
∆ SPX
∆ T Bill T–1
200120001999199819971996199519941993199219911990198919881987198619851984198319821981198019791978197719761975197419731972197119701969196819671966196519641963
–13.0%–10.1%19.5.526.7%31.0%20.3%34.1%–1.5%7.1%4.5%26.3%–6.6%27.3%12.4%2.0%14.6%26.3%1.4%17.3%14.8%–9.7%25.8%12.3%1.1%–11.5%19.1%31.5%–29.7%–17.4%15.6%10.8%0.1%–11.4%7.7%20.1%–13.1%9.1%13.0%18.9%
10.8%18.9%–15.9%2.8%1.4%–10.3%84.3%–2.6%–20.0%–41.0%–13.5%–6.8%42.6%2.3%–21.5%–9.6%–12.8%13.3%–28.5%–24.4%21.1%30.7%51.1%41.2%–16.5%–23.1%–9.7%45.7%39.7%–24.1%–38.8%29.2%21.6%4.8%7.1%17.5%8.2%21.7%5.8%
Figure 8.2 Percentage Change in T-Bill Rates and S&P 500 Futures in Next Year.Source: Standard & Poor’s Security Price Index Record.plotted each pair in the graph in Figure 8.2. The arrows going from the numbers in the chart to the scatter diagram show you how to plot the variables.The arrows point to the values for 1995 and 1980, when interest rates andstock returns were both exceptionally high.It is customary to place the thing that is thought to be predicted or influenced on the vertical axis, while the predicting or influencing variable goeson the horizontal axis. The joint values are indicated by dots, X’s, checkmarks, or letters.Visually, if the swarm of points lies from lower left to upper right, thenthe association is positive. That would indicate:
192 Practical Speculation• High values of one variable are accompanied by high values of theother.• Middle values of one variable are accompanied by middle values of theother.• Low values of one variable are accompanied by low values of the other.If the swarm of points goes from upper left to bottom right, the association is negative; high values of one variable would be accompanied by lowvalues of the other, and so on. If the points fall in a shapeless mass, then thereis little or no association.In this case, the diagram shows a negative association between interestrates in one year and stock prices in the next.The scatter plot provides a first impression of a relation between twothings. But the analysis can be taken much further. The next step is to perform a series of calculations based on the departure (or variance, or deviation)of each observation from the mean. The goal is to determine how strong therelation is.To start, the values can be grouped in a 2 × 2 table according to whetherthey lie above or below the mean—in this case, 8.9 percent for stock returnsand 5.2 percent for interest-rate changes (see Table 8.1).Notice that there are 15 matches where both are above or both below, and24 misses where one is above and the other is below. A good quick-and-dirtyestimate of the degree of relation is the number of matches less the numberof misses normalized by the total. In this case, that would come to:
(15 − 24) = − 0.2339
That number, −0.23, is a rough measure of correlation, +1 being the mostdirect relation where there are 100 percent matches, and −1 being the most inverse relation where all the observations were misses.
Table 8.1 2 × 2 Table of T-Bill Rate Changes in OneYear and S&P 500 Return the Next YearS&P 500 Return Next YearT-Bill rate changesBelow meanAbove mean
Below Mean
Above Mean
710
148
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The negative correlation indicates an inverse relation between interestrates and S&P changes.To calculate the 2 × 2 table, draw a vertical line at the average of the horizontal numbers and a horizontal line at the average of the vertical numbers.The two lines divide the graph into four quadrants. Then, count the numberof entries in each of the quadrants and place them in the table.
A More Accurate MeasureBut much information is lost when using numbers that merely relate to theirposition above or below the mean. To remedy this, we can use a more accurate and sophisticated correlation coefficient that weights the matches andmisses by their magnitudes. The one most often used is called the PearsonProduct Moment correlation.The correlation coefficient between two variables is calculated from thecovariance between them. The correlation-covariance relationship is:Γyx
COVyxSySx
Where Γyx stands for the correlation coefficient and S is standarddeviation.The covariance between Y and X is defined as:COVyx =
Σ (Yi − Y ) ( Xi − X )n−1
The Y and X refers to the mean, or average, of the individual Ys and Xs.Thus, we are computing the average product of how much each pair jointlyvaries from their means. If one is above and the other is below, it reduces thecovariance. (N is the sample size, and it is reduced by one to give the properdegrees of freedom.) In this case, the Y variable—the dependent variable—isthe S&P 500 percentage change in the next year, and the X variable is thechange in T-bill rates in the current year.Here are the steps in calculating the covariance between the two variables:1. Calculate the mean of all the values in X. Do the same for Y.2. Subtract the mean from each individual observation.3. Multiply each pair of differences (variances), and calculate the average.
