Overestimating or underestimating risk?

Question

This question might be silly, but I want to be sure of myself.

If one has Value-at-Risk forecasts and there are zero VaR breaches (i.e. no return value is smaller than or equal to the VaR value) then the risk is said to be overestimated right?

So in a time series we would get zero observations for which holds $$r_t\leq -VaR_t.$$

In this particular case, I guess the risk is said to be overestimated. Hope someone can confirm this and if this case is called underestimation I would like to hear the reason for why this case is called underestimation.

Thanks

Answer 1

Yes, it is correct.

1. Underestimation: you under-estimate the risk, so you have more VaR violations than what your model predicts. Ex: With 100 observations, and a 99% VaR, you expect 1 violation but you observe 5 violations.
2. Overestimation: you over-estimate the risk, i.e the risk is less important that you expect. You observe less VaR violations that you expect accordingly your VaR forecast. Ex: With 100 observations, and a 99% VaR, you expect 1 violation but you observe 0 violation.

Please note that here we only focus on the so called “unconditional coverage hypothesis” (the frequency of VaR violations). You should also check that the VaR violations are independently distributed (the independence hypothesis) .

See Christoffersen, P. (1998). Evaluating interval forecasts. International Economic Review, 39(4), 841–862. Retrieved from http://www.jstor.org/stable/2527341

see also here on ssrn