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One way to this is the following (you can code all these constraints if you use the right software, I am doing such things using mathematica) You define $w_{i,j}$ which is the weight of asset $j$ in subportfolio $i$, furthermore you define $w =(w_j)_{j=1}^{\text{no of assets}}$ the total weight of the portfolio in asset $j$. the objects for the ...


The problem as you formulate it above already allows for short-selling. You just have to add the constraint: $$ \theta_i \ge l $$ where $l$ is the lower bound. This is equivalent to $$ -\theta_i \le -l $$ which if often the way linear constraints are formulated. Any solver that is able to work with box-constaints can solve this.


There is one minor mistake: If you compute sum(mean.var) you'll obtain $-1$ instead of $1$. So it should be mean.var<-xt/sum(xt) in order to ensure that the weights sum up to one. The remainder is correct. Incorporating a risk aversion parameter into the framework requires the solution to the minVar problem (See for example here). Therefore, dividing ...


@Richard I assume your $V$ stands for variance so that your formula is correct. The question was about standard deviation, though, and there the square root needs to be taken.


The first part of the question is correct. The second is wrong: If you model daily log returns: $$ r_t = \log(P_t)-\log(P_{t-1} $$ then your yearly return $r_y$ is just $$ \sum_{t=0}^{250} r_t, $$ assuming $250$ days. Then $$ E[r_y] = 250 E[r_t], $$ and $$ V[r_y] = 250 V[r_t] $$ if we assume that returns are uncorrelated. In the case of arithmetic ...


It depends on the distribution of the returns. If you assume that it's roughly normally distributed, then you have a ~68% chance for a return in the range of 1 standard deviation, ~95% chance for 2 standard deviations, and so on.

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