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1、Good is good, but better carries it.精益求精，善益求善。EdSeykota谈风险管理-EdSeykota谈风险管理RiskManagement(c)EdSeykota,2003RiskRISKisthepossibilityofloss.Thatis,ifweownsomestock,andthereisapossibilityofapricedecline,weareatrisk.Thestockisnottherisk,noristhelosstherisk.Thepossibilityoflossistherisk.Aslongasweownthest
2、ock,weareatrisk.Theonlywaytocontroltheriskistobuyorsellstock.Inthematterofowningstocks,andaimingforprofit,riskisfundamentallyunavoidableandthebestwecandoistomanagetherisk.RiskManagementTomanageistodirectandcontrol.Riskmanagementistodirectandcontrolthepossibilityofloss.Theactivitiesofariskmanageraret
3、omeasureriskandtoincreaseanddecreaseriskbybuyingandsellingstock.TheCoinTossExampleLetssaywehaveacointhatwecantossandthatitcomesupheadsortailswithequalprobability.TheCoinTossExamplehelpstopresenttheconceptsofriskmanagement.ThePROBABILITYofaneventisthelikelihoodofthatevent,expressingastheratioofthenum
4、berofactualoccurrencestothenumberofpossibleoccurrences.Soifthecoincomesupheads,50timesoutof100,thentheprobabilityofheadsis50%.Noticethataprobabilityhastobebetweenzero(0.0=0%=impossible)andone(1.0=100%=certain).Letssaytherulesforthegameare:(1)westartwith$1,000,(2)wealwaysbetthatheadscomeup,(3)wecanbe
5、tanyamountthatwehaveleft,(4)iftailscomesup,weloseourbet,(5)ifheadscomesup,wedonotloseourbet;instead,wewintwiceasmuchaswebet,and(6)thecoinisfairandsotheprobabilityofheadsis50%.Thisgameissimilartosometradingmethods.Inthiscase,ourLUCKequalstheprobabilityofwinning,or50%;wewillbelucky50%ofthetime.OurPAYO
6、FFequals2:1sincewewin2forevery1webet.OurRISKistheamountofmoneywewager,andthereforeplaceatrisk,onthenexttoss.Inthisexample,ourluckandourpayoffstayconstant,andonlyourbetmaychange.Inmorecomplicatedgames,suchasactualstocktrading,luckandpayoffmaychangewithchangingmarketconditions.Tradersseemtospendconsid
7、erabletimeandefforttryingtochangetheirluckandtheirpayoff,generallytonoavail,sinceitisnottheirstochange.Theriskistheonlyparametertheriskmanagermayeffectivelychangetocontrolrisk.Wemightalsomodelmorecomplicatedgameswithamatrixoflucksandpayoffs,toseearangeofpossibleoutcomes.Seefigure1.LuckPayoff10%lose2
8、20%lose130%breakeven20%win110%win210%win3FixedBet$10Fixed-FractionBet1%Start10001000Heads10201020Tails10101009.80Heads10301030Tails10201019.70Heads10401040.09Tails10301029.69Heads10501050.28Tails10401039.78Heads10601060.58Tails/font10501049.97Noticethatbothsystemsmake$20.00(twicethebet)onthefirsttos
9、s,thatcomesupheads.Onthesecondtoss,thefixedbetsystemloses$10.00whilethefixed-fractionsystemloses1%of$1,020.00or$10.20,leaving$1,009.80.Notethattheresultsfromboththesesystemsareapproximatelyidentical.Overtime,however,thefixed-fractionsystemgrowsexponentiallyandsurpassesthefixed-betsystemthatgrowsline
10、arly.Alsonotethattheresultsdependonthenumbersofheadsandtailsanddonotatalldependontheorderofheadsandtails.Thereadermayprovethisresultbyspreadsheetsimulation.%BetStartHeadsTailsHeadsTailsHeadsTailsHeadsTailsHeadsTails01000.001000.001000.001000.001000.001000.001000.001000.001000.001000.001000.0051000.0
11、01100.001045.001149.501092.031201.231141.171255.281192.521311.771246.18101000.001200.001080.001296.001166.401399.681259.711511.651360.491632.591469.33151000.001300.001105.001436.501221.031587.331349.231754.001490.901938.171647.45201000.001400.001120.001568.001254.401756.161404.931966.901573.522202.9
12、31762.34251000.001500.001125.001687.501265.631898.441423.832135.741601.812402.711802.03301000.001600.001120.001792.001254.402007.041404.932247.881573.522517.631762.34351000.001700.001105.001878.501221.032075.741349.232293.701490.902534.531647.45401000.001800.001080.001944.001166.402099.521259.712267
