With the AMH, Lo attempts to reconcile the Efficient Markets Hypothesis with behavioral models, which often seem to contradict each other. The AMH is based on “Darwinian” evolutionary principles:

Based on evolutionary principles, the Adaptive Markets Hypothesis implies that the degree of market efficiency is related to environmental factors characterizing market ecology such as the number of competitors in the market, the magnitude of profit opportunities available, and the adaptability of the market participants.

Behavioral biases are, in fact, consistent with an evolutionary model of individuals adapting to a changing environment via simple heuristics.

The EMH model can be seen as the equilibrium state in a perfect/ideal world where market efficiency runs at 100%. However, reality is often more complex than a simple theoretical model. Because of ever-changing factors, real market efficiency oscillates towards and away from the “perfect” EMH model, without necessarily converging towards it.

The framework that Lo describes is mostly qualitative and as such its applications might not be immediate.

The Behavioral Side

Lo covers some well-known behavioral biases to illustrate how “investor idiosyncrasies” contradict the EMH assumption: investors are not always rational.

Examples of behavioral biases used for illustration are:

• Over-confidence
• Probability assessment
• Risk aversion

These behavioral biases are presented as being investor heuristics:

Within this paradigm, behavioral biases are simply heuristics that have been taken out of context.

Given enough time and enough competitive forces, any counterproductive heuristic will be reshaped to better fit the current environment. The dynamics of natural selection and evolution yield a unifying set of principles from which all behavioral biases may be derived.

The EMH Response

The way EMH proponents address these questions raised by behavioralists is by affirming that markets as a whole gravitate towards efficiency. All “small” inefficiencies are arbitraged away and their effect counteract each other. The argument is that behavioral biases are negligible and irrelevant.

But this last conclusion relies on the assumption that market forces are sufficiently powerful to overcome any type of behavioral bias, or equivalently, that irrational beliefs are not so pervasive as to overwhelm the capacity of arbitrage capital dedicated to taking advantage of such irrationalities.

Lo then uses anecdotal evidence in the form of various classic financial manias and panics (tulip mania, South Sea Bubble, Dot-Com Crash, etc.) as examples that sometimes, forces of irrationality can dominate the forces of rationality.

A Look from the Neuroscience Angle

Behavioral finance falls into the field of psychology rather than economy and Lo takes a look at neuroscience to get a better understanding of behavioral biases.

EMH proponents sometimes criticize the behavioral literature as primarily observational, an intriguing collection of counterexamples without any unifying principles to explain their origins. To a large extent, this criticism is a reflection of the differences between economics and psychology.

The field of psychology has its roots in empirical observation, controlled experimentation, and clinical applications. From the psychological perspective, behavior is often the main object of study, and only after carefully controlled experimental measurements do psychologists attempt to make inferences about the origins of such behavior.

In contrast, economists typically derive behavior axiomatically from simple principles such as expected utility maximization, resulting in sharp predictions of economic behavior that are routinely refuted empirically.

In this section, Lo gives us (succinct) explanation of how the brain works and how it can be split in three parts: reptilian, mammalian and hominid brains, all traces of our evolutionary past. These “three” brains react to and manage situations differently. The way they interact when presented with “emotional distress” (fear and greed for example) is likely the root of behavioral biases: Emotion is at the heart of irrational decision.

The Hypothesis and its Implications

Lo’s theory falls in the “Darwinian alternatives to the EMH”, arguing that individual investors develop heuristics to solve various economic challenges, based on their experience. They learn by receiving positive or negative reinforcement from the outcomes. If the environment remains stable, heuristics will tend towards optimal solutions. However, with changing market environments, heuristics become unsuited and need to adapt: this is when behaviors can appear irrational.

This means:

1. Individuals act in their own self-interest.
2. Individuals make mistakes.
3. Individuals learn and adapt.
4. Competition drives adaptation and innovation.
5. Natural selection shapes market ecology.
6. Evolution determines market dynamics.

Any market will be more or less efficient depending on its ecology (ie number and variety of market participants, availability of profit opportunities). However, convergence to equilibrium is neither guaranteed nor likely to occur at any point in time (as per concepts of evolutionary biology).

Lo concludes with applications of the AMH, to try and render his model more practical.

