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]]>Financial Engineering has long been a staple application of Mathematica, an area in which is capabilities in symbolic logic stand out. But earlier versions of the software lacked the ability to work with Ito calculus and model stochastic processes, leaving the user to fill in the gaps by other means. All that changed in version 9 and it is now possible to provide the complete framework of modern financial theory within the context of the Wolfram Language.

The advantages of this approach are considerable. The capabilities of the language make it easy to provide interactive examples to illustrate theoretical concepts and develop the student’s understanding of them through experimentation. Furthermore, the student is not limited merely to learning and applying complex formulae for derivative pricing and risk, but can fairly easily derive the results for themselves. As a consequence, a course in stochastic calculus taught using Mathematica can be broader in scope and go deeper into the theory than is typically the case, while at the same time reinforcing understanding and learning by practical example and experimentation.

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]]>The post Applications of Graph Theory In Finance appeared first on QUANTITATIVE RESEARCH AND TRADING.

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In this example I have selected a universe of the Dow 30 stocks, together with a sample of commodities and bonds and compiled a database of daily returns over the period from Jan 2012 to Dec 2013. If we want to look at how the assets are correlated, one way is to created an adjacency graph that maps the interrelations between assets that are correlated at some specified level (0.5 of higher, in this illustration).

Obviously the choice of correlation threshold is somewhat arbitrary, and it is easy to evaluate the results dynamically, across a wide range of different threshold parameters, say in the range from 0.3 to 0.75:

The choice of parameter (and time frame) may be dependent on the purpose of the analysis: to construct a portfolio we might select a lower threshold value; but if the purpose is to identify pairs for possible statistical arbitrage strategies, one will typically be looking for much higher levels of correlation.

Reverting to the original graph, there is a core group of highly inter-correlated stocks that we can easily identify more clearly using the Mathematica function FindClique to specify graph nodes that have multiple connections:

We might, for example, explore the relative performance of members of this sub-group over time and perhaps investigate the question as to whether relative out-performance or under-performance is likely to persist, or, given the correlation characteristics of this group, reverse over time to give a mean-reversion effect.

An obvious application might be to construct a replicating portfolio comprising this equally-weighted sub-group of stocks, and explore how well it tracks the Dow index over time (here I am using the DIA ETF as a proxy for the index, for the sake of convenience):

The correlation between the Dow index (DIA ETF) and the portfolio remains strong (around 0.91) throughout the out-of-sample period from 2014-2016, although the performance of the portfolio is distinctly weaker than that of the index ETF after the early part of 2014:

Another application might be to construct robust portfolios of lower-correlated assets. Here for example we use the graph to identify independent vertices that have very few correlated relationships (designated using the star symbol in the graph below). We can then create an equally weighted portfolio comprising the assets with the lowest correlations and compare its performance against that of the Dow Index.

The new portfolio underperforms the index during 2014, but with lower volatility and average drawdown.

Graph theory clearly has a great many potential applications in finance. It is especially useful as a means of providing a graphical summary of data sets involving a large number of complex interrelationships, which is at the heart of portfolio theory and index replication. Another useful application would be to identify and evaluate correlation and cointegration relationships between pairs or small portfolios of stocks, as they evolve over time, in the context of statistical arbitrage.

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]]>The post Metal Logic appeared first on QUANTITATIVE RESEARCH AND TRADING.

]]>Precious metals have been in free-fall for several years, as a consequence of the Fed’s actions to stimulate the economy that have also had the effect of goosing the equity and fixed income markets. All that changed towards the end of 2015, as the Fed moved to a tightening posture. So far, 2016 has been a banner year for metal, with spot prices for platinum, gold and silver up 26%, 28% and 44% respectively.

So what are the prospects for metals through the end of the year? We take a shot at predicting the outcome, from a quantitative perspective.

Source: Wolfram Alpha. Spot silver prices are scaled x100

One of the key characteristics of metals is the very high levels of price-correlation between them. Over the period under investigation, Jan 2012 to Aug 2016, the estimated correlation coefficients are as follows:

A plot of the join density of spot gold and silver prices indicates low- and high-price regimes in which the metals display similar levels of linear correlation.

Levels of correlation that are consistently as high as this over extended periods of time are fairly unusual in financial markets and this presents a potential trading opportunity. One common approach is to use the ratios of metal prices as a trading signal. However, taking the ratio of gold to silver spot prices as an example, a plot of the series demonstrates that it is highly unstable and susceptible to long term trends.

A more formal statistical test fails to reject the null hypothesis of a unit root. In simple terms, this means we cannot reliably distinguish between the gold/silver price ratio and a random walk.

Along similar lines, we might consider the difference in log prices of the series. If this proved to be stationary then the log-price series would be cointegrated order 1 and we could build a standard pairs trading model to buy or sell the spread when prices become too far unaligned. However, we find once again that the log-price difference can wander arbitrarily far from its mean, and we are unable to reject the null hypothesis that the series contains a unit root.

We can hope to do better with a standard linear model, regressing spot silver prices against spot gold prices. The fit of the best linear model is very good, with an R-sq of over 96%:

A trader might look to exploit the correlation relationship by selling silver when its market price is greater than the value estimated by the model (and buying when the model price exceeds the market price). Typically the spread is bought or sold when the log-price differential exceeds a threshold level that is set at twice the standard deviation of the price-difference series. The threshold levels derive from the assumption of Normality, which in fact does not apply here, as we can see from an examination of the residuals of the linear model:

Given the evident lack of fit, especially in the left tail of the distribution, it is unsurprising that all of the formal statistical tests for Normality easily reject the null hypothesis:

However, Normality, or the lack of it, is not the issue here: one could just as easily set the 2.5% and 97.5% percentiles of the empirical distribution as trade entry points. The real problem with the linear model is that it fails to take into account the time dependency in the price series. An examination of the residual autocorrelations reveals significant patterning, indicating that the model tends to under-or over-estimate the spot price of silver for long periods of time:

As the following chart shows, the cumulative difference between model and market prices can become very large indeed. A trader risks going bust waiting for the market to revert to model prices.