194
11.8
Mean2.8
10.818.95−15.92.81.4−10.384.3−2.6−20.0−41.0
− 13.0 − 11.8 = −24.9− 10.0 − 11.8 = −22.019.5 − 11.8 = 7.726.7 − 11.8 = 14.831.0 − 11.8 = 19.220.3 − 11.8 = 8.434.1 − 11.8 = 22.3− 1.5 − 11.8 = −13.47.1 − 11.8 = −4.84.5 − 11.8 = − 7.4
−13.0−10.119.526.731.020.334.1−1.57.14.5
2001200019991998199719961995199419931992Mean
Xi(%)
Yi −Y(%)
Yi(%)
Xi −X(%)
Covariance
10.8 − 2.8 = 8.018.9 − 2.8 = 16.1− 15.9 − 2.8 = −18.72.8 − 2.8 = 0.01.4 − 2.8 = −1.4− 10.3 − 2.8 = −13.184.3 − 2.8 = 81.5− 2.6 − 2.8 = 5.4− 20.0 − 2.8 = 22.8− 41.0 − 2.8 = −43.8
Covariance
Date
Table 8.2
14.86
−1.98−3.53−1.44−0.01−0.28−1.1118.140.731.093.23
(Xi −X )(Yi −Y )(%)
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Table 8.2 shows how it works with the most recent 10 years of data on interest rates and S&P 500 returns.Thus, the covariance between Y and X is 14.86, divided by 10 years minus1, or 9 (referred to as nine degrees of freedom):
COVyx
14.86= 1.65%9
To convert this covariance into a correlation coefficient, calculate the standard deviations of Y and X. Square the differences of each observation fromthe mean to remove the + and − signs. Then, total them up (see Table 8.3).The variances are the totals divided by n − 1:25.7= 2.86%999.6= 11.07%9For the standard deviation, take the square root of these numbers. Thesquare root of 2.86 percent is 16.9 percent. The square root of 11.07 percent is33.3 percent.
Table 8.3
Correlation Coefficient
Yi −Y(%)
(Yi −Y)2(%)
Xi −X(%)
(Xi −X)2(%)
−24.9−22.07.714.819.28.422.3−13.4−4.8−7.4
6.24.80.62.23.70.75.01.80.20.5
8.016.1−18.70.0−1.4−13.181.5−5.4−22.8−43.8
0.62.83.50.00.01.766.40.35.219.2
Sum
25.7
99.6
196 Practical SpeculationFinally, simply plug those numbers into the correlation formula:Γyx =Γyx =
COVyxSySx1.65%
(16.9%)(33.3%)
= 0.29344
Here is a more elegant way of expressing it:Γxy =
Σ (Zx × Zy )n−1
Xi − XSxYi − Yand Zy =Sy
where Zx =
As with the rough estimate we made from the 2 × 2 table, departuresfrom the mean were the basic elements in computing the numerator and thedenominator of the preceding formula.
