itudethan 2.576(cid:2). By the same token, the more complex your portfolio is (e.g.,the higher the incidence of options and other derivatives), the greater is
TABLE 4.1
Percentage of ReturnsAssociated with StandardDeviation Intervals
Percentage of Normal Curve Area
Mean (cid:1) Number of Standard Deviations
50.0%68.390.095.095.498.099.099.7
.674(cid:2)
1.0001.6451.9602.0002.3262.5763.000
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the likelihood that extreme observations will quantify themselves at sur-prising multiples of (cid:2).
It is therefore a valuable rule of thumb that the more mathematicallycomplicated the portfolio and the better the VaR measurement tools, themore  useful  it  will  be  to  apply  higher  confidence  intervals.  However,because most portfolios are simple and because most VaR systems lack thesophistication to identify the nuanced exposures at the tails of a distribu-tion, there is generally only marginal benefit to be gained by using thesemore  conservative  estimates.  As  for  me,  I  prefer  measuring  volatility  interms of a single standard deviation because it allows for the most directcomparison  to  the  volatility  of  individual  financial  instruments  andbecause, as we will see in the next several chapters, it can be used, througha modification of the Sharpe Ratio, as a means of sizing investment levelsand volatilities to specific, risk-adjusted return targets.
Setting of VaR confidence intervals becomes most relevant when theVaR  estimates  are  used  as  something  other  than  a  general  indication  ofexposure levels. The most prominent example of this is the use of VaR asa benchmark against which limits or credit lines are set, as happens withperhaps  most  professional  traders  operating  in  the  setting  of  large  andsmall financial institutions. Typically, these institutions will impose restric-tions on traders based on what they view to be acceptable levels of expo-sure as estimated by their in-house VaR programs. 
When a capital provider sets trading/risk limits on the basis of a VaRanalysis, the specific VaR methodology used in conjunction with the cal-culation  looms  much  larger  in  importance.  In  my  experience,  fortunateindeed is the market investment professional whose VaR limits exist in har-mony with her most rational view of the exposures she encounters in themarketplace.  Unfortunately,  everyone  else  will  have  to  improvise,  com-pounding  the  already  considerable  difficulties  of  maximizing  profits  bybeing forced to rationalize limits with an unclouded view of the true expo-sures in the portfolio. The only way I have found to accomplish these con-flicting objectives consistently (as I will touch on shortly) is to reduce yourrisk  taking  to  levels  that  fit  comfortably  within  both  the  risk  systemsimposed  on  you  and  the  amount  of  “true”  exposure  you  are  willing  toaccept in the markets. This will probably cost you money, as it will impedeyou from the assumption of maximum exposure profiles—particularly athigh-conviction points when you feel that the exposures are justified. 
STARLESS AND BIBLE BLACK
I want you to consider, for a moment, the hypothetical example of a bank—alarge international bank—with big, far-flung trading operations and (of course)
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an  unshakeable  commitment  to  risk  management.  How  committed  was  thisbank to risk management? Very committed to risk management. It said so rightin the brochure. 
The risk departments of this bank, for obvious reasons, were sequesteredin  the  deepest  recesses  of  the  organization’s  glittering  corporate  headquar-ters, located in the center of the financial district of a major Western city. It isin these highfalutin’ environs that the keepers of the sacred risk managementflame set about the task of creating their masterwork—the infamous Black Box(Boitte Noire). This goes to show that, among other things, black boxes, RubeWalker  machines,  and  other  dubious  risk  management  devices  exist  on  bothsides of the Atlantic Ocean. Perhaps it was the thinness of the air on the upperfloors; perhaps it was a desire to outshine their inferiors at competing organi-zations. But one way or another, they decided that any prudent risk manage-ment system must take into account not some of the risk but, gosh almighty,“all the risk there ever was.” 
