In financial markets, it is sometimes impractical to invest in all of the assets that are incorporated into a specified target portfolio (such as market portfolio or a given efficient portfolio), simply because the number of total assets N could be too large. One alternative is to find the portfolio, made up of a given subset of n assets, that tracks the specified portfolio most closely - in the sense of minimizing var (r_{p}-r_{M}) the variance of the difference in returns, aptly named tracking error. Suppose that the target portfolio has a random rate of return rM and n assets you are considering for your portfolio have randomrates of return ri and weights\alphai for i = 1, 2, ...,n where rP = \alpha1r1+....+\alphanrn.
a) Define the other parameters as needed and formulate the optimization problem with the appropriate objective function and the constraint(s).
b) Find a set of equations for the \alphai ’s implied by the first order conditions.
c) Suppose that the subset of assets you are considering for your portfolio is not given. You are asked to determinewhich subset of at most n out of N possible assets to include in your portfolio in order to minimize the tracking error. Formulate the optimization problem you have to solve to achieve this.
d) Suppose you are asked to numerically solve the problem in part c. How do you approach solving it efficiently?