## Correlation in practice â€“ equity curve synchronisation

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damian
Roundtable Knight Posts: 815
Joined: Tue Apr 15, 2003 8:43 pm
Location: dusseldorf

### Correlation in practice â€“ equity curve synchronisation

I have found some pairs show low correlation (monthly, last 5 years). An example is SI and HG at 0.34 based on futures or 0.37 based on spot physical. I am using Bloomberg data and function to calculate.

I know intuitively that a system I am researching produces synchronised equity curves for both product. They move in the same directions at roughly the same time. Big winners begin and end at the roughly the same time, along with whipsaw losers.

"The Same Time" is a phrase that is loosely prepared based on eyeball observation. It might mean SI exits a big winner a within a week or two of HG. Or in March they both had the same 1R loser, same again in April and one more time in July.

I find that a mathematical calculation of correlation based on price series sometimes misses these clearly synchronised equity curve pairs. Attempting to calculate mathematical correlation using the equity curves produces at times a less reliable result as well.

How to define and quantify 'similarity' or 'synchronised'?

Some terms that could be incorporated:

(a) Count the number of two-week periods where HG and SI both enter a trade of any direction.
(b) Count the number of times HG and SI exit a trade within a two week period, where both are either profitable or both unprofitable.
(a+b) a count of trading synchronisation
(c) ratio of (a+b) to total number of trades in HG and SI
(d) create a â€˜weightingâ€™ multiplier using the ratio of total trades in SI to total trades in HG. A ratio of 1 leads to a stronger equity curve correlation.

The synchronisation ranking could range between 0 and 1.

Example1:

(a) 20
(b) 20
(a+b) = 40

(c) 40/40 = 1

(d) 20/20 = 1

(c x d) = 1 x 1 = 1 therefore the equity curves are strongly synchronised.

Example 2:

(a) 10
(b) 10
(a+b) = 20

(c) 20/40 = 0.5

(d) 20/20 = 1

(c x d) = 1 x 0.5 = 0.5 therefore the equity curves are mildly synchronised.

Example 3:

(a) 15
(b) 15
(a+b) = 30

(c) 30/35 = 0.86

(d) 15/20 = 0.75

(c x d) = 0.75 x 0.86 = 0.65 therefore the equity curves are between mild and strongly synchronised.

Why should ex. 3 produce a higher synchronised rating than ex. 2? I donâ€™t think that it should.

Example 4:

(a) 10
(b) 2
(a+b) = 12

(c) 12/65 = 0.19

(d) 15/50 = 0.3

(c x d) = 0.19 x 0.3 = 0.06 therefore the equity curves are not synchronised.

There are clearly still lots of problems, not the least of which is defining (a) and (b) to account for instances when multiple trades are entered and exited in the same 2 week period in both symbol.
rabidric
Roundtable Knight Posts: 243
Joined: Mon Jan 09, 2006 7:45 am
it would seem fairly obvious that uncorrelated outputs(i.e. equity curves) is the holy grail of portfolio balancing.

since LTTF is essentially long delta and gamma, then you could argue that equity curves are a derivative of volatility shifts over certain time horizons. so instead of focussing on return correlations of products, maybe you should look at volatility correlations...

just a thought sluggo
Roundtable Knight Posts: 2987
Joined: Fri Jun 11, 2004 2:50 pm
rabidric wrote:it would seem fairly obvious that uncorrelated outputs(i.e. equity curves) is the holy grail of portfolio balancing.
Um, I don't think you mean uncorrelated outputs. I believe you mean negatively correlated outputs is the holy grail. Negative correlation decreases the variance of the combination. With less variance, the combination is a smoother ride, so those trading system evaluation statistics which prize smoothness (Sharpe Ratio, R-Squared, Lake Ratio, Ulcer Index, Return Retracement Ratio) will improve.