On the wings of this lark, they flew right past the one, two, and three stan-dard deviation thresholds, beyond the 95th, 99th, and even 99.99th percentileconfidence  intervals,  all  the  way  to  the  honest-to-goodness-no-turning-back100th percentile observation. Off some glib tongue came the suggestion to usethe 200th percentile; but another in the room had enough presence of mind topoint out the oxymoronic nature of the concept, rendering at least one factioncrimson with anger and embarrassment. In a face-saving gesture, rather thanlooking  back  at  the  worst  outcome  over,  say,  the  past  year,  they  took  the(extraordinarily patriotic, in my view) step of going back to the very largest pricechange that had taken place in each market over the past decade. So enthusi-astically  was  this  proposal  received  that  a  particularly  ambitious  fellow  in  thegroup suggested that even 10 years was not a sufficient interval of analysis andthat  the  only  truly  inspired  answer  lay  in  comparing  today’s  positions  to  theworst price movements over the century just prior to the 21st (“Schizoid, man!!!”said the one trader present; but no one knew what he was talking about). I’mquite sure this suggestion would have carried the day had not one of the dourrealists in the crowd pointed out the difficulties of obtaining 100 years of dailydata for markets that in many cases hadn’t existed even a generation earlier.
So they went forward with the 10-year plan. This meant that every positionin equity markets had risk measured as though it lost as much as it might haveon  October  19,  1987;  every  interest  rate  position  was  mapped  against  thefunding crisis in France in early 1994; and so on. Everyone congratulated every-one else on having created the most thorough risk system on the planet. And,in a spirit of true generosity (spurred in part by a threat of a mass walkoutfrom  traders on the desk), knowing that the Black Box was throwing off riskestimates that were five times as high as was the case for VaR programs oper-ating with a 95th confidence interval parameter, the powers that be promptlyquintupled everyone’s risk limit.
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The net result was a system where the risk estimates and associated lim-its  were  five  times  higher  than  those  inherent  in  the  previous  structure.  Thebank  was  happy  because  it  had  demonstrated  to  tout  le  monde its  single-minded  commitment  to  prudent  risk  management.  Traders  were  happybecause for every dollar of risk they could take under the previous regime, theywere now permitted to take five dollars. If some suspected that they had donenothing  but  effect  some  “limit  inflation”—increasing  both  sides  of  the  limitequation by equal amounts and leaving portfolio managers in exactly the samespot at which they had started—they did so quietly. The Crimson Kings of RiskControl had their Black Box, and it was their Epitaph. 
Then, one day in the fall of 1997, something very interesting happened.On October 20, the 1987 crash data point (being exactly 10 years and 1 dayold) rolled right out of the equity data series. The 10-year, worst-case loss inequity markets was reduced by more than 60 percent, in the process more thandoubling the risk capacity for the bank’s equity traders. Celebrations took var-ious  forms,  including  a  couple  of  industrious  fellows  who  decided  to  doubledown on stocks they were holding. Exactly one week later, October 27, 1997,the Dow experienced a drop even steeper than the one that had occurred 10years  (and  1  week)  earlier.  Those  guys  weren’t  seen  much  around  the  bankafter that episode, but no one noticed. Meanwhile, the Black Box remains. 
Use of VaR Calculation in Portfolio Management
Most  professionals  who  trade  the  capital  of  financial  institutions  will  besubject to some sort of VaR regime for the setting of their limits. However,even those who trade for themselves are subject to VaR governance, oftenin hidden ways. For example, futures and options exchanges use a modi-fied version of VaR to calculate margin requirements for their products andinsist  that  the  brokerage  firms  that  clear  their  contracts  do  the  same.Therefore,  unless  you  are  doing  nothing  but  trading  equity  securities  foryour own account, you are likely to be subject to some sort of VaR calcu-lation on at least a subset of your portfolio.
