equal. This is a special case and we’ll see further down that these are usually different values, especially in trading.

Adapting this example to a single Trend Following Wizard track record, we can establish a posteriori what the optimal f would have been, in order to maximize the growth of the equity curve.

Here is the f curve for Dunn’s track record:

The shape of the curve is fairly similar, it peaks at around f=0.5.
Note that here, the optimal f does not represent the actual leverage or fraction of capital to apply. This is because of the nomenclature in the formulas used by Vince, where each “leveraged” return (or HPR: Holding Period Return) is expressed as:

In effect, the leverage applied to each periodic (monthly) return is:

In the case of Dunn, the biggest monthly loss is 30.68%. An f of 1 would equate to leveraging each period return by a factor of 1/0.3068 = 3.26, which is the maximum leverage that can be achieved before hitting a zero final equity (the HPR would be equal to 0 as soon as the biggest loss occurs). The actual leverage amount is independent of the biggest loss, but expressing it that way bounds the value for f between 0 and 1. This is really just a notation.

The optimal leverage for the Dunn track record can be derived from Optimal f = 0.5: Optimal leverage = 0.5/0.3068 = 1.63. This effectively means that an investor would have achieved the highest possible final equity investing in Dunn by resetting the notional account size to 163% of the actual account size, every month (this is theoretical as it ignores the (im)practicality of this and impact of fees, etc.).

Any other value, higher or lower would have resulted in a lower equity curve. At 100% (no leverage), the monthly geometric mean return is 1.11%; at 163% leverage, the mean return becomes 1.29% (which is the maximum value possible). Of course the drawdown would also increase (around 60% MaxDD at 100% leverage vs. around 83% MaxDD at 163% leverage).

Multi-Component Scenarios with Coin Tosses

Following on with our coin-toss example, let’s now consider the case of two simultaneous coin-tosses.

The method requires a discrete set of outcomes and associated probabilities. In case of the simple two-coin toss example, these are easy to identify:

• 2 Tails: lose 2, 25% probability
• 1 Tail and 1 Head: gain 1, 50% probability
• 2 Heads: gain 4, 25% probability

Using this set of outcomes with the same concept as above for f optimization, this now gives us a 3-dimensional curve displaying each possible f-combination (each simultaneous coin toss has its own f). Each combination generates its corresponding growth rate (which is related to the Terminal Wealth Relative, TWR). The f-combination that generates the highest TWR is the “optimal” solution.

In the case of the two-coin toss, the optimal f-combination is (0.23, 0.23) – meaning that staking 23% of the capital on each simultaneous coin toss (46% in total for each period) would result in the highest growth rate (over time). This is simply where the curve peaks in the chart above.

Adding a third simultaneous coin toss would simply generate a 4-dimensional curve (with each point representing the growth rate output of the f-triplet) and so on: the curve is always [N+1]-dimensional, where N is the number of components.

Multi-Component Scenarios with Trading Data

Let’s now look at our test case with the four Trend Following Wizards.

Similarly, we need a discrete set of outcomes and associated probabilities for our input . For this, we need to bin the data distribution and create the Joint-Probability Table (which holds each possible outcome combination and its associated probability – similarly to the 3 possible outcomes identified above for the coin-toss example). This is effectively how Vince does away with the concept of correlation input that can be used in other models such as Mean-Variance Optimization.

The pseudo-code to build the Joint-Probability Table (JPT) is as follows:

• Bin each component’s stream of returns (ie. each Wizard set of returns)
• Calculate a single outcome for each bin (for example the mean return of all instances falling in that bin).
• Loop through each period and:
• For each component, determine which bin the period’s return fall into; and assign the bin outcome to that component, for that period.
• Record the combination of all bin outcomes (ie for all components) for that period and assign it the probability: 1 / number of periods
• If different periods have the same combination, these can be grouped together (by summing the individual probabilities – as in the two-coin toss example where the Head-Tail and Tail-Head combinations are grouped into one at 50% probability).
• The JPT is the full list of outcome combination and associated probability.

This is a bit tricky to explain in a concise manner, and it is developed in further detail in Vince’s book. However, the files attached at the end of this post in the “technical” appendix should help you retrace this logic.

Once the JPT is built, it can be used as an input for the f optimization.

