Does Trend Following Work?

I’m not sure how I came across it, but I have had Jez Liberty’s Au.Tra.Sy blog in my reader since around 2009.

Since then, he has tracked well-known trend following systems and reported monthly performance figures. These are things like moving average crossovers, Bollinger band breakouts and stuff like that.

The systems had a very good month in November, and a very good year, up ~45% YTD.

I thought I would take a look and how things have gone over the time it has been tracked. Helpfully, Jez posts annual summaries for each system and a composite average, which is what I used. You can find the links to specific posts in the script linked under the table below. N.B the first two years are the average of actual trend following funds not the generic strategies.

There are results for 6 years, from 2009 to 2014, using the YTD November figure for 2014. We end up with a series like this, starting from 100.

Source [R]

We can see that 4 of the 6 years ended under water. Over the same time there has been a huge surge in US equities. A simple buy and hold in 2009 would have more than doubled equity.  

The pain of losing

I have been following Jez posting results every month for 5 – 6 years now. When I think back to where I was in 2009 versus where I am today, five years is a really, really long time.

You can get historical data from 1920s and even further back in time. If you run a backtest over 80 – 90 years of data, a 3 – 5 year period of underperformance is barely noticeable on an equity curve.

Sometimes I see people get excited about their backtests even though they include these periods of poor performance. There is plenty of research and evidence that humans in general find it very hard to stick with a losing system.

How seriously would you take someone touting a system that is underwater after five years, but they assured you a 45% year was just around the corner?

Professional CTA

I believe the Au.Tra.Sy data is based on commodities and futures trading, so thought I would take a look at the Barclay CTA Index as a comparison. The results seem somewhat similar, it has struggled somewhat since 2010, though both the upside and downside returns are smaller. 

In the 29 years from 1980 to 2008 inclusive, there were only 3 down years in total, none of them consecutive.

In the period we are examining here, the 6 years from 2009 – 2014, four of them have been down years, including three down years in a row (2011 - 2013).

Proselytizing

In some ways, you can think of mechanical trend following as being long tail risk. Effectively it wants a big sustained move in either direction. If you think about the distribution of returns, the big moves that are “pay days” are out in the tails.

I don’t follow commodities vol at all, but apparently it has been low for several years, according to this article from September, 2014. It probably shouldn’t be a surprise that strategies dependent on big moves have not performed well in an environment of low volatility. 

I personally dislike these indicator-based trend following systems, even though I think they give entries as good as any. The problem is they are too slow to close positions and give back too much profit.

They only really make serious money when a megatrend eventuates, and these are relatively rare and IMO getting rarer. In the mean time they can get chopped around and experience significant drawdowns.

I don’t want to draw any final conclusions about classical trend following, but I know I would have a lot of trouble sticking to systems like these in practise, even though a long term backtest might look really nice.

However, if you understand the conditions under which your system does well or is likely to underperform, it is a lot easier to stick with it during periods of underperformance, which are inevitable over the long run.

It is worth spending the time thinking about the conditions in which your system does well and does poorly. 

Trading in a low vol world

I wanted to take a look at what works in low vol environments, such as we are currently experiencing. I am open to the idea we have entered a period of structurally low volatility due to increased regulatory burden and flow on effects from the decline of institutional FICC trading. Or it may just be a function of QE, and post-tapering we will see a return to higher levels.

The plan

The main idea is to compare mean reversion (MR) vs. follow through (FT). For simplicity I define mean reversion as an up day being followed by a down day, and a down day being followed by an up day. Conversely, follow through sees an up day followed by another up day, and a down day followed by a down day.

I took a look at the major US equity indices, SPX (GSPC), NDX and RUT.

For each series we calculate daily log returns for the current period and shift the forward to get the return for the next period. Then we calculate realized volatility (RV) and split the data set into "low volatility" and "high volatility", by looking at median realized vol for the whole series.

Then, for each series, we use bootstrapped samples to simulate a number of trajectories/equity curves for each strategy (MR/FT) under the two classes of RV. Finally we take an average of the total return of each trajectory to get a ballpark idea of how they went.

Results

The data is from the start of 1999 to the present, so roughly 15 years. Each run generates 1000 trajectories with a sample size of roughly 950.

For the low vol case, the results are unfortunately ambiguous. Follow through in a low vol environment seemed to do well for NDX and RUT, but the opposite was the case for SPX.

The TR column is the sum of the series over the whole period for the volatility class (i.e. a simple long only strategy), giving an idea of a directional bias that may be present in the sampling.

In the high vol environment, mean reversion was a clear winner, and consistent over the different underlyings.

The results seem relatively stable across trajectory size/sample size.

Outro

I'm not really sure what is going on SPX. My intuition was that FT would do well in low vol environments, but that doesn't seem to be the case, at least not for SPX.

