E2)C5: (‘Stock 1’!E2/‘Stock 2’!E2)D5: (‘Stock 1’!H2-‘Stock 2’!H2)Step 7. Highlight cells B5, C5, and D5; then press Control C (Command Cfor Mac users) to copy.Step 8. Using your mouse, highlight cells B6, C6, and D6 all the waythrough cells B400, C400, and D400; then press Control V (Command V
FIGURE 12.1 Pairs Data Spreadsheet
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for Mac users) to paste the data from Step 7 into all of the highlightedcells. You have now not only imported data into an Excel spreadsheet, butyou have also begun your analysis! Figure 12.1 shows an example of whatyour Pair Data sheet should look like when you have completed Step 8.A few quick Excel tips: If you prefer using the keyboard over yourmouse, you can highlight cells B6, C6, and D6, then hold the shift key downand press the down arrow key. The three columns of cells will highlight allother cells in the three columns as you continue pressing the down arrowkey. To highlight entire pages of cells, highlight cells B5, C5, and D5, thenhold the shift key down and press the end key. Finally, if you highlight cellsB6, C6, and D6, then scroll down in the spreadsheet without clicking onanother cell, you can then hold down the Control and Shift keys while clicking on cells B400, C400, and D400. As a result, all cells between B5 throughD400 should then be highlighted. Use these functions as shortcuts to copyand paste data between Excel spreadsheets.
AVERAGE, MEDIAN, MODE, AND STANDARD DEVIATIONNow that we have our Pairs Data spreadsheet formatted, let’s put some simple descriptive statistics into our Main spreadsheet. The Main spreadsheetshould be all ready to go, but double-check it using Figure 12.2, just to be sure.Excel is straightforward in the use of terminology, so there shouldn’t bemuch confusion. Here are the Excel commands that we will use and a briefexplanation of each:• AVERAGE: Returns the average (arithmetic mean) of the data.• MEDIAN: Returns the median of the given numbers. The median is thenumber in the middle of a set of numbers; that is, half the numbers havevalues that are greater than the median, and half have values that are less.• MODE: Returns the most frequently occurring, or repetitive, value in anarray or range of data. Like MEDIAN, MODE is a location measure.• AVDEV: Returns the average of the absolute deviations of data pointsfrom their mean. AVEDEV is a measure of the variability in a data set.• STDEV: Estimates standard deviation based on a sample. The standarddeviation is a measure of how widely values are dispersed from theaverage value (the mean).• STDEVP: Calculates standard deviation based on the entire populationgiven as arguments. The standard deviation is a measure of how widelyvalues are dispersed from the average value (the mean).
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FIGURE 12.2 Main Spreadsheet
Now, let’s plug in our actual Excel formulas in the Main sheet.Step 1. In cells B8, B9, and B10, put in the formulas: AVERAGE(‘PairData’!$B$5:$B$195), MEDIAN(‘Pair Data’!$B$5:$B$195), andMODE(‘Pair Data’!$B$5:$B$195). In cells C8, C9, and C10, type in:AVERAGE(‘Pair Data’!$C$5:$C$195), MEDIAN(‘PairData’!$C$5:$C$195), and MODE(‘Pair Data’!$C$5:$C$195). (Note: CellC10 will return a N/A value; this is okay.) In cells D8, D9, and D10, enter:AVERAGE(‘Pair Data’!$D$5:$D$195), MEDIAN(‘PairData’!$D$5:$D$195), and MODE(‘Pair Data’!$D$5:$D$195).Step 2. In cells B12, B13, and B14, enter the following formulas,respectively: AVEDEV(‘Pair Data’!$B$5:$B$195), STDEV(‘PairData’!$B$5:$B$195), and STDEVP(‘Pair Data’!$B$5:$B$195).
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Step 3. In cells C12, C13, and C14, type in: AVEDEV(‘PairData’!$C$5:$C$192), STDEV(‘Pair Data’!$C$5:$C$192), andSTDEVP(‘Pair Data’!$C$5:$C$192).Step 4. Finally, in cells D12, D13, and D14, type in: AVEDEV(‘PairData’!$D$5:$D$192), STDEV(‘Pair Data’!$D$5:$D$192), andSTDEVP(‘Pair Data’!$D$5:$D$192). (Again, here you can make yourdata look all the way through the end of the worksheet if you wish tostudy a longer time period).At this point, we have some basic descriptive statistics entered into Excel,and we are beginning to paint a picture of what is happening within our pair.Follow these steps to see where the pair is:Step 1. Label cell A18 Current Data, B19 Differential, C19 Ratio, D18 Closing, and D19 Spread.
