Why Portfolio Variance Is a Quadratic Form
Summary
The document explains why portfolio variance depends quadratically on asset weights and why covariances enter the calculation. For one asset, scaling its return by a weight scales its variance by the square of that weight; extending this relationship to several weighted returns produces a quadratic form involving the covariance matrix.
It also compares two equivalent ways to calculate risk from historical return scenarios: first combine asset returns within each scenario and then measure the portfolio variance, or calculate the asset covariance matrix and apply it to the weight vector. Correlated assets can move together and raise portfolio variance, while negatively correlated assets can offset one another. The discussion gives intuition rather than a worked numerical example, and it assumes the same return sample and covariance calculation are used in both methods.
Key ideas
- Variance scales with the square of an asset's weight.
- Portfolio variance includes covariance terms as well as individual asset variances.
- Scenario-by-scenario portfolio returns and the covariance-matrix calculation yield the same variance.
- Positive covariance can increase portfolio risk, while negative covariance can reduce it.
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# Why is the variance of a portfolio a quadratic form?
# Why is the variance of a portfolio a quadratic form?
I was reading about MPT http://en.wikipedia.org/wiki/Modern_portfolio_theory and notices that the total variance of a portfolio is $x' \Sigma x$, where x is the weighting of the assets and $\Sigma$ the covariance of returns.
I'm trying to gain some intuition on why this is a quadratic for and not simply the sum of the individual assets risks multiplied by their weighting.
## Answer by Mark Joshi (score 7, accepted)
https://quant.stackexchange.com/a/17637
if you take the variance of a single asset it scales as a quadratic, $$ var(\lambda X) = \lambda^2 var(X) $$ so it's not surprising that the general case gives a quadratic form.
## Answer by Richi Wa (score 2)
https://quant.stackexchange.com/a/17645
If you take the a sample of historical asset returns as model for the risk then you can do two things:
- You calculate $r_j = \sum_{i=1}^n w_i r_i^j$ thus for each scenario $j$ you aggregate the individual asset returns to get a scenario for the portfolio. Then you can calculate $Var(r_j)$ the variance of the sample of portfolio returns. This is the same as
- you calculate the covariance matrix $\Sigma$ on the individual asset returns and then perform $x' \Sigma x$.
The result will be the same. In procedure 1) you will see that assets with positive correlation tend to increase or decrease the aggregate portfolio return in scenarios $j$ (positive covariance) - others will offset each other (negative covariance). This is how how covariance influences the variance of the portfolio return. In 1) you can see it plainly in each scenario (on average) and in 2) you have it translated into $\Sigma$.
This should answer why covariance enter the expression for the variance. For why it is a quadratic form: as Mark Joshi puts it: it is a straight forward generalization of the univariate situation.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.