your own proprietary capital, the risk managementcall will ultimately belong to someone else. However, this does not implythat the exercise of determining appropriate exposure ranges is one thatyou can avoid, as I believe that it is an important one for virtually all port-folio managers—whether you are self-funded or otherwise.
This is true because the guidelines that your funding entity is likely toprovide will offer little in the way of details beyond setting an upper boundof  acceptable  portfolio  exposure;  and  these  figures  may  or  may  not  beentirely  rational  from  a  risk  management  perspective.  Moreover,  whileyour  management  will  almost  certainly  be  excessively  vigilant  (if  notentirely sensible) in establishing and enforcing risk limits at the upper endof the range, they are likely to provide you with very little guidance as towhat an acceptable lower bound for your risk ought to be. This concept ismore important than most traders realize, as they are often conditioned tobelieve that less risk is always better than more risk. However, this is oftenfar from the case. And I wish I had a dime for every trader I know who wasprofitable on a unit basis but simply did not take on enough risk to gener-ate  a  revenue  stream  large  enough  to  cover  the  fixed  costs  of  his  or  heraccounts or to meet minimally acceptable rates of return on invested cap-ital.  Sadly,  even  tragically,  the  talents  of  these  traders  go  fallow,  simplybecause they never feel comfortable enough to take the risks necessary toachieve the absolute revenues that are fully within their power to achieve.The same dynamic may also apply if you are self-funded; I have encoun-tered many individual traders who make money consistently but at the endof the day aren’t properly compensated for the time and effort they put intotrading activities because they simply aren’t taking enough risk.
For all these reasons, I strongly recommend that you take the troubleto  establish  and  routinely  review  exposure  ranges  that  fit  your  circum-stances in terms of both minimally acceptable risk levels and the more intu-itive upper bounds. This section will offer guideline on setting these ranges.In order to arrive at the appropriate figures, let’s draw from that rel-atively small but often untapped resource—common sense. Clearly, we
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want to set our exposure ranges such that when we operate on the lowerend of the spectrum, they still provide us with a sportsman’s chance toreach our profitability objectives, while, when operating at the other endof the range, they offer an acceptably low probability of hitting what wehave defined as our largest tolerable economic loss (whether stipulatedby  an  external  funding  entity  or  self-imposed).  In  other  words,  wealways want to take enough risk to enable us to achieve our goals whilenot  extending  beyond  a  point  where  the  likelihood  of  exhausting  (or,heaven forbid, exceeding) our risk-taking resources becomes a danger-ous possibility.
While there is no question that setting these parameters is a subjectiveprocess,  limited  in  its  accuracy  by  our  universal  inability  to  predict  thefuture, our statistical tool kit provides us with the basis for setting theseranges in a manner that should positively impact our ability to achieve ourdual  objective  of  reaching  our  goals  without  assuming  undue  exposure.Specifically,  the  model  features  a  dual  methodology  that  draws  on  suchconcepts as Portfolio Volatility, Sharpe Ratio, Targeted Return, and Maxi-mum Acceptable Loss in order to identify the exposure level sufficient toreach our targets while minimizing both the probability and the magnitudeof loss-threshold violations.
Method 1: Inverted Sharpe Ratio
We’ll start the process by focusing on the less intuitive task of developinga methodology for setting a sensible lower bound for portfolio exposure. Ifwe accept the critical notion that risk and return are interrelated, then weought to be able to express one in units of the other. Indeed, as we havealready covered, some of the most important work done in the era of finan-cial engineering is the expression of return in units of risk. Most famously,and most relevantly for our purposes, these concepts are embodied in theSharpe Ratio calculation, which you may recall is a ratio of returns to theirassociated volatility.