FIGURE 12.3 Spreadsheet with Statistics
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Step 2. In cell B20, put in: (‘Pair Data’!B5); in C20, type: (‘PairData’!C5); and in cell D20, enter: (‘Pair Data’!D5)Figure 12.3 shows what our spreadsheet should now look like.
UNDERSTANDING THE PRESENT DATATaking a brief moment to pause and understand the data that we have created,we can begin to see where our pair is headed. First, as we know from our Mainpage, the mean of the differential is 8.86, denoting that during the period wehave studied, the two stocks traded (on average) 9 points apart. The data thatwe are studying goes back just under 200 days, which equates to slightly underone year’s worth of actual trading sessions. Next, the pair is (in the periodstudied) trading just over two standard deviations from the mean. Thus, thestatistics are telling us that that 95 percent the data falls within our presentrange. If we were to actually trade based on the data, we would short KLACand go long NVLS—though the data presented does not necessarily present agood trading situation.
HOW DO YOU KNOW WHICH STOCK TO BUY OR SHORT?To explain buy and sell sides of a pair it may be a good idea to first painta picture on our spreadsheet so that we may get a visual idea of what ishappening. Taking a look at the Main spreadsheet, we should have ourbasic statistics already formatted. The sheet should include our twocorrelations; the mean, median, and mode for the differential; ratio andclosing spread; and average deviation, standard deviation, and populationstandard deviation. Figure 12.3 depicts a picture of a formatted version ofour present spreadsheet. Incidentally, at this point you can always use thedisk in the back of the book to load this actual spreadsheet onto yourcomputer.To further understand buy and sell sides, our goal is to add/subtractthree standard deviations on either side of the mean to see where the pair isin regard to the mean.Step 1. As a picture tells a thousand words, label cells F6 through I7 asdepicted in Figure 12.4.Step 2. Copy cells B8, C8, and D8 into cells G11, H11, and I11.Step 3. In cell G10, enter the following formula: $G$11$B$13, so that wehave the mean plus one standard deviation. In cell G9, enter the formula
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FIGURE 12.4 Spreadsheet Main Page
$G$11(2*$B$13). This gives us two standard deviations from the mean.In cell G8, do the same, except multiply cell B13 times three instead of two.Step 4. Repeat steps 1 through 3 for all of the rest of the cells in this box,while remembering to subtract the standard deviation from the lowerhalf of the cells. As a result, your standard deviations box should looksimilar to Figure 12.5. Keep in mind that columns G, H, and I are using thevalues from row 13 of B, C, and D. I know this is confusing, so you canalways reference the spreadsheet in the CD-ROM. However, if you canstruggle through this and build the sheet, you will only empower yourselflater on.It is important to always keep track of which stock came first in yourdata analysis. This is because when we refer to buying and selling a spread,we are assuming that there is a set structure for each of the equitiesinvolved. And indeed there is! Because KLAC comes first in our worksheets
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FIGURE 12.5  3 Standard Deviations in Spreadsheet
and all of our pairs calculations, the terms buy and sell will relate directly toKLAC. To be brief, when we refer to “buying or selling” a spread, we are making direct reference to the direction of the first stock in our pair. If the differential were above the mean, we would say “Sell the pair”; conversely, if the differential were below the mean, we would “buy the pair.” Thus, as KLAC/NVLSis above the mean, we would refer to the pair as being on the “sell side.” On anormal distribution, the buy side of the pair would be on the left of the mean,while the sell side of the pair would be on the right. In essence, the higher thevalue, or more overbought a pair is, the further up (or right) it would be on sellside of the statistical mean. Figure 12.6 displays the buy and sell side of themean on our spreadsheet and in reference to the Gaussian curve.At present, we may assume that the pair has made significant divergenceover the last three months. How do we know that the pair has been diverging over the last three months? We can do one of two things to find this
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FIGURE 12.6 Buy/Sell Spreads and Gaussian Curve
information. First, we can look through the information on the Pair Datasheet, where our manual investigation will allow us to see the actual historical numbers diverging, or we can create a chart that will allow us to quicklydecipher the present and historical trends.
