00
500,000
400,000
300,000
200,000
100,000
0
60
50
40
30
20
10
0
2
−80or <
51
43 43
35
22
18
2
0
5
3
1
10
8
2 3 2 1 1
−79to −70
−69to −60
−59to −50
−49to −40
−39to −30
−29to −20
−19to −10
−9to −1
0
to 
9
10to
19
20to
29
30to
39
40to
49
50to
59
60to
69
70to
79
80to
89
>90
FIGURE 3.7 Sample consolidated statistical profile.
providers of capital, or by other entities on a need-to-know basis. It alsoillustrates the point with which I’d like to leave you concerning this topic:Avoid the practice of viewing each individual statistic in isolation; giventheir  interdependencies,  they  are  instead  best  interpreted  “holistically,”with an eye toward identifying the factors that reside on the periphery ofyour trading approach and (more important) that are true drivers of rela-tive performance.
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CHAPTER  4
The RiskThe RiskComponents ofManagementan IndividualInvestmentPortfolio
A man with a watch knows what time it is. A manwith two watches is never sure.
—Segal’s Law
Our focus thus far has been on reviewing and understanding overall
portfolio performance from a statistical/risk management perspec-tive.  Specifically,  we  have  introduced  methodologies  that  seek  tocharacterize  the  risk  components  of  trading  accounts,  as  though  theseaccounts were the most granular units of risk assumption that we couldanalyze.  Of  course,  as  I’ve  already  mentioned,  this  is  not  strictly  true.  Infact, as we will discuss at length in this chapter, it is within the power ofthe portfolio manager to engineer not only the overall level of exposure inthe  portfolio  but  also  the  characteristics  of  its  subcomponents,  such  asdistribution  across  markets  and  specific  levels  of  exposure  to  such  riskfactors  as  position  concentration,  implied  leverage,  and  correlation  tounderlying  markets.  This  chapter  describes  the  analytical  tools  thatprovide the basis for such control. Understanding and applying these toolseffectively in the portfolio management process may very well representthe next step in your risk management evolution. 
It is important to point out that when we speak here about risk control,we are not in all cases implying risk reduction. The techniques on which Iwill focus are designed to allow you to modify your exposures as you see fit:increasing  them,  reducing  them,  or  redistributing  them,  as  circumstancesdemand.  As  may  have  already  occurred  to  you,  sound  risk  managementoften demands adjustments that either leave overall exposure levels flat or
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actually increase them at times. If you can master these concepts and com-bine them with knowledge you may already have gained as a trader, partic-ularly as to the optimal timing for “exposure adjusting transactions,” you willhave  gained  a  significant  competitive  advantage  over  many  other  marketparticipants who, for the most part in my estimation, tend to be very passiveagents in the risk control process. 
In order to move from an enlightened understanding of the magnitudeand the distribution of the risks we are taking to a program that actuallyallows us to modify these exposures, we must aim our statistical tool kit ata  specificity  level  deeper  than  that  of  overall  portfolios.  Specifically,  wemust focus now on the portfolio building blocks: the universe of financialinstruments that we trade. 
The same analytics that apply to entire portfolios can be used to evalu-ate individual financial instruments for their applicability to a specific set ofrisk  management  objectives  based  on  factors  such  as  current  portfoliocomposition and market conditions. Therefore, in the pages that follow, youwill  notice  some  repetition  of  concepts  introduced  earlier.  However,  themanner  in  which  we  will  apply  these  concepts  differs  from  their  earliertreatment  in  subtle  yet  important  ways;  and  I  encourage  you  to  take  thetrouble to evaluate these distinctions, as well as the associated commonal-ities, as carefully as possible. By performing the analysis at both levels (i.e.,that of the portfolio as well as of its subcomponents), I believe you can gainperspectives that are analogous to the insights one might achieve from, say,looking at a statue up close (i.e., in three dimensions) versus looking at aphotograph  of  one  (i.e.,  in  two  dimensions).  Another  advantage  of  per-forming these types of analyses at the individual position level is that thepractice is actually geared toward figuring out how best to adjust your risklevels  once  you  have  decided  such  a  change  makes  sense.  For  example,while the correlation analysis covered in Chapter 3 might argue for reduc-ing, say, long market exposure, the individual position analysis would notonly help select the best securities for doing so (given the specific charac-teristics of your portfolio) but also could assist in setting the size and thenmeasuring the likely impact on portfolio performance.
