Jonathan, Author at QUANTITATIVE RESEARCH AND TRADING

September 17, 2018September 18, 2018

Beating the S&P500 Index with a Low Convexity Portfolio

What is Beta Convexity?

Beta convexity is a measure of how stable a stock beta is across market regimes. The essential idea is to evaluate the beta of a stock during down-markets, separately from periods when the market is performing well. By choosing a portfolio of stocks with low beta-convexity we seek to stabilize the overall risk characteristics of our investment portfolio.

A primer on beta convexity and its applications is given in the following post:

In this post I am going to use the beta-convexity concept to construct a long-only equity portfolio capable of out-performing the benchmark S&P 500 index.

The post is in two parts. In the first section I outline the procedure in Mathematica for downloading data and creating a matrix of stock returns for the S&P 500 membership. This is purely about the mechanics, likely to be of interest chiefly to Mathematica users. The code deals with the issues of how to handle stocks with multiple different start dates and missing data, a problem that the analyst is faced with on a regular basis. Details are given in the pdf below. Let’s skip forward to the analysis.

Portfolio Formation & Rebalancing

We begin by importing the data saved using the data retrieval program, which comprises a matrix of (continuously compounded) monthly returns for the S&P500 Index and its constituent stocks. We select a portfolio size of 50 stocks, a test period of 20 years, with a formation period of 60 months and monthly rebalancing.

In the processing stage, for each month in our 20-year test period we calculate the beta convexity for each index constituent stock and select the 50 stocks that have the lowest beta-convexity during the prior 5-year formation period. We then compute the returns for an equally weighted basket of the chosen stocks over the following month. After that, we roll forward one month and repeat the exercise.

It turns out that beta-convexity tends to be quite unstable, as illustrated for a small sample of component stocks in the chart below:

A snapshot of estimated convexity factors is shown in the following table. As you can see, there is considerable cross-sectional dispersion in convexity, in addition to time-series dependency.

At any point in time the cross-sectional dispersion is well described by a Weibull distribution, which passes all of the usual goodness-of-fit tests.

Performance Results

We compare the annual returns and standard deviation of the low convexity portfolio with the S&P500 benchmark in the table below. The results indicate that the average gross annual return of a low-convexity portfolio of 50 stocks is more than double that of the benchmark, with a comparable level of volatility. The portfolio also has slightly higher skewness and kurtosis than the benchmark, both desirable characteristics.

Portfolio Alpha & Beta Estimation

Using the standard linear CAPM model we estimate the annual alpha of the low-convexity portfolio to be around 7.39%, with a beta of 0.89.

Beta Convexity of the Low Convexity Portfolio

As we might anticipate, the beta convexity of the portfolio is very low since it comprises stocks with the lowest beta-convexity:

Conclusion: Beating the Benchmark S&P500 Index

Using a beta-convexity factor model, we are able to construct a small portfolio that matches the benchmark index in terms of volatility, but with markedly superior annual returns. Larger portfolios offering greater liquidity produce slightly lower alpha, but a 100-200 stock portfolio typically produce at least double the annual rate of return of the benchmark over the 20-year test period.

For those interested, we shall shortly be offering a low-convexity strategy on our Systematic Algotrading platform – see details below:

Section on Data Retrieval and Processing

Data Retrieval

September 16, 2018

Pattern Trading

Summary

Pattern trading rules try to identify profit opportunities, based on short term price patterns.
An exhaustive test of simple pattern trading rules was conducted for several stocks, incorporating forecasts of the Open, High, Low and Close prices.
There is clear evidence that pattern trading rules continue to work consistently for many stocks.
Almost all of the optimal pattern trading rules suggest buying the stock if the close is below the mid-range of the day.
This “buy the dips” approach can sometimes be improved by overlaying additional conditions, or signals from forecasting models.

Trading Pattern Rules

From time to time one comes across examples of trading pattern rules that appear to work. By “pattern rule”, I mean something along the lines of: “if the stock closes below the open and today’s high is greater than yesterday’s high, then buy tomorrow’s open”.

Trading rules of this kind are typically one-of-a-kind oddities that only work for limited periods, or specific securities. But I was curious enough to want to investigate the concept of pattern trading, to see if there might be some patterns that are generally applicable and potentially worth trading.

To my surprise, I was able to find such a rule, which I will elaborate on in this article. The rule appears to work consistently for a wide range of stocks, across long time frames. While perhaps not interesting enough to trade by itself, the rule might provide some useful insight and, possibly, be combined with other indicators in a more elaborate trading strategy.

The original basis for this piece of research was the idea of using vector autoregression models to forecast the daily O/H/L/C prices of a stock. The underlying thesis is that there might be information in the historical values of these variables that, combined together, could produce more useful forecasts than, say, using close prices alone. In technical terms, we say that the O/H/L/C price series are cointegrated, which one might think of as a more robust kind of correlation: cointegrated series tend to continue to move together for some underlying economic reason, whereas series that are merely correlated will often see that purely statistical relationship break down. In this case the economic relationship between the O/H/L/C series is clear: the high price will always be greater than the low price, and the open and close prices will always lie between the two. Furthermore, the prices cannot drift arbitrarily far apart indefinitely, since volatility is finite and mean-reverting. So there is some kind of rationale for using a vector autoregression model in this context. But I don’t want to dwell on this idea too much, as it turns out to be useful only at the margin.

To keep it simple I decided to focus attention on simple pattern trades of the following kind:

If Rule1 and/or Rule2 then Trade

Rule1 and Rule2 are simple logical statements of the kind: “Today’s Open greater than yesterday’s Close”, or “today’s High below yesterday’s Low”. The trade can be expressed in combinations of the form “Buy today’s Open, Sell today’s Close”, or “Buy today’s Close, Sell tomorrow’s Close”.

In my model I had to consider rules combining not only the O/H/L/C prices from yesterday, today and tomorrow, but also forecast O/H/L/C prices from the vector autoregression model. This gave rise to hundreds of thousands of possibilities. A brute-force test of every one of them would certainly be feasible, but rather tedious to execute. And many of the possible rules would be redundant – for example a rule such as : “if today’s open is lower than today’s close, buy today’s open”. Rules of that kind will certainly make a great deal of money, but they aren’t practical, unfortunately!