Leverage Space Trading Model: Optimizing f with R

The java app developed by Vince creates the Joint-Probability Table from the import of the equity curve for each component.

The “meat” of the leverage space model code is contained in the R implementation by Joshua Ulrich (from FOSS Trading). This is where the optimization is actually run (Vince’s java app also implements the optimization but the R implementation is much faster).

Note that Josh has a blog post on how to create the JPT, so you might be able to only use R, should you want to experiment with the Leverage Space Model (I am not sure the java app is freely available).

The R LSPM package needs the JPT as an input, as well as the optimization parameters. Note that this is my first foray in R – so you definitely do not need to be any expert at it to run this sort of test. The fact that the java app generates the R commands directly was helpful, but you can probably “learn by example” by checking the R session file at the end of the post (there is more doc available on Josh’s blog or the LSPM project page anyway).

Using the JPT as an input, the LSPM package runs the optimization, which estimates the peak of the 5-dimensional curve. A genetic algorithm is used for the optimization; and after a specific number of iterations, the optimal set of f-values is output by the program.

For the four wizards, the respective optimal f values are as follows:

• Campbell: 0.050767
• Dunn: 0.000000
• JWH: 0.322954
• Millburn: 0.375109

Again, these are f values, which only relate to the leverage to be applied, via each component’s biggest loss:

Leverage values:

• Campbell: 0.050767 / 16.7% = 0.304
• Dunn: 0.0 / 30.68% = 0
• JWH: 0.322954 / 27.32% = 1.182
• Millburn: 0.375109 / 14.12% = 2.657

One interesting thing to note is the fact that some f-values can be assigned to 0, which basically means that the component does not add value to the portfolio with regards to the growth rate, which maximum is attained when excluding it.

An investor would have maximized the geometric growth of their capital by resetting every month the notional account size allocated to each manager according to the three leverage factors identified above. For example, starting with $100 million total capital, the first month would see $30.4M allocated to Campbell, $118.2M to JWH and $265.7M to Millburn. After the first month, the equity would have increased by 22.9% to $122.9M, which would then be re-allocated according to the same leverage ratios.

After 25 years of repeating the same process monthly, the $100M would become a theoretical figure of… $411.4 Billion thanks to a monthly geometric mean return of 2.81% (whereas an unleveraged equal-split across four managers – with monthly rebalancing – would result in a “paltry” $3.9 Billion). Of course, this improvement is only possible “in hindsight”.

The other implication from this sample application is that the optimal f value can dictate a leverage value which can be higher than the maximum allowed (be it by margin requirements or stock trading in a cash account, etc.). The Leverage Space Model caters for this, with the possibility of adding margin constraints (I have not looked into this yet but this post on FOSS trading talks about it).

Drawdown Constraints

One of the main problem usually raised with the concept of optimal f is that trading for growth rate optimization is often not realistic, as it generates untenable levels of drawdown and volatility. Most investors, traders or managers would happily give up some return to stay in their acceptable levels of volatility and drawdown.

I will not detail the formula here (I’ll refer you to the book again) but Vince presents a way of calculating the probability of a specific drawdown. The main idea is to introduce a risk constraint to the model, so that instead of solely optimizing for maximum growth rate, one can optimize with a constraint on drawdown. For example:

Find the optimal f values for which the probability of a 30% drawdown over 12 periods does not exceed 50%. The n-dimensional curve is constructed in the exact same way, but any f-values that result in a probability of drawdown over the constraint threshold are ignored – this would usually result in all values around the peak being discarded.

This is exactly what I ran on the same four Wizard track records and the f-values obtained were as follows:

• Campbell: f=0.053919, leverage=0.3229
• Dunn: f=0.002451, leverage=0.007987
• JWH: f=0.297744, leverage =1.08984
• Millburn: f=0.237712, leverage =1.68351

The monthly geometric mean return drops to 2.68% (final equity of $282.7B). The Max Drawdown figure drops from 79% to 68%.

Slow despite the Snow

The main problem of adding drawdown constraint to the optimization is the dramatic increase in computing time. Whereas the first optimization for the simple growth-optimal f values took seconds for 100 or 1,000 iterations of the algorithm, adding the drawdown constraint has a significant impact on the computation time.

