The mix of a fixed set of assets that had the lowest volatility over a measured period. It is the left end of the efficient frontier.
The minimum variance portfolio is the mix of a given set of assets with the lowest Volatility. It sits at the left end of the Efficient Frontier. To find it, a solver chooses weights that make the portfolio variance wᵀΣw as small as possible, with the weights adding up to 100%. A long-only version also keeps every weight at zero or above.
Unlike the rest of the frontier, the minimum variance portfolio needs no estimate of expected return. It depends only on the covariance between the assets. This matters because past returns are a poor guide to future returns, while volatility and correlation are more stable from one period to the next. For this reason, many low-volatility index funds use a version of this calculation. NSE's Nifty 100 Low Volatility 30 index takes a simpler approach: it picks the 30 least volatile stocks of the Nifty 100.
The result often puts large weights on a few assets with low volatility and low correlation to the others, such as gold or a debt fund in a mixed book. The estimate of the covariance matters a lot. A shrinkage method, such as Ledoit-Wolf, gives a more stable result than the raw sample covariance when the window is short or the number of assets is large.
The minimum variance portfolio describes the past window that it was measured on. It does not account for the investor's goals, horizon, taxes or costs, and the lowest past volatility does not mean the lowest future loss.
Formula
Minimise wᵀΣw subject to Σwᵢ = 1 (and wᵢ ≥ 0 for long only)