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you caught these similarities, you should congratulate yourself. You wereindeed paying attention and have latched on successfully to at least someof the critical themes I have tried to impart. Moreover, I believe that any-one who is using either optimal f or risk of ruin mathematics in his or herown portfolio decision making is on the right track. However, as I hope tomake clear in this Appendix, they are at least in some ways less applicableto the holistic task of portfolio risk management than the analogous mech-anisms we have covered in this book. Let’s take a brief look at each con-cept  so  that  we  understand  its  application  to  portfolio  risk  managementand  how  it  works  vis  à  vis  the  methodologies  I  have  recommended  toaccomplish the same tasks.
OPTIMAL f
In  his  book  Portfolio  Management  Formulas:  Mathematical  TradingMethods  for  Futures,  Options,  and  Stock  Markets (John  Wiley,  1989),Ralph Vince laid out a formula that computes the optimal transaction sizefor  risk  takers  given  specific  information  regarding  the  likely  range  ofreturn outcomes and associated probabilities. I’ve done my best to avoidan excessive use of equations throughout the body of this text, and I don’tmean  to  break  my  streak  now;  so  exclude  the  exact  formula  from  myanalysis. Suffice to say that Vince sets fopt (the optimal transaction size) asa function of the ratio of a given transaction’s expected return to its asso-ciated likely worst-case outcome.
That  the  optimal  f  concept,  which  added  a  useful  element  of  mathe-matical precision to the sizing of transactions, is a worthy advancement inthe field of portfolio science is a matter of very little dispute. However, themethodology, as Vince himself points out, is rife with shortcomings. In thefirst instance, the concept assumes that we understand more than is the lotof mere mortals to know about the return distribution of our transactions.Specifically, in order to calculate optimal f, it is necessary to input actualreturn information into the equation; and by now, you should understandthat if you actually have this data at hand, there’s not much point in both-ering about concepts such as risk management. Instead, I recommend thatyou simply plug this information into your fopt machine and let it work itsoptimization magic.
Again, Vince recognized this paradox, stating it succinctly in the follow-ing  few  sentences  taken  directly  from  a  later  work,  The  Mathematics  ofMoney Management (John Wiley, 1992): “In other words, it doesn’t matterhow profitable your trading system is on a one-contract basis, so long as itis profitable, even if marginally so. If you have a system that makes $10 per
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contract per trade . . . , you can use money management to make it far moreprofitable than a system that shows a $1,000 average trade. . . . What matters,then, is not how profitable your system has been, but rather how certain isit that the system will show at least a marginal profit in the future. Therefore,the most important preparation a trader can do is to make as certain as pos-sible that he has a positive mathematical expectation in the future.”
I completely concur with these observations, which, in fact, point usdirectly to the second basic problem with a strict reliance on optimal f asour exclusive means of sizing exposures. Of course, our universal inabilityto predict the future, that ubiquitous stumbling block on the road towardportfolio management nirvana, trips us up yet again. For anyone who does,in fact, have reasonably accurate estimates of distribution of future trans-actions returns, optimal f will indeed point you toward return maximiza-tion.  This  is  one  reason  why  “clinical”  risk-taking  environments,  such  asthose  associated  with  coin  flipping  and  the  purchase  of  lottery  tickets,offer the most elegant examples of its application.  However, in our imper-fectly constructed trading universe, the mean return to individual transac-tions  is  a  great  imponderable.  Of  course,  the  most  rational  source  ofpredictive information regarding future returns is probably historical data;and  if  you  want  to  be  an  optimal  f-er,  it  is  my  recommendation  that  youstart there.
The other recognized problem with the optimal f calculation is that itviews the world from the perspective of a single transaction, exclusivelycontemplating  the  issue  of  appropriate  sizing  of  individual  positions toachieve the objective of maximum transactions-level profitability. The fol-lowing problem then emerges: What if, even if I have accurately estimatedthe  range  of  transactions-level  returns,  I  hit  a  bad  streak  and  hit  orapproach my worst-case scenario on contemporaneous transactions? This(again  the  careful  reader  of  the  main  body  of  the  text  will  remember)  isvery close to the concept of drawdown, which, if not managed carefully,may exhaust risk capital before the optimal f machine can even begin toconfer its advertised benefit. Under a strict adherence to the methodology,as  Vince  himself  recognized,  there  is  virtually  no  limit  to  the  size  of  thedrawdown  that  a  portfolio  can  experience  if  it  applies  the  optimal  fmethodology  without  taking  into  account  the  possibility  that  a  string  ofconsecutive  losses  (or,  indeed,  even  losses  in  close  proximity  to  oneanother) can exhaust even the largest reserve of risk capital.
