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Thread: Finding minimum variance portfolio of n stocks

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    Finding minimum variance portfolio of n stocks

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

    Consider a portfolio K_{V}=\sum w_{i}K_{i} where 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 \sum w_{i}=1 and 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.

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

    How does the lagrange setup help in this situation?


    So i know that \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.
    Last edited by LawnChair; Oct 22nd 2016 at 07:56 PM.
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