 reserve  established  during  the  run-up.  Asperformance  begins  to  turn  negative,  the  portfolio  manager  in  question
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demonstrates  the  ability  to  reduce  P/L  volatility  until  such  time  as  hebegins to enjoy a turnaround, at which point he feels comfortable onceagain  in  increasing  his  exposures.  A  pattern  such  as  this  one,  in  myexperience, can generate profitable, scalable returns over an extendedtime period; and if this trader can continue to exercise the basic disciplineof  risk  management,  I  am  confident  in  his  ability  to  continue  to  operatesuccessfully.
Contrast this to the story told by Figure 3.2. In addition to the fact thatthe general trend is negative, there is discernable effort to tie the level ofvolatility to performance success. As events continue to worsen, volatility(as  depicted  by  the  vertical  bars)  remains  constant  or  even  increases.Eventually, the portfolio shifts into risk-reduction mode, but by then it istoo  late.  The  opportunity  to  recoup  losses  and  to  reach  new  plateaus  ofprofitability has been severely compromised, if not lost altogether. 
For  my  tastes,  there’s  simply  nothing  in  the  entire  tool  kit  that  is  asuseful as these graphs. Once you get comfortable reading them, you canpretty much see what’s going on in a portfolio in a heartbeat. What is more,if  you  are  oriented  toward  technical  analysis,  you  can  apply  the  samemethodologies  to  the  review  of  these  diagrams  as  you  use  in  looking  atstock or commodities charts. To reinforce this point, think of the two port-folios  represented  as  individual  securities.  In  the  case  of  the  first,  per-formance fluctuates; but every time it suffers, there is a nice basing patternthat forms —each time at levels comfortably within the established range.The trend over the entire period is upward, and it looks like this trend willcontinue.
Now take another look at the other portfolio. Here, the trend is down-ward,  and  performance  is  deteriorating  at  what  looks  like  an  increasingrate. If this portfolio were a stock, it would be one of the last ones in theworld you would want to own. In fact, as any chartist would be happy totell you, the only possible trade here is to be short.
Histogram of P/L Observations
Another visual tool that is well worth a thousand statistics is the graphicpresentation of returns in the form of a P/L histogram. The creation of ahistogram  involves  grouping  individual  P/L  observations  into  size  cate-gories and identifying the number of observations within each increment.The ranges of these increments can be expressed in terms of dollars or per-centages. When the histogram is plotted in bar chart form, it provides anillustrative description of your P/L. 
Consider Figure 3.3, the histogram associated with the portfolio firstintroduced to us in Figure 3.1: If you tilt your head at a 45-degree angle andkind of squint out of your left eye, you can see what looks a little bit like a
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60
50
40
30
20
10
0
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51
43 43
35
22
18
2
2
0
5
3
1
10
8
2 3 2 1 1
−80or <
−79to −70
−69to −60
−59to −50
−49to −40
−39to −30
−29to −20
−19to −10
−9to −1
0 
to 
9
10
to 
19
20
to 
29
30
to 
39
40
to 
49
50
to 
59
60
to 
69
70
to 
79
80
to 
89
>90
FIGURE 3.3 Sample histogram showing daily P(cid:1)L.
bell shape, with most observations centered around the mean and decreas-ing numbers of data points situated within the outer brackets. This, as wewill discuss shortly, is very typical for P/L distributions and other randomdata sets—a fact that, as we will also cover, is more important than per-haps it ought to be. You will also notice that there are more observationsat the extreme ends of the distribution than might be entirely intuitive; butthis too, as we will review, is to be expected. 
The graph of the histogram offers an immediate visual image as to themanner  in  which  portfolio  returns  distribute  themselves,  but  it  also  pro-vides the basis for meaningful further analysis. In particular, take a closelook at the observations in both tails of the distribution. Try to draw com-parisons here. Are the big gains part of the same market sequence, or dothey at least occur during markets with similar characteristics? Are theytied  to  the  same  financial  instruments?  If  they  are,  is  it  because  theseinstruments are inherently volatile, because you traded them during a par-ticularly volatile period, because of position sizing, or some combinationthereof?  Once  you  have  taken  the  effort  to  fully  reconstruct  the  portfo-lio/market characteristics associated with your extreme observations, it isuseful to attempt to reengineer associated trading decisions as well. Forthe  negative  observations,  were  there  any  mistakes  you  made  that  wereobjectively avoidable? Similarly, for those days of extreme profitability, didyou  do  everything  in  your  power  to  maximize  on  this  favorable  trend?