The LSPM uses another R package: snow, in order to leverage multi-processors with parallel computing to speed things up. Info for the techies: I am running an Intel Core 2 Quad Processor @2.40GHz and I allocated three processors to the optimization process – but the running time came at a disappointing three hours for 100 iterations. This is only for four components and 300 monthly periods. I initially had the idea of running the daily equity curve of a few hundred components through the LSPM package but that will probably have to wait for the technology to improve.

I understand that this is mostly due to the heavy computational costs of evaluating the probability of drawdown. Maybe a less costly risk computation might make the running time more manageable.

First Impressions

This is a bit of an extended post (probably the longest on the blog so far) but it hopefully provides a decent step-by-step illustration of some of the concepts and how to apply them practically (there are more details in the “technical” appendix below).

The model is really an alternative portfolio allocation method and I can not see how it could be applied directly to determine position sizing for a trading system. This is primarily because all component returns have to be split across identical periods, whereas trades from a single system do overlap. It might well be possible to use each instrument’s equity curve when traded through a specific system. Something to investigate – but with the high computation costs, running the optimization over a large diversified portfolio might be all but impossible.

On the other hand, I can see how the model might be useful to the manager running a programme made up of several systems and wanting to optimize the allocation to each system. Alternatively, one could split the system portfolio into several asset classes (Financials, Currencies, Energies, etc.) and optimize the allocation to each asset class.

Another aspect worth looking into is how useful the model is in a forward-looking mode (ie to determine optimal f/leverage to apply to each component for the next periods) and how this can be used/configured (over the whole history available at that time or over a rolling optimization window? Which length of data to use in that window?). This would obviously be dependent on how stable the component returns are over time (like for any aspect of back-testing).

Technical Appendix

Below is some additional material to support the explanations in this post and illustrate the step-by-step process.

I used the java app to generate both the Joint-Probability Table and the R commands to run the optimization.

The app requires a file per component, containing the equity curve with constant position sizing (ie. no reinvestment). For the four Wizard track records, the equity curve is simply the cumulative sum of the monthly percentage returns (representative of a constant position size of 100). Below are the four CSV files:

Campbell.csv
Dunn.csv
JWH.csv
Millburn.csv

When importing the files, the app bins the period returns and generates the JPT:

The optimization parameters can be configured via the second half of the screen below:

The parameters of the optimization defined above are: maximum probability of a 30% drawdown over 12 periods must be less than 50%. The Maximum Calculated Cycles of 8 is a number used in the drawdown probability calculation. As I understand it, the drawdown probability is extrapolated off an 8-period calculation and the computation time goes up exponentially with this number.

The R button generates the R instructions to run with the LSPM package. Below is a file containing the R session that I used to run the example in the posts. The first run is a straight optimal f calculation with 100 iterations. The second run is the same with 1,000 iterations and the third run is the optimization with drawdown constraint.

You can check the joint-probability table (contained in the outcomes and probs arrays) generated from the track record input files as well as the command to run in R.

R-Session

If you want to download R: http://cran.r-project.org/bin/, and the R LSPM package

I have been travelling in the last three weeks. One of my destinations was Tampa, Florida, where I attended Ralph Vince‘s Risk-Opportunity Analysis course, over a weekend (check here for a presentation of the course).
My main objective for attending was to strengthen my knowledge of Money Management/Position Sizing and my understanding of the concepts presented in Vince’s books.
The secondary objective was to gain an understanding of how to practically apply these concepts to Trend Following systems, and check what improvements the Leverage Space Model framework could provide.

This is not a full review of the course but more of a “teaser” post giving you a “heads-up” that several posts covering this topic will be coming on the blog – as I explore and test the concepts on real-life trading systems.

The Course

The course was dense and covered a lot of material with all of the mathematical formulas behind the concepts explained in the Leverage Space paper that was discussed on this blog. In a way, there is a large overlap with the contents of his Leverage Space Trading Model book, which expands on the concepts in the paper, with the Mathematics behind the concepts. The main ideas and concepts are the same.

The pivotal concept in the framework that Vince introduces is the multi-dimensional leverage terrain, which draws the return of a portfolio based on the leverage (position size) applied to each instrument.

In this example the model builds the terrain for 2 simultaneous coin-toss with a payoff of 2:1. The x and y axis represent the respective f-values (leverage) for each of the bets/trades - while the z-axis (vertical) represents the expected return

The maximum portfolio growth is located at the peak of the terrain, resulting from the specific corresponding leverage (or f-values) combination. The terrain construction does not take into account correlation between the instruments – instead, the model uses the joint probability of two scenarios occurring simultaneously, dictated by the price data history.