To keep the number of possibilities to a workable number, I restricted the trading rule to the following: “Buy today’s close, sell tomorrow’s close”. Consequently, we are considering long-only trading strategies and we ignore any rules that might require us to short a stock.

I chose stocks with long histories, dating back to at least the beginning of the 1970’s, in order to provide sufficient data to construct the VAR model. Data from the period from Jan 1970 to Dec 2012 were used to estimate the model, and the performance of the various possible trading rules was evaluated using out-of-sample data from Jan 2013 to Jun 2014.

For ease of illustration the algorithms were coded up in MS-Excel (a copy of the Excel workbook is available on request). In evaluating trading rule performance an allowance was made of $1c per share in commission and $2c per share in slippage. Position size was fixed at 1,000 shares. Considering that the trading rules requires entry and exit at market close, a greater allowance for slippage may be required for some stocks. In addition, we should note the practical difficulties of trading a sizeable position at the close, especially in situations where the stock price may be very near to key levels such as the intra-day high or low that our trading rule might want to take account of.

As a further caveat, we should note that there is an element of survivor bias here: in order to fit this test protocol, stocks would have had to survive from the 1970’s to the present day. Many stocks that were current at the start of that period are no longer in existence, due to mergers, bankruptcies, etc. Excluding such stocks from the evaluation will tend to inflate the test results. It should be said that I did conduct similar tests on several now-defunct stocks, for which the outcomes were similar to those presented here, but a fully survivor-bias corrected study is beyond the scope of this article. With that caveat behind us, let’s take a look at some of the results.

Trading Pattern Analysis

Fig. 1 below shows the summary output from the test for the 3M Company (NYSE:MMM). At the top you can see the best trading rule that the system was able to find for this particular stock. In simple English, the rule tells you to buy today’s close in MMM and sell tomorrow’s close, if the stock opened below the forecast of yesterday’s high price and, in addition, the stock closed below the midrange of the day (the average of today’s high and low prices).

Fig. 1 Summary Analysis for MMM

Source: Yahoo Finance.

The in-sample results from Jan 2000, summarized in left-hand table in Fig. 2 below, are hardly stellar, but do show evidence of a small, but significant edge, with total net returns of 165%, profit factor of 1.38 and % win rate of 54%. And while the trading rule is, ultimately, outperformed by a simple buy-and-hold strategy, after taking into account transaction costs, for extended periods (e.g. 2009-2012), investors would have been better off had they used the trading rule, because it successfully avoided the worst of the effects of the 2008/09 market crash.

Out-of-sample results, shown in the right-hand table, are less encouraging, but net returns are nonetheless positive and the % win rate actually increases to 55%.

Fig 2. Trade Rule Performance

Source: Yahoo Finance.

I noted earlier that the first part of our trading rule for MMM involved comparing the opening price to the forecast of yesterday’s high, produced by the vector autoregression model, while the second part of the trading rule references only the midrange and closing prices. How much added value does the VAR model provide? We can test this by eliminating the first part of the rule and considering all days in which the stock closed below the midrange. The results turn out to as shown in Fig. 3.

Fig. 3 Performance of Simplified Trading Rule

Source: Yahoo Finance.

As expected, the in-sample results from our shortened trading rule are certainly inferior to the original rule, in which the VAR model forecasts played a role. But the out-of-sample performance of the simplified rule is actually improved – not only is the net return higher than before, so too is the % win rate, by a couple of percentage points.

A similar pattern emerges for many other stocks: in almost every case, our test algorithm finds that the best trading rule buys the close, based on a comparison of the closing price to the mid-range price. In some cases, the in-sample test results are improved by adding further conditions, such as we saw in the case of MMM. But, as with MMM, we often find that the additional benefit derived from use of the autoregression model forecasts fails to improve trading rule results in the out-of-sample period, and indeed often makes them worse.

Conclusion

In general, we find evidence that a simple trading rule based on a comparison of the closing price to the mid-range price appears to work for many stocks, across long time spans.

In a sense, this simple trading rule is already well known: it is just a variant of the “buy the dips” idea, where, in this case, we define a dip as being when the stock closes below the mid-range of the day, rather than, say, below a moving average level. The economic basis for this finding is also well known: stocks have positive drift. But it is interesting to find yet another confirmation of this well-known idea. And it leaves open the possibility that the trading concept could be further improved by introducing additional rules, trading indicators, and model forecasts to the mix.

September 14, 2018

More on Strategy Robustness

Commentators have made the point that a high % win rate is not enough.

Yes, you obviously want to pay attention to other performance metrics also, such as profit factor. In fact, there is no reason why you shouldn’t consider an objective function that explicitly combines various desirable performance measures, for example:

net profit * % win rate * profit factor

Another approach is to build the model using a data set spanning a different period. I did this with WFC using data from 1990, rather than 1970. Not only was the performance from 1990-2014 better, so too was the performance during the OOS period 1970-1989. Profit factor was 2.49 and %Win rate was 70% across the 44 year period from 1970. For the period from 1990, the performance metrics increase to 3.04 and 73%, respectively.

So in this case, it appears, a most robust strategy resulted from using less data, rather than more. At first this appears counterintuitive. But it’s quite possible for a strategy to be over-condition on behavior that is no longer relevant to the market today. Eliminating such conditioning can sometimes enable strategies to emerge that have greater longevity.

September 14, 2018

Optimizing Strategy Robustness

Below is the equity curve for an equity strategy I developed recently, implemented in WFC. The results appear outstanding: no losing years in over 20 years, profit factor of 2.76 and average win rate of 75%. Out-of-sample results (double blind) for 2013 and 2014: net returns of 27% and 16% YTD.

So far so good. However, if we take a step back through the earlier out of sample period, from 1970, the picture is rather less rosy:

Now, at this point, some of you will be saying: nothing to see here – it’s obviously just curve fitting. To which I would respond that I have seen successful strategies, including several hedge fund products, with far shorter and less impressive back-tests than the initial 20-year history I showed above.

