Question

What happens to the standard error of a VaR estimate that uses historical simulation when we decrease the VaR confidence level? Why?

Answer 1

When we decrease the VaR confidence interval in historical simulation the standard error diminishes .

BECAUSE as we decrease the confidence interval more and more returns of the security will lie outside the confidence interval and will increase if we increase the confidence since there is very scope of deciation of returns

EXAMPLE

As a historical example, let's look at the Nasdaq 100 ETF, which trades under the symbol QQQ (sometimes called the "cubes"), and which started trading in March of 1999.2 If we calculate each daily return, we produce a rich data set of more than 1,400 points. Let's put them in a histogram that compares the frequency of return "buckets." For example, at the highest point of the histogram (the highest bar), there were more than 250 days when the daily return was between 0% and 1%. At the far right, you can barely see a tiny bar at 13%; it represents the one single day (in Jan 2000) within a period of five-plus years when the daily return for the QQQ was a stunning 12.4%.

Notice the red bars that compose the "left tail" of the histogram. These are the lowest 5% of daily returns (since the returns are ordered from left to right, the worst are always the "left tail"). The red bars run from daily losses of 4% to 8%. Because these are the worst 5% of all daily returns, we can say with 95% confidence that the worst daily loss will not exceed 4%. Put another way, we expect with 95% confidence that our gain will exceed -4%. That is VAR in a nutshell. Let's re-phrase the statistic into both percentage and dollar terms:

• With 95% confidence, we expect that our worst daily loss will not exceed 4%.
• If we invest $100, we are 95% confident that our worst daily loss will not exceed $4 ($100 x -4%).

You can see that VAR indeed allows for an outcome that is worse than a return of -4%. It does not express absolute certainty but instead makes a probabilistic estimate. If we want to increase our confidence, we need only to "move to the left" on the same histogram, to where the first two red bars, at -8% and -7% represent the worst 1% of daily returns:

• With 99% confidence, we expect that the worst daily loss will not exceed 7%.
• Or, if we invest $100, we are 99% confident that our worst daily loss will not exceed $7.

AS WE CAN SEE THERE IS A POSTIVE RELATION BETWEEN CONFIDENCE INTERVAL AND VAR THEREFORE WE DECREASE IN CONFIDENCE INTERVAL STANDARD ERROR DECREASES