13、.481360.492448.881469.33451000.001900.001045.001985.501092.032074.851141.172168.221192.522265.791246.18501000.002000.001000.002000.001000.002000.001000.002000.001000.002000.001000.00551000.002100.00945.001984.50893.031875.35843.911772.21797.491674.74753.63601000.002200.00880.001936.00774.401703.6868
14、1.471499.24599.701319.33527.73651000.002300.00805.001851.50648.031490.46521.661199.82419.94965.85338.05701000.002400.00720.001728.00518.401244.16373.25895.80268.74644.97193.49751000.002500.00625.001562.50390.63976.56244.14610.35152.59381.4795.37Ata0%betthereisnochangeintheequity.Atfivepercentbetsize
15、,webet5%of$1,000.00or$50.00andmaketwicethatonthefirsttoss(heads)sowehaveandexpectedvalueof$1,100,showningray.Thenoursecondbetis5%of$1,100.00or$55.00,whichwelose,sowethenhave$1,045.00.Notethatwedothebestata25%betsize,showninred.Notealsothatthewinningparameter(25%)becomesevidentafterjustonehead-tailcy
16、cle.Thisallowsustosimplifytheproblemofsearchingfortheoptimalparametertotheexaminationofjustonehead-tailcycle.Noticethattheexpectedvalueofthesystemrisesfrom$1000.00withincreasingbetfractiontoamaximumvalueofabout$1,800ata25%betfraction.Thereafter,withincreasingbetfraction,theprofitabilitydeclines.This
17、curveexpressestwofundamentalprinciplesofriskmanagement:(1)TheTimidTraderRule:ifyoudontbetverymuch,youdontmakeverymuch,and(2)TheBoldTraderRule:Ifyoubettoomuch,yougobroke.Inportfoliosthatmaintainmultiplepositionsandmultiplebets,werefertothetotalriskastheportfolioheat.Note:NotethechartillustratestheExp
18、ectedValue/BetFractionrelationshipfora2:1payoffgame.Foragraphofthisrelationshipatvaryingpayoffs,seeFigure8.TheKellyFormulaK=W-(1-W)/RK=FractionofCapitalforNextTradeW=HistoricalWinRatio(Wins/TotalTrials)R=WinningPayoffRate-Forexample,sayacoinpays2:1with50-50chanceofheadsortails.Then.K=.5-(1-.5)/2=.5-
19、.25=.25.Kellyindicatestheoptimalfixed-fractionbetis25%.Thisgraphshowstheoptimalbetfractionforvariousvaluesofluck(Y)andpayoff(X).Optimalbetfractionincreaseswithincreasingpayoff.Forveryhighpayoffs,optimalbetsizeequalsluck.Forexample,fora5:1payoffona50-50coin,theoptimalbetapproachesabout50%ofyourstake.
20、Thisgraphshowsoptimalexpectedvalueforvariousvaluesofluckandpayoff,givenbettingattheoptimalbetfraction.Thehigherthepayoff(X:1:1to5:1)andthehighertheluck(Y:.20to.70),thehighertheexpectedvalue.Forexample,thehighestexpectedvalueisfora70%winningcointhatpays5:1.Thelowestexpectedvalueisforacointhatpays1:1(
21、evenbet).Thisgraphshowstheexpectedvalueofa50%lucky(balanced)coinforvariouslevelsofbetfractionandpayoff.Theexpectedvaluehasanoptimalbetfractionpointforeachlevelofpayoff.Inthiscase,theoptimalbetfractionfora1.5:1payoffisabout15%;ata2:1payofftheoptimalbetfractionisabout25%;ata5:1payoff,theoptimalbetfrac
22、tionisabout45%.Note:Figure4aboveisthecrosssectionoffigure8,atthe2:1payofflevel.StockPrice/ShareSharesValueAB$100C$200$50,000StockPrice/ShareRisk/ShareSharesRiskValueAB$100$10C$200$51000$5,000$200,000Figure1:ALuck-Payoffmatrix,showingsixoutcomes.Thismatrixmightmodelaput-and-takegamewithasix-sidedspin
23、ningtop,oreventrading.Fornow,however,wereturntoourbasiccoinexample,sinceithasenoughdimensionstoillustratemanyconceptsofriskmanagement.Weconsidermorecomplicatedexampleslater.OptimalBettingInourcointossexample,wehaveconstantluckat50%,constantpayoffat2:1andwealwaysbetonheads.Tofindariskmanagementstrate