During the course Vince showed us several ways to “navigate” the terrain and determining the corresponding leverage factors, based on different objectives: geometric return, risk of drawdown and probability of being profitable in the next “period”. Another concept developed was the idea of investment horizon.

The Software

The other aspect covered in the course was the software that implements the Leverage Space Model calculations, which you would typically use to evaluate the optimal combination of leverage for your portfolio based on your criteria/objective function.

We walked through several simple examples from the course using both software packages that implement the Leverage Space Model concepts

Vince used to make available on his homepage a java application that implemented the model. This is proprietary though and when I last quickly checked I could not see the link for it any more. You might have some luck obtaining a copy by emailing him.

However, Josh Ulrich, the author of the FOSS Trading blog and several R packages has implemented the Leverage Space Model in a dedicated R package. The implementation is actually faster than the Java software (so much faster that Vince’s java app now generates R commands to run the LSPM package functions instead of its own optimizer).

Josh was there in Tampa and it was great meeting him too. He definitely has a great handle on the Leverage Space Model and I strongly recommend following his blog if you are interested in this material. Take a look now if you are after more specific examples on how to get started with the package (which can be found on R-forge repository).

Next Steps

If I have a criticism about the course, it would be regarding the format. I would have liked to spend less time on the Maths behind the concepts (most of it is covered in the book anyway) and spend more time in a “workshop” format where we could have investigated the software with some of our specific cases.
I do not mind doing some homework on this after the course, and the fact that there is a google group where attendees can ask questions is useful, but being able to do this “in person” is probably better.

It has left me hungry for more (in a good way) and I am keen to explore some of the concepts more in detail. I anticipate several posts looking into this from a practical point of view in the next few weeks/months.

In the mean time, if you fancy getting started on this topic, you could take a quick look at Vince’s paper – in which he has managed to distill his ideas in a short, manageable 30 pages. – or a more in-depth look by getting his latest book on the Leverage Space Model. In any case you might want to get started on the software side of things by downloading the R package. It is very easy to get started with it.

As the cliche goes:

Diversification is the only free lunch on Wall Street.

This is a concept equally shared by Modern Portfolio Theorists and Trend Following Wizards, who usually emphasise the concept – and are often quoted as trading around 100 different types of instrument, if not more.

The State of Trend Following report contains a decent level of diversification with around 50 instruments and I wanted to use this as a base to check the impact of diversification on performance.

The idea is to run the same strategy using a subset of the portfolio (ie less instruments = less diversification) and see how it performs.

The problem, though, in selecting a subset of instruments out of the 51 in the original portfolio is that it could affect performance in the same way as any portfolio selection can (ie you could obtain vastly different results in trading the same system with two different sets of instruments, just by virtue of a “lucky” pick of strong performers).

Historical Performance

First, let’s get a reference point and look at the historical performance of the system chosen for this test: the 20-50 Moving Average system. Below is the performance back-test of that system over the last 20 years with the original portfolio:

Tests with Less Diversification

As mentioned below, the idea is to work on a subset of instruments and compare the results with the initial portfolio. To avoid any sort of data mining/hindsight bias in the portfolio selection, I decided to run a Monte-Carlo-like approach to test the system with multiple instrument subset combinations: instead of picking a single portfolio subset of 25 instruments, I’ll run the system over 1,000 different sub-portfolios, chosen randomly.

In order to get an idea of how gradually diversification affects the performance, I ran the test in three steps:

• sub-portfolio of 15 instruments
• sub-portfolio of 25 instruments
• sub-portfolio of 40 instruments

All instruments are picked at random from the list of 51 instruments in the original portfolio.

Each of the 3,000 runs generated a full system performance record. Below are plotted the CAGR and Max Drawdown for each instance:

Click to zoom in

The original system is also represented as the yellow dot.

Note that the “portfolio randomizer” did not account for any logic in terms of sector allocation. The original portfolio is balanced over several sectors (currencies, energies, rates, agriculturals, etc.) and there is no account for correlation between the different instruments (obviously correlation plays a big role in diversification: there is not much point in having dozens of instruments if they are all strongly correlated). However, over the large number of simulations, the main ideas of the test still come through.

