Reading 36: Multifactor Models

Page No. 11Date / /Reading 36 : Multijactor ModelsHistory1958 Markowitz Mean - Variance Frameworkfolio Concept:- DiversificationDiver sification if # returncorrelation <+1folio returns: Multivariate normal distributionIEIR) Expected return202 Return variance3P Return correlationMorden portfolio theory1964 Sharpe's Capital Asset pricing Model (CAPM)Based on mean variance framworkAssets return based on Rf rate & pricied risk &Stock specific, unsystematic risk Can be diversified (non priced)Systematic risk (non diversifiable) priced riskSingle factor = market folioB measures of stocks exposure to the factorsPricied RiskA risk that does not affect many assets (ie , unsy stematic)Can be diver sified and is not pricedInvestors should not be rewarded (i.e no return) for nonpriced riskRisks that effect all assets are systematic risk factors &Should be pricedInvestors should earn a return for non diversifiable priced liskfactor risk) Page No.DateNo arbitrage opportunity availcArbitrage Pricing Theory (RP+)aK-factor model1976 Ross APTECR) linear functions of a set of factors Capturing systematicIdenty and no. of factors not identified (unlike (APM)Multiple priced factors (CAPM single -priced factor)Alternative to CAPMLinear model with factor that capture systematic riskNo Risk fixed no- of factorsDescribes the equil brium relationship bet" expected returnsfor well - diversifiedportfolio and their multiple sources ofsystematic riskfolic factor betas or loadingE(Rp) = RF + Bp,(1) + Bp, t (12)++Bex (AK)Risk premPure factor portfolio:Rp = RFR + B,XFactor Risk premUnlike CAPM APT does not req one of the factors to be ERPCAPM :- Make many (implausible) assumption (homagenousexpectations and the marketfelio) Page No. 13Date / /IMP. is a equilibrium pricing modelRoll Ross 2001APT Assumptions:-1-3% of portfolio variance due toReturns derived from multifactor process nonsystematic factorUnsystematic risk can be diversified awayNo arbitrage opportunity Casset returns Correctly reflects risk)Large no. of asset are available A, 7 Asset return are described byArbitrage Opportunityk -factormadePortfolio R,B,C = Single factor portfoliosfor PortfolioECR)Factor SensitivityA10%1.0B20%2.0C13%1.5Create portfolio (D) equally weighted to A &BECRo) = 15%Factor Sensitivety (B) = 1.5( 2 20% )(1x))+(1**)Arbitrage opportunity:Portfolio C does not generateLong D, short (:sufficient return for its factorProfit = (CRo) - E(R) = 2%risk (market not in equilibrium)Example 2 PortfolioFactor BECR)PerBA220%BI12%12) 107. ] LongC1.514%9.3% -Short Page No. 19Date/W, x 2tW2XI=1.52\W, = 50% W2 = 50%1.50.5050.5 x 20% + 0.5X 12%= 16%o s 0.5TakelongpositiononportfolioA&BandshortpositiononportfolioC to earnanarbitrageof2%Example APTAninvest"firmemployeesatwo-factorAPTmodel.Risk - free rate of 5%FactorFactor IFactor 2Invest Fund factor betas1.502.0Factorriskprem0.030.0125The expetedreturnfor theinvest"fund(IF)equals:-ECRIF)=0.05+5(0-03)t2(0-0125) = 0.12= 12%Example2StockFactor betaAExpected returnX01 0.9Yol 17 7.BX02 1.7402 25%CalculateRFRandfactorrisk premLUseDATAEnterfunction(2nd7)Set.IN2Xvaluesasbeta&Yvaluesas3Use STATexpectedreturnfunction (2nd 8)a = 8% (RFR)UseDATAinCalci Page No. 15Date//Expected return = RFR + (BXFRP)17=8+0.9XFRPFRP = 10%Three Types of Multifactor ModelsIMacfo economic FactorSuprices in macro variables that explain diff instock returnInt rates inflation , biz cycle credit spread , etc. + FactorSupries :- Actual value - Est valueIntercept 'ECR.) is derived - from APT modelExpected Return = Interceptfirm -specificEquation:- Ra = ECR. t B,F, t PzF2 + + P.F.+EsupriseII] FundamentalAttributes of stocks that are important in explaining Cross - sectionaldiff in return Factor are firm. Specific & stated returnFactor:- PLE P/B Mcap T f leverage etcBeta is standardstandarized B = (Actual value- - Mean Value)bStandardization allows as to use fundamental factors measured indiff unit in the Same factor model(geristalIntercept is not interpreted as the E(R) Page No.16Date//III ] Statistical factorprinciple Components modelsFactorsarefoliosofsecthatbestreproducehistoryretainvarianyFactor analysis modelFactor are portfulio s of Sec that based reproduce historicreturn convaciancesIssue = attaching eco meaning to Stat factor difficult.Note: Statistical methods are applied to historical retern todetermine factor that explain Covaliance in retainLess. assumptionhard to intelpertMacro eco Factor model - factor sensitivity are est lastEx -ante returnSensitivity toInt rate suppulise actual -(no suprise)WT suprisesconsensus predictedTwo factor eg Ri = E(R) + bil FGOP t biz FINT + ESensitivity