24、gy,wehavetofindawaytomanagethebet.Thisissimilartotheproblemconfrontingariskmanagerinthebusinessoftradingstocks.Goodmanagersrealizethatthereisnotmuchtheycandoaboutluckandpayoffandthattheessentialproblemistodeterminehowmuchtowageronthestock.Webeginourgamewith$1,000.HunchesandSystemsOnewaytodetermineab
25、etsizeisbyHUNCH.Wemighthaveahunchandandbet$100.Althoughhunch-centricbettingiscertainlypopularandlikelyaccountsforanenormousproportionofactualrealworldbetting,ithasseveralproblems:thebetsrequiretheconstantattentionofanoperatortogeneratehunches,andinterpretthemintobets,andthebetsarelikelytorelyasmucho
26、nmoodsandfeelingsasonscience.Toimproveonhunch-centricbetting,wemightcomeupwithabettingSYSTEM.Asystemisalogicalmethodthatdefinesaseriesofbets.Theadvantagesofabettingsystem,overahunchmethodare(1)wedontneedanoperator,(2)thebettingbecomesregular,predictableandconsistentand,veryimportantly,(3)wecanperfor
27、mahistoricalsimulation,onacomputer,toOPTIMIZEthebettingsystem.Despitealmostuniversalagreementthatasystemoffersclearadvantagesoverhunches,veryfewriskmanagersactuallyhaveadefinitionoftheirownriskmanagementsystemsthatisclearenoughtoallowacomputertoback-testit.Ourcoin-flipgame,howeverisfairlysimpleandwe
28、cancomeupwithsomebettingsystemsforit.Furthermore,wecantestthesesystemsandoptimizethesystemparameterstofindgoodriskmanagement.FixedBetandFixed-FractionBetOurbettingsystemmustdefinethebet.Onewaytodefinethebetistomakeitaconstantfixedamount,say$10eachtime,nomatterhowmuchwewinorlose.ThisisaFIXEDBETsystem
29、.Inthiscase,asinfixed-bettingsystemsingeneral,our$1,000EQUITYmightincreaseordecreasetothepointwherethe$10fixedbetbecomesproportionatelytoolargeorsmalltobeagoodbet.Toremedythisproblemoftheequitydriftingoutofproportiontothefixedbet,wemightdefinethebetasasFIXED-FRACTIONofourequity.A1%fixed-fractionbetw
30、ould,onouroriginal$1,000,alsoleadtoa$10bet.Thistime,however,asourequityrisesandfalls,ourfixed-fractionbetstaysinproportiontoourequity.Oneinterestingartifactoffixed-fractionbetting,isthat,sincethebetstaysproportionaltotheequity,itistheoreticallyimpossibletogoentirelybrokesotheofficialriskoftotalruini
31、szero.Inactualpractice,howeverthedisintegrationofanenterprisehasmoretodowiththepsychologicalUNCLEPOINT;seebelow.SimulationsInordertotestourbettingsystem,wecanSIMULATEoverahistoricalrecordofoutcomes.Letssaywetossthecointentimesandwecomeupwithfiveheadsandfivetails.Wecanarrangethesimulationinatablesuch
32、asfigure2.Figure2:SimulationofFixed-BetandFixed-FractionBettingSystems.PyramidingandMartingaleInthecaseofarandomprocess,suchascointosses,streaksofheadsortailsdooccur,sinceitwouldbequiteimprobabletohavearegularalternationofheadsandtails.Thereis,however,nowaytoexploitthisphenomenon,whichis,itselfrando
33、m.Innon-randomprocesses,suchasseculartrendsinstockprices,pyramidingandothertrend-tradingtechniquesmaybeeffective.Pyramidingisamethodforincreasingaposition,asitbecomesprofitable.Whilethistechniquemightbeusefulasawayforatradertopyramiduptohisoptimalposition,pyramidingontopofanalready-optimalpositionis
34、toinvitethedisastersofover-trading.Ingeneral,suchmicro-tinkeringwithexecutionsisfarlessimportantthanstickingtothesystem.Totheextentthattinkeringallowsawindowforfurtherinterpretingtradingsignals,itcaninvitehunchtradingandweakenthefabricthatsupportsstickingtothesystem.TheMartingalesystemisamethodfordo
35、ubling-uponlosingbets.Incasethedoubledbetloses,themethodre-doublesandsoon.Thismethodisliketryingtotakenickelsfrominfrontofasteamroller.Eventually,onelosingstreakflattenstheaccount.Optimizing-UsingSimulationOnceweselectabettingsystem,saythefixed-fractionbettingsystem,wecanthenoptimizethesystembyfindi
