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CORRELATION
Once again extrapolating from our P/L analysis tool kit, the concept of cor-relation can be applied to individual positions as easily as it can to entireportfolios. For example, every equity security has a beta statistic, which isthe  functional  equivalent  of  a  correlation  coefficient  between  the  instru-ment and its associated benchmark index. In addition, you can always cal-culate  the  correlations  between  any  two  instruments  (whether  they  areequity securities or not) in your portfolio using spreadsheet tools such asExcel.
By being constantly mindful of the correlations across the individualpositions in your account, you stand to learn a great deal about the diver-sification characteristics of your portfolio. You can determine whether, forexample, long/short combinations are providing the appropriate exposureoffsets. For positions on the same side of the market, correlation analysiswill  help  you  identify  whether  the  positions  are  redundant  or  offer  truediversification. Beyond this, if you are able to capture this data not only forpositions that you may be holding at any given time but also for the otherinstruments  you  are  evaluating  and  may  hold  at  some  future  time  in  theinvestment cycle, you will place yourself in a better position to select themost  appropriate  securities  to  add  to  your  portfolio  in  order  to  achievewhatever modification to your overall risk profile you desire. 
For example, if you are holding a given stock position that you want toretain but feel it is vulnerable to a decline in the overall market, rather thansell it outright, you might choose to sell short a security that has a high cor-relation  to  the  one  you  are  holding,  thus  reducing  your  exposure  whileretaining your core position. The same general effect can also be achievedby purchasing a security that has a low or even a negative correlation tothe stock whose risk you are trying to hedge. This isn’t likely to be a reve-lation to you if you already engage in this type of activity routinely. I sim-ply add that the use of sound correlation data in the management of theseactivities can only add to the efficiency of the process. 
As with everything else, correlation data is time specific; and in calcu-lating  this  statistic  across  different  time  spans,  you  are  likely  to  achievedifferent results. Moreover, historical correlation data only describes pastpricing  patterns,  which  cannot  necessarily  be  extrapolated  forward  withfull  confidence.  Indeed,  particularly  in  times  of  market  stress,  historicalcorrelation data is likely to be of minimal use, as markets have a tendencyto  move  in  lockstep  with  one  another  under  these  circumstances  (i.e.,correlations converge to (cid:7)100%). As is the case elsewhere, this limitationcontains  a  hidden  opportunity:  I  believe  it  is  useful  to  compare  correla-tions between securities across different time spans so as to gain a better
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understanding of interactive pricing dynamics across the fullest range ofavailable market conditions. 
VALUE AT RISK (VaR)
Through  the  efforts  of  modern-day  financial  engineers,  a  new  paradigmhas emerged: It is now possible, nay, even fashionable, to combine the con-cepts of volatility and correlation into a single, portfolio-based exposureestimate. This work, most of which has been conducted over the past 15 orso years, is most broadly synthesized under the heading of Value at Risk,which is now thought of as the standard methodology for risk managementin the financial services industry. The underlying objective is to aggregateall risks in a given portfolio in such a way as to produce a volatility statis-tic at the portfolio level boiled down to a single number that will charac-terize  overall  portfolio  exposure.  This  number  is  intended  to  predict  P/Lvolatility, such that it identifies the size of P/L swings that are tied back toa  specified  confidence  interval.  For  example,  if  the  confidence  intervalused in the VaR calculation is 95% (as is often the case), then the risk num-ber produced is designed to identify the P/L threshold such that a port-folio experienced a fluctuation in excess of this only about 5% of the time, oronce in 20 days. Similarly, if the confidence interval is set at the 99% level,it implies that the portfolio in question should only experience a P/L obser-vation in excess of this figure approximately once in every 100 days.
