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The concept behind showing volatility vs expected return is that a risk averse investor will wish to minimise risk, and maximise return. However, how good a proxy is volatility for risk? Given a normal distribution, risk and return (sigma and mu) alone will suffice. But especially for non-normally distributed returns (as alluded to by the mention of higher ...

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The sign of the portfolio weight $h_n$ assigned to one asset cannot be solely determined by the sign of that asset's expected return, $\alpha_n$, due to the model having to also take as an input that asset's dependence structure with the other $N-1$ assets being considered, captured within the covariance matrix $V$. Those other assets are also vying for ...

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The Markowitz mean-variance model is the basis for many extensions and portfolio solutions that have been discovered over the years: The standard model (Markowitz, 1952, 1959) originally only considered: Constrained model where short sales are forbidden Only risky assets considered for investment (no risk-free asset) Scenarios that the mean-variance ...

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I have actually considered the problem that you are working on, though configured somewhat differently. There isn't going to be a universal answer to your question. See, in particular, Holland, Paul W. Covariance Stabilizing Transformations. Ann. Statist. 1 (1973), no. 1, 84--92. Nonetheless, there are answers, some already mentioned. I would argue ...

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For each of the examples provided in the question, if you had some view as to how X changes Y, you would update your $\mu$, not your $\Sigma$. Your covariance matrix by definition is meant to measure RANDOM noise between two variates. Your sports examples are just as applicable to stocks. In the tech industry, company A manufactures parts overseas, but ...

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This is not a complete answer, more a different perspective to the answers already given. If you have some a-priori knowledge about the covariance structure and about the factors influencing it, you should try to reflect this in your statistical model. Three ideas: Divide your sample into subpopulations with identical factor values and estimate separately. ...

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