That said, would you be willing to take the risk of trading a strategy such as this one? I would not: at the back of my mind would always be the concern that the market might easily revert to the conditions that applied during the 1970s and 1980’s. I expect many investors would share that concern.

But to the point of this post: most strategies are designed around the criterion of maximizing net profit. Occasionally you might come across someone who has considered risk, perhaps in the form of drawdown, or Sharpe ratio. But, in general, it’s all about optimizing performance.

Suppose that, instead of maximizing performance, your objective was to maximize the robustness of the strategy. What criteria would you use?

In my own research, I have used a great many different objective functions, often multi-dimensional. Correlation to the perfect equity curve, net profit / max drawdown and Sortino ratio are just a few examples. But if I had to guess, I would say that the criteria that tends to produce the most robust strategies and reliable out of sample performance is the maximization of the win rate, subject to a minimum number of trades.

I am not aware of a great deal of theory on this topic. I would be interested to learn of other readers’ experience.

September 13, 2018

Enhancing Mutual Fund Returns With Market Timing

Summary

In this article, I will apply market timing techniques to several popular mutual funds.

The market timing approach produces annual rates of return that are 3% to 7% higher, with lower risk, than an equivalent buy and hold mutual fund investment.

Investors could in some cases have earned more than double the return achieved by holding a mutual fund investment over a 10-year period.

Hedging strategies that use market timing signals are able to sidestep market corrections, volatile conditions and the ensuing equity drawdowns.

Hedged portfolios typically employ around 12% less capital than the equivalent buy and hold strategy.

Background to the Market Timing Approach

In an earlier article, I discussed how to use marketing timing techniques to hedge an equity portfolio correlated to the broad market. I showed how, by using signals produced by a trading system modeled on the CBOE VIX index, we can smooth out volatility in an equity portfolio consisting of holdings in the SPDR S&P 500 ETF (NYSEARCA:SPY). An investor will typically reduce their equity holdings by a modest amount, say 20%, or step out of the market altogether during periods when the VIX index is forecast to rise, returning to the market when the VIX is likely to fall. An investment strategy based on this approach would have avoided most of the 2000-03 correction, as well as much of the market turmoil of 2008-09.

A more levered version of the hedging strategy, which I termed the MT aggressive portfolio, uses the VIX index signals to go to cash during high volatility periods, and then double the original equity portfolio holdings (using standard Reg-T leverage) during benign market conditions, as signaled by the model. The MT aggressive approach would have yielded net returns almost three times greater than that of a buy and hold portfolio in the SPY ETF, over the period from 1999-2014. Even though this version of the strategy makes use of leverage, the average holding in the portfolio would have been slightly lower than in the buy and hold portfolio because, in a majority of days, the strategy would have been 100% in cash. The result is illustrated in the chart in Fig. 1, which is reproduced below.

Fig. 1: Value of $1,000 – Long-Only Vs. MT Aggressive Portfolio

Source: Yahoo Finance.

Note that this approach does not entail shorting any stock. And for investors who prefer to buy and hold, I would make the point that the MT aggressive approach would have enabled you to buy almost three times as much stock in dollar terms by mid-2014 than would be the case if you had simply owned the SPY portfolio over the entire period.

Market Timing and Mutual Funds

With that background, we turn our attention to how we can use market timing techniques to improve returns from equity mutual funds. The funds selected for analysis are the Vanguard 500 Index Admiral (MUTF:VFIAX), Fidelity Spartan 500 Index Advtg (MUTF:FUSVX) and BlackRock S&P 500 Stock K (MUTF:WFSPX). This group of popular mutual funds is a representative sample of available funds that offer broad equity market exposure, with a high degree of correlation to the S&P 500 index. In what follows, we will focus attention on the MT aggressive approach, although other more conservative hedging strategies are equally valid.

We consider performance over the 10-year period from 2005, as at least one of the funds opened late in 2004. In each case, the MT aggressive portfolio is created by exiting the current mutual fund position and going 100% to cash, whenever the VIX model issues a buy signal in the VIX index. Conversely, we double our original mutual fund investment when the model issues a sell signal in the VIX index. In calculating returns, we make an allowance for trading costs of $3 cents per share for all transactions.

Returns for each of the mutual funds, as well as for the SPY ETF and the corresponding MT aggressive hedge strategies, are illustrated in the charts in Fig. 2. The broad pattern is similar in each case – we see significant outperformance of the MT aggressive portfolios relative to their ETF or mutual fund benchmarks. Furthermore, in most cases the hedge strategy tends to exhibit lower volatility, with less prolonged drawdowns during critical periods such as 2000/03 and 2008/09.

Fig. 2 – Value of $1,000: Mutual Fund Vs. MT Aggressive Portfolio January 2005 – June 2014

Source: Yahoo Finance.

Looking at the performance numbers in more detail, we can see from the tables shown in Fig. 3 that the MT aggressive strategies outperformed their mutual fund buy and hold benchmarks by a substantial margin. In the case of VFIAX and WFSPX, the hedge strategies produce a total net return more than double that of the corresponding mutual fund. With one exception, FUSVX, annual volatility of the MT aggressive portfolio was similar to, or lower than, that of the corresponding mutual fund, confirming our reading of the charts in Fig. 2. As a consequence, the MT aggressive strategies have higher Sharpe Ratios than any of the mutual funds. The improvement in risk adjusted returns is significant – more than double in the case of two of the funds, and about 40% higher in the case of the third.

Finally, we note that the MT aggressive strategies have an average holding that is around 12% lower than the equivalent long-only fund. That’s because of the periods in which investment proceeds are held in cash.

Fig. 3: Mutual Fund and MT Aggressive Portfolio Performance January 2005 – June 2014

Source: Yahoo Finance.