toGDP suiptise actual - -GDP SupriseConsensus predictedCompany -specific 2 jossurparise returnAssuming all sources of Common risk Crisk factor.) have been includedIntercept = APT Page No. 17Date / /Fundamental Factor Model - factor sens est firstFactors are returns , not SuprisesIntercept # Expected returnFactor sensitivities are Standardized :2 -stat alikeBi = value of attribute K for asset i - avg value of attribute Kb (values of attributes K)Scaling allows all factor sensitivity to be interpreted similarlyregardless of units of measurement.Two - factor example : Ri = aj + big FPLE + biz Fsize + Eiai - interceptBil = PLE, - PTEBil - Standardized PLE for stock iPIEFPIE - Return diff bet high & low PLE stockFsize - Return diff bet" high and low Size stocksBiz - Return diff bet high and tow Size stocksMacro Vs Fundamental ModelsMacro Factor ModelFundamental Factor ModelRegressionTime Series of sulpriseCross sectional asset returnFactor Sentivity (B)Regression basedStandarized from attribute dataFactor return (F)Suprises in macro variableComputed from multiple regressionInterreptExpected returnundefined Page No. 18Date / /Multifactor Models in Return AttributionFundamental Factor ModelUse = decompose Sources of an asset manager return relative to abenchmarkModel uses easily understand & communicated factorsCan express invest" style choices and SOC Characterstics in detailMore Commonly used than macro or statistically factor models.Active Risk & ReturnActive Return is the diff bet" the portfolio return (P) andits benchmark (B)Rp - RBActive Risk (tracking error / risk) is the standard deviation of theactive return. SD (of act AKA tracking error or tracking riskTE = OR(RP) = E(Rpt - RB+2n-1Inder IR is 0 : active return is 0Info ratio : Active ReturnActive RiskActive return per unit of active risk (standardized measures)Measures manager consistency in generating active returnsIR= (Rp - RB)b (Re - Ra)Similar to Sharpe Ratio for hedging Risk factor isEPage No. 19Date / /Sources of Active ReturnActive Retuin = factor return + sec selection returnKK=1(Portfolio B - Benchmark B) X Ar + Security selectionproduct of managers factorFactor ReturnManager's skill In individualtitsasset selection (residual)Active Risk DecompositionActive Risk Squared:CRp -RB) = active factor risk t active specific riskActive risk attributable toActive risk attribututublefactor f tiltsto stock selectionActive Specific Risk=E(We - WBi) o Ei Z1=1Residual risk of theith assetActive factor Risk = Residualset of factor sensetivitiesFactor Models in portfolio Constiuction designed to replicate thetracking Portfolio factor exposures of a benchmai indexIPassive Mary" : Use multifactor models to ensure portfolioreplicates an index's factor exposuresSensitivities of 0 to Remaining - Factor portfolio- sensitively I (Risk factol2Active Mag" Rely on factor multifactor models to predict alpha generatedfrom active bets on certain factors while hedgieyother. In constructing part/olio, multifactor modds areused to esd desired risk profiles". Enhanced indexing by matching primary risk factor could beimplimented by creating tracking folio with the same PageNo.20factor sensitivity as the index but with diff set of Date//bonds3Rules - based algorithmic active many" Calternative indicies): Userules to make factor fits low cost transparent Captureof some alphaStrategic Portfolio DecisionsTwo questions for investors:-1What Kind of risk do I have a Competitive advantage inbearning2What kind of risk do I have a Competitive disadvantagein bearning ?Example : Pension funds have long investment hurizons and as aresult are not exposed to liquidity riskAn employed worker reliant on income is exposed to biz cycleitCAPM leaves the investor with a choice bet" a nskfreeasset and the market portfolio.Multifactor models :- describles portfolio characterstic-Better- Provide better diversification Irisk mang" possibilitity- Provide more efficient portfolio if asset returns arebetter describes by multifactor models. Page No. 21Date / /Uses of Multifactor ModelsPerformance attributionRisk AnalysisReturn attributionRisk attributionFactorSecurityActive factorActive specificReturn SelectionRiskRiskMultilactor models are useful in both active & passive invest"Set of specific factorPassive Mang"Tracking portfolioPassive MangnFactor portfolioonly one factorAn investor Can achieve better diversified and efficient postfolio byusing SMB HML & WML factorsPotential benefits for investors In using a mdeffector modelGenerally , an investor gains from accepting above - - avg (below-avg)risk that he has Comparative advantage Comparative disadvatage)in bearing