MIT18_S096F13_CaseStudy1.pdf - Regression Analysis Case Study 1 Dr Kempthorne Contents 1 Linear Regression Models for Asset Pricing 1.1 CAPM Theory 1.2

MIT18_S096F13_CaseStudy1.pdf - Regression Analysis Case...

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Regression Analysis: Case Study 1 Dr. Kempthorne September 23, 2013 Contents 1 Linear Regression Models for Asset Pricing 2 1.1 CAPM Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Historical Financial Data . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Fitting the Linear Regression for CAPM . . . . . . . . . . . . . . 9 1.4 Regression Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . 10 1.5 Adding Macro-economic Factors to CAPM . . . . . . . . . . . . 16 1.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1
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1 Linear Regression Models for Asset Pricing 1.1 CAPM Theory Sharpe (1964) and Lintner (1965) developed the Capital Asset Pricing Model for a market in which investors have the same expectations, hold portfolios of risky assets that are mean-variance efficient, and can borrow and lend money freely at the same risk-free rate. In such a market, the expected return of asset j is E [ R j ] = R riskfree + β j ( E [ R Market ] - R riskfree ) β j = Cov [ R j , R Market ] /V ar [ R Market ] where R Market is the return on the market portfolio and R riskfree is the return on the risk-free asset. Consider fitting the simple linear regression model of a stock’s daily excess return on the market-portfolio daily excess return, using the S&P 500 Index as the proxy for the market return and the 3-month Treasury constant maturity rate as the risk-free rate. The linear model is given by: R j * ,t = α j + β j R M * arket,t + j,t , t = 1 , 2 , . . . where j,t are white noise: WN (0 , σ 2 ) Under the assumptions of the CAPM, the regression parameters ( α j , β j ) are such that β j is the same as in the CAPM model, and α j is zero. 1.2 Historical Financial Data Executing the R-script“fm casestudy 1 0.r”creates the time-series matrix casestudy 1 .data 0 . 00 which is available in the R-workspace “casestudy 1 0.Rdata”. > library("zoo") > load("casestudy_1_0.RData") > dim(casestudy1.data0.0) [1] 3373 12 > names(casestudy1.data0.00) [1] "BAC" "GE" "JDSU" "XOM" "SP500" [6] "DGS3MO" "DGS1" "DGS5" "DGS10" "DAAA" [11] "DBAA" "DCOILWTICO" > head(casestudy1.data0.00) BAC GE JDSU XOM SP500 DGS3MO DGS1 DGS5 DGS10 2000-01-03 15.79588 33.39834 752.00 28.83212 1455.22 5.48 6.09 6.50 6.58 2000-01-04 14.85673 32.06240 684.52 28.27985 1399.42 5.43 6.00 6.40 6.49 2000-01-05 15.01978 32.00674 633.00 29.82252 1402.11 5.44 6.05 6.51 6.62 2000-01-06 16.30458 32.43424 599.00 31.36519 1403.45 5.41 6.03 6.46 6.57 2
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2000-01-07 15.87740 33.69002 719.76 31.27315 1441.47 5.38 6.00 6.42 6.52 2000-01-10 15.32631 33.67666 801.52 30.83501 1457.60 5.42 6.07 6.49 6.57 DAAA DBAA DCOILWTICO 2000-01-03 7.75 8.27 NA 2000-01-04 7.69 8.21 25.56 2000-01-05 7.78 8.29 24.65 2000-01-06 7.72 8.24 24.79 2000-01-07 7.69 8.22 24.79 2000-01-10 7.72 8.27 24.71 > tail(casestudy1.data0.00) BAC GE JDSU XOM SP500 DGS3MO DGS1 DGS5 DGS10 DAAA 2013-05-23 13.20011 23.47254 13.17 91.79 1650.51 0.05 0.12 0.91 2.02 3.97 2013-05-24 13.23009 23.34357 13.07 91.53 1649.60 0.04 0.12 0.90 2.01 3.94 2013-05-28 13.34001 23.41301 13.37 92.38 1660.06 0.05 0.13 1.02 2.15 4.06 2013-05-29 13.46991 23.45269 13.56 92.08 1648.36 0.05 0.14 1.02 2.13 4.04 2013-05-30 13.81965 23.41301 13.73 92.09 1654.41 0.04 0.13 1.01 2.13 4.06 2013-05-31 13.64978 23.13523 13.62 90.47 1630.74 0.04 0.14 1.05 2.16 4.09 DBAA DCOILWTICO 2013-05-23 4.79 94.12 2013-05-24 4.76 93.84 2013-05-28 4.88 94.65 2013-05-29 4.88 93.13 2013-05-30 4.90 93.57 2013-05-31 4.95 91.93 We first plot the raw data for the stock GE , the market-portfolio index SP 500 , and the risk-free interest rate. 3
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> library ("graphics") > library("quantmod") > plot(casestudy1.data0.00[,"GE"],ylab="Price",main="GE Stock") 2000 2002 2004 2006 2008 2010 2012 5 10 15 20 25 30 35 40 Index Price GE Stock 4
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> plot(casestudy1.data0.00[,"SP500"], ylab="Value",main="S&P500 Index") 2000 2002 2004 2006 2008 2010 2012 800 1000 1200 1400 1600 Index Value S&P500 Index 5
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> plot(casestudy1.data0.00[,"DGS3MO"], ylab="Rate" , + main="3-Month Treasury Rate (Constant Maturity)") 2000 2002 2004 2006 2008 2010 2012 0 1 2 3 4 5 6 Index Rate 3-Month Treasury Rate (Constant Maturity) Now we construct the variables with the log daily returns of GE and the SP500 index as well as the risk-free asset returns > # Compute daily log returns of GE stock > r.daily.GE<-zoo( x=as.matrix(diff(log(casestudy1.data0.00[,"GE"]))), 6
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