MachineLearningforTradingSoftwarePlatform

Introduction&Installation

Projects(upcoming)

•  AssessPortfolio•  AssessLearners&DefeatLearners•  MarketSimulator

TowardsSharpeRatioIntuition

•  Considersourreturninthecontextofrisk

•  Riskisvolatile(standarddeviation)

•  Adjustourreturninreturnfortherisk

•  Volatility– Measuredbystandarddeviation

SharpeRatio

•  Considersourreturninthecontextofrisk

•  Riskisvolatile(standarddeviation)

•  Adjustourreturninreturnfortherisk

•  Volatility– Measuredbystandarddeviation

1

2

3

SharpeRatio

1.  HigherReturnsisBetter2.  LessVolatility/LessRisk

isBetter3.  NotEnoughInformation

–  Returns:ABC>XYZ–  VolatilityABC>XYZ–  ABCishigherreturns,

butmorerisk

1

2

3

SharpeRatio•  Adjuststhe

–  returnforrisk•  Aquantitativewaytoassessaportfolio

–  1.ABCisbetterbecauseithasthesamevolatilitybuthigherreturns

–  2.samereturnsbutXYZhaslowerrisksoXYZisbetter–  3.AquantitysuchastheSharpeRatiomaygiveyouanumbertodeterminewhichisbetter

•  Sharperatioalsoconsiders(comparative)–  Riskfreerateofreturns

•  ExampleBankaccountortreasurenote–  Latelyriskfreereturnis0,bankinterestrateis0,orcloseto0–  Caveat:WhencomputingSharpeRationeedtobecarefuloftheriskfreereturnsrate:

•  Annual,weekly,daily

Intuition:WhichFormulaisBest?

ConsiderParameterstoincludeinformula:•  Rp:PortfolioReturn•  Rf:RiskFreeRateofReturn•  σp :StandardDeviationofPortfolioReturna)  Rp–Rf+σp

b)  Rf/Rf-σp

c)  (Rp–Rf)/σp

GeneralFormoftheSharpeRatio

HeuristicsSharpeRatio

•  SharpeRatioAnswers:Istheportfolioreturnduetointelligentinvestmentdecisionsortheresultofexcessiverisktakenbytheinvestor?

•  Thegreateraportfolio’sSharperatio:

•  Thebetteritsrisk-adjustedperformancehasbeen.

•  AnegativeSharperatioindicatesthatarisk-lessasset(riskfree)wouldperformbetterthantheportfolio(return)beinganalyzed.”

ComputingSharpeRatio

•  Expectedreturn:Canusemean()ofdailyreturns(accessibleviayahoofinance).

•  StD()ofdailyreturns.•  StD()oftheReturnoftheRiskFreeRate

–  Standarddeviationofaconstantis0(canremove).•  Whatistheriskfreerate?

–  LIBOR(LondonInterBankOfferRate)–  InterestRate:3monthsTreasuryBill(Standard)–  0%!ShortCut.–  Ifyouusedailyreturns,thentheriskfreerateneedstobeatthesamesamplingrate:

•  Soifyouarereferencinganannualratethenyouwillneedtoconvertittoadailyriskfreerate.

RiskFreeRate

•  IfyouhaveannualrateandwantEffectiveDailyRatecompoundeddaily.

•  Caveat:Compoundinterestisnotlinearwithtime.

Example:2.5%AnnualReturn-Goingtoeffectivedailyreturn–242tradingdaysinayear.

=POWER(1+0.025,1/242)-1

€

1+ Annualized ReturnNth−1

1+0.025242 −1

•  SharpeRatioitselfisanannualmeasure,soifwearesamplingatadifferentratee.g.daily,needanadjustmentfactortoconvertittothecorrectperiod– SRannualized=k*SR

SampleRate K(converttoannual)

Daily

Weekly

Monthly

SR=

ComputingSRSummary

•  2thingstoconsiderwhencomputingSharpeRatio:– 1)ConvertRiskFreeRatetosameperiodasexpectedreturnperiod.

– 2)ConvertSRtoannualSR.(usingadjustmentfactork).

Quiz:WhatistheSharpeRatio?•  60DaysofData•  BPS–Basispointsaretenthsofpercentages.•  AverageDailyReturn=10bps=0.001•  DailyRiskFreeRate=2bps=0.0002•  StandardDeviationDailyReturn=10bps=0.001

SR=SQRT(Period)*mean(Rportfolio-RRiskFree)--------------------------------Std(DailyReturn)SR=SQRT(Period)*(10-2)-------10

Quiz:WhatistheSharpeRatio?•  60DaysofData•  BPS–Basispointsaretenthsofpercentages.•  AverageDailyReturn=10bps=0.001•  DailyRiskFreeRate=2bps=0.0002•  StandardDeviationDailyReturn=10bps=0.001