However,  whether  you  find  your  portfolio  subject  to  an  externallyimposed VaR calculation, I submit that there is significant benefit to exe-cuting and analyzing a VaR methodology for your trading because it offersthe best means of isolating the exposure characteristics of your combinedposition  profile  at  any  given  time.  In  addition,  VaR  serves  as  one  of  thebest  indicators  of  actual  level  of  market  participation  available  to  thetypical investor—a subtle point, but one that bears pausing over in orderto  ensure  a  full  understanding.  Consider  two  portfolios  that  each  have$1,000,000  invested  in  the  markets.  One  portfolio  is  full  of  high-flyingstocks;  the  other  has  boring  dividend-paying  assets.  Although  the  same
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dollars  are  committed  to  the  market  from  each  portfolio,  I  would  arguethat  the  high-flying  portfolio  is  more  fully  invested  and  has  a  higherdegree of market participation, largely due to the fact that it will experi-ence more acute financial impacts to any level of price action in the mar-kets  than  will  the  dividend  portfolio.  VaR  has  no  purpose  other  than  tomeasure these concepts, and I particularly endorse your taking the stepsnecessary to understand the time series dynamics of the VaR calculation,starting  with  a  routine  periodic  measurement  of  its  predictive  accuracy(i.e., the back-test). Study its ability to adhere to the confidence intervalassumptions you have specified (as described in the previous Testing VaRAccuracy section).
Once you have achieved a sufficient level of accuracy in this regard,you  can  then  use  the  VaR  statistic  as  yet  another  means  of  ensuring  theefficient use of your capital. If your VaR numbers are increasing, it meansthat your exposure is on the rise; and (rationally speaking) this should onlytake place when (1) there is sufficient risk capital available to increase thebet  and  (2)  profit  opportunities  are  rising  in  a  manner  that  justifies  theincreased exposure. For this reason your VaR should be positively corre-lated to your overall performance. If it isn’t, it may be due to a failure onyour part to manage your risk capital efficiently and to identify your bestopportunities on a “look-ahead” basis.
Wherever and whenever it is feasible to do so, I also recommend thatyou review the “intermediate” calculations associated with your VaR pro-gram in order to identify the elements of your portfolio that are contribut-ing  most  strongly  to  your  overall  exposure.  This  can  be  done  in  mostprograms by comparing the risks projected for a single position across allnames held in your portfolio. As is the case across time, the greatest con-tributors  to  your  overall  exposure  should  be  those  that  are  tied  to  yourbest opportunities, and these ideas should generate your greatest profitsover the longer haul. 
If there is significant mismatch between the positions that are produc-ing  your  greatest  overall  exposure  according  to  the  VaR  calculation  andthose that are most directly responsible for your P/L, take it as an indica-tion that you could be managing your overall position more effectively andthat greater profitability can be achieved by reallocating your risk capitalin such a way that the lion’s share of your revenue derives from the tradesfor which you feel the highest level of conviction.
Once your VaR program is up, running, and in stable production, thinkof it as a barometer of your risk. Set ranges of acceptable VaR outcomesand stick to them. Make sure that when you do find yourself operating inthe upper portions of the range, you have done so because you believe thatsuch additional risk assumption is justified by the quality of associated mar-ket opportunities and does not derive from “investment creep,” boredom,
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or any of the other factors that can cause investors to initiate and maintainpositions in their portfolios for reasons that even they cannot explain. 
To  summarize,  VaR,  with  all  its  limitations  and  prone  though  it  is  to  car-toonish overemphasis, gets a bad rap. For one thing, when portfolios aretruly  complex—in  particular  involving  multiple-asset  and  -instrumentclasses, such as stocks, bonds, and options—it’s difficult to contemplate amethod for aggregating exposures that doesn’t operate under the basic VaRframework. The same can be said, to a lesser but still considerable extent,for  portfolios  with  long/short  orientations  or  even  for  one-sided,  single-instrument-class  portfolios  that  make  an  effort  at  diversification.  In  eachcase, a well-framed VaR calculation ought to be a valuable tool for the siz-ing of exposures at both the individual and the portfolio levels. I can’t thinkof a single situation (polytechnicians included) where a trading operationisn’t better off with some sort of VaR–based model than without one. As arisk manager, I shudder at the thought of operating without VaR, or at leastsomething  that  closely  resembles  it.  And  in  the  more  complex  environ-ments I’ve had to manage, proceeding without a risk aggregation tool suchas VaR would be laughable. As an investor, if you take the trouble to under-stand the strengths and weaknesses of the VaR model that applies to youraccount, the methodology can provide all kinds of additional useful insightsinto both the internal and the market-related dynamics of your account. 
