证券投资学英文课件：07 Factor Models

1、InvestmentsLecture 7Factor ModelsnIt is difficult, sometimes even impossible, to construct the efficient frontier without making further assumptions about the return-generating process.nOne process weve already seen is the market model (index model):iIIiIiIirrFactor modelnA small number of underlyin

2、g basic sources of randomness: factorsnNote that the factors can be anythingnThey can be macroeconomic variables (inflation, GDP-growth, )nThey can be average industry returnsnThey can be statistical factors, which do not have any economic meaningSingle-factor model0)(0IIeEeEwhereeIbariiiiiiIraSlope

3、=b222222/ ),cov( ,IiiIjiijeIiiiiiIrbjibbbIbariMultifactor modelnFactors could be correlated.0)( , 0)( .2211jjijiininiiiiIIeEjieeEeIbIbIbaR21221112222211),cov(),cov(IIiIiiIIiIiibbIrbbIrStock Return Characteristics and 2-Factor ModelsnExpected ReturnnVariance)()()(2211IEbIEbrEiiii221212222212),cov(221

4、ieiiIiIiiIIbbbbStock Return Characteristics),cov()(21122122221121IIbbbbbbbbjijiIjiIjiij CovariancePortfolio CharacteristicsnFactor loadings, etc. are weighted average of component loadings:NiijipjX1ExampleCalculate the expected return and standard deviation of the following portfolio:Factor 1 (2) ha

5、s an expected value of 15% (4%) and a standard deviation of 20% (5%). The factors are uncorrelated.SecurityZero FactorFactor 1 SensitivityFactor 2 SensitivityNonfactor RiskProportionA2%.32196.7B3.51.8100.3Selection of factorsnExternal factors: GDP, CPI, nExtracted factors:nConstructed from returns o

6、f securitiesnIndustrial factorsnComplex waysnFirm characteristics: effective additionsExample:3-Factor Modeln3-Factor Model (Fama-French)n1st factor: Market Premium (rM rF )nThe difference between the market return and the risk-free raten2nd factor: Small-Stock Premium (rS rL )nThe difference betwee

7、n the return of small and large companies, measured according to their market capitalizationn3nd factor: Value-Stock Premium (rV rG )nThe difference between the return of value and growth stocks. Value stocks are mature companies and growth stocks are companies with large growth potentialExample:3-F

8、actor ModelnYou estimate the 3-Factor model for a mutual fund. You get the following betas:nMarket beta: M = 1.30nSize beta: S = 0.45nValue beta: V = - 0.25nWhat investment strategy does the fund follow?Example:3-Factor ModelnIn one month the fund has an excess return of 12% above the risk-free inte

9、rest ratenThe factor returns are as follows:nMarket premium: rM rF = 10%nSmall-stock premium: rS - rL = -2%nValue-stock premium: rV rG = -3%nWhat is the abnormal return (the alpha) of the fund?Why are Factor Models Useful?nImposing a factor structure makes estimating the efficient frontier possible.

10、nIf you can identify portfolios that have only factor risk in them, they can be used to build the efficient frontier.nThe returns on these portfolios are called factor mimicking returns.Example: Building “factor mimicing portfolios”nSuppose three investments x, y, and z have the following returns: n

11、To form the factor 1 portfolio solve the system:nThis gives:nHence:and the factor premium is:xxFFr213208. 0yyFFr212310. 0zzFFr215310. 03/4; 3/1; 2zyxXXX105231332zyxzyxzyxXXXXXXXXXfrF 1106. 0211006. 0FFRpBuilding the Efficient Frontier using Factor-Mimicking PortfoliosnWhen the efficient frontier has

12、 only factor risk, we can restrict our attention to the factor-mimicking returns only.CAPM as a factor modelnThe assumption of efficiencyAnother Way of Writing Factor ModelnThis means that, instead of defining an f directly as economic growth, we would have to define it as the deviation of economic

13、growth from what was expected.0)(.)(22,11 ,jiiiiifEefbfbrErArbitrage Pricing Theory (APT)nArbitrage: the law of one pricenLess strict assumptions:nUtility function: not necessary nMore than mean and variancenStill homogeneous beliefnReturn generating processnInfinite number of securitiesExampleAPT0)

