• Magnitude  of  Subsequent  Incremental  Gains. As  just  indicated,  adrawdown is not over until the account in question has achieved a newhigh-water mark. It is useful to then measure the amount by which thenew high-water mark exceeds the old. This figure can be expressed asa ratio, with the drawdown statistic residing in the denominator, andwill give you some idea of the trade-offs in your markets between riskand return. In particular, for some longer-term strategies (e.g., privateequity investments), trade execution considerations and liquidity con-straints may be such that you may have to suffer significant drawdownin order to achieve the desired returns. In such a case, it is instructiveto understand the associated magnitudes. By contrast, short-term trad-ing  of  liquid  securities  in  relatively  frictionless  markets  should  notdemand that enormous pain be endured in order to achieve your tar-geted objectives.
• Return  over  Maximum  Drawdown  (ROMAD). This  statistic,  inwhich annual returns are expressed as a percentage of the worst draw-down experienced over a specified period of analysis, actually repre-sents  what  many  people  believe  to  be  the  most  accurate  availablemeasure of risk-adjusted return. This is true primarily because draw-down  is  arguably  the  most  accurate  measure  of  portfolio  risk.  Bytracking your drawdowns over, say, a number of years and particularlyacross a wide range of market conditions, you begin to get an idea ofthe  full  range  of  negative  outcomes  that  can  occur  in  your  trading,along with the associated impacts. By comparing the worst of these to
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your  annual  returns,  you  provide  yourself  with  a  uniquely  effectivemanner  in  which  to  express  the  risks  and  rewards  in  your  portfolio,one that does not rely in any way, shape, or form on assumptions aboutthe distribution of your returns.
For  many  professional  trading  accounts,  including  those  tied  to  fee-based structures typically associated with the hedge funds and managedfutures,  portfolio  managers  are  only  paid  for  incremental  performance.One  very  meaningful  implication  of  this  is  that  between  periods  wherereturns are calculated, if you are operating under this framework, you willnot be paid for the revenues you have earned in recovering from a draw-down. This, in turn, has important implications for portfolio management.First, and perhaps needless to say, it means that you want to do everythingin your power to minimize the magnitude of drawdowns as they take shapein their formative stages. However, once they have reached certain levels,the associated recovery process will require an investment in time on yourpart  that  may  be  accompanied  with  financial  constraints  that  transcendyour professional activities and impact your personal life. This may temptyou toward portfolio management options that are not entirely consistentwith the decision path you would take if personal considerations weren’t afactor, and the manner in which you conduct yourself at these junctureswill  have  implications  for  your  long-term  well-being  as  a  portfolio  man-ager.  If  you  become  impatient  and  try  to  make  back  your  losses  tooquickly, you may very well find yourself in a deeper hole, in the processthreatening  the  viability  of  your  entire  trading  program.  However,  inalmost every case, it will not be practical to put off the paydays you seekforever.
For these reasons, I feel that using the drawdown thresholds as thebasis for risk reduction is an essential component of any risk managementprogram. Do yourself a favor and identify these thresholds at the begin-ning of a trading cycle. Then stick to this commitment. For example, youmay decide to reduce your exposure by, say, 50% if you experience a draw-down  of  10%.  If  you  are  able  to  do  so,  this  will  achieve  two  importantresults. First, it will go a long way toward reducing the magnitude of anyfurther drawdowns in your account. Second, it may, over the long term,give  you  the  confidence  to  take  larger  risks  in  aggregate,  as  you  haveestablished a framework that transforms the concept of drawdown froma mysterious and evil phenomenon that is lurking to do you no good at anymoment into a controllable dynamic that is actually part of a sound risk-control program.
The major benefit of such an approach is that you avoid placing your-self in a position where your drawdown is too deep to allow you to reviveyour trading fortunes. The trade-off is that for any level of unit success you
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have subsequent to your drawdown, you can expect the associated recov-ery period to be significantly longer than it might have been had you nottried to make up your losses in the quickest possible fashion. However, Istrongly believe that the extra investment of time is one that will pay off inmyriad ways.
In  general,  by  examining  your  drawdown  patterns  as  part  of  a  com-prehensive  statistical  review  process,  you  gain  unique  insights  into  yourspecific portfolio dynamics. You will better understand what went wrong,why it went wrong, and what it will take to correct the situation. Moreover,if you use drawdown as an “inverse barometer” of the amount of exposureacceptable for your account—reducing risk when significant drawdownsoccur  and  increasing  your  exposure  only  when  they  are  substantiallyerased—you stand to retain much more explicit control over your tradingfortunes than you would if you operated in a vacuum with respect to thiscritical information metric.
