Case Study 1: Linear Regression Models for Asset Pricing

Regression Analysis: Case Study 1 Dr. Kempthorne September 23, 2013 Contents 1 Linear Regression Models for Asset Pricing 1.1 CAPM Theory . . . . . . . . . . . . . . . . . 1.2 Historical Financial Data . . . . . . . . . . . 1.3 Fitting the Linear Regression for CAPM . . . 1.4 Regression Diagnostics . . . . . . . . . . . . . 1.5 Adding Macro-economic Factors to CAPM . 1.6 References . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 2 9 10 16 21 1 1.1 Linear Regression Models for Asset Pricing 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[Rj ] = Rriskf ree + βj (E[RM arket ] − Rriskf ree ) βj = Cov[Rj , RM arket ]/V ar[RM arket ] where RM arket is the return on the market portfolio and Rriskf ree 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: ∗ Rj∗,t = αj + βj RM arket,t + j,t , t = 1, 2, . . . where j,t are white noise: W N (0, σ 2 ) Under the assumptions of the CAPM, the regression parameters (αj , βj ) are such that βj is the same as in the CAPM model, and αj is zero. 1.2 Historical Financial Data Executing the R-script“fm casestudy 1 0.r”creates the time-series matrix casestudy1.data0.00 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" [6] "DGS3MO" [11] "DBAA" "GE" "JDSU" "DGS1" "DGS5" "DCOILWTICO" "XOM" "DGS10" "SP500" "DAAA" > head(casestudy1.data0.00) 2000-01-03 2000-01-04 2000-01-05 2000-01-06 BAC 15.79588 14.85673 15.01978 16.30458 GE 33.39834 32.06240 32.00674 32.43424 JDSU 752.00 684.52 633.00 599.00 2 XOM 28.83212 28.27985 29.82252 31.36519 SP500 DGS3MO DGS1 DGS5 DGS10 1455.22 5.48 6.09 6.50 6.58 1399.42 5.43 6.00 6.40 6.49 1402.11 5.44 6.05 6.51 6.62 1403.45 5.41 6.03 6.46 6.57 2000-01-07 15.87740 33.69002 719.76 31.27315 1441.47 2000-01-10 15.32631 33.67666 801.52 30.83501 1457.60 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 5.38 6.00 6.42 5.42 6.07 6.49 6.52 6.57 > tail(casestudy1.data0.00) 2013-05-23 2013-05-24 2013-05-28 2013-05-29 2013-05-30 2013-05-31 2013-05-23 2013-05-24 2013-05-28 2013-05-29 2013-05-30 2013-05-31 BAC GE 13.20011 23.47254 13.23009 23.34357 13.34001 23.41301 13.46991 23.45269 13.81965 23.41301 13.64978 23.13523 DBAA DCOILWTICO 4.79 94.12 4.76 93.84 4.88 94.65 4.88 93.13 4.90 93.57 4.95 91.93 JDSU 13.17 13.07 13.37 13.56 13.73 13.62 XOM 91.79 91.53 92.38 92.08 92.09 90.47 SP500 DGS3MO DGS1 DGS5 DGS10 DAAA 1650.51 0.05 0.12 0.91 2.02 3.97 1649.60 0.04 0.12 0.90 2.01 3.94 1660.06 0.05 0.13 1.02 2.15 4.06 1648.36 0.05 0.14 1.02 2.13 4.04 1654.41 0.04 0.13 1.01 2.13 4.06 1630.74 0.04 0.14 1.05 2.16 4.09 We first plot the raw data for the stock GE, the market-portfolio index SP 500, and the risk-free interest rate. 3 > library ("graphics") > library("quantmod") > plot(casestudy1.data0.00[,"GE"],ylab="Price",main="GE Stock") 25 20 15 10 5 Price 30 35 40 GE Stock 2000 2002 2004 2006 Index 4 2008 2010 2012 > plot(casestudy1.data0.00[,"SP500"], ylab="Value",main="S&P500 Index") 1200 1000 800 Value 1400 1600 S&P500 Index 2000 2002 2004 2006 Index 5 2008 2010 2012 > plot(casestudy1.data0.00[,"DGS3MO"], ylab="Rate" , + main="3-Month Treasury Rate (Constant Maturity)") 3 0 1 2 Rate 4 5 6 3−Month Treasury Rate (Constant Maturity) 2000 2002 2004 2006 2008 2010 2012 Index 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 + order.by=time(casestudy1.data0.00)[-1]) > dimnames(r.daily.GE)[[2]]<-"r.daily.GE" > dim(r.daily.GE) [1] 3372 1 > head(r.daily.GE) r.daily.GE 2000-01-04 -0.0408219945 2000-01-05 -0.0017376199 2000-01-06 0.0132681098 2000-01-07 0.0379869230 2000-01-10 -0.0003966156 2000-01-11 0.0016515280 > > + > > # Compute daily log returns of the SP500 index r.daily.SP500<-zoo( x=as.matrix(diff(log(casestudy1.data0.00[,"SP500"]))), order.by=time(casestudy1.data0.00)[-1]) dimnames(r.daily.SP500)[[2]]<-"r.daily.SP500" dim(r.daily.SP500) [1] 3372 1 > head(r.daily.SP500) r.daily.SP500 2000-01-04 -0.0390992269 2000-01-05 0.0019203798 2000-01-06 0.0009552461 2000-01-07 0.0267299353 2000-01-10 0.0111278213 2000-01-11 -0.0131486343 > > > > > + > > > > > > > # Compute daily return of the risk-free asset # accounting for the number of days between successive closing prices # apply annual interest rate using 360 days/year (standard on 360-day yearsince the pr r.daily.riskfree<-log(1 + .01*coredata(casestudy1.data0.00[-1,"DGS3MO"]) * diff(as.numeric(time(casestudy1.data0.00)))/360) dimnames(r.daily.riskfree)[[2]]<-"r.daily.riskfree" # Compute excess returns (over riskfree rate) r.daily.GE.0<-r.daily.GE - r.daily.riskfree dimnames(r.daily.GE.0)[[2]]<-"r.daily.GE.0" r.daily.SP500.0<-r.daily.SP500 - r.daily.riskfree dimnames(r.daily.SP500.0)[[2]]<-"r.daily.SP500.0" 7 > > > + > # Merge all the time series together, # and display first and last sets of rows r.daily.data0<-merge(r.daily.GE, r.daily.SP500, r.daily.riskfree, r.daily.GE.0, r.daily.SP500.0) head(r.daily.data0) r.daily.GE r.daily.SP500 r.daily.riskfree r.daily.GE.0 2000-01-04 -0.0408219945 -0.0390992269 0.0001508220 -0.0409728165 2000-01-05 -0.0017376199 0.0019203798 0.0001510997 -0.0018887196 2000-01-06 0.0132681098 0.0009552461 0.0001502665 0.0131178433 2000-01-07 0.0379869230 0.0267299353 0.0001494333 0.0378374897 2000-01-10 -0.0003966156 0.0111278213 0.0004515647 -0.0008481802 2000-01-11 0.0016515280 -0.0131486343 0.0001508220 0.0015007061 r.daily.SP500.0 2000-01-04 -0.0392500488 2000-01-05 0.0017692801 2000-01-06 0.0008049796 2000-01-07 0.0265805020 2000-01-10 0.0106762566 2000-01-11 -0.0132994562 > tail(r.daily.data0) 2013-05-23 2013-05-24 2013-05-28 2013-05-29 2013-05-30 2013-05-31 2013-05-23 2013-05-24 2013-05-28 2013-05-29 2013-05-30 2013-05-31 r.daily.GE r.daily.SP500 r.daily.riskfree r.daily.GE.0 -0.008417558 -0.0029281358 1.388888e-06 -0.008418947 -0.005509656 -0.0005514968 1.111110e-06 -0.005510767 0.002970509 0.0063209120 5.555540e-06 0.002964954 0.001693481 -0.0070728921 1.388888e-06 0.001692092 -0.001693481 0.0036635956 1.111110e-06 -0.001694592 -0.011935351 -0.0144105503 1.111110e-06 -0.011936462 r.daily.SP500.0 -0.0029295247 -0.0005526079 0.0063153565 -0.0070742809 0.0036624845 -0.0144116614 Now we plot the excess returns of GE vs those of the SP500: 8 0.15 0.10 0.05 0.00 −0.10 −0.05 r.daily.GE.0 −0.10 −0.05 0.00 0.05 r.daily.SP500.0 1.3 Fitting the Linear Regression for CAPM The linear regression model is fit using the R-function lm(): > > lmfit0<-lm(r.daily.GE.0 ~ r.daily.SP500.0, data=r.daily.data0) names(lmfit0) #element names of list object lmfit0 9 0.10 [1] "coefficients" "residuals" [5] "fitted.values" "assign" [9] "xlevels" "call" > "effects" "qr" "terms" "rank" "df.residual" "model" summary.lm(lmfit0) #function summarizing objects created by lm() Call: lm(formula = r.daily.GE.0 ~ r.daily.SP500.0, data = r.daily.data0) Residuals: Min 1Q Median -0.153166 -0.005605 -0.000334 3Q 0.005560 Max 0.137232 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -0.0001334 0.0002376 -0.561 0.575 r.daily.SP500.0 1.1843613 0.0177920 66.567 <2e-16 Residual standard error: 0.0138 on 3370 degrees of freedom Multiple R-squared: 0.568, Adjusted R-squared: 0.5679 F-statistic: 4431 on 1 and 3370 DF, p-value: < 2.2e-16 > Note that the t-statistic for the intercept αGE is not significant (-0.5613). 1.4 Regression Diagnostics Some useful R functions • anova.lm(): conduct an Analysis of Variance for the linear regression model, detailing the computation of the F -statistic for no regression structure. • inf luence.measures(): compute regression diagnostics evaluating case influence for the linear regression model; includes ‘hat’ matirx, case-deletion statistics for the regression coefficients and for the residual standard deviation. > # Compute influence measures (case-deletion statistics) > lmfit0.inflm<-influence.measures(lmfit0) > names(lmfit0.inflm) [1] "infmat" "is.inf" "call" > dim(lmfit0.inflm$infmat) [1] 3372 6 10 > head(lmfit0.inflm$infmat) dfb.1_ dfb.r..S dffit cov.r cook.d 2000-01-04 0.006987967 -0.0207373156 0.021908094 1.003354 2.400416e-04 2000-01-05 -0.004808670 -0.0006547183 -0.004850631 1.000850 1.176753e-05 2000-01-06 0.015354314 0.0009828679 0.015382160 1.000420 1.183126e-04 2000-01-07 0.008170676 0.0161694450 0.018089276 1.001941 1.636488e-04 2000-01-10 -0.016729492 -0.0133945658 -0.021391856 1.000525 2.288100e-04 2000-01-11 0.021629043 -0.0215352003 0.030579350 1.000239 4.674667e-04 hat 2000-01-04 0.0028508517 2000-01-05 0.0003020630 2000-01-06 0.0002977757 2000-01-07 0.0014754368 2000-01-10 0.0004878168 2000-01-11 0.0005883587 > head(lmfit0.inflm$is.inf) 2000-01-04 2000-01-05 2000-01-06 2000-01-07 2000-01-10 2000-01-11 dfb.1_ dfb.r..S dffit cov.r cook.d hat FALSE FALSE FALSE TRUE FALSE TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE > # Table counts of influential/non-influential cases > # as measured by the hat/leverage statistic. > table(lmfit0.inflm$is.inf[,"hat"]) FALSE 3243 TRUE 129 11 > > > > > + > > > > > + > > > + # Re-Plot data adding # fitted regression line # selective highlighting of influential cases plot(r.daily.SP500.0, r.daily.GE.0, main="GE vs SP500 Data \n OLS Fit (Green line)\n High-Leverage Cases (red points)\n H abline(h=0,v=0) abline(lmfit0, col=3, lwd=3) # Plot cases with high leverage as red (col=2) "o"s index.inf.hat<-which(lmfit0.inflm$is.inf[,"hat"]==TRUE) points(r.daily.SP500.0[index.inf.hat], r.daily.GE.0[index.inf.hat], col=2, pch="o") # Plot cases with high cooks distance as big (cex=2) blue (col=4) "X"s index.inf.cook.d<-which(lmfit0.inflm$is.inf[,"cook.d"]==TRUE) points(r.daily.SP500.0[index.inf.cook.d], r.daily.GE.0[index.inf.cook.d], col=4, pch="X", cex=2.) 