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w1 Cov(r1,r1) Cov(r1,r2) . . . Cov(r1,rGM) . . . Cov(r1,rn) w2 Cov(r2,r1) Cov(r2,r2) . . . Cov(r2,rGM) . . . Cov(r2,rn)


wGM Cov(rGM,r1) Cov(rGM,r2) . . . Cov(rGM,rGM) . . . Cov(rGM,rn) wn Cov(rn,r1) Cov(rn,r2) . . . Cov(rn,rGM) . . . Cov(rn,rn)     Recall that we calculate the variance of the portfolio by summing over all the elements of the covariance matrix, first multiplying each element by the portfolio weights from the row and the column. The contribution of one stock to portfolio variance therefore can be ex- pressed as the sum of all the covariance terms in the row corresponding to the stock, where each covariance is first multiplied by both the stocks weight from its row and the weight from its column.6 For example, the contribution of GMs stock to the variance of the market portfolio is wGM[w1Cov(r1,rGM) w2Cov(r2,rGM)  wGMCov(rGM,rGM)  wnCov(rn,rGM)] (9.3) Equation 9.3 provides a clue about the respective roles of variance and covariance in de- termining asset risk. When there are many stocks in the economy, there will be many more covariance terms than variance terms. Consequently, the covariance of a particular stock with all other stocks will dominate that stocks contribution to total portfolio risk. We may summarize the terms in square brackets in equation 9.3 simply as the covariance of GM       6 An alternative approach would be to measure GMs contribution to market variance as the sum of the elements in the row and the column corresponding to GM. In this case, GMs contribution would be twice the sum in equation 9.3. The approach that we take in the text allocates contributions to portfolio risk among securities in a convenient manner in that the sum of the contribu- tions of each stock equals the total portfolio variance, whereas the alternative measure of contribution would sum to twice the port- folio variance. This results from a type of double-counting, because adding both the rows and the columns for each stock would result in each entry in the matrix being added twice. III. Equilibrium In Capital Markets 9. The Capital Asset Pricing Model