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Calculating the mean-variance efficient frontier For this exercise, we will use the same industry portfolios that you considered in section 2 for the industry momentum strategy. (1) Calculate monthly gross excess returns for these portfolios by subtracting the risk-free rate, then load the data on historical excess returns into your statistics package. (2) Calculate the vector of means and the variance-covariance matrix, let's denote these by r (since these are net returns) and x. (There is no need to report anything here!) (3) Trace out the hyperbola which characterizes the set of portfolios which minimize variance subject to matching a particular level of expected return. To do this, recall the two-fund theorem from lecture 10, which showed that any portfolio on the hyperbola satisfies w° = a + so you can trace out the entire plot by varying a from a fairly large negative number (e.g. -3) and a fairly large positive number (e.g., 3). You can compute means and variances using the formulas from class: mean = W*F std deviation = (4) To the existing plot, overlay the capital allocation line connecting the tangency portfolio (just assumerg the net risk-free rate, equals 1) and scatter plots with the individual means and standard deviations of the individual assets. Congrats! You will have successfully reproduced a figure that appears in every finance textbook. 7

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(1)
data<-read.csv("Industry_Momentum_Data.csv",header=TRUE)
data.adj <- apply(data[,2:13],2,function(c) {c-data[,17]})

(2)
r <- as.matrix(apply(data.adj,2,mean))
Sigma <- cov(data.adj)...

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