# VAR - Effect of probability

Here’s another bummer… Is VAR higher at 1% or at 5% - I haven’t reasoned out the meaning of this, but according to my math: VAR @ 1% significance = [mean - 2.33 * std dev]* Portfolio value -----equation 1 VAR @ 5% significance = [mean - 1.65 * std dev]* Portfolio value -----equation 2 May be I need to sleep for a while, but isn’t equation 2 bigger ?

equation 2 gives a greater number than equation 1 in the sense that equation 1 says with 1% probability you will have a p/L as bad as -\$1M but with 5% probability you will have a P/L as bad as -\$750,000.

this has been discussed as well before, and JoeyD’s right. equation 1 gives a larger value for VAR. its counter intuitive.

Let’s assume that equation 1==> VAR: -100,000 equation 2==> VAR: -75,000 Yes, equation 2 has a higer number (mathmatically) than equation 1, however, in VAR world, it is the magnitude matters (absolute value).

Thanks guys… I feel pathetic today… one thing after another, it seems so straightforward when someone explains it… I think I am getting into a panic mode that I am so behind…

Yeah, think of it in terms of which number is a greater loss. VaR is kind of assumed to be a loss so when you say the VaR is bigger, that means it is a bigger loss.

an easy approach is to look at VAR as an absolute number, That way 1 M is > 750 K. I had tough time understanding this one too.

or simply look at VAR = z x std deviation

VAR means Value At Risk. If you are talking about the bottom 1% of the tail, you are going to be talking about cases where you (on average) lose more than the bottom 5% of the tail. So there is a larger value at risk in that case. Note that the average amount you would lose, assuming you are in the bottom X% of the tail is called the Conditional Value at Risk (i.e. if you know you’re in that bottom X%, how much are you expecting to lose). However, it’s the most intuitive way to explain why - other things equal - VAR goes up when you bring X% down. Personally, I think conditional value at risk makes a lot more sense to use, since VAR becomes most important when the !\$%# hits the fan, but it’s a little harder to compute. The regular (unconditional) Value at Risk basically says “if you know you *aren’t* in the bottom X%, what’s the maximum you can lose.” That’s useful for saying “this is the maximum you can expect to lose, unless things get really really weird”

The problem with that conditional VaR is that it’s harder to estimate than VaR and nobody ever believed my VaR estimates. The conditional VaR requires that you estimate the VaR and then condition on that estimate (probably). At the end of that two stage estimation project, way out in the tail how much do you believe? There might be some cool ways around that though.

think about the shape of a bell curve: + probability increases from left to right + if you go from left to right, you are going from bad returns to better returns hope this helps

I agree Joey. VAR is pretty straightforward when you’re assuming normal distributions. I’ve always had a problem with the fact that when you really need VAR to work, you’re usually in a situation where assuming a normal distribution is a really bad idea, and I don’t know of a sensible way of figuring out a PDF for situations like that other than just assuming something that could be equally implausible. My first interview when coming to this field was for a position where I’d be calculating VAR for a fund-of-funds. I think I flubbed it up because even though I knew more or less how to do it, I figured that it was silly to be assuming normal distributions out there on the tail end, and I didn’t know how else to manage it; this came through in the interview as “he doesn’t think he can do the job.” Only years later do I realize that a large number of FoF shops just assume normal distributions and run with that, and that I should have just kept my mouth shut on all those complications.

I think many people confuse VaR with centralized VaR. Continuing the example where equation 1==> VAR: -100,000 equation 2==> VAR: -75,000 If expected value is 50,000, then the centralized VaR’s are 1==> centralized VAR: 150,000 2==> centralized VAR: 125,000 Centralized VaR is very frequently called “VaR”, but sloppily so. People find centralized VaR intuitive because it behaves like other measures of dispersion: it usually increases with volatility. (Unlike VaR, for positive values, like the example above.) There is also (analogous to coefficient of variation) the “proportional, centralized VaR”, which divides centralized VaR by the mean.

this of it this way: the last 1% loss will be higher than the last 5% loss - in terms of percentile.