Again,  optimal  f-ers,  including  Vince  himself,  understood  the  limita-tions of the approach and have offered elegant remedies to address thesecritical  shortcomings.  Most  notably,  they  have  created  a  concept  calledsecure f, which utilizes the essence of the optimal f calculation but which(1) relies on historical return information to determine return sequencesand (2) features a very useful maximum drawdown constraint as an input
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to the calculation. These are undoubted improvements, which render themethodology, already a useful metric if applied effectively, that much moreapplicable.
I should point out, before explaining why I prefer the Inverted Sharpemethodology described in Chapter 5, that it suffers from some of the samelimitations as optimal f. Most notably, it requires the user to provide someinputs as to what future returns might look like (through the SustainableSharpe  concept)  that  are,  by  definition,  somewhat  subjective.  Moreover,there is nothing inherent in the calculation, when measured in its “static”form,  to  preclude  single-minded  individuals  from  burning  through  all  oftheir risk capital, and (perhaps) then some. With all of this in mind, here’swhere I feel the advantages to the Inverted Sharpe methodology lie:
• It utilizes a “portfolio” approach, as opposed to a transactions-levelmodel, to size exposures. While optimal f will give you some notionof transaction sizing and its attendant impacts on performance, whathappens at the individual transactions level is, in my view, entirely lessimportant than what occurs with respect to the overall portfolio. Veryfew market participants use a methodology under which their fortunesare  tied  either  to  a  single  transactions  or  to  series  of  transactions,which  take  place  sequence.  This  is  the  implied  setting  for  optimal  fresearch and is another reason why much of the analysis appears to bemore applicable to games of chance like coin flips or dice rolls. Port-folio  management,  by  contrast,  involves  maintaining  an  inventory  offinancial instruments, some designed to drive profit/loss (P/L), othersto provide some sort of diversification benefit, and still others to act ashedges of other exposures. Optimal f will not assist you in your effortsto  size  overall  exposures  at  the  portfolio  level,  taking  into  accountthese subcomponents; Inverted Sharpe will.
• The  projected  return  elements  of  the  Inverted  Sharpe  calculation  arebased on more realistic inputs than those associated with optimal f.On  the  whole,  I  am  troubled  by  the  notion  of  placing  too  much  cre-dence in the estimation of expected return at the individual transac-tions level—even when, as is the case with the secure f, the inputs arebased  on  historical  volatility  data.  I  wonder,  for  instance,  how  onegoes about selecting the entry and exit points. Moreover, there’s stillenough  of  the  University  of  Chicago  boy  left  in  me  to  feel  that  theexpected return on a given, single transaction ought to be somewherearound zero.
By  using  the  Sustainable  Sharpe  component  of  the  InvertedSharpe ratio methodology, by contrast, we are basing our return esti-mates on (1) portfolio-level data (which I have argued in the immedi-ately  preceding  discussion  is  more  reliable  than  transactions-level
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data) and (2) empirical information that derives directly from our ownperformance. Moreover, as careful readers of Chapter 5 will recall, theInverted Sharpe methodology does not call for the use of your actualSharpe ratio in the setting of exposure parameters, but rather suggestsyou set this input at a comfort level that you can sustain across most,if not all, market conditions. Prudent portfolio managers will set theirSustainable Sharpe ratio inputs at levels below their actual Sharpes soas to render them entirely consistent with an approach that uses pastperformance data as a means of establishing conservative estimates offuture performance
• The Inverted Sharpe methodology is designed to work hand in handwith the 10% of Trading Capital Rule to ensure that risk taking is nei-ther too high nor too low—given reasonably established objectives andconstraints.If you remember what we covered in Chapter 5, the ideaof the Inverted Sharpe/10% Rule is intended to achieve the objective notof optimization but rather of rationalization. The whole idea here is toset your risk levels neither too low to reduce your portfolio managementefforts to little more than spinning your wheels, nor too high to impedeyour  ability  to  manage  your  risk  capital  effectively.  In  this  way,  themethodology  establishes  what  I  believe  to  be  an  effective  and  highlyapplicable upper bound and lower bound to exposure assumption. Theranges between the two, which for most portfolio managers will be sub-stantial, allow for a healthy dose of that most critical component of mosteffective  portfolio  management:  judiciously  applied  discretion.  Everysituation you face as a trader will be different, and from this perspectivetwo trades are about as likely to be identical as two snowflakes. How-ever, if you set your risk taking at ranges that are consistent with theresults of the Inverted Sharpe/10% Rule, you are in a great position if notto maximize returns in every instance, then at least to ensure that scarceresources such as risk capital are never foolishly squandered.