Conclusion

The aim of market timing is to smooth out the returns by hedging, and preferably avoiding altogether periods of market turmoil. In other words, the objective is to achieve the same, or better, rates of return, with lower volatility and drawdowns. We have demonstrated that this can be done, not only when the underlying investment is in an ETF such as SPY, but also where we hold an investment in one of several popular equity mutual funds. Over a 10-year period the hedge strategies produced consistently higher returns, with lower volatility and drawdown, while putting less capital at risk than their counterpart buy and hold mutual fund investments.

September 12, 2018

How to Bulletproof Your Portfolio

Summary

How to stay in the market and navigate the rocky terrain ahead, without risking hard won gains.

A hedging program to get you out of trouble at the right time and step back in when skies are clear.

Even a modest ability to time the market can produce enormous dividends over the long haul.

Investors can benefit by using quantitative market timing techniques to strategically adjust their market exposure.

Market timing can be a useful tool to avoid major corrections, increasing investment returns, while reducing volatility and drawdowns.

The Role of Market Timing

Investors have enjoyed record returns since the market lows in March 2009, but sentiment is growing that we may be in the final stages of this extended bull run. The road ahead could be considerably rockier. How do you stay the course, without risking all those hard won gains?

The smart move might be to take some money off the table at this point. But there could be adverse tax effects from cashing out and, besides, you can’t afford to sit on the sidelines and miss another 3,000 points on the Dow. Hedging tools like index options, or inverse volatility plays such as the VelocityShares Daily Inverse VIX Short-Term ETN (NASDAQ:XIV), are too expensive. What you need is a hedging program that will get you out of trouble at the right time – and step back in when the skies are clear. We’re talking about a concept known as market timing.

Market timing is the ability to switch between risky investments such as stocks and less-risky investments like bonds by anticipating the overall trend in the market. It’s extremely difficult to do. But as Nobel prize-winning economist Robert C. Merton pointed out in the 1980s, even a modest ability to time the market can produce enormous dividends over the long haul. This is where quantitative techniques can help – regardless of the nature of your underlying investment strategy.

Let’s assume that your investment portfolio is correlated with a broad US equity index – we’ll use the SPDR S&P 500 Trust ETF (NYSEARCA:SPY) as a proxy, for illustrative purposes. While the market has more than doubled over the last 15 years, this represents a modest average annual return of only 7.21%, accompanied by high levels of volatility of 20.48% annually, not to mention sizeable drawdowns in 2000 and 2008/09.

Fig. 1 SPY – Value of $1,000 Jan 1999 – Jul 2014

Source: Yahoo! Finance, 2014

The aim of market timing is to smooth out the returns by hedging, and preferably avoiding altogether, periods of market turmoil. In other words, the aim is to achieve the same, or better, rates of return, with lower volatility and drawdown.

Market Timing with the VIX Index

The mechanism we are going to use for timing our investment is the CBOE VIX index, a measure of anticipated market volatility in the S&P 500 index. It is well known that the VIX and S&P 500 indices are negatively correlated – when one rises, the other tends to fall. By acting ahead of rising levels of the VIX index, we might avoid difficult market conditions when market volatility is high and returns are likely to be low. Our aim would be to reduce market exposure during such periods and increase exposure when the VIX is in decline.

Forecasting the VIX index is a complex topic in its own right. The approach I am going to take here is simpler: instead of developing a forecasting model, I am going to use an algorithm to “trade” the VIX index. When the trading model “buys” the VIX index, we will assume it is anticipating increased market volatility and lighten our exposure accordingly. When the model “sells” the VIX, we will increase market exposure.

Don’t be misled by the apparent simplicity of this approach: a trading algorithm is often much more complex in its structure than even a very sophisticated forecasting model. For example, it can incorporate many different kinds of non-linear behavior and dynamically adjust its investment horizon. The results from such a trading algorithm, produced by our quantitative modeling system, are set out in the figure below.

Fig. 2a -VIX Trading Algorithm – Equity Curve

Source: TradeStation Technologies Inc.

Fig. 2b -VIX Trading Algorithm – Performance Analysis

Source: TradeStation Technologies Inc.

Not only is the strategy very profitable, it has several desirable features, including a high percentage of winning trades. If this were an actual trading system, we might want to trade it in production. But, of course, it is only a theoretical model – the VIX index itself is not tradable – and, besides, the intention here is not to trade the algorithm, but to use it for market timing purposes.

Our approach is straightforward: when the algorithm generates a “buy” signal in the VIX, we will reduce our exposure to the market. When the system initiates a “sell”, we will increase our market exposure. Trades generated by the VIX algorithm are held for around five days on average, so we can anticipate rebalancing our portfolio approximately weekly. In what follows, we will assume that we adjust our position by trading the SPY ETF at the closing price the day following a signal from the VIX model. We will apply trading commissions of $1c per share and a further $1c per share in slippage.

Hedging Strategies

Let’s begin our evaluation by looking at the outcome if we adjust the SPY holding in our market portfolio by 20% whenever the VIX model generates a signal. When the model buys the VIX, we will reduce our original SPY holding by 20%, and when it sells the VIX, we will increase our SPY holding by 20%, using the original holding in the long only portfolio as a baseline. We refer to this in the chart below as the MT 20% hedge portfolio.

Fig. 3 Value of $1000 – Long only vs MT 20% hedge portfolio

Source: Yahoo! Finance, 2014

The hedge portfolio dominates the long only portfolio over the entire period from 1999, producing a total net return of 156% compared to 112% for the SPY ETF. Not only is the rate of return higher, at 10.00% vs. 7.21% annually, volatility in investment returns is also significantly reduced (17.15% vs 20.48%). Although it, too, suffers substantial drawdowns in 2000 and 2008/09, the effects on the hedge portfolio are less severe. It appears that our market timing approach adds value.

The selection of 20% as a hedge ratio is somewhat arbitrary – an argument can be made for smaller, or larger, hedge adjustments. Let’s consider a different scenario, one in which we exit our long-only position entirely, whenever the VIX algorithm issues a buy order. We will re-buy our entire original SPY holding whenever the model issues a sell order in the VIX. We refer to this strategy variant as the MT cash out portfolio. Let’s look at how the results compare.

