Problems for Monday posted, now including the extra problem
I have posted problems for Monday, but from last year's problem set, a couple of questions had to be dropped. I will therefore shortly edit this message, with an extra question for you.
Edit: here is the problem.
Consider the mean–variance trade-off problem for the greedy but not necessarily risk-averse agent, who for whatever portfolio variance level Q2 chosen, will maximize expected (excess) return ?Tx - and consider the two problems (i) with and (ii) without the no-riskless-opportunity constraint 1Tx = w (the agent's wealth; here, 1T = (1,...,1) is the vector of ones). That is:
maximize ?Tx subject to xTMx = Q2 and for problem (ii) ALSO subject to 1Tx = w.
Throughout, assume M positive definite, and let Λ denote the Lagrange multiplier associated to the xTMx = Q2 constraint. Answer the following for each of problems (i) and (ii), each value of Q and of w:
- Could the constraint qualification possibly fail? If so, when?
- Under what conditions is it mathematically possible that the Lagrange conditions hold with Λ = 0?
- ?Interpret these conditions you found in the previous item.
- Explain why the Lagrange conditions cannot hold with Λ < 0 except possibly in a degenerate case where the agent has no choice.