Correlation is a term from linear regression analysis that describes the strength of the relationship between a dependent variable and an independent variable. Central to pairs trading is the idea that if the two stocks (or other instruments) are correlated enough, any changes in correlation may be followed by a reversion to the pair’s mean trend, creating a profit opportunity. For example, stock A and stock B are highly correlated. If the correlation weakens temporarily – stock A moves up and stock B moves down – a pairs trader might exploit this divergence by shorting stock A (the over-performing issue) and going long on stock B (the under-performing issue). If the stocks revert to the statistical mean, the trader can profit.

The importance of correlation
Correlation measures the relationship between two instruments. We can see from Figure 1 that the e-mini S&P 500 (ES, in red) and e-mini Dow (YM, in green) futures contracts have prices that tend to move together, or that are correlated.

Figure 1 This daily chart of the ES and YM e-mini futures contracts shows that prices tend to move together. Image created with TradeStation.

• Identify relationships between two instruments;
• Determine the direction of the relationship; and
• Execute trades based on the data presented.

The correlation between any two variables – such as rates of return or historical prices – is a relative statistical measure of the degree to which these variables tend to move together. The correlation coefficient measures the extent to which values of one variable are associated with values of another. Values of the correlation coefficient range from -1 to +1, where:

• Perfect negative correlation (-1) exists when the two securities move in opposite directions (i.e., stock A moves up while stock B moves down);
• Perfect positive correlation (+1) exists if the two securities move in perfect unison (i.e., stock A and stock B move up and down at the same time); and
• No correlation (0) exists if the price movements are completely random (stock A and stock B go up and down randomly).

-1                                                   0                                         +1
Perfect negative correlation           No correlation                    Perfect positive correlation

Pairs traders seek instruments whose prices tend to move together; in other words, whose prices are correlated. In reality, it would be difficult (and highly improbable) to achieve sustained perfect positive correlation with any two securities: that would mean prices exactly mimicked one another. Instead, pairs traders look for securities with a high degree of correlation so that they can attempt to profit when prices behave outside this statistical norm. Correlations of 0.8 or above are often used as a benchmark for pairs traders (a correlation less than 0.5 is generally described as weak). Ideally, good correlation presents over multiple time frames.

Why is correlation important to pairs trading? If the two instruments were not correlated to begin with, any divergence and subsequent convergence in price might, in general, be less meaningful. As an example, let’s consider a main road along a river. In general, the road follows the river very closely. Occasionally, the road must diverge away from the river due to terrain or development (comparable to the “spread” in price). Each time this happens, however, the road eventually reverts to its spot parallel to the river.

In this example, the road and the river have a correlated relationship. If we compare the river to another nearby dirt road, however, with no definable correlation to the river (i.e., their movements are completely random), it would be futile to predict how the two would behave relative to one another. The positive correlation between the main road and the river, however, is what makes it reasonable to anticipate that the main road and the river will eventually reunite. The same logic holds true for pairs trading: by identifying correlated securities, we can look for periods of divergence, try to figure out why price is separating and attempt to profit through convergence.

Note: A different approach is to attempt to profit through additional divergence (referred to as divergence trading). Here, we will focus on strategies that attempt to profit through convergence, or a reversion to the mean (known as convergence trading).

Determining correlation
The first step in finding suitable pairs is to look for securities that have something in common, and that trade with good liquidity and can be shorted. Because of similar market risks, competing companies within the same sector make natural potential pairs and are a good place to start. Examples of potentially correlated instruments might include pairs such as:

• Coca-Cola and Pepsi
• Dell and Hewlett-Packard
• Duke Energy and Allegheny Energy
• E-mini S&P 500 and E-mini Dow
• Exxon and Chevron
• Lowe’s and Home Depot
• McDonald’s and Yum! Brands
• S&P 500 ETF and SPDR DJIA ETF.