Because this is such an important concept, we will cover it over threechapters.  In  the  first  installment,  we  will  apply  the  tools  we  covered  inChapter 3 (in addition to some new ones of a similar nature) to individualsecurities so that you can make individual transaction decisions in a waythat is mindful of their associated impact on your overall risk profile. Onceyou have learned how to include this type of thinking into your investmentdecision-making  process,  we  will  have  completed  our  description  of  theportions of the tool kit explicitly designed to measure exposure. This putsus  in  a  position  to  introduce  and  incorporate  mechanisms  designed  todetermine  the  levels  of  exposure  that  are  most  appropriate  for  portfolio
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managers, specifically sizing them at levels high enough to achieve targetreturn objectives while, at the same time, ensuring that they minimize theprobability of exceeding critical, predefined thresholds of profit/loss (P/ L)stop-out. It is interesting that the process of setting appropriate exposureparameters,  which  is  the  essence  of  Chapter  5,  is  one  that  applies  manynow-recognizable  elements  of  the  statistical  tool  kit,  including  volatility,Sharpe Ratio, and a number of others.
With the statistical tools firmly in place, aimed at both the overall port-folio (Chapter 3) and its various subcomponents (Chapter 4) and appliedas a means of setting risk levels in a manner consistent with both objec-tives and constraints (Chapter 5), we will turn our attention to the criticaltopic  of  identifying  and  evaluating  the  trading  tools  available  to  actuallycontrol and adjust volatility (Chapter 6). Here, we will focus on the specificalternatives available to you should you want to either adjust your risk pro-file (upward or downward) or redistribute your exposures for other strate-gic  purposes.  Again,  the  mastery  of  these  concepts  may  be  all  you  everneed (and may perhaps be all you ever get) in terms of a risk managementedge.
In  the  meantime,  we  return  to  a  critical  question:  Which  statisticaltools are most applicable to the task of risk management at the individualposition level? Volatility and correlation (both old friends by now) cometo mind, as you can not only calculate them for individual securities butalso, as we will demonstrate, use them as a sizing technique for individualpositions. For instance, if you are holding a $50,000 position in an instru-ment for which the volatility is 50%, all things being equal, this is roughlyequivalent  to  maintaining  a  trade  of  $100,000  in  a  security  for  which  thevolatility is 25%. In a very meaningful (though imperfect) way, these posi-tions have similar risk characteristics; and it becomes a relatively simplemathematical exercise to view your positions not just in terms of numberof shares or contracts (bad), but in terms of dollar value invested (better)or units of expected volatility (better still). As we will discuss later in thebook, the primary sets of circumstances under which these relationshipsbreak  down  are  those  involving  large  position  sizes,  where  transactioncosts begin to increase due to liquidity concerns. These concerns certainlyarise with greater frequency in large institutional portfolios but should notdeter anyone from understanding the volatility-adjusted exposures tied tothe individual positions in their accounts.
You can generate similar incremental benefits by introducing the con-cept of correlation into the mix, which will give you some insight as to howmuch diversification positions on the same side of the market are generat-ing for your portfolio and how much direct exposure offset you are achiev-ing  with  positions  on  opposite  sides  of  the  market.  With  volatility  andcorrelation firmly established as measures of individual position exposure,
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it is also useful to measure the interplay between the two—a concept cap-tured perhaps most rigorously through Value at Risk (VaR) methodologies,developed  by  the  money  management  industry  over  the  past  couple  ofdecades to add precision to their own risk-estimation efforts. In this chap-ter, we will cover not only the benefits of VaR, but also its limitations—thelatter of which, in part, has caused the money management industry to aug-ment  these  methodologies  with  more-targeted  tools,  mostly  capturedunder the heading of scenario analysis.
For those wishing to delve still deeper into these methodologies, thereis an entire field of market evaluation, referred to commonly (if somewhatcryptically)  as  technical  analysis,  which  is  devoted  to  the  science  ofstudying  the  ramp-up/drawdown  characteristics  of  individual  securitiesover time. 
Following is a brief description of the potential application of statisticaltools  in  the  measurement  of  the  risk  characteristics  of  individual  marketpositions.
HISTORICAL VOLATILITY
I hope you have been practicing your (cid:1)s because you’re going to need themagain here as we reintroduce the concept of standard deviation, this time asit applies to individual securities. These statistics are widely available for vir-tually  every  security  on  systems  such  as  Bloomberg  and  Reuters  (in  fact,unless you are an expert in the securities in question, it might be a good ruleof thumb to avoid trading any market where the historical daily price changetime series is not readily available). You can also calculate these volatilitieson your own by simple entering the daily pricing history for the instrumentin question, computing a daily price change, and performing a standard devi-ation calculation using a program such as Microsoft Excel. If you choose themanual method, remember to annualize this statistic, typically by multiply-ing the daily figure by the square root of the number of trading days (250or so) in a year.