I encourage anyone who uses the Inverted Sharpe methodology to con-stantly be reviewing the actual Sharpe and aggressively reducing the Sus-tainable Sharpe in the event that the former falls to levels below the latter.Inverted Sharpe, like every other element of the Trading  Risk statisticaltool kit, is a diagnostic tool designed to characterize the qualitative aspectsof your risk taking. Nothing in it implies the need or, in fact, the wisdom ofattempting  to  precisely  calibrate  exposures  on  the  basis  of  its  results.Again, it is simply there to tell you (1) whether your risk taking is consis-tent with your objectives and (2) what level of exposure is roughly consis-tent with the goals you have set for yourself.
The  key  to  using  these  figures  effectively,  of  course,  is  to  ensure  thatyour Sustainable Sharpe is a number on which you can comfortably bank.
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Therefore, in order to use the methodology effectively, it is necessary to takeroutine periodic checks to ensure that your actual Sharpe doesn’t slip mate-rially below the figure you set as your Sustainable Sharpe. If this happens,you must make the appropriate adjustments to your objectives and risk lev-els or to some combination of the two.
Otherwise, I’d say you’re on your way.Of course, it is not my intention to imply that optimal f and InvertedSharpe are in competition with one another. Moreover, there’s no reasonthat you can’t apply both methodologies contemporaneously, with optimalf applied in the sizing of individual transactions and Inverted Sharpe usedas a means of establishing exposure bands at the portfolio level. However,I do caution against the use of optimal f, or even secure f, as your exclu-sive tool for risk management.
RISK OF RUIN
This is another concept that takes its origins from the universe of gam-bling.  Of  course,  for  centuries,  speculators  have  been  trying  to  deter-mine  the  probabilities  of  blowing  their  whole  load,  so  to  speak;  but  abook by Alan N. Wilson, The Casino Gamblers Edge, perhaps best syn-thesizes these efforts. Wilson, a very cerebral fellow who once workedon the staff of Owen Chamberlain, 1959 Nobel laureate in physics, andlater spent 30 years at General Dynamics, as perhaps the defense indus-try’s leading expert in the field of random-number generation, sought toanswer the following question: In a game of chance, what is my risk oflosing  my  entire  bankroll  before  doubling  it?  He  ultimately  arrived  atthe  following  equation,  solving  specifically  for  r(x),  the  probability  oflosing x:
r (x) (cid:3) A (cid:5) B (1/S)^x
where p (cid:3) probability of winning on a single play
q (cid:3) probability of losing on a single playS (cid:3) p/q
A and B (cid:3) arbitrary constants that depend on (1) the initial capitalof the player, (2) the amount the player is willing to lose,and (3) the amount the player wishes to win
x (cid:3) the amount of capital the player has at any given time
As is the case with optimal f, the equation simply synthesizes such con-cepts as (1) how much risk capital the risk taker has and (2) the success-to-failure ratio, into an estimate of what is likely to happen in the tails of a
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return distribution. Risk of ruin is a very useful means of sizing individualtransactions such that there is an upper bound to worst-case outcomes. Inorder  to  do  so,  it  is  necessary  to  scale  transactions  sizes  to  probability-adjusted  return  streams,  reducing  them  as  remaining  risk  capital  erodesand increasing them when it builds.
