 magnitudes,  the  correlation  coefficient  has  a  maximumvalue of 1.0, or 100%, indicating perfect correlation (e.g., the temperaturein Toronto as measured in Fahrenheit and Celsius), and a minimum valueof (cid:4)1.0, or (cid:4)100%, indicating perfect negative correlation (e.g., the priceof a zero-coupon bond and its yield). All values in between are valid, andthe  process  lends  itself  to  all  the  subjectivity  that  the  human  mind  canmuster. However, you may find the following (admittedly simplistic) rulesof thumb to be useful:
Value of Correlation CoefficientLess than (cid:4)50%
Interpretation
High negative correlation—merits
full investigation.
Between (cid:4)50% and (cid:4)10%
Material negative correlation—take
Between (cid:4)10% and (cid:5)10%Between (cid:5)10% and (cid:5)50%Greater than (cid:5)50%
a look.
Statistically insignificant—ignore.Material correlation—take a look.High correlation—merits full 
investigation.
The  following  topics  constitute  a  subset  of  the  types  of  correlationanalyses that you can calculate between your own portfolio patterns andyour profitability.
Number of Daily Transactions
This  analysis  will  give  you  some  idea  as  to  whether  your  profitabilityincreases or decreases with your level of trading activity. In my experience,the outcome will vary from account to account and across such factors asbasic  trading  strategy  and  external  market  conditions.  If  the  number  ismaterially  positive,  it  suggests  that  for  the  period  of  analysis  your  activeengagement  in  the  markets  (as  we  have  argued  earlier  can  only  be  fullymanifested through actual trading) is a positive contributor to your overallsuccess. I suspect that this is true for most market participants who tend tobe less active in their actual trading than is absolutely conducive to theirmaximum success. Thus, I believe that in most cases a sustained, positive,and  material  correlation  between  the  number  of  trades  you  do  and  yourprofitability should induce you to trade more. Get active in the markets, act
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on your instincts when putting your positions on, and be quick to reverseyour  course  when  you’re  wrong.  In  addition  to  potentially  increasing  thenumber of profitable trades you experience, I’m guessing that the effort willalso  produce  benefits  in  terms  of  enhancing  your  market  insight,  honingyour executions and risk-control edge, and in general improving your port-folio management skills.
However, there are points of diminishing and even negative returns tothis process that become apparent through negative correlations betweentrading activity and P/L. If this applies to your portfolio management activ-ities, it gives rise to the possibility that you are overtrading. Typically, thisonly happens in the case of truly hyperactive accounts, but it is a conditionto guard against nevertheless. Day traders (online or otherwise) might beparticularly susceptible to this and should consider performing this analy-sis in particular when trading at a very high velocity but underperformingin  terms  of  the  bottom  line.  When  this  happens,  I  suggest  you  take  theopportunity to slow down a bit (how’s that for an intellectually daring rec-ommendation?). Review your activity on a trade-by-trade basis. I’m guess-ing that a whole bunch of superfluous transactions are staring you right inthe  face.  From  there,  the  next  logical  step  is  simply  to  place  a  slightlytighter  filter  on  what  constitutes  an  actionable  idea.  You’ll  know  prettyquickly if the filter program is working by virtue of (1) whether the nega-tive correlations patterns diminish, and, more important, (2) whether yourprofitability rises.
Capital Invested
This correlation statistic measures the level at which your profitability iseither a positive or a negative function of the actual amount of dollars youare putting to work in the marketplace. In many ways, this is among themost important measures of your overall success: While a positive corre-lation  between  capital  utilization  and  P/L  suggests  that  you  are  doing  agood  job  of  using  your  financial  resources  effectively,  based  on  marketopportunities, any other outcome suggests portfolio inefficiencies. At min-imum, a nonmaterial correlation indicates that you are not doing as gooda  job  as  you  might  of  putting  capital  to  work  when  opportunities  are  attheir greatest. What is worse, if the correlation is meaningfully negative, itsuggests that the more capital you deploy in the markets, the worse yourperformance. This is plainly a less-than-desirable outcome that bears full-scale examination.