Correlations
The final core element of our introductory statistical tool kit is correlationanalysis.  You  ought  to  be  at  least  nominally  familiar  with  this  concept,which  involves  identifying  the  extent  to  which  two  or  more  data  seriesdynamically exhibit similar characteristics, most notably, for our purposes,across time. Correlation coefficients can range from  (cid:5)100% to (cid:4)100% but(unless  data  series  are  simply  disguised  representations  of  a  single  con-cept,  for  example,  the  yield  on  a  given  bond  and  its  price)  typically  fallsomewhere in between.
By  performing  correlation  analysis  on  the  time  series  of  portfolioreturns, traders stand to gain unique and specific insights into underly-ing portfolio economics. For example, you may find yourself highly cor-related to some benchmark stock index such as the Standard & Poor’s(S&P)  500,  the  Dow  Jones  Industrial  Average  (the  Dow,  DJIA),  or  theNasdaq Composite. More likely than not, this will be indicative of a longbias in your portfolio, which predisposes your account to perform moreeffectively when markets are rallying. However, this is merely one typeof  correlation  analysis  that  can  be  applied  to  great  effect  to  your  P/Ltime series.
Following is a summary of some of the standard categories of correla-tion analysis that you may find useful in identifying the drivers of relativeperformance for your portfolio.
Correlation against Market Benchmarks. This is the general caseassociated with the example provided prior, under which you might calcu-late “correlation coefficients” between your returns and the performance
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of various market indexes. Here, I recommend that you begin the processby simply identifying, in an anecdotal sense, the market indexes that mightbest capture the essence of your trading and then running some introduc-tory correlations there. For example, if you are trading U.S. equities, youmight begin with the S&P, the Dow, or the Nasdaq Composite. (This con-cept  is  largely  analogous  to  that  of  a  “beta”  for  an  individual  security.Therefore,  when  you  calculate  your  correlation  to  the  standard  equitybenchmark for the markets you are trading, it is not incorrect to think ofthis as your portfolio beta. If you wish to be creative, it might be interest-ing to compare your correlation to the equity markets to the average betaof the securities held in your portfolio.)
Similarly,  if  you  are  trading  U.S.  Fixed  Income,  the  Treasury  bond(long term), or the Eurodollar (short term), futures contracts might be thebest place to start. With these basics in place, I encourage you to be cre-ative,  running  correlations  between  your  P/L  and  as  many  market  timeseries  as  you  can  think  of—whether  they  make  intuitive  sense  or  not.While  you  may  not  find  any  surprising  interdependencies,  you  may  gainone or two insights into the external market patterns that are likely to havethe most dramatic impact on your performance. This is particularly true ifyou  perform  the  analysis  over  multiple  time  periods,  which  I  highly  rec-ommend. Doing so will give you a sense of how stable your correlationsare, how they are changing over time and across market conditions, and soon—which  in  turn  can  be  analyzed  for  insight  into  where  your  greatestopportunities and associated exposures reside.
As  you  review  the  dynamics  across  the  different  time  periods,  youshould  think  qualitatively  about  external  conditions  at  that  time.  Forexample, you may wish to determine whether your correlations tend toincrease or to decrease during market rallies, breaks, and so on. More-over, particularly during periods of market turmoil (e.g., third quarter of1998), the markets themselves tend to correlate strongly to one another;and I believe it is extremely instructive to determine whether and by howmuch your own performance tends to converge or to diverge from mar-ket trends under these conditions. If you can avoid falling into volatilityvortexes during these periods of disruption, you can take great comfortfrom  a  risk  management  perspective.  Conversely,  if,  as  is  typically  thecase, your correlation to the markets increases along with their correla-tions  to  one  another,  you  should  be  mindful  of  this  and  look  to  reduceyour exposure when these types of extreme conditions begin to presentthemselves.
When  calculating  your  correlation  against  market  benchmarks,  it  isimportant  to  ensure  that  you  are  applying  your  P/L  against  the  preciselyappropriate  data  set,  and  this  can  be  a  tricky  process.  First,  in  order  todetermine  whether  your  performance  is  correlated  to  the  markets,  it  is
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critical  to  recognize  the  need  to  run  your  performance  series  against  thedaily net change in the market, not its absolute level. This is true becausewhat we care about in this regard is whether you make money when themarket is going up or going down, not whether it is at a high or a low pricelevel.  In  addition,  if  you  are  trading  international  markets,  you  may  verywell find that the most relevant correlation data involves lagging one timeseries  by  a  day  to  account  for  time  differential  (e.g.,  correlate  yourP/L against  the  previous  day’s  price  change  in  the  appropriate  market).Also, if the appropriate benchmark happens to be a futures contract, it isimportant to correct for the differential associated from the rollover fromone delivery month to another, which, on the day it occurs, will generate aprice change that may not be consistent with pricing patterns in the under-lying markets.