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Other  questions  of  a  similar  nature  should  come  to  you  also—a  processyou should nurture. Do your best to answer the questions. You may learnsomething in the process.
Finally, if you find that over extended periods of time you continue toexperience larger numbers of negative extreme data points than are consis-tent with a bell-curve distribution, you must take the time to examine thesedynamics and to form a plan for effective management of associated expo-sures. Absent a fundamental flaw in the way you are approaching risk man-agement, excessive losses are more than likely due to the presence of somekurtosis  (i.e.,  overabundance  of  extreme  observations)  in  the  underlyingmarkets that you are trading. Many markets are characterized by kurtosis—particularly over very long time horizons—most notably global equities andmost emerging markets. Test this hypothesis for your markets, if you can, byplotting the distribution of daily returns for the securities in which you aremost active. Do they lend themselves to occasional abrupt departures fromnormal  pricing  patterns?  If  so,  it  will  be  important  to  size  your  associatedposition  taking  in  a  manner  that  accounts  appropriately  for  the  additionalexposure.
In any event, if extreme negative-return events continue to persist, itwill be necessary for you to reduce your risk profile relative to what wouldbe acceptable if your portfolio had the similar risk characteristics, withoutthe excessive presence of negative outliers. While it will require consider-able  amounts  of  discipline  to  do  so,  the  payoff  will  come  in  the  form  ofdampening  the  magnitude  and  associated  impact  of  the  worst  points  inyour trading cycle.
STATISTICS
We could spend a month drawing pictures and interpreting them for funand profit. Certainly Dr. Freud would endorse this approach, and it mighteven pay us over the long run. But now, having put off the matter longerthan I had intended, I find I must revert to the numeric. Statistics I havepromised, and statistics I will deliver.
A Tribute to Sir Isaac Newton
To fortify us in this exceedingly daunting enterprise, I will call on the spiritof Sir Isaac Newton. It seems to me that Newton would have made a greattrader;  history  tells  us  he  had  all  of  the  tools.  He  spent  his  early  daysobserving certain patterns of motion in the universe, generalized them intoa theory that is certainly the basis for all modern physics, and invented the
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mathematical science of calculus as a means of explaining his theories.1 Asthe world eventually found out, this latter discovery was not only perfectlyapplicable to the description of the physical universe, but also proved itselfto be enormously handy in modeling the social sciences, most notably (atleast for our purposes) economics. 
Rumor has it that before they hoisted him up to his final resting placein Westminster Abbey, he amassed a considerable fortune; and no doubtthis is the case. After all, Queen Anne saw fit to knight him; and up till thattime, this sort of thing had been the exclusive domain of the aristocracyand military types (and was certainly unheard of for a scientist). When hetired of academic life, he moved seamlessly into government affairs, serv-ing in what one can only assume was among the most lucrative bureau-cratic posts ever devised, the Master of the Royal Mint. By all accounts,he prospered for most of the nearly seven decades of his adult life. How-ever,  his  record  as  an  investor  is,  to  be  sure,  somewhat  mixed.  Mostnotably, he is known to have lost a bundle in the infamous South Sea Bub-ble  episode,  a  bizarre  scheme  under  which  the  British  governmentappears to have created a full boom/bust cycle in the single, fateful yearof 1720, by issuing on no less than four occasions a form of convertibledebt—without ever bothering to inform the investing public what the con-version ratio was. 
The  South  Sea  escapade  was  quite  a  hoot—even  by  today’s  somewhatjaded post-dot-com bubble standards. The whole country knew it would endbadly; but this didn’t stop everyone who was in a position to do so (includingmany members of parliament, Chancellor of the Exchequer John Aislabie andhis staff, such literary luminaries as Daniel Defoe and Alexander Pope, andKing George I himself) from grabbing every share they could get their handson. During the tumultuous period between spring and autumn of 1720, someshares  in  South  Sea  and  related  enterprises  increased  10-fold,  attractinginvestors from such remote locales (particularly from an eighteenth-centuryperspective) as America and the Far East, before collapsing entirely and in
1Purists will be quick to point out that the German mathematician Gottfried Leibnizhas at least as plausible a claim as the father of calculus as does our friend Newton.I am told that he published his theories a couple of decades before the appearanceof the latter’s landmark Principia Mathematica. To be perfectly honest, I have noinformed  opinion  whatsoever  as  to  who  got  there  first.  Here,  as  elsewhere,  I’llclaim author’s privilege, as Newton is clearly the more appealing hero for this par-ticular digression. However, let me do what I can to correct this by giving Leibnizhis props as a righteously clever dude and someone who also would probably makea great trader.
