# VaR Backtesting. High frequency of exceedances

I'm preparing for thesis defense and I've got simple question connected with Value at Risk backtesting. Portfolio VaR was calculated using historical simulation approach (250 days and 500days) and backtested with Kupiec test. In those portfolios where VaR was not properly estimated, there was always high frequency of exceedances so the Value at Risk was underestimated.

Why there was so many exceedances? The statistical important information was included in tails of distribution, because of lack of these tails the VaR was not properly estimated?

What should be done to improve the precision of estimation?

• What kind of products does your portfolio contain? If there are non-linear instruments (e.g. options), then the historical VaR might not be suitable to capture the non-linearity of options payoff. Jul 5, 2017 at 8:32
• Portfolios contain only shares. Jul 5, 2017 at 8:36
• Hi Jeje. Few things that are nice to know. So you have 500 numbers, i.e. the daily returns, for how many shares? So are you using the T squared law to scale the time horizon? What is the confidence level you are using? Jul 5, 2017 at 9:25
• I got time series 1200 observations, based on 250 or 500 quotations. I used both 0.9 and 0.95 confidence intervals. Every portfolio contains 5 different companies. I know there will be only theoretical question how to lower the number of exceedances :/ but my supervisor didn't help me. He said only something about lack of distribution tails. Jul 5, 2017 at 9:31
• All right Rafal, sorry I called you Jeje. Sorry about your supervisor. Well you need to have good consistent data if you want to have good consistent results. Make sure all your data lines up in chronological order. Just to make sure, I assume you would not make that kind of rookie mistake. So how many shares do you have in your data? You say you have 1200 observations, and you only use 250 or 500 of them, why is that, why not 600 to 1200 then? Jul 5, 2017 at 9:41

I assume that the problem can be boiled down into:

SAMPLE SIZE

If I am right, this problem should be less where you are using 500 daily returns in you VaR calculation instead of 250. Then the reason is that the statistical parameters become fuzzier and fuzzier with fewer samples.

A simple way to make it better is to make it grow i.e. make it remember. Allow it to grow never to shrink, then it remembers volatile times.

• Thank you, I also thought about sample size. But I think that, I should also take into consideration the simple optimization routine in Excel. Because the case where I was using 500days for VaR calculations produced better estimation (less rejections in Kupiec test). Supervisor told me that the simple solver-optimization will be fine. But now I'm thinking that there is also probably also a problem. For single stocks, the model based on 250days was much better, but for optimized portfolios better was the model based on 500days Jul 5, 2017 at 22:28
• Could you tell me, what do you mean by "make it remember'? Jul 6, 2017 at 9:09
• Lets say if you make the 250 day window grow for each day, so for the first observation it is 250 days, the second it is 251 days, and third observation it is 252 you get the point, so you only add historical information to it and never take it, that is one way to make it remember. Jul 6, 2017 at 9:22
• Ok :) Now I understand, so after 250+X days, the VaR will be calculated based on historical data from 250+X days Jul 6, 2017 at 9:26
• Yes, and please accept my answer above if you think it is correct. Jul 6, 2017 at 9:33