Next, we need to determine how correlated they are. We can measure this using a correlation coefficient (described above), which reflects how well the two securities are related to each other. The specific calculations behind the correlation coefficient are somewhat complicated and fall outside the scope of this tutorial; however, traders have several options for determining this value:

• Most trading platforms provide some type of technical indicator that can be applied to the two securities, performing the math functions automatically and plotting the results on a price chart.
• Traders who do not have access to this particular technical indicator can perform an Internet search “correlation coefficient calculator” to access online tools that perform the calculations.
• Traders can enter the price data in Excel and use its “CORREL” function to perform the calculations, as shown in Figure 2:

Figure 2 Excel can be used to calculate a pair’s correlation coefficient.

After the correlation coefficients have been determined, the results can be used as a filter to find the pairs that show the most potential.

Price ratio
Once we find correlated pairs, we can determine if the relationship is mean reverting; that is, when price does diverge, will it revert to its statistical norm? We can establish this by plotting the pair’s price ratio. Like the correlation coefficient, most trading platforms come equipped with a technical indicator (perhaps named price ratio or spread ratio) that can be applied to a chart to plot the price ratio of two instruments, which essentially provides a visible and numeric representation of the price of one instrument divided by the price of the other:

Price ratio = Price of Instrument A / Price of Instrument B

If traders do not have access to this type of analysis in a trading platform, the price data can be entered into Excel, as shown in Figure 3:

Figure 3 Excel can be used to calculate a pair’s price, or spread, ratio.

If we add standard deviation lines, we can gain insight into how far away from the mean the price ratio moves. Standard deviation (calculated as the square root of variance) is a statistical concept that illustrates how a specific set of prices is divided or spread around an average value. A normal probability distribution can be used to compute the probability of occurrence of any particular outcome; in normal distribution:

• 68.26 percent of the data will fall within +/- one standard deviation of the mean;
• 95.44 percent of the data will fall within +/- two standard deviations of the mean;
• 99.74 percent of the data will fall within +/- three standard deviations of the mean.

Applying this data, we wait until the price ratio diverges “x” number of standard deviations – such as +/- two standard deviations – and enter a long/short trade based on the information (the number of standard deviations selected is determined through historical analysis and optimization). If the pair reverts to its mean trend, the trade can be profitable.

Events that trigger weakness in correlation
When two instruments are highly correlated, certain events can cause a temporary weakness in correlation. Because many factors that would cause price movements would affect correlated pairs equally (such as Federal Reserve announcements or geopolitical turmoil), events that trigger weakness in correlation are generally limited to things that primarily impact only one of the instruments. For example, divergence can be the result of temporary supply and demand changes within one stock, such as when a single large investor changes positions either through buying or selling in one of the securities represented in a pair.

Note: All U.S.-listed companies must notify the listing exchange (e.g., NYSE or Nasdaq) about any corporate developments that have the potential to affect trading activity in that stock before making the announcement public. Examples of developments include:

• Changes related to the company’s financial health;
• Restructuring or mergers;
• Significant information about its products (whether positive or negative);
• Changes in key management; and
• Legal or regulatory issues that could affect the company’s power to conduct business.

U.S. stock exchanges are authorized to issue a trading halt – a temporary suspension of trading activity – based on their evaluation of an announcement. In general, the more likely the announcement is to have an effect of the stock’s price, the greater the likelihood that the exchange will call for a trading halt until the news is disseminated to the public.

Additionally, if a U.S.-listed stock’s price changes significantly within any five-minute period, a short-term trading pause may be issued. A pause lasts five minutes unless there is still a significant imbalance between the security’s buy and sell orders after that period. The price moves that trigger a pause are:

• 10 percent price movement for securities in the S&P 500, Russell 1000 Index and some exchange-traded products;
• 30 percent price movement for other stocks priced \$1 or above; or
• 50 percent price movement for other stocks priced below \$1.

Weakness can also be caused by internal developments – or events that occur within companies – such as mergers and acquisitions, earnings reports, dividend changes, the development/approval of new products, and scandal or fraud. Particularly if an internal event is unexpected, the involved company’s stock price can experience rapid and dramatic price fluctuations. Depending on the event, the price change can be very short-term or can result in a trend change.

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