A Workhorse EquationThe Pearson Product Moment correlation coefficient defined previously is theworkhorse for studies of relations that appear on scatter plots. It measures thelinear degree of association between two things by showing how closely the observed values fit around a straight line. In simple terms, if the vertical distanceof the dots from the slope line is small, then the correlation is high.In the case of Treasury bills and S&P 500 returns, the correlation is 0.29for the 10-year example shown in Tables 8.2 and 8.3, but it is –0.07 for the full39-year data series displayed in Table 8.1. This shows instability all too typicalof correlations between economic time series. The culprit in this particularcase was just one year, 1995, when high values of interest rates and stockappreciation jumped the correlation up from a small negative to a high positive. (We thank Steve Stigler for kindly bringing this outlying data point toour attention.) In the world of stock market speculation, it would not be significant enough to trade on. In practice, we have found any number above0.10 or below −0.10 based on 100 or more observations to be useful. Thegeneral formula we follow for usefulness is that the correlation coefficient
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times the number should be greater than 10. Thus, for 50 observations, a correlation of 0.20 would be useful and for 20 observations, a correlation of 0.50would be useful.The information provided by the correlation coefficient is useful to investors in many ways. For example, it can tell you approximately how much ofthe movement in the Y variable—in this case, the S&P 500—can be explainedby the X variable.Another way of putting this is that the square of the correlation coefficient is the amount by which a linear predictor of one variable by the other isnot attenuated by a regression back to its mean value.The correlation coefficient cannot be used to directly predict the amountof change in the explained variable that a change in the independent variablescreates. For that, a regression equation is necessary. A regression equationconsists of a straight line that fits the points on a scatter diagram. It requires aslope and a y intercept. The slope is the correlation coefficient multiplied by afactor of the standard deviation of the y variable divided by the standard deviation of the x variable.In the case of the interest rate return data, the regression equation is:S&P 500 return = 2.3 percent less one-tenth of a percentage changein Treasury bill ratesThe most frequently used regression equation is fitted by minimizingthe squared distance of the values in the graph from a straight line. The linealways goes through the point corresponding to the mean of y and the meanof x. To compute the regression line by hand from a scatter diagram, calculate that point of intersection and then draw a straight line from that point toyour estimate of the y intercept (i.e., the average of the y values when x is approximately 0.) If you want to gain a measure of the variability about thatline for prediction or control purposes, draw two parallel lines at an equaldistance from the line so that the space between them includes about 95 percent of the points.Scatter plots have many uses in addition to revealing relations. Probablythe most useful is the visual observation that leads to discovery of possibleoutliers caused by either an error in data collection or a changed relation.Enhancements of scatter diagrams are available and useful. Colors, symbols,or letters for various groupings can assist evaluation. The colors green andyellow and the letters H, Q, and Z are all distinctive and are often used. Circling groups of observations that share similar characteristics may highlightimportant relations among the points that would not be clear from the usualsummaries that come out of standard computer programs. New scatter
198 Practical Speculationdiagrams can be created by varying the tallies and shadings of the line. Detailed discussions of these and other enhancements can be found on the Internet; the search engine google.com alone has 1,580 entries on “scatterdiagram enhancements.”Our favorite way to enhance a scatter diagram is to show the path of atime series. All that is required is to draw arrows between the dots in the sequence they occur.
Avoid Fallacious ConclusionsWhatever you do, never place too much reliance on scatter diagrams, correlations, or regression formulas. They are associations, not a proof of causation.Seemingly statistically significant correlations are often due to chance or hidden factors.Graphs are particularly vulnerable to misinterpretation. A famous example of seeing patterns that indicated causal relations that were, in fact,mere random fluctuations occurred in the bombing of London duringWorld War II. Newspapers published maps pinpointing where German aircraft dropped bombs, and people scrutinized the maps to see which areaswere left untouched, in the belief that the patterns revealed the residences ofGerman spies. Subsequent analysis showed that the bombing patterns werecompletely consistent with chance. A discussion of the tendency to infercauses from random data, such as the geographic incidence of cancer, iscontained in Rational Choice in an Uncertain World, by Reed Hastie andRobin M. Dawes.2The most serious problem in this regard is the drawing of conclusions oncausality based on spurious or bogus correlations. Such false conclusions areusually built on one or more of the following flaws:• Failure to account for the possibility of randomness.• Omission of a third variable that causes the other two.• Failure to consider mobility, or changes in the population itself.• The fallacy of aggregation.Sadly, most studies of market relations suffer from all these fallacies simultaneously. For a great case study, we recommend many of the academicworks previously cited in this book. However, an even better introduction isprovided by Purdue University’s “Spurious Correlations Contest.” The 1998winner is timeless and ineffable. Ron Malzer’s entry is based on the observed