If the results of this analysis are favorable, it should boost your confi-dence to assume risk in situations where your conviction about your posi-tions is high. One way or another, you are reading the markets with somedegree  of  accuracy,  and  this  is  something  you  can  build  on.  Trust  your
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instincts here by increasing your size in your favorite trades. The data indi-cates  that  the  outcomes  will  be  favorable  on  the  whole.  Similarly,  youshould  take  the  positive  correlation  statistic  as  an  indication  that  if  youdon’t see sufficient opportunity in the markets, it’s probably a good idea tobe underinvested. Again, the data indicates that your instincts are servingas your ally in this regard.
By  contrast,  if  the  correlations  are  low  or  negative,  then  it’s  time  toreevaluate the way in which you are sizing your positions. If this patternoccurs  during  periods  of  profitability,  it  suggests  that  you  are  capturingsome market inefficiencies and simply not doing a great job of identifyingthe  relative  quality  of  the  opportunities  you  are  trading.  By  definition,  itimplies that the correct calls you made were ones where you didn’t havemaximum dollar investments involved and that your biggest winners werein names where you had a good deal of capital invested but the positiondidn’t actually move that much. I suggest that you carefully examine theindividual trades that comprise this dichotomy. I’m guessing that you willsee patterns that are correctible in terms of the sizing of your bets.
If  you  are  experiencing  the  worst  possible  combination  of  negativereturns  that  are  negatively  correlated  to  the  amount  of  capital  you  aredeploying,  it’s  probably  an  indication  that  something  is  fundamentallywrong with your entire trading/investment program. Perhaps the marketshave  changed,  or  perhaps  your  portfolio  management  dynamics  are  notconducive to profitability. Take a good, hard look here because, needlessto say, this is not a pattern that you will be able to sustain for long periodsof  time—unless  you  have  infinite  amounts  of  capital  at  your  disposal(which you do not).
Net Market Value (Raw)
In  addition  to  calculating  correlations  between  your  P/L  and  variousbenchmark indexes, such as the S&P 500, you can determine the extentto which your fortunes are tied to those of the underlying markets by cal-culating  a  correlation  between  your  returns  and  the  net  market  value  inyour portfolio at any given time. If this correlation is strongly positive, itsuggests that you do better when your portfolio has most of its positionson  the  long  side  of  the  market  and  that  your  bias  is  toward  the  netpurchase  of  securities  (as  opposed  to  their  net  sale).  By  contrast,  if  thecorrelation is strongly negative, it indicates that you have a short bias, pre-ferring to operate with short positions as your key themes. Finally, if thiscorrelation  result  is  statistically  insignificant,  it  means  that  there  are  nospecific directional biases in your portfolio strategy.
As you may well surmise, the zero/low correlation outcome is proba-bly the one that is most conducive to stable, scalable profitability across
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full  ranges  of  market  conditions.  While  you  should  never  orient  yourmethodology toward any goal other than risk-adjusted return, obtaining abenign correlation between your P/L and portfolio net market value maybe a useful subobjective. If you have high correlations to net market value,understand  that  your  strategy  is  geared  toward  directional  market  bets,which, as a general rule, tend to be more volatile than balanced portfoliostrategies. Take a look at the side of the market that is opposite from theone that the analysis suggests is your strong side, and see if there are anyopportunities to incorporate positions without damaging your profitability.If you can find these, your largest losses are likely to be lesser in terms ofmagnitude and frequency. Further, as this phenomenon evolves, it placesyou in a more confident position to deploy more capital. Remember thatportfolio strategies that dampen direct market exposure are almost alwaysmore  scalable  than  those  that  embrace  a  market  bias.  Whether  it’s  yourown  capital  or  someone  else’s,  the  more  neutrality  you  can  achieve,  themore your business can grow.
