# How does Cornish-Fisher VaR (aka modified VaR) scale with time?

I have already posted this question in the quant section, maybe the statistics community is more familiar with the topic:

I am thinking about the time-scaling of Cornish-Fisher VaR (see e.g.page 130 here for the formula).

It involves the skewness and the excess-kurtosis of returns. The formula is clear and well studied (and criticized) in various papers for a single time period (e.g. daily returns and a VaR with one-day holding period).

Does anybody know a reference on how to scale it with time? I would be looking for something like the square-root-of-time rule (e.g. daily returns and a VaR with $d$ days holding period). But scaling skewness, kurtosis and volatility separately and plugging them back does not feel good. Any ideas?

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I have studied this topic in my thesis, “Cornish-Fisher Expansion and Value-at-Risk method in application to risk management of large portfolios” and it is currently available as pdf, you will need to download it. For time scaling, after modifying the variance formula you will just need to multiply by the time factor square root of estimation time, since time decay is independent of the parameters: mean, variance, skewness, etc.

I have the formula in the thesis, hope it will help. Just search by the thesis name, you will find the pdf in diva portal.

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Welcome to the site, @ozlemaktas. We appreciate your contributing an authoritative answer to the OP's question. One of our goals is to provide a permanent repository of statistical information, & as such, we prefer answers are given in complete sentences w/o texting-style abbreviations, eg. These topics & more are discussed in our FAQ, which you may want to read. –  gung Nov 3 '12 at 16:40
@ ozlem aktas: I browsed through your thesis and I could not find the section where you adress the issue of time scaling of CFVaR. I have posted the question in the quant section of stackexchange and there you find the correct answer. It is not only scaling by the square root. Mean, variance, skewness and kurtosis all scale differently if you aggregate returns over time. Skewness and ex. kurtosis tend to zero (at different rates) which is called aggregational Gaussanity. –  Richard Nov 5 '12 at 10:35