How does one remedy this? The shortcoming of the simple linear model is that, while it captures the interdependency between the price series very well, it fails to factor in the time dependency of the series. What is required is a model that will account for both features.

Rather than modeling the metal prices individually, or in pairs, we instead adopt a multivariate vector autoregression approach, modeling all three spot price processes together. The essence of the idea is that spot prices in each metal may be influenced, not only by historical values of the series, but also potentially by current and lagged prices of the other two metals.

Before proceeding we divide the data into two parts: an in-sample data set comprising data from 2012 to the end of 2015 and an out-of-sample period running from Jan-Aug 2016, which we use for model testing purposes. In what follows, I make the simplifying assumption that a vector autoregressive moving average process of order (1, 1) will suffice for modeling purposes, although in practice one would go through a procedure to test a wide spectrum of possible models incorporating moving average and autoregressive terms of varying dimensions.

In any event, our simplified VAR model is estimated as follows:

The chart below combines the actual, in-sample data from 2012-2015, together with the out-of-sample forecasts for each spot metal from January 2016.

It is clear that the model projects a recovery in spot metal prices from the end of 2015. How did the forecasts turn out? In the chart below we compare the actual spot prices with the model forecasts, over the period from Jan to Aug 2016.

The actual and forecast percentage change in the spot metal prices over the out-of-sample period are as follows:

The VAR model does a good job of forecasting the strong upward trend in metal prices over the first eight months of 2016. It performs exceptionally well in its forecast of gold prices, although its forecasts for silver and platinum are somewhat over-optimistic. Nevertheless, investors would have made money taking a long position in any of the metals on the basis of the model projections.

Another way to apply the model would be to implement a relative value trade, based on the model’s forecast that silver would outperform gold and platinum. Indeed, despite the model’s forecast of silver prices turning out to be over-optimistic, a relative value trade in silver vs. gold or platinum would have performed well: silver gained 44% in the period form Jan-Aug 2016, compared to only 26% for gold and 28% for platinum. A relative value trade entailing a purchase of silver and simultaneous sale of gold or platinum would have produced a gross return of 17% and 15% respectively.

A second relative value trade indicated by the model forecasts, buying platinum and selling gold, would have turned out less successfully, producing a gross return of less than 2%. We will examine the reasons for this in the next section.

If we re-estimate the VAR model using all of the the available data through mid-Aug 2016 and project metal prices through the end of the year, the outcome is as follows:

While the positive trend in all three metals is forecast to continue, the new model (which incorporates the latest data) anticipates lower percentage rates of appreciation going forward:

Once again, the model predicts higher rates of appreciation for both silver and platinum relative to gold. So investors have the option to take a relative value trade, hedging a long position in silver or platinum with a short position in gold. While the forecasts for all three metals appear reasonable, the projections for platinum strike me as the least plausible.

The reason is that the major applications of platinum are industrial, most often as a catalyst: the metal is used as a catalytic converter in automobiles and in the chemical process of converting naphthas into higher-octane gasolines. Although gold is also used in some industrial applications, its demand is not so driven by industrial uses. Consequently, during periods of sustained economic stability and growth, the price of platinum tends to be as much as twice the price of gold, whereas during periods of economic uncertainty, the price of platinum tends to decrease due to reduced industrial demand, falling below the price of gold. Gold prices are more stable in slow economic times, as gold is considered a safe haven.

This is the most likely explanation of why the gold-platinum relative value trade has not worked out as expected hitherto and is perhaps unlikely to do so in the months ahead, as the slowdown in the global economy continues.

We have shown that simple models of the ratio or differential in the prices of precious metals are unlikely to provide a sound basis for forecasting or trading, due to non-stationarity and/or temporal dependencies in the residuals from such models.

On the other hand, a vector autoregression model that models all three price processes simultaneously, allowing both cross correlations and autocorrelations to be captured, performs extremely well in terms of forecast accuracy in out-of-sample tests over the period from Jan-Aug 2016.

Looking ahead over the remainder of the year, our updated VAR model predicts a continuation of the price appreciation, albeit at a slower rate, with silver and platinum expected to continue outpacing gold. There are reasons to doubt whether the appreciation of platinum relative to gold will materialize, however, due to falling industrial demand as the global economy cools.

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]]>The post Dynamic Time Warping appeared first on QUANTITATIVE RESEARCH AND TRADING.

]]>You certainly wouldn’t know it from a reading of the CBOE S&P500 Volatility Index (CBOE:VIX), which printed a low of 11.44 on Friday, but there is a great deal of uncertainty about the prospects for the market as we move further into the third quarter, traditionally the most challenging period of the year. Reasons for concern are not hard to fathom, with the Fed on hold and poised to start raising rates, despite anemic growth in the economy; gloom over the “earnings recession”; and an abundance of political risk factors in play, not least of which is the upcoming presidential election.

At times like these investors need a little encouragement to stay the course – and where better to look for it than in the history books. More specifically, the question is whether the past has anything to teach us about the prospects for the market, going forward. Academic theory says no; but Wall Street traders controlling trillions of dollars of investments believe that, on the contrary, history contains valuable information that can be helpful in predicting the likely future outcome for the market.

There are several difficulties in making historical comparisons. Firstly, while it maybe intuitively obvious what one means when speaking about a “comparable” historical period, it is not as easy as one might think to translate that concept into mathematical form. And it does need to be translated, because unless you plan to study market history comprising tens or hundreds of thousands of historical patterns by hand (eye), you are going to have to rely on computers to do the heavy lifting.

