Minimum-Variance Portfolio

Of all the portfolios you can build from a set of assets, exactly one has the lowest possible volatility. That's the minimum-variance portfolio (MVP), and it sits at the far-left tip of the The Efficient Frontier — the point where the curve of risk-versus-return stops moving left. What makes it special isn't just that it's the calmest portfolio available; it's how little you need to know to build it. Every other point on the frontier requires you to forecast expected returns, the single hardest and least reliable input in all of investing. The MVP needs only the covariance matrix.

The weights that minimise variance have a clean closed form:

$$w_{\text{MVP}} = \frac{\Sigma^{-1}\mathbf{1}}{\mathbf{1}^\top \Sigma^{-1}\mathbf{1}} .$$

$\Sigma$ is the covariance matrix of asset returns, $\Sigma^{-1}$ its inverse, and $\mathbf{1}$ is a column of ones. The numerator $\Sigma^{-1}\mathbf{1}$ tilts weight toward assets that are low-volatility and low-correlation — the ones that shrink total risk. The denominator is just a normaliser that forces the weights to sum to 1. No expected returns appear anywhere.

For just two assets the formula unpacks into something you can compute by hand:

$$w_A = \frac{\sigma_B^2 - \rho\,\sigma_A\sigma_B}{\sigma_A^2 + \sigma_B^2 - 2\rho\,\sigma_A\sigma_B} .$$

Slide the weight below to walk along the two-asset frontier and find its leftmost point — that's the MVP. Then drag the correlation $\rho$ down: as the two assets become better hedges for each other, the frontier bows further left and the minimum-variance point drops to a lower and lower risk.

Worked example

Two assets: a calm one A with volatility 10%, a racy one B with volatility 20%, correlation $\rho = 0.2$.

$$w_A = \frac{0.20^2 - 0.2(0.10)(0.20)}{0.10^2 + 0.20^2 - 2(0.2)(0.10)(0.20)} = \frac{0.04 - 0.004}{0.01 + 0.04 - 0.008} = \frac{0.036}{0.042} = 0.857 .$$

So the MVP holds 85.7% in A and 14.3% in B. Its variance is

$$0.857^2(0.10^2) + 0.143^2(0.20^2) + 2(0.857)(0.143)(0.2)(0.10)(0.20) = 0.00914,$$

giving a volatility of $\sqrt{0.00914} = 9.6\%$. Look closely: 9.6% is lower than either asset alone — lower even than the 10% of the calm asset A. That's the magic of diversification: mixing in a little of the riskier B, which doesn't move in perfect lockstep with A, actually lowers total risk below what you could get holding only the safest asset. The MVP finds the exact blend where that benefit is maximised.

Where it misleads

• $\Sigma^{-1}$ amplifies noise. Inverting the covariance matrix magnifies estimation error, so a naive MVP often concentrates in a handful of "low-vol" names that just looked calm and uncorrelated in your sample. Shrinkage (Ledoit-Wolf Covariance Shrinkage) or weight constraints are close to mandatory in practice (Pitfalls of Mean-Variance Optimization).
• Low variance is not low risk of ruin. The MVP minimises volatility, which treats up-moves and down-moves alike and assumes the covariances are stable. In a crisis correlations jump toward 1 and the "diversified" MVP can lose its cushion exactly when you need it.
• It's a corner, not the whole answer. The MVP ignores return entirely, so it's not where you'd want all your money if some assets genuinely offer more reward. It's the robust anchor of the frontier, often blended with a return-seeking portfolio.