Investments, tenth edition

  

  Debt  

  

  

  Equity  

 Expected return,  E ( r )  

 8% 

 

 

  

 13% 

 Standard deviation,  s  

 12%  

 

  

 20% 

 Covariance, Cov( r 

  D 

 ,  r  

 E 

 )  

 

 72  

 

  

 Correlation coefficient,  r  

 DE 

  

  

  

 .30  

 

 Table 7.1 

 Descriptive statistics 

for two mutual funds 

  

2

 See Appendix B of this chapter for a review of portfolio statistics. 

bod61671_ch07_205-255.indd   208

bod61671_ch07_205-255.indd   208

6/18/13   8:11 PM

6/18/13   8:11 PM

Final PDF to printer

---

  C H A P T E R  

7

  Optimal Risky Portfolios 

209

In words, the variance of the portfolio is a weighted sum of covariances, and each weight 

is the product of the portfolio proportions of the pair of assets in the covariance term. 

  Table 7.2  shows how portfolio variance can be calculated from a spreadsheet. Panel A 

of the table shows the  bordered  covariance matrix of the returns of the two mutual funds. 

The bordered matrix is the covariance matrix with the portfolio weights for each fund 

placed on the borders, that is, along the first row and column. To find portfolio variance, 

multiply each element in the covariance matrix by the pair of portfolio weights in its row 

and column borders. Add up the resultant terms, and you have the formula for portfolio 

variance given in Equation 7.5.  

 We perform these calculations in panel B, which is the  border-multiplied   covariance 

matrix: Each covariance has been multiplied by the weights from the row and the column 

in the borders. The bottom line of panel B confirms that the sum of all the terms in this 

matrix (which we obtain by adding up the column sums) is indeed the portfolio variance 

in Equation 7.5. 

 This procedure works because the covariance matrix is symmetric around the diagonal, 

that is, Cov( r  

 D 

 ,  r  

 E 

 )  5  Cov( r  

 E 

 ,  r  

 D 

 ). Thus each covariance term appears twice. 

 This technique for computing the variance from the border-multiplied covariance matrix 

is general; it applies to any number of assets and is easily implemented on a spreadsheet. 

Concept Check 1 asks you to try the rule for a three-asset portfolio. Use this problem to 

verify that you are comfortable with this concept.