Fig. 4 Value of $1,000 – Long only vs MT cash out portfolio

Source: Yahoo! Finance, 2014

The MT cash out portfolio appears to do everything we hoped for, avoiding the downturn of 2000 almost entirely and the worst of the market turmoil in 2008/09. Total net return over the period rises to 165%, with higher average annual returns of 10.62%. Annual volatility of 9.95% is less than half that of the long only portfolio.

Finally, let’s consider a more extreme approach, which I have termed the “MT aggressive portfolio”. Here, whenever the VIX model issues a buy order we sell our entire SPY holding, as with the MT cash out strategy. Now, however, whenever the model issues a sell order on the VIX, we invest heavily in the market, buying double our original holding in SPY (i.e. we are using standard, reg-T leverage of 2:1, available to most investors). In fact, our average holding over the period turns out to be slightly lower than for the original long only portfolio because we are 100% in cash for slightly more than half the time. But the outcome represents a substantial improvement.

Fig. 5 Value of $1,000 – Long only vs. MT aggressive portfolio

Source: Yahoo! Finance, 2014

Total net returns for the MT aggressive portfolio at 330% are about three times that of the original long only portfolio. Annual volatility at 14.90% is greater than for the MT cash out portfolio due to the use of leverage. But this is still significantly lower than the 20.48% annual volatility of the long only portfolio, while the annual rate of return of 21.16% is the highest of the group, by far. And here, too, the hedge strategy succeeds in protecting our investment portfolio from the worst of the effects of downturns in 2000 and 2008.

Conclusion

Whatever the basis for their underlying investment strategy, investors can benefit by using quantitative market timing techniques to strategically adjust their market exposure. Market timing can be a useful tool to avoid major downturns, increasing investment returns while reducing volatility. This could be especially relevant in the weeks and months ahead, as we may be facing a period of greater uncertainty and, potentially at least, the risk of a significant market correction.

Disclosure: The author has no positions in any stocks mentioned, and no plans to initiate any positions within the next 72 hours. The author wrote this article themselves, and it expresses their own opinions. The author is not receiving compensation for it (other than from Seeking Alpha). The author has no business relationship with any company whose stock is mentioned in this article.

September 11, 2018

How Not to Develop Trading Strategies – A Cautionary Tale

In his post on Multi-Market Techniques for Robust Trading Strategies (http://www.adaptrade.com/Newsletter/NL-MultiMarket.htm) Michael Bryant of Adaptrade discusses some interesting approaches to improving model robustness. One is to use data from several correlated assets to build the model, on the basis that if the algorithm works for several assets with differing price levels, that would tend to corroborate the system’s robustness. The second approach he advocates is to use data from the same asset series at different bars lengths. The example he uses @ES.D at 5, 7 and 9 minute bars. The argument in favor of this approach is the same as for the first, albeit in this case the underlying asset is the same.

I like Michael’s idea in principle, but I wanted to give you a sense of what can all too easily go wrong with GP modeling, even using techniques such as multi-time frame fitting and Monte Carlo simulation to improve robustness testing.

In the chart below I have extended the analysis back in time, beyond the 2011-2012 period that Michael used to build his original model. As you can see, most of the returns are generated in-sample, in the 2011-2012 period. As we look back over the period from 2007-2010, the results are distinctly unimpressive – the strategy basically trades sideways for four years.

How do Do It Right

In my view, there is only one, safe way to use GP to develop strategies. Firstly, you need to use a very long span of data – as much as possible, to fit your model. Only in this way can you ensure that the model has encountered enough variation in market conditions to stand a reasonable chance of being able to adapt to changing market conditions in future.

Secondly, you need to use two OOS period. The first OOS span of data, drawn from the start of the data series, is used in the normal way, to visually inspect the performance of the model. But the second span of OOS data, from more recent history, is NOT examined before the model is finalized. This is really important. Products like Adaptrade make it too easy for the system designer to “cheat”, by looking at the recent performance of his trading system “out of sample” and selecting models that do well in that period. But the very process of examining OOS performance introduces bias into the system. It would be like adding a line of code saying something like:

IF (model performance in OOS period > x) do the following….

I am quite sure if I posted a strategy with a line of code like that in it, it would immediately be shot down as being blatantly biased, and quite rightly so. But, if I look at the recent “OOS” performance and use it to select the model, I am effectively doing exactly the same thing.

That is why it is so important to have a second span of OOS data that it not only not used to build the model, but also is not used to assess performance, until after the final model selection is made. For that reason, the second OOS period is referred to as a “double blind” test.

That’s the procedure I followed to build my futures daytrading strategy: I used as much data as possible, dating from 2002. The first 20% of the each data set was used for normal OOS testing. But the second set of data, from Jan 2012 onwards, was my double-blind data set. Only when I saw that the system maintained performance in BOTH OOS periods was I reasonably confident of the system’s robustness.

This further explains why it is so challenging to develop higher frequency strategies using GP. Running even a very fast GP modeling system on a large span of high frequency data can take inordinate amounts of time.

The longest span of 5-min bar data that a GP system can handle would typically be around 5-7 years. This is probably not quite enough to build a truly robust system, although if you pick you time span carefully it might be (I generally like to use the 2006-2011 period, which has lots of market variation).

For 15 minute bar data, a well-designed GP system can usually handle all the available data you can throw at it – from 1999 in the case of the Emini, for instance.

Why I don’t Like Fitting Models over Short Time Spans

The risks of fitting models to data in short time spans are intuitively obvious. If you happen to pick a data set in which the market is in a strong uptrend, then your model is going to focus on that kind of market behavior. Subsequently, when the trend changes, the strategy will typically break down.
Monte Carlo simulation isn’t going to change much in this situation: sure, it will help a bit, perhaps, but since the resampled data is all drawn from the same original data set, in most cases the simulated paths will also show a strong uptrend – all that will be shown is that there is some doubt about the strength of the trend. But a completely different scenario, in which, say, the market drops by 10%, is unlikely to appear.

