Minimum Variance Portfolios Explained

A minimum variance portfolio is the combination of assets that has the lowest variance, which is the lowest overall return volatility, of any mix that can be built from a given set. It sits at the far left of the curve that Harry Markowitz described in his 1952 paper “Portfolio Selection”. Finding it needs two inputs for every asset: how volatile it is, and how it moves with each of the others. The sections below explain the idea, show a small example and describe the practical limits of the approach. It is education only, not advice.

Where the idea comes from

Markowitz introduced modern portfolio theory in a 1952 paper titled “Portfolio Selection”. In it, the efficient frontier is the upper edge of the set of possible portfolios in risk and return space: each portfolio on it offers the lowest risk for a given expected return, or the best expected return for a given risk. The minimum-variance portfolio is the leftmost point of that curve, the global minimum-variance portfolio (Wikipedia: Modern portfolio theory). Wikipedia notes that Markowitz and William Sharpe were awarded the Nobel Memorial Prize in Economic Sciences together in 1990.

The key point is that portfolio variance is not just an average of individual variances. With weights w and a covariance matrix Σ, portfolio variance is w′Σw, so how assets move together matters as much as how each one moves alone.

A two-asset example

Suppose two assets with these hypothetical figures:

	Asset A	Asset B
Annual volatility	20%	10%
Correlation between A and B	0.3

For two assets, the minimum variance weight in A is (σB² − ρσAσB) ÷ (σA² + σB² − 2ρσAσB).

1. Numerator: 0.01 − 0.3 × 0.20 × 0.10 = 0.004
2. Denominator: 0.04 + 0.01 − 2 × 0.3 × 0.20 × 0.10 = 0.038
3. Weight in A ≈ 10.5%, weight in B ≈ 89.5%

The resulting portfolio volatility is about 9.8%, slightly lower than the 10% of the calmer asset held alone. A 50/50 mix, by comparison, would have a volatility of about 12.4%. The example shows that adding a small amount of a more volatile asset can lower overall risk when the two do not move in lockstep. The numbers are invented to show the arithmetic and say nothing about real securities.

The general formula

With more assets, the minimum variance weights come from the inverse of the covariance matrix: the weight vector is proportional to Σ⁻¹ multiplied by a vector of ones, scaled so the weights sum to 1. In practice, software solves this. Expected returns are not needed, which is one reason the method appeals to people who distrust forecasts of returns, since returns are much harder to estimate than volatilities and correlations.

Why real portfolios add constraints

An unconstrained solution can give large negative weights (short positions) and heavy concentration in a few assets. Practical approaches usually add rules such as:

• no short selling
• a maximum weight for any one stock
• limits on sector or country exposure
• limits on turnover, so the portfolio does not change too much from one rebalancing to the next

A 2003 paper by Ravi Jagannathan and Tongshan Ma, cited in Wikipedia’s article on the low-volatility anomaly, is titled “Risk Reduction in Large Portfolios: Why Imposing the Wrong Constraints Helps”, a hint that some constraints can improve out-of-sample results even if they are not “correct” in theory.

Estimation is the weak point

Wikipedia’s account of modern portfolio theory notes that, in practice, investors substitute historical estimates of returns and volatility, and that those estimates may not reflect new circumstances. In a minimum variance setting, the covariance matrix is the input: with 500 stocks there are 125,250 distinct variances and covariances to estimate, and small errors can change the weights a lot. Common remedies include using longer histories, shrinking estimates toward simpler structures, and limiting position sizes.

The model also treats up and down moves as equally risky, and critics point out that investors often care more about losses than gains.

How it differs from simply buying “low volatility” stocks

Ranking stocks by their own volatility and buying the calmest ones ignores how they move together. A minimum variance approach uses the correlations too, so it can include a more volatile stock if it diversifies the others. The Wikipedia article on the low-volatility anomaly notes that the minimum-variance approach is applied to sets of stocks, unlike absolute-volatility and low-beta approaches. Several index providers and fund managers use optimized or rules-based versions, and the site lists examples on its low volatility funds and institutional managers pages, which are historical listings and not recommendations.

What a minimum variance portfolio does not do

• It does not guarantee gains or protect against losses. Stocks can still fall together in a downturn.
• It is not market-neutral. Results depend on the market’s direction and on factors like sector tilts that the optimizer can introduce.
• It reflects past relationships, which can change.

Frequently asked questions

Is a minimum variance portfolio the same as a low-risk portfolio?

It is the portfolio with the lowest variance given the inputs and constraints used. That is lower risk by one measure, but not risk-free.

Do you need expected returns to find it?

No. Only volatilities and correlations (the covariance matrix) are needed.

Why do the weights change over time?

Because covariances change, and the estimates are updated. Constraints on turnover are often added to limit this.

Is this a recommendation?

No. This article describes a concept. For decisions about your own money, speak with a qualified financial professional.