6.7 THREEFIRM CASE: CML
The objective is to find the line with the largest slope connecting the riskfree asset with points on the minimumvariance frontier. This line will be tangent to the minimumvariance frontier. You can apply CAPM Tutor to find this point of tangency and see the CML using the default data set. If you change the riskfree interest rate, you will see how the point of tangency and the slope of the CML also change.
We start by considering three lines that intersect the frontier. These are portfolios having different target returns:
Target Return 

Portfolio I: 
0.158 

Portfolio II: 
0.159 

Portfolio III: 
0.160 
You learned the method for determining the portfolio weights that correspond to a particular target return E in the ThreeFirm Case. The equations for the portfolio weights as a function of E are:
Firm 1 = 6.288E  0.460
Firm 2 = 1.153E + 0.338
Firm 3 = 5.075E + 1.122
Substituting our three targets, 0.158, 0.159, and 0.160, for E yields the results in Table 6.3.
Table 6.3 
Target Returns and Portfolio Weights 
Portfolio I 
Portfolio II 
Portfolio III 

Target Expected Return 
0.158 
0.159 
0.160 
Firm 1 weight 
0.524 
0.530 
0.536 
Firm 2 weight 
0.156 
0.155 
0.154 
Firm 3 weight 
0.320 
0.315 
0.310 
To calculate the riskiness of each portfolio, recall the variancecovariance matrix for the ThreeFirm Case:
0.961 
0.063 
0.582 
0.063 
0.928 
0.359 
0.582 
0.359 
0.727 
The diagonal terms are the security return variances for securities 1, 2, and 3, respectively. The offdiagonal terms are the covariances. For example, the variance of return for security 2 is 0.928 and the covariance between securities 1 and 2 is 0.063.
The standard deviation of the portfolios can be calculated using the formula
The resulting standard deviations for Portfolios I, II, and III are:
Table 6.4 
Portfolio Standard Deviations and Slopes 
Portfolio I 
Portfolio II 
Portfolio III 

Target Expected Return 
0.158 
0.159 
0.160 
Standard Deviation 
0.328 
0.337 
0.346 
Slope 
0.115751 
0.115755 
0.115752 
Finally, the slope of the line between the riskfree security and the portfolio is computed by taking the vertical rise over the horizontal run. The vertical rise is the difference between the target return and the riskfree rate. The horizontal run is the portfolio's standard deviation (because the standard deviation for the riskfree security is zero).
Using CAPM Tutor you can verify that this slope of Portfolio I is 0.115751, as in Table 6.4.
The maximum slope is attained by the Portfolio II, with a target return equal to 0.159. This is actually the point of tangency between the capital market line and the efficient frontier. Therefore, the line passing through the riskfree rate and Portfolio II represents the capital market line for the ThreeFirm Case.
The CML describes a linear risk/return tradeoff for the market as a whole, but what type of tradeoff is implied for individual securities? This question is answered in topic 6.8, titled Security Market Line.
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