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the process ruining many a fortune and poisoning the continental investmentclimate for many years after.
In terms of Newton’s involvement, he apparently used his considerableconnections to grab a sweet slice of Round 1; but recognizing the unsoundbasis of the enterprise, he sold his early shares at a sizable profit, famouslyuttering, “I can calculate the motions of heavenly bodies, but not the mad-ness of people.” However, for reasons indeterminate, he dove back on induring the summer months and got slaughtered with the rest of the massesas the leaves began to turn on his country estates, losing his entire originalinvestment and perhaps a lot more. 
If it were possible to discuss the incident with Newton, he would nodoubt agree that even perfect mathematical and statistical insight cannotsave  you  from  the  caprices  of  the  marketplace.  That  said,  it  is  my  beliefthat Newton’s most famous efforts are even better oriented to the marketsthan they are to the physical sciences. For generations after he created hislaws of motion, people believed that they perfectly described the dynam-ics  of  the  universe.  Then,  early  in  the  twentieth  century,  Einstein  camealong  and  pointed  out  that  these  laws  don’t  apply  at  high  speeds  and  inother nooks and crannies of the cosmos. This was part of the process ofequating  matter  to  energy,  which,  though  arguably  as  important  as  New-ton’s key revelations, was known by even Einstein to fall short of explain-ing in a comprehensive fashion how the universe came into being and whyit does what it does. 
Over time, a Newton replaces a Galileo, an Einstein replaces a Newton;and bit by bit, we learn more, but not everything, about the inner workingsof the cosmos. So it goes in the markets as well, which I would argue ebband flow in a manner much more consistent with Newtonian theories thanwith those of any of his forerunners or successors. As traders, we live in aworld where the natural state of prices are either at rest or in motion, whereit takes an external stimulus to change this balance and where actions aretypically met with equal and opposite reactions. For example, one very use-ful way of viewing the market downturn in the first decade of the twenty-first century is as a Newtonian response to the well-documented excessesof the last decade of the twentieth century. Yes, there are patterns of mar-ket  behavior  that  more  directly  reflect  Einstein’s  view  of  the  universe—where  instantaneous  changes  in  core  conditions,  particularly  as  theyapproach  zero  or  infinity,  generate  market  events  that  are  the  financialequivalent of atom splitting. However, if one is careful, commits to clean liv-ing,  and  only  involves  oneself  in  markets  where  one  understands  pricingpatterns,  then  one  stands  every  chance  of  trading  for  an  entire  lifetimewithout ever encountering conditions that even remotely resemble nuclearfusion/fission. If Einstein and his cohorts like Fermi ever got into the busi-ness of trading, they would probably prefer complex instruments, such as
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derivatives. No doubt they would do well; but even here, they’d be the firstto acknowledge their heavy debt to Newton.
It  is  therefore  to  the  memory  of  that  learned  second  holder  of  theLucasian chair at Cambridge University (a spot currently occupied by thatmiracle of modern science, Stephen Hawking) that I dedicate my statisti-cal  tool  kit.  If  we  can  look  at  the  market  universe  with  a  measure  ofNewton’s intellectual efficiency, we may live to avoid if not the big crunch,then at least at least the apples that seem to be forever falling on the headsof unwitting market participants. 
Before we turn to the numbers, one additional word of caution is per-haps required. Please note that the statistical section and, indeed, much ofthe rest of the book contains numerous examples. Some of these use verylarge numbers; others apply more modest figures. Please do not be overlyimpressed by the former or depressed by the latter. The point here is thatthe same concepts pretty well apply whether you are managing $50 thou-sand worth of capital or $50 million.
Average P/L
This is a basic calculation of the mathematical mean of your daily returns,as derived by dividing your cumulative performance by the number of peri-ods (e.g., days) the performance covers. You may find it useful to performthis calculation in terms of both dollars and percent of capital, the latter ofwhich will help you to evaluate your performance across various levels ofcapital committed to the markets. For instance, if your account balance is,say, $50,000 at the start of a given period and your average daily return is$500,  this  would  represent  a  1%  daily  return.  However,  if  over  time  youraccount balance grew to, say, $100,000, the $500 would represent only halfthis amount when expressed in percentage terms.