While VaR has inspired some well-deserved cynicism over the years,it’s  hard  to  argue  against  its  conceptual  merits.  Think  of  it  as  a  form  ofalchemy  that  transforms  portfolios  with  multifaceted  exposure  elementsinto a single exposure-bearing unit. If you have little faith in the existenceof alchemists (and let’s face it, who doesn’t these days?), it may be usefulto think of your portfolio as being the equivalent of one giant security, theGeneral Electric Corporation (GE), for example. Like your portfolio, GEhas multiple business lines across virtually every important market sector,each of which must deal with unique and diverse risks on a daily basis. Butalso like your portfolio, there’s not necessarily a great deal that ties thesebusinesses together. While I’m sure that all of the unit managers enjoy theperiodic  schmooze  fests  up  at  the  corporate  headquarters  in  Stamford,Connecticut, it’s hard to see, other than common ownership, what bondsexist  among  say,  the  presidents  of  the  medical  devices  unit,  the  aircraftengine division, and the National Broadcasting Company.
Ah, but they do have that common ownership thing working, and thoseowners have the additional advantage of being able to own these businessesthrough the simplicity of a single security with a single price. And though
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it may not have occurred to you, dear reader, consider this: What is yourinvestment account if not a mini-me version of a GE–type structure? Lookat your next monthly statement and take pride in the diverse businessesyou  own:  the  world’s  leading  software  manufacturer  here,  an  entertain-ment  conglomerate  built  around  an  anthropomorphic  rodent  there,  per-haps a virtual bookstore (and who among us hasn’t hunkered to own oneof those?), or an Internet search engine—maybe you even own GE stockitself. Why, you’re a veritable Jack Welch! And why not have what he has(no, not the company-subsidized Manhattan duplex or the steamy romancewith a former Harvard Business Review editor, but something even bet-ter): an ability to aggregate exposures into a single volatility unit that canbe  tracked  holistically  with  an  eye  toward  making  appropriate  adjust-ments. This is what we risk managers had in mind when we created VaRfor you all those years ago. 
Much has been written and said about VaR over the years, and I willresist the temptation to attempt to add significantly to this august body oferudition. However, as I think that VaR is a useful and effective portfoliodiagnostic  (and  because  most  professional  traders  are  likely  to  be  gov-erned by some sort of VaR calculation), it is important to have a generalidea of the basic purpose and underlying justification of the approach, aswell as some knowledge as to how the statistic is calculated. 
Justification for VaR Calculations
The purpose of VaR is to provide an estimate of future P/L volatility, which,as I have argued earlier, is the essence of financial risk. Therefore, whilethe statistical tool kit described in the preceding chapter offers means ofevaluating financial  risk  (i.e.,  after  the  fact),  VaR  methodologies  aredesigned to predict this exposure. The evolution of VaR, which took placecontemporaneously across the most sophisticated segments of the finan-cial markets over the past couple of decades, can be traced to the desire ofthe financial community to improve risk control through the use of limits(for  direct  market  exposure),  collateralization  (for  counterparty  expo-sure), and other remedies that would either help minimize loss or at leastmitigate its associated impacts. 