SCENARIO ANALYSIS
I have spoken of VaR as a form of alchemy, through which those wishingto predict future P/L dispersion for portfolios with multiple and complexfinancial instruments seek to express that dispersion in a single numericalestimate.  Hopefully,  I  have  convinced  you  that  imprecise  though  themethodology might be, there is considerable merit in the effort. In order toaddress the imperfections associated with the aggregation process, as wellas  those  tied  to  the  methodology’s  inability  to  elegantly  express  certainspecific types of exposures, the risk-obsessed among us have created theconcept of scenario analysis.
As implied in the name, scenario analysis is designed to measure theexposure associated with specific, user-defined contingencies in the mar-kets that are not well captured by other methodologies. Most typically andperhaps most effectively, scenario analysis is applied to various arbitrageportfolios where the interplay between individual instruments is less eas-ily captured by VaR’s more brute force mapping of positions in an accountinto easy-to-comprehend risk factors.
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For purposes of illustration, I offer some examples of the application
of scenario analysis to three specific types of arbitrage portfolios: 
1. Relative  Value  Fixed  Income. This  strategy,  rendered  somewhatinfamous  by  the  Long-Term  Capital  Management  episode  of  a  fewyears  back,  involves  the  simultaneous  purchase  and  sale  of  fixed-income  instruments  with  similar  but  nonidentical  economic  charac-teristics  that  exhibit  pricing  divergence,  in  the  hopes  that  thisdivergence disappears over the life of the transaction. Often, this mayfeature the purchase and sale of bonds with similar credit qualities acrossdiffering maturities or of bonds with differing credit qualities across thesame maturity. In the case of the former, a prudent risk manager mightwant to know what happens to this portfolio if interest rates continueto move in nonintuitive ways that are adverse to the portfolio, andhe or she might design a set of scenarios to cover the following suchcontingencies:
• Interest rates change by the same magnitude across the yield curve
(called a parallel shift).
• Rates at earlier maturities increase at a faster rate than those at fur-ther  out  maturities  (flattening  twist),  or  vice  versa  (steepeningtwist).
• Rates  at  both  ends  of  the  yield  curve  remain  relatively  constant,while those in between these endpoints either increase or decrease.
By using scenario analysis, one can determine the exposures asso-ciated  with  these  types  of  subtle  price  movements  with  a  great  dealmore precision than by using other methods of exposure estimation.
In the case of relative value plays involving instruments of differ-ing credit quality, the designers of scenario analysis routines will typi-cally try to estimate the impact of changes in credit spreads, definedas the premium that lenders will demand of borrowers of lesser creditquality. Like the yield curve manipulations mentioned immediately ear-lier, these exercises are likely to capture risks that are assumed awayby the aggregations embedded in the VaR calculation. 
2. Convertible Bond Arbitrage. The typical configuration of a convert-ible bond arbitrage portfolio is one under which the portfolio containsinventories  of  bonds  that  are  convertible  into  stock,  as  hedged  byshort positions in the cash equity securities of the same corporations.Because these bonds are typically highly correlated to their associatedstocks,  most  VaR  programs  would  fail  to  register  much  exposure.However,  there  are  scenarios  under  which  these  portfolios  (perhapsdue to the demands of the firms that provide leverage to convertible
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arbitrage portfolio managers) might suffer from the worst of all possi-ble combinations: The bonds drop in value while the equities experi-ence a contemporaneous rise. It behooves those who either manage orfund convertible arbitrage portfolios to have a clear understanding ofthe risks associated with these worst-case scenarios. 
