3 million. Under these circumstances, a one standard deviationfigure  would  be  approximately  0.75%,  and  the  95%  confidence  intervalwould  be  located  between (cid:4)1.47%  and (cid:5)1.47%.  In  turn,  this  means  thatonly  once  a  month  would  we  expect  a  daily  P/L  observation,  positive  ornegative, to exceed approximately 1.47% of capital, and this is one mean-ingful way of expressing portfolio risk.
Note, however, that we do not arrive at these numbers by simply divid-ing the dollar volatility ($25,349) by the introductory capital figure ($3 mil-lion)  and  scaling  up  to  make  the  confidence  interval  calculation—aprocess that would return a higher result of 0.85% for one standard devia-tion  and  (cid:7)1.66%  for  a  95th  percentile  confidence  interval.  This  is  truebecause as profits and losses accumulate (in the case of this sample port-folio,  I  am  happy  to  say  they  are  profits),  the  capital  base  of  $3  millionchanges accordingly. In order to calculate the most accurate version of thestandard deviation (which, as we should bear in mind, is merely an esti-mate of  portfolio  exposure  and,  as  such,  will  never  be  entirely  accurateunder  any  methodology),  it  is  necessary  to  adjust  the  capital  figure  on  adaily basis to reflect this P/L impact. However, for most purposes, you areprobably just as well off keeping the capital figure constant.
Applications  of  the  Standard  Deviation  Statistic.It should beclear to the careful reader by now that I really like this statistic. For onething,  it’s  typically  represented  in  mathematical  shorthand  by  the  Greekletter sigma, as applied in the lowercase: (cid:2). I find its curves to be very sen-sual and much easier on the eyes than its boxy, overmasculine uppercasecounterpart: (cid:6).
More important, I believe that it is the most concise measurement ofexposure to the markets. I use it as the primary unit of risk measurementin my own personal tool kit (yes, I have one), and I encourage readers todo  the  same.  As  is  the  case  with  many  such  measurements,  its  absolutemagnitude  is  of  less  importance  than  the  manner  in  which  it  changesacross time, market conditions, financial environment, and so on.
Note  that  the  standard  deviation  statistic  is  highly  sensitive  to  thenumber of observations used to calculate it and that you are almost certainto derive different answers across different time spans. For example, thefigure that you derive by using an entire year’s worth of history may differsignificantly from the result obtained by reviewing data over, say, the past
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20  days.  (Due  to  the  principal  of  statistical  significance  discussed  else-where,  I  recommend  against  performing  the  calculation  on  any  data  setconsisting of less than 20 observations.) Do not be surprised by these dis-parities,  which  in  fact  can  offer  useful  insight  into  the  risk  patterns  thatyour portfolio is displaying. Indeed, I believe that you should remain at alltimes mindful of your standard deviation data, across various time spans,perhaps relying on rolling monthly (20-day), quarterly (60-day), and annual(250-day)  windows  as  your  primary  guide.  Compare  and  contrast  thesenumbers. Try to tie back the highest and the lowest to specific incidents inyour market experience. If there is significant disparity between the peri-ods, you may want to investigate the sources of the higher number. Was itthe  result  of  extreme  volatility  in  the  markets?  Did  it  correspond  to  aperiod  when  you  (for  whatever  reason)  decided  to  increase  your  overallmarket participation levels? Perhaps some combination of the two?
A  continuous  review  of  this  information  will  help  you  in  the  criticaltask  of  determining  your  maximum  threshold  for  performance  volatility.How big a standard deviation is too big? Of course, this will vary from port-folio to portfolio and from situation to situation. As a general rule, I believethat  the  figure  should  not  exceed  10%  of  the  amount  of  capital  you  canafford to lose from the point of the calculation. For example, if your max-imum  tolerance  for  losses  is,  say,  $100,000,  I  would  recommend  againsttrading  in  such  a  manner  that  the  daily  standard  deviation  of  your  P/Lexceeds  $10,000.  I  believe  (and  here  purists  will  argue  with  me  that  thenumber is too small to generate a reliable result) that the best interval overwhich to calculate this is probably the rolling 20-day time span, as this isthe one most sensitive to daily fluctuations. Moreover, if you don’t exceedthis 10% threshold in any given 20-day span, you place yourself in the bestpossible position for preservation of capital over longer time spans.