In my view, you’re best off expressing this statistic in percentage termsso  as  to  render  it  fully  comparable  to  statistics  for  other  securities.  Forexample,  if  you  simply  calculate  the  standard  deviation  from  the  priceitself, the underlying unit of account will be share price points. You mayfind that you have calculated an annualized standard deviation for a givensecurity of, say, 10 points. However, this figure has significantly differentimplications for a security that is trading at 20 than for one whose price is100. By using percentage terms, we would find a 50% volatility for the firstcase and a 10% volatility for the second and should by now understand that
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the former security is probably dramatically more risky than the latter. Inorder to calculate volatility in percentage terms, simply express the pricechange in percentages by dividing this figure by the current price level. Forexample, a daily price change from 50.0 to 49.0 can be viewed as a changeof either 1.0 points or 2.0%. Then, perform the standard deviation calcula-tion on the percentage series.
Once  you  have  calculated  the  percentage-based  volatility  for  a  givensecurity, you can convert this to dollars of risk by multiplying the percent-age term by the dollar size of your investment. The effect of this process isto express risk in volatility-adjusted terms. The good news here is that thisconcept  lends  itself  entirely  to  comparison  across  securities.  Thus,  onecan  think  of  a  $50,000  trade  in  a  security  with  a  10%  volatility  as  havingroughly  the  same  risk  characteristics  as  a  $25,000  investment  in  a  20%volatility security.
Because  different  financial  instruments  may  have  vastly  divergentvolatility characteristics, the implications of this concept on portfolio riskmanagement are hard to overestimate. Indeed, it is hardly a stretch to saythat they provide the backdrop for the development of risk managementtheory and practice in the institutional segment of the market (where it allbegan). Indeed, it was the realization that one could not measure risk with-out taking into account the single-variable and multivariate volatility char-acteristics of a securities portfolios that led to the development of the VaRmodels that drive much of the risk management focus of the professionaltrading universe and that we will discuss shortly. 
In the meantime, however, you can apply the concept in a much moresimplistic form by sizing your individual transactions in terms of the asso-ciated volatility you are willing to accept for each trade. For example, youmay wish to establish as a guideline that no individual transaction shouldhave an annualized volatility associated with it of more than, say, $1,000.Using this simple formula, you would know how much you would be will-ing to spend on a security that had a 10% annual volatility ($10,000), 25%volatility ($4,000), or even 100% volatility ($1,000) by simply carrying for-ward with the algebra. If you have a shorter-term trading orientation, youmay wish to express volatility in terms of one-day standard deviation—afeat that you can accomplish by dividing the annualized volatility statisticby the square root of the number of observations in a year. Specifically, ifwe assume 250 trading days in a year, the factor, which converts daily sta-tistics into annual ones, is the square root of 250, or 15.81.
As  a  final  point  regarding  historical  volatility,  it  is  important  to  notethat the results you get will vary with the length of the data set you use. Toelaborate,  it  is  entirely  possible  to  calculate  an  annualized  or  daily  stan-dard deviation of returns for a given security, using, say, 20, 50, 100, 1,000,or  indeed  any  number  of  daily  points  of  observation  (due  to  issues  of
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statistical significance, I do not recommend calculating a standard devia-tion statistic for any data set containing fewer than 20 independent obser-vations). However, the results you get for each individual series are likelyto differ materially. While this reality introduces an additional layer of com-plexity  to  your  analytical  efforts,  it  also  offers  an  opportunity  to  betterunderstand your exposures. For example, if the standard deviation statisticfor a security calculated over the past 20 days is significantly higher thanthe  associated  250-day  (i.e.,  one-year)  figure,  it  indicates  a  condition  ofaccelerating volatility in this financial instrument, which should be investi-gated further. In these cases, it bears asking why volatility has picked upand whether this condition is likely to persist indefinitely. For example, ifthe volatility spike can be traced to increased price action leading up to ascheduled event, such as an earnings announcement, one might choose toview the added exposure as a temporary phenomenon. Conversely, if thesecurity in question is more volatile over the past month or so because ofsome more embedded change, such as deregulation, then perhaps it makessense to err on the side of caution and use the higher number to calibrateyour exposure on individual transactions. In any event, it should come asno  surprise  that  I  think  that  you  should  look  at  volatility  across  differenttime increments, at minimum focusing on the 20-day (one-month), 60-day(one-quarter), and 250-day (one-year) series as a means of obtaining a mul-tidimensional view of the security’s price dynamics. 