3. Options Volatility Arbitrage. For those intrepid few wishing to cap-italize  on  subtle  and  theoretically  unsustainable  discrepancies  in  thevolatility pricing of options with the same or highly similar underliers,it is prudent to create a scenario analysis that specifically targets con-ditions under which these mispricings extend themselves. Due to thenonlinear  and  (often)  leveraged  nature  of  options  portfolios,  thisprocess can often identify exposures that a VaR program or other risk-estimation methodology might otherwise miss. 
It is also not unusual for risk managers with oversight responsibility oflarge portfolios involving multiple asset classes (e.g., global macro portfo-lios)  to  contrive  scenarios  under  which  all  financial  instruments  eitherdecouple  from  each  other  or  begin  to  move  in  lockstep—in  dramaticadverse fashion. These so-called event-risk scenarios are very prevalent inthe risk modeling of large financial institutions and provide the consider-able benefit of informing all involved parties (including high-level decisionmakers who are perhaps somewhat removed from the day-to-day portfoliomanagement process) as to the institutions’ best estimates of worst-caseloss. However, as a day-to-day risk management tool, I find them to be oflimited use.
The main point here regarding scenario analysis is that particularly forcertain  complex  portfolio  types,  it  will  often  behoove  the  portfolio  man-ager to move beyond widely used instruments of portfolio risk estimationinto  something  much  more  targeted  to  their  specific  exposures.  If  thisapplies to you, you probably already know it and are aware that you areprobably the best engineer of the appropriate scenario analysis. However,if  your  portfolio  involves  nothing  more  than  directional  speculations  inspecific markets, scenario analysis is probably overkill. 
TECHNICAL ANALYSIS
Most traders are familiar with the time series evaluation of the historicalpricing patterns of individual financial instruments, commonly referred toas technical  analysis. Over  the  past  few  decades,  an  entire  science  oftechnical  analysis  has  emerged  that  involves  myriad  methodologies  thatare mind-boggling in their number, range, scope, and complexity. It is not
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my intention here to delve too deeply into these mysteries (we’ve enoughground to cover as it is); however, I do believe that the rudiments of tech-nical analysis can be effectively applied as a risk management tool.
Specifically, it is important to remain mindful of the “technical” char-acteristics  of  the  financial  instruments  you  are  trading.  This  is  true  bothprospectively for positions you are adding and in terms of the managementof the positions you are maintaining on your sheets at any given time. Inorder to accomplish this in even a basic way, it is necessary to introducethe concepts of support and resistance. You probably deal with these con-cepts  every  minute  of  every  day,  and  I  don’t  mean  to  insult  your  intelli-gence by providing basic definitions. However, on the odd chance that youare either unacquainted with or confused by them, the following generaldescriptions may be of use: 
• Support  Level. The  price  for  a  given  security  that  represents  arepeated lower bound of a trading range. Financial instruments that doviolate support levels are often considered to have more risk for down-side  price  movement,  as  the  violation  indicates  that  the  market  mayhave revised downward its perception about the core valuation levelthat exists. 
• Resistance Level. The price for a given security that represents theupper  bound  of  a  trading  range.  Financial  instruments  that  violateresistance  levels  are  often  considered  to  have  more  opportunity  forupside  price  movement,  as  the  move  above  resistance  may  suggestthat the market may have revised upward its perception as to the max-imum level of valuation that exists. 
There are a number of methodologies for determining the support andthe  resistance  levels  for  the  securities  you  are  trading,  many  of  whichinvolve the use of moving averages and other simple statistical techniques.In addition, a careful review of time series charts will provide you with ageneral idea as to where the support and resistance levels are for the secu-rities you are trading. Once you have estimated these pricing points, theyrepresent ideal levels of review for position initiation/liquidation. 
