1. Consider the following portfolio optimization problem:

where  is the expected return vector,  is the return covariance matrix, and p is a target level of expected portfolio return. Assume that the random return vector r follows a simplified factor model of the form

whereis a factor loading matrix, is given, and    is such that

The above optimization problem is a convex quadratic problem that involves n decision variables. Explain how to cast this problem into an equivalent form that involves only k decision variables. Interpret the reduced problem geometrically. Find a closed-form solution to the problem.

2. Consider the following variation on the previous problem:

where  is a tradeoff parameter that weights the relevance in the objective of the risk term and of the return term. Due to the presence of the constraint x ≥ 0, this problem does not admit, in general, a closed-form solution.

Assume that r is specified according to a factor model of the form where F, / , and / are as in the previous point, and e is an idiosyncratic noise term, which is uncorrelated with  and such that

Suppose we wish to solve the problem using a logarithmic barrier method of the type discussed in Section 12.3.1. Explain how to exploit the factor structure of the returns to improve the numerical performance of the algorithm.

p* = min x st: f x > ,