Another point is that the only difference between the different runs were regarding the position sizing of each trade (fixed fractional), which were adjusted to obtain results of similar magnitude in each test (a portfolio with less instruments will require a slightly higher position size to match the return/drawdown rate of a portfolio with more instruments).

Looking at the plot chart, there are two main observations:

• We can see the gradual effect of diversification improving the system results by “moving” the cloud of performance points towards the left (less drawdown) and up (more return).
• The other observation is that the more diversification there is, the lower the deviation in the system results – therefore providing more robustness and less chance of data mining impact from portfolio selection on your back-tests.

Diversification or Why the Coffee Cup Never Jumps

That last point makes me think of an example discussed by Nassim Taleb in his Black Swan explaining the averaging of randomness:

Yet physical reality makes it possible for my coffee cup to jump – very unlikely, but possible. Particles jump around all the time. How come the coffee cup, itself composed of jumping particles, does not? The reason is, simply, that for the cup to jump would require all of the several trillion particles to jump in the same direction, and do so in lockstep several times in a row. This is not going to happen in the lifetime of this universe

Every trade/instrument can be seen as a particle composed of a (large) random element and a smaller edge that we try to extract via a mechanical system.

A portfolio composed of too few instruments would be like drinking your coffee or tea from a cup made up of only a few particles: cups would be jumping around everywhere, making coffee drinking a perilous venture. Same concept applies to trading.

This is the way I see diversification: by adding a large number of mostly random elements, you can ensure that random moves have some cancelling effect on each other so that your “trading cup” never jumps. All that is left is to collect the small edge from all the instruments via your preferred trading strategy(ies).

In effect, this is how casinos operate – and with diversification you somehow get to be the house!
 
 
Credits: The use of a portfolio randomizer and the display of results in a CAGR/MaxDD scatterplot was directly inspired from user sluggo on this Trading Blox forum thread.
 
 

• position sizing
• notional funding

These were discussed more in detail in a “how to apply leverage?“.

Today, I just want to revisit this concept with a classic system: the Donchian Channel.

As Ralph Vince’s Optimal f tells us: there is an optimal “bet size” (our fraction of capital risked on every trade) to achieve maximum “Terminal Wealth” (i.e. maximum CAGR).

This can be better illustrated with this chart generated by stepping through different position sizes for the Donchian system:

One can see that past a certain point (2% position size), the extra leverage is detrimental to the actual return. On the other hand the Max Drawdown figure keeps on rising.

To apply leverage, we are free to slide along the curves to reach a desired level, but there is no point going past the optimal f. A 0.50% position-sized system will have less return and less volatility than a 1.50% one, but a 3.5% one will have less return and more volatility.

Notional Funding

I went into more detail comparing the notional funding leverage to the position sizing one in that post mentioned at the start. A couple of problems with notional funding (which can be thought of as trading an account with part-real funds, part-imaginary funds to reach a desired notional account size) are:

• its impact dilutes over time (unless some re-balancing occurs)
• its results are dependent on the equity curve path

The second point is easily illustrated by this example: consider a system with a CAGR of 30% and MaxDD of 50%.
If the notional funded account (trading a 100K notional account with real funds of 50K) reaches its MaxDD immediately, the loss would be 50% of 100K = -50K, which wipes out the real funds.
On the other hand, if the account gets the opportunity to grow sufficiently before hitting the MaxDD, it will have built a “cushion” allowing it to better withstand the MaxDD.

With this in mind, I still wanted to check the impact that notional-funding leverage would have on the performance figures from the system.

I chose to leverage the system(s) tested under different position sizes with a 100% notional-funding levered one and a 50% de-levered one (think notional funding in reverse: trading a 200K account as if it was 100K):

The chart above shows the impact that both types of leverage have on CAGR and MaxDD (note that position size is not charted, but as we saw in the first chart, it increases with MaxDD).

Why Notional Funding does not affect Max Drawdown

The CAGR figures (for a given position size) increase with the leverage provided by notional funding. However, the MaxDD figures seem little changed by this type of leverage.

The track record used for testing was fairly long (15 years) and this is one of the reasons why MaxDD figures seem barely affected:
If the MaxDD occurs after years of compounding at high double-digit rates, the account will have grown by a large factor compared to the initial account and the effect of notional funding will be diluted to next to zero.