I heartily recommend that you calculate and compare these averagesacross different segments of your portfolio, over different time horizons,and so on. For example, you may want to calculate your average returnon  longs  versus  shorts  in  order  to  determine  whether  there  is  a  dis-cernible bias in your market orientation. Similarly, you can compare youraverages  across  market  sectors  (for  equities),  underlying  markets  (forfutures), segments of the yield curve (for fixed income trading), and soon.  Look  carefully  here  for  differences  in  your  unit  performance.  Arethey  tied  to  external  factors  such  as  market  conditions  or  perhaps  toareas  of  expertise  or  trading  comfort?  By  calculating  averages  acrossthese factors, you can begin to form an idea of what is working in yourportfolio and what isn’t.
Once you have mastered this relatively simple concept, you may wishto  manipulate  the  calculation  in  such  a  way  as  to  provide  additional
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insights akin to those just described. One such adjustment is to separateyour profitable and unprofitable days (or winning and losing trades) intoseparate data pools and to calculate the associated averages of these P/Lpools. This will enable you to begin to compare your performance in win-ning situations to those in losing ones. In turn, this will be a critical themethat we revisit throughout the book, as I will argue throughout that no ele-ment of your trading success is more important than making sure you haveextracted more from the market during profitable intervals than you havegiven back when events are working against you.
Standard Deviation
This second core element of statistical analysis is a bit elusive to define,but it is one of the most important components of our tool kit. As such, itbears  spending  some  time  and  effort  on  standard  deviation  to  obtain  afairly precise understanding so it can be efficiently applied. In order to doso, we must first introduce the concept of a normal distribution, which,stated simply, is a characteristic of most random data sets, ranging frompeople’s heights, to their bowling scores, to their performance in the mar-kets over time. There are very statistically complex definitions and associ-ated tests respecting the normality of distributions; but, of course, the sim-plest and safest way to think of them is as those data sets under which, bygrouping the range of outcomes by their magnitude, one observes a largeconcentration of data points around the mean, with the number of obser-vations  decreasing  sequentially  in  either  direction  from  the  mean  datapoint. If you plot this number of outcomes (y) against their absolute mag-nitude (x), you end up with a graph whose shape looks something like abell, as depicted in Figure 3.4.
y
0
FIGURE 3.4 Graph of a standard normal distribution.
x
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y
Mean
0
0
x
FIGURE 3.5 Normal distribution with standard deviation bands.
If  we  assume  that  a  distribution  is  in  fact  normal  (and  this  is  anassumption, as we will reveal later, that you undertake with some degreeof hazard), then it is entirely defined by its mean (defined earlier), whichprovides  information  regarding  the  magnitude  of  specific  observations,and standard deviation, which is a measure of dispersion around this meanobservation.
The  standard  deviation  of  a  normal  distribution  is  indicated  by  the
width of the shaded bands illustrated in Figure 3.5.
The  standard  deviation  will  thus  be  measured  in  units  of  the  x-axis,which, for our purposes, may be thought of as dollars or percent of dailyreturn. The magnitude of this figure will become our main unit for meas-uring the riskiness in the portfolio.
Figures 3.6(a) and 3.6(b) illustrate normal distributions that are narrow(carrying  minimal  risk)  and  wide  (indicating  higher  levels  of  exposure),respectively.
Figure 3.6(a) offers an example of a normally distributed data set witha small standard deviation (as measured by the width of the bands that areclosest to the mean along the x-axis). A low standard deviation is one wheremost of the observations in a given distribution are close to the mean datapoint. If we think in terms of daily P/L performance, a low standard devia-tion would imply that the daily returns to a given portfolio resided largelywithin a narrow band of dollar-based or percentage-based gains and losses.In turn, this would be indicative of a low risk profile. By contrast, the large
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y
Mean
0
x
(a)
Mean
(b)
x
y
0
FIGURE 3.6 Examples of (a) small and (b) large standard deviations.
amount of x-axis space occupied by the distribution in Figure 3.6(b) sug-gests a much wider range of P/L outcomes, as would be consistent with ahigher risk profile. Over the course of the next few chapters, you will begin,I hope, to understand just how powerful the standard deviation concept isin measuring the volatility of portfolio returns, which is exactly the notionwe are trying to capture in measuring risk, or portfolio exposure.