If we can estimate our Sharpe Ratio with relative accuracy, we have apretty good notion of the kind of return we can expect for a given level ofperformance volatility. But what if we are looking to ask the opposite ques-tion? What if instead of wondering what return we can expect, given theriskiness of our portfolio, we want to figure out how much exposure weneed in order to generate the level of performance we desire? As it turnsout, with a quick sleight of algebraic hand, we can manipulate the SharpeRatio equation to determine both what a given level of volatility will pro-duce in terms of returns and (more important for our purposes here) whatlevel of volatility is consistent with a given performance target. In order tounderstand this critical concept as thoroughly as its importance demands,
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let us take the steps involved one at a time. Begin by resurrecting the basicSharpe Ratio equation: 
Sharpe Ratio (cid:3) (Return (cid:4) Risk-Free Rate)(cid:1)Portfolio Volatility
The Sharpe Ratio can be thought of as a scorecard of your perform-ance as a portfolio manager from a risk-adjusted-return perspective. It alsois a measure of the amount of return you are likely to generate for a givendollar of risk, which, as argued earlier, is a concept that is expressed mostsuccinctly in the portfolio volatility statistic. Thus, once you have carefullyanalyzed your Sharpe Ratio and are comfortable that the number you havecalculated  is  one  that  you  can  sustain,  you  can  invert  this  equation  todetermine what level of return you are likely to generate for a given levelof risk assumption. Note that for the purposes of this discussion, we willdesignate  this  figure  to  be  the  “Sustainable  Sharpe.”  It  differs  from  the“actual” Sharpe Ratio derived for selected time sequences by containing aqualitative overlay that bases the figure on a conservative assessment ofwhat you are likely to achieve in the future, given the dynamic nature ofmarket conditions. The “actual” Sharpe Ratio is based solely on your trad-ing history. Our “Sustainable Sharpe” is one that uses the historical Sharpe(and perhaps other inputs) to derive a lower bound for what we can con-fidently achieve as a Sharpe Ratio on a going forward basis. The results of this algebraic maneuvering are as follows:
1. (Return (cid:4) Risk-Free Rate) (cid:3) Sustainable Sharpe (cid:11) Portfolio Volatility2. Portfolio Volatility (cid:3) (Return (cid:4) Risk-Free Rate)/Sustainable Sharpe
These equations define a concept that I will refer to as the Inverted SharpeMethodology for setting exposure levels. The first equation is designed toprovide you with some idea of the amount of return you can expect givenyour  Sustainable  Sharpe  and  your  current  level  of  volatility.  The  secondwill tell you what kind of portfolio volatility you should target in order toyield a specific return (again given a Sustainable Sharpe). 
Both equations can be extremely useful in the determination of appro-priate target ranges of exposure. To illustrate, consider a portfolio with thefollowing characteristics:
Target Return:Risk-Free Rate:Return (cid:4) Risk-Free Rate:Sustainable Sharpe:
25%5%20%2.0
We  take  the  target  return,  the  risk-free  rate,  and  the  SustainableSharpe as constants in this analysis, meaning in the case of the Sustainable
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Sharpe that the account in question believes it can generate more than $2 inreturn for $1 of exposure it takes. The variable in these equations is portfoliovolatility, which, as you will recall, is our proxy for risk. Now let’s plugvarious volatility scenarios into the inverted equations in order to deter-mine  what  they  say  about  performance.  In  order  to  do  so,  however,  wemust take a careful look at our options for characterizing volatility. In theSharpe Ratio calculation, volatility is represented as the standard devia-tion of portfolio returns. So, whatever statistic you are currently employ-ing as your benchmark exposure measurement, it must be mapped backinto  standard  deviation  units  in  order  to  produce  meaningful  outcomes.Perhaps the best alternative at your disposal in this regard is the resultsof a Value at Risk (VaR) calculation, which have the advantage of beingbased  on  current  portfolio  characteristics.  If  you  have  access  to  a  VaRcalculation, it is therefore possible to substitute this figure into the denomi-nator of the Sharpe Ratio, as long as you take care to scale down the confi-dence interval statistic to the one standard deviation level (for example, ifyou are using a 95th percentile VaR, you can map it back into a one standarddeviation figure by dividing by 1.96). If you choose not to use a VaR approach,the  best  alternative  is  simply  to  calculate  a  one  standard  deviation  P/Lvolatility.
