stock having zero exposure to this factorIQ,r ²R³iFβmktRMRF³βsizeSMB³βvalueHML³βliqLiiii
Example: Fama–French Modelmktsizevalueir ²R³βRMRF³βSMB³βHML²3%³1.20(5%)³0.12(2.2%)³0.34(3.8%)²10.56%iFiiPositive size beta•Small-cap stockbecause stock has positive exposure to the sizepremium.•Negative beta would indicate alarger than average stockPositive value beta• Value stock because stock haspositive exposure to the valuepremium• A negative beta would indicate agrowth stockExamining the sign of the betas, helps ascertain the stock’s characteristics1b|Risk-free rate3.0%Equity risk premium5.0%Beta1.20Size premium2.2%Size beta0.12Value premium3.8%Value beta0.34Positive value beta•Value stockbecause stockhas positive exposure•to the valuepremium•A negative beta would indicatea growth stock
ANOTHER Example: Fama–French ModelRisk-free rate3.0%Equity risk premium5.0%Beta1.20Size premium2.2%Size beta0.12Value premium3.8%Value beta0.34
4-FACTOR MODELSSelecting the PortfoliosFama-French-Carhart (FFC) Factor Specifications11 ( )  ²³´³³³MktSMBsfsMktfsSMBHMLPR YRsHMLsPR YRE RrE RrE RE RE Rbbbb
EXAMPLE: 4-FACTOR MODELSFFC Portfolio Average Monthly Returns, 1926–2008
Multi-factor models (3/3) – Macro economic model (APT)1bFactor sensitivity is asset’s sensitivity to a particular factor (holding all other factors constant) Factor risk premium is the factor’s expected return in excess of the risk-free rateArbitrage Pricing Theory (APT)Given N factor portfolios with returns RF1, . . . , RFN, the expected return of asset sis defined as:|εFβFβFβRfRSSGNPGNPII³³³³²β1…. βNare the factor betas.
Arbitrage Pricing Theory (APT)For example, suppose we have identified three systematic risks: Inflation, GNPgrowth, and the dollar-euro spot exchange rate, S($,€).The model is:betarateexchangespot theisbetaGNPtheisbetainflationtheisSGNPISSGNPGNPIIβββFβFβFβRfRεmRfR³³³²³³²
Arbitrage Pricing Theory (APT)Suppose we have made the following estimates:1.bI= -2.302.bGNP= 1.503.bS= 0.50εFβFβFβRfRSSGNPGNPII³³³³²%1²εSGNPIFFFRfR¶³¶³¶´²50.050.130.2
Arbitrage Pricing Theory (APT)We must decide what surprisestook place in the systematic factors. If it were the case that the inflation rate was expected to be 3%, but in fact was 8% during the time period, then: FI = Surprise in the inflation rate = actual – expected= 8% – 3% = 5%SGNPIFFFRfR¶³¶³¶´²50.050.130.2SGNPFFRfR¶³¶³¶´²50.050.1%530.2
Arbitrage Pricing Theory (APT)If it were the case that the rate of GNPgrowth was expected to be 4%, but in fact was 1%, then: FGNP= Surprise in the rate of GNPgrowth= actual – expected = 1% – 4% = – 3%SGNPFFRfR¶³¶³¶´²50.050.1%530.2SFRR¶³´¶³¶´²50.0%)3(50.1%530.2
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