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high correlation of the population of Oldenburg, Germany, and the number ofstorks sighted there: Oldenburg, Germany, has the world’s highest proportionof adults who read the works of Immanuel Kant, and each year itspopulation is increased due to the immigration of Kantonians fromother places. As Kant discussions increase, there is pressure on thecooks to prepare ever-increasing quantities of wurst. As quantitiesincrease, quality decreases. Bad wurst has been known to lead to thought disorders. This leads to serious misinterpretations of Kant’s work, called unkantverstehenlassenhummels. Written discourse about the error leads to disorders of theeyes just from having to read such a long word. The resulting double visionleads to spurious sightings of storks, with data reported at least double the actual numbers involved.3Let’s examine the problem of specious correlations systematically.The Error of RandomnessThe failure to impute chance or randomness as an explanation for a supposed market relation is by far the most common problem in analyzing correlations. Almost everything you read in the financial pages about what isgoing on in markets is mainly attributable to chance. The most common example here is the breathless reporting of some mutual fund manager’s recommendations on individual stocks or future market direction. “Mr. X couldforecast the exact high. He bought at the exact low and wrote a book. Hebought some ABC before the recent run-up and sets a target for the stock of$100.” Yes, of course. There always are going to be managers and advisorsfrom among the tens of thousands who called the last up wave right. But didtheir success come through superior insight? A much more likely explanation is chance.A classic investment scam capitalizes on this failure to consider the roleof chance. A prediction is mailed out to half the people on a list of “targets.”The other half receives the opposite prediction. The ones who receivedwhichever prediction turned out to be correct are then split into two groups,half receiving one prediction and the other half receiving the opposite. Afterfive or so iterations, you have a small group of followers who think you are agenius. Then you ask for big bucks.The same failure plagues discussions of correlation between technical indicators and market performance. So many technical indicators are out there,as discussed earlier in this book, that there will always be one or another indicator, one or another practitioner, one or another seminar speaker or promoter, with a hot hand.
200 Practical SpeculationA discussion of this in Chapter 3 of Vic’s previous book, The Education ofa Speculator, illustrates how even a very long string of consecutive price movesin the same direction is consistent with randomness. In 1995, the yen declined 16 Fridays in a row. Yet taking account of all the different commoditiesand days, such a run was not improbable at all. Similarly, random moves explain almost every old adage that you will hear about the market. High on thelist are the Super Bowl indicator and the January barometer. This list of examples could be vastly extended.There is a great antidote for drawing spurious relations from randomphenomena. The antidote is to read a statistics book on multiple comparisonsor a chapter on the subject in any of the multivariate statistics books. A goodbook here is Multiple Comparison Procedures by Yusef Hochberg and AjitTamhane (John Wiley & Sons, 1987). The gist of all such work is that if yousearch for a thousand different relations or look back on the winners of thousands, then you have to adjust the actual odds against the phenomena by asimilar factor of—say—a thousand.The Error of Omission, or the Missing Third VariableThe second most common cause of errors leading to faulty conclusions aboutstocks is mobility in conjunction with the omission of a third variable. Let’sturn again to those messengers of fecundity, the Scandinavianstorks. Data from Germany, Denmark, and Norway show that thegreater the number of storks, the greater the number of births. InThe New Statistical Analysis of Data, T. W. Anderson and Jeremy D.Finn explain the correlation in a nice way: “Districts with large populationshave a large number of births and also have many buildings, in the chimneysof which storks can nest.” Thus, it is population that explains both the frequency of births and sightings of storks. They illustrate with a diagram, asseen in Figure 8.3.It would be a good idea for all students of market relations to post Figure 8.3 somewhere near their work desk as a reminder of this pitfall.Examples of omitted third variables abound in market studies. Typicalin this research are the studies of the validity of technical indicators basedon sentiment. Almost all of these show that after sentiment reaches a certainoptimistic level, the market becomes bearish; and after the swing to unduepessimism, the market becomes bullish. As shown in the technical indicators chapter, there is a strong negative correlation between the recent movein the market and subsequent performance. But all sentiment indicators,
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Manybabies born
LargepopulationManybuildings
Manystorks
Figure 8.3
Many Storks, Many Babies. Source: The New StatisticalAnalysis of Data, Springer-Verlag, 1996. Used with permission.
