Cursed by Randomness

25 February 2015 / Dr Ewan Kirk


For all investors there has always been a strong desire to “pick winners” or beat the market. In the presence of randomness this can be extremely challenging.
Investors are driven, whether they be high net worth individuals or institutional pension funds, to focus a great deal on the trades, positions or underlying investment thesis of a potential investment. Quite naturally, this results in analysis of the past performance of a manager or similar trades in the hope that this will dimly illuminate the uncertain future.
Given our focus on systematic and statistical analysis, we would be the last people to be dismissive of analysis of historical data. After all, it is our "thing”. However, the tools of historical analysis are frequently used by market participants without a clear understanding of the limitations of the data set. And sometimes it is those who think they are the most sophisticated and clever statisticians who are prone to these errors – perhaps more so than investors who take a strong dose of common sense with their porridge in the morning.
The limitations of data and analysis rear their heads in a variety of situations. Investors want to answer questions such as:
…and so on. An investor’s life is full of difficult questions and, given there is so much data around, it is natural to seek answers to these through statistical analysis.
All of the questions above (and many others that are asked by managers, investors and the media) boil down to one question: "What is going to happen in the future?". It is a clichéd old chestnut but Niels Bohr was right: “prediction is very difficult, especially about the future”.
Furthermore, few of the questions that investors seek to answer have simple binary answers. The answers that managers give need to be (and should be) hedged with phrases such as “on balance…”, “it is likely that…” or “over the long term…”.
Outside finance, this is not unusual in the real world. Smoking cigarettes has a relatively low probability of causing lung cancer (about 1 in 5) but a very high certainty that the life time risk increases if you smoke (from 1 in 100 to 1 in 5). The LHC spent two years collecting data to be 99.99997% certain that the Higgs Boson exists. But there is still a small chance it doesn’t exist. There are some things which truly are binary: “Can I beat Sir Bradley Wiggins in a bike race?”(footnote) but in reality many of the answers to difficult real world questions are really statements about relative probabilities.
It’s therefore surprising(footnote) that, despite randomness, uncertainty and error bars in finance being larger than in almost any other field of human endeavour, there are such high expectations of generating insight from superficial analysis. There is a widespread belief that a transparent window into the future is opened through statistically suspect analysis of the performance of existing investments.
I’m sure we will return to this theme in future pieces but at the moment we’re going to discuss one particular sub-issue: the desire investors have to deeply understand the constituents of their managers’ portfolios. Investors want to have position-level transparency to aid this understanding. Occasionally this can be a source of friction between managers and investors. In some very specific cases (distressed debt or other illiquid investments, or some forms of activist investing) telling your investors what your positions are could be an issue. In most cases, however, investors will be given information about positions and trades and they want to know more about what happened to these positions. This is entirely justified – it is, after all, their money.
But how much can an investor (or indeed the manager themselves) glean from this information about positions, trades and what they have done in the past? Sadly, the answer is “not much”.
Those of you who have met me probably realise that I have a penchant for “toy models”. The financial world is a hugely complex place with an uncountable number of factors which could affect the solution to a problem or the answer to a question. Simplified models can often allow us to gain insight into a complex process.
This is not unlike the situation facing physicists. The real world is complex. Newton wouldn’t have made much progress attempting to explain gravity by trying to model the motions of every single molecule inside the apple, the effect of air resistance on the falling apple, the Coriolis effect from the Earth's rotation and the minute general relativistic effects where time appears to slow for the apple as it drops into the Earth's gravitational well.(footnote) No, Newton considered two idealised point masses and came up with this equation:
$$ F=G \frac{m_1 m_2}{r^2} $$
Newton’s toy model was so good that it was 229 years before Einstein published a better one.
I’m always keen to come up with simple toy models which explain (or rather “illuminate”) a problem so let’s look at the problem of understanding the returns of managers by analysing the returns of their individual assets.
Let’s imagine that we have a set of assets. In the case of a managed futures or CTA style strategy, this could range from say 70 to 250 futures contracts and foreign exchange forwards. For a systematic equities stat arb fund, it could be many thousands of individual stocks. But in our case, we are going to assume that our simple manager has 100 assets, because it’s a nice round number and it makes arithmetic easy.
Let’s also assume our manager truly has a Sharpe ratio of 1.0 after fees. Whilst there are managers who have consistently outperformed this, it’s fair to assume that this is a reasonable ex-ante expectation. Despite the well-known (and often overblown) concerns regarding using Sharpe ratio as a measure, it is going to work fairly well for this simple example. As we all know, volatility is something you choose and if we’ve fixed the Sharpe Ratio, we can choose a volatility at will.
The maths gets even easier if you choose a “unit volatility”(footnote). However there is something different about the statement “the asset will make a 100% return with a 100% risk” and the statement “the asset will make 10% return with a 10% risk”(footnote) so let’s choose a 10% volatility.
We can summarise our Toy CTA as a manager with 100 assets, targeting a 10% annualised volatility and it should make 10% per annum over the long run(footnote).
How can we characterise this manager’s positions? Well, sadly for us all 100 assets doesn’t mean 100 uncorrelated assets. For example, the US 30 Year contract looks an awful lot like a less liquid, higher vol version of the 10 year contract. There is some very clever and sophisticated analysis that one can do of portfolios to identify how many truly uncorrelated assets there are in a real world portfolio but (since we’re in The Land Of Toy Models) let’s assume that although the manager has 100 assets, this really corresponds to 10 truly independent assets(footnote).
Assuming our manager has positions in these 10 underlying assets (or “eigen-assets” to coin a phrase), what is the average expected return and risk of each of these eigen-assets if our manager is going to have a 10% per annum return with a 10% per annum volatility? Some more simple statistics (or simulation in Excel) will give us the result that the 10 eigen-assets will have an average return of 1% per annum and each one will have a risk(footnote) of 3.3%; equivalently, we can say that each eigen-asset will have a Sharpe ratio of 0.33. In any individual year, approximately four of our eigen-assets will lose money and six will make money(footnote): All because of random noise.
In the simulation below, we see 10 assets each of which has a Sharpe ratio of 0.33 being simulated for a year. Each month's return is a draw from a random process with a mean of $\frac{0.033}{12}$ and a standard deviation of $\frac{0.10}{\sqrt{12}}$. The sum of these 10 uncorrelated assets will have a Sharpe ratio of 1.0 over a long period of time and despite the performance of each asset being largely uninformative, the sum of the returns of the individual assets would be an excellent investment.

