# Tag Info

You do note require a sum up constraint that gives you that the weights sum up to 1? Then the problem is equivalent to a maximization without constraints: $$Z(\omega)=w'\mu - \frac{\gamma}{2}w'Vw$$ then it holds that $$\frac{dZ}{d\omega}=\mu-\gamma V\omega\overset{!}{=}0\\ \Leftrightarrow \frac{1}{\gamma}\mu=V\omega^*\\ \Leftrightarrow\omega^* = ... 1 This is a strange question. Usually questions about expected utility involve some uncertainty about the future wealth of the investor. If there is no uncertainty in the outcome and the investor is not doing anything which might change his or her future wealth then the expectation of utility is a constant, that is E[U(w)] = U(w). 1 To answer your first question: Under the necessary assumptions, the Markowitz portfolio optimization framework can be used to obtain the minimum variance portfolio for a given level of return. Together all the portfolio with a minimum variance for a specified level of return are (or span) the efficient frontier. By definition it is not possible to get ... 3 If you can add linear constriants (as you can do in quadprog) then you can formulate w \mu = c_1 as linear constraint, no matter what \mu is (and first delete it from the objective by setting the parameter to zero. The only problem is the one norm. Let my clarify, this is:$$ \sum_{i=1}^n |w_i| < c_2 $$Thus you allow for short sales but you want to ... 4 I wonder if it's possible to use solve.QP from quadprog by using dummy variables. One dummy variable y_i would be used for each w_i, each y_i would be constrained to be greater than zero, and the leverage constraint would be applied to the sum of the y_i. Problem formulation would look like$$ \text{min } w^tΣw $$subject to the ... 2 If you take the a sample of historical asset returns as model for the risk then you can do two things: You calculate r_j = \sum_{i=1}^n w_i r_i^j thus for each scenario j you aggregate the individual asset returns to get a scenario for the portfolio. Then you can calculate Var(r_j) the variance of the sample of portfolio returns. This is the same as ... 3 if you take the variance of a single asset it scales as a quadratic,$$ var(\lambda X) = \lambda^2 var(X)  so it's not surprising that the general case gives a quadratic form.