CHARTING THE DATAFollow these steps to chart the data:Step 1. On the bottom of the spreadsheet, create a new worksheet (byright clicking on the spreadsheet tabs and then highlighting the “Insert”option) and label it Pairs Charts.Step 2. In cell A2, key in the title Differential.Step 3. Highlight cell A3, then click on the chart wizard icon on the top ofthe screen. Choose the B&W Line–Timescale chart under the CustomTypes tab. Press Next.Step 4. Click the Series tab on the top of the chart option screen, andthen choose “Add” to insert the data to be charted—only if there is nota blue highlighted bar named Series to the left. If there is, disregard thisstep. (Note: If there is any other data already selected, delete it by highlighting the unwanted series and then clicking the Remove key.) Oncethe Series One data is highlighted, then label the data in the Name boxas Differential Chart.
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FIGURE 12.7 Format Axis Box
Step 5. In the Values field, either highlight the first 195 days ofdifferential data in the Pair Data spreadsheet, or type in ‘PairData’!$B$5:$B$400.Step 6. On the bottom of the present chart wizard screen, you will seeanother field titled Category (X) axis labels. In this field, enter the datarange that will correspond with the differential data within our newchart. To do so, either highlight cells A3 through A195, or type in‘Stock 1’!$A$3:$A$397. Click Next, Next, and Finish to complete thechart.Step 7. From here you might need to resize and move your chart to get itinto place. To do so, highlight the entire chart and simply drag and dropthe borders into place.Step 8. We might now need to reformat our X axis to make our datesappear properly on the chart. On the lower portion of the chart, rightclick on the date range, then select Format Axis. Under the Scale tab,uncheck the Major Unit and Minor Unit boxes. Then, for the Major Unit
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and Minor Unit boxes, replace the numerical value with 20. Figure 12.7displays the box.Step 9. Next, click on the Alignment tab at the top of the Format Axisscreen. You will see a text orientation dial on the left portion of thescreen. With your mouse, grab the actual dial and point it upward so thatthe text reads vertically from the bottom to top. You have just createdyour first pairs chart! Figure 12.8 displays how the new chart shouldappear in the spreadsheet.Now we can visually see where the differential has been for the lastyear. What may jump out at us right away is that the differential has beenexpanding for quite some time. Then, if we hop over to the Pair Data sheet,we can see what the actual high value was for the period we see on thechart!
FIGURE 12.8 Differential Chart
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The highest value for the differential was (for the period studied) onFebruary 6th, 2002 (cell 377), when the two stocks were trading 18.04 pointsapart. Further scanning the data chronologically backwards, we can see thatthe spread hit a low of 3.19 on January 31, 2003. Thus, if history and statistics are any indication of what could happen in the future, we may deducethat the differential of 15.85 on July 31 may be nearing an actual tradingpoint. One quick word of caution: Even though our initial data is pointingtoward a nearing sell signal, we still have more work to do!
CHAPTER 13
Differentialversus RatioDifferentialRatio
efore getting too far into analyzing the present data, I will take amoment and make sure that we have a full understanding of how weare gauging our pairs. We are presently using the two simplest methods: differential and ratio. Each has its own merit for the pairs analyst,though the bulk of the pairs community seems to prefer a ratio over the differential. In this book, we are mainly using the differential for the sake ofsimplicity, though as you begin to move further forward in pairs research, itmay be a good idea to switch over to using a ratio. In the next few chapters,we quickly go over both in hope of grasping a deeper understanding of eachmethod.