Therefore, whether or not a position made or lost money last year is entirely uninformative (and please don’t get me started on the information content of returns from shorter periods of time like months or weeks!): mostly it’s just noise. Indeed, in our toy example, every asset has the same probability of making money the following year.
Unsurprisingly, outside the Land of Toy Models, it becomes a lot more difficult. Each eigen-asset is a combination of imperfectly correlated assets with their own distributions. The distributions aren't stationary and positions change either because the signal changes or the weighting given to the position changes. For more sophisticated investment processes, trades may be relative value or hedges may be employed to remove or reduce unwanted risks. The hedges may lose a lot of money but if the desired position makes more money then the trade or idea was a good one. The ever present randomness and the exponentially increasing complexity soon makes it impossible to construct a simple narrative to explain the past. Furthermore, the whole point of explaining the past is to potentially forecast the future and if it's impossible to understand the past, there is absolutely no way of forecasting the future. The investor has no way of answering her fundamental question.
So why does everybody focus on which assets or securities or trades have performed well or performed badly? We should all be very wary of people who think they have the answer to these sorts of questions, but... here’s the answer. It is the “story bias”. As humans, we have a strong bias to understanding the world through stories. Stories are visceral: they get lodged in our basic, ancient, reptilian brains. We overweight the relevance of stories. And when trying to explain complicated and sometimes random events, stories make them comprehensible, if not any more explicable. Indeed, as humans, we find random or near random events particularly difficult to comprehend. One only has to think of the irrational reasoning which has often been employed through the ages to explain gambling and the weather, both of which are largely driven by random factors.
Finance is particularly cursed with randomness and our natural bias to construct stories causes us to want to make up stories about which assets have “worked” and which haven’t “worked”. Then we can construct stories about why they have or haven”t worked.
What is more dangerous is that we believe that these stories inform or illuminate what might happen in the future. As scientific and statistical investors, we are also influenced by this story bias (although in a more subtle way) just as discretionary traders are.
What should we do about this? Once again, you should always be wary of anybody who tells you that they have a solution to a difficult problem, but... here are some potential solutions. This is how we try to understand the curse of randomness and reduce the effects of the story bias.
Random noise is not a great story. It”s hard to imagine headlines in the press about the "market action" yesterday being mostly random noise and that watching TV, reading newspapers or superficially analysing data to gain insight into the future is probably futile. Many public figures appear to believe that they have insight and therefore investors feel that by obsessively analysing the details the insight will appear. Yet, attempting to assign too much meaning to all this noise can be unhelpful or, worse, misleading(footnote).
Truly the financial world is cursed by randomness.


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