Returning to our example, let us first assume that by our best estimatethe portfolio is managed such that its projected, annualized volatility is 7%.Inserting this figure, along with the Sustainable Sharpe and Risk-Free Rateparameters into equation 1, and solving for Return, we find that this port-folio is likely to produce returns of approximately 19%—respectable, butsignificantly below our target return of 25%.
This analysis begs the question of what portfolio volatility is consistentwith a 25% return target, again assuming the portfolio in question is able tosustain  a  Sharpe  Ratio  of  2.0.  We  can  arrive  at  this  answer  quite  simplyusing  the  second  equation  by  inputting  Return,  Risk-Free  Rate,  and  Sus-tainable Sharpe, and solving for Portfolio Volatility. Carrying through thesesteps with respect to the sample portfolio profile generates a result of 10%,implying that a portfolio with a Sustainable Sharpe of 2.0 should generateabout 10% annualized volatility in order to produce returns of 25%.
In considering these concepts, it is important to understand that thealgebraic manipulations of the Sharpe Ratio that are embodied in the twoequations are nothing more than rough approximations of the amount ofvolatility that is consistent with a given target return, based on historicalrisk-adjusted  performance.  The  algebraic  results  will  never  be  entirelyaccurate because, in the first place, the figure you select as your Sustain-able Sharpe Ratio will almost certainly deviate from both your historicalperformance and from your best guess as to what actual Sharpe you canproduce in the future.
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Moreover,  I  must  caution  against  too  literal  an  interpretation  of  anyresults  you  produce  in  this  type  of  analysis,  which,  among  other  thingsmight indicate that those who have lower Sharpes should assume higherrisk profiles in order to achieve specified return thresholds (instead, theresponsible  solution  involves  targeting  a  lower  return  profile).  However,when viewed from the proper perspective, the Inverted Sharpe Equationsoffer useful insights into the appropriate sizing of your risk profile. This isparticularly  true  in  cases  where  (as  often  occurs  in  my  experience)  thevolatility characteristics and target returns are entirely inconsistent withone another.
For  example,  on  the  one  hand,  the  sample  portfolio  in  our  examplemay target 10% annualized volatility and still come nowhere near the 25%return target. On the other hand, we can state with a great deal of confi-dence that if this account produces volatility in the low single digits, say,3%, it has very little chance of ever hitting the 25% objective (indeed, it canonly  do  so  if  the  actual  Sharpe  Ratio  is  more  than  triple  the  estimatedSustainable Sharpe). By the same token, if this account trades to an annu-alized volatility profile of, say, 30%, the projected returns would be 75%—afigure so far in excess of the targeted objective as to raise questions abouteither the appropriateness of the target or the level of risk assumption, orsome combination of the two. The point here is that by combining infor-mation  about  your  Sharpe  Ratio  with  expectations  of  return,  you  beginto create  a  picture  of  what  type  of  risk  levels  are  appropriate  to  theseobjectives;  volatility  levels  significantly  below  these  ranges  will  almostguarantee  a  failure  to  achieve  associated  targets,  whereas  figures  vastlyabove  these  thresholds  give  rise  to  potential  inconsistencies  betweenreturn targets and risk tolerances.
Method 2: Managing Volatility as a Percentageof Trading Capital 
As  you  may  have  surmised,  the  Inverted  Sharpe  Ratio  method  is  bestadapted  to  determining  the  lower  bound  of  exposure  that  is  consistentwith  reaching  your  targeted  objectives.  The  main  insight  that  it  will  pro-vide you in determining the appropriate associated upper bound is that ifyour volatility is too high, it will project out a return that far exceeds yourobjectives.  While  this  information  may  be  useful,  it  doesn’t  address  thebinding constraint to risk assumption, which, as we all know by now, is thelimited availability of risk capital. Clearly, in determining an upper boundof acceptable exposure, we would want to be motivated by a methodologythat both minimizes the probability of our reaching our largest allowableloss and ensures that if, in fact, we do experience hard times, we don’t endup with a number that considerably exceeds this figure.