Perhaps the most well-known measure of “similarity” is correlation; but this is entirely unsuited to the task at hand. The reason is that correlation removes the very thing that traders are most interested in: the trend, or drift. Let me illustrate the point with a simple example. The following chart illustrates two, perfectly correlated series (technically, Wiener processes) that simulate the evolution of returns in two stocks.

Source: Wolfram Research

Even though the processes are perfectly correlated, they end up producing very different outcomes, chiefly because the red process has a strongly positive drift, or trend, while the green process has a strongly negative drift. It is highly implausible that the path realized by the green process would be regarded as similar to that of the red process, despite their correlation. The moral is simple: whatever else we mean by “similar”, it must entail a comparable result in terms of the overall appreciation (or depreciation) of the stock; i.e. the two periods must show a similar degree of trending behavior.

Now let’s consider a second illustrative example. Here, not only are the two processes perfectly correlated, they also have an identical drift (14%).

Source: Wolfram Research

Despite their commonalities, it is again very unlikely that we would regard the behavior of the green process as being sufficiently “similar” to that of the red process. Here the problem is not the difference in drift, but in volatility, which is ten times higher for the green process than for the red process. Even though both processes are expected to appreciate at the same average rate, the high volatility of the green process means that it may appreciate at a much faster, or slower, rate than the mean. We say that the volatility “drowns out” the drift in the process, at least over the short term.

So, in conclusion, what we are looking for in our quest for “similarity” is a period in which the process has displayed, not only a similar trend, but also a comparable level of volatility.

The challenges don’t end there, however. Let’s say we find a period in history in which the overall appreciation and volatility of the process is “similar” to that of the period of interest, but the historical period is shorter, or longer, than the period we are evaluating. Does that matter? Well to some degree, it must: if during the historical period the stock appreciated by 50% in 3 days, vs the same percentage over a term of 3 months during the current period, it is again unlikely that we would regard the historical period as similar. On the other hand, if the two periods differed in length by, say, only 4-5 days out of a total of 30, would that be enough to make you discard the historical precedent? Probably not, other things being equal. So there has to be scope for some, but not unlimited, flexibility in making historical comparisons using periods of slightly differing durations.

Having presented the problem, let me now turn to the solution.

Dynamic Time Warping (DTW) is a machine learning algorithm originally developed for speech recognition that aims two align two sequences by warping the time axis iteratively until a match is found between the two sequences.

Consider two sequences

The two sequences can be arranged on the sides of a grid, with one on the top and the other up the left hand side. Both sequences start on the bottom left of the grid.

Inside each cell a distance measure is used to compare the corresponding elements of the two sequences. To find the best match or alignment between these two sequences one needs to find a path through the grid which minimizes the total distance between them. The procedure for computing this overall distance involves finding all possible routes through the grid and computing the overall distance for each one. The overall distance is the minimum of the sum of the distances between the individual elements on the path divided by the sum of the weighting function.

A great many measures of “distance” (the inverse of “similarity”) have been developed over the years, in fields as disparate as image analysis, signal processing and machine learning. In what follows we shall be using Euclidean Distance as our preferred distance metric, but there are a large number of alternative options:

The DTW algorithm optimizes the distance, constrained by limits on acceptable paths through the grid:

: the path will not turn back on itself, both the__Monotonic condition__*i*and*j*indexes either stay the same or increase, they never decrease.: the path advances one step at a time. Both__Continuity condition__*i*and*j*can only increase by at most 1 on each step along the path.: the path starts at the bottom left and ends at the top right.__Boundary condition__: a good path is unlikely to wander very far from the diagonal. The distance that the path is allowed to wander is the window width.__Warping window condition__: The path should not be too steep or too shallow. This prevents short sequences matching too long ones. The condition is expressed as a ratio__Slope constraint condition__*p/q*where*p*is the number of steps allowed in the same (horizontal or vertical) direction. After*p*steps in the same direction is not allowed to step further in the same direction before stepping at least*q*time in the diagonal direction.

These constraints allow to restrict the moves that can be made from any point in the path and so limit the number of paths that need to be considered. The power of the DTW algorithm lies in that instead finding all possible routes through the grid which satisfy the above conditions, it works by keeping track of the cost of the best path to each point in the grid. During the calculation process of the DTW grid it is not known which path is minimum overall distance path, but this can be traced back when the end point is reached.

Let’s apply the DTW approach to find a similar period in the history of the SPDR S&P 500 ETF (NYSEArca:SPY).

We download data for the SPY ETF and look for periods in the history of the series from 1995 that are similar to how the process has evolved in Q3 2016. Note that we are using log-returns rather than raw prices. This is because what we care about is that the rate at which the ETF has appreciated relative to its initial starting value should be comparable for both periods.

The algorithm picks out a 21-day period from 3/6/1995 to 4/7/1995 as being the most similar to the way the SPY ETF has progressed in Q3 2016. Let’s take a look at the two periods and assess their similarity. Note that we set the initial starting price at 100 in each case.

Source: Author

It is clear that the SPY appreciates by approximately the same percentage over the two periods, but the way in which it evolves looks rather different in each case. That’s because the earlier period covers only 25 business days, compared to the 51 days from the start of Q3 2016 period. If you can imagine dilating the shorter time span of the orange line in the above chart to cover the same span as the blue line, the patterns look very similar indeed.

We are going to use the historical period to forecast how the ETF is likely to evolve of the next 35 days to the end of Q3. We do this by mapping the evolution of the ETF over the 35 days following the end of the historical period on 4/7/95 and projecting that pattern onto the end of the current period, from 8/12/16. The result is as follows.

