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?


1 Answer 1


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.


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