B
DIFFERENTIALFirst and foremost, it may be a good idea to simply use the differential astrend confirmation of the ratio. While the differential is useful in gaugingwhere two stock prices are closing, it also lacks the ability to encompass valuation calculations like the ratio. Usually, when I am looking at a differential, I am only scanning a historical pairs chart to see just where the twostocks have traded in recent history. Generally, I can usually get a good ideawhere the visual mean is just by taking a moment to look over the chart. I donot want to discourage you entirely from using the differential as a way ofreading your pairs data; however, there might be added benefits when usinga ratio. For the purposes of this book, we will mainly concentrate on the differential, but will use the ratio as confirmation of our feelings.87
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RATIOWhen using a ratio, we may obtain a better comprehension of the actual valuation of each stock. How is this so? If we use exterior data like earnings,beta, float, and dividends paid in our ratio calculation, the resulting numberbecomes more than simple division. For example, if one stock is trading at$100 and another at $50, the ratio is 2.0. This information indicates that foreach single share of the $100 stock that we buy, we may want to purchasetwo shares of the lesser-valued security. On paper, the simple ratio appearsto be a good idea, but unfortunately, we still leave out further valuation.To get a better idea of how we could weight our pair, we can add infundamental concepts to tweak our position valuation. For example, whenadding volatility into our equation, we can divide each price by beta and thendivide the results from one another. As an example, on any random date,suppose stock A is trading at $45, while stock B is trading at $60. Calculating a simple ratio (dividing Stock A by Stock B), we come up with 0.75. Whatthis is telling us is that for every one share of the less expensive stock, wewould purchase .75 shares of the more costly security. However, to measureStandard & Poor’s (S&P) 500 volatility, we can add in the beta numbers tofind an entirely new ratio. Assume that the beta for Stock A is 1.43, while thebeta for Stock B is 1.62. Our new equation would be [(45/1.43) / (60/1.60)],which equals 0.83. Remember that we are always using the denominator asthe subject stock. Thus, incorporating volatility, we would actually purchaseslightly more of Stock B than with the simple ratio division model.
CHAPTER 14
ApplyingStatisticsto PairsDensity CurveUnderstanding the Data
ext, we will work on further breaking down our data, where wewill attempt to zero the mean. This is the most important part ofthe book and will be the ultimate “trigger” for where we enterand exit our trades. Our goal is to try and pinpoint entry and exit pointsfor our pair, while taking little pain and having tight stop loss points. Thetool that we will use is called a density curve, the formula is shown inFigure 14.1.
N
DENSITY CURVEBy definition, a density curve is “the graph of a nonnegative functioncalled a density function. The area between the graph of the function andthe x axis is 1.” Generally, normal density curves have many differentfunctions, including characterizing probabilities; however, we will use itin a different fashion. A density curve looks similar to a bell curve, withthe data evenly distributed around the mean. However, our curve willactually show up as more of a moving histogram, depicting where our current and historical differential or ratio are in regard to the mean. The density curve itself will yield a value between zero and positive one. Our buysignals will typically (though this changes from pair to pair) be triggeredwhen the pair is within 0.003 thousandths of zero. Conversely, the densitycurve will indicate a potential selling point when our data nears 0.997thousands of positive one. Let’s set up the normal density curve in ourspreadsheet.89
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f(x, m, s) ⫽
122ps
e ⫺ A(x⫺m) >2s B1
1
FIGURE 14.1 Normal Density
Step 1. In your Pairs Data spreadsheet, label cell E3 Density and E4 Curve.Step 2. Then, we will need to import a little data from the Main sheet tomake this work. This might get a little confusing, so I’ll try and explain itvery plainly. Click once in cell B1 of the Pair Data sheet. By doing thiswe have merely selected the cell.Step 3. In cell B1, then hit the ⫽ (equal) key.Step 4. Without hitting enter or clicking on another cell in the sheet, clickon the Main sheet tab to revert to our first page. Click once in cell B8 tohighlight the cell, and then hit Enter. The sheet should automatically flipback to the Pairs Data sheet, and the value in Main sheet B8 should nowbe in the B1 cell.Step 5. Now, do the same for cell C1, except use the Main sheet cell B13instead of B8.Step 6. In cell E5, put in the formula ⫽NORMDIST(B5,$B$1,$C$1,TRUE).Step 7. Copy and paste