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Even while using historical simulation VaR, 1 day VaR is converted into 10 day VaR by multiplying 1 day VaR by Sqrt(10) for regulatory reporting purposes.

What are the underlying assumptions for doing this and how can those assumptions be tested statistically?

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What are the underlying assumptions for doing this

Assumption: Historical returns are lognormally distributed with no autocorrelation.

can those assumptions be tested statistically

Testing: $\sqrt{xy} = \sqrt{x} \sqrt{y}$

Substitute time $t$ and variance $\sigma^2$ for $x$ and $y$ respectively

$\sqrt{t\sigma^2} = \sqrt{t} \sqrt{\sigma^2} = \sigma\sqrt{t}$

Some links for you to check out if you would like to investigate further:

https://eprints.lse.ac.uk/24827/1/dp439.pdf

Square root of time

https://www.investopedia.com/articles/04/101304.asp

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Practically, I can tell you the sqare root assumption doesn't actually hold in practice--vol is not actually homoskedastic as a result of underlying returns not being iid (the scale tends to fall just short of the square of 12 in equities as a result of heterskedasticity).

A quick google turned up this, which seems to walk through precisely what you're asking about. Would probably be as good as any place to start.

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