Suppose that it is impractical to use all the assets that are incorporated into a specified portfolio (such as a given efficient portfolio). One alternative is to find the portfolio, made up of a given set of $n$ stocks, that tracks the specified porfolio most closely-in the sense of minimizing the variance of the difference in returns.

Specifically, suppose that the target portfolio has (random) rate of return $r_{M}$. Suppose that there are $n$ assets with (random) rates of return $r_{1}, r_{2}, \ldots, r_{n}$. We wish to find the portfolio rate of return

\[r=\alpha_{1} r_{1}+\alpha_{2} r_{2}+\cdots+\alpha_{n} r_{n}\]

(with $\sum_{i=1}^{n} \alpha_{i}=1$ ) minimizing var $\left(r-r_{M}\right)$.

(a) Find a set of equations for the $\alpha_{i}$ 's.

(b) Although this portfolio tracks the desired portfolio most closely in terms of variance, it may sacrifice the mean. Hence a logical approach is to minimize the variance of the tracking error subject to achieving a given mean return. As the mean is varied, this results in a family of portfolios that are efficient in a new sense-say, tracking efficient. Find the equation for the $\alpha_{i}$ 's that are tracking efficient.