While  the  10%  threshold  may,  at  first  glance,  appear  to  be  somewhatarbitrary, let me assure you that I have developed this guideline very care-fully, over a number of years and over wide ranges of market performanceand conditions. While we will discuss the details of this at greater length ina couple of chapters, I have based it on the indisputable fact that marketsoften move by factors significantly greater than one standard deviation andthat these large market movements often take place in direct succession.Therefore, the once-a-year conditions that cause prices to change by threestandard  deviations  or  more  are  events  that  would  put  a  10%  portfolio,under the methodology set forth earlier, in considerable jeopardy of hittingassociated maximum loss thresholds as the result of unforeseen volatilitythat operates as a perpetual threat to each and every risk-bearing portfolio.By operating below the 10% threshold, you place yourself in the best posi-tion to minimize the probability of avoiding the maximum loss level, aroundwhich any well-thought-out risk management program operates.
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There are many other potential applications of the standard deviationstatistic, some of which are best discovered through experimentation withthis and other related statistics, many of the most important of which aredescribed in the coming paragraphs. If you are willing to play with thesenumbers and apply them creatively to different elements of your perform-ance data, you stand to benefit.
At  this  early  stage  of  the  discussion,  I  will  introduce  only  one  moreconcept for your consideration: the use of standard deviation in the calcu-lation of risk-adjusted return. Specifically, it is common practice for port-folio  managers  and  allocators  of  capital  to  express  returns  as  units  ofvolatility, with volatility being measured by the standard deviation of port-folio  returns.  For  example,  reverting  back  to  our  sample  portfolio  fromFigure 3.1, the mean daily P/L for this account is $2,456, or 0.07%; and itsstandard deviation over the period is about $25,349, or 0.75%. By express-ing these two statistics as a ratio, we begin to derive some notion of risk-adjusted return. Specifically, we can state that for the period in question,75 basis points of daily risk generates 7 basis points of average daily return.While at this stage it is difficult to have an informed opinion as to whetherthis is good or bad performance, you should conceptually understand thatthis  performance  is  better  than  that  of  a  portfolio  that  yields  the  samereturn at a higher level of volatility. And you should understand that fromthis “risk-adjusted  return  perspective,”  we  might  be  indifferent  as  aninvestor between this portfolio and one that was twice as profitable andtwice as volatile.
Later  on  in  this  chapter,  we  will  formalize  the  calculation  of  risk-adjusted return in a number of important ways, most notably by introduc-ing  such  calculations  as  the  Sharpe  ratio  and  the  Return  over  MaximumDrawdown (ROMAD) statistic.
Standard Deviation Statistic for Nonnormal Distributions. Asindicated earlier, the standard deviation statistic is one that, in the strictestsense, can only apply to distributions that are characterized as normal, withmost  observations  centering  around  an  arithmetic  mean  and  with  a  dimin-ishing number of data points pushing away, in a bell-shaped pattern, from thiscenter point. In market economics, this assumption of normality is often chal-lenged, almost always on one of the two types of observable anomalies:
1. The distribution is materially skewed to one side of the mean.2. Extreme observations tend to occur with more frequency than is con-
sistent with the normality assumption (or kurtosis).
Purists will tell you that when a given distribution deviates from pat-terns of normality, the use of a standard deviation statistic to describe such
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concepts as volatility and dispersion becomes an increasingly questionableanalytical methodology. Indeed, for distinctly nonnormal distributions, thestandard deviation statistic may be entirely inapplicable.
