Market Risk - Chapter 6, page 121, 10-day VaR

[Kjetil]

Hi
On page 121 in the market risk book,
"The number $1.7486 million is the 10-day VaR on a 99% confidence level. This means that on average once in a hundred 10-day periods (so once every 1,000 days) this VaR number of $1. 7486 million will be exceeded. If we have roughly 250 trading days in a year, the company is expected to exceed the VaR about once every four years. "

Does it mean that the total loss during a 10 days period is expected to be less than $1,7486 with 99 % confidence?
Or is it referring to a loss of $1,7486 happening during one day? And this is expected to happen once every 1000 days?

 

[David Harper CFA FRM]

Hi @Kjetil It refers the former (i.e., "total loss during a 10 days period") not the latter (not "happening during one day"). In the expression "10-day VaR on a 99.0% confidence level," the 10 days is the time horizon. If the 1-day C% VaR is $1.0 million, then assuming i.i.d. we might scale this to a 10-day VaR of $1.0*sqrt(10) = $3.16 and we can interpret this as "an (1-C)% probability that our 10-day loss will exceed $3.16 million." That would be a cumulative loss over the ten-day period, or really, we could call it a 10-day holding period (as a financial term). That is, if we today start with $10.0, where is our worst expected position at the end of the 10-day horizon?

But this does not necessarily imply non-overlapping periods. Notice it says "if we have roughly 250 trading days in a year, the company is expected to exceed the VaR about once every four years". I don't have the reading in front of me, but as (250 trading days * 4 years = 1,000 trading days), that would imply non-overlapping periods: 250 * 4 / 10 = 100 non-overlapping periods, because (1 - 99%) = 1.0% of those is only one day. But we could have 1,000 overlapping 10-day periods (with technical issues associated ...). I hope that's helpful, I always appreciate questions like this because they are fundamental yet they have follow-on nuances such that it's important to get the definitions right, or at least be aware of them!

 

[poojanmehta1]

Hi @David Harper CFA FRM ,

I Am trying to understand once in every four years statement. Isn't 1% * 250 = 2.5days when this VAR will get exceeded in a year and for 4 years means in a total of 10 days.
What am I missing here?

 

[David Harper CFA FRM]

Hi @poojanmehta1 Yes, if we refer to a 1-day VaR, then we expect a 99.0% VaR to be exceeded 1.0%*250 = 2.5 days per year, and 1.0%*(250 *4) = 10 days per four years, exactly as you say. Clearly, GARP's text does not refer to a 1-day VaR, but rather to a 10-day VaR. We can further infer that GARP refers to non-overlapping periods:

Jan 1st to Jan 10th (inclusive): first 10-day period
Jan 11th to 20th: second 10-day period

... see how there are 250/10 = 25 10-day periods in the year. And therefore there are (250*4)/10 = 100 non-overlapping 10-day periods over four years such that we expect the 99.0% 10-day VaR to be exceeded once. I'm just inferring non-overlapping from the quoted text. I hope that's helpful,