In general, the number of ways in which it is possible to derive inter-esting information from this type of correlation analysis is virtually limit-less, and I encourage you to approach this task with a sense of creativity,adventure, and discovery. 
Cross  Correlation  Analysis. This  type  of  correlation  analysisinvolves the calculation of the statistic across different trading accounts.Needless to say, it is most relevant when the trading environment featuresmultiple portfolios, either separate books managed by an individual deci-sion maker or some group of independent portfolio managers operating incooperation (or at minimum proximity) to one another. In these cases, it isinstructive to learn how the different accounts correlate with one another,again across different time intervals, market conditions, and so on. Thesetypes of analyses can serve to either corroborate or refute any notions thatmay exist regarding the extent to which multiple accounts are truly inde-pendent or are simply nominally different manifestations of a single trad-ing strategy. In turn, this can only assist the process of understanding thedrivers of trading results and the identification of the frameworks that aremost maximally organized for market success.
From a capital allocation perspective, correlation analysis is a criticaltool in managing risk-adjusted return, as portfolio theory has long recog-nized the benefits of diversification in generating stable returns. In orderto achieve such diversification, those in the business of funding portfoliostrategies  are  naturally  predisposed  to  minimize  the  correlations  acrossthe accounts to which they are allocating capital. Therefore, unless you arefunding your own book, even if you don’t care a whit about the manner inwhich your returns correlate with those of your peers, it’s a pretty fair betthat your supervisor does. Accordingly, I urge you to be mindful of thesefactors even if you have neither the ability nor the inclination to calculatethe actual correlations themselves.
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If you do find yourself highly correlated to those around you—par-ticularly  if  these  correlations  persist  over  time—it  is  unlikely  that  it  isthe result of coincidence. Take a careful look at your trading patterns aswell as at those of your colleagues. More likely than not, there is an ele-ment  of  shared  decision  making.  Try  to  determine  in  these  instanceswhether all of the good ideas are emanating from a single source. Be asobjective as possible in this process. If you conclude that there is a true,multilateral sharing of ideas, it can often represent the best of all possi-ble  trading  environments.  Conversely,  if  the  idea  generation  processcenters  around  one  individual—whether  it  is  you  or  one  of  your  col-leagues—be aware that this condition is highly unstable, lending itself toresentment  and  other  negative  dynamics.  I  recommend  that  you  seekways  to  remedy  the  situation,  either  by  being  more  proactive  in  yourcontribution to the idea generation process or (if you are the Alpha ofthe  group)  by  insisting  that  others  step  up  their  contributions.  If  thevalue proposition between you and your partners cannot be normalized,then  it  behooves  you  to  ensure  that  the  economic  split  of  the  profitsaccurately reflects the associated value contribution of the participants.Unless you can make this type of partnership effective and equitable forall concerned, it is likely to end badly, both in professional and perhapsin personal terms.
Serial  Correlation. This  final  correlation  concept  is  one  that  isslightly more complex; and unless you’re one of us who find this sort ofthing  aesthetically  pleasing,  you  can  probably  survive  without  goinganywhere near this murky subject. However, the topic can have impor-tant implications for risk-adjusted profitability, so I’ll say a quick wordabout it. 
Serial  correlation is  a  concept  that  measures  the  extent  to  whichobservations in a time series are influenced by earlier observations in thatsame  series.  To  use  a  non-market-based  example,  one  might  apply  thismethodology  to  determine  whether  your  golf  score  today  is  somehowimpacted by your performance on the course yesterday. This is certainlytrue  in  my  case,  but  I’ve  played  exactly  one  hole  of  golf  in  the  past  twodecades, and this only at my dad’s insistence. My attention span is painfullyshort, and my mind wanders if forced to focus on any effort for more thana very small number of minutes. Golfers, on the other hand, appear to neednothing so much as the patience and the presence of mind to endure a rit-ual that seems to me to go on for much too long and to proceed at muchtoo slow a pace.
I am happy to report, though, that I did manage to avoid utter embar-rassment  on  the  one  hole  I  did  play,  scoring  a  5  on  a  par  3  and  in  theprocess amazing my dad, who has never been the biggest admirer of my
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athletic prowess and who has long been incredulous that I am able to getthrough such trivial tasks as shaving and riding the subway without doingmyself bodily harm. 