Although very few markets or trading portfolios are skewed signifi-cantly  to  one  side  or  another  of  their  mean  return  profile  (indeed,portfolios with high degrees of skewness on either side are difficult tosustain, due to financial constraints on the negative side and to the ten-dency  for  the  market  to  quickly  arbitrage  away  “superprofits”  on  thepositive  side),  many  tend  to  have  more  extreme  observations  than  isperfectly  consistent  with  the  concept  of  normality.  In  these  cases,  thepresence of outliers will cause the standard deviation to be higher thanit would be if their magnitude or frequency adhered better to the normaldistribution  profile.  However,  the  standard  deviation  is,  simply  stated,not  a  precise  enough  tool  to  identify  these  conditions,  much  less  toquantify them.
I must tell you that the whole business of determining whether a dis-tribution is actually normal or not gives me a headache, so I’m not going tospend any more time on the subject. What is more, I don’t think you shouldtrouble yourself much on this score either. In fact, my best advice to you(though it be blasphemy if uttered in the church of the pointy-headed) is togo  ahead  and  calculate  the  standard  deviations,  without  worrying  aboutthe validity of the normality assumption. When you do, try to write thoselowercase sigmas with flourish and flair aplenty. 
As Sly Stone said at Woodstock, it’ll do you no harm. 
Sharpe Ratio
Now that we have (more or less) thoroughly explored the concepts of bothmean  return  and  associated  standard  deviation,  we  can  unite  these  con-cepts  into  what  has  become  the  industry  standard  for  calculating  risk-adjusted  return:  the  Sharpe  Ratio.  This  ratio  is  designed  to  normalizereturns  against  their  associated  volatility,  the  idea  being  that  a  unit  ofreturn can be judged qualitatively in inverse proportion to the amount ofvolatility required to produce it. For example, consider two accounts: onethat returned 25% in one year, with the first generating a standard devia-tion of 10%, and one that returned 20%. Under the Sharpe Ratio measure-ment, the first portfolio would be the better performer, because it ostensi-bly operated under a reduced risk profile while producing the same overallperformance.
There are many variations to the Sharpe Ratio calculation, but all of
them attempt to capture the following concept:
Sharp Ratio (cid:3) (Return (cid:4) Risk-Free Return)/Standard Deviation of Return
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Note that the right side of the equation can be expressed in terms ofdollars  or  percentages,  as  long  as  the  same  convention  applies  to  eachside.  A  few  words  about  the  individual  terms,  all  of  which  are  bestexpressed on an annual basis:
• Return. The amount you earn on assets.• Risk-Free  Return. This  is  the  amount  of  money  you  can  expect  toearn on assets that are classified as “riskless” in the field of economicanalysis, on amount of capital equivalent to the one that you are com-mitting to the markets you trade. For all but a few specified situations,the appropriate rate of return is that of U.S. Treasury instruments. Inthe  Sharpe  Ratio  calculation,  the  risk-free  return  is  subtracted  fromthe total return of the portfolio in order to isolate that portion of theperformance that is tied to the assumption of market exposure. Onerather elegant outcome of this is that an individual who takes capitaland invests it in Treasuries earns exactly the risk-free rate and, there-fore, generates a Sharpe Ratio of precisely 0, while portfolios that failto  earn  even  this  modest  level  of  return  actually  have  negativeSharpes.  Therefore,  it  is  only  for  performance  above  the  minimumbenchmark  of  government  coupons—and  as  such  is  deemed  to  beassociated with risk-bearing market activities—that the Sharpe Ratiobegins to recognize a positive, risk-adjusted return.
