Trading Pairs

by Douglas S. Ehrman

Article Contributed by Active Trader Magazine  

Pairs trading is a non-directional strategy that identifies two companies (or futures contracts) with similar characteristics whose price relationship is outside of its historical range. The strategy simply buys one instrument and sells the other in hopes that relationship moves back toward normal. The idea is the price relationship between two related instruments tends to fluctuate around its average in the short term, while remaining stable over the long term.

For example, the price relationship between Bank of America and U.S. Bancorp (BAC/USB) has remained relatively stable at a ratio of 1.5 over the past four years. During that period, its price ratio has been as high as 1.8 and as low as 1.35. The goal is to identify these stable relationships and buy the stock lagging the historical average and sell short the one leading it.

When trading pairs, you’re not concerned with how the individual stocks perform, but with their relative performance. As the market rises, both long and short stocks appreciate, so the pair’s overall value remains constant. Similarly, if the market declines, each stock drops. However, the pair profits as long as the long stock outperforms the short one.

Simply going long (or short) limits forecasting to a single direction, which means you can’t take advantage of the relative
values of all the instruments you track. Assume, for example, that you expect two stocks in the same industry to perform
much differently — one is expected to have a higher-than-average return and the other a lower-than-average return.

If you’re limited to going long, you’d buy the more attractive stock and ignore the one that’s expected to underperform.
In a pairs trade, however, you could enter a long position in the attractive stock and enter a short position in the less attractive one, which uses all available information and exploits the relative performance of both stocks.

Matching stocks

Consider a group of stocks within the semiconductor industry. Historically, each stock trades at a specific price ratio to one another as well as to the overall industry index. The long-term stability of this relationship is intuitive, because the index — comprised of the stocks in that industry — is directly affected by the performance of its individual components.

Figure 1 shows the daily price ratio between Intel and the semiconductor index (INTC/SMH) over the past four years.
While the price relationship between the index and one of its components stays near its average value in the long term, the stock tends to trade above and below that average value in the short term. Each stock will either outperform its index or lag it at different times.

Figure 2 shows the same relationship between INTC and SMH from late July 2005 to January 2006. If you compare Figures 1 and 2, you’ll notice the ratio appears to be flat over the long term, but it often fluctuates from day to day, providing an array of trading opportunities.

The same behavior exists between two stocks within the industry. Figure 3 shows the price ratio between Intel and Analog Devices (INTC/ADI) in the past four years; the trading range of the pair (0.53 to 0.81) appears flat. However, Figure 4 shows a more volatile relationship between Intel and Analog Devices over the past year, despite a smaller price range (0.64 to 0.74).

You can identify the same price relationships by calculating the correlation coefficient between two instruments. Correlation measures the strength of the price relationship between two stocks, which pair traders use to match stocks. Correlation ranges from +1 to -1, with +1 representing a perfect positive relationship (for every 1-percent increase in A, B will increase by 1 percent) and -1 representing a perfect negative relationship (for every 1-percent increase in A, B will decrease by 1 percent).

Pairs vs. spreads

Many traders think of a pair as a “spread” trade, but this comparison is not quite accurate. A spread trade creates either net long or net short exposure, but a properly executed pairs trade is dollar-neutral. By maintaining a market-neutral position, the effects of market direction can be largely eliminated from the trade.

Consider the following comparison of a spread trade vs. a pairs trade:

Stock A: $20 per share
Stock B: $10 per share

Spread trade
Long 100 shares of stock A: $2,000
Short 100 shares of stock B: $1,000

Net long: $10 per share ($1,000)
This is a hedged, bullish position.

Pairs trade
Long 100 shares of Stock A: $2,000
Short 200 shares of Stock B: $2,000

Net long/short: $0
This is a true market-neutral position.

Scenario 1
Both stocks rise 50 percent.
Stock A: $30
Stock B: $15

Scenario 2
Both stocks fall 50 percent.
Stock A: $10
Stock B: $5

A spread trade is a market bet with a built-in hedge, while a pairs trade is a market-neutral position. In the first scenario’s bull market, the spread trade gains $500 (Stock A’s $1,000 profit - Stock B’s $500 loss), and the pairs trade is flat (Stock A’s $1,000 profit - Stock B’s $1,000 loss).

In the second scenario’s bear market, however, the spread trade loses $500 (Stock A’s $1,000 loss - Stock B’s $500 profit) as the pairs trade stays flat. Here, the spread trade loses money despite both stocks dropping by an equal percentage. In both scenarios, the specifics of either stock had no effect on price — the entire move is explained by the broader market fluctuations.

The trade must be market-neutral to ensure it won’t lose money unless there’s a change in relative performance (i.e., one stock performs better than the other).

Similarly, if both stocks dropped by $5, the spread trade would remain flat even though Stock A outperformed Stock B (Stock A loses 25 percent, while Stock B loses 50 percent). A trade can only capture this relative performance if the trade is dollar-neutral.

Calculating pair statistics in Excel Most analysis software programs will allow you to export price data into Excel. In other cases, you may be able to simply copy the price data and paste it into Excel. On the bottom of an Excel spreadsheet, you will see separate individual worksheet tabs. By placing the historical data for each stock in the latter two worksheets, you can reserve the first sheet for your statistical (correlation coefficient, in this case) studies. Label worksheets 2 and 3 “Stock 1” and “Stock 2,” respectively, and then import the historical price data for the first stock into Stock 1 and the second stock into Stock 2. Figure A shows the different worksheet tabs in Excel, and what historical data imported into a worksheet would look like. Notice the closing prices are in column E. The correlation function name in Excel is “CORREL.” Enter the correlation formula in worksheet 1. For 250 trading days of data, the correlation study would be: =CORREL(‘Stock 1’!E2:E252, ‘Stock 2’!E2:E252). This compares the closing prices of the two stocks and determines their correlation over 250 days of data. You can calculate the pair’s value, its average, and standard deviation with similar formulas. First, insert a new worksheet by right-clicking the worksheet tabs on the bottom of the Excel window and selecting the “Insert” option. You can also create a new worksheet by selecting the “Insert” menu and clicking on “Worksheet.”

Name the new worksheet “Pair Data” and label columns A and B “Date” and “Pair Value” (see Figure B). Next, copy and paste the dates from the Stock 1 worksheet into the “Date” column of the Pair Data sheet. The Pair Value column will contain the daily relationship between the two stocks — i.e., the closing price of Stock 1 divided by the closing price of Stock 2. Name the new worksheet “Pair Data” and label columns A and B “Date” and “Pair Value” (see Figure B). Next, copy and paste the dates from the Stock 1 worksheet into the “Date” column of the Pair Data sheet. The Pair Value column will contain the daily relationship between the two stocks — i.e., the closing price of Stock 1 divided by the closing price of Stock 2. The formula is: =(‘Stock 1’!E2/‘Stock 2’!E2). Drag and drop this formula throughout the rest of the cells in the Pair Value column. On the “Correlations” worksheet, label two cells “Mean” and “Standard Deviation,” respectively. To calculate the mean and standard deviation, enter the following formulas: =AVERAGE(‘Pair Data’!B2:B252) and =STDEVA(‘Pair Data’!B2:B252).

Making a trade Creating a pairs trade involves the following four steps: 1. Perform correlation analysis to match stocks. 2. Apply technical and statistical indicators. 3. Perform fundamental and technical tests to confirm relationship. 4. Manage and exit the trade. First, use correlation analysis to match stocks. After identifying stocks (usually within the same industry) that seem to have traded in line with each other in the past, test each relationship’s strength by calculating the correlation coefficients between each pair. The pair should have a correlation of greater than 0.7. It is important to measure correlation over different time periods to see if the relationship between the two stocks has changed over time. Short-term traders should measure 30-, 90-, and 180-calendar-day correlations; longer-term traders should consider 90-, 180-, and 365-day correlations. Major differences between these values often indicate when the stock prices diverged, a useful hint to the expected length of the trade. The longest measurement period is the most important because it indicates the longer-term strength of the relationship. You should consider the pair as a single instrument (Stock A/Stock B) as you apply technical and statistical indicators to it. Next, calculate the pair’s average (mean) and standard deviation values for the longest correlation period (See “Variance and standard deviation.”)

Pairs trading is a mean-reverting strategy, which means the ratio may trade above or below its average value, but it eventually tends to move toward it. Enter the trade when the pair’s current value diverges at least two standard deviations from its mean.

Technically, the pair should be treated as any individual stock. You can apply overbought/oversold indicators such as the Relative Strength Index (RSI) and stochastics as well as any other indicators (Bollinger Bands, moving averages, etc.) that could confirm the pair’s strength.

The third step is to examine each stock’s fundamentals. The goal isn’t to become an expert on each company, but rather to check that the fundamentals do not contradict the trade’s premise. Buy the stock with a lower P/E, price-to-book, and market capitalization as well as a higher dividend yield, cash position, and short-interest ratio.

Also, check the recent and anticipated news for each stock to make sure a news-driven price move is not responsible for the pair diverging from its average value. Red flags include announcements of earnings, litigation, regulation, or major management changes. These events could change the pair’s relationship in a fundamental, permanent way, and the pair’s current value may not revert back to its mean.

Finally, you should analyze each stock’s technical and sentiment outlook for warning signs. For example, a pair’s 14-day RSI should be oversold or overbought (under 30 or over 70). You should reject the trade if the long candidate’s RSI is below 40 and the short candidate is over 60. This suggests the pair will continue to diverge from its mean instead of revert towards it.

Enter the pair on the short side first to avoid runs caused by the up-tick rule. Also, set a profit objective and stop-loss upon entry. The profit objective should be two standard deviations above the entry (near the mean), and the stop should be 1.5 standard deviations below the entry (3.5 total standard deviations of divergence).

It’s a good idea to exit the pairs trade if it achieves 50 percent of its profit target in any one day.

Trade example

Figure 5 (above) shows an example of a pairs trade on Jan. 10, 2006: long Wells Fargo vs. short Washington Mutual (WFC/WM). In the preceding six months, the pair traded in a well-defined range (between 1.42 and 1.53) before dipping
below its lower threshold at the beginning of 2006. The pair’s 14-day RSI is also oversold (30.17).

Table 1’s statistics confirm the pair’s attractiveness. There is a high degree of correlation between these two stocks, but the longer-term correlation has been reduced by shorter-term fluctuations.A one-year standard deviation is used because Figure5 suggests that it takes several months for the pair to move fromone end of the range to another, making a longer-term historical average more appropriate. Finally, the pair has dropped the required two standard deviations below its average.

No major news announcements affected this trade. Table 2 shows both companies’ fundamentals are similar. While Table 2’s values actually lean toward buying Washington Mutual instead of Wells Fargo, they are very close, and we only expect to hold a trade a few weeks (not months), so the trade is still viable. However, skip the trade if the companies’ fundamentals don’t roughly match or if you intend to hold it for longer periods.