Standard  Deviations  and  Confidence  Intervals. As  I  have  sug-gested  earlier,  the  standard  deviation  statistic,  while  not  lending  itselfto simple  definitions,  tells  us  what  level  of  distributional  dispersion  is
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TABLE 3.1
Percentage of ReturnsAssociated with StandardDeviation Intervals
Percentage of Normal Curve Area
Mean (cid:1) Number of Standard Deviations
50.0%68.390.095.095.498.099.099.7
.674(cid:2)
1.0001.6451.9602.0002.3262.5763.000
consistent  with  a  specified  percentage  of  total  observations  being  con-tained within the associated range, with one standard deviation on eitherside  of  the  mean  (indicated  in  Figures  3.5  and  3.6  by  the  innermost  por-tions of the graphs) accounting for approximately 68.3% of all the data inthe sample. By using multiples of the statistic, we can identify the magni-tudes that would be associated with greater percentages of the distributioncurve:  with  two  standard  deviations  (indicated  by  the  sum  of  the  twoinnermost  areas  together  in  Figures  3.5  and  3.6)  representing  more  than95%  of  all  observations,  three  standard  deviations  (covering  the  threeinnermost areas) representing more than 99%, and so on—again assumingthat the data set in question is indeed distributed normally.
Table 3.1, which may come in handy for you in the future, maps spe-cific  multiples  of  standard  deviations  into  corresponding  percentages  ofthe area underneath the bell curve in a normal distribution. 
It is important to note that the tools of mean and standard deviationare  typically  viewed  in  financial  markets,  as  elsewhere,  as  measures  ofestimation and prediction. In this sense, the left-hand column of Table 3.1represents a confidence interval associated with a given number of stan-dard  deviations.  For  example,  we  would  be  68.3%  confident  that  a  givenobservation  will  fall  within  plus-or-minus  one  standard  deviation  of  themean,  95.4%  confident  that  an  observation  will  fall  within  plus-or-minustwo standard deviations of the mean, and so on. 
The concept of a confidence interval in this sense is only as valid as isthe  tendency  of  historical  distribution  patterns  to  continue  to  replicatethemselves (i.e., the past is a good predictor of future behavior). However,because  (typically  due  to  the  introduction  of  new  factors  or  informationsets)  the  future  will  always  deviate  somewhat  from  the  past,  your  confi-dence interval statistic will never be entirely accurate. For example, you
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are likely to experience a P/L fluctuation that you have identified as repre-senting a 95%, more frequently or less frequently than the precise one dayeach month, which is implied in the calculation. Despite these shortcom-ings,  the  confidence  interval  concept  is  very  important  to  risk  managersand other control entities as a general indicator of the size of their currentexposure. Be that as it may, for our purposes (and for reasons I will explainlater),  we  will  often  be  more  inclined  to  maintain  our  focus  on  standarddeviations as the purest measure of volatility.
How  to  Calculate  the  Standard  Deviation  Statistic. As  thisbook is a practical guide to trading and not a treatise in statistical analysis,I will try to avoid the posting of mathematical formulas. The standard devi-ation statistic is as ubiquitous to the world of statistics as body tempera-ture or blood pressure is to the field of medical science and can be calcu-lated using embedded formulas in spreadsheet products such as MicrosoftExcel or in any calculator with even basic statistical functions.
For the mathematically rigorous among you, and in the interest of pub-
lic service, I offer the formula in its arithmetic form:
s (cid:3) 1var (cid:3)
(cid:6)1X (cid:4) X 22N (cid:4) 1
B
where s (cid:3) standard deviationvar (cid:3) portfolio variance
X (cid:3) individual observationXN (cid:3) number of observations
(cid:3) sample mean
Also in the interest of public service and full disclosure, I should tell youthat  some  of  the  world’s  finest  statisticians  (certainly  some  of  the  finest  Iknow) couldn’t regurgitate this formula if their lives depended on it. So don’tfeel bad when you end up doing this the easy way and plugging it into Excel.
Sample  Standard  Deviation  Calculation.In  order  to  maintainsome consistency in the use of examples, I ask you to return to the sampleportfolio depicted in Figure 3.1. If you were to go to the trouble of calculat-ing a standard deviation for this portfolio for the year 2002, you would derivea result of $25,349. Applying the logic we are using throughout this discus-sion, such a statistic implies that 68.3% of all daily P/L observations will fallsomewhere between (cid:4)$25,349 and (cid:5)$25,349. Further, using the data fromTable 3.1, we can establish a 95% confidence interval for future observationsby  simply  multiplying  the  standard  deviation  statistic  by  the  appropriatescale-up figure, in this case 1.96. Thus, the 95% confidence interval for this
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particular portfolio is (cid:7)$49,684, and we would expect no more than 1 out of20 observations (about one trading day per month) to fall outside of thesebands.
In addition to expressing this statistic in dollar terms, it is also usefulto  apply  it  against  a  percentage  of  capital.  For  simplicity’s  sake,  assumethat  the  account  in  question  was  trading  with  an  introductory  capitalamount of $