Equation for the capital market line?

Question

Assume that two securities constitute the market portfolio. Those securities have the following expected returns, standard deviations, and proportions:

Security A

Expected return 10%

Standard Deviation 20%

Proportion .4

Security B

Expected return 15%

Standard Deviation 28%

Proportion .6

Based on this information, and given a correlation of .30 between the two securities and a risk-free rate of 5%, specify the equation for the capital market line.

Answer 1

The capital market line (CML) will have the form:

E(r_c) = r(f) + stdev(c) * [ E(r_m) - r(f) ] / stdev(m)

where:

E(r_c) = the expected return on portfolio c

r(f) = the risk-free rate

stdev(c) = the standard deviation of portfolio c

E(r_m) = the expected return on the market portfolio m

stdev(m) = the standard deviation of the market portfolio m

From here, we just have to solve or plug in the values for the following variables:

r(f), E(r_m), and stdev(m)

r(f) = 0.05 = 5.0%

(this was given in the information above);

E(r_m) = (0.1*0.4) + (0.15*0.6) = 0.13 = 13.0%

(calculated as a weighted average of the securities' expected returns by their proportions in the market portfolio)

stdev(m) = ((0.4^2)*(0.2^2) + (0.6^2)*(0.28^2) + (2*0.4*0.6*0.2*0.28*0.3))^0.5 = 0.2066 = 20.66%

(calculated by taking the square root of portfolio m's variance; portfolio m's variance is calculated as a weighted average of the securities' covariances)

Now that the relevant variables have been calculated, you can just plug them in:

E(r_c) = 5.0% + stdev(c) * [ 13.0% - 5.0% ] / 20.66%

Therefore, the CML you're looking for is:

E(r_c) = 5.0% + stdev(c) * 0.3872

As in investor, you simply choose how much risk (i.e. stdev(c) ) you're willing to accept, and you can use the CML equation to calculate your expected return for that level of accepted risk.

Another way to think about it is if an investor has a predetermined level of expected returns he/she wants to obtain, the CML will tell her/him how much risk (i.e. stdev(c)) will be involved.