SR=SQRT(Period)*mean(Rportfolio-RRiskFree)--------------------------------Std(DailyReturn)SR=SQRT(Period)*(10-2)-------10

ProjectDemo

•  Computeport_valandstatistics…

GettheDailyTotalValueofthePortfolio•  Step1:PricesDataFrameindexbydates

–  pricesisaframe.•  Step2:NormalizebyFirstRow

–  normed=prices/prices[0]•  Step3:Multiplybyallocation(avector)

–  allocated=normed*allocs•  Step4:Positionvalues=wortheachday

–  pos_vals=allocated*start_val•  Step5:DailyTotalValueofPortfolio

–  port_vals=pos_vals.sum(axis=1)

prices normed allocated pos_vals port_vals

Given:

start_val = $1,000,000start_date = 2011-01-01end_date = 2011-12-31symbols =[‘SPY’,’XOM’, ’GOOG’, ‘GLD’]allocs = [0.4,0.4,0.1,0.1]

prices/prices[0] normed/allocs

allocated

*start_val

pos_vals.sum(axis=1

)

port_vals=get_portfolio_value(prices,allocs,start_val)

ComputeStatistics•  AverageDailyReturn

–  Intuition:•  Today/Yesterday-1•  Remove1strowsincenowitisall0s

–  ExcelacellG10=E10/E9-1•  E10=current’srowadjustedclose•  E09=previousday’sadjustedclose

–  Pandas:•  daily_rets=port_val/port_val.shift(X)-1•  #whatisX?

•  Cumulativefrombeginningtoend–  >=periodthandailyreturn.

•  AverageandStandardDeviationoftheDailyReturn.mean(),.std()

•  SharpeRatio(Annual)

Given:1)   port_vals: portfolio

daily return values 2)   rfr: risk free rate3)  sample_frequency: Sample

Frequency – Example, daily 252 in a year.

cr,adr,sddr,sr=get_portfolio_stats(port_val,risk_free_rate,sample_frequency)

Optimization

•  Whatisanoptimizer?1.  Findminimumvaluesoffunctions

•  Example:f(x)=x2+x3+…+1

2.  Findparametersfromdata•  Enables:buildingparameterizedmodelsbasedondata•  How:polynomialfittodata

3.  Find(refine)allocationofstocksinaportfolio• Whatpercentageshouldbeallocatedtoeachstocktomaximizetheportfolioreturn(partoftheproject).

FindMinimaofaFunction

•  Howtouseanoptimizer:1)  Provideafunctiontominimize:

•  Example:f(x)=x2+0.5

2)  Provideaninitialguess:•  Example=5(generatedbyarandomizer)

3)  Calltheoptimizerwiththeparametersabove

Example:Minimization1)  Functionprovided:

–  f(x)=(x–1.5)2+0.52)  Provideaninitialguess:3.03)  CallOptimizerwithparameters

definedabove.–  Onemethod:

•  Gradientdescendtonarrowinonthesolution.

•  Experimentwithothermethods.

•  Next:LookatCode(provided):–  pdf-code-finance/001-minimizer.py

http://www.wolframalpha.com/min(x-1.5)^2 +0.5)

f(x):Y = (X - 1.5)**2 + 0.5 # parabola atX = 1.5 Y = 0.5min_result=spo.minimize(f,Xguess,method='SLSQP',options={'disp':True})

Exercise:Minimization1)  Functionprovided:

–  f(x)=(x1)2+(x2)22)  Provideaninitialguess:[1,1]3)  CallOptimizerwithparameters

definedabove.–  Onemethod:

•  Gradientdescendtonarrowinonthesolution.

•  Experimentwithothermethods.

•  CodeSnippets:

http://www.wolframalpha.com/min(X1^2 + X2^2)

x1,x2=paramsZ=x1**2+x2**2Xguess=[1,1]min_result=spo.minimize(f,Xguess,method='SLSQP',options={'disp':True})

optimize1D2.py

Exercise:Minimization1)  Functionprovided:

–  f(x)=(x1)2+(x2)22)  Provideaninitialguess:[1,1]3)  CallOptimizerwithparameters

definedabove.–  Onemethod:

•  Gradientdescendtonarrowinonthesolution.

•  Experimentwithothermethods.

•  CodeSnippets:

http://www.wolframalpha.com/min(X1^2 + X2^2)

x1,x2=paramsZ=x1**2+x2**2Xguess=[1,1]min_result=spo.minimize(f,Xguess,method='SLSQP',options={'disp':True})

optimize1D2.py

Exercise

#min (X1+0.5)^2 + (X2-0.25)^2+sin(3X1)+sin(5X2)

Whichfunctionsarechallengingtosolve(fortheminimizer)?

•  A

•  B

•  C

•  D

Whichfunctionsarechallengingtosolve(fortheminimizer)?

•  A–flatareasdon’thaveagradient.

•  B-convexproblems

•  C–severalglobalminima

•  D–discontinuity(andaflatarea).