12 GE vs SP500 Data OLS Fit (Green line) High−Leverage Cases (red points) High Cooks Dist (blue Xs) 0.15 o o oo oo o oooooo o oo oo o o o oo oo o o oo o oo o o oooooooooo ooo o o o 0.05 0.00 −0.05 −0.10 r.daily.GE.0 0.10 o oo o o o −0.10 o oo o ooo oo o oooo o o o ooo oooo o o ooo o o ooooo oo o o o oo o o o o o o oo o o o o −0.05 Xo 0.00 r.daily.SP500.0 13 o 0.05 0.10 > lmfit0.leverages<-zoo(lmfit0.inflm$infmat[,"hat"], order.by=time(r.daily.SP500.0)) > chartSeries(lmfit0.leverages) lmfit0.leverages [2000−01−04/2013−05−31] Last 0.000639405004758484 0.020 0.015 0.010 0.005 0.000 Jan 04 2000 Jul 01 2003 14 Jan 03 2007 Jul 01 2010 The R function plot.lm() generates a useful 2x2 display of plots for various regression diagnostic statistics: > layout(matrix(c(1,2,3,4),2,2)) # optional 4 graphs/page > plot(lmfit0) 0.00 3.0 2.0 1.0 0.10 −0.10 0.00 0.10 Residuals vs Leverage 2008−10−13 −2 0 2 1 0.5 Cook's distance 2008−10−13 0.000 Theoretical Quantiles 15 0.5 0 5 2009−03−10 2008−12−02 −10 5 2008−10−10 2009−02−09 10 Normal Q−Q Standardized residuals Fitted values 10 Fitted values 0 −10 Standardized residuals −0.10 2008−10−13 2008−10−10 2009−02−09 0.0 2008−10−13 Scale−Location Standardized residuals 0.00 0.10 2008−10−10 2009−02−09 −0.15 Residuals Residuals vs Fitted 0.010 Leverage 1 0.020 1.5 Adding Macro-economic Factors to CAPM The CAPM relates a stock’s return to that of the diversified market portfolio, proxied here by the S&P 500 Index. A stock’s return can depend on macro-economic factors, such commodity prices, interest rates, economic growth (GDP). > # The linear regression for the extended CAPM: > lmfit1<-lm( r.daily.GE.0 ~ r.daily.SP500.0 + r.daily.DCOILWTICO, data=r.daily.data00) > summary.lm(lmfit1) Call: lm(formula = r.daily.GE.0 ~ r.daily.SP500.0 + r.daily.DCOILWTICO, data = r.daily.data00) Residuals: Min 1Q Median -0.152977 -0.005567 -0.000260 3Q 0.005589 Max 0.133583 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -0.0001216 0.0002373 -0.512 0.608532 r.daily.SP500.0 1.1972374 0.0181296 66.038 < 2e-16 r.daily.DCOILWTICO -0.0342538 0.0096188 -3.561 0.000374 Residual standard error: 0.01378 on 3368 degrees of freedom (1 observation deleted due to missingness) Multiple R-squared: 0.5692, Adjusted R-squared: 0.5689 F-statistic: 2225 on 2 and 3368 DF, p-value: < 2.2e-16 The regression coefficient for the oil factor (r.daily.DCOILW T ICO) is statistically significant and negative. Over the analysis period, price changes in GE stock are negatively related to the price changes in oil. Consider the corresponding models for Exxon-Mobil stock, XOM > # The linear regression for the simple CAPM: > lmfit0<-lm( r.daily.XOM.0 ~ r.daily.SP500.0 , data=r.daily.data00) > summary.lm(lmfit0) Call: lm(formula = r.daily.XOM.0 ~ r.daily.SP500.0, data = r.daily.data00) Residuals: Min 1Q Median -0.085289 -0.005788 -0.000009 3Q 0.006230 Coefficients: 16 Max 0.113614 Estimate Std. Error t value Pr(>|t|) (Intercept) 0.0002968 0.0002105 1.41 0.159 r.daily.SP500.0 0.8299221 0.0157595 52.66 <2e-16 Residual standard error: 0.01222 on 3370 degrees of freedom Multiple R-squared: 0.4514, Adjusted R-squared: 0.4513 F-statistic: 2773 on 1 and 3370 DF, p-value: < 2.2e-16 > # The linear regression for the extended CAPM: > lmfit1<-lm( r.daily.XOM.0 ~ r.daily.SP500.0 + r.daily.DCOILWTICO.0, data=r.daily.data00) > summary.lm(lmfit1) Call: lm(formula = r.daily.XOM.0 ~ r.daily.SP500.0 + r.daily.DCOILWTICO.0, data = r.daily.data00) Residuals: Min 1Q -0.085977 -0.005564 Median 0.000010 3Q 0.005765 Max 0.105583 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 0.0002520 0.0002029 1.242 0.214 r.daily.SP500.0 0.7823785 0.0155009 50.473 <2e-16 r.daily.DCOILWTICO.0 0.1324461 0.0082237 16.105 <2e-16 Residual standard error: 0.01178 on 3368 degrees of freedom (1 observation deleted due to missingness) Multiple R-squared: 0.4906, Adjusted R-squared: 0.4903 F-statistic: 1622 on 2 and 3368 DF, p-value: < 2.2e-16 The R-squared for XOM is lower than for GE. Its relationship to the market index is less strong. The regression coefficient for the oil factor (r.daily.DCOILW T ICO) is statistically significant and positive. 