Long-Only Tangent Portfolios in Mean-Variance Optimization

Summary

The document asks how to construct the tangent portfolio when short selling is prohibited. With short sales allowed, it gives the familiar solution proportional to the inverse covariance matrix times expected excess returns, then normalizes the risky-asset weights. Those unconstrained weights can be positive, negative, or zero, so the question is whether the long-only constraint makes every weight positive.

The text poses the issue but provides no answer, derivation, numerical example, or evidence about conditions that guarantee positive weights. A useful treatment would require solving the constrained mean-variance problem; simply truncating negative unconstrained weights is not generally enough. The document therefore identifies a portfolio-optimization question rather than teaching a complete method, and its assumptions include a risk-free asset and risky returns with correlations that are not perfect.

Key ideas

• The unconstrained tangent portfolio weights are proportional to inverse covariance times expected excess returns.
• Short-sale constraints turn the problem into a constrained portfolio optimization.
• A long-only constraint alone does not establish that every asset receives a strictly positive weight.
• The document asks for positivity conditions but does not provide an answer or derivation.

Tags

# Tangent portfolio weights without short sales?


# Tangent portfolio weights without short sales?












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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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.