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You've highlighted one of the weaknesses of using beta, despite it being a major part of mean variance analysis. You could focus on long-term beta to reduce the impact of recent volatility. Or use Kalman filters, but understand the inherent complexity/costs you're getting into there.

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Most of the literature in Finance assumes continuous hedging which is just practically impossible. Minimum variance hedge ratio assumes the same. Unless you’re a bank or HF that can cost effectively re-hedge portfolios regularly, you’ll never get anything near a perfect hedge. For a retail investor I would say to avoid time evolving betas like with Kalman ...

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Without a risk free investment, the efficient frontier is described by a hyperbola, as you have already suggested. Efficient Frontier: Tour de force Given asset covariance matrix $\Sigma$ and the full-investment condition $w_1+\ldots w_N=1$, it can be traced out by optimising  \min_w w^T\Sigma w \quad s.t. \quad w^T\mathbf{1}=1 \quad \mathrm{and} \quad ...

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