• Standard Deviation of Return. This is our old friend/nemesis, whichwe  thought  was  beaten  to  death  already,  resurrected  to  apply  as  therisk component in a risk-adjusted return calculation. Note that here itis extremely important to express this statistic in the appropriate timespans—ideally,  as  indicated  earlier,  one  year.  Due  to  the  specifics  ofthe calculation (under which the figure varies directly with the squareroot of the number of data points), this requires either multiplicationor division of the square root of the number of observations. For exam-ple, assume you have a year’s worth of daily data that indicates a dailystandard deviation of, say, $10,000, or 1.0% (assume $1 million of cap-ital).  In  order  to  figure  out  the  annualized standard  deviation,  wewould have to multiply this figure by the square root of the number oftrading days in a year. When weekends and holidays are removed fromthe calendar, this number converges to 250 (give or take a day or two),with  the  associated  square  root  being  about  15.9.  Therefore,  anaccount with a daily standard deviation of $10,000, or 1.0%, would havean annualized standard deviation of approximately $159,000, or 15.9%.
It is necessary to achieve this normalization of time spans across theSharpe Ratio equation in order to derive results that do not fall into the cat-egory of nonsensical. Note that the equation allows for adjustments that
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account  for  such  factors  as  partial  data  sets  (e.g.,  half  a  year’s  worth  ofdata) and for time periods other than daily. However, I will defer to my pro-fessional statistician friends for explanations of these mystic phenomena. By now, you have probably rushed out to calculate your Sharpe and arewondering if you should be proud or ashamed. As a rule of thumb, I believethat  individuals  should  almost  always  strive  for  Sharpe  Ratios,  calculatedunder the methodology just described, equal to or in excess of 1.0. For exam-ple, if we assume a 5% risk-free rate and a 15% annualized standard devia-tion of returns, the portfolio in question would have to return at least 20% inorder to achieve this threshold—(20% Return (cid:4) 5% Risk-Free Return)/15%Standard Deviation (cid:3) 1.0. It is certainly possible to achieve significant finan-cial  objectives  over  extended  time  spans  with  Sharpe  Ratios  below  thisbenchmark; however, the attractiveness of the returns from a risk-adjustedperspective  quite  naturally  diminishes  accordingly.  In  these  cases,  theproviders of capital (whether it be you or some other economic entity) mayvery well decide there are better places to risk their money. At the other endof the spectrum, I have seen cases where portfolios can achieve Sharpes of5.0,  10.0,  or  better  for  sustained  periods  of  time.  These  are  rather  uniqueextremes and can only be reflective of either an uncommon market edge orsome  risks  not  well-identified  by  the  standard  deviation  calculation;  and  Iurge you to evaluate these situations with extreme care.
This  brings  us  to  the  final  element  of  our  discussion  on  the  SharpeRatio, namely, its limitations, which are largely tied to the accuracy of thestandard deviation calculation as a proxy for exposure and to the appli-cability of historical returns and volatility distributions as a predictor offuture  performance.  As  indicated,  the  limitations  of  the  standard  devia-tion calculation derive from its assumption that portfolio returns are nor-mally  distributed;  and,  as  we  have  demonstrated,  this  is  not  always  thecase.  Moreover,  past  volatility  patterns  may  not  repeat  themselves—particularly  wherever  and  whenever  volatility  is  calculated  over  shortertime spans.
To illustrate the kind of problems that these limitations can present,consider  a  portfolio  that  does  nothing  but  sell  deep  out-of-the-moneyoptions, close to expiration. Because these options pay off in all but themost remote (in a probabilistic sense) outcomes, portfolio managers whoemploy  these  strategies  can  enjoy  stable  returns  with  low  volatility  forextended  periods  of  time—often  ranging  into  years.  The  associated  highSharpe  Ratio,  however,  disguises  the  fact  that  every  now  and  again,  anabrupt market move will cause a substantial loss in these portfolios. Whenthis occurs, we see both the limitations of (cid:2) in calculating exposures andthe hazards associated with using historical returns as a proxy for futureexposures. For these reasons, and a number of others, although the SharpeRatio remains an important benchmark for risk-adjusted return, it is best
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used in conjunction with analytics that don’t rely on standard deviations astheir exclusive measure for risk, such as Return over Maximum Drawdown(which we’ll get to in a moment).