Hopefully, this anecdote will have convinced you that the calculationof  serial  correlations  is  a  dangerous  business.  However,  if  you  insist  onproceeding,  you  should  know  that  these  statistics  fall  into  the  followingtwo categories: 
1. Autocorrelation. This  concept  measures  the  levels  at  which  today’sabsolute  performance  is  tied  to  absolute  performance  in  the  recentpast.  For  example,  in  a  stock  portfolio  using  a  momentum  strategy,we might expect that yesterday’s performance could greatly impact ourresult  for  today,  as  successful  themes  build  on  themselves,  and  (con-versely)  breakdowns  in  momentum  create  multiday  losses  until  newpricing patterns present themselves. By contrast, in an arbitrage strat-egy,  one  might  expect  yesterday’s  losses  to  be  a  reasonably  accurateindicator of likely strong performance today, and vice versa.
2. Autoregression. Autoregressive  time  series  are  those  for  which  themain  predictor  is  the  amount  that  previous  observations  deviate  fromthe  mean  of  the  data  set.  For  instance,  suppose  you  are  trading  anaccount that has an average daily P/L of, say, $10,000. The account wouldbe considered autoregressive if it tended to perform excessively well orbadly on a routine basis on days after it deviated from this mean signif-icantly  (say,  making  more  than  $20,000  or  losing  more  than  $5,000).Portfolios that are deemed to be autoregressive are ones that are basedon either breakout strategies (futures system trading) or some type ofarbitrage.
Note that serial correlation can be positive or negative and that it oftenhas  a  dynamic  “lag”  element,  meaning  that  the  appropriate  relationshipmay not be to the previous day’s time series but rather to series that areseveral days old.
The science of determining the relative presence of serial correlationis  applied  heavily  in  the  evaluation  of  individual  financial  instrumentsunder the conceptual heading of technical analysis. The associated math-ematical methodologies are rather complex and are certainly beyond thescope of this discussion. For more information on this topic, refer to dis-cussions of ARIMA (Autoregressive Integrated Moving Average) and Box-Jenkins analysis, available in most standard statistical textbooks.
Suffice it to say that if you notice repeating patterns in your P/L timeseries, it may be indicative of serial correlation; and based on simple read-ings of the charts, you should be in a good position to take the appropriatecorrective risk control measures.
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“Kitchen Sink” Correlations. Okay, so I lied. I have one more typeof correlation analysis I need to cover. As “kitchen sink” implies, it involvescorrelating any two things that move against one another. Once you gatherup a healthy head of steam in this regard, the process can be quite addic-tive. If you’ve followed the script, you may by now be gathering up untoldnumbers  of  time  series,  covering  specific  aspects  of  portfolio  dynamics.You  may,  for  instance,  have  accumulated  data  on  such  factors  as  capitalusage, net market value, long/short P/L, number of daily transactions, aver-age transaction size, average holding period, et cetera, et cetera, et cetera.While it’s easy to get carried away, it is also not much of a stretch to saythat there is potential benefit from calculating a correlation between anytwo such time series and, more specifically, by giving some thought to theimplication of any significant result. For example, you may find that yournet market value might be negatively correlated to your holding periods,which might reinforce the notion that it is easier to hold for extended peri-ods of time positions with small directional overhang and more difficult toretain  position  profiles  that  have  large  net  market  exposures.  Similarly,correlation  patterns  between  transaction  size  and  capital  usage  mightreveal interesting insights into factors such as liquidity, transaction costs,and other dynamics that will be pertinent to portfolio performance.
Be  creative.  No  matter  what  two  factors  you  try  to  correlate  againsteach other, I promise you that there have been sillier ones based on lessintelligent premises.
BABY’S ON FIRE
It is very easy to take correlation analysis to extremes. I once worked with aguy who was obsessed with this and who would seek to find correlations in thestrangest of places. At various points, he was pretty sure his performance wascorrelated  to  factors  ranging  from  the  Consumer  Price  Index  to  the  ratio  ofinvestment on put options to call options.
While many of his colleagues found his methodologies difficult to under-stand, no one disputed the fact that he had traded profitably over broader andmore diverse market conditions than virtually anyone in the firm. This individ-ual and I shared a number of interests, and we bonded on such diverse topicsas history, literature, and (naturally) rock and roll. Moreover, I was fascinatedby his focus on correlation analysis and volunteered to try to help him unearthuseful patterns from the data available to us. 
