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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):Factor modelnA small number of underlying basic so
2、urces 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 modelIraSlope=bMultifactor modelnFactors could be
3、 correlated.Stock Return Characteristics and 2-Factor ModelsnExpected ReturnnVarianceStock Return CharacteristicsCovariancePortfolio CharacteristicsnFactor loadings, etc. are weighted average of component loadings:ExampleCalculate the expected return and standard deviation of the following portfolio
4、:Factor 1 (2) has 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:nConstructe
5、d from returns of 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 d
6、ifference between 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 pote
7、ntialExample:3-Factor 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 th
8、e risk-free interest 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 fr
9、ontier possible.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 foll
10、owing returns: nTo form the factor 1 portfolio solve the system:nThis gives:nHence:and the factor premium is:Building the Efficient Frontier using Factor-Mimicking PortfoliosnWhen the efficient frontier has only factor risk, we can restrict our attention to the factor-mimicking returns only.CAPM as
11、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 growth from what was expected.Arbitrage Pricing Theory (APT)nArbitrage: the law of one pricenL
12、ess strict assumptions:nUtility function: not necessary nMore than mean and variancenStill homogeneous beliefnReturn generating processnInfinite number of securitiesExampleAPTSimple APTnAn idealized special casenConstruct a portfolionPrice of risknFactor priceGenerally in equilibrium we haveArbitrag
13、enAn 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 StrategynSu
14、ppose 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?Trading StrategynBuy S and short-sell DnIf we ignore margin requirements, then you do not need to put any
15、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:Example of Trading StrategynThe expected return is:nThe variance and standard deviation are:n
16、Note that the variance of the firm-specific risk is given by the variance of the individual stocks minus the systematic variance:Example of Trading StrategynNote that this strategy is very risky with an expected return of 10% and a standard deviation of 49%nWhat happens to the risk if we include add
17、itional 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 portfolio has the following returns: Trading StrategynBy introducing additional assets we can decrease the firm-specific risk thr
18、ough diversificationnBy short-selling 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 asse
19、ts is:nThus, as the number of assets increases, the risk of our portfolio will go towards zeroStandard Deviation of Trading StrategyTrading StrategynIf we have 10,000 stocks of each type, then the standard deviation of the portfolio decreases to just 0.5%nNow, this strategy looks very attractive and
20、 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 many investors follow such trading strategies?Arbitr
21、age PricingnThe alpha of a large number of assets can not deviate from zero in equilibriumnThis implies that almost all assets have zero alphas:FE(r)(%)PortfolioFE(r)(%)Individual SecurityPortfolio & Individual Security ComparisonE(r)%Beta for F1076Risk Free = 4ADC.51.0Disequilibrium ExampleDisequil
22、ibrium 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 is a multi-factor model:The Arbitrage Pricing TheorynNote that the fac
23、tors 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-Factor Modeln3-Factor Model (Fama-French)n1st factor: Market Premium (rM rF )nThe difference betwee
24、n 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 factor: Value-Stock Premium (rV rG )nThe difference between the return of value and growth stocks.
25、 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 in expected inflation, as measured by the change in rTBill (DEI).nUnexpected inflation (UI)nUnant
26、icipated 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. (UTS). This is often called the “Term Spread”Example: one-factor modelnDo these expected returns an
27、d factor sensitivities represent an equilibrium?iEribiStock 115%0.9Stock 2213.0Stock 3121.8Arbitrage portfolioAnother wayIn equilibriumnSuppose nWhat are the equilibrium prices of 3 stocks? E(r)%Beta for F1076Risk Free = 4ADC.51.0Another Way of Writing Factor ModelnThis means that, instead of defini
28、ng an f directly as economic growth, we would have to define it as the deviation of economic growth from what was expected.A Simple ExamplenRemember that Expected Return is precisely equivalent to the Discount Rate that investors are applying to the expected cash flows.Second SecuritynThe price inve
29、stors will pay for DELL will be less than $100, even though DELLs expected 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
30、 IBMs expected return of 20%.nLets assume that investors are only willing to 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
31、indicator is one (at the end of the next year) if the economy is in an expansion, 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
32、 of 0.5 = 1-0.5 if the economy booms, and -0.5 = 0-0.5 if the economy goes 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), estimatin
33、g the coefficients is the same as finding the coefficients that fit the equations 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
34、DELL gives E(rDELL) = 0.3333 and bDELL,BC = 0.8889.Factor Risk PremiumnThe 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
35、evaluate the APT Pricing Equation.nThe factor risk premium is a measure of 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 ri
36、sk inconsistently across securities, then, assuming these securities are well 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
37、 that we can create a synthetic risk-free asset with a return of 9.09%.nSo 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
38、 risk-free.InterpretationTwo issuesnFirst, define a qualified modelnSecond, 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
39、 forecasting factor is to calculate a history of factor returns and take their average.nA factor relationship is more stable than a stock relationshipnFactor forecasts are difficult. Structure may help.ApplicationsnStructure model 1: Given exposures, estimate factor returns:nBARRA modelnStructure mo
40、del 2: Given factor returns, estimate exposures:nThe index modelnStructure model 3: combine model 1 and 2. An examplenFirst, we classify stocks by their industry membershipnThen choose four attributes:nA forecast of earnings growth based on the consensus forecast and past realized earnings growthnBo
41、nd beta: sensitivity to bond indexnSize: log of equity capitalizationnReturn on equity (ROE): earnings/booknIndustry ThennStandardize the exposure by subtracting the market average from each attribute and dividing by the standard deviation.nIn this way, the average exposure will equal zero, roughly
42、66% of the stocks will have exposures running from -1 up to +1, and only 5% of the stocks will have exposures above +2 or below -2Factor forecastsnThe factor forecasts are 2% for growth, 2.5% for bond beta, -1.5% for size and 0% for ROE. n8% for the chemical industry, and 6% for all the other indust
43、ries.nCAPM forecast is 6% market excess returnnE.g. IBM:nFactor model: 0.06+0.51* 0.02+(-0.61*0.025 )+1.16*(-0.015) -0.62*0=3.7%BARRA modelHaugen and Baker (1996)Application (contd)nStatistical model 1: Principal components analysisnStatistical model 2: maximum likelihood factor analysisnStatistical model 3: the dual of statistical model 2nConnor and Korajczyk (1988)
