Delta-Normal VaR and the Return Distribution Quantile
Summary
The document identifies the described portfolio risk calculation as Delta-Normal Value at Risk. It combines portfolio weights with a forecast covariance matrix to obtain predicted portfolio variance, then scales a distribution quantile by the resulting volatility to estimate VaR at a selected tail level.
The quantile is taken from the assumed distribution of standardized, scale-free returns. The answer says the method is typically based on normally distributed returns, and warns that substituting a Student t distribution can make the interpretation less straightforward when the scaling parameter is not the distribution's standard deviation. “Cumulative” refers to the cumulative distribution function, whose inverse supplies the requested percentile. The explanation is brief and does not assess VaR backtesting, tail-model alternatives, or the accuracy of the normality assumption.
Key ideas
- The portfolio variance forecast is calculated from weights and the conditional covariance estimate.
- Delta-Normal VaR scales a return-distribution quantile by forecast portfolio volatility.
- The quantile applies to standardized returns after removing the volatility scale.
- The method commonly assumes normally distributed returns, and other distributions can complicate interpretation.
- Cumulative refers to the cumulative distribution function used to define the percentile.
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# What is the name of this VaR calculation strategy?
# What is the name of this VaR calculation strategy?
Here's a question on a passage from this paper I'm reading. Here's the quote:
> Given the vector of portfolio weights $w$, and the estimate of the conditional variance, $\Sigma_{t,k}$, the predicted portfolio variance is $\hat{\sigma}_{p,t,k} = w'\Sigma_{t,k}w$. The VaR at the $1\%$ and $5\%$ level is computed for each portfolio using the predicted portfolio variance as $$ \text{VaR}_{p,t-1,k}(\alpha) = \sqrt{\hat{\sigma}_{p,t,k}}F^{-1}(\alpha) $$ where $F^{-1}(\alpha)$ is the $\alpha$-th percentile of the cumulative one-step-ahead distribution assumed for portfolio returns.
Question 1: is there a name for this calculation strategy here? Something I can google would be nice.
Question 2:
When they say "percentile of the cumulative one-step-ahead distribution assumed for portfolio returns", do they mean a distribution for the scale free random variable?
Say $y_{t}$ is the return, and say it can be written as $\sqrt{\sigma_{t,k}}z_{t}$. Is $F$ the CDF of $Z$? It gets a little weird with t random variables because the scale factor isn't the standard deviation. Here's why I think this: $P[y_{tp} < \text{VaR}_{p,t-1,k}(\alpha)] = P[y_{tp} < \sqrt{\hat{\sigma}_{p,t,k}}F^{-1}(\alpha)] \approx P[z_{tp} < F^{-1}(\alpha)] = F[F^{-1}(\alpha)] = \alpha$ .
I ask because it seems like it would be strange to have a parametric model, strip out the variance predictions, and then make up another probability distribution to to calculate this.
Question 3: why do they use the word "cumulative?" What is cumulative about this?
## Answer by Tim Wilding (score 2, accepted)
https://quant.stackexchange.com/a/38669
The strategy is typically known as Delta Normal VaR, which you should be able to Google. Yes, they mean a distribution for the scale-free random variable. This method is typically based around the assumptions that returns are Normally distributed, and this is why using the t-distribution may give you strange results. They use cumulative because they are referring to the Cumulative Distribution Function for the Normal distribution.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.