Solution for Markowitz problem with Safety-First Ratio

What is the solution for the following markowitz unconstrained problem? The sum of the entries of the weights vector $w$ should always be requied to sum one? Or if we use the risk-free asset it can be let unconstrained? I thought we should always have the constraint $1^{T}w = 1$. Am I wrong? Anyway I would like the solution for the unconstrained and the constrained problem.

• Yes, still need $1^T w =1$ (usually only 1 $w_i$ will be 1, the others 0). "Roy's Safety First Criterion" ams.jhu.edu/~daudley/447/jhuonly/550.447%20SP13%20Wk4A.pdf Page 13, 14 – noob2 Mar 23 '17 at 15:06
• @noob2 would you have a derivation of the solution? – John Mar 24 '17 at 9:36