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The  challenge,  therefore,  is  to  identify  a  threshold  of  exposure  thatexpends, without exhausting, the full measure of risk capital allocated tothe account. Reviewing this concept intuitively, the brain (well, at least mybrain) moves fluidly to the following premise: In a rational portfolio riskmanagement  program,  maximum  exposure  threshold  should  be  tied  toP/L performance over routine and identifiable cycles (e.g., one year). As amatter of good business judgment, I’m sure that you’ll agree that your risk-taking prerogatives are much broader when you have accumulated somebankable  P/L  to  work  with  than  they  are  when  you’re  down  money  andstruggling to trade on.
The  logical  extension  of  this  argument  is  that  your  maximum  risktolerance  is  fungible  against  your  P/L,  and  the  process  of  establishingan upper  bound  for  your  exposure  begins  with  quantifying  this  interplay.Our methodological approach begins by introducing the concept of tradingcapital, which  is  defined  simply  as  the  sum  of  your  maximum  loss  toler-ance and your period-to-date P/L:
Trading Capital (cid:3) Risk Capital (cid:7) P/L
where Risk Capital (cid:3) largest allowable loss for the period
This simple equation defines the current amount of “losable” dollars asso-ciated with a given account until it hits the previously defined maximumloss threshold. For example, if an institutional trader has been given theright to trade until he or she has lost $5 million, then the trading capital onthe first day of the year is $5 million. If the account subsequently makes $2million, trading capital increases to $7 million. Similarly, if this account loses$2 million, trading capital is diminished to $3 million.
With  the  concept  of  trading  capital  established,  we  can  apply  it  as  ameasure of risk-taking capacity. As the figure rises, we have more roomfor  exposure  and  can  expand  our  risk  taking  accordingly:  by  taking  onlarger  position  sizes  or  by  deploying  larger  amounts  of  capital  into  newideas, or some combination thereof. By the same token, as trading capitalis  diminished,  so  too  is  our  ability  to  bear  the  negative  consequences  ofrisk taking, and our upper bound for market risk assumption must contractaccordingly.
The next step is to formalize this into an equation that expresses risk-taking capacity as a function of trading capital. While some may argue forcomplex formulas that may (or may not) bring some added precision, it ismy  experience  that  you  can  operate  quite  effectively  by  simply  fixingupper exposure bands as a fixed percentage of trading capital. Once youhave framed the issue in this manner, the exercise of establishing a maxi-mum is reduced to one of selecting the appropriate percentage threshold.Here,  again,  there  is  room  for  divergent  opinion;  but  based  on  years  of
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experience  in  monitoring  traders  in  drawdown,  I  think  that  this  numbershould  not  exceed  10%.  In  other  words,  as  a  rule,  maximum  volatility(expressed  in  one-day,  one-standard-deviation  terms)  should  not  exceed1/10 of trading current capital. In mathematical terms:
(cid:6)
P/L
(cid:9) Trading Capital (cid:11) 10%
where (cid:6)
P/L
(cid:3) VaR  calculation  at  one-standard-deviation  confidence
level, or 20 standard deviations of P/L.
Again, if you have access to a VaR program and can convert the outputto a one-standard-deviation confidence interval, it represents probably thebest proxy for (cid:6)P/L, as it is tied directly to the risks that currently charac-terize your portfolio. However, the 20-day average volatility is a very goodproxy of the kinds of risk that you are currently taking. In either case, here’smy rationale behind establishing 10% as an appropriate upper bound:
If (cid:6)
P/L reaches 10% of the amount of capital you have left to lose at anygiven time, then we can expect with relative frequency (one observationout of six, or better, if we’ve figured it right) a loss that meets or exceeds10% of your remaining risk-taking capacity. While a loss of this size wouldclearly be acceptable from time to time, we must be mindful of the fact thataccording  to  our  standard  deviation  theory,  market  moves  of  two,  three,and even four standard deviations occur routinely throughout the tradingcycle.  Indeed,  most  market  mavens  agree  that  these  types  of  outlyingobservations happen with more frequency than is indicated by the proba-bilities  typically  associated  with  a  normal  distribution.  As  we  have  dis-cussed  briefly,  this  phenomenon  is  commonly  referred  to  as  kurtosis,  or“fat tails” for the more pedestrian among us. Like it or not, fat tails are areality  in  the  markets.  And  I  for  one  am  glad  for  their  presence,  for  theworld would certainly lose some if its charm without them.
