Correlations in Parametric Value at Risk

Summary

The document addresses a misconception about parametric value at risk: that a parametric model necessarily assumes portfolio assets are independent and therefore uncorrelated. Parametric describes a model built around parameters governing a joint probability distribution; those parameters can include cross-asset correlations. A correlation matrix is therefore relevant when the modeled assets may move together.

Portfolio variance can be expanded into individual variances plus covariance terms whether or not the assets are independent. If independence is assumed, the covariance terms are zero, so the expression reduces to the sum of the individual variances. The answers provide conceptual clarification rather than a VaR calculation or evidence from model performance. A model could impose zero correlations, but that is a restrictive modeling choice, not a general requirement of parametric VaR; the document does not discuss distributional choices, estimation error, or validation.

Key ideas

• Parametric models use parameters to describe a joint probability distribution.
• Cross-asset correlations can be among the parameters in a parametric VaR model.
• The portfolio variance formula includes covariance terms without requiring dependence.
• Under independence, covariance is zero and portfolio variance reduces to summed variances.
• Assuming zero correlation is possible but imposes a restrictive model specification.

Tags

# Parametric VaR assumption question


# Parametric VaR assumption question












Why do you have to make a correlation matrix when calculating the parametric value at risk, if one of the assumptions for this method to work is that the assets of the portfolio must be independently distributed (i.e. their correlation must be equal 0)? Furthermore, $Var(X+Y) = Var(X) + Var(Y) + 2\cdot Cov(X,Y)$ is used when expanding the variance to get the portfolio variance, but this can only be used also if $X$ and $Y$ are not independent.

## Answer by Attack68 (score 1)

https://quant.stackexchange.com/a/41681

Parametric simply means that a set of parameters govern the nature of the (joint) probability distribution of assets, some of those parameters being the correlations. It is not true in general to state that a parametric VaR model has cross-correlation of assets as zero.

I have never used a model that specifically precludes correlations. But if you defined one as such then your equations would be reduced as you state, but it is a very stringent assumption.

## Answer by Andrew (score 0)

https://quant.stackexchange.com/a/41682

Under your assumptions the Cov(X,Y) expression is zero. The equation is still valid, just its result is determined by the Var(...) terms.

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