• Investor’s preferences matter and need to be managed to meet their objectives.
• Risk/Reward relations are likely to evolve over time.
• Arbitrage opportunities do exist from time to time. The market does not necessarily tend to higher efficiency but is subject to more complex market dynamics with cycles and market trends.
• Specific investment strategies’ profitability evolves over time
• A better way to achieve a consistent level of expected returns is to adapt to changing market conditions

As Lo warned us in the introduction, this work is still at a level of qualitative framework of thoughts, and as such still needs to be developed if it wants to compete with the EMH theory – which, despite being flawed, provides more practical applications for an investor.

The main practical application from the above points hints at the idea of using regime switching to adapt a trading strategy to the changing market environment.

Following the previous post on backwardation, contango and crude oil, let’s look at how several factors can play a part in the overall trend following trading system performance.

I have previously referenced the study by EDHEC Risk, which shows the different sources of return of a Trend Following strategy.

The authors build the Mt. Lucas Management (MLM) index, which applies a 200-day moving average Trend Following strategy to a diversified range of 25 futures markets, rebalanced monthly. The performance of the index is separated in three periods and further broken down by the three individual sources of return identified by the authors:

1. T-Bill returns on marginable assets
2. Static returns from trend-following futures markets
3. Rebalancing gains

The chart below shows the representation of each component’s participation to the overall return:

You might find it surprising how relatively small the impact of the actual Trend Following strategy return is (I know I did…)

Further breakdown

The previous post highlighted that the performance of any Crude Oil futures contract was unable to match the performance of the spot price (despite a fairly high beta of 0.83). This is mostly because holding commodities via futures contracts expose you to the spot price beta move (as mostly anticipated) but also to the yield curve of that futures contract.

Over time, the impact of the backwardation or contango in the yield curve (roll yield) can create a large drift between the actual spot price performance and the equivalent futures contract.

This is very apparent in a simple “buy-and-hold” example (as in the Crude Oil case) but the same would apply to any trading strategy using futures instruments. Any futures trade could be broken down into two “components”:

• spot price beta move
• roll yield

And therefore the sum of all the trades resulting from a trading strategy (such as Trend Following) could fall into these two same categories, giving us now four overall components.

1. T-Bill returns on marginable assets
2. Returns from trend-following spot price beta moves
3. Roll yield
4. Rebalancing gains

Roll Yield: a misnomer?

Note that Roll yield is not aptly named; it implies that the act of rolling contracts yield a specific return. However, the return resulting from contango or backwardation is merely “crystallized” at the time of rolling contracts: the contango premium (backwardation discount) actually deflates over time gradually.

For example, think of buying a Crude Oil contract at the beginning of the month with a premium over spot of $1 per barrel (at $76). If the spot price does not change during the whole month, the price of the future contract simply converges to the spot price ($75) – the premium simply deflates. When selling the contract at the end of the month, you will have lost $1 without doing a contract roll and with the price having stayed stable.
 
 
Breaking down the actual components of a global strategy might be helpful in order to understanding the sources of returns and focus on a specific one to possibly improve it and enhance the overall system. For example, a more sophisticated approach to rolling contracts might enhance the overall roll yield produced by the trading strategy.

I have recently been browsing the great resource that is the SSRN (Social Science Research Network), looking for some Managed Futures/CTA/Trend Following papers.

Here is a list of some I have bookmarked:

Is Managed Futures an Asset Class? The Search for the Beta of Commodity Futures
Abstract:
We hypothesize that the classic arbitrage pricing theory contains circular logic, and as a consequence, its natural state is disequilibrium, not equilibrium. We extend this hypothesis to suggest that the term structure of the futures price curve, while indicative of a potential roll return benefit, in fact implies a complex series of roll yield permutations. Similarly, the hedging response function elicits a behavioral risk management mechanisms, and therefore, corroborates social reflexivity. Such models are inter-related and each reflects certain qualities and dynamics within the overall futures market paradigm.

With respect to managed futures, it is an observable materialization of behavioral finance, where risk, return, leverage and skill operate un-tethered from the anchor of an accurate representation of beta. In other words, it defies rational expectations equilibrium, the efficient market hypothesis and allied models – the CAPM, arbitrage pricing theory or otherwise – to single-handedly isolate a persistent source of return without that source eventually slipping away.