Net Market Value (Absolute Value)
As a slight wrinkle to the net market value correlation analysis, I have foundit  useful  to  calculate  a  correlation  between  P/L  and  absolute  net  marketvalue, defined as your net position in dollar terms with the sign conventionremoved. For example, a net market value of $1 million, whether positiveor negative, would be expressed in the time series as $1 million.
Your  P/L  correlation  to  absolute  net  market  value  will  give  you  aslightly different insight into portfolio dynamics than your correlation toraw net market value will. Specifically, rather than telling you whether youdo better or worse depending on how bullish or bearish you might be, ittells you the extent to which you perform better or worse depending on thelevel of balance that exists in your portfolio. If the correlation is high, itsuggests that you do better when your entire portfolio is a directional bet(long or short) on the markets. A negative correlation suggests that yourperformance improves along with its level of market neutrality. A low cor-relation of either sign indicates that these factors don’t have a great impactone way or another.
Because directional bets tend to carry more event risk than portfoliosthat retain a measure of balance do, it is my view that negative correlationis probably the best outcome. Such a result suggests that your exposuresare more targeted to specific ideas you have about the securities you aretrading,  and  these  types  of  edges  are  the  ones  that  tend  to  be  the  mosteffective  of  them  all.  However,  beyond  this,  it  also  means  that  you  aredoing  well  under  portfolio  constructs  that  are  likely  to  minimize  yourexposure to market crashes or frenzied rallies such as those that drove the
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path  of  the  Nasdaq  Composite  to  above  5000  early  in  the  year  2000.  Youmay still get hurt when the market moves dramatically in one direction orthe other, but the impact is likely to be dampened relative to that of a port-folio that is entirely loaded up on one side or the other. Moreover, if youput the effort into portfolio construction that features two-sided positionselection  (as  opposed  to  one  side  being  primarily  comprised  of  hedgingpositions), then the protection you achieve from sharp market correctionsis not likely to come at the expense of portfolio returns. Finally, balancedportfolios  are  entirely  more  scalable  than  are  directional  ones  becausethere is a much more binding constraint to the amount of risk the capitalprovider  should  be  willing  to  take  in  accounts  whose  fortunes  are  sodirectly tied to the idiosyncrasies of the market.
The purpose of correlation analysis is to provide insights into dynam-ics that aren’t immediately observable, not to create objectives or to divertyour goals away from risk-adjusted return. If you happen to have a high netmarket value correlation statistic and things are working well on a consis-tent basis, do not be too greatly troubled on this score. Be mindful of thecondition, but don’t reorient an approach that is otherwise succeeding toachieve  a  different  result  on  a  secondary  statistic.  If  things  start  goingbadly for you and the condition persists, perhaps there’s an opportunity todig deeper into the net market value correlation. Otherwise, don’t fix whatain’t broke.
Number of Positions
As one measure of the effectiveness of your diversification program, youmay wish to examine the extent to which your portfolio performs better orworse  as  the  number  of  names  you  have  on  the  sheets  increases  anddecreases. As follows the pattern observed with other elements of corre-lation analysis, a high positive correlation implies that you do better withmore investments on the books rather than fewer; a high negative correla-tion suggests the opposite; and a modest result of either sign indicates thatyour portfolio is not impacted by the number of investments you are hold-ing at any given time.
As you may very well surmise, there is no ideal number of positionsother  than  the  one  that  is  most  conducive  to  your  own  maximum  prof-itability. For this reason, the number of positions correlation statistic maybe of most use to you in trying to pinpoint some of the causes for periodsof underperformance. When things are not going well in your account, youmay wish to see if there is a trend away from the equilibrium number ofnames in the portfolio, as well as an associated opportunity to apply cor-rective  techniques  to  help  stem  the  tide  of  reversals.  This  could  meanincreased portfolio decentralization. Your risks may be concentrated too
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dramatically  in  a  handful  of  positions—particularly  during  periods  ofhighly volatile pricing conditions where even your best ideas are subject tothe whims of the broader market. Or you might discover that a large num-ber of the positions you are holding in your account are doing very littlebut  taking  up  space  on  your  blotter  and  using  up  capital  that  might  bebetter either held in cash (either earning interest or saved for future oppor-tunities) or used to increase your size in the positions where you bear ahigher degree of conviction. Correlation analysis can help uncover thesepatterns,  but  the  manner  in  which  you  react  to  either  take  advantage  ofthem or adjust them is the main challenge that confronts you. Try small,controlled experiments when adjusting your portfolio strategies, and usestatistics  to  measure  the  results.  This  may  cost  you  a  bit  of  money  inthe short run, but it will stretch you as a portfolio manager in many invalu-able ways over the longer term.