One possible answer to that problem, recommended by some system developers, is simply to rebuild the model when a breakdown is detected. While it’s true that a product like MSA can make detection easier, rebuilding the model is another question altogether. There is no guarantee that the kind of model that has worked hitherto can be re-tooled to work once again. In fact, there may be no viable trading system that can handle the new market dynamics.

Here is a case in point. We have a system that works well on 10 min bars in TF.D up until around May 2012, when MSA indicates a breakdown in strategy performance.

So now we try to fit a new model, along the pattern of the original model, taking account some of the new data. But it turns out to be just a Band-Aid – after a few more data points the strategy breaks down again, irretrievably.

This is typical of what often happens when you use GP to build a model using s short span of data. That’s why I prefer to use a long time span, even at lower frequency. The chances of being able to build a robust system that will adapt well to changing market conditions are much higher.

A Robust Emini Trading System

Here, for example is a GP system build on daily data in @ES.D from 1999 to 2011 (i.e. 2012 to 2014 is OOS).

September 10, 2018

A Primer on Genetic Programming

Posted by androidMarvin:

Genetic programming is an approach to letting the computer generate its own program code, rather than have a person write the program. It doesn’t specifically “find patterns” or rules within data structures. It starts with a number of randomly-constructed (as long as they are mathematically valid) sample programs, evaluates how close each one is to achieving what the desired result program should achieve, then steadily modifies the best matches to the desired target program in order to improve their match to the desired target; the original random attempts “evolve” towards a better match by natural selection, the best ones being selected to act as the basis for the next generation of attempts.

A tree representing a candidate formula could be represented as follows:

It basically shows the mathematical operations that will be used in the formula, the order in which they are applied, and what values they act on. When the EL Verifier is analysing a statement like

value1 = sin( X ) / a + b * cos( X )

it has to see work out what order the parts of the statement should be evaluated in, which a person sees immediately; effectively, the Verifier constructs the tree diagram above, so that it knows that it has to generate code to make the computer :

take the value of variable X and pass it through a call to the sin() functio
take that result, and divide it by the value of a
take the value of variable X and pass it through a call to the cos() functio
take that result and multiply it by the value of variable
take the result of step 2 and the result of step 4 and add the
that result is the value of Y for the input value of X

Tradestation optimiser would take a single such tree, defining a fixed formula, and attempt to fit it to the data by varying the values of variables a and b. A Genetic Programming optimiser could do the same, but it also has the freedom to change the mathematical operators and the merge points in the tree, and change the shape of the tree to make the formula more or less complex as well; it can adjust both the parameters to the equation and the equation itself in order to evolve it to a better result.

For a mathematical curve fit, a GP optimiser would evaluate each individual tree by applying all the measured X values to the tree’s inputs, compare each output to the measured Y values, and sum a measure of the error over all the data; that sum would be the measure of how well the current tree matches the measure data. The “genetic” part of the name derives from the way it tries to evolve the population of trees its using to find the best.

The main evolution technique is “crossover”. When two parent animals create offspring, each offspring will get part of its DNA from one parent and part from the other; improvement of the species happens if some of the offspring get DNA component combinations that suit the environment better than their parents are suited. The GP optimiser emulates this process by selecting two parent trees, and swapping a section of one of those trees with a section of tree from the other parent, to create two offspring. Eg given parent trees

representing equations

value1 = sin( X )/a + b * cos( X )

and

value1 = cos( X ) / a + b * sin( X )

the offspring might be

representing equations

value1 = sin( X )/cos( X ) + b * cos( X )

and

value1 = a / a + b * sin( X )

Those specific changes are unlikely to both be an improvement, but that’s the way with random processes; the changes made aren’t guided by any sort of principle, its just a case of “change something, anything, and see if its any better”.

A secondary change process that can be used is “mutation”, in which something about a single tree is simply changed, not swapped. This is intended to introduce diversity, so that if none of the current trees is a particularly good performer, there’s a chance that something radically better might be brought into the pool.

The push trying to steer the evolution towards a better result comes from deciding which parents are allowed to create offspring. The original idea was that all the current trees were ranked in sorted order of their fitness, the worst ones were removed from the population to be replaced by new offspring, amd the trees that were the best performing are selected to be parents – so the weak die, and the strongest breed, hoping their offspring will be at least as good as the parents.

One reservation I have about a product like Adaptrade Builder is that it doesn’t follow this original pattern. It chooses “a few” (2 by default) trees to be considered as parents, by entering a “tournament” and the best tree in the tournament is selected as a parent. This seems to me to reduce the bias towards breeding strength with strength, but I’m no expert.

Rather than being simply mathematical, Builder seems to generate tests for entry and exit orders. It takes arithmetic and comparison operators for granted, and allows trees to be built from technical indicators rather than mathematical functions like sin() and cos(). So where an EL programmer might write

if average( Close, fast ) crosses above Average( ( High + Low )/2, slow ) and CCI( length ) > overbought then buy

Builder would have a tree

from which an offspring might be generated as :

to use a Buy test

if average( ( High + Low )/2, fast ) crosses above Average( Close, slow ) and CCI( length ) > overbought then buy

The structure of the test to go long has changed, but in a random rather than the guided way a human might do when trying to develop a strategy.

September 9, 2018

Developing High Performing Trading Strategies with Genetic Programming

One of the frustrating aspects of research and development of trading systems is that there is never enough time to investigate all of the interesting trading ideas one would like to explore. In the early 1970’s, when a moving average crossover system was considered state of the art, it was relatively easy to develop profitable strategies using simple technical indicators. Indeed, research has shown that the profitability of simple trading rules persisted in foreign exchange and other markets for a period of decades. But, coincident with the advent of the PC in the late 1980’s, such simple strategies began to fail. The widespread availability of data, analytical tools and computing power has, arguably, contributed to the increased efficiency of financial markets and complicated the search for profitable trading ideas. We are now at a stage where is can take a team of 5-6 researchers/developers, using advanced research techniques and computing technologies, as long as 12-18 months, and hundreds of thousands of dollars, to develop a prototype strategy. And there is no guarantee that the end result will produce the required investment returns.

