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Part VI: More SPLIT/APPLY/COMBINE Suppose that Boeing has a fixed amount of money to invest into itself (Y) and another company (X). We must decide what fraction (a) of money to invest in Y = BA and (1-a) in stock X. The total returns (W) are modeled by the relation W = ax + (1 -a)Y. 6 To determine the optimal percentage a, we minimize the variance of W, i.e., minimize the expression var(W)=n'o + (1 -a)poz +20(1 - a)cov(X,Y). The univariate derivative of total variance var(W) with respect to a is: d var(W)= 200 +20o - 203 +2cou(X,Y) - 4acov(X,Y) da Consequently, the optimal value of a is o - Cov(X,Y) o toz - 2c00(X,Y) Hence, we can choose a reasonable estimator of a as a - Cv(X,Y) of +- 2cv(X,Y) In R, you can easily estimate a using: #(var(Y)-cov (X,Y))/(var (X) +var(Y)-2+cov (X,Y)) Problem 15 Compute the estimated a based on Y=BA versus X=AXP. Conclude what percentage to invest in each stock. Solution ### Solution goes here Problem 16 Now suppose that Boeing is interested in a stock buy-back but also wants to invest in one other company. To determine which company to invest in, the data scientist compares the sample correlations of B A versus all other tickers in the DJIA. Further, you also store the estimated a value for each DJIA ticker verses BA. Define a function BA_relationships with inputs: ticker, BA and data. The BA relationships function should output a vector of length 2 with the sample correlation and estimated alpha value. Test BA_relationships on the ticker AXP. Solution ### Solution goes here Problem 17 Run a SPLIT/APPLY/COMBINE procedure to store sample correlations of B A verses all stocks and the corresponding estimated a values. Display the head and dimension of your resulting dataframe after sorting its rows by correlation. Note that your resulting dataframe should have dimension (29 x 2) and GS has the highest correlation with BA among all tickers in the DJIA. Solution ### Solution goes here

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