Median P/L
The statistical median is defined as the observation within the data set forwhich half of the data points are of a higher magnitude and half are lower. Inperfectly symmetrical distributions, the mean and the median are identical(or  nearly  identical).  However,  in  my  experience,  few  performance-relateddata sets are perfectly symmetrical. Therefore, I suggest that you comparethe mean and the median P/L observations in your portfolio across varyingtime  horizons.  Wherever  the  mean  significantly  exceeds  the  median,  it  ismore than likely due to the fact that a handful of your most profitable daysare  bringing  up  your  average.  Similarly,  where  the  median  is  significantlyhigher than the mean, it is probably due to the impact of a number of extremenegative outcomes on the average across the data set.
In either case, if the median and the mean diverge significantly fromone another, it represents yet another opportunity to further explore thedynamics and the associated impact of activity at the tails of your P/L dis-tribution.  I  highly  recommend  that  you  take  this  opportunity,  by  furtherexploring the associated causes, and so on. 
Percentage of Winning Days
This is simply a straightforward percentage representation of the numberof days where your trading ended profitably. While, at first glance, it wouldseem  somewhat  intuitive  that  a  percentage  target  of  50%  represents  theminimum for profitable trading over the long term, I have found this notnecessarily to be the case. As long as your unit returns on profitable daysare sufficiently higher than they are on losing days, I have observed that anactual percentage of winning days as low as 40% can be conducive to prof-itable trading. The important thing here is to recognize, as I will elaborateon  next,  that  this  diagnostic  combined  with  the  “performance  ratio”  dis-cussed in the next section represents a critical first-order P/L evaluationtool, as success in at least one of these two ratios is a necessary compo-nent of a profitable portfolio. 
As with virtually every other element of our tool kit, the review of the%  winning  days  statistic  is  most  useful  as  a  metric,  which  is  trackedacross multiple time periods and against such factors as external marketconditions, level of capital deployed in the market, and seasonality. I inviteyou to drill as deeply as possible into your P/L data set in order to derive
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the best possible understanding of the conditions that are the greatest con-tributors  to  the  probability  of  your  ending  the  day  ahead  of  the  game,rather than a few steps back. 
Performance Ratio, Average P/L, Winning Daysversus Losing Days
My first law of trading dynamics: The key to risk-adjusted profitability isto extract more from the market when you are right than you give backwhen you are wrong. There are many ways to measure your performancein this regard, perhaps the most important of which, as we will addressin  subsequent  chapters,  is  transactions-level  analysis.  However,  astransactions-level data can be inaccurate, too voluminous to process, orsimply  unavailable,  it  is  important  to  review  this  from  a  daily  P/L  per-spective as well, which, in the overwhelming majority of cases, will be aclose reflection of what happens on a trade-by-trade basis. As previouslyindicated, this “performance ratio” goes hand in hand with the % winningdays statistic as a critical, first-order benchmark of trading performance.Look at it this way: In order to be successful as a trader, you either haveto be right more than 50% of the time or make more on winners than yougive  back  on  losers—or  (ideally)  some  combination  of  the  two.  There-fore, when an account is having difficulties, one of the first things I con-sider is whether the problem lies within the % winning days (indicating,perhaps,  suboptimal  selection)  or  in  the  performance  ratio  (indicatingpotential  problems  in  the  actual  trading  habits).  In  turn,  this  providesinsights  into  potential  inefficiencies  and  is  helpful  in  prescribing  theappropriate countermeasures.
The  %  winning  days  statistic  and  the  performance  ratio  are  interre-lated. But make no mistake: The performance ratio is the more importantof the two. This difference in their levels of criticality is dependent on anumber of identifiable factors but most notably is governed by where theportfolio lies on the trading/investment spectrum. For long-term holders ofpositions, intuition suggests that accurate selection looms larger in impor-tance then does any factor tied to the unit profitability on given days. Bycontrast, those who trade very actively will have the luxury of living withquite a lot of losers, so long as they are monetizing their winners at a suf-ficient premium. However, for virtually any successful account, it will benecessary to make more on your up days than you give back on your downdays. Unless you operate in ways consistent with this objective, you willfind your portfolio paddling up the performance stream.