Source: Author

In other words, if the closely parallel history of the series from 1995 were to repeat itself in 2016, we might expect a further gain of perhaps 3.5% to 4% in the SPY ETF between now and the end of September.

Whether you hold this approach as valid or not depends largely on how credible you regard the thesis that history is inclined to repeat itself. There are significant differences in fundamental factors that are likely to affect the outcome over the next several months, compared to the similar period in 1995.

The early part of 1995 marked the beginning of a powerful market surge driven by an economic boom that lasted until late 1999. By contrast, in the more recent period, the economy has been emerging very slowly from the long-running credit crisis commencing in 2008. Likewise, in contrast to 1995 when the Fed was at the early stage of reducing interest rates, the Fed is now more likely to start raising rates at some point before the end of 2016.

While these and other aspects of the economic landscape are fundamentally different, there are parallels. The recent slowdown in productivity, for example, often cited as a cause for concern, mirrors a similar slowdown in 1995. As Bloomberg points out, such pauses are not uncommon and in fact the plateaus in 1993-1996 and 2003-2005 occurred at a time when the economy was pretty healthy. Conversely, several of the biggest productivity jumps have happened during recessions, in 2001 and 2009.

Source: Bloomberg

Doom-mongering is at record levels these days, as the market continues to make new highs. The Wall Street Journal, for example, recently ran a piece entitled “This Tech Bubble Is Bursting”, drawing parallels with the dot-com bust of 2000.

They could be right.

On the other hand, if history repeats itself we could see the market 4% higher going into the final quarter of 2016.

Sakoe,H. and Chiba, S. Dynamic programming algorithm optimization for spoken word recognition. IEEE Trans. on Acoust., Speech, and Signal Process., ASSP 26, 43-49 (1978).

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]]>The post Machine Learning Trading Systems appeared first on QUANTITATIVE RESEARCH AND TRADING.

]]>In addition to the open/high/low and close prices of the preceding day session, we have selected a number of other plausible variables to build out the feature vector we are going to use in our machine learning model:

- The daily volume
- The previous day’s closing price
- The 200-day, 50-day and 10-day moving averages of the closing price
- The 252-day high and low prices of the SPY series

We will attempt to build a model that forecasts the overnight return in the ETF, i.e. [O(t+1)-C(t)] / C(t)

In this exercise we use daily data from the beginning of the SPY series up until the end of 2014 to build the model, which we will then test on out-of-sample data running from Jan 2015-Aug 2016. In a high frequency context a considerable amount of time would be spent evaluating, cleaning and normalizing the data. Here we face far fewer problems of that kind. Typically one would standardized the input data to equalize the influence of variables that may be measured on scales of very different orders of magnitude. But in this example all of the input variables, with the exception of volume, are measured on the same scale and so standardization is arguably unnecessary.

First, the in-sample data is loaded and used to create a training set of rules that map the feature vector to the variable of interest, the overnight return:

In Mathematica 10 Wolfram introduced a suite of machine learning algorithms that include regression, nearest neighbor, neural networks and random forests, together with functionality to evaluate and select the best performing machine learning technique. These facilities make it very straightfoward to create a classifier or prediction model using machine learning algorithms, such as this handwriting recognition example:

We create a predictive model on the SPY trainingset, allowing Mathematica to pick the best machine learning algorithm:

There are a number of options for the Predict function that can be used to control the feature selection, algorithm type, performance type and goal, rather than simply accepting the defaults, as we have done here:

Having built our machine learning model, we load the out-of-sample data from Jan 2015 to Aug 2016, and create a test set:

We next create a PredictionMeasurement object, using the Nearest Neighbor model , that can be used for further analysis:

There isn’t much dispersion in the model forecasts, which all have positive value. A common technique in such cases is to subtract the mean from each of the forecasts (and we may also standardize them by dividing by the standard deviation).

The scatterplot of actual vs. forecast overnight returns in SPY now looks like this:

There’s still an obvious lack of dispersion in the forecast values, compared to the actual overnight returns, which we could rectify by standardization. In any event, there appears to be a small, nonlinear relationship between forecast and actual values, which holds out some hope that the model may yet prove useful.

There are various methods of deploying a forecasting model in the context of creating a trading system. The simplest route, which we will take here, is to apply a threshold gate and convert the filtered forecasts directly into a trading signal. But other approaches are possible, for example:

- Combining the forecasts from multiple models to create a prediction ensemble
- Using the forecasts as inputs to a genetic programming model
- Feeding the forecasts into the input layer of a neural network model designed specifically to generate trading signals, rather than forecasts

In this example we will create a trading model by applying a simple filter to the forecasts, picking out only those values that exceed a specified threshold. This is a standard trick used to isolate the signal in the model from the background noise. We will accept only the positive signals that exceed the threshold level, creating a long-only trading system. i.e. we ignore forecasts that fall below the threshold level. We buy SPY at the close when the forecast exceeds the threshold and exit any long position at the next day’s open. This strategy produces the following pro-forma results:

The system has some quite attractive features, including a win rate of over 66% and a CAGR of over 10% for the out-of-sample period.

Obviously, this is a very basic illustration: we would want to factor in trading commissions, and the slippage incurred entering and exiting positions in the post- and pre-market periods, which will negatively impact performance, of course. On the other hand, we have barely begun to scratch the surface in terms of the variables that could be considered for inclusion in the feature vector, and which may increase the explanatory power of the model.

In other words, in reality, this is only the beginning of a lengthy and arduous research process. Nonetheless, this simple example should be enough to give the reader a taste of what’s involved in building a predictive trading model using machine learning algorithms.

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]]>The post The Swan of Deadwood appeared first on QUANTITATIVE RESEARCH AND TRADING.

