Suppose that you back-test a VaR model using 1,000 daily stock returns for the following (a), (b) and (c). The VaR confidence level is 99% and you observe 16 exceptions.

(a)  Specify the null hypothesis that you wish to test and select an alternative hypothesis to be accepted if the null is rejected.

(b) Using a one-tailed Z-test with the 5% significance level (or 95% confidence level), back-test the VaR model and interpret the result.

 (c) Kupiec's one-tailed test can be used for the back-test of the VaR model. From the properties of the binomial distribution, show that how you calculate the probability of 16 or more exceptions out of 1,000 days if you use Excel's function.

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(d)  Suppose you back-test a VaR model with Kupiec's one-tailed test. The VaR confidence level is 99% and you observe one exception for 600 daily stock returns. From the properties of the binomial distribution, show how you calculate the probability with Excel's function (this is a separate question from the above).

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(e) If the VaR model is reasonably calibrated, the number of observations falling outside VaR should be in line with the confidence level. What are the consequences of too many exceptions and too few exceptions? Discuss them in terms of under- or over-estimating the risk