14、( , 0)( .2211jjijiininiiiiIIeEjieeEeIbIbIbaRSimple APTnAn idealized special casenConstruct a portfolioIbarIbarjjjiiiIbwwbawwarrwwrrjijiji)1 ()1 ()1 (cbacbababbbabarbbbwiiiijjijijjiijj0000iiiiiiibrIbcbIbar00nPrice of risknFactor priceGenerally in equilibrium we havepremiumriskifactorbbbrininiii :.221

15、10ArbitragenAn arbitrage is a trading strategy that generates only positive cash flowsnNo initial investment is required upfrontnStrictly positive cash flows occur either today or sometimes in the futurenUnder no conditions can you lose any moneynArbitrages follow from mispricings of assetsTrading S

16、trategynSuppose asset returns are given by:nYou have the following assets:nAsset S: E(rS)=15% S=1 S=40%nAsset D: E(rD)=5% D =1 D=40%nMarket: E(rM)=10% M =1 M=20%nT-Bills:rF=2% F =0 F=0%nWhat could you do?iFMiiFirrrrTrading StrategynBuy S and short-sell DnIf we ignore margin requirements, then you do

17、 not need to put any money down. You just use the proceeds from the short-sale of D to buy SnWhat is the expected return and the variance of your portfolio?Trading StrategynBuy S and short-sell DnThe return of this portfolio is:DSDSDSDSFMDSDSDSPrrrrr1 . 0Example of Trading StrategynThe expected retu

18、rn is:nThe variance and standard deviation are:nNote that the variance of the firm-specific risk is given by the variance of the individual stocks minus the systematic variance: %10DSPrE 12. 02 . 014 . 0222222MiiVar 24. 02VarVarrVarDSP %4924. 0PPrVarExample of Trading StrategynNote that this strateg

19、y is very risky with an expected return of 10% and a standard deviation of 49%nWhat happens to the risk if we include additional stocks?nSuppose we have two stocks of type S and two stocks of type DnBuy half of each S-stock and short-sell half of each D-stockExample of Trading StrategynThis portfoli

20、o has the following returns: 2,1 ,2,1 ,2,1 ,2,1 ,5 . 05 . 05 . 0DDSSDSDDSSPrrrrr %10DSPrE 12. 0425. 025. 05 . 02,1 ,2,1 ,2,1 ,2,1 ,VarVarVarVarrVarDDSSDDSSP %3512. 0PPrVarTrading StrategynBy introducing additional assets we can decrease the firm-specific risk through diversificationnBy short-selling

21、 asset D and by buying asset S we completely eliminate systematic risknThus, the total risk of this trading strategy approaches zero if the number of available stocks increasesExample of Trading StrategynYou can show that the variance of the portfolio with N assets is:nThus, as the number of assets

22、increases, the risk of our portfolio will go towards zero VarNrVarP2Standard Deviation of Trading Strategy00.40.5110100100010000Number of StocksStandard Deviation of StrategyTrading StrategynIf we have 10,000 stocks of each type, then the standard deviation of the portfolio decreases to jus

23、t 0.5%nNow, this strategy looks very attractive and many investors will undertake this trading strategynWhat happens to the prices of the stocks and their alphas?Arbitrage PricingnIn equilibrium, such attractive trading strategies are not possiblenWhat happens to the stock prices and the alphas if m

24、any investors follow such trading strategies?Arbitrage PricingnThe alpha of a large number of assets can not deviate from zero in equilibriumnThis implies that almost all assets have zero alphas:iFMiFirrrrFE(r)(%)PortfolioFE(r)(%)Individual SecurityPortfolio & Individual Security ComparisonE(r)%

25、Beta for F1076Risk Free = 4ADC.51.0Disequilibrium ExampleDisequilibrium ExamplenShort Portfolio CnUse funds to construct an equivalent risk higher return Portfolio DnD is comprised of A & Risk-Free AssetnArbitrage profit of 1%The Arbitrage Pricing TheorynThe CAPM is a one factor model:nThe APT i

26、s a multi-factor model:iFMiFirrrriFNNiFiFirrrrrr11The Arbitrage Pricing TheorynNote that the factors can be anythingnThey can be macroeconomic variables (inflation, GDP-growth, )nThey can be average industry returnsnThey can be statistical factors, which do not have any economic meaningExample:3-Fac

27、tor Modeln3-Factor Model (Fama-French)n1st factor: Market Premium (rM rF )nThe difference between the market return and the risk-free raten2nd factor: Small-Stock Premium (rS rL )nThe difference between the return of small and large companies, measured according to their market capitalizationn3nd fa

