Portfolio Variance with Short Positions and Negative Weights

Summary

The document asks whether the standard covariance-matrix formula for portfolio variance remains valid when some portfolio weights are negative, as they are for short positions. It presents a futures portfolio with both positive and negative contract quantities as its motivating example and asks whether taking the square root of the resulting variance gives portfolio risk.

The core concept is that portfolio variance is computed from the full weight vector and covariance matrix, so signs matter through the cross-products between positions. The standard quadratic form accommodates negative weights; the square root gives portfolio standard deviation when the inputs are consistently scaled. The source provides the question and formula but no answer or worked calculation, so it does not clarify how to translate contract counts into exposure weights or whether the resulting risk measure should be adjusted for futures-specific units and valuation conventions.

Key ideas

• The portfolio variance formula uses the complete vector of portfolio weights and the covariance matrix.
• Negative weights can represent short positions within the same quadratic-form calculation.
• The square root of portfolio variance is its standard deviation when weights and covariances use consistent units.
• Contract counts may need conversion into comparable exposures before applying the formula.

Tags

# How to compute the portfolio risk when weights are negative?


# How to compute the portfolio risk when weights are negative?












In QMiF (p. 239) , the variance of a portfolio is defined as:

V(R) = w'Vw = w'DCDw = x'Cx

Does this formula hold if the weights are negative (i.e., short)?

For example, if I have a 5x5 covariance matrix of various futures contracts and my position is:

```
n_01: -100
n_02: 230
n_03: -140
n_04: -79
n_05: 290
```

Can I simply use the formula above and arrive at the portfolio risk if i take the square root of the variance equation above?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.