I am currently analyzing the Kupiec test used for backtesting $VaR$. Suppose that I backtest a $VaR$ system for $n$ days (for example 250), with a confidence interval of $1-\alpha$ (for example a $1-\alpha =0.99$, thus $\alpha = 0.01$). According to Kupiec test (and using the $VaR$ definition) we know that the probability of having $x$ exceedances is given by a Binomial distribution with parameters $n$ and $\alpha$.

In this formulation, however, the holding period of the VaR does not appear as a parameter. In other words, if I backtest a 1-day $VaR$ or a 5-day $VaR$ with same $n$ and $\alpha$, the probability of the exceedances is always given by the same binomial distribution.

Is there a way to introduce the VaR holding period as a parameter of the Kupiec test?

  • $\begingroup$ $n$ is not the number of days, but the number of independent VaR observations. If you have 250 days of data you have 250 observations of daily VaR, but only 50 = 250/5 independent observations of 5-day VaR. $\endgroup$ – noob2 Feb 25 '20 at 16:50
  • $\begingroup$ Thanks for your answer! In general, if I backtest a m-day VaR for n days, thus the independent observations are given by n / m? Is this correct? $\endgroup$ – Tommaso Ferrari Feb 26 '20 at 7:35
  • $\begingroup$ In any case, I do not agree with you. I can run every day a 5-day VaR: considering only business days, if I run a 5-day VaR on Monday I'll get the expected loss in 5 days (the next Monday); if I run a 5-day VaR on Tuesday I'll get the expected loss in 5 days (the next Tuesday). So, I obtain a sequence of 250 5-day VaR and all of these are independent observations. Is it not true? $\endgroup$ – Tommaso Ferrari Feb 27 '20 at 7:38
  • $\begingroup$ No, they are overlapping observations, which are not statistically independent. $\endgroup$ – noob2 Feb 27 '20 at 9:35
  • $\begingroup$ Can you please tell me why they are overlapping? I run a 5-day VaR on Monday, that gives me the expected loss for the next Monday. Then on Tuesday I run the same 5day VaR, that gives me the expected loss for the nex Tuseday and so on. I can not understand why they are overlapping. $\endgroup$ – Tommaso Ferrari Feb 27 '20 at 9:37

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