The following problem is from Tijms's Understanding Probability.
The annual rates of return on the three stocks A, B, and C have a trivariate normal distribution. The rate of return on stock A has expected value 7.5% and standard deviation 7%, the rate of return on stock B has expected value 10% and standard deviation 12%, and the rate of return on stock C has expected value 20% and standard deviation 18%. The correlation coefficient of the rates of return on stocks A and B is 0.7, the correlation coefficient is -0.5 for the stocks A and C, and the correlation coefficient is -0.3 for the stocks B and C.
An investor has \$100,000 in cash. Any cash that is not invested in the three stocks will be put in a riskless asset that offers an annual interest rate of 5%.
(a) Suppose the investor puts \$20,000 in stock A, \$20,000 in stock B, \$40,000 in stock C, and \$20,000 in the riskless asset. What are the expected value and the standard deviation of the portfolio’s value next year?
(b) Can you find a portfolio whose risk is smaller than the risk of the portfolio from Question (a) but whose expected return is not less than that of the portfolio from Question (a)?
My attempt.
(a) We would like to know the expected value of the random variable: $$ X = \frac15 A + \frac15 B + \frac25 C + \frac15 D, $$ where we know that $E(D) = 0.05$, $\sigma_D = 0$, and $A,B,C$ have a trivariate normal distribution with mean vector: $$ \mu = (0.075, 0.1, 0.2) $$ and covariance matrix $$ \mathbf{C} = \begin{bmatrix} \left(\frac{0.07}{0.075}\right)^2 & 0.7 & -0.5 \\ 0.7 & \left(\frac{0.12}{0.1}\right)^2 & -0.3 \\ -0.5 & -0.3 & \left(\frac{0.18}{0.2}\right)^2 \end{bmatrix}. $$
If A, B, and C were independent, I guess that by the central limit theorem we would have that $X$ has a normal distribution with mean $$ \mu_X=\frac{0.075}{5} + \frac{0.1}{5} + \frac{2\cdot 0.18}{5} + \frac{0.05}{5} $$ and variance $$ \sigma^2_X = \frac15 \left(\frac{0.07}{0.075}\right)^2 + \frac15 \left(\frac{0.12}{0.1}\right)^2 + \frac25 \left(\frac{0.18}{0.2}\right)^2, $$ but in this case I am unsure on how to proceed.
The covariance of the random variables X and Y in terms of the correlation coefficient and standard deviation is $cov(X,Y)=\rho_{X,Y}\cdot \sigma_X\cdot \sigma_Y$ and not the other way round. That means that the variance-covariance matrix is
$$ \mathbf{C} = \begin{bmatrix} 0.07^2 & 0.7\cdot 0.07\cdot 0.12 & -0.5\cdot 0.07\cdot 0.18 \\ 0.7\cdot 0.07\cdot 0.12& 0.12^2 & -0.3\cdot 0.12\cdot 0.18 \\ -0.5\cdot 0.07\cdot 0.18 & -0.3\cdot 0.12\cdot 0.18 & 0.18^2 \end{bmatrix}. $$
And the Variance of the weighted sum (portfolio) is $Var\left( \frac15 A + \frac15 B + \frac25 C + \frac15 D \right)$
$=\frac{1}{25}\cdot Var(A)+\frac{1}{25}\cdot Var(B)+\frac{4}{25}\cdot Var(C)+2\cdot \frac{1}{5} \cdot \frac{1}{5}\cdot cov(A,B)$
$+2\cdot \frac{1}{5} \cdot \frac{2}{5}\cdot cov(A,C)+2\cdot \frac{1}{5} \cdot \frac{2}{5}\cdot cov(B,C)$
We don´t have to regard $D$ here, since $Var(D)=cov(D,X)=0$, where $X\in \{A,B,C\}$
Remark:
Your calculation for the expected value and my calculation for the variance refer to the $\texttt{return rate}$ of the portfolio.
To obtain the expectation of the $\texttt{portfolio´s value}$ you multiply your result by $100,000$ and then add $100,000$. If you want the variance of the portfolio you multiply my result by $100,000^2$