When  a  fat-tailed  event,  say,  to  the  tune  of  three  or  four  standarddeviations does occur in the markets, if our volatility assumptions hold, aportfolio that has volatility of 10% of its trading capital may be exposed tolosses  that  are  between  one-quarter  and  one-half  of  its  remaining  riskcapacity. Needless to say, this is a potentially very serious erosion of riskcapital.  Moreover,  as  you  have  no  doubt  experienced,  these  types  ofextreme market movements tend to happen on multiple days in succession(or at least in very close proximity to one another), meaning that marketsthat have three- or four-standard-deviation moves in a single day can oftenexperience much larger fluctuations before the dust settles and the marketstabilizes.  Thus,  at  volatility  levels  much  above  10%  of  trading  capital,traders can literally find themselves one adverse market event away fromhaving blown through the lion’s share of their reserve. It is for this reasonthat I view 10% as an appropriate upper bound.
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Of course, portfolio construction plays a major role in the percentageselection process. For portfolios operating in liquid instruments for whichthe P/L tends to be in proportion to moves in underlying markets, there isarguably  some  breathing  room  here.  By  contrast,  liquidity-constrainedportfolios, and/or those where the P/L impact may be orders of magnitudeof the price movement in underlying markets (e.g., portfolios with a greatdeal  of  optionality),  it  may  be  necessary  to  be  even  more  conservative.However, while we may quibble with the applicable percentages, I see nocompelling  arguments  against  the  notion  that  trader  volatility  should  bescaled to trading capital, and I encourage you to adopt this approach in asthorough and as timely a manner as you can.
There are a couple of very important practical implications to tradingunder  a  discipline  that  ties  P/L  volatility  to  trading  capital.  First,  on  theupside,  it  implies  that  as  your  profits  build,  so  does  your  risk-takingcapacity. Among other alternatives, you can use these additional resourcesin ways that may spur your development as a trader—by trading in largersizes;  by  experimenting  with  new  asset  classes,  sectors,  or  instrumenttypes; by modifying your long/short profile; and so on. Ultimately, by mak-ing a long-term commitment to the continued expansion of your portfoliomanagement horizons, you stand the best chance of maximizing your suc-cess over time. However, in order to do so, you will routinely be compelledto operate in ways that will deviate from your normal trading habits; and byfar  the  best  time  to  do  this  is  when  your  trading  capital  has  increasedthrough profit accumulation. The extra cushion this provides offers you theopportunity  to  assume  new  kinds  of  portfolio  profiles  at  magnitudes  ofexposure that will not eat significantly into your risk capital.
Let’s say you start out the year with risk capital of $1 million. Underour  trading  capital  approach,  we  would  not  recommend  that  your  dailyvolatility, as expressed in units of standard deviation, extend much beyond$100,000. Now let’s assume that you make an additional $1 million duringthe  first  quarter  of  the  year.  Your  trading  capital  has  now  expanded  to$2 million, and your maximum recommended exposure has similarly dou-bled,  to  $200,000.  The  additional  risk  capacity  can  then  be  deployedentirely  or  in  part  in  the  development  of  complementary  trading  disci-plines  that  will  enhance  your  arsenal.  This  is  the  ideal  situation  underwhich to either try new strategy variants or increase your risk by scalingup your trading sizes. 