CTA/Managed Futures Strategy Benchmarks: Performance and Review
Abstract:
In this paper we provide: 1) a brief synopsis of the benefits of managed futures investment; 2) a short review of manager based CTA benchmark construction; and 3) an empirical analysis on the relative performance of various CTA benchmarks (non-investible manager based indices, investible manager based indices, and passive security based indices). In this analysis the various CTA indices are also compared on a zero risk (e.g., Treasury Bill), total risk (Sharpe Ratio), market factor risk (e.g., S&P 500) and strategy risk (e.g., passive futures based CTA index) and peer group basis (investible and noninvestible manager based indices). Lastly, for a selected set of CTAs with full data over the period of analysis an example of the use of various CTA benchmarks in determining excess peer group return, and zero risk, total risk, market risk or strategy (futures based) risk excess return is provided.

Fooling Some of the People All of the Time: The Inefficient Performance and Persistence of Commodity Trading Advisors
Abstract:
nvestors face significant barriers in evaluating the performance of hedge funds and commodity trading advisors (CTAs). The only available performance data comes from voluntary reporting to private companies. Funds have incentives to strategically report to these companies, causing these data sets to be severely biased. And, because hedge funds use nonlinear, state-dependent, leveraged strategies, it has proven difficult to determine whether they add value relative to benchmarks. We focus on commodity trading advisors, a subset of hedge funds, and show that during the period 1994-2007 CTA excess returns to investors (i.e., net of fees) averaged 85 basis points per annum over US T-bills, which is insignificantly different from zero. We estimate that CTAs on average earned gross excess returns (i.e., before fees) of 5.4%, which implies that funds captured most of their performance through charging fees. Yet, even before fees we find that CTAs display no alpha relative to simple futures strategies that are in the public domain. We argue that CTAs appear to persist as an asset class despite their poor performance, because they face no market discipline based on credible information. Our evidence suggests that investors’ experience of poor performance is not common knowledge.

Performance Persistence for Managed Futures
Abstract:
Past literature on managed futures funds has found little evidence that the top performing funds can be predicted. But, the past literature has used small datasets and methods which had little power to reject the null hypothesis of no performance persistence. The objective of this research is to: determine whether performance persists for managed futures advisors using large datasets and methods which have power to reject the null hypothesis.
We use data from public funds, private funds, and commodity trading advisors (CTAs). The analysis proceeds in four steps. First, a regression approach is used to determine whether after adjusting for changes in overall returns and differences in leverage that funds all have the same mean returns. Second, we use Monte Carlo methods to demonstrate that Elton, Gruber, and Rentzler’s methods have little power to reject false null hypotheses and will reject true null hypotheses too often. Third, we conduct an out-of-sample test of various methods of selecting the top funds. Fourth, since we do find some performance persistence, we seek to explain the sources of this performance persistence by using regressions of (a) returns against CTA characteristics, (b) return risk against CTA characteristics, (c) returns against lagged returns, and (d) changes in investment against lagged returns.
Performance persistence could exist due to either differences in cost or differences in the skill of the manager. Our results favor skill as the explanation since returns were positively correlated with cost. The performance persistence is statistically significant, but is small relative to the variation in the data (only 2-4% of the total variation). But, the performance persistence is large relative to the mean. Monte Carlo methods showed that the methods used in past research could often not reject false null hypotheses and would reject true null hypotheses too often.
Out-of-sample tests confirmed that there is some performance persistence, but it is small relative to the noise in the data. A return/risk measure showed more persistence than either of the return measures. Picking CTAs based on returns in the most recent year may even be worse than a strategy of randomly picking a CTA.

Managed Futures and Hedge Funds: A Match Made in Heaven
Abstract:
In this paper we study the possible role of managed futures in portfolios of stocks, bonds and hedge funds. We find that allocating to managed futures allow investors to achieve a very substantial degree of overall risk reduction at limited costs. Apart from their lower expected return, managed futures appear to be more effective diversifiers than hedge funds. Adding managed futures to a portfolio of stocks and bonds will reduce that portfolio’s standard deviation more and quicker than hedge funds will, and without the undesirable side-effects on skewness and kurtosis. Overall portfolio standard deviation can be reduced further by combining both hedge funds and managed futures with stocks and bonds. As long as at least 45-50% of the alternatives allocation is allocated to managed futures, this again will not have any negative side-effects on skewness and kurtosis.

