Minimum variance portfolio

The idea in plain words

Expected returns are hard to estimate; risk is much easier. The minimum-variance portfolio uses only the covariance matrix and finds the left tip of the efficient frontier. It favours low-volatility tickers and those that hedge each other.

How it is calculated

Minimise the portfolio variance with the weights adding up to 1 (long only by default, capped by the maximum weight).

minimise  wᵀΣw
subject to  Σwᵢ = 1,  0 ≤ wᵢ ≤ w_max
Unconstrained: w = Σ⁻¹1 ÷ (1ᵀΣ⁻¹1)

How to use it

• As a defensive core when you care about smoothness more than return.
• Cap the maximum weight; otherwise it often puts most of the money in the single calmest ticker.
• Compare its OOS volatility with its in-sample one: it usually holds up well.

Drawing conclusions

• High weight in one ticker is normal for this model; it is not a recommendation for that ticker, only its lowest-risk role.
• Low expected return is the expected price of low risk.

Watch out for

• Concentration: the effective number of tickers here is only 1.3.
• Low volatility in the past does not guarantee it in the future (a calm currency can devalue).

A little history

Implicit in Markowitz (1952). It gained a following after Haugen and Baker (1991) and Clarke, de Silva and Thorley (2006) found that minimum-variance stock portfolios had matched or beaten the market with much less risk, the "low-volatility anomaly".