17 For the extended model, we use the R function plot.lm() to display regression diagnostic statistics: > layout(matrix(c(1,2,3,4),2,2)) # optional 4 graphs/page > plot(lmfit1) 0.00 3.0 Scale−Location 1.0 2.0 2000−03−07 2001−01−03 2008−10−16 0.0 2001−01−03 −0.05 0.05 −0.05 0.00 0.05 Residuals vs Leverage 2001−01−03 −2 0 2 0 5 2008−10−13 2008−10−15 Cook's distance 0.000 Theoretical Quantiles 0.010 Leverage The high-leverage cases in the data are those which have high Mahalanobis distance from the center of the data in terms of the column space of the independent variables (see Regression Analysis Problem Set). 18 0.5 2008−10−16 −5 0 5 2000−03−07 2008−10−16 10 Normal Q−Q Standardized residuals Fitted values 10 Fitted values −5 Standardized residuals Standardized residuals 0.00 2000−03−07 2008−10−16 −0.10 Residuals 0.10 Residuals vs Fitted 0.5 0.020 We display the data in terms of the independent variables and highlight the high-leverage cases. > > > + + > # Refit the model using argument x=TRUE so that the lm object includes the # matrix of independent variables lmfit1<-lm(r.daily.XOM.0 ~ r.daily.SP500.0 + r.daily.DCOILWTICO, data=r.daily.data00, x=TRUE) names(lmfit1) [1] [5] [9] [13] "coefficients" "fitted.values" "na.action" "model" "residuals" "assign" "xlevels" "x" "effects" "qr" "call" "rank" "df.residual" "terms" > dim(lmfit1$x) [1] 3371 3 > head(lmfit1$x) 2000-01-05 2000-01-06 2000-01-07 2000-01-10 2000-01-11 2000-01-12 (Intercept) r.daily.SP500.0 r.daily.DCOILWTICO 1 0.0017692801 -0.036251729 1 0.0008049796 0.005663446 1 0.0265805020 0.000000000 1 0.0106762566 -0.003232326 1 -0.0132994562 0.038893791 1 -0.0045473564 0.023467128 We now compute the leverage (and other influence measures) with the function inf luence.measures() and display the scatter plot of the independent variables, highlighting the high-leverage cases. > > > > > > + + > lmfit1.inflm<-influence.measures(lmfit1) index.inf.hat<-which(lmfit1.inflm$is.inf[,"hat"]==TRUE) par(mfcol=c(1,1)) plot(lmfit1$x[,2], lmfit1$x[,3],xlab="r.daily.SP500.0", ylab="r.daily.DCOILWTICO.0") title(main="Scatter Plot of Independent Variables \n High Leverage Points (red o s)") points(lmfit1$x[index.inf.hat,2], lmfit1$x[index.inf.hat,3], col=2, pch="o") 19 Scatter Plot of Independent Variables High Leverage Points (red o s) 0.15 o 0.10 o oo oo o o oo o o o o oo oo o oo o o oo o o o o o o o o oo ooooo o ooo o o o o o o oo o oo ooo o o oo oo o o o o o ooo oo ooo o o o o o oo o ooo o o o oo o oo oo o o o o o o o oooo o o ooo oo o oo oo o o o o oo o 0.05 o 0.00 −0.15 −0.10 −0.05 r.daily.DCOILWTICO.0 o o o o o o o o o o o −0.10 −0.05 0.00 r.daily.SP500.0 20 0.05 0.10 1.6 References Lintner, J. (1965). “The Valuation of Risky Assets and the Selection of Risky Investments in Stock Portfolio and Capital Budgets,” Review of Economics and Statistics, 47: 13-37. Sharpe, W. (1964). “Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk,” Journal of Finance, 19: 425-442. 21