I was actually getting consistent votes for FT in the low vol case, then restarted R to run with a clean environment and started getting the above instead. You can't spell argh without R it seems.

Source is up here. As always you can find me on twitter here. Thanks for stopping by.

What can we use the VIX for?

In part 1, we took a look at VIX and the relationship it had between historical volatility and realized volatility.

Continuing on, I thought I would take a look at next day returns and the VIX. There is a relationship between SPX and VIX in that when SPX drops, VIX typically rises. This leads the question: does a high VIX serve as a useful indicator of what might happen to SPX the next day?

A problem in answering this is quantifying what we mean by a "high" VIX. A VIX of 25 might be high if it has been under 20 for the last 3 months, but might be considered low if it has been around 30.

To deal with this I use a proportional measure based on its previous trading range. If it is at the top of its range, it gets a 1, and at the bottom it gets a 0. Anywhere in between it gets a number between 0 and 1.

I used data from 2000-2013 and a 252 period lookback, which is roughly 1 year, recording the next day return of SPX. I split this data into 5 groups, the first with relative VIX readings up to 0.2, the second between 0.2 and 0.4, and so on up to 1.0. Then I plotted the next day returns as a boxplot, giving the distribution of returns for a VIX reading in its relative quintile.

The box on the far left corresponds to the lowest relative VIX levels, while the box on the far right corresponds to the highest.

One thing that jumps out to me is that the mean of all these returns is more or less zero. Using this relative measure of VIX has no value as a predictor of next day returns, as whatever measure we use, a mean return of zero will see us end up flat (or in practice, probably down a bit). If there was a real edge, we would expect the mean return to be somewhere other than zero.

The second thing that I notice is the distribution of returns is a lot wider for the right most two boxes, which corresponds to relative VIX measure above 0.60, i.e. relatively higher volatility. Now this makes sense, as we would reasonably expect periods of higher volatility to have higher volatility.

It seems that high readings of VIX can serve as a useful predictor of higher periods of next day volatility, and for someone trying to implement a volatility filter, that could be quite useful.

Conclusion

To recap, we have seen that:

1) VIX is not very good at forecasting 30 day realized volatility.
2) Historical volatility can be a reasonable proxy for IV/VIX.
3) Higher periods of volatility do portend continued higher volatility, at least for the short term (1 day).

Note that 3 does contradict the mean reversion of volatility, because it is looking at the next day returns only. Over a longer period we would certainly expect it mean revert, otherwise we'd be experiencing some very, very bumpy rides ...

Also, I did similar analysis using RUT & RVX and NDX & VXN, both of which had very similar results. We will take a look at them in the next post where we use the findings from this post and the previous to implement a volatility filter.

Code for this post and the previous is up: here

Digging into the VIX

I wanted to revisit using some sort of volatility filter for systematic trading. In particular, if we are trading SPX, can we somehow use the VIX to produce better risk adjust returns? This is not about trading volatility, but more about using additional "out of band" data in our systems.

I thought I would take a look at the VIX, what is it, and can it help us?

In theory, the VIX is the market consensus of what future 30 day realized volatility of SPX will actually be, as derived from option prices of SPX. Anyone who watches or trades SPX has seen that a drop in SPX usually results a rise in VIX, or the IV in SPX option chains.

Occasionally this relationship breaks down, which can be a useful as an indicator in itself, but I had a sneaking suspicion that the VIX would have more in common with historical vol than what realized vol turns out to be over the next 30 calendar days. That is, the VIX is not particularly useful as a longer term forward looking indicator of volatility, at least no more than current historical volatility.

I did a bit of data gathering and calculated realized and historical volatility for SPX. I used 21 periods when calculating vols as the VIX covers 30 calendar days, which is roughly a month, and there are roughly 21 trading days in an average month.

First of all, we can take a look in hindsight at how well the VIX forecast of volatility matched the actual realized volatility over the next 21 days.

In this chart, the red line is what realized volatility turned out to be, going forward from a given day. It is impossible to know this in advance, so we are engaging in some serious data snooping, for educational purposes only. The teal line is the VIX. Below them is a plot of VIX - SPX realized vol, and flagrant abuse of gradients.

You can see that the VIX gives an estimate that is typically higher than what actually eventuated, a well-known phenomenon, which for convenience I will put in "The Volatility Risk Premium" basket.

There is one big divergence around the end of July, 2011, when the VIX was significantly underestimating what realized vol would actually turn out to be.  In August 2011 there was a large selloff and you can see the VIX jumped to a rather ebullient level.

If we moved the red line forward 21 periods, we would get historical volatility, that is, the actually volatility that occurred over the last 21 trading periods.