A Quantitative Analysis of CTA Funds
Abstract:
Our research studies various properties of commodity trading advisors (CTAs) from a quantitative point of view. Our investigation is based on a commercial database of 549 funds and focuses on the period 1990 to present.

Firstly, CTAs’ return distributions are analyzed and strong evidence of non-normality is found, stressing the need for portfolio allocation techniques which take into account higher-order moments.

Secondly, relative persistence in return distribution parameters is studied. We find strong persistence for volatility, but fail to find significant persistence for average return or higher order moments.

Thirdly, we review the major benchmarks available to the industry and build new benchmarks from our dataset. This allows us to infer the magnitude of various biases. We study homogeneity of 2 CTA subsets, namely trend-followers and non-trend-followers, and study the diversification possibilities in a CTA portfolio.

In the second part of the study, we focus on linking CTAs returns with that of traditional assets. After showing that a buy and hold multi-factor linear model fails to explain CTAs returns, we point out the presence of option-like payoffs in CTAs return patterns. Trend-following CTAs exhibit straddle-like payoffs, while non-trend-followers’ return patterns that are reminiscient of a long call option.

Lastly, using simple trading algorithms based on moving averages, we propose a linear market model in which factors capture the dynamic nature of CTA managers’ strategies. Our model leads to significant improvements over the classical model. Notably, we show that our model is able to closely replicate a broad index of CTAs for long out-of-sample periods.
 
 
Have a great weekend!
 
 
^{Picture credits: rugosa rosa@flickr}

I recently found a 30-page paper introducing the ideas and principles of his Leverage Space Model. I thought reading it might be a good way to get back into Vince’s material.

Follows a summary of the paper (PDF download link), which is maths-less (only concepts and principles are discussed). It is a good introduction to Ralph Vince theories.

Vince’s Optimal f

This is what Vince is famous for. It is basically a way to determine trading quantity (aka leverage) using the probability distributions of the trade outcomes.

If f represents the fraction of capital to wager (risk) on each bet (trade), the optimal value is the one which optimises the geometric growth of the bankroll (account balance).

In his previous books, Vince has defined the formula to determine the optimal f. The first part of the paper discusses the optimal f concept and is a good introduction for the non-initiated (showing how over-betting on a game with positive expectancy can and will result in a loss).

Leverage Space Model promises

The optimal f section discusses a single-component approach whereas the Leverage Space Model deals with multiple components portfolio.

It is presented as an improvement on the Modern Portfolio Theory, briefly discussed. This is based on the following advantages:

1. Risk is defined as drawdown (instead of variance in the MPT)
2. The fallacy and danger of correlation is eliminated
3. Valid for any distributional form – Fat-tails are addressed
4. The model is all about Leverage, which is not addressed in the MPT model.

Return aspect

The model starts by building a multi-dimensional terrain, drawing the overall expected return, based on multiple combinations of components in the portfolio and their respective f-values.

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 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.

The Risk Aspect

So far, the model has only looked at returns. To introduce the risk component, you must determine your maximum allowed drawdown. This is a hard and fast rule: no combination should breach that limit.

Using a derivation of the risk of ruin, the model computes the risk of maximum drawdown for each set of f-values (for a specific timeline – as, in the long run, the risk of drawdown tends to 100%). If the risk of drawdown is too high, the specific f-values combination is ignored.

In practice, the initial terrain is truncated: by removing all points breaching the maximum drawdown threshold.

The terrain has been truncated: all areas deemed too risky, from a drawdown perspective, have been removed.

The Algorithm

Vince implements a genetic algorithm to calculate the terrain, by initially calculating the expected return for each set of f-values, and secondly by running the maximum drawdown test on this same set. Once the whole set combination has been run through, the terrain is built (including truncations). The the aim is then to find the optimal set (highest return with lowest f-values).