28、ctor: Value-Stock Premium (rV rG )nThe difference between the return of value and growth stocks. Value stocks are mature companies and growth stocks are companies with large growth potentialChen, Roll and Ross (1986)nMonthly and annual unanticipated growth in industrial production (YP, MP)nChanges i

29、n expected inflation, as measured by the change in rTBill (DEI).nUnexpected inflation (UI)nUnanticipated changes in risk premiums, as measured by rBaa - rAAA (UPR). This is often called the “Default Spread”nUnanticipated changes in the slope of the term structure, as measured by rT-Bond-rT-Bill. (UT

30、S). This is often called the “Term Spread”Example: one-factor modelnDo these expected returns and factor sensitivities represent an equilibrium?iEribiStock 115%0.9Stock 2213.0Stock 3121.8Arbitrage portfolio000332211332211321XrXrXrXbXbXbXXX)175. 0 ,075. 0 , 1 . 0( :tryAnother way130312021101)?(?brChe

31、ckbrbrSolveIn equilibriumnSuppose nWhat are the equilibrium prices of 3 stocks? %4%810E(r)%Beta for F1076Risk Free = 4ADC.51.0Another Way of Writing Factor ModelnThis means that, instead of defining an f directly as economic growth, we would have to define it as the deviation of economic growth from

32、 what was expected.0)(.)(22,11 ,jiiiiifEefbfbrErA Simple ExamplenRemember that Expected Return is precisely equivalent to the Discount Rate that investors are applying to the expected cash flows.Second SecuritynThe price investors will pay for DELL will be less than $100, even though DELLs expected

33、cash flows are the same as IBMs. nThe discount rate investors will apply to DELLs cash flows will be higher than the 20% applied to IBMs cash flows.nThe expected return investors will require from DELL will be higher than the IBMs expected return of 20%.nLets assume that investors are only willing t

34、o buy up all of DELLs shares if the price of DELL is $90, or, quivalently, that the discount rate that they will apply to DELL is 33.33%How this relates to APTnCalculating the business cycle factor fBC in the two states. The indicator is one (at the end of the next year) if the economy is in an expa

35、nsion, and zero if the economy is in a recession.nAssuming there is a 50%/50% chance that we will be in an expansion/recession, the expected value of the indicator is 0.5.nThis means that the business-cycle factor has a value of 0.5 = 1-0.5 if the economy booms, and -0.5 = 0-0.5 if the economy goes

36、bust.Factor LoadingnThe way of doing this is just running a time-series regression of the returns of IBM on the factor.nHere, since there are only two things that can happen at each point in time (a boom or a bust), estimating the coefficients is the same as finding the coefficients that fit the equ

37、ations in the boom, and in the bust.ResultsnSolving these gives E(rIBM) = 0.20 (which we already knew) and bIBM,BC =0.4.nSince DELLs returns in the boom and bust are 77.78% and -11.11%, respectively, similar calculations for DELL gives E(rDELL) = 0.3333 and bDELL,BC = 0.8889.Factor Risk PremiumnThe

38、RGP tells us nothing about why investors are discounting the cash flows from the different securities at different rates.nTo determine how investors view the risks associated with each of the risks in the economy, we have to evaluate the APT Pricing Equation.nThe factor risk premium is a measure of

39、how much more investors discount a stock as a result of having one extra unit of risk relating to the BC factor.Finding ArbitragesnArbitrage arises when the price of risk differs across securities.nIf investors are pricing risk inconsistently across securities, then, assuming these securities are we

40、ll diversified, arbitrage will be possible.nLets extend the example to include a risk-free asset which we can buy or sell at a rate of 5%.ArbitragenSince lamda0 = 0.0909, we know that we can combine DELL and IBM in such a way that we can create a synthetic risk-free asset with a return of 9.09%.nSo

41、we borrow money at 5% (by selling the risk-free asset) and lend money at 9.09%Building PortfolionTo determine how much of IBM and DELL we buy/sell, we solve the equation for the weights on IBM and DELL in a portfolio which is risk-free.InterpretationTwo issuesnFirst, define a qualified modelnSecond,

42、 find the correct set of factor forecastsA qualified modelnAny factor model that is good at explaining the risk of a diversified portfolio should be (nearly) qualifeid as an APT model.Factor ForecastsnThe simplest approach to forecasting factor is to calculate a history of factor returns and take their average.nA factor relationship is more stable than a stock relationshipnFactor fo