The lengthening lead times and rising cost and risk of strategy research has obliged trading firms to explore possibilities for accelerating the R&D process. One such approach is Genetic Programming.

Early Experiences with Genetic Programming
I first came across the GP approach to investment strategy in the late 1990s, when I began to work with Haftan Eckholdt, then head of neuroscience at Yeshiva University in New York. Haftan had proposed creating trading strategies by applying the kind of techniques widely used to analyze voluminous and highly complex data sets in genetic research. I was extremely skeptical of the idea and spent the next 18 months kicking the tires very hard indeed, of behalf of an interested investor. Although Haftan’s results seemed promising, I was fairly sure that they were the product of random chance and set about devising tests that would demonstrate that.

One of the challenges I devised was to create data sets in which real and synthetic stock series were mixed together and given to the system evaluate. To the human eye (or analyst’s spreadsheet), the synthetic series were indistinguishable from the real thing. But, in fact, I had “planted” some patterns within the processes of the synthetic stocks that made them perform differently from their real-life counterparts. Some of the patterns I created were quite simple, such as introducing a drift component. But other patterns were more nuanced, for example, using a fractal Brownian motion generator to induce long memory in the stock volatility process.

It was when I saw the system detect and exploit the patterns buried deep within the synthetic series to create sensible, profitable strategies that I began to pay attention. A short time thereafter Haftan and I joined forces to create what became the Proteom Fund.

That Proteom succeeded at all was a testament not only to Haftan’s ingenuity as a researcher, but also to his abilities as a programmer and technician. Processing such large volumes of data was a tremendous challenge at that time and required a cluster of 50 cpu’s networked together and maintained with a fair amount of patch cable and glue. We housed the cluster in a rat-infested warehouse in Brooklyn that had a very pleasant view of Manhattan, but no a/c. The heat thrown off from the cluster was immense, and when combined with very loud rap music blasted through the walls by the neighboring music studios, the effect was debilitating. As you might imagine, meetings with investors were a highly unpredictable experience. Fortunately, Haftan’s intellect was matched by his immense reserves of fortitude and patience and we were able to attract investments from several leading institutional investors.

The Genetic Programming Approach to Building Trading Models

Genetic programming is an evolutionary-based algorithmic methodology which can be used in a very general way to identify patterns or rules within data structures. The GP system is given a set of instructions (typically simple operators like addition and subtraction), some data observations and a fitness function to assess how well the system is able to combine the functions and data to achieve a specified goal.

In the trading strategy context the data observations might include not only price data, but also price volatility, moving averages and a variety of other technical indicators. The fitness function could be something as simple as net profit, but might represent alternative measures of profitability or risk, with factors such as PL per trade, win rate, or maximum drawdown. In order to reduce the danger of over-fitting, it is customary to limit the types of functions that the system can use to simple operators (+,-,/,*), exponents, and trig functions. The length of the program might also be constrained in terms of the maximum permitted lines of code.

We can represent what is going on using a tree graph:

In this example the GP system is combining several simple operators with the Sin and Cos trig functions to create a signal comprising an expression in two variables, X and Y, which may be, for example, stock prices, moving averages, or technical indicators of momentum or mean reversion.
The “evolutionary” aspect of the GP process derives from the idea that an existing signal or model can be mutated by replacing nodes in a branch of a tree, or even an entire branch by another. System performance is re-evaluated using the fitness function and the most profitable mutations are retained for further generation.
The resulting models are often highly non-linear and can be very general in form.

A GP Daytrading Strategy
The last fifteen years has seen tremendous advances in the field of genetic programming, in terms of the theory as well as practice. Using a single hyper-threaded CPU, it is now possible for a GP system to generate signals at a far faster rate than was possible on Proteom’s cluster of 50 networked CPUs. A researcher can develop and evaluate tens of millions of possible trading algorithms with the space of a few hours. Implementing a thoroughly researched and tested strategy is now feasible in a matter of weeks. There can be no doubt of GP’s potential to produce dramatic reductions in R&D lead times and costs. But does it work?

To address that question I have summarized below the performance results from a GP-developed daytrading system that trades nine different futures markets: Crude Oil (CL), Euro (EC), E-Mini (ES), Gold (GC), Heating Oil (HO), Coffee (KC), Natural gas (NG), Ten Year Notes (TY) and Bonds (US). The system trades a single contract in each market individually, going long and short several times a day. Only the most liquid period in each market is traded, which typically coincides with the open-outcry session, with any open positions being exited at the end of the session using market orders. With the exception of the NG and HO markets, which are entered using stop orders, all of the markets are entered and exited using standard limit orders, at prices determined by the system

The system was constructed using 15-minute bar data from Jan 2006 to Dec 2011 and tested out-of-sample of data from Jan 2012 to May 2014. The in-sample span of data was chosen to cover periods of extreme market stress, as well as less volatile market conditions. A lengthy out-of-sample period, almost half the span of the in-sample period, was chosen in order to evaluate the robustness of the system.
Out-of-sample testing was “double-blind”, meaning that the data was not used in the construction of the models, nor was out-of-sample performance evaluated by the system before any model was selected.

Performance results are net of trading commissions of $6 per round turn and, in the case of HO and NG, additional slippage of 2 ticks per round turn.

(click on the table for a higher definition view)

The most striking feature of the strategy is the high rate of risk-adjusted returns, as measured by the Sharpe ratio, which exceeds 5 in both in-sample and out-of-sample periods. This consistency is a reflection of the fact that, while net returns fall from an annual average of over 29% in sample to around 20% in the period from 2012, so, too, does the strategy volatility decline from 5.35% to 3.86% in the respective periods. The reduction in risk in the out-of-sample period is also reflected in lower Value-at-Risk and Drawdown levels.

A decline in the average PL per trade from $25 to $16 in offset to some degree by a slight increase in the rate of trading, from 42 to 44 trades per day, on average, while daily win rate and percentage profitable trades remain consistent at around 65% and 56%, respectively.

Overall, the system appears to be not only highly profitable, but also extremely robust. This is impressive, given that the models were not updated with data after 2011, remaining static over a period almost half as long as the span of data used in their construction. It is reasonable to expect that out-of-sample performance might be improved by allowing the models to be updated with more recent data.