What  is  an  appropriate  benchmark  objective  for  the  performanceratio?  My  experience  tells  me  that  somewhere  in  the  range  of  125%  is
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entirely achievable, and I have worked with guys who have done much bet-ter,  some  of  them  exceeding  that,  some  consistently  achieving  200%(cid:5)  inthis regard. Play around with these statistics, and find the conditions underwhich  you  can  maximize  your  performance  ratio  results.  If  these  condi-tions happen to be scalable, you’re on your way.
Drawdown
If you stick with this business of trading and investment, know this: Youare going to spend most of your time in an unhappy state known as draw-down. This statistical concept, which is pervasive to the evaluation of alltypes of portfolio managers, has a very specific but often misunderstooddefinition. Specifically, drawdown refers to the difference between a port-folio’s highest valuation over a given period and its lowest subsequent val-uation. This means that your portfolio either is at a new record high for theperiod  or,  by  definition,  is  in  drawdown.  Moreover,  your  status  doesn’tchange until you have again reached a new high. Therefore, if you are man-aging, say, $10 million, earn $1 million over the next month, lose $50,000 onthe first day of the following month, another $25,000 on the next day, andthen make back the $50,000 on the third, you are experiencing a drawdownof  $75,000,  which  won’t  be  officially  over  until  you  make  back  another$25,000  and  your  account  balance  reaches  at  least  $11,000,001.  Becausetraders very seldom operate at new high account balances, they are almostalways in at least a nominal condition of drawdown.
A thorough analysis of your drawdown patterns will offer you perhapsthe most objective and comprehensive window into the risks inherent inyour portfolio of any element of our tool kit. This is true for a couple of rea-sons. First, by using actual data as opposed to such statistical estimationtools as standard deviation, you eliminate the need for assumptions suchas the implied normality of your returns. Your drawdown history is yourdrawdown history; and it doesn’t matter whether it is based on a data setthat is skewed, has excessive presence of observations in the tails of thedistribution, or (alternatively) results in a portfolio histogram that takes onsuch a bell-like shape that you can hear it chime on Sunday mornings. Sec-ond,  while  other  statistics  we  describe  are  typically  single-point  descrip-tions  of  complex,  multifactored  phenomena,  the  drawdown  conceptencompasses  the  depth,  temporal  interval,  and  associated  recovery  pat-terns of your actual exposure. 
Because by my estimate and according to the definition specified youwill spend at least 80% of your trading career in a condition of at least nom-inal drawdown, the concept lends itself to continual, broad-ranging analy-sis.  Among  the  different  drawdown-related  factors  of  which  you  shouldtrack for each individual drawdown are: 
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• Magnitude. Of  course,  you  will  want  to  keep  track  of  the  value  ofyour  drawdowns,  in  terms  of  both  dollars  and  percentage  of  capitalallocated or deployed. Note that these figures lend themselves exten-sively to additional analysis. How are these drawdowns changing overtime? Are they increasing or decreasing on an absolute ($) and relative(%)  basis?  Are  they  tied  to  specific  market  conditions?  How  do  theycorrelate  with  your  profitability  over  specified  longer  time  periods(e.g.,  a  year)?  All  of  these  questions  can  and  should  be  addresseddynamically, always providing the potential for revealing new insightsinto your trading behavior. 
• Recovery  Period.
It is also naturally very important to have a goodsense of the timing dynamics of your drawdowns, which can best beexpressed in terms of associated recovery periods. Here it is importantto once again bear in mind the definition of drawdown, which is anynegative  P/L  that  takes  an  account  down  from  its  historical  “high-water” performance mark. The Recovery Period statistic is thus a sim-ple  measure  of  the  time  it  takes  to  reestablish  a  new  high.  Like  thedrawdown statistic, it lends itself to considerable additional analysisover multiple time periods. 