Comments and Extra Info

The paper is rather short and does not deal with any of the maths behind the models. For this you’d have to get yourself a copy of the Handbook of Portfolio Mathematics which introduces the model in more detail or Vince’s latest book, dedicated to the Leverage Space Model, which has had a not-so-positive review by Max Dama.

The ideas in the paper are an interesting take on position sizing. Vince uses a simple objective/bliss function (CAGR with a binary risk/drawdown filter) to evaluate all possible scenarios of portfolio allocation/leverage. It might be interesting to use the concepts of the model with your own bliss function recipe.

One of Vince’s claim that the MPT does not address leverage sounds a bit simplistic – surely the percentage of Cash as an asset in the portfolio is an implicit measure of leverage. On the other hand, the approach on correlation/joint probability of scenarios sounds interesting and seems to go in the right direction. As Vince says:

Counting on correlation fails you when you need it the most.

Another point that seems missed out is how the model handles non-stationarity of the market. Vince mentions the chronomorphism of market prices distributions (i.e. they change with time) and even draws a betting comparison with blackjack – in which the optimal f curve changes for each card dealt. However there is no mention of how the model takes an adaptive approach to these chronomorphic distributions.

Vince’s homepage contains a link to the java software that implements his model (needs to register/leave email to download) and another one with a spreadsheet example. I have not had time to take a serious look at all those. Please let me know your feedback if you do.

Joshua Ulrich – blogger and reader of this blog (hello there: finally got round to adding you to the blogroll!) – is collaborating with Ralph Vince to port the Leverage Space Model to the R platform. His FOSS Trading blog is definitely worth a read too.

A research paper (summarised below) was recently published by AQR, explaining some concepts of trend following.

Clifford Asness, AQR Managing & Founding Principal, was also invited to speak about it with his good friends at CNBC (he is also an ex-Goldman, so he surely has lots of connections with the media and government). The video is not that interesting but here it is below, anyway. If you’re short of time (aren’t we all?), I recommend you skip to the paper (8 pages) or the summary, which yield more interesting insights.

Summary of Paper: Understanding Managed Futures

Click to download paper

- They start with a chart displaying the Managed Futures “smile”, basically a scatter plot of Trend Following strategy return vs. S&P 500 total return – making the point that Trend following performs best in case of extreme stock market moves (one of the main points they want to drive home throughout the paper is that Trend Following is a great portfolio diversificator with low correlation to other assets). They further empirically demonstrate this by pointing out Trend Following’s performance in Q4 2008 (strongly positive) in contrast to the global crash.

- The paper uses a hypothetical Trend following strategy for comparison and analysis. This strategy trades 60 liquid futures markets divided in the four asset classes defined by AQR (equities, commodities, bonds and currencies). To determine the trend, the strategy considers the excess return over cash of each instrument for the prior 12 months (a positive return results in a long position and a negative return results in a short position). The portfolio is equal-risk-weighted (i.e. normalised for annualised volatility) across the instruments and rebalanced every month.

- The second part breaks down the three parts of a trend and some behavioral biases or technical explanations for them:

• Start of the trend and under-reaction, due to anchor and insufficient adjustment to new conditions (i.e. news, supply shock, etc.), disposition effect (selling winners too early) and market particpants fighting trends (central banks or commodity hedgers)
• Trend continuation and over-reaction, due to herding and feedback trading as well as confirmation bias (similar to the reflexivity concept explained by George Soros in The Alchemy of Finance) and Risk Management practices (stop losses being triggered generate more losses, etc.)
• End of the trend where the market comes to the realisation that prices have gone too far

- The analysis of the strategy looks at the performance of each market and compares it to the overall strategy performance – noting the effect of the free-lunch that is diversification: The Sharpe ratio of the overall strategy is 1.3, higher than any of the individual market Sharpe ratio (all between 0 and 1).

- Another observation is the low correlation between the individual markets (average pair-wise correlation of 0.08) as well as between the overall strategy and various asset classes (Equities, Bonds and Commodities).

