# Leptokurtic vs Normal Distribution

Hi

Would someone please be kind enough to clear up the below?

At higher levels of significance (1% VAR), a leptokurtic distribution will have a lower standard deviation and therefore a higher VAR estimate than a normal distribution.

At lower levels of significance (5% VAR), a leptokurtic distribution will have a higher standard deviation and therefore a lower VAR estimate than a normal distribution.

Overall, a leptokurtic distribution will have fatter tails and therefore higher probability of more extreme values than a normal distribution.

Where did you get this?

The first two statements are complete garbage. A leptokurtic distribution (or any distribution, for that matter) has a standard deviation; its standard deviation doesn’t change simply because you change the level of significance in your VaR calculation.

Kurtosis and standard deviation are statistically independent of each other. The author of those sentences is either smoking ro drinking something, or he’s an idiot. Maybe both.

I saw this from a friend if mine, confused the living daylights out of me! thankfully like you say I can forget it!

will relay the message! Haha

You’re quite welcome.

I get what your friend is trying to say, but yeah he’s explained it all wrong.

A leptokurtic distribution will over estimate VaR at low confidence levels compared to a normal distribution and will under estimate VaR at high confidence levels compared to a normal distribution.

If you draw out a leptokurtic distribution on top of a normal distribution and then see where the values lie for the usual 1.65 and 2.33 standard deviations away, you will see firstly that the fat tails of the leptokurtic distribution are above the normal tails and therefore the actual VaR is under estimated compared to the normal (that is in reality you will experience more extreme losses than predicted by the analytical VaR method due to the fat tails). The peakdness of the leptokurtic distribution means that the distribution lies below the normal distribution at the 1.65 standard deviation level meaning that VaR is overestimated compared to the normal. (That is you will experience fewer breaches of the VaR level than predicted by the analytical VaR method)

Ive used the 1.65 and 2.33 standard deviations levels for illustration purposes, the levels at which the distributions actually cross over each other depends of just how leptokurtic the distribution is.

Remember when you use the analytical/parametric VaR method you are always assuming normality of returns and your “estimated” VaR is that based on a normal distribution. If the returns are actually leptokurtic, then the explanation above holds in terms of over/under estimation.

Perhaps I haven’t explained this any better, but there you go…