Security Market LineRelative Risk, I. IntroductionII. Capital Asset Pricing ModelIII. Ordinary least squaresOct. 16, 2008 LEC #7 ECON 240A-1 L. PhillipsBivariate RelationshipsI. IntroductionIn much of economics we are interested in the relationship between one variable and another. For example, is the demand for tea sensitive to the price of coffee? How sensitive?We will look at an example that comes from the capital asset pricing model. How much of the variation in the monthly rate of return for the UC stock index fund is explained by variation in the monthly rate of return in the Standard and Poor’s Index? That is, how much does the rate of return on this specific asset, the UC stock index fund, depend on the market? What accounts for the variation that is not explained by the market?To answer these questions in this bivariate example, we will first rely on exploratory graphical analysis and look at a scatter plot of the data. Then we will see howwell a linear model fits the data. This will introduce us to ordinary least squares, and the estimation of the linear returns generating process for an individual security. The estimated slope of this linear model is the famous beta, an indicator of whether the UC stock index fund is more volatile or less volatile than the market. In this example, the estimated slope or beta is specific to the UC index fund. Since stock index funds are designed to match market behavior, the slope should be fairly close to one.In exploring the linear model various issues will arise. Is the relationship between the dependent variable and the independent variable linear, quadratic, or some other functional form? Often the exploratory graphical analysis provides a clue.How well does the linear model fit the data? We will develop measures of goodness of fit. Using our example of a returns generating process, we will estimate beta.Oct. 16, 2008 LEC #7 ECON 240A-2 L. PhillipsBivariate RelationshipsWhat is the expected value of this estimate? What is its variance? Is beta significantly different from zero? Is beta significantly different from one? We will develop hypothesis tests for this slope parameter estimated for the linear model.II. Capital Asset Pricing ModelThe data for the monthly rates of return for the UC stock index fund and for the Standard and Poor’s Index of 500 stocks is reproduced in Table I, along with the rate of return on the 30 day Treasury bill. -----------------------------------------------------------------------------------------Table I: Monthly Rate of Return, UC StockIndex Fund, S&P 500, 30 Day Treasury BillDate UC Stock Index S&P 500 30 Day TreasuryAugust 99 -2.46 -0.50 0.39September 99 -2.44 -2.74 0.39October 99 7.48 6.23 0.39November 99 3.79 2.03 0.36December 99 5.48 8.21 0.43January 2000 -1.95 -5.02 0.43February 2000 2.67 -1.89 0.43March 2000 8.78 9.78 0.47April 2000 1.45 -3.01 0.46May 2000 -0.56 -2.05 0.50June 2000 1.97 2.47 0.40July 2000 -2.03 -1.56 0.48A scatterplot of the monthly rate of return on the UC stock index fund versus the monthlyrate of return on the Standard and Poor’s Index of 500 stocks is illustrated in Figure 1.---------------------------------------------------------------------------------------------------Oct. 16, 2008 LEC #7 ECON 240A-3 L. PhillipsBivariate Relationships-------------------------------------------------------------------------------------------A plot offers a visual indication of the relationship between two variables, whether it is positive or negative, linear or nonlinear, and whether the relationship is tightor not, i.e. is the goodness of fit high or low. In this example, note that for ten of the twelve observations, the two variables have the same sign. Thus there is a strong indication of a relationship with a positive slope. There is no visual indication of nonlinearity. The data points do not lie along a straight line so the goodness of fit is not perfect.A scatterplot often offers a great deal of insight into the relationship between two variables. Furthermore, exploratory data analysis can be extended to look at the relationship between the dependent variable, y, and the explanatory variable, x, Figure 1: Scatterplot of Monthly Rates of Return-4-20246810-6 -4 -2 0 2 4 6 8 10 12Standard and Poor's Index of 500 StocksUC Stock Index FundOct. 16, 2008 LEC #7 ECON 240A-4 L. PhillipsBivariate Relationshipscontrolling for a third factor, w. this can be accomplished in various ways. One possibilityis to use different symbols for the data points. For example, in Figure 2, the data points plotted in Figure 1, are differentiated, using triangles for 1999 and squares for 2000.----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------There are probably too few data points to distinguish whether the slope or other aspects of the relationship between these two variables has changed between 1999 and 2000. We will return to the topic of graphical multivariate data analysis.The specification for the returns generating process for the UC stock index fund, as developed in the capital asset pricing model is:rUC (t) – rf (t) = + [rSP(t) – rf (t)] + eUC(t), (1)Figure 2: Scatterplot of Monthly Rates of Return, Triangles-1999, squares-2000-4-20246810-6 -4 -2 0 2 4 6 8 10 12Standard and Poor's Index of 500 StocksUC Stock Index FundOct. 16, 2008 LEC #7 ECON 240A-5 L. PhillipsBivariate Relationshipswhere rUC(t) is the monthly rate of return on the UC stock index fund, and rf(t) is the risk free rate, proxied by the 30 day Treasury Bill rate, and presumed known at the beginning of each month. The monthly rate of return for the market is measured in this example by rSP(t), the return for the Standard and Poor’s Index of 500 stocks. The error term, eUC(t), is a source of variation in the rate of return on the UC stock index fund that is specific to that asset. The parameter is the intercept of the linear relationship, i.e. the value of the dependent variable when the explanatory variable is zero, assuming an error of zero. In equilibrium, for the returns generating process, the intercept is expected to be zero. If the intercept were negative for example, the UC stock index fund would be in disequilibrium,and the expected return for this asset would be too low, as a consequence of being negative and dragging the return down. So, for example, if we
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