Value of a Portfolio

What are the expected return and standard deviation of an equal-weight portfolio of two assets, A and B, with the following characteristics?

Expected return of A = 10%
Expected return of B = 5%
Standard deviation of A = 20%
Standard deviation of B = 15%
The correlation between A and B is 0.5

The expected return of the portfolio,
E[r_p] = E[0.5 r_A + 0.5 r_B]
E[r_p] = 0.5 E[r_A] + 0.5 E[r_B]
E[r_p] = 0.5 x 0.1 + 0.5 x 0.05 = 0.075 = 7.5%

The standard deviation of the return
To calculate the standard deviation, std, we first calculate the variance, std².
var[r_p] = var[0.5 r_A + 0.5 r_B]
var[r_p] = 0.5² x var[r_A] + 0.5² var[r_B] + 2 x 0.5 x 0.5 cov[A,B]
cov[A,B] = corr(A,B) x std_A x std_B
var[r_A] = std_A²
var[r_B] = std_B²
var[r_p] = 0.5² x std_A² + 0.5² x std_B² + 2 x 0.5 x 0.5 corr(A,B) x std_A x std_B
var[r_p] = 0.5² x 0.2² + 0.5² x 0.15² + 2 x 0.5 x 0.5 x 0.5 x 0.2 x 0.15
var[r_p] = 0.023125
std_rp = sqrt(var[rp]) = sqrt(0.023125) = 0.152 = 15.2%

The portfolio’s expected return is midway between the lower and the higher; the risk (the standard deviation) is closer to the lower.