In  seeking  to  better  apply  these  concepts,  the  providers  of  capitaland credit recognized and presided over what I believe to be a rationalevolution  (yes,  occasionally  even  these  knuckleheads  deserve  madprops)—from methodologies that dealt with individual instruments sep-arately to an approach that was based on the level of actual performancevolatility  for  the  overall  portfolios  in  question.  However,  because  per-formance  volatility  (much  like  performance  itself)  is  not  known  until
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after  the  fact,  the  focus  turned  toward  approaches  that  would  predictthis variable; and the result is the development and institutionalization ofValue  at  Risk.  Although  VaR  can  be  both  mystifying  and  maddening  tothose  whose  actions  are  governed  by  it,  the  methodology  represents  asignificant upgrade over its predecessor approaches under which actualdollars  of  investment  were  used  to  set  trading  parameters.  Under  thisless-sophisticated  approach,  many  limit  and  collateralization  regimesprovided counterintuitive incentives that penalized investors for certainrisk-reducing  trades  they  might  make  and  rewarded  them  (often  inap-propriately)  for  the  assumption  of  additional  exposure.  For  example,  atrader whose activities were governed by the amount of dollars deployedin the market might actually be charged a higher amount against his orher limit or might face a higher collateral requirement for a spread posi-tion involving longs and shorts in highly correlated securities than he orshe would for holding only one leg of this spread. This is in spite of thefact that overwhelming empirical evidence indicates that the full spreadposition will be far less risky than a directional bet of the same size. Ofcourse, it would be a stretch to suggest that spread trading in every situ-ation carries less exposure than outright speculation. The case of Long-Term  Capital  Management  (LTCM)  reinforces  the  notion  that  at  certainlevels  of  investment  there  is  virtually  no  limit  to  the  amount  of  riskinherent in a spread portfolio. The key distinctions here involve liquidityconstraints  and  the  use  of  leverage.  Although  no  one  would  seriouslyargue that $1 invested in a long/short strategy involving highly correlatedsecurities is much riskier than a $2 or even a $1 investment in either leg,spread strategies achieve the higher exposure through the use of lever-age, often employing strategies whose dollar values are 50 or 100 (in thecase of LTCM) times greater than the associated capital invested. More-over,  these  situations  are  certainly  exacerbated  (again  as  occurred  inthe LTCM crisis) when position sizes exceed the ability of the market toprovide sufficient liquidity to allow for orderly liquidation.
Indeed,  despite  the  routine  inconsistency  with  formulas  that  equateexposure  to  gross  dollars  invested,  gross  dollar  investment  collateraliza-tion/limit regimes are alive and well even in today’s environment. Case inpoint: your securities portfolio, which allows you to purchase $2 worth ofa single stock for every $1 of capital but would not allow you to hold morethan $1 long versus $1 short for the same level of capital—even though thelatter profile is likely, under most circumstances, to be far less risky. VaRprograms (unless they’re written by madmen) recognize this disparity. Forreasons  such  as  this,  most  everyone  I  have  encountered  whose  tradingactivities are governed by limits will attest to the fact that a rational VaRsystem  is  far  more  conducive  to  the  production  of  risk-adjusted  returnthan many other methodologies available in the market.
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Also, if you’ll allow a moment’s sympathy to us poor, beleaguered, and,yes, misunderstood risk managers, perhaps you’ll not judge us too harshlyfor at least trying to create an effective exposure-measurement machine. Ihave had to deal with very complex portfolios, involving long and short posi-tions in stocks, bonds, convertibles, options, and other types of derivativesall at the same time. In addition, these positions were typically denominatedin as many as a dozen different currencies. How the heck was I supposed tomake any judgment about exposure levels unless I did something to trans-late this rubric into (reasonably) homogeneous risk units? VaR may be sillyand stupid and is certainly applied in ridiculous ways by people who have nobusiness occupying the same galaxy as that inhabited by rational risk takers.But the only thing that I can think of that might be sillier or more stupid isto try to perform risk management on complex portfolios without somethingthat was if not VaR then at least its functional equivalent. 
Types of VaR Calculations
Over  the  years,  three  basic  methodologies  for  calculating  VaR  haveemerged:
1. Variance/Covariance. This  is  perhaps  the  most  widely  used  VaRmethodology,  under  which  the  program  gathers  volatility  (variance)and  correlation  (covariance)  statistics  across  all  individual  positionsin the portfolio and combines them, using matrix algebra, into a singlecalculation of portfolio volatility. 