Holding Periods
Another extremely interesting correlation pattern is the one between P/L andholding periods. Do you tend to do better when you are maintaining positionsover longer time horizons, or do extended holding periods work against you?Correlation analysis can help you answer this question and perhaps provideopportunities for improved portfolio efficiency. Positive correlation suggeststhat  your  performance  improves  the  longer  you  hold  onto  positions,  whilenegative correlation indicates the precise opposite relationship.
Because the timing of entry and exit points is perhaps as difficult asany  of  the  portfolio  management  arts,  this  is  one  of  the  more  importantcorrelation statistics to analyze. In fact, in my experience I have found thatit is almost impossible to perfectly time the markets on a consistent basisand  that  even  the  best  traders  in  the  world  have  to  work  very  hard  toapproach  maximum  efficiency  in  this  aspect  of  their  activities.  It  is  veryeasy to be either too quick or too slow (and sometimes both) with your liq-uidation,  and  correlation  analysis  can  help  you  pinpoint  these  patterns.For example, day traders often fail to capture the full set of opportunitiesavailable  to  them—due  to  a  desire  to  take  quick  profits  and  use  tradingvelocity to minimize losses. While day trading can be a very effective strat-egy,  those  who  engage  in  it  may  well  benefit  by  an  understanding  of  themagnitudes  of  their  holding  periods  and  of  associated  impacts  on  prof-itability.  If  you  measure  your  holding  period  in  minutes  (as  many  daytraders do), you may not be giving yourself enough time in your positionsto  capture  the  price  movement  you  anticipate.  This  would  be  consistentwith one of a high, positive correlation to your holding periods. If you findthis type of pattern in your own account, then I strongly encourage you toseek  methodologies  for  extending  the  amount  of  time  you  maintain  key
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positions. For most, the major obstacle faced is a laudable desire to con-trol exposures by adhering to a discipline that implies position liquidationin the face of adverse price fluctuation of any magnitude. In order to allowfor sufficient time to capture positive trends while at the same time main-taining this discipline, I suggest you measure your holding periods in termsof price movement, rather than hours, minutes, or days. Commit to a stopand an objective for each transaction, and try holding on until one or theother  price  target  is  breached.  By  doing  so,  you  will  have  limited  yourdownside to an amount that you specified beforehand as being the maxi-mum level of pain you are willing to take in the trade. Similarly, if the trans-action works out the way you would have hoped, by liquidating when thesecurity  reaches  your  objective  price  target,  you  will  lock  in  profits  thatmay  elude  traders  and  investors  who  for  whatever  reason  cannot  bringthemselves to liquidate positions in a timely fashion.
I highly recommend that you size positions such that the dollar lossassociated with an adverse move is an amount that you are comfortablelosing. If your position reaches either its objective or its stop level, be verycompulsive  about  liquidating  it.  Because  most  rational  traders  will  putpositions on where the realistic target gains exceed their stop-associatedlosses,  I’m  guessing  that  if  you’re  right  even  close  to  half  the  time,  theoverall result will be positive. Finally, it is important to keep track of yourperformance on transactions to which you have applied this methodology.I suspect that your results will be favorable. But if they’re not, then by allmeans  establish  an  amount  you  are  willing  to  lose  under  this  modifiedtrading approach; and if you hit this level, go back to what was workingbest.