- Finally, they compare a 60/40 portfolio performance (60% Equities, 40% Bonds) with a hybrid portfolio (80% 60/40 portfolio and 20% Managed Futures) and show that return, standard deviation, Sharpe ratio, worst month and worst drawdown are all improved under the second scenario. I believe this is how they intend to market their new trend following funds: as a portfolio diversificator improving its overall variance-adjusted return

- In conclusion they highlight some of the risks (range-bound periods, high turnover and trading costs as well as manager fees) and finally (it is a marketing paper after all!) some of the value-add that a fund like theirs can provide (advanced strategies using rigorous quantitative methods over different time horizons, sophisticated risk management systems, portfolio optimisation and smart order execution algorithms, etc.)

The trend following space is just getting a bit more crowded…

Following the article, a reader of the blog (Alex) kindly forwarded me a research paper trying to identify which moments of a distribution (mean/drift, variance, skew, kurtosis) affect the returns of a trend-following strategy. This is an interesting read (10 pages, not too mathematically challenging) which I encourage you to read:

Click to download paper

Summary of the paper

The authors are mostly interested in currencies and in order to free themselves of historical data limitation, they generate artificial price data to simulate different types of price distributions by varying the different underlying moments.

They then apply a standard Triple moving Average Trend Following system to the different time series generated and measure the annualized gross profit for the simulation (over 5,000 trading days) for each type of distribution.

The other parameter measured is the Trading Frequency, from which they derive the auto-correlation characteristic of the underlying data (by applying the logic that a trend following system will trade in and out more frequently in a mean-reverting environment and vice-versa).

Conclusions

By applying some regression analysis to the various results observed, the authors arrive to the equation predicting the return of thee Trend following system:

TMA Result = 38.88Stdev(1 – 6.77TFrq + 0.0392Skew – 0.010Kurtosis + Drift(65.65 + 324,600Drift))

with a standard estimation error of 0.3%

The interpretations are that:

Market volatility (38.88 Stdev) determines the profit (or loss) potential of the trend-following strategy. This relationship is direct, so if market volatility doubles,so does the expected TMA result. Accordingly, it is no longer surprising that trend-following models tend to show the best results across the major currency blocks with high market volatility.

A high Tfreq will have a negative impact on trend model performance.

Skew will enhance performance, while the opposite is true for kurtosis. Drift will increase the value of the equation and thereby contribute positively to the TMA model result.

So it appears that the kurtosis (the source of fat-tails) actually has a negative effect on a trend following model (contrary to that earlier post) and in a relatively large way:

The currency path (auto-correlation/trading frequency) is the most important factor in determining performance (91%). The impact from kurtosis (68%) and drift (56%) is also significant. Skewness is less significant, but still explains 26% of variance on its own. Volatility has no importance at all (0.4%). This might initially come as a surprise, but as illustrated in Equation, it is a multiplication variable and so does not in itself generate trend model profitability (or loss where the path characteristic is unfavorable).

Comments

Mathematical theory is not my strongest suit (despite studying over 10 hrs of Maths per week in my prime!) and I am definitely thinking of getting some refresher training on that. But there are a few points that bother me in that research paper. To the more knowledgeable readers: “Please chime in and tell me where I might be wrong:

Simulated Data
This tends to make me a bit sceptical of the results especially with the authors’ random walk and efficient markets hypothesis (EMH) assumptions. After reading Taleb and especially Mandelbrot (I really enjoyed reading The (mis)behavior of Markets with its debunking of the EMH and alternative explanation of financial markets), I have been converted to the Dark Side of the force: I am not sure we really know how to modelise the data theoritically and this random simulation might be missing specific price data characteristics (granted, the authors empirical verification seems to confirm the assumptions…).

Measure of auto-correlation by proxy
The auto-correlation of the underlying data is actually derived from the trading frequency of the trading system itself. This does not sound right.
Whereas it seems intuitive that the more a trend following strategy trades in and out the least profitable, I am not sure this necessarily and direclty implies auto-correlation in the underlying data.

Only one model tested
Testing the performance of trend following by using one model only might not bee so representative. It would be interesting to see how the results translate to a collection of trend following systems (different styles, different timeframes, additional markets, etc.)

In any case, the finding that kurtosis has a negative effect on trend following returns seems to contradict my earlier post. As per the points above I’ll take the paper’s findings with a pinch of salt – but it might be a useful tool in determining when or not to use trend following strategy (by measuring the characteristics of the price distribution).

Whether trading using these market regimes identification is a valid and robust approach is a different question…