Let’s imagine an extreme scenario for illustration purposes:
The strategy returns 100% a year for 10 years (no volatility, drawdown: just smooth straight line), then it drops by 50% (MaxDD) in the next year.

• If you start trading with 100, you end up with 100 x 2^10 = 102,400 after 10 years and 51,200 (102400 x 0.5) after the eleventh year: CAGR = (51200/100)^(1/11) – 1 = 76.32%, MaxDD = 50%.
• If you start trading with notional account size of 100 but real funds of 50, you end up with 100 x 2^10 – 50 = 102,350 after 10 years and 51,175 after the eleventh year: CAGR = (51175/50)^(1/11) – 1 = 87.78%, MaxDD = 50%.
• If you start trading with notional account size of 100 but real funds of 200, you end up with 100 x 2^10 +100 = 102,500 after 10 years and 51,250 after the eleventh year: CAGR = (51250/200)^(1/11) – 1 = 65.57%, MaxDD = 50%.

If one rebalances regularly (ever year, every 5 years, every x% increase) to keep the level of leverage significant, we would expect to keep a similar impact on CAGR but a stronger impact on MaxDD.
In the chart above, we would expect the 100% leverage curve to move towards the right (i.e. higher drawdowns) and the -50% one to move towards the left.

A Couple of Points

• It does not make sense to leverage a strategy using position size past its “optimal bet size” (how to evaluate it is probably the topic for another post a book like Vince’s)
• It does not make sense to de-lever using notional funding as it will mainly reduce CAGR but not the drawdown
• If using notional funding, re-balancing would affect MaxDD

This is just an intro point on how money management is an essential part of any system…
 
 
Note: that the figures are dependent on the number of portfolio instruments, length of track record, length of breakout, etc. so do not attach too much importance on the actual figures. The main point is how these figures are relative to each other.

Looking over the drawdown cliffs

in this world nothing can be said to be certain, except death and taxes.

Arguing against Ben Franklin, I would say it is certain your (and my) Max Drawdown so far – whether in real trading or backtest – will be surpassed in the future. The maths even say that the probability of any drawdown in the future is 100% (although this might require a very long future, in a similar logic to the Infinite Monkey Theorem).

Dave Harding on Drawdown

I came across this paper (PDF download) from Winton Capital, co-authored by David Harding, which discusses the pros and cons of drawdown as a statistical measure, whether used to evaluate managers or trading systems.

The main points of the paper are:

• Unlike volatility, drawdown represents a physical reality: the magnitude of loss that an investor could have suffered – this is probably why it is a popular statistic to evaluate systems and funds performance/risk.

• Drawdown is not such a good indicator of quality for a system or manager – at least not as straightforward as usually assumed.

• A couple of charts illustrate how expected maximum drawdown increases with volatility, track record and reporting frequency. On the other hand, drawdown decreases as mean return increases. Nothing too surprising.

• Maximum drawdown is a a single number derived from a single string of data: it is going to have a large error associated with it. Any extrapolation of future performance will therefore be highly error-prone. Even with adjustments to equalise the volatility of track records, maximum drawdown is a poor statistic for making inferences about future reward/risk ratio or even future drawdown. Averaging worst drawdowns would be less error-prone, statistically speaking.

The paper concludes with:

Drawdown may have a role in manager risk control, but it should be used with caution, and should be calculated with reference to probability (95%, 99% confidence level) from the characteristics of the underlying process rather than purely from the historical track record.

Risk of Drawdown

A couple of volumes on my bookshelf discuss drawdowns and how to calculate their probability.

Balsara, in Money Management Strategies for Futures Traders, publishes tables of calculated risk of ruins based on different parameters.

However risk of ruin is different from risk of drawdown. Ruin is usually defined as a fixed capital level, representing a large percentage loss on initial capital. For example, a risk of ruin at 60% is the probability that your equity falls to 40% of your startuing capital. As the equity grows, the risk of hitting that “ruin threshold” decreases.