]]>So it was that I recently tuned in to watch Deadwood, a masterpiece of modern drama written by the talented David Milch, of NYPD Blue fame. The setting of the show is unpromising: a mud-caked camp in South Dakota around the turn of the 19th century that appears to portend yet another formulaic Western featuring liquor, guns, gals and gold and not much else. The first episode appeared at first to confirm my lowest expectations. I struggled through the second. But by the third I was hooked.

What makes Deadwood such a triumph are its finely crafted plots and intricate sub-plots; the many varied and often complex characters, superbly played by Ian McShane (with outstanding performances by Brad Dourif, Powers Boothe, amongst an abundance of others, no less gifted); and, of course, the dialogue.

Yes, the *dialogue*: hardly the crowning glory of the typical Hollywood Western. And here, to make matters worse, almost every sentence uttered by many of the characters is replete with such shocking profanity that one is eventually numbed into accepting it as normal. But once you get past that, something strange and rather wonderful overtakes you: a sense of being carried along on a river of creative wordsmith-ing that at times eddies, bubbles, plunges and roars its way through scenes that are as comedic, dramatic and action-packed as any I have seen on film. For those who have yet to enjoy the experience, I offer one small morsel:

https://www.youtube.com/watch?v=RdJ4TQ3TnNo

Around the start of Series 2 a rather strange idea occurred to me that, try as I might, I was increasingly unable to suppress as the show progressed: that the writing – some of it at least – was almost Shakespearian in its ingenuity and, at times, lyrical complexity.

Convinced that I had taken leave of my senses I turned to Google and discovered, to my surprise, that there is a whole cottage industry of Deadwood fans who had made the same connection. There is even – if you can imagine it – an online quiz that tests if you are able to identify the source of a number of quotes that might come from the show, or one of the Bard’s many plays. I kid you not:

Intrigued, I took the test and scored around 85%. Not too bad for a science graduate, although I expect most English majors would top 90%-95%. That gave me an idea: could one develop a machine learning algorithm to do the job?

Here’s how it went.

We start by downloading the text of a representative selection of Shakespeare’s plays, avoiding several of the better-known works from which many of the quotations derive:

For testing purposes, let’s download a sample of classic works by other authors:

Let’s build an initial test classifier, as follows:

It seems to work ok:

So far so good. Let’s import the script for Deadwood series 1-3:

DeadWood = Import[“……../Dropbox/Documents/Deadwood-Seasons-1-3-script.txt”];

Next, let’s import the quotations used in the online test:

etc

We need to remove the relevant quotes from the Deadwood script file used to train the classifier, of course (otherwise it’s cheating!). We will strip an additional 200 characters before the start of each quotation, and 500 characters after each quotation, just for good measure:

and so on….

Now we are ready to build our classifier:

And we can obtain some information about the classifier, as follows:

Let’s see how it performs:

Or, if you prefer tabular form:

**The machine learning model scored a total of 19 correct answers out of 23, or 82%.**

Let’s take a look as some of the questions the algorithm got wrong.

**Quotation no. 13** is challenging, as it comes from *Pericles*, one of Shakespeare’s lesser-know plays and the idiom appears entirely modern. The classifier assigns an 87% probability of Milch being the author (I got it wrong too).

On **Quotation no. 15** the algorithm was just as mistaken, but in the other direction (I got this one right, but only because I recalled the monologue from the episode):

**Quotation no.16** strikes me as entirely Shakespearian in form and expression and the classifier thought so too, favoring the Bard by 86% to only 14% for Milch:

**Quotation no. 19** had the algorithm fooled completely. It’s a perfect illustration of a typical kind of circumlocution favored by Shakespeare that is imitated so well by Milch’s Deadwood characters:

The model clearly picked up distinguishing characteristics of the two authors’ writings that enabled it to correctly classify 82% of the quotations, quite a high percentage and much better than we would expect to do by tossing a coin, for example. It’s a respectable performance, but I might have hoped for greater accuracy from the model, which scored about the same as I did.

I guess those who see parallels in the writing of William Shakespeare and David Milch may be onto something.

The Hollywood Reporter recently published a story entitled

“How the $100 Million ‘NYPD Blue’ Creator Gambled Away His Fortune”.

It’s a fascinating account of the trials and tribulations of this great author, one worthy of Deadwood itself.

A silver lining to this tragic tale, perhaps, is that Milch’s difficulties may prompt him into writing the much-desired Series 4.

One can hope.

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]]>The post Outperforming Winton Capital appeared first on QUANTITATIVE RESEARCH AND TRADING.

]]>The performance of the program over the last 19 years has been impressive, especially considering its size, which now tops around $13Bn in assets.

Source: CTA Performance

With that background, the idea of improving the exceptional results achieved by David Harding and his army of quants seems rather far fetched, but I will take a shot. In what follows, I am assuming that we are permitted to invest and redeem an investment in the program at no additional cost, other than the stipulated fees. This is, of course, something of a stretch, but we will make that assumption based on the further stipulation that we will make no more than two such trades per year.

The procedure we will follow has been described in various earlier posts – in particular see this post, in which I discuss the process of developing a Meta-Strategy: Improving A Hedge Fund Investment – Cantab Capital’s Quantitative Aristarchus Fund

Using the performance data of the WDP from 1997-2012, we develop a meta-strategy that seeks to time an investment in the program, taking profits after reaching a specified profit target, which is based on the TrueRange, or after holding for a maximum of 8 months. The key part of the strategy code is as follows:

If MarketPosition = 1 then begin

TargPrL = EntryPrice + TargFr * TrueRange;

Sell(“ExTarg-L”) next bar at TargPrL limit;

If Time >= TimeEx or BarsSinceEntry >= NBarEx1 or (BarsSinceEntry >= NBarEx3 and C > EntryPrice)

or (BarsSinceEntry >= NBarEx2 and C < EntryPrice) then

Sell(“ExMark-L”) next bar at market;

end;

It appears that by timing an investment in the program we can improve the CAGR by around 0.86% per year, and with annual volatility that is lower by around 4.4% annually. As a consequence, the Sharpe ratio of the meta-strategy is considerably higher: 1.14 vs 0.78 for the WDP.