such as the percentage of bullish advisors and the flow of money to equityfunds, are highly positively correlated with the recent market move. Thus,when the market is up big over the previous two weeks, sentiment tends tobe optimistic, which is bearish. Conversely, when the market has been downa lot, sentiment is negative, which is bullish. Sentiment has an indirect relation to the future market movements, but only because of its relation withthe recent market move.The Fallacy of MobilityCorrelations reported in the market literature often suffer from failing to takeinto account a change in the nature of one of the variables. Helen Walker, astatistics professor at Columbia University, came up with the following spurious correlation to illustrate this fallacy. Young womenare taught to walk by placing one foot before the other along animaginary line, a practice that results in pleasantly swaying hips.Older women, perhaps less preoccupied with the need to appear alluring,tend to walk with their feet on either side of that invisible line. However, ageis not caused by a wide gait. A more plausible hypothesis is that age causeswide stepping, but that would fail to take into account the fact that olderwomen grew up at a time when girls were taught to toe out, whereas today’sgirls are encouraged to keep their feet parallel.Studies relating longevity to occupation are particularly vulnerable to spurious correlations. In Statistics on the Table, Steven Stigler discusses an 1835Swiss study that found natural philosophers lived to 75 while the average student died at 21. A child contemplating what he wants to be when he grows up
202 Practical Speculationmight well be advised to place more weight on the nature of the job than toany possible benefits in life expectancy, however, given mobility between occupations. After all, most students don’t die students; if they can make itthrough those dangerous student years, they turn into natural philosophers,sculptors, or clerks.A study on the life expectancy of conductors nicely illustrates how omission of a third variable—the achievement of living to a certain age or not—biases most studies of longevity based on occupation or lifestyle. On average,conductors live to age 74. Poets live much shorter lives. Yet, that does notmean prospective artists could improve their longevity by choosing conducting over poetry. To become a conductor, extensive training is necessary; 35would be a reasonable expected minimum age for inclusion in the fraternity.Those who died as students and those who died in childhood have no chanceof making the mark. Once this is factored in, the apparent longevity of conductors disappears. The average life expectancy of males who reach the age of35 is about 74, the same as conductors.This type of fallacy in correlation analysis is well known, yet this bias isoften present in Wall Street studies. Using samples of stocks that were createdand weighted with surviving companies, much work tends to conclude thatvalue stocks are supreme. Careful analysis, however, often reveals that the results are skewed by the construction of the files. Small companies are includedin the growth files only on attaining a certain size; as a result, many fast-growingcompanies are omitted. The value files, on the other hand, include manycompanies experiencing financial difficulties. If the company’s financial difficulties worsen to the point that it goes belly up, it is not represented in thefiles. If it survives, it is included.Much of the superior retrospective performance attributed to value stocksis attributable to this survivorship bias. Except for this flaw, the studies ofJames P. O’Shaughnessy are a model of good research. Note, however, thatO’Shaughnessy decided to leave the mutual fund business after a value mutualfund based on the models in his 1998 book, What Works on Wall Street, suffered from dismal performance.4 A subsequent venture that sold customizedbaskets of stocks over the Internet closed its Web site in September 2001. Almost all other mutual funds based on such correlations have had similar results. A good case in point is the Rea-Graham Fund, which was first to apply thecorrelations discovered by Benjamin Graham between value stocks and returnsfor the benefit of investors. The fund’s performance since inception earnedit the lowest Morningstar ranking possible in all categories. But it looked goodon paper.
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The Fallacy of AggregationIf low P/Es are associated with high subsequent returns for individual stocks,then a low P/E for the market is associated with higher subsequent market returns—this is an example of what statisticians call the fallacy of aggregation.
