4

In the traditional academic model, the sum of absolute weights adds up to 1. The investor is assumed to have X (usually 100) to invest, some of which goes into cash versus "the market portfolio", of which some goes into bonds versus stock, of which some goes to this stock and some goes that stock etc. For any fund manager whose performance is measured and ...


3

It would have been helpful had you provided links to those papers. But in general, you need to distinguish between the optimisation model, and the numerical technique used to solve the model. Suppose you wanted to estimate a linear regression, with the mean squared residual as the criterion of fit, and without further constraints. This is a model. Now you ...


3

Ok, I found a solution ! So, we are starting from $(x_i\sigma_i - x_j\sigma_j)((x_i\sigma_i + x_j\sigma_j)(1 - \rho) + \rho\sum_k x_k \sigma_k) = 0 $ and we will show that the elements in the second parenthesis is greater than $0$. We have: $(x_i\sigma_i + x_j\sigma_j)(1 - \rho) + \rho\sum_k x_k \sigma_k = (x_i\sigma_i + x_j\sigma_j) + \rho(\sum_k x_k \...


2

What you want is to design a risk budgeting portfolio. If your constraints are only $\mathbf{1}^T\mathbf{w}=1$ and $\mathbf{w} \geq \mathbf{0}$, then the correct way to do it is to use the formulation proposed by Spinu [1]: $$\begin{array}{ll} \underset{\mathbf{w}}{\textsf{minimize}} & \frac{1}{2}\mathbf{w}^{T}\Sigma\mathbf{w} - \sum_{i=1}^{N}b_i\log(w_i)...


2

A positive coefficient means you are going long; a negative coefficient means you are going short. Think for example of a portfolio where you borrow 1000\$ (i.e. you go short a 1000\$ bond) and you use it to buy 1000\$ worth of shares XYZ. You haven't invested any money from your pocket, hence the sum of coefficients is 0.


1

Portfolios with weights summing to zero are known as "arbitrage portfolios" or "zero net investment portfolios". They are relatively common in the literature. The idea is that the funds necessary to buy the assets with positive weights are supplied by selling short the assets with negative weights. At first this may seem impossible, since the value of the ...


1

In the US, Prime Brokers will generally follow either Reg T rules or Portfolio Margining rules. For Portfolio Margining accounts, assuming the account is somewhat diversified (not everything in one stock), they will generally allow 4 times gross leverage on the overall portfolio ($\sum_i |w_i|<=4$). This is negotiable and you may be able to get a higher ...


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This is a difficult question per se, but people in the literature have tried different ways of dealing with the time-inconsistency of the mean variance problem. Basak and Chabakauri (2010) is one of the seminal references.From their paper: "In this article, we solve the dynamic asset allocation problem of a meanvariance optimizer in an incomplete-market ...


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