Risk of drawdown, on the other hand, stays constant regardless of how high the equity grows, because the drawdown “capital barrier” keeps moving up in line with the equity. Vince expands the concept of risk of ruin by modifying the calculation to derive the risk of drawdown (in The Handbook of Portfolio Mathematics)

Both risk of drawdown and risk of ruin increase as the track period or backtest length increases. However, the risk of drawdown tends to 100% as track period length increases, whereas the risk of ruin is bounded at a value determined by the characteristic of the trading system results (probability of win, payoff, trade risk, etc.).

Monte-Carlo simulation allows for estimating the risks of drawdown and ruin by iterating a random process governed by characteristics such as probability of win, payoff ratio, percentage of capital risked on each trade.

Risk of Ruin formula

Perry Kaufman, in New Trading Systems and Methods, presents a formula to calculate the risk of ruin. This is more convenient than having to run a Monte-Carlo simulation but it does not allow for calculating a risk of drawdown.

The formula is as follows:

Calculate your risk of Drawdown/Ruin

Below is a calculator that implements risk of ruin or risk of drawdown calculations based on the two methods described above (the risk of ruin is calculated from both a Monte-Carlo simulation and from the formula).

Just fill in the stats of the trading system, the test length and the level of drawdown/ruin to be tested and hit the Calculate button. Note that both calculated values can diverge significantly (as in the pre-populated example) if the number of periods is relatively low.

In case you find this tool useful, it has been added under the resources page. Disclaimer: I have not done full-proof 100% testing on it but playing with it seems to give good results.

Note that this method is probably not ideal as it only considers the average trade statistics and simulates sequential trading (whereas real-life systems will have multiple trades on the go at the same time). Also importantly, it completely ignores any time-dependence in the stream of results (such as auto-correlation, etc.).

The Trading Blox approach

Trading Blox runs Monte-Carlo simulations for any system backtested and produces some useful charts and information from it.

The Monte-Carlo simulation is different as it is applied to the daily equity curve returns – which is probably more realistic (although it still removes the time-dependence of the return stream).

One of the standard charts produced analyses drawdowns and attempts to give confidence levels as discussed in the Harding paper:

picture credits: WilWheaton@flickr

Several CTAs or fund managers offer a standard version of their fund, along with a leveraged version (called enhanced risk, 2x or 3x fund, etc.). However a simple performance comparison of the leveraged option against the standard option usually makes it obvious that the former does not offer a simple performance multiplier of latter.

I was looking at Conquest Capital Group and their MFS fund (based on the Trend Following benchmark used in their paper discussed here), which offers two such options: their standard fund and a 3x option.

The main fund, since inception (2004), returns 56% (net of fees) with a max drawdown of 14%. Their 3x leveraged option, over the same period, returns 85% with a max drawdown of 47%. The risk is indeed tripled, but the reward falls fairly short of it. When removing the fees on both funds (1% for the standard fund and 3% on the leveraged fund), the performances are 66% vs. 120%. This is just an example. Most CTAs are similar in that respect and it highlights some effects of leverage.

Leverage is not a magic bullet

A similar concept has become more mainstream with the recent apparition of leveraged ETFs and the realisation by some investors of the large volatility decay that they incur (when they are held on a long term-basis, i.e. anything longer than intra-day). Check the long-term charts of FAS and FAZ (3x leveraged bullish and bearish financial ETFs): the only long-term profitable trade would be to short both, to cash in on the volatility decay while being hedged (since Nov 08 they are down respectively 58% and 98%).

This is simply due to the design of these instruments, which aim to replicate and triple the daily returns of the tracked benchmark. This highlights a fact about leverage: multiplying the arithmetic returns will not result in the same multiplication of the geometric returns.

Leverage with Position Sizing

A typical way of leveraging a trading strategy is to increase the position size, and the resulting risk and return for each trade.

Assuming a fractional betting strategy for position sizing, the leverage dictated by the value of the fraction of capital risked on each trade will impact the overall average geometric return. Theories describing this phenomenon are the Kelly criterion and Ralph Vince’s Optimal f.

One of the main implications of this type of Money Management is that, for every strategy, there is an optimal fractional position size to maximize the (past) geometric return (and future if you assume that the past is a good representation of the future). Any fractional position size higher than this optimal value would result in a lower return.

Any strategy (with its resulting stream of returns) has an embedded maximum leverage, which can be defined with the optimal f.

