The minimum-variance portfolio

Allocation as a bead on a rail; short selling as a repulsive spring.

Hold two assets, fully invested: $w$ in asset 1 and $1-w$ in asset 2, with volatilities $\sigma_1, \sigma_2$ and covariance $c = \rho\,\sigma_1\sigma_2$. Half the portfolio variance decomposes exactly into two spring energies plus a constant that no force ever sees:

$$ \tfrac12\,\mathrm{Var}(w) \;=\; \tfrac12\,(\sigma_1^2 - c)\,w^2 \;+\; \tfrac12\,(\sigma_2^2 - c)\,(1-w)^2 \;+\; \tfrac{c}{2}. $$

So the fully-invested portfolio is a bead on the allocation rail, tied to the corner portfolio “all asset 2” ($w=0$) by a spring of stiffness $\sigma_1^2 - c$ and to “all asset 1” ($w=1$) by one of stiffness $\sigma_2^2 - c$. Release it and it settles at the global minimum-variance weight $w^\ast = (\sigma_2^2 - c)\,/\,(\sigma_1^2 + \sigma_2^2 - 2c)$, the bottom of the variance bowl. Note which way the springs pull: asset $i$'s variance is the stiffness dragging the bead away from it — risk repels. With uncorrelated assets the equilibrium is $w^\ast = \sigma_2^2/(\sigma_1^2+\sigma_2^2)$, i.e. weights proportional to $1/\sigma_i^2$: inverse-variance weighting, the centre-of-mass rule yet again.

Now raise the correlation. Both springs slacken together, and when $c > \sigma_2^2$ (i.e. $\rho > \sigma_2/\sigma_1$) the spring tied to the safer corner turns repulsive — negative stiffness, drawn red — and the equilibrium is pushed out of $[0,1]$ into the shaded zone: the minimum-variance portfolio shorts the riskier asset. Short selling is not an add-on here; it is what a repulsive spring does. The total stiffness $\sigma_1^2+\sigma_2^2-2c = \mathrm{Var}(r_1 - r_2)$ stays positive, so the bead always settles.

vol $\sigma_1$ vol $\sigma_2$ correlation $\rho$

$w^\ast$ / min vol stiffnesses ,

Two deeper connections. First, the minimum-variance weights are $w^\ast \propto \Sigma^{-1}\mathbf{1}$ — precision row sums — which makes the minimum-variance portfolio literally the GLS estimator of a common mean: the same masses as the bid-ask bounce demo, wearing a finance hat. Second, for many assets the corner portfolios become blocks, eliminating a block is a Schur complement, and damping that complement interpolates hierarchical and minimum-variance allocation — the γ-bridge of schur.microprediction.org, whose scalar shadow is the Schur damping demo.