Benefits and Risks of the GP Approach to Trading System Development
The potential benefits of the GP approach to trading system development include speed of development, flexibility of design, generality of application across markets and rapid testing and deployment.

What about the downside? The most obvious concern is the risk of over-fitting. By allowing the system to develop and test millions of models, there is a distinct risk that the resulting systems may be too closely conditioned on the in-sample data, and will fail to maintain performance when faced with new market conditions. That is why, of course, we retain a substantial span of out-of-sample data, in order to evaluate the robustness of the trading system. Even so, given the enormous number of models evaluated, there remains a significant risk of over-fitting.

Another drawback is that, due to the nature of the modelling process, it can be very difficult to understand, or explain to potential investors, the “market hypothesis” underpinning any specific model. “We tested it and it works” is not a particularly enlightening explanation for investors, who are accustomed to being presented with a more articulate theoretical framework, or investment thesis. Not being able to explain precisely how a system makes money is troubling enough in good times; but in bad times, during an extended drawdown, investors are likely to become agitated very quickly indeed if no explanation is forthcoming. Unfortunately, evaluating the question of whether a period of poor performance is temporary, or the result of a breakdown in the model, can be a complicated process.

Finally, in comparison with other modeling techniques, GP models suffer from an inability to easily update the model parameters based on new data as it become available. Typically, as GP model will be to rebuilt from scratch, often producing very different results each time.

Conclusion
Despite the many limitations of the GP approach, the advantages in terms of the speed and cost of researching and developing original trading signals and strategies have become increasingly compelling.

Given the several well-documented successes of the GP approach in fields as diverse as genetics and physics, I think an appropriate position to take with respect to applications within financial market research would be one of cautious optimism.

September 8, 2018

A Scalping Strategy in E-Mini Futures

This is a follow up post to my post on the Mathematics of Scalping. To illustrate the scalping methodology, I coded up a simple strategy based on the techniques described in the post.

The strategy trades a single @ES contract on 1-minute bars. The attached ELD file contains the Easylanguage code for ES scalping strategy, which can be run in Tradestation or Multicharts.

This strategy makes no attempt to forecast market direction and doesn’t consider market trends at all. It simply looks at the current levels of volatility and takes a long volatility position or a short volatility position depending on whether volatility is above or below some threshold parameters.

By long volatility I mean a position where we buy or sell the market and set a loose Profit Target and a tight Stop Loss. By short volatility I mean a position where we buy or sell the market and set a tight Profit Target and loose Stop Loss. This is exactly the methodology I described earlier in the post. The parameters I ended up using are as follows:

Long Volatility: Profit Target = 8 ticks, Stop Loss = 2 ticks
Short Volatility: Profit Target = 2 ticks, Stop Loss = 30 ticks

I have made no attempt to optimize these parameters settings, which can easily be done in Tradestation or Multicharts.

What do we mean by volatility being above our threshold level? I use a very simple metric: I take the TrueRange for the current bar and add 50% of the increase or decrease in TrueRange over the last two bars. That’s my crude volatility “forecast”.

The final point to explain is this: let’s suppose our volatility forecast is above our threshold level, so we know we want to be long volatility. Ok, but do we buy or sell the ES? One approach ia to try to gauge the direction of the market by estimating the trend. Not a bad idea, by any means, although I have argued that volatility drowns out any trend signal at short time frames (like 1 minute, for example). So I prefer an approach that makes no assumptions about market direction.

In this approach what we do is divide volatility into upsideVolatility and downsideVolatility. upsideVolatility uses the TrueRange for bars where Close > Close[1]. downsideVolatility is calculated only for bars where Close < Close[1]. This kind of methodology, where you calculate volatility based on the sign of the returns, is well known and is used in performance measures like the Sortino ratio. This is like the Sharpe ratio, except that you calculate the standard deviation of returns using only days in which the market was down. When it’s calculated this way, standard deviation is known as the (square root of the) semi-variance.

Anyway, back to our strategy. So we calculate the upside and downside volatilities and test them against our upper and lower volatility thresholds.

The decision tree looks like this:

LONG VOLATILITY
If upsideVolatilityForecast > upperVolThrehold, buy at the market with wide PT and tight ST (long market, long volatility)
If downsideVolatilityForecast > upperVolThrehold, sell at the market with wide PT and tight ST (short market, long volatility)

SHORT VOLATILITY
If upsideVolatilityForecast < lowerVolThrehold, sell at the Ask on a limit with tight PT and wide ST (short market, short volatility)
If downsideVolatilityForecast < lowerVolThrehold, buy at the Bid on a limit with tight PT and wide ST (long market, short volatility)

NOTE THE FOLLOWING CAVEATS. DO NOT TRY TO TRADE THIS STRATEGY LIVE (but use it as a basis for a tradable strategy)

1. The strategy makes the usual TS assumption about fill rates, which is unrealistic, especially at short intervals like 1-minute.
2. The strategy allows fees and commissions of $3 per contract, or $6 per round turn. Your trading costs may be higher than this.
3. Tradestation is unable to perform analysis at the tick level for a period as long at the one used here (2000 to 2014). A tick by tick analysis would likely show very different results (better or worse).
4. The strategy is extremely lop-sided: the great majority of the profits are made on the long side and the Win Rates and Profit Factors are very different for long trades vs short trades. I suspect this would change with a tick by tick analysis. But it also may be necessary to add more parameters so that long trades are treated differently from short trades.
5. No attempt has been made to optimize the parameters.
6 This is a daytading strategy that will exit the market on close.

So with all that said here are the results.

As you can see, the strategy produces a smooth, upward sloping equity curve, the slope of which increases markedly during the period of high market volatility in 2008.
Net profits after commissions for a single ES contract amount to $243,000 ($3.42 per contract) with a win rate of 76% and Profit Factor of 1.24.

This basic implementation would obviously require improvement in several areas, not least of which would be to address the imbalance in strategy profitability on the short vs long side, where most of the profits are generated.