The basic equation under which the variance/covariance approachto VaR is derived is a calculation of the standard deviation of the port-folio  under  the  assumption  that  its  distribution  can  be  described  asmultivariate normal; that is, the individual variable components (i.e.,trading positions) in the portfolio will behave in such a way as to pro-duce a normal distribution when viewed in aggregate. Under these cir-cumstances, the portfolio’s standard deviation ((cid:2)), can be calculatedusing the following formula: 
(cid:2) (cid:3)
B ai
i (cid:5) 2 a(cid:2)2i (cid:9)
aj
(cid:2)i(cid:2)j(cid:8)i, j
(cid:3) standard deviation of individual portfolio instruments(cid:3) covariance between instruments i and j
where (cid:2)
i, (cid:2)(cid:8)
j
i, j
Do not be overly troubled by any failure on your part to fully compre-hend the finer points of this nasty equation. For one thing, you have to
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be  a  bit  skeptical  about  any  mathematical  expression  that  includesboth capital sigma (which I have to admit is cooler when it has thoselittle i’s and j’s underneath it, and even better if you can throw in—asoften happens—a (cid:10) symbol on top) and small sigma. I’ll make a littleconfession here: I’ve been at this risk management game for nearly twodecades, and I can only recall one instance where I actually had to yankout this dubious formula to mollify some very fastidious Japanese guys.On the whole, I think you can pretty safely remove the formula fromyour memory bank—that is, if you’ll promise me to take into consider-ation the following:
Using  a  combination  of  volatility  and  correlation  statistics  and(you’ll have to trust me a bit here) a simple matrix-based expansion ofthe  basic  standard  deviation  calculation,  we  can  derive  a  standarddeviation  estimate  for  complex  portfolios,  which  should  be  more  orless  as  accurate  and  applicable  as  those  that  we  have  calculated  forindividual securities.
Due  in  part  to  its  relative  simplicity,  the  variance/covarianceapproach  to  VaR  has  become  the  most  frequently  used  of  the  VaRmethodologies. It is widely available in risk management software pack-ages and can actually be calculated, with relative ease, using spreadsheetprograms—provided you have the proper volatility and correlation data.In  turn,  you  can  either  obtain  this  data  directly  from  a  number  ofproviders (such as RiskMetrics) or calculate it on your own, using histor-ical pricing information for each position associated with the portfoliosin question and converting these to a variance/covariance matrix. How-ever, I must caution do-it-yourselfers away from underestimating the taskof  creating  this  matrix,  the  scope  and  complexity  of  which  will  varyexponentially with the number of individual positions in the portfolio.
For instance, a matrix for a portfolio of, say, 10 securities will con-tain 100 observations (10 (cid:11) 10). However, if you double the numberof positions in the portfolio, the matrix size increases fourfold to 400(20 (cid:11) 20).  Indeed,  by  the  time  you  reach  100  names,  the  matrix  willhave 10,000 individual data points; and if you ever were so ambitiousas to model, say, the Standard & Poor’s (S&P) 500 variance/covariancematrix, the number would rise to 250,000! For these reasons, I stronglyrecommend the use of external VaR programs for all but the most data-intensive traders; and if you must calculate your own VaR, you shouldat least purchase the variance/covariance matrix from an external dataprovider.
The variance/covariance approach to VaR is widely used becauseamong VaR alternatives it offers the least calculation-intensive meansof identifying a portfolio’s exposure. However, for these reasons it isalso the methodology that bears the greatest set of limitations. 
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First and foremost, it suffers from all of the potential inaccuraciesassociated with the key inputs to the calculation: volatility and corre-lation data, which may not be accurate predictors of future pricing pat-terns for the portfolios in question. Second, the methodology has notruly  effective  means  for  dealing  with  complex  derivatives,  such  asoptions,  which  cannot  be  plugged  neatly  into  a  variance/covariancematrix. Indeed, most VaR programs use a simple adjustment, such as adelta-mapping  for  options,  to  convert  these  positions  into  fungibleunits of the underlying instrument. For example, a long call option for$1 million on stock XYZ that had a delta of .5 would receive identicaltreatment as a position of $500,000 in the cash market. For these rea-sons, the results must be interpreted with extreme caution. However,in spite of these limitations, the variance/covariance approach proba-bly offers the best combination of simplicity and accuracy available tomost investors (i.e., the benefits of its simplicity more than offset thecosts associated with reduced accuracy) and is on this basis arguablythe most applicable of VaR approaches. 