We can see the relationship is much closer. The VIX still generally reflects a premium over historical volatility, but big moves in the underlying SPX correlate to big moves in VIX/IV.

Relations

The correlations between these three measures, IV, RV and HV are worth taking a look at. This is using data from January 2000 up to about the end of January 2013:

There is a 91% correlation between historical volatility and VIX. This is higher than the correlation between HV and RV, and HV is really just RV shifted back 21 periods. It seems the relation between VIX and HV is something quite strong.

Now because these are all measures of very similar things, we might expect high correlation. Another metric we could use too look at the relationship is R-Squared.

We can see a relatively strong relationship between historical vol and IV/VIX, at 0.60. The relationship between IV and realized vol is looking a lot weaker.

Conclusion

From all of this, I concluded that the VIX is not a particularly useful indicator of 30 day future realized volatility. However, it is a reasonable estimate of what historical volatility was.

As it turns out, this is not particularly useful, as we can easily calculate historical volatility. It does mean that we can flip things around, and use historical volatility as a reasonable proxy for VIX or market expectations of future realized volatility.

This will turn out to be quite useful, as we may find ourselves trading instruments that do not have nice, readily available VIX equivalents, but that is a topic for a subsequent post.

So ends part 1. These posts turned out very long so I decided to split them up, part 2 should be available here.

The R code for this and the next post is up here.

Another cut at market randomness

I have some background in computer security and one day found myself tasked with assessing the quality of randomness for session id tokens generated by popular web frameworks (namely Java and .NET). As it turns out, NIST have developed a series of tests for just this purpose detailed here.

As a non-believer in the absolute randomness of markets, I thought I might take a look at the series of returns from SPX/GSPC to see how they held up.

About the tests

The NIST tests are designed to be run on bit streams from a random number generator, so some liberties had to be taken. I am mainly interested in the predictability of up days vs down days, so I first generated a series of returns using quantmod then encoded them using a 1 for an up day and -1 for a down day and run the tests on the encoded stream. The NIST tests use -1 in place of a 0.

The main test is called the monobit test, to quote from the document:

The focus of the test is the proportion of zeroes and ones for the entire sequence.  The purpose of this test is to determine whether the number of ones and zeros in a sequence are approximately the same as would be expected for a truly random sequence.  The test assesses the closeness of the fraction of ones to ½, that is, the number of ones and zeroes in a sequence should be about the same.  

The tests are done using a significance level of 0.01, so come with a good degree in confidence assuming the underlying method is sound. 

One caveat is the length of the runs compared and how it relates to the distributions used to model the results. For the monobit test, the suggested input size is 100, requiring 101 days of data to determine 100 up or down days. If we were looking at 32 bit integers, 100 bits would only be 3 "full" random numbers, so arguably we would want to look at shorter time periods (e.g. 3-5 days of data). Given the difficulties around distributions which require a large n, I thought I would vary the significance level instead of a lower n, as our requirements are not as stringent as those for cryptographic random numbers.   

Results

At a basic level, this series does appear to be random, at least the vast majority of the time with n = 100 and alpha = 0.01. My confirmation bias was very upset with this. 

However, if we plot the proportion of runs deemed random vs the significance level, we see the proportion rising as one might expect. One thing that remains unexplained is why this appears to rise in steps rather than something more linear, though I expect this to be a side affect of either methodology or the normalisation done by the tests. I also took a look at the weekly data, which tends to a greater proportion of non-random runs quicker than daily data.

I am interested in the applications of machine learning to financial markets. Close to close returns that we have been looking at here are not the only information we have available, nor are they what I trade on a personal level. Also this is only one price series, and one could argue that in practise it is not actually a tradable series. 

Close to close returns are very useful in lots of applications, but if we are trying to build some predictive model we might need to look for more predictable pastures. Machine learning algorithms are great, but can only do so much. Finding some better potential inputs is what I will take a look at next.

The test code is available on github here: R Monobit test. Would be very interested to hear if anyone else takes a look.

I also took a visual and binomial look at randomness of the series in this post: A visual look at market randomness.

Oh and in case you were wondering, the web session tokens all turned out to be very strong.

Trading Mean Reversion with Augen Spikes

One of the more interesting things I have come across is the idea of looking at price changes in terms of recent standard deviation, a concept put forward by Jeff Augen. The gist is to express a close to close return as a function of the standard deviation of recent price moves. If returns were normally distributed you would expect moves of less than 1 standard deviation approximately 68% of the time. It's probably no surprise that actual returns don't appear to be normally distributed, so larger spikes are relatively frequent.