OPTIONS IMPLIED VOLATILITY
The critical drawback of using historical data to measure the volatility of agiven security is that history is not always the best predictor of future pric-ing  behavior.  For  example,  a  security  with  relatively  benign  historicalvolatility  characteristics  may  become  extremely  active  in  advance  of  aproduct launch; and, all other things being equal, there would be nothingin the historical time series data, per se, to anticipate and account for this.Thus, while the historical time series volatility analysis has been a main-stay  of  our  statistical  tool  kit,  we  must  now  recognize  that  its  ability  tomodel  future  pricing  dispersion  is  only  as  effective  as  past  volatilitypatterns  are  in  predicting  associated  dynamics  in  the  future.  However,there  are  relatively  simple  methodologies  available  to  overcome  theselimitations that offer insights into future volatility patterns that transcendwhat can be inferred directly by historical volatility analysis.
One such methodology derives from the concept of implied volatilityembedded in the price of every option. Implied volatility is a measure of theamount of price dispersion expected in a given security on a prospective
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basis that is implied in the price of a given option. If you are already famil-iar with the rudiments of options pricing theory, you will understand thisconcept; if not, it merits a quick review of its components and associatedapplications.
An  option  is  typically  defined  in  terms  of  its  underlying  instrument,strike price, expiration date, and buy/sell orientation (i.e., call options arethose  with  a  buy  orientation;  put  options  bestow  selling  prerogatives  ontheir  holders).  These  are  the  instrument’s  static  parameters;  and  oncethey  are  established,  its  value  is  driven  by  two  dynamic  factors:  (1)  theprice of the underlying instrument and (2) the expected volatility of thisinstrument. Strictly speaking, there is a third dynamic factor that impactsoptions pricing: the applicable interest rate, which establishes the oppor-tunity  cost  of  holding  the  amount  of  your  premium  in  option  value,  asopposed  to  investing  it  in  interest-bearing  instruments.  However,  unlessthe option in question is (1) deep-in-the-money, (2) has a long time to gobefore  expiration,  or  (3)  some  combination  of  the  two,  interest  rate  willnot be a driving force in valuation; and we can ignore it for the purposes ofthis discussion. 
To illustrate the interaction between underlying price and volatility inoptions  pricing,  consider  the  following  extreme  case:  No  rational  eco-nomic  entity  would  purchase  a  call  option  at  a  strike  above  the  currentunderlying price, or a put option below this price, if the price for the under-lying instrument was fixed and, thus, the expected volatility in the instru-ment was zero. This is true, of course, because it implies a 0% chance ofthe option finishing with some value at the time of expiration. Conversely,options for a security with an infinite volatility over a given period wouldtheoretically be worth an infinite amount, as there is a theoretical oppor-tunity to liquidate them at infinite intrinsic value. 
The  implied  volatility  pricing  for  all  options  with  economic  meaningresides somewhere in between these extremes, and it is not an exaggera-tion to say that a preferred method of expressing the price of an option—particularly for sophisticated users of options products—is in terms of itsvolatility.  Options  traders  will  often  speak  in  terms  of  purchasing/sellingthis or that option at, say, 10%, meaning that this is the implied volatility atwhich they were able to execute an options transaction. To elaborate fur-ther  on  the  associated  pricing  implications,  the  execution  of  an  optionstrade  at  a  10%  volatility  implies  that  the  parties  in  question  expect  theunderlying  security  to  exhibit  an  annualized  standard  deviation  of  10%over the life of the option.
How  does  all  of  this  pertain  to  the  applicability  of  options  impliedvolatility  to  our  efforts  to  estimate  portfolio  risk?  As  it  turns  out,because implied volatility is quoted in the exact manner that historicalvolatility  is  expressed  (i.e.,  in  annualized  terms),  for  the  purposes  of
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estimating  price  dispersion,  the  terms  can  be  used  interchangeably.  Infact, implied volatility can be viewed very explicitly as the options mar-ket’s attempt to predict what historical volatility will be in the future. Wecan  therefore  substitute  options  volatilities  for  historicals  in  order  tosize our associated exposure. Perhaps surprisingly, we can do so with-out ever trading a single option. For example, just as is the case withhistorical  volatility,  if  we  take  a  $50,000  bet  in  a  given  security  with  a10% implied volatility, we would expect the position in question to fluc-tuate at an annualized, one-standard-deviation rate of $5,000. Again, wedon’t  have  to  trade  options  to  arrive  at  this  estimate  but  can  simplyderive data from the options markets and apply it against our position incash securities.
