Thread: Finding minimum variance portfolio of n stocks

1. Finding minimum variance portfolio of n stocks

Assume the returns $\displaystyle K_{i}$ of n stocks are random variables of the form $\displaystyle K_{i}=X_{i}+B_{i}Z$, where $\displaystyle X~\aleph(\mu_{i},\sigma_{i}^2)$ and $\displaystyle Z~\aleph(0,1)$ and B is a real number.

Consider a portfolio $\displaystyle K_{V}=\sum w_{i}K_{i}$ where $\displaystyle w_{i}$ are the weights.

b. Write down the Lagrange multipliers system for finding the minimum variance portfolio with covariance matrix C subject to the constraints $\displaystyle \sum w_{i}=1$ and $\displaystyle dK_{V}/dZ=B$.

c. Solve the system found in b in terms of C and B (do not use the explicit form of C).
I am unsure of the Lagrange multiplier setup.

$\displaystyle L(K,\lambda)=\sum w_{i}K_{i}-\lambda(\sum w_{i}-1)$
$\displaystyle dL/dK_{x}}=w_{x}$ ???
$\displaystyle dL/d\lambda=1-\sum w_{i}=0$

How does the lagrange setup help in this situation?

So i know that $\displaystyle \sigma_{V}^2 = w^T*C*w$ is needed in order to find the minimum variance portfolio, but I don't know how to multiply this out while keeping C intact. Any help will be appreciated. Thanks in advance.