Notional funding: another type of Leverage

Notional funding can be compared to trading with imaginary money. Typically, CTAs also offer notional funding: they trade your account based on an agreed notional amount, which is different from the actual funds in the account. For example, you could send 100k to your CTA and instruct them to trade it as if it were a 300k account. Or you could do the same thing with your own trading system.

One could intuitively think that this is exactly the same thing as tripling the position size but there is a difference.

Let’s consider this simple example with a strategy producing two consecutive trades of +10% and -5%:

The notional-funded account would close at 300 x 1.10 x 0.95 = 313.50. This would result in a gain of 13.50 on actual funds of 100: +13.50%, which is exactly triple the return one would get on a fully-funded, unleveraged account (100 x 1.10 x 0.95 = 104.50 for a 4.5% return)

Leveraging by tripling the position size on an account of 100 would result in two consecutive trades of +30% and -15%. Final balance would be: 100 x 1.30 x 0.85 = 110.50 – or a +10.50% return.

Initially, notional funding appears a better solution to the leverage issue as it does not suffer from the volatility decay or erosion of returns introduced by increasing position size.

Issues with Notional Funding: Cost

But funding always comes at a cost, pretty much in the same way as an overdraft gets charged by your bank. Another way to look at notional funding is to compare it to borrowing trading capital. The higher ratio of your notional account you borrow (higher leverage), the more the cost of borrowing will impede your trading returns.

In practice, you do not really borrow any funds (only by the power of imagination) but this practice still ends up impacting your return when you start taking into consideration the return on margin.

Impact of Margin Interest on Trend Following Returns

This paper from EDHEC Risk (PDF) looks at Trend Following/Managed Futures performance and presents an interesting breakdown of the overall return of such strategy. Assuming interest being paid at T-bill rates on the margin used for trading, it represents the bulk of the returns compared to actual Trend Following gains and rebalancing gains.

This is one of the reasons for the hidden cost of notional funding: a notional-funded account would enjoy the same Trend Following and rebalancing gains as a fully-funded account of the same size, but only a fraction of the return on margin (which is based on the actual account size).

To illustrate that point, below is a chart showing the impact of fees and margin interest on an arbitrary performance curve:

Notice the fairly sizable difference between taking return on margin into account or not (red v. blue curves). The green curve adds fees at 1% annually. The return on margin was assumed to be the T-bill rate for that month.

Risk and Margin Requirements

Another issue of Notional funding used as leverage are the constraints imposed by the underlying strategy.

Any strategy requires a minimum margin commitment to support the positions. Effectively the Margin-to-Equity ratio will dictate the maximum leverage one can use to trade a strategy.

The Drawdown is also a leverage-limiting factor: the more risk the strategy generates (higher drawdowns), the less leverage can be applied. If a strategy regularly exhibits drawdowns at the 40% level, there is a always the possibility that such drawdown appears at the beginning of trading the strategy. Based on the level of leverage, a notional-funded account might not have the time to build enough equity cushion to withstand the drawdown, which would result in the actual account funds going to 0 or a margin call.

Note that in this case, the sequence of trade returns has an impact on the outcome: if the strategy produces gains of 100% followed by a drawdown of 40% a 3x leverage via notional funding would still result in a gain of 60% (300 x 2 x 0.6 = 360, or a profit of 60 with actual funds of 100). If the 40% drawdown appears first, the notional account would shrink to 180, which would result in a negative actual account balance (trading would have to be stopped before then).

Leverage Comparisons

With these considerations and the benefit of hindsight, here is a comparison between the same arbitrary strategy as above (fees and interest included) traded with:

• no leverage
• 6x leverage using notional funding
• 6x leverage using position sizing

The notional funding appears to be the better option. However the position sizing leverage is independent of the order in the sequence of returns – as opposed to the notional funding leverage. If that nasty current drawdown (which is the largest one historically) had appeared at the beginning of trading and carried on further to reach 17%, the notional-funded account would have been wiped out.

The next chart highlights the impact of leverage on the difference between the two types of leverage for that specific strategy:

The target curve represents a simple multiplier (equal to the leverage factor) of the unleveraged performance (which, as we have shown is not attainable). The position sizing leverage clearly exhibits the behaviour discussed in Optimal f theories with the return breaking down past the optimal value.

Hopefully, this gives you a few ideas about how to work leverage in your trading strategies or when sending funds to CTAs. Taking the time to look into it has cleared up a few points for me.