We were soon correlating his performance to everything we could think of;and,  like  most  fishing  expeditions,  we  came  up  empty  more  often  than  not.However, we did hit on a couple of important themes over the course of this
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collaboration,  most  notably  that  he  had  an  extreme  positive  correlation  withtwo key external market factors: (1) the level of volume in the markets and (2)the  level  of  volatility.  Intuitively,  this  made  sense:  He  had  long  been  aware(albeit  in  somewhat  of  a  subconscious  manner)  that  he  did  his  best  tradingwhen the markets were moving and when there was generally a good deal oftrading action taking place therein.
Was this a revelation in his trading? Hardly. However, I like to think that itplaced him in a position to size his market participation to the opportunitiesat hand. I saw some anecdotal evidence that he did indeed save his risk capi-tal  for  active,  fast-moving  markets  and  that  when  things  were  slow,  heresponded by trading less. Correlation analysis thus helped him make betteruse of his resources.
I was less successful in convincing him that the Beach Boy’s “Pet Sounds”is  the  most  overrated  album  in  the  history  of  popular  music  and  that  BrianEno’s “Here Come the Warm Jets” is the most underrated. 
“Baby’s on fire, better throw her in the water.”
I can’t close this discussion without referencing an old platitude thatcautions against confusing “correlation with causation.” While correlationanalysis  can  be  extremely  useful  in  understanding  patterns,  providinginsights, and offering clues as to what is driving performance, it is crucialto resist the temptation of reading too much into associated outcomes. Ide-ally, like all other statistics discussed in this chapter, the calculation of cor-relations will evoke as many questions as answers. It will then be up to youto  derive  the  applicable  inferences  and  to  make  the  appropriate  adjust-ments with respect to your trading.
PUTTING IT ALL TOGETHER
Figure 3.7. is a consolidated report of many of the statistics discussed inthis chapter. As can be seen, the report summarizes in graphical/tabularform  a  wide  range  of  information  pertaining  to  the  performance  of  theaccount, in terms of daily P/L time series, capital usage, and so on. In laterchapters, we will incorporate some individual transactions-level analysis.The data is presented, wherever possible and/or appropriate, in terms ofboth dollars and percentages. Such an overall snapshot, synthesizing thestatistical concepts that I have found are most important, creates a usefulframework  for  further  critical  analysis  of  portfolio  performance—whether  undertaken  by  the  individual,  by  current  or  prospective
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7/1/02–9/30/02
10/1/02–12/31/02
1/1/02–12/31/02
Daily P/L
Cumulative P/L
Buying Power ($)
3,000,000
3,000,000
3,000,000
Return StatsTotal P/L  $Total Return(% Buy. Pwr.)Avg P/L (Daily $)Avg P/L (Daily %)Sharpe Ratio (Daily)
361,39412.05%5,6470.16%0.20
19,0000.63%2970.00%0.00
619,00020.63%2,4560.07%0.10
Volatility StatsStd. Dev. (Daily P/L $)Std. Dev. (Daily % of Buy. Pwr.)Maximum (Daily $)Minimum (Daily $)Maximum / 1 Std. Dev. P/LMinimum / 1 Std. Dev. P/LDownside Deviation ($)
28,0900.82%90,962(60,587)3.24(2.16)14,130
32,5500.91%82,000(122,000)2.52(3.75)25,452
25,3490.75%90,962(122,000)3.59(4.81)17,498
Daily P/L StatsNumber of DaysPercentage of Up DaysPercentage of Down DaysAverage Up DayAverage Down DayWin / Loss Ratio
6450%50%27,122(15,828)1.71
6450%50%24,063(23,469)1.03
25254%46%19,439(17,139)1.13
150,000
100,000
50,000
0
(50,000)
(100,000)
(150,000)
1 1/1 4/2 0 0 21 0/3 1/2 0 0 21 2/1 3/2 0 0 29/5/2 0 0 29/1 9/2 0 0 26/1 1/2 0 0 21 0/1 7/2 0 0 21 1/2 9/2 0 0 23/1 5/2 0 0 24/1 5/2 0 0 21/2/2 0 0 21 0/3/2 0 0 25/1 3/2 0 0 23/1/2 0 0 28/7/2 0 0 25/2 8/2 0 0 21/3 1/2 0 0 21 2/3 0/2 0 0 28/2 1/2 0 0 21/1 6/2 0 0 27/1 0/2 0 0 22/1 4/2 0 0 26/2 5/2 0 0 27/2 4/2 0 0 24/2 9/2 0 0 24/1/2 0 0 2
Daily P/L
Cumulative P/L
800,000
700,000
600,0