Why Long-Only Minimum-Variance Portfolios Can Be Concentrated
Summary
The document explains why a long-only minimum-variance optimizer may assign positive weights to only a few assets, even when the investment universe contains many ETFs. The question concerns a portfolio built from monthly returns for U.S. equity and bond funds, with weights constrained to be nonnegative and sum to one. The answers say that concentration is a common outcome: the optimizer selects assets that reduce measured portfolio variance, and the constrained optimum may lie at a corner of the feasible weight set.
The result reflects the objective and constraints, not a complete assessment of risk. Volatility minimization does not account for other concerns such as default risk. Concentration is especially likely when asset volatilities differ substantially or correlations are high. One suggested way to make the allocation denser is to add a constant to the covariance matrix diagonal, which shifts the solution toward equal weights. The document offers intuition rather than empirical comparisons or guidance on choosing that adjustment, so its effect should be evaluated in the context of the investor’s broader objectives.
Key ideas
- Long-only minimum-variance optimization can produce portfolios with only a few nonzero weights.
- The optimizer may concentrate in assets that reduce measured variance, particularly when volatilities differ or correlations are high.
- A minimum-variance objective measures volatility but omits risks such as default risk.
- Adding a constant to the covariance matrix diagonal can make weights denser and move them toward equal weighting.
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# Help on minimum variance optimization on U.S. Equity/Bond ETFs - Intuition # Help on minimum variance optimization on U.S. Equity/Bond ETFs - Intuition I run a MVP on 10 ETFs: SPY, SDY, IWB, XLP, VGT, BND, XLF, IJR, XLY, XLI from 2008 to 2016 on monthly return data. The weighs array (I am using a MATLAB function "Portfolio" - constraints are simple: no shorts + weights sum to 1) gives me only 1-3 of the 10 ETFs only - the rest are zero weight. Does this seem intuitive? Thanks, Andrew ## Answer by kannitverstan (score 3, accepted) https://quant.stackexchange.com/a/33148 The long only minimum variance portfolio is either equal to the unconstrained minimum variance portfolio, which is usually dense, or it is guaranteed to be sparse - in the sense that there are only few nonzero coefficients. This is a consequence of the fact that the minimum is either achieved at the global minimum or at one of the corners of a convex polyhedron. A typical example for a dense long only portfolio result is from a diagonal covariance matrix. At first this looks counterintiutive, as one intuitively equates low variance with low risk, and low risk and a sparse portfolio seems like a contradiction. But this method only looks at volatility - other aspects like default risk or so are ignored. The usual trick is to increase the density of the resulting portfolio is to add a constant to the diagonal of the covariance matrix, thus making it more diagonal. The effect is to make the resulting portfolio more similar to the equal weight portfolio and thus less sparse. ## Answer by Enrico Schumann (score 1) https://quant.stackexchange.com/a/33136 The output you have given looks more like an 'efficient frontier' than a minimum-variance portfolio, but in general, long-only minimum-variance portfolios tend to be concentrated in only a few assets, in particular when i) the marginal volatilities in the asset universe are very different (then, the low-vol assets will be included) and ii) when correlation between the assets is high.
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