Consider a mean-variance investor in a world with a risk-free asset.

Let $R_f>0$ be the return of the risk-free asset, $\mathbb{E}(R_i)>R_f$ the expected return of the risky asset $i$ and $SD(R_i)$ the standard deviation of the return of the risky asset $i$ for $i=1,...,N$.

Let $V$ be the variance-covariance matrix of the returns of all risky assets and $\bar{R}$ be their expectation.

Returns of risky assets can be positively correlated but not perfectly.

The weight of risky asset $i$ in the tangent portfolio is $x_i^\star=\frac{y_i}{\sum_{k=1}^N x_k}$ with $y_i=\{V^{-1}(\bar{R}-R_f)\}_{ii}$ if we allow for short sales. Hence, $x^\star_i$ can be strictly positive or strictly negative or zero (right?).

Question: which are the weights of risky assets in the tangent portfolio if we do not allow for short sales? Are they all strictly positive? If not, under which additional restrictions are they all strictly positive?


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