For  example,  many  traders  use  the  violation  of  support/resistancethresholds as the catalyst for either position initiation or liquidation. How-ever, even if you choose not to adopt this approach, it is still important tobe  mindful  of  these  technical  thresholds  because  there  is  a  heightenedprobability that once violated, the associated position is likely to experi-ence  an  increase  in  its  volatility.  Also  note  that  this  volatility  can  go  ineither direction. Technicians make a living by buying financial instrumentsthat have penetrated resistance and selling them when they break throughsupport. Oftentimes they are right, but they are also often wrong. The main
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point is that once these levels, which represent perceived equilibrium ofsupply and demand for economic assets, are well known by markets andonce they are breached, it is typically an indication of heightened differ-ences of opinion as to the proper valuation. In my experience, these eventsare often signals that new intervals of volatility are indeed on the horizon.If you wish to probe deeper into the concept of technical analysis, youwill find a limitless array of tools at your disposal. For example, technicalmodels  that  possess  a  higher  level  of  sophistication  than  simple,support/resistance  algorithms  often  feature  in-depth  analysis  of  the  rela-tionship  between  volume  and  pricing,  taking  such  forms  as  momentumanalysis and relative strength indicators. Again, in each instance, there areassociated risk management implications that can be harnessed in a con-structive manner. However, I caution you against overdoing it in this regard;it is easy to fall into deep analytical traps using technical analysis that arenot at all guaranteed to enhance your portfolio management skills on anylevel. My best advice is to start slowly, understanding the areas of historicalprice stability for the securities you are trading, and to have some sort ofgame plan when pricing patterns violate these equilibrium levels. 
Measures  of  volatility,  correlation,  VaR,  technical  analysis,  and  scenarioanalysis can all be useful in helping you determine and manage the specificlevels  of  exposure  in  your  account.  Each  can  be  distilled  into  intuitive,mathematically uncomplicated, quantitative descriptions of either portfo-lio risk characteristics or exposures in individual positions that an investormight hold. In my experience the use of these techniques has both directand indirect benefits, with the latter taking the form of unearthing a deepwell of potential knowledge about the markets in which investors operatethat can be mined in a multitude of beneficial ways, some of which we willdiscuss in subsequent chapters.
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CHAPTER  5
SettingThe RiskAppropriateManagementExposure LevelsInvestment(Rule 1)
He wanna play chords to the chicks in the street,He wanna play words to a world that don’t speak,He wanna play people, who play hide and seek 
when they talk.
He wanna play riff to the man with the wires,He wanna play lead, but his hand’s getting tired,He just wanna play, but don’t know how to say, 
STOP!!!
—Ian Hunter
“Marionette” by Mott the Hoople
A t this point, it can be hoped that you have established your method-
ology  for  quantifying  your  portfolio  exposure  in  terms  of  both  itshistorical characteristics and its future consequences. Empoweredwith these tools, a couple of critical questions regarding portfolio manage-ment should come to mind.
1. How can I establish appropriate exposure ranges for my trading suchthat my risk is neither excessive nor too low to attain my objectives?
2. What should I do to adjust my risk profile if I feel it is either too high
or too low?
In this chapter, we will focus on the first of these questions, while leavingthe second question to be the central issue attacked in Chapter 6. Althoughthere are certainly limitations as to how much control you can directly exertover  your  profit/loss  (P/L)  volatility,  the  methodologies  that  I  describe  are
109
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designed to increase such control considerably and in the process to placeyou  in  the  best  possible  position  to  achieve  your  performance  objectiveswhile minimizing the damage if events play out that you did not anticipate.
DETERMINING THE APPROPRIATE RANGES OF EXPOSURE
Read this next statement very carefully, kids, because it’s important: It isup to the capital provider to determine the appropriate risk ranges forthe trading portfolios that it is funding. This means that if you are trad-ing with anything but