In either case, it is important to undertake these developmental effortsat limited levels of exposure. If you do try some new, different, or scaled-up activities, determine in advance the amount of risk capital you wish toallocate  to  this  exercise.  Perhaps,  carrying  forward  with  the  precedingexample,  the  appropriate  figure  is  $500,000,  or  half  of  your  accumulatedprofits  for  the  period  in  question.  It  is  very  important  to  adhere  to  this
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discipline  and  to  terminate  the  developmental  activities  if  you  subse-quently lose an amount equal to your specified threshold. The very processof stretching your portfolio activities at a point when you have reached abenchmark  level  of  profitability,  investing  a  portion  of  these  profits  intothe further development of your portfolio management skills, and then ter-minating this activity, if necessary, at specified levels of loss can be enor-mously empowering. Arguably, you’ll have renewed opportunities for suchexperimentation in the future (such as when you have rebuilt your P/L tolevels where the risk is justified); and in the meantime, you have demon-strated the ability to manage the associated outcomes of broadening yourtrading horizons to acceptable levels of loss. 
Now let’s examine what happens when your trading capital contracts.At volatility levels below the introductory threshold of 10% of your tradingcapital, the risk of a single event or series of events pushing you toward theprecipice is minimal. However, as we have said, this is not the case whenyou  have  generated  a  significant  negative  P/L  for  the  period  in  question.Under these circumstances, the percentage-based impact of material neg-ative P/L observations on your loss reserve increases exponentially, and itbecomes imperative for you to scale down your risks accordingly. 
To illustrate, let’s return to our earlier example of a portfolio that hasbeen allocated $1 million of risk capital/potential losses. Let’s assume thatafter  the  first  quarter,  this  portfolio  suffers  losses  of  $500,000,  in  theprocess reducing trading capital by 50% to $500,000. If you continue to gen-erate (cid:6)P/L of  $100,000  (as  would  be  consistent  with  our  model  for  fullinvestment at a cumulative P/L of $0 for the period), you have increasedthe probability of meeting or exceeding your stop-out level by an order ofmagnitude as (due once again to the fat-tailed nature of the markets) a sin-gle 5(cid:2) market event; or consecutive price movements on the order of halfthis  size  are  much  more  commonplace  than  most  investors  are  aware.Moreover, because these types of market swings can happen abruptly andwithout  warning,  a  failure  of  this  portfolio  to  scale  back  its  exposure  toreflect its reduced ability to absorb losses creates a material positive prob-ability that the account not only reaches its $1 million loss threshold butexceeds this level by a significant amount. 
Note that this is particularly true for portfolios that operate in marketscharacterized  by  liquidity  constraints  because  in  these  cases  the  veryefforts  undertaken  to  reduce  exposure  may  exacerbate  volatility—in  theprocess causing negative price action that can push valuations to levels notjustified by the underlying economics of the situation—simply because themarket is aware of the portfolio manager’s acute need (as well as those ofperhaps others in a similar situation) to liquidate his or her position. Notthat it’s overly relevant to this discussion, but in these cases you may findthat in addition to being compelled to confront the consequences of havinghit your economic stop-out, you may be subject to the further indignity of
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watching positions you just liquidated under duress bounce back to neu-tral  or  even  positive  P/L  levels  once  the  market  understands  that  it  hassqueezed out all those who are operating near their financial constraints.
IF A TREE FALLS IN THE FOREST . . .
The late 1990s offer a treasure trove of case studies of what to do and what notto do when managing liquidity-constrained portfolios under conditions of draw-down. Many of these lessons emanate from the hedge fund community (wheremorality tales are sometimes more plentiful than investor returns), and I ask youto consider the sad ending of a hypothetical hedge fund, once pre-eminent butnow  defunct,  whose  name  I  can’t  state  specifically  (even  hypothetical  hedgefunds are notoriously publicity-shy) but I can tell you that it derives from one ofthe sturdiest and tallest trees in the forest, the sequoia dendron gigantum, orredwood tree. For nearly 2 decades, this mighty colossus towered over its com-petition, in terms of both