Like most trend-following CTA strategies, Winton’s WDP has positive skewness, an attractive feature that means that the strategy has a preponderance of returns in the positive right tail of the distribution. Also in common with most CTA strategies, on the other hand, the WDP suffers from periodic large drawdowns, in this case amounting to -25.73%.

The meta-strategy improves on the baseline profile of the WDP, increasing the positive skew, while substantially reducing downside risk, leading to a much lower maximum drawdown of -16.94%.

Despite its stellar reputation in the CTA world, investors could theoretically improve on the performance of Winton Capital’s flagship program by using a simple meta-strategy that times entry to and exit from the program using simple technical indicators. The meta-strategy produces higher returns, lower volatility and with higher positive skewness and lower downside risk.

The post Outperforming Winton Capital appeared first on QUANTITATIVE RESEARCH AND TRADING.

]]>The post Falling Water appeared first on QUANTITATIVE RESEARCH AND TRADING.

]]>A 120 foot high band of rock, bleached white by the water, and known as the “bathtub ring” encircles the lake, a stark reminder of the water crisis that has enveloped the surrounding region. The Colorado River takes a 1,400 mile journey from the Rockies to Mexico, irrigating over 5 million acres of farmland in the Basin states of Wyoming, Utah, Colorado, New Mexico, Nevada, Arizona, and California.

The Colorado River Compact signed in 1922 enshrined the States’ water rights in law and Mexico was added to the roster in 1994, taking the total allocation to over 16.5 million acre-feet per year. But the average freshwater input to the lake over the century from 1906 to 2005 reached only 15 million acre-feet. The river can’t come close to meeting current demand and the problem is only likely to get worse. A 2009 study found that rainfall in the Colorado Basin could fall as much as 15% over the next 50 years and the shortfall in deliveries could reach 60% to 90% of the time.

With an average of only 4 inches of rain a year, and a daily high temperatures of 103 ^{o} F during the summer, Las Vegas is perhaps the most hard pressed to meet the demand of its 2 million residents and 40 million visitors annually.

Despite its conspicuous consumption, from the tumbling fountains of the Bellagio to the Venetian’s canals, since 2002, Las Vegas has been obliged to cut its water use by a third, from 314 gallons per capita a day to 212. The region recycles around half of its wastewater which is piped back into Lake Mead, after cleaning and treatment. Residents are allowed to water their gardens no more than one day a week in peak season, and there are stiff fines for noncompliance.

Historically, two intake pipes carried water from Lake Mead to Las Vegas, about 25 miles to the west. In 2012, realizing that the highest of these, at 1050 feet, would soon be sucking air, the Southern Nevada Water Authority began construction of a new pipeline. Known as the Third Straw, Intake No. 3 reaches 200 feet deeper into the lake—to keep water flowing for as long as there’s water to pump. The new pipeline, which commenced operations in 2015, doesn’t draw more water from the lake than before, or make the surface level drop any faster. But it will keep taps flowing in Las Vegas homes and casinos even if drought-stricken Lake Mead drops to its lowest levels.

The monthly reported water levels in Lake Mead from Feb 1935 to June 2016 are shown in the chart below. The reference line is the drought level, historically defined as 1,125 feet.

One statistical technique widely applied in hydrology involves fitting a Kumaraswamy distribution to the relative water level. According to the Arizona Game and Fish Department, the maximum lake level is 1229 feet. We model the water level relative to the maximum level, as follows.

The fit of the distribution appears quite good, even in the tails:

ProbabilityPlot[relativeLevels, edist]

Since water levels have been below the drought level for some time, let’s instead consider the “emergency” level, 1,075 feet. According to this model, there is just over a 6% chance of Lake Mead hitting the emergency level and, consequently, a high probability of breaching the emergency threshold some time over before the end of 2017.

One problem with this approach is that it assumes that each observation is drawn independently from a random variable with the estimated distribution. In reality, there are high levels of autocorrelation in the series, as might be expected: lower levels last month typically increase the likelihood of lower levels this month. The chart of the autocorrelation coefficients makes this pattern clear, with statistically significant coefficients at lags of up to 36 months.

ts[“ACFPlot”]

An alternative methodology that enables us to take account of the autocorrelation in the process is time series analysis. We proceed to fit an autoregressive moving average (ARMA) model as follows:

tsm = TimeSeriesModelFit[ts, “ARMA”]

The best fitting model in an ARMA(1,1) model, according to the AIC criterion:

Applying the fitted ARMA model, we forecast the water level in Lake Mead over the next ten years as shown in the chart below. Given the mean-reverting moving average component of the model, it is not surprising to see the model forecasting a return to normal levels.

There is some evidence of lack of fit in the ARMA model, as shown in the autocorrelations of the model residuals:

A formal test reveals that residual autocorrelations at lags 4 and higher are jointly statistically significant:

The slowly decaying pattern of autocorrelations in the water level series suggests a possible “long memory” effect, which can be better modelled as a fractionally integrated process. The forecasts from such a model, like the ARMA model forecasts, display a tendency to revert to a long term mean; but the reversion process is dampened by the reinforcing, long-memory effect captured in the FARIMA model.

Taking the view that the water level in Lake Mead forms a stationary statistical process, the likelihood is that water levels will rise to 1,125 feet or more over the next ten years, easing the current water shortage in the region.