Let me say right here and now that I have considerable respect for therisk  of  ruin  approach  to  portfolio  management  because  it  fully  contem-plates  all  of  the  lessons  I’ve  tried  to  impart  about  increasing  risk  duringtimes of success while reducing risk during more difficult intervals. How-ever, it does suffer from the same problems that plague optimal f, namelyits focus on the individual transaction level and its reliance on very sub-jective  inputs  as  to  what  success  ratios  are  likely  to  be  in  the  future.  Asindicated  earlier,  the  latter  of  these  challenges  is  a  bit  ubiquitous  in  theportfolio-sizing game, so we don’t want to be too explicit in our criticismshere.
About  the  only  direct  issue  I  have  with  the  methodology  is  that  itpoints  at  individual  transactions  rather  than  at  portfolio  volatility  as  awhole. However, it is certainly much easier to convert risk of ruin calcula-tions into portfolio measures than it is for optimal f. In fact, you can thinkof the 10% Rule covered in Chapter 5 as nothing more than a risk of ruincalculation applied at the portfolio level.
In closing, I’d say that the inclusion of risk of ruin dynamics into theportfolio decision-making process is a good idea. I would only caution thatyou  not  get  too  myopic  here.  By  ensuring  that  your  exposures  are  sizedappropriately at the portfolio level, you stand the best chance of preserv-ing capital for its most effective applications.
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IndexContents
Absolute net market value, 173–174Accuracy ratio, 184–186Active trading, 186, 203–205, 221Arbitrage, 65, 105–106Asset classes, 106, 145, 147, 200Asymmetric payoff functions, 130, 132,
150–151
At-the-margin, behavior, 16, 236–237Autocorrelation, 77Autoregression, 77Average holding period, individual
trades, 164–165
Average P/L, 56–57, 69, 163–164,
186–188Averaging, 137–138
Balanced portfolio, 135–140Bias, implications of, 172–173.See also Directional biasBid/offer spreads, 143, 153, 221, 223Break-even position, 200–201, 203Breakout strategies, 77Brokerage firms, functions of, 165, 199Budgets/budgeting, 239Budget revenues, 24Business cycles, 145Buy-and-hold strategy, 204
Capital allocation, 75, 226Capital investment, 171–172Capital preservation, 12, 123, 215, 231Capital providers, risk tolerance,
121–122, 228Capital utilization, 171Charts/charting, benefits of, 107, 196
Collateralization, 92Commissions, 143, 153, 198, 236Confidence intervals, 59–62, 100Consolidated statistical profile, 79–80Contrarian investing, 128Convertible bond arbitrage, 105–106Correlation. See specific types of
correlations
analysis, 73–78, 168–181capital utilization, 171–172with causation, 79characteristics of, 73coefficient, 73, 170defined, 73, 169holding periods, 175–177individual trades, 168–170, 179–181kitchen sink, 78against market benchmarks, 73–75net market value, 172–174number of daily transactions,
170–171
number of positions, 174portfolio diversification and, 145risk exposure and, 83–84, 90–91VaR, 178–179volatility, 177–179Credit spreads, 105Cross-collateralization, 198Cross correlation analysis, 75–76
Daily net change, 75Daily transactions, number of, 
170–171
Day traders/day trading, 143–144, 170,
176
253
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INDEX
Decision-making process, influentialfactors, 8, 10, 82, 220–221
Delta-neutrality, 153Derivatives, 56, 134, 148, 235–236, 241Directional bias, 135–141, 166Direct market exposure, 92Discretionary capital, 29Diversification, 144–146Dollar investment, individualsecurities, 160–161
Drawdown, 50, 70–73
netting risk, 129–131risk exposure and, 120–125, 233
Earnings announcements/releases, 43,
143, 236
Edge, in trading. See Trading with an
edge
Effective risk management, 124–125Efficient Market Hypothesis, 219Equilibrium, 108, 219Equity markets, 135Equity traders, 139, 145, 147, 164–167Event risk, 106, 223Exchange rate, 120Execution management, 193–199Exit strategy, 9.  See also Stop-out levelExposure adjustment strategies:
directional bias, 135–141diversification, 144–146leverage, 146–148optionality, 148–153position level volatility, 141–142position size, 134–135time horizon, 142–144
Exposure range determination:
components of, 110–111inverted Sharpe Ratio, 111–114,
248–250
volatility management, as tradingcapital percentage, 114–126
Fat tails, 116, 118Federal Reserve Regulation T, 148Financial statements, 185, 225Fixed-income market, 145Fixed-income portfolio, 147
Fixed-income trading, 56, 74, 105Front-running, 196–197Fully invested risk position, 227Futures/futures market, 56, 185, 239–240
Games of chance, 247–248“Going to the beach,” 32–36Gross market value, 161
Harvard Matrix, 226Hedge funds, 41, 119–120, 166, 194–195Hedge-oriented trades, 139–140Hedging, 151–152, 202High-risk profile, 59High-velocity trading strategies, 193High-water approach, 71, 121Histograms, 51–53Historical volatility, 84–88, 96–97Holding period:
correlation analysis, 175–177individual trades, 162–165time horizon and, 142–144trading with an edge, 222
“House,” being the, 223Humor, importance of, 243–244
Impact ratio, 186–188Implied volatility, 86–87, 89Incremental risk, 126, 141–142Individual portfolio, risk components:
correlation, 90–91historical volatility, 84–88, 96–97options implied volatility, 86–89scenario analysis, 104–106technical analysis, 106–108value at risk (VaR), 91–104
Individual trades, risk components of:core transactions-level statistics,
161–168, 209, 211
correlation analysis, 168–181influential factors, generally,
208–211
performance ratio improvement
methods, 189–208
performance success metrics,
184–189
transaction performance, 156–161
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Ineffective risk management, 
124–125
Information processing, 220–221Institutional investments/traders, 29,
193
Instrument classes, 200Interest rates, as influential factor, 87,
105
In-the-money option, 150Intraday prices, 39Intraday P/L, 42Intraday trading, 162Intrinsic value, 87, 150Inverted Sharpe Ratio, 111–114,
248–250
Macroeconomics, 137, 241Margin requirements, 148Market benchmarks, 73–74Market conditions, impact of, 20, 32,
74, 78, 158, 173, 179, 200, 214, 223,234, 236Market cycles, 44Market downturn, 55, 165Market economy, 241–242Market environment, 29Market liquidity, defined, 207.  See also
Liquidity
Market makers, 197, 221–222Market opportunities, 14, 103–104Market participation, 102–103, 
Investment. See Risk Management
242–244
InvestmentInvestment plan:
adjustments to, 218–219at the margin performance, 236–237expectations and, 234–236importance of, 214–218nonmonetary benefits, 237–242play your P/L, 226–234trading with an edge, 219–225
Kitchen sink correlations, 78Kurtosis, 53, 64, 116, 201
Lagging, 178Leverage, 146–148, 151–153Limits, 10. See also Stop-out levelLiquidation, 29, 107, 118, 160, 175, 177,186, 191, 193, 204, 206, 217–218
Liquidity, 89, 159, 203Liquidity providers, 221–223Long call options, 96, 135, 153Long-side P/L, 166–168Long-Term Capital Management
(LTCM), 228–232
Losing days, percentage of, 69–70Losing trades, impact ratio, 186–188,
191
Loss(es),  recovery from, 33, 125–126,
227–228Lotteries, 247Low-risk profile, 58
Market proximity, 221–222Market rallies, 74, 158, 173Market risk, 1–2, 152Market technician, functions of, 185Market trends, 22–23Market volume, 79, 108Mean, defined, 6Median P/L, 68Momentum strategy, 77Momentum trading, 206–207Moneyness, 149–150Monte Carlo simulation, 97–98Monthly P/L, 41–42, 44Motivation, importance of, 16–18Moving averages, 107, 218Multistrategy trading teams, 129–130Multivariate normal distribution, 94
Negative correlation, 172–173, 175, 179Negative P/L, 71, 228Negative return events, 53Net market value, 161, 172–174Netting risk, 129–130Newton, Sir Isaac, 53–5690/10 rule, 200–208Nonlinear pricing, 149, 235Nonmonetary benefits, 237–242Nonnormal distribution, 64–65Normal distribution, 57–58, 64, 116Number of positions correlation, 161,
174–175
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