•  Convexfunctionsareguaranteedtofindaminima(onlyone).

•  But–otheralgorithmsforspecificissues,youcanexperimentwithdifferentonesonyourown.

ConvexProblems

•  Convexfunction•  Wikipedia:"...areal-valuedfunctionf(x)definedonanintervaliscalledconvexifthelinesegmentbetweenanytwopointsonthegraphofthefunctionliesabovethegraph..."

Parameterizedmodelsfromdata•  Example:f(x)=mx+b

–  c1x+c0–  c3x3+c2x2+c1x+c0

•  Goal1:Findparametersofthelinec0,c1,wherec0isthey-intercept,andc1isslopethatbestfitsthedata

•  Goal2:Howdowereframetheproblemsothatitmakessensetotheminimizer?

•  Whatshouldwebeminimizing?

humidity

rain

003-parameters-data.py

WhichMetricsaregoodforfittingdata?

•  Σ ei•  Σ abs(ei)•  Σ ei2

WhichMetricsaregoodforfittingdata?

•  Σ eiü Σ abs(ei)ü Σ ei2

BothareOKSquareErrorisastandardmethod

Steps:

•  Expressafunction‘error()’thatreturnsanerrormeasurebetween:– Agivenline,and– DataPoints.

•  deferror(data,line)

003-parameters-data.py

(data− line)2∑

(Ydata−Yline)2∑

RunningtheCode

•  Horizontallineistheinitialguess.•  Minimizestheerrorbetweenthelineanddata.

BonusExercise

•  Howwouldyoudocurvefitting?– Fitdatatoacurve?

HowtoOptimizeaPortfolio?(futureactivity)

•  Exercise/Activity:Maximizeperformanceofaportfolio

•  Criteria(maximization)?Whichiseasiest?– Cumulativereturn– VolatileReturn– RiskAdjustedReturn(SharpeRatio)

http://quantsoftware.gatech.edu/Optimize_something

HowtoOptimizeaPortfolio?(futureactivity)

•  Exercise/Activity:Maximizeperformanceofaportfolio•  Problem:Needtofind‘coefficients’ofstocksinaportfolio

(theallocationofeachstockinportfolio]–  [‘GOOG’,‘GLD’,’APPL’,’XOM’]–  [0.25,0.25,0.25,0.25]=equal–  [0.0,0.0,0.0,1.0]=justAPPL?–  Whichisbetter?

•  Criteria(maximization)?Whichiseasiest?–  Cumulativereturn(trivialjustthemaximumreturn)–  VolatileReturn–  RiskAdjustedReturn(SharpeRatio)

•  OptimizingforVolatilityorSRyouhavetoevaluatevariouscombinationsofstocksandcheckresult.

http://quantsoftware.gatech.edu/Optimize_something

FramingtheProblem(andRecap)(similartopreviousexamples)

•  Inordertouseanoptimizerthatminimizesweneedtodothefollowing:– Provideafunctiontominimizef(x)– ProvideaninitialguessforX– Calltheoptimizer

•  NEW:– Constraint:allallocationsneedtosumto1.0– WhataboutSharpeRatio

•  Arewelookingforaminimaormaxima?

•  WewanttomaximizeSR,higherisbetter.•  Sincewehaveaminimizerweshouldmakethemaximaaminima

ActivityHints

•  f(x)=SR*(-1)•  Provideaninitialguess•  LimitsonvaluesforX•  Constraints–allstocksaddsto1.

MoreDetailed(Hints)•  F(allocs)WewanttooptimizeforSharperatio,

–  Wewanttooptimizefor-1*SRto‘maximizeit’–  Functioninoptimizer

•  get_portfolio_value()•  SR=get_porftolio_stats()•  Return–SR

•  Imaginethatthearethepercentageofeachofthefundsallocated•  constraints=({'type':'eq','fun':lambdaallocs:1.0-

np.sum(allocs)})#allocationsmustsumto1(sameas1-sum=0)•  result=spo.minimize(

f,initial_allocs,method='SLSQP',bounds=bounds,constraints=constraints)

Constraints:•  Xaretheallocationswearelookingor

– Needtoadduptoamaximumof1.– Limitandconstraints

•  Wewanttheoptimizertotrydifferentallocationstodiscoverbestsetofallocations

•  SharpeRatio?Volatility?

Example:EqualAllocation

•  .25GOOG•  .25AAPL•  .25GLD•  .25XOM

Example:SharpeRatioOptimization

•  .00GOOG•  .40AAPL•  .60GLD•  .00XOM

Challenge:Bonus•  Pickasetof4stocks•  Pickaperiodtotestover.•  Optimize

–  SR–  Volatility

•  Dotheydiffer?–  Discuss.

•  Analyzeperformanceofresultingportfolio–  Howdoesitperformcomparedtothemarket.–  Usingtheseallocationstestitoveradifferentperiod

•  Howdoesitperform?