2. Historical  Volatility. Under  the  historical  volatility  approach,  theVaR program actually recalculates a P/L for the existing portfolio, eachday going back into the past in order to determine what the accountwould have made or lost over each day in the sequence. The programthen ranks these P/L observations in ascending order and assigns theVaR  number  that  corresponds  to  the  loss  associated  with  the  confi-dence interval specified by the user. For example, if the historical VaRprogram you were using had the ability to look back one year, it wouldtake your current portfolio, calculate a P/L for each day going back thefull 12 months, and then rank these statistics. If you set your VaR con-fidence  interval  to  be  95%,  the  program  would  then  throw  out  theworst 5% of your losses (over a 250-day trading year, this would meaneliminating the worst 12 or 13 observations) and set your risk at theloss level corresponding to the next highest observation. The impliedassumption is that you ought to be 95% sure, on a going-forward basis,that the daily losses should not exceed this figure.
Historical  volatility  is  a  significantly  process-intensive  VaRmethodology  because  it  requires  the  repricing  of  every  single  instru-ment  over  every  single  day  covering  the  historical  time  intervalselected. However, I believe that if it is possible to execute such a cal-culation, it may well be worth doing so, as historical volatility is gen-erally  viewed  to  be  the  most  accurate  of  the  VaR  methodologies,  atleast for the vast majority of portfolios. A careful review of the under-lying methodologies helps reinforce its basic advantages. 
First, rather than relying on correlation data to impute the mannerin  which  multiple  securities  will  price  themselves  in  relation  to  one
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another,  historical  VaR  actually  utilizes  historical  pricing  to  estimatethe magnitude of associated exposure. This can be particularly impor-tant  during  times  of  market  stress,  when  even  static  correlationpatterns tend to break down. Second, unlike variance/covariance, his-torical  VaR  extends  seamlessly  (provided  that  the  user  meticulouslyincorporates the full set of inputs) to cover such derivative productsas options—often with results that elegantly illustrate precisely why itis the preferred methodology among many risk managers who are in aposition to choose their optimal VaR approach. Finally, historical VaRoffers all sorts of opportunities for qualitative analysis against identifi-able market episodes derived from real-world experience. If you seesomething in the results that interests you, it is very useful to look atthe specific dates that generate these outcomes. This is an analyticaltool that the other forms of VaR do not offer. Try to avail yourself of itsbenefits by evaluating both the likelihood for recurrence and perhapseven by formulating an associated contingency plan. 
The  binding  limitations  associated  with  historical  VaR  are  thosethat  pervade  the  entire  business  of  statistically  evaluating  portfolioexposure. As we’ve discussed, these derive from the inability of pasthistory to reliably predict future outcomes. In addition, as we’ve alsocovered, it is calculation intensive; and, as such, beyond the practicaldifficulties of executing the routine, it is probably more prone to errorthan  other  models.  Individual  bad  data  points  can  throw  off  resultsmaterially even when every other element of the calculation (involvingperhaps  hundreds  or  thousands  of  individual  steps)  is  entirely  accu-rate. However, if you are in a position to overcome these problems, Ihighly recommend its application wherever the data and systems arein place to render it feasible. 
3. Monte Carlo Simulation. The Monte Carlo approach to VaR is simi-lar  in  methodology  to  that  of  Monte  Carlo  pricing  in  the  world  ofoptions  trading.  Under  this  method,  the  program  regenerates  largenumbers  of  random,  feasible  price-change  outcomes  for  individualsecurities  (based  on  user-specified  ranges  of  scenarios)  and  ranksthem statistically according to their probability. It then assigns the VaRnumber  tied  to  the  “probability  outcome”  that  corresponds  to  theselected confidence interval. For example, a Monte Carlo VaR programdesigned  to  capture  risk  at  the  95th  percentile  confidence  intervalmight reprice entire portfolios up to several million times in order toidentify the full range of potential P/L outcomes and then set the VaRnumber such that 95% of the observations were below the figure andthe remaining 5% were above it.