Options  implied  volatility  thus  has  the  advantage  over  historicalvolatility of encompassing not only historical time series information butalso “qualitative” data and prospective economic inputs into its estimatesof price dispersion. The measure gains further credibility through the factthat  options  traders  are  actually  risking  financial  capital  on  the  basis  ofimplied volatility valuations. Take my word for it—this type of reality dosedoes wonders for the accuracy of financial models. From these perspec-tives,  implied  volatility  arguably  is  the  superior  measure.  However,  likeeverything  else  in  our  statistical  tool  kit,  it  has  shortcomings.  First,because the implied volatility statistic is derived entirely from the mannerin  which  options  are  priced,  it  is  subject  to  the  same  idiosyncrasies  thatcharacterize  the  options  markets  themselves.  Particularly  during  volatileintervals and periods of diminished liquidity, options pricing may deviatefrom what would be expected from the core economic characteristics ofthe security in question. Moreover, because volatility can be viewed as oneform of expression of an options price, these anomalies will have a directimpact on the accuracy of the statistic. 
For  example,  during  periods  of  extreme  market  duress,  prices  andvolatilities for all types of options are often bid up to levels beyond whatwould be justified by the underlying economic data; and the use of impliedvolatility  as  an  exposure  input  might  lead  to  inaccuracies  in  theseinstances. In addition, there is a well-known concept within the universeof  option  trading,  referred  to  as  the  volatility  skew,  or smile,  thatdescribes  the  tendency  for  out-of-the-money  options  to  trade  at  higherimplied volatilities than those that are at, near, or in the money. It is there-fore necessary to understand the relationship between the strike price andthe  underlying  market  price  (i.e.,  the  “moneyness”)  for  the  option  onwhose  implied  volatility  you  are  relying.  As  a  practical  matter,  I  recom-mend the volatility for the at-the-money strike as the best approximationof future price dispersion, as this is the part of the options market that isleast impacted by market inefficiencies.
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Second,  options  pricing  is  also  significantly  impacted  by  the  widelyranging liquidity characteristics of the market, and this too can interferewith the applicability of implieds as an indicator of exposure for underly-ing instruments. The “liquidity factor” is particularly acute either for smalltransaction  sizes,  which  do  not  generate  enough  competitive  biddingamong  market  makers  to  fully  reflect  current  market  conditions,  or  forvery large orders, where the current volume cannot efficiently accommo-date  the  transaction  size.  In  these  cases,  options  pricing  patterns  candiverge  materially  from  those  that  would  prevail  under  ideal  liquidityconditions (i.e., those resulting from transaction sizes large enough to cap-ture the attention of the market, while small enough to operate within itsliquidity constraints). Wherever and whenever this occurs, it will lead toimplied volatility outcomes that differ, to varying degrees, from what canbe best described as the rational equilibrium. In addition, options for thesame underlier may have different volatility magnitudes for different expi-ration dates, particularly if, for instance, one option expires before a keyinformation  event  (e.g.,  corporate  earnings),  and  the  other  expires  afterthis occasion.
For these reasons, while implied volatility offers a unique and crit-ical  insight  into  the  likely  price  dispersion  characteristics  of  a  givenfinancial instrument, it must be viewed, as with other elements of ourtool kit, with a critical eye and a healthy degree of skepticism (which,like  Chastity,  to  paraphrase  that  righteous  man  of  letters  GeorgeSantayana,  should  not  be  relinquished  too  readily).  I  believe  that  itmakes sense to utilize both the implied volatility statistic and historicalvolatility  figures,  with  both  statistics  calculated  over  multiple  timespans. Line these up side by side, and examine the extent to which theydeviate from one another. See if you can theorize why one figure mightbe  significantly  higher  than  the  others.  Is  there  pending  news  (whichmight  push  the  implieds  to  levels  well  above  the  associated  histori-cals)?  Or  are  we  in  a  period  of  anticipated  calm  after  a  particularlyvolatile interval (which might have precisely the opposite effect)? Onceyou have determined what is driving the disparity across volatility esti-mates,  you  can  use  these  inputs  to  compliment  one  another  in  yourevaluation  of  position  exposures  (they  may  also  help  you  sharpen any  hypothesis  you  may  be  forming  about  what’s  happening  in  themarkets).
One  final  note:  You  can  find  extensive  implied  volatility  data  setsthrough  news  services  such  as  Reuters  and  Bloomberg  or  in  periodicalssuch  as  Barron’s.  You  can  also  calculate  them  (if  you  have  the  option’sprice  and  static  characteristics)  either  with  simple  options  pricing  pack-ages available on the Internet or with financial calculators with basic levelsof options functionality. 
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