(Mathematically, it's the covariance of the two equity curves that's actually the lynchpin of wonderfulness. But since the correlation coefficient has the same sign as the covariance, and since people are more comfortable talking about correlation than talking about covariance, the former is more widely discussed than the latter.)
rabidric
Roundtable Knight Posts: 243
Joined: Mon Jan 09, 2006 7:45 am
no , i did mean UNcorrelated.

20 products say, in a portfolio, all negatively "volatility" correlated is somewhat unrealistic. and implies that many are likely to be positively "Volatility" correlated with others...
nodoodahs
Roundtable Knight Posts: 218
Joined: Wed Aug 09, 2006 4:01 pm
A portfolio of two negatively correlated assets would increase the volatility in a long/short portfolio, as both would be gaining equity when trending (you're long one and short the other), and both would be whipsawing at the same time.

A portfolio of two negatively correlated assets would, if traded long-only or short-only, be pretty smooth when both were trending, as you would only have a position in the positive trend but would be out of the negative trend. However, when both were whipsawing at the same time, you would have increased downside volatility.

A portfolio of two UNcorrelated assets would be the smoothest. The odds are that while one is whipsawing the other is trending, and vice versa.
damian
Roundtable Knight Posts: 815
Joined: Tue Apr 15, 2003 8:43 pm
Location: dusseldorf
Earlier I just asked three late workers in my office to guess the correlation of the blue and grey equity curves. Two said negatively correlated, one said zero.

Grey and blue equity curves have a correlation of +0.68 even though they never move in the same direction on the same day. And the longer those two equity curves continue like that, the closer the correlation moves towards 1.0

I am posting this as like me a several years ago, I am sure others who are reading this thread think that two equity curves that always move in the opposite direction on the same day are negatively correlated. It is a common perception error and also very misleading.

I now hand back to the smarter guys already active on this thread.

ps - having only just seen the correlation thread recently started by Chris67, I feel a bit silly for starting this strange one on the same topic
Attachments chart.jpg (69.87 KiB) Viewed 9270 times
damian
Roundtable Knight Posts: 815
Joined: Tue Apr 15, 2003 8:43 pm
Location: dusseldorf
Having extended the chart and plotted progressive correlation in pink (and broken a few rules).
Attachments chart3.jpg (44.08 KiB) Viewed 9259 times
Last edited by damian on Wed Nov 29, 2006 6:45 pm, edited 1 time in total.
nodoodahs
Roundtable Knight Posts: 218
Joined: Wed Aug 09, 2006 4:01 pm
Talk about a straw man example. The two curves have a strong correlation to a THIRD variable - TIME. If you adjust the formula so that either of those lines has a zero slope regression line to time, your correlation goes bye-bye.

They are not moving in opposite directions and equal magnitudes; when one is uptrending, the other is downtrending, but with less momentum (price movement over time). If you adjust the formula so that these lines always move in opposite directions on the same day, and the moves are of equal magnitude, you will have perfect negative correlation.

Cute visual trick. Kind of like the old line length trick

>---< versus <--->
damian
Roundtable Knight Posts: 815
Joined: Tue Apr 15, 2003 8:43 pm
Location: dusseldorf
Well, you might be clever, but you're not much fun. Spoil sport. And hence the value of calculating correlation of de-trended series.

I very quickly threw the only Excel de-trend addin that I have at the data and detrended one series, correlation 0.10. My results are rubbish
Attachments chart4.jpg (31.55 KiB) Viewed 9250 times
damian
Roundtable Knight Posts: 815
Joined: Tue Apr 15, 2003 8:43 pm
Location: dusseldorf
On the topic of system equity curve correlations (not individual instrument correlations):

You can only be long or flat a trading system. You canâ€™t go short the equity curve. You are invested or you are not.

So since we are restricted to long-only system investments, I would want to allocate funds to systems with negatively correlated equity curves.

SYSTEMS:
- Equity curves from trading systems should be negatively correlated for reduced variance.