The Curse of Specious CorrelationsIt has been shown that shoe size is highly correlated with readingskills, but this does not mean that buying bigger shoes would helpkids read better or that reading more will make the feet grow. Instead, a third unmeasured factor is involved: age. As children getolder, their reading abilities grow along with their feet.While every student is taught to be careful about inferring causation fromcorrelation, it is amazing how rarely this is taken into account in the numerical work of system developers and quants of all stripes. Invariably they comeup with a relation, the correlation between this indicator and that, and thesubsequent superior performance. If they could only remember the studies ofstorks, shoe size, conductors et al., and pay more attention to the scatter plots,they would waste much less time chasing after spurious systems.In Gilbert and Sullivan’s comic opera Rudiggore, Mad Margaret loses hersanity because the baronet she loves is doomed by a curse to commit a crimeonce a day. Only the enunciation of the word “Basingstoke” can bring herback temporarily to her senses.A solution is found, the curse is finally broken, and the entire ensembletoddles off happily to settle in the bucolic Hampshire town of Basingstoke.Back when Laurel was at the wire service editing stock market stories,Vic often cautioned her about attributing price moves to anecdotal examplesof ephemera, especially those involving earnings woes or up/down volumein individual stocks. Eventually, Vic shortened his messages to the word“Basingstoke.”It is a much more serious error to attribute causal relations to two thingsthat are attributable, as is so often the case, to purely spurious factors. In viewof the seriousness of this crime, we recommend to readers that wheneverthey feel the tendency to succumb to such an attribution, they repeat theword unkantverstehenlassenhummels three times. The rewards for readers whodiligently apply this technique will be early retirement to a splendid mansionand numerous wurst dinners in Oldenburg, Germany.
9
THE FUTUREOF RETURNS
The source of all market triumphs is not bullishness orbearishness, but the investigating human mind.—Correspondence from Brett Steenbarger, PhD, June 23, 2002
T
he history of science is replete with examples of monumental collectionsof data that sparked revolutions in knowledge. Tycho Brahe’s careful andextensive astronomical calculations, based on observations made with his innovative telescopes, led to the development of Johann Kepler’s laws on planetarymotion. The data on plants collected by Carl Linnaeus in the early eighteenthcentury laid a base for Charles Darwin’s discoveries and groupings. Painstakingly exact and detailed measurements by Henry Cavendish and AntoineLavoisier on the various chemical elements led to Dmitri Mendeleyev’s discovery of the periodic table.A similar advancement in knowledge in the field of investments cameduring the 1960s when the University of Chicago Center for Research in Securities Prices published the first reliable and comprehensive database of dailyprices of U.S. stocks, from 1926 to 1960. The science of securities analysis205
206 Practical Speculationexploded shortly thereafter. The ensuing contributions in the fields of portfolioanalysis, capital structure of corporations, option pricing, the efficient marketshypothesis, the theory of behavioral finance, and rational expectations havebeen well chronicled and memorialized with numerous Nobel Prizes. Accounts can be found in any standard college finance text.It took four more decades to complete a comparable database for allcountries. Elroy Dimson, Paul Marsh, and Mike Staunton of the Universityof London Business School undertook this immense project. The professorspublished their findings in a 2002 book, Triumph of the Optimists: 101 Years ofGlobal Investment Returns. Within Triumph’s pages, an investor may find definitive information on inflation-adjusted returns for stocks versus bondsand Treasury bills, real dividends, correlation between markets worldwide,and the relative performance of value and growth stocks.Unlike most books written by academics, Triumph avoids hasty generalizations and biased sampling procedures. The authors rightly fault earlier investment studies for arbitrary selection of starting and stopping points, thetendency to include the good and exclude the bad, and the exclusion of all but asmall slice of the global picture. Their work epitomizes outstanding investmentresearch.Great works can be created in humble circumstances. Shakespeare wasan actor and entrepreneur who reworked old plots so that his companycould make a buck. Cervantes wrote a parody of the mania for knight errantry books to repay his debts. Rabelais wrote humor to help cure the painof his sick patients. Dimson told us that he and his colleagues thoughtof Triumph as “a labor of love, just a small contribution that could lead to apaperback meant for light reading on planes.” He added, “Our familieswould be less kind about our fixation.” Staunton, who collected the data,prefers to gather statistics by himself from original sources at specialized libraries instead of delegating the work. He wrote his doctoral thesis on airline ticket prices.The main conclusion of Triumph is that a random selection of U.S. stocksreturned 1,500,000 percent in the tw