Portfolio 1: Minimum Variance Portfolio (Fully Invested)
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There are two assets and we wish to solve for the portfolio (this is a vector of weights) that minimizes the portfolio's risk. Risk is the standard deviation of the time series of returns on the portfolio, which is a weighted average of the individual risks on the two assets and their covariances. That is, we want to minimize the scalar quantity (by scalar, I mean a single value, that is w′Vw is a number; a 1 × 1 matrix) given by:
Expanding this as we did in Chapter 5 is equivalent to optimizing the following objective function:
Written out, the summation is:
This is the portfolio's risk for a given set of weights and covariances. We want to minimize this quantity subject to the constraint that the portfolio is fully invested, that is, that the weights sum to unity,
. We set this up as a Langrangian:
Write out the summation and take derivatives with respect to the wi to get the first ...
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