MARKETS:
- Within a trading system that has long and short positions, traded products should display zero correlation.

Example of system correlation:

correlation(System A, System B) = -0.87

You are invested long in each system (strategy, CTA, Fund)

Thanks to corr = -0.87, when A zigs, B zags, thus to some degree negating each other and smoothing the summed curve. Thatâ€™s a good thing!

Both systems are winners (have upward sloping equity curves) and you will make the sum of the curves over time. Thatâ€™s another good thing!

You make A+B and get a smoother ride thanks to the negative correlation. Two good things!!

Can your leverage, or percent allocation, to each strategy work for or against you?

What about measuring equity curves from individual products (System A on the price of Gold and System A on the price of Silver)? Where the position sizes in GC and SI always volatility matched and did that volatility matching remain constant over the life of the trades?
Last edited by damian on Thu Nov 30, 2006 6:14 am, edited 2 times in total.
damian
Roundtable Knight Posts: 815
Joined: Tue Apr 15, 2003 8:43 pm
Location: dusseldorf
Traveling with the idea a little further (but possibly not arriving anywhere in particular):

System A trades Futures x, y and z. The individual equity curves from each product, derived from the one system, System_A, could be seen as 3 separate investments and you can only be long each equity curve, or flat. You can't be short the equity curve of Futures y. It is either in the portfolio or it is out of the portfolio. At first, a great idea might arrive in your head: populate your portfolio with futures where the individual product equity curves from System_A display a strong negative correlation and of course both make money in the system. This lets you benefit from both the profit of each curve (futures product) and the offsetting zigs and zags: You make the summed profit with a smoother ride.

So which Futures applied to Sytem_A produce de-trended equity curves with strong negative correlations?

Consider the Futures price correlations

xy +0.95
xz 0
zy -0.95

Futures x and Futures y in System_A will make and lose money at the same time. They _usually_ enter and exit long(short) trades in the same week and exit at much the same time. And they share a synchronized profit from each trade. They also suffer sharp reversals in winning trends at the same time. They will produce a positively correlated pair of equity curves.

Almost the same of Futures z and y, except when System_A enters long z, it will _usually_ in the same week enter short y. As for xy, zy exit at much the same time, they share a synchronized profit from each trade. They also suffer sharp reversals in winning trends at the same time. They will produce a positively correlated pair of equity curves.

What about Futures xz with zero price correlation? This is the only pair that has a chance of producing individual de-trended product equity curves that are negatively correlated. So, the original proposition is satisfied with Futures that have zero price correlation:

â€œpopulate your portfolio with futures where the de-trended individual product equity curves from System_A display a strong negative correlation and of course both make money in the system. This lets you benefit from both the profit of each curve (futures product) and the offsetting zigs and zags: You make the summed profit with a smoother ride.â€
nodoodahs
Roundtable Knight Posts: 218
Joined: Wed Aug 09, 2006 4:01 pm
When I type of being long or short two asset classes or issues, based on their correlation, I am typing about the correlation of their price movements, not the correlation of equity curves created by your trading system, and typing about being long or short as a position of your trading system on that asset or issue.

Assuming the same or a similar LTTF system is applied to multiple issues, the equity curve will be smoothest when the issues are closest to UNcorrelated. That is the example I typed earlier.

Assuming your LTTF system is profitable on each issue, the non-de-trended equity curves cannot be negatively correlated, as they will all be positively correlated to the variable T (time). However, if the issues have price movements that are close to UNcorrelated, the correlation of the equity curves will be lower than it would for issues having price movements that were positively or negatively correlated.

[Edit added]Assuming you use the same or a similar system on two or more asset classes or issues, and assuming you find classes that are UNcorrelated, there should be no reason to test the equity curves of the system on those classes, as they should have a low correlation.

IF you use different systems on the different classes, THEN in my opinion you should de-trend ALL the equity curves to eliminate the impact of time, and then test for correlation between them.