On the other hand, there are good reasons to believe that there are exogenous (deterministic) factors in play, specifically the over-consumption of water at a rate greater than the replenishment rate from average rainfall levels. Added to this, plausible studies suggest that average rainfall in the Colorado Basin is expected to decline over the next fifty years. Under this scenario, the water level in Lake Mead will likely continue to deteriorate, unless more stringent measures are introduced to regulate consumption.

The drought in the South West affects far more than just the water levels in Lake Mead, of course. One study found that California’s agriculture sector alone had lost $2.2Bn and some 17,00 season and part time jobs in 2014, due to drought. Agriculture uses more than 80% of the State’s water, according to Fortune magazine, which goes on to identify the key industries most affected, including agriculture, food processing, semiconductors, energy, utilities and tourism.

In the energy sector, for example, the loss of hydroelectric power cost CA around $1.4Bn in 2014, according to non-profit research group Pacific Institute. Although Intel pulled its last fabrication plant from California in 2009, semiconductor manufacturing is still a going concern in the state. Maxim Integrated, TowerJazz, and TSI Semiconductors all still have fabrication plants in the state. And they need a lot of water. A single semiconductor fabrication plant can use as much water as a small city. That means the current plants could represent three cities worth of consumption.

The drought is also bad news for water utilities, of course. The need to conserve water raises the priority on repair and maintenance, and that means higher costs and lower profit. Complicating the problem, California lacks any kind of management system for its water supply and can’t measure the inflows and outflows to ground water levels at any particular time.

The Bureau of Reclamation has studied more than two dozen options for conserving and increasing water supply, including importation, desalination and reuse. While some were disregarded for being too costly or difficult, the bureau found that the remaining options, if instituted, could yield 3.7 million acre feet per year in savings and new supplies, increasing to 7 million acre feet per year by 2060. Agriculture is the biggest user by far and has to be part of any solution. In the near term, the agriculture industry could reduce its use by 10 to 15 percent without changing the types of crops it grows by using new technology, such as using drip irrigation instead of flood irrigation and monitoring soil moisture to prevent overwatering, the Pacific Institute found.

We can anticipate that a series of short term fixes, like the “Third Straw”, will be employed to kick the can down the road as far as possible, but research now appears almost unanimous in finding that drought is having a deleterious, long term affect on the economics of the South Western states. Agriculture is likely to have to bear the brunt of the impact, but so too will adverse consequences be felt in industries as disparate as food processing, semiconductors and utilities. California, with the largest agricultural industry, by far, is likely to be hardest hit. The Las Vegas region may be far less vulnerable, having already taken aggressive steps to conserve and reuse water supply and charge economic rents for water usage.

** **

** **

**Modeling Water Levels in Lake Mead**

The monthly reported water levels in Lake Mead from Feb 1935 to June 2016 are shown in the chart below. The reference line is the drought level, historically defined as 1,125 feet.

One statistical technique widely applied in hydrology involves fitting a Kumaraswamy distribution to the relative water level. According to the Arizona Game and Fish Department, the maximum lake level is 1229 feet. We model the water level relative to the maximum level, as follows.

The fit of the distribution appears quite good, even in the tails:

ProbabilityPlot[relativeLevels, edist]

Since water levels have been below the drought level for some time, let’s instead consider the “emergency” level, 1,075 feet. According to this model, there is just over a 6% chance of Lake Mead hitting the emergency level and, consequently, a high probability of breaching the emergency threshold some time over before the end of 2017.

One problem with this approach is that it assumes that each observation is drawn independently from a random variable with the estimated distribution. In reality, there are high levels of autocorrelation in the series, as might be expected: lower levels last month typically increase the likelihood of lower levels this month. The chart of the autocorrelation coefficients makes this pattern clear, with statistically significant coefficients at lags of up to 36 months.

ts[“ACFPlot”]

An alternative methodology that enables us to take account of the autocorrelation in the process is time series analysis. We proceed to fit an autoregressive moving average (ARMA) model as follows:

tsm = TimeSeriesModelFit[ts, “ARMA”]

The best fitting model in an ARMA(1,1) model, according to the AIC criterion:

Applying the fitted ARMA model, we forecast the water level in Lake Mead over the next ten years as shown in the chart below. Given the mean-reverting moving average component of the model, it is not surprising to see the model forecasting a return to normal levels.

There is some evidence of lack of fit in the ARMA model, as shown in the autocorrelations of the model residuals:

A formal test reveals that residual autocorrelations at lags 4 and higher are jointly statistically significant:

The slowly decaying pattern of autocorrelations in the water level series suggests a possible “long memory” effect, which can be better modelled as a fractionally integrated process. The forecasts from such a model, like the ARMA model forecasts, display a tendency to revert to a long term mean; but the reversion process is dampened by the reinforcing, long-memory effect captured in the FARIMA model.

**The Prospects for the Next Decade**

Taking the view that the water level in Lake Mead forms a stationary statistical process, the likelihood is that water levels will rise to 1,125 feet or more over the next ten years, easing the current water shortage in the region.

On the other hand, there are good reasons to believe that there are exogenous (deterministic) factors in play, specifically the over-consumption of water at a rate greater than the replenishment rate from average rainfall levels. Added to this, plausible studies suggest that average rainfall in the Colorado Basin is expected to decline over the next fifty years. Under this scenario, the water level in Lake Mead will likely continue to deteriorate, unless more stringent measures are introduced to regulate consumption.