The plan

Augen uses it to gain an edge in detecting where option volatility is mispriced, but I believe the concept is very useful and can be used in other areas. I originally thought it might be useful as a volatility filter by highlighting "abnormal" spikes in the VIX but I'm yet to have much success with that. Another idea was that it might be useful for swing trading in these mean-reverting times, which turned out to be more fruitful.

The basic signal is to go short when an upwards spike greater than 1 standard deviation occurs, and to go long when a downwards spike greater than 1 occurs (i.e spike values greater than 1 or less -1 respectively). For reference I compared it to trading daily mean reversion (going short after an up day, going long after a down day), and fading RSI(2) extremes of 90/10.

How'd it go?

Overall it performed well, with the caveat it was only effective in markets where mean reversion strategies in general are effective. Interestingly, using the default parameters of a 20 day standard deviation with a 1 day lookback has outperformed both Daily Mean Reversion and RSI(2) over the last few years, both of which have fallen a bit flat during the same period.

Scaling into positions based on the size of the spike also seemed to be effective in producing good returns in absolute terms.

Final thoughts

Like most MR strategies it has an abysmal backtest if you start from the 70's. I am still fascinated by the regime change that took place between follow through and mean reversion around 1999/2000, which Marketsci has covered in some depth. Maybe one day I will figure it out.

A comparison of the equity curves is below, along with the code. I think this is a very useful way of looking at price changes, and will continue to investigate ways it can be put to good use.

require(quantmod)
require(PerformanceAnalytics)

slideapply <- function(x, n, FUN=sd) {
    v <- c(rep(NA, length(x)))
    for (i in n:length(x) ) {
        v[i] <- FUN(x[(i-n+1):i])
    }
    return(v)
}

augenSpike <- function(x, n=20, k=1) {
    prchg <- c(rep(NA,k), diff(x, k))
    lgchg <- c(rep(NA,k), diff(log(x), k))
    stdevlgchg <- slideapply(lgchg, n, sd)
    stdpr <- x * stdevlgchg
    #shuffle things up one
    stdpr <- c(NA, stdpr[-length(stdpr)])
    spike <- prchg / stdpr
    return(spike)
}

perf <- function(x) { print(maxDrawdown(x)); table.AnnualizedReturns(x) }
retsum <- function(x) { return(exp(cumsum(na.omit(x))))}

getSymbols("SPY", from="2000-01-01")
spy <- SPY["2000/2011"]
spy$up_day <- ifelse(ROC(Cl(spy)) >=0, 1, 0 )
sig <- Lag(ifelse(spy$up_day == 1, -1, 1))
ret <- ROC(Cl(spy)) * sig
dmr_eq <- retsum(ret)
plot(dmr_eq)
perf(ret)
#[1] 0.3217873
#                          SPY.Close
#Annualized Return            0.1131
#Annualized Std Dev           0.2203
#Annualized Sharpe (Rf=0%)    0.5137

spy$rsi2 <- RSI(Cl(spy), 2)
sig <- Lag(ifelse(spy$rsi2 < 10, 1, ifelse(spy$rsi2 > 90, -1, 0)))
ret <- ROC(Cl(spy)) * sig
rsi_eq <- retsum(ret)
plot(rsi_eq)
perf(ret)
#[1] 0.1819428
#                          SPY.Close
#Annualized Return            0.0963
#Annualized Std Dev           0.1198
#Annualized Sharpe (Rf=0%)    0.8038

aus <- augenSpike(as.vector(Cl(spy)), k=1)
spy$spike <- aus
sig <- Lag(ifelse(spy$spike > 1, -1, ifelse(spy$spike < -1, 1, 0)))

ret <- ROC(Cl(spy)) * sig
k1_eq <- retsum(ret)
plot(k1_eq)
perf(ret)
#[1] 0.1379868
#                          SPY.Close
#Annualized Return            0.1066
#Annualized Std Dev           0.1152
#Annualized Sharpe (Rf=0%)    0.9256

aus <- augenSpike(as.vector(Cl(spy)), k=2)
spy$spike <- aus
sig <- Lag(ifelse(spy$spike > 1, -1, ifelse(spy$spike < -1, 1, 0)))

ret <- ROC(Cl(spy)) * sig
k2_eq <- retsum(ret)
plot(k2_eq)
perf(ret)
#[1] 0.2134826
#                          SPY.Close
#Annualized Return            0.1091
#Annualized Std Dev           0.1433
#Annualized Sharpe (Rf=0%)    0.7608

aus <- augenSpike(as.vector(Cl(spy)), k=1)
spy$spike <- aus
sig <- Lag(ifelse(spy$spike > 1, (-1 * spy$spike), ifelse(spy$spike < -1, abs(spy$spike), 0)))

ret <- ROC(Cl(spy)) * sig
k1_scaled_eq <- retsum(ret)
plot(k1_scaled_eq)
perf(ret)