While day traders tend to experience positive correlations with theirholding periods, most of us who are not engaged perpetually in the mar-kets  fall  into  the  opposite  prototype.  Typically,  we  hold  onto  our  posi-tions—particularly  losers—for  too,  too  long  and  tend  to  bleed  cash(through inertia) that could be better deployed anywhere else but in posi-tions that are working against us for an extended period. This is even thecase with top-tier portfolio managers who may have great portfolio ideasbut who, if not careful, run the risk of neglecting longer-term positions forwhich, best case, the arguments in favor of the original theme have deteri-orated. Moreover, even your profitable trades can turn into mediocre oneswhen you don’t find the appropriate exit point.
In general, when your returns have a negative correlation to your hold-ing periods, it is an indication that your positions are no longer subject tothe underlying economics that caused you to put the trade on in the firstplace.  In  these  cases,  the  appropriate  question  to  ask  yourself  is  notwhether  the  trade  has  a  chance  to  work  as  you  initially  anticipated,  butinstead whether you would initiate the position at prevailing price levels
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today with your unutilized capital. If the answer is yes, by all means main-tain the position; however, if the answer is no, liquidate it.
The  use  of  the  stop/objective  methodology  for  all  transactions  can  beimportant  in  the  management  of  the  position  liquidation  process  for  thosewho suffer from a negative P/L holding-period correlation. They should estab-lish a target price on the positive side and a stop-out level on the downsidefor every transaction they execute, just as was the case with the day traders.As previously indicated, the target gains should in every case exceed the max-imum  allowable  loss,  or  the  trade  is  probably  not  worth  doing  (unless,  ofcourse, a high probability of success changes this relationship significantly).Stick to the discipline and liquidate positions if either level is breached. Notethat you may still believe in the idea in these instances—that you were righteither about a loser that you are sure will turn around or about a winner thatstill has upside beyond your original target. In these cases, it is your preroga-tive  to  simply  reestablish  the  position.  This  will  increase  your  transactioncosts, but you stand to recoup these incremental expenses many times overby ensuring that every dime of the capital you are deploying is in positionsthat apply directly to your investment hypotheses.
Finally, if you are a trader with long-term holding horizons, you maywish to establish an absolute time limit on certain positions to supplementyour  stop/objective  regime.  If  a  position  operates  inbetween  your  targetpricing  band  for  excessive  periods  of  time,  then  the  likelihood  of  yourhypotheses working as you expected them to is pretty minimal in my view.Over time, the purging of such aged position inventory will allow you todeploy your capital more efficiently.
Volatility/VaR
One very efficient way of determining whether you are effectively readingthe  ebbs  and  flows  of  the  market  is  to  measure  the  correlation  betweenyour P/L and the volatility in your account. All things being equal, positivecorrelation would be a desirable outcome because, in the best of circum-stances, you will want to have your maximum risk position in place duringmarket intervals that coincide with the greatest opportunities. Therefore,a positive correlation with volatility becomes a barometer of your abilityto identify optimal opportunities and to seize on them through activity inyour  portfolio.  However,  as  long  as  your  account  performs  well  from  arisk-adjusted  return  perspective,  you  shouldn’t  be  too  troubled  if  yourreturns are negatively correlated to measures of volatility because, amongother things, this may simply reflect a tendency to perform better duringexternally stable market conditions.
As indicated in previous chapters, there are a number of ways to meas-ure portfolio volatility, with the most precise of these (at least in my view)
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being the actual standard deviation of daily returns. In order to determinethe extent to which higher levels of volatility measured in these terms coin-cide with intervals of outsized profitability, it may be useful to introduce alagging  component  into  the  mix.  Lagging is  a  common  statistical  tech-nique,  which  is  used  extensively  in  correlation  analysis  and  predictivemodeling, under which observations from one time series are not matcheddirectly with data points that correspond in time with the other, but ratherare  lined  up  against  subsequent  points  in  the  data  series.  For  example,when determining the impact of your caloric intake with your weight, youmay  want  to  correlate  yesterday’s  calories  with  today’s  readings  on  thescale, and so on. Of course, my weight is positively correlated to virtuallyevery random variable in the universe—lagged or otherwise—so this maynot be an ideal example.