** **

**Economic Impact**

The drought in the South West affects far more than just the water levels in Lake Mead, of course. One study found that California’s agriculture sector alone had lost $2.2Bn and some 17,00 season and part time jobs in 2014, due to drought. Agriculture uses more than 80% of the State’s water, according to Fortune magazine, which goes on to identify the key industries most affected, including agriculture, food processing, semiconductors, energy, utilities and tourism.

In the energy sector, for example, the loss of hydroelectric power cost CA around $1.4Bn in 2014, according to non-profit research group Pacific Institute.

Although Intel pulled its last fabrication plant from California in 2009, semiconductor manufacturing is still a going concern in the state. Maxim Integrated, TowerJazz, and TSI Semiconductors all still have fabrication plants in the state. And they need a lot of water. A single semiconductor fabrication plant can use as much water as a small city. That means the current plants could represent three cities worth of consumption.

The drought is also bad news for water utilities, of course. The need to conserve water raises the priority on repair and maintenance, and that means higher costs and lower profit. Complicating the problem, California lacks any kind of management system for its water supply and can’t measure the inflows and outflows to ground water levels at any particular time.

The Bureau of Reclamation has studied more than two dozen options for conserving and increasing water supply, including importation, desalination and reuse. While some were disregarded for being too costly or difficult, the bureau found that the remaining options, if instituted, could yield 3.7 million acre feet per year in savings and new supplies, increasing to 7 million acre feet per year by 2060. Agriculture is the biggest user by far and has to be part of any solution. In the near term, the agriculture industry could reduce its use by 10 to 15 percent without changing the types of crops it grows by using new technology, such as using drip irrigation instead of flood irrigation and monitoring soil moisture to prevent overwatering, the Pacific Institute found.

**Conclusion**

We can anticipate that a series of short term fixes, like the “Third Straw”, will be employed to kick the can down the road as far as possible, but research now appears almost unanimous in finding that drought is having a deleterious, long term affect on the economics of the South Western states. Agriculture is likely to have to bear the brunt of the impact, but so too will adverse consequences be felt in industries as disparate as food processing, semiconductors and utilities. California, with the largest agricultural industry, by far, is likely to be hardest hit. The Las Vegas region may be far less vulnerable, having already taken aggressive steps to conserve and reuse water supply and charge economic rents for water usage.

The post Falling Water appeared first on QUANTITATIVE RESEARCH AND TRADING.

]]>The post The Amazon Killer appeared first on QUANTITATIVE RESEARCH AND TRADING.

]]>Source: Yahoo! Finance

It’s hard to argue with success, but could you have done better? The answer, surprisingly, is yes – and by a wide margin.

I am going to reveal this mystery stock in due course and, I promise you, the investment recommendation is fully actionable. For now, let’s just refer to it as AMZN+.

A comparison between investments made in AMZN and AMZN+ over the period from January 2010 to June 2016 is shown the chart following.

Visually, there doesn’t appear to be much of a difference in overall performance. However, it is apparent that AMZN+ is a great deal less volatile than its counterpart.

The table below give a more telling account of the relative outperformance by AMZN+.

The two investments are very highly correlated, with AMZN+ producing an extra 1% per annum in CAGR over the 6 ½ year period.

The telling distinction, however, lies on the risk side of the equation. Here AMZN+ outperforms AMZN by a wide margin, with an annual standard deviation of only 21% compared to 29%. What this means is that AMZN+ produces almost a 50% higher return than AMZN per unit of risk (information ratio 1.51 vs. 1.07).

The more conservative risk profile of AMZN+ is also reflected in lower maximum drawdown (-13.43% vs -23.74%), semi-deviation and higher Sortino Ratio (see this article for an explanation of these terms).

Bottom line: you can produce around the same rate of return with substantially less risk by investing in AMZN+, rather than AMZN.

There is another way to make the comparison, which some investors might find more appealing. Risk is, after all, an altogether more esoteric subject than return, which every investor understands.

So let’s say the investor adopts the same risk budget as for his original investment in AMZN, i.e. an annual volatility of just over 29%. We can produce the same overall level of risk in AMZN+, equalizing the riskiness of the two investments, simply by leveraging the investment in AMZN+ by a factor of 1.36, using margin money. i.e. we borrow $360 and invest a total of $1,360 in AMZN+, for each $1,000 we would have invested in AMZN. Look at the difference in performance:

The investor’s total return in AMZN+ would have been 963%, almost double the return in AMZN over the same period and with a CAGR of over 44.5%, more than 13% per annum higher than AMZN.

Note that, despite having an almost identical annual standard deviation, AMZN+ still enjoys a lower maximum drawdown and downside risk than AMZN.

Ok, so what is this mystery stock, AMZN+? Actually it isn’t a stock: it’s a simple portfolio, rebalanced monthly, with 66% of the investment being made in AMZN and 34% in the Direxion Daily 20+ Yr Trsy Bull 3X ETF (NYSEArca: TMF).

Well, that’s a little bit of a cheat, although not much of one: it isn’t too much of a challenge to put $667 of every $1,000 in AMZN and the remaining $333 in TMF, rebalancing the portfolio at the end of every month.

The next question an investor might want to ask is: what other stocks could I apply this approach to? The answer is: a great many of them. And where you end up, ultimately, is with the discovery that you can eliminate a great deal of unnecessary risk with a portfolio of around 20-30 well-chosen assets.

We saw that AMZN incurred a risk of 29% in annual standard deviation, compared to only 21% for the AMZN+ portfolio. What does the investor gain by taking that extra 8% in annual risk? Nothing at all – in fact he would have achieved a slightly worse return.

The key take-away from this simple example is the fundamental law of modern portfolio theory:

The market will not compensate an investor for taking diversifiable risk

As they say, diversification is the only free lunch on Wall Street. So make the most of it.

The post The Amazon Killer appeared first on QUANTITATIVE RESEARCH AND TRADING.

]]>