36、ngthePARAMETERSthatyieldthebestEXPECTEDVALUE.Inthecointosscase,ouronlyparameteristhefixed-fraction.Again,wecangetouranswersbysimulation.Seefigures3and4.Note:Thecoin-tossexampleintendstoilluminatesomeoftheelementsofrisk,andtheirinter-relationships.Itspecificallyappliestoacointhatpays2:1witha50%chance
37、ofeitherheadsortails,inwhichanequalnumberofheadsandtailsappears.Itdoesnotconsiderthecaseinwhichthenumbersofheadsandtailsareunequalorinwhichtheheadsandtailsbunchuptocreatewinningandlosingstreaks.Itdoesnotsuggestanyparticularriskparametersfortradingthemarkets.Figure3:Simulationofequityfromafixed-fract
38、ionbettingsystem.Figure4:Expectedvalue(endingequity)fromtentosses,versusbetfraction,foraconstantbetfractionsystem,fora2:1payoffgame,fromthefirstandlastcolumnsoffigure3.Optimizing-UsingCalculusSinceourcoinflipgameisrelativelysimple,wecanalsofindtheoptimalbetfractionusingcalculus.Sinceweknowthatthebes
39、tsystembecomesapparentafteronlyonehead-tailcycle,wecansimplifytheproblemtosolvingforjustoneofthehead-tailpairs.Thestakeafteronepairofflips:S=(1+b*P)*(1-b)*S0S-thestakeafteronepairofflipsb-thebetfractionP-thepayofffromwinning-2:1S0-thestakebeforethepairofflips(1+b*P)-theeffectofthewinningflip(1-b)-th
40、eeffectofthelosingflipSotheeffectivereturn,R,ofonepairofflipsis:R=S/S0R=(1+bP)*(1-b)R=1-b+bP-b2PR=1+b(P-1)-b2PNotehowforsmallvaluesofb,Rincreaseswithb(P-1)andhowforlargevaluesofb,Rdecreaseswithb2P.Thesearethemathematicalformulationsofthetimidandboldtraderrules.WecanplotRversusbtogetagraphthatlookssi
41、milartotheonewegetbysimulation,above,andjustpickoutthemaximumpointbyinspection.Wecanalsonoticethatatthemaximum,theslopeiszero,sowecanalsosolveforthemaximumbytakingtheslopeandsettingitequaltozero.Slope=dR/db=(P-1)-2bP=0,therefore:b=(P-1)/2P,and,forP=2:1,b=(2-1)/(2*2)=.25Sotheoptimalbet,asbefore,is25%
42、ofequity.Optimizing-UsingTheKellyFormulaJ.L.Kellysseminalpaper,ANewInterpretationofInformationRate,1956,examineswaystosenddataovertelephonelines.Onepartofhiswork,TheKellyFormula,alsoappliestotrading,tooptimizebetsize.Figure5:TheKellyFormulaNotethatthevaluesofWandRarelong-termaveragevalues,soastimego
43、esby,Kmightchangealittle.Reference:http:/www.racing.saratoga.ny.us/kelly.pdfSomeGraphicRelationshipsBetweenLuck,PayoffandOptimalBetFractionTheOptimalBetFractionIncreaseswithLuckandPayoffFigure6:Optimalbetfractionincreaseslinearlywithluck,asymptoticallytopayoff.TheExpectedValueoftheProcess,attheOptim
44、alBetFractionFigure7:Theoptimalexpectedvalueincreaseswithpayoffandluck.FindingtheOptimalBetFractionfromtheBetSizeandPayoffFigure8:Forhighpayoff,optimalbetfractionapproachesluck.Non-BalancedDistributionsandHighPayoffsSofar,weviewriskmanagementfromtheassumptionthat,overthelongrun,headsandtailsfora50-5
45、0coinwillevenout.Occasionally,however,awinningstreakdoesoccur.Ifthepayoffishigherthan2:1forabalancedcoin,theexpectedvalue,allowingforwinningstreaks,reachesamaximumforabet-it-allstrategy.Forexample,fora3:1payoff,eachtossyieldsanexpectedvalueofpayoff-times-probabilityor3/2.Therefore,theexpectedvaluefo
46、rtentossesis$1,000x(1.5)10orabout$57,665.Thissurpasses,byfar,theexpectedvalueofabout$4,200fromoptimizinga3:1cointoabouta35%betfraction,withtheassumptionofanequaldistributionofheadsandtails.AlmostCertainDeathStrategiesBet-it-allstrategiesare,bynature,almost-certain-deathstrategies.Sincethechanceofsur
47、vival,fora50-50coinequals(.5)NwhereNisthenumberoftosses,aftertentosses,thechanceofsurvivalis(.5)10,oraboutonechanceinonethousand.Sincemosttradersdonotwishtogobroke,theyareunwillingtoadoptsuchastrategy.Still,theexpectedvalueoftheprocessisveryattractive,sowewouldexpecttofindthesysteminuseincaseswheredeathcarriesno