The  advantage  of  Monte  Carlo  simulation  is  that  by  rigorouslyextrapolating  market  history  across  a  huge  number  of  results,  it  is
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possible to obtain a better picture of the full, prospective price distribu-tion for the portfolio in question and thus have more information as tothe probabilities associated with a wide range of outcomes. Moreover,Monte Carlo can be a useful methodology for portfolios bearing a highdegree  of  optionality  because  it  allows  for  the  evaluation  of  virtuallyunlimited  combinations  of  underlying  prices  and  volatilities  that  com-bine to determine the value of the options in a given portfolio, includingthose  associated  with  acute  market  conditions  that  can  generate  dra-matic  portfolio  swings.  However,  for  most  simple  portfolios,  MonteCarlo  simulation  probably  represents  a  bit  of  overkill,  with  any  incre-mental accuracy to be gained in the process offset by the level of com-plexity and calculation intensity needed to execute the methodology. 
It is worth noting that for all but the most complex of portfolios withextremely  nuanced  pricing  characteristics,  VaR  results  should  be  prettyconsistent  across  the  three  approaches  just  defined.  I  think  it’s  moreimportant to use your time incorporating VaR than squandering resourcesselecting  which  method  to  use.  Pick  one  that  makes  sense  to  you,  givenyour portfolio characteristics and resource constraints, and incorporate itinto your decision making. The rest, as they say, is irrelevant. 
Testing VaR Accuracy
VaR models, as we will discuss shortly, vary in terms of their actual math-ematical construction; but they are all designed, as we’ve covered, to pre-dict performance volatility associated with a specified confidence interval.Specifically, all models will yield results that have a probabilistic compo-nent, each producing a number that serves as a threshold that the port-folio  volatility  should  violate  with  a  specified  level  of  probability.  Assuch, the accuracy of the model can be back-tested by checking how manyP/L observations actually exceed the VaR figure over a given period of timeand then tying the resulting percentage back to the confidence interval. Ifthe two figures are consistent, the model can be said to be in good order;if not, then the model’s assumptions and inputs must be examined in orderto determine the source of the inconsistency (more about this later). Thistype  of  backtest  is  one  of  the  simplest  mathematical  exercises  we  willcover in this book, and I urge you to periodically check to see if your actualP/L volatility is consistent with the model and parameters you are using.Some  purists  will  argue  that  comparing  actual  portfolio  volatility  to  VaRresults is not an effective backtest because while the VaR calculation is basedon the current portfolio as measured against historical price action, the testmeasures  the  accuracy  of  this  calculation  against  future price  fluctuation.
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However, I would ignore this distinction for the simple reason that if youractual P/L volatility differs substantially from the results implied in the VaRcalculation, it is failing to tell you anything—irrespective of the fact that itmay be faithfully adhering to its core assumption base. 
Setting VaR Parameters
At this very moment, you may be asking yourself which confidence inter-val is most appropriate for your VaR analysis. The bad news is there’s noeasy answer; the good news is that as long as you’re consistent in its appli-cation, you pretty much can’t go wrong.
First  of  all,  if  you  use  the  variance/covariance  approach  to  VaR,  itdoesn’t matter what confidence interval you use because the numbers willsimply increase linearly, according to the scale-up factors set forth in Table4.1, which I have resurrected Lazarus-like, for the purposes of this discus-sion,  from  Chapter  3.  Thus,  under  the  variance/covariance  approach,  a95th percentile confidence interval will equate, forever and anon, to 1.960times a 1.000(cid:2) projection; a 99th percentile confidence interval to a 2.576(cid:2);and so on. 
Matters  get  a  bit  more  interesting  under  both  historical  VaR  andMonte  Carlo  simulation,  both  of  which  contemplate  universes—remotethough  they  may  be—where  portfolio  returns  distribute  in  ways  otherthan normally. For example, historical volatility analysis will capture thefact  that  equity  market  distributions  tend  to  have  fatter  tails  than  otherfinancial assets; and if you are using historical volatility, you may find thatthe  99th  percentile  observation  is  of  a  significantly  greater  magn