In terms of the relationship between volatility and return, it may beuseful to calculate a correlation coefficient between yesterday’s measureof  volatility  and  today’s  returns.  The  argument  here  is  that  there  is  anominal time lag between the point at which you increase the risk of yourportfolio  and  the  time  when  you  can  measure  corresponding  impactson portfolio return. Stated in an alternative fashion, there’s likely to beless of an impact on today’s P/L to changes you make in your risk profilethan  there  is  on  tomorrow’s  P/L.  Apart  from  applying  a  time  lag  to  thedata,  it  is  also  possible  to  capture  the  sequential  impact  of  risk  adjust-ments on returns by using not the standard deviation of returns as yourbenchmark,  but  rather  statistics  produced  by  a  Value  at  Risk  (VaR)model.
Recall from the earlier discussion on VaR that its precise intent is topredict the volatility of portfolios based on the way that their pricing pat-terns have shown them to react over historical market intervals. Therefore,today’s VaR (assuming the program is performing its core tasks with a min-imal competence) is a measure of today’s risk, and this eliminates the needfor applying a lagging operation on the data series.
Unlike the case with several of the other correlation statistics coveredin this chapter, I can’t offer very many useful strategies for adjusting yourportfolio management activities in order to address a suboptimal correla-tion pattern between P/L and volatility. Strong, positive correlation resultssuggest that you are doing a very good job of allocating your risk capitalto  those  situations  that  offer  the  best  opportunities.  However,  it’s  notentirely certain that it implies the need for dramatic adjustments to yourportfolio management program. It may simply be that calmer markets andlower risk profiles are just a better proposition for you. The best option,in my view, is to use correlation analysis to increase your understandingof risk and reward as it pertains specifically to your portfolio. If the cor-relations in this regard are high, perhaps you should use this as a pretext
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for  increasing  your  confidence  (and  perhaps  your  market  participation)during highly volatile market intervals where you have strong convictionabout  your  ideas.  History  shows  you  are  likely  to  do  pretty  well  here.Conversely, if your correlations are strongly negative (and your P/L is stillpositive),  it  argues  for  capping  your  risk  levels  and  for  adjusting  themdownward when market conditions cause volatility to drift upward inde-pendent of any portfolio activity on your account. Finally, if your correla-tions are negative along with your P/L, it’s probably time to take a criticallook at your overall trading program. Be careful here because somethingvery basic is not working, and the worst thing you can do is apply morerisk to markets for which the data is giving you clear signals that you don’tunderstand.
Other Correlations
There are many supplementary correlation analyses that you can performagainst  your  trading  statistics  that  will  offer  insights  similar  to  thoseavailable  through  the  analyses  we’ve  covered:  P/L  to  average  positionsize,  to  commission  dollars  paid,  to  the  value  of  unrealized  P/L  in  youraccount at any given point in time, and so on. These and others may offerinteresting  perspectives  not  discussed  in  this  chapter,  and  I  hope  you’lltry  a  few—just  for  kicks.  Think  carefully  about  the  implications  of  theresults  as  in  most  cases  intuition  will  be  your  best  guide  as  to  what  tomake of them.
Final Word on Correlation
I caution you, yet again, against reading too much into the implications ofthe results. Correlation analysis is a very useful descriptive statistic, but itis  an  imprecise  predictive  mechanism.  As  such,  I  can’t  stress  stronglyenough the age-old adage admonishing us not to confuse correlation withcausation. The goal here is to gain insight into those elements of your rou-tine trading program that are most likely to bring you success in your questfor risk-adjusted return and into those that are causing inefficiencies thatcan at least be managed, if not altogether corrected.
Remember that, due to the extraordinary amount