Minimum variance answers one question: of all the ways to weight this set of stocks, which combination has the lowest expected volatility? It is the leftmost point on Markowitz’s efficient frontier, and it is the only point on that frontier that requires no return forecast at all.
The problem is:
minimise wᵀ Σ w subject to Σ w_i = 1, w_i ≥ 0
where Σ is the covariance matrix of returns. The long-only constraint w_i ≥ 0 matters for Indian retail portfolios, since short selling in the cash segment is restricted to intraday for most participants.
Because it uses covariances rather than volatilities alone, minimum variance rewards two things: low standalone volatility and low correlation to the rest of the book. A moderately volatile stock that moves independently of everything else can receive a larger weight than a calmer stock that tracks the index closely.
How it differs from its neighbours
| Scheme | Inputs | Optimiser | Typical outcome |
|---|---|---|---|
| Inverse volatility | Volatilities only | None, closed form | Spread across calm names |
| Risk parity | Full covariance | Yes | Balanced risk contributions |
| Minimum variance | Full covariance | Yes | Concentrated in the calmest, least correlated names |
The concentration problem
Left unconstrained, minimum variance tends to pile into a small number of positions. Run it over the Nifty 100 and it will frequently allocate the bulk of the portfolio to a handful of low-beta FMCG, pharma and utility names, because those are the stocks that happen to have the lowest estimated covariance with everything else. That is a live sector bet arriving through the back door of a risk calculation.
Practical fixes are constraints rather than a different objective: a maximum position weight, a maximum group weight per sector, and a minimum position weight to stop the optimiser from producing 40 holdings of which 32 are rounding errors.
Estimation error is the real risk
The covariance matrix for a 100-stock universe carries 5,050 parameters. Estimating them from 250 trading days of history leaves each one noisy, and a variance minimiser is precisely the machine that seeks out the smallest estimated numbers. Underestimated variances and understated correlations get the largest weights, so the optimiser systematically overweights its own errors. Shrinkage estimators of the Ledoit-Wolf family are the standard correction and materially reduce out-of-sample variance.
Low forecast variance also does not mean low drawdown. A minimum-variance portfolio of Indian financials would have looked calm through 2017 and still absorbed the full 2020 fall, because correlations converged when it mattered. Test the realised drawdown, not just the realised volatility, before trusting the label.