the data through cell E400. Your Pairs Data sheetshould mirror Figure 14.2.Step 8. Add a graph of the data in the Pairs Charts spreadsheet.As a quick explanation of the formula we have just entered into ourspreadsheet, what we have done is entered three variables with a condition.We made the spreadsheet look for the first variable (our differential) chronologically through all 400 cells. The second and third variables are the meanand standard deviation from our Main spreadsheet.What we are now discovering (as we visually look at our chart) is thatthe present value of the density curve is quite elevated, yielding 0.982. Whilethis seems very high, it is not within the aforementioned 0.003 thousandthsof positive one. Perhaps looking back through history will give us a betteridea of why we are seeking such extreme values.To do this accurately, we will need to look further back into history thanjust 400 days. Thus, go back to your spreadsheet and import 1,000 days of
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FIGURE 14.2 Pairs Data
data (up to July 31) for each stock. If you prefer not to take the time, you willfind the worksheet on the CD-ROM labeled Example_pairs_sheet2.Historically, we see that the density curve traded within 0.003 thousandthsof one in February 2002. To be precise, the density curve hit a high of 0.997 onFebruary 6. At the same time, our pair hit an almost exact near-term high of$18.04. What is truly remarkable about this data is that the spread did notdiverge any further (during this time period) and then yielded a gain of 10points over the next few months. What becomes even more amazing is that ifyou look farther back through the data to see if the pair ever yielded anotherbuy or sell signal, we unearth more startling information.In February 2002, the density curve hit 0.997 when the pair was trading18.04 points apart. The following day, it collapsed to 0.995, with KLAC/NVLSconverging to post a differential of 17.34. The gain equated to 70 cents, andthe spread continued to buckle over the next month. By May, the differentialhad fallen to below 10. Farther down the road, the pair continued to declineall the way to just above 3 the following year. Clearly, the trade would have
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FIGURE 14.3 Density Curve
been a winner. However, don’t get too optimistic yet—there is plenty ofroom for error, as you will soon see. (Examine Figure 14.3.)
UNDERSTANDING THE DATAIn the above examples, if we had bought and sold the pair when the densitycurve pulled within 0.003 thousandths of zero and one, we would have hada large winner. However, I would like to again reiterate that pairs trading isnot this simple! The fact of the matter is that we are dealing with movingnumbers. Our mean, median, and mode all shift as each trading day presentsadditional data. This causes our standard deviations, charts, and densitycurve to also change positions. For example, on August 1, 2003, our densitycurve was at 0.982. However, if we update the data through August 31, thedensity curve for August 1 then shifts to 0.985. We must understand that thedata is constantly changing, making it a necessity to translate all numberssurfacing each day. Simply put, if we are to trade pairs with real money, wewill not have the luxury of looking back into history to see whether we wereright or wrong. Because of this truth, it is important to have several parameters that constitute a trade instead of one or two simple numbers.If we revert to our data ending on July 31 and analyze the pair datasheet, we can make a few decisions. First, we know the density curve is at
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0.982, not quite within the 0.003 thousandths of one that we would like toinstitute a trade. However, at the same time, the differential is trading at15.85, coming off a near-term high of 17.17 on July 29. For all practical purposes, our gut instinct may be telling us that the pair has expanded. In fact,if we look at the differential chart, it appears as if KLAC/NVLS may definitely be at a potential sell signal. We infer this visually, with the differentialchart indicating that the pair is at an extreme point of expansion. What’smore, the ratio chart appears overbought, and is also displaying a possiblesell signal as well.The pair is currently at 15.85, trading over two standard deviations fromthe mean at 8.86. If the empirical rule has any merit, then we are presentlytrading with 95 percent of the total data. While the data seems exciting atfirst glance, we must ask ourselves about that other 5 percent—or at leastthe other 4.7 percent that could pull the spread apart over three more pointsbefore we hit the third standard deviation at 18.81! The point of the entirelast paragraph is that regardless of how close our density curve is getting tozero or one, we may want to double-check our other descriptive statistics foradditional confirmation before we take a position.
CHAPTER 15
Taking a StepBack for aMoment
e have built our spreadsheet up to where we have a useful tool inanalyzing our pair. What’s more, we also have some fundamentaland technical information to also compare each individual stockwith, just to double-check the statistics that have been presented. So nowwe ask, what constitutes a decent convergence pair’s setup?We would like to see our pair trade very close to (if not at) three standarddeviations, while our charts confirm extreme expansion at the same moment.I usually only take a position if the differential is within 10 percent of the thirdstandard deviation. Second, ideally the density curve will be within 0.003 thousandths of positive one, thus inferring that we are very close to maximumspread expansion. You must change the density curve entry signals for the typeof pairs you are trading. What I mean here is that if you are trading KLAC-NVLS,I would actually recommend waiting until the density curve is pegged out at 0or 1. The reason is that the entire sector and two stocks are much more volatilethan other, slower-moving equities like energy stocks. When trading slower oilstocks, I will use the 0.003 rule, but only after I have scanned the historicaldata to see if entries of 0.003 thousandths of zero or one proved to be profitable. The pairs information contained in this book leaves much to the decision of the individual trader, and there is no single hard number that worksin every case. This is very important to remember: If you simply build aspreadsheet and wait for a 0.003 or 0.997 entry without doing technical, fundamental, and commonsense analysis, you will lose.I would like to take a step back and briefly discuss a losing situation withour present data. If we take another look at our Pair Data spreadsheet, wewill find some disturbing information. Peeping farther back into history to
W
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April 2000, we see that the density curve was trading on the upper end of thescale pegged near 1.00. Keep in mind that, at this time, the market was completely wacko, and it would have been a good idea not to trade at all, but forthe sake of example, I will use this time period to completely dissuade youfrom trading pairs. On March 10, 2000, the density curve hit 1.00, a dead-ontrade trigger. On that day, the differential closed at 21.18. The following day,the density curve began to drop very fast, bobbling around and showingextremely erratic trading. Ideally, we would have taken a position on the13th, when the differential was at 17.91. The result would have been a gigantic losing trade, as the pair opened up another 10 points. In fact, on April 7,the differential hit a high of 38.94 before falling. I think it is safe to say thatif you did not have a money management plan in place, you could have losta ton of money. It would have been very important to double-check allevents surrounding the trade, while also taking notice of all descriptive statistics. I cannot reiterate how important it is to have a solid money management plan in place with tight stop losses before a trade is ever entered! Wewill go over money and risk management later in the book, but for nowplease just keep in mind that no system is completely foolproof and, thus,we must always keep the worst-case scenario in mind. What’s even moreamazing, as the bear market began to ravage portfolios when the bubbleburst, the density curve quickly sank to 0.001 on May 4. Had we taken a position, we might have made a little money if we had scalped, but more thanlikely, we would have lost our shirts. By October, the differential had fallento under 10. This is just food for thought. Though the market and pairs havecalmed down considerably, it is very important to note the possibility of losing megabucks if tight stops are not in place.
CHAPTER 16
Moving Averageand NormalizedStandardDeviationDifferential Moving AverageNormalized Standard Deviation
DIFFERENTIAL MOVING AVERAGEMuch like technical analysis, pairs trading benefits from using movingaverages to uncover trend-trading possibilities. As mentioned earlier inChapter 4, a simple moving average (MA) is the average of a security’sprice over a specified period of time, represented in the form of a line ona stock’s chart. Thus, for further statistical analysis, we will determine a90-day MA, and then calculate standard deviations from the mean. Theprinciple behind calculating an MA is so that we can gain additional statistical confirmation when entering a position. If you imagine the differentialdrawn as a thick line on a chart, the MA would then be a more erratic linetrading around the differential. Figure 16.1 depicts the differential and MAtogether on a chart.What we are looking for is a standardized sell signal on the MA at thesame time as the longer-term statistics (differential or ratio) are alsoindicating that the pair is nearing a sell point. Conversely, we would likethe MA to be on the buy side when we are purchasing the pair. The reason we are trying to line the two up is that it is possible to get a movingaverage buy signal when the longer-term statistics are indicating a sellsignal. When convergence trading, we want to make sure that all facetsof the rubber band’s expansion are on our side before we take a position.Figure 16.2 depicts how we can be on the wrong side of a trade if we buya pair based on short-term statistics, even though the long-term datapoints toward selling.97
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FIGURE 16.1 Moving Average and Differential
FIGURE 16.2 Moving Average and Differential with Buy/Sell Explanations
Moving Average and Normalized Standard Deviation
99
NORMALIZED STANDARD DEVIATIONIn The Complete Arbitrage Desk Book, Stephane Reverre describes how to normalize “the value of the difference.” (EN 16.1) By normalizing a value, Reverreis referring to changing data to a point where it is later represented with a basevalue. In essence, we will take our moving average data and “normalize” theresult on a histogram with a base of zero. Thus, we will be able to measure themovements of the standard deviations as a histogram. Our goal is to use thisdata as an overlapping indicator and confirmation of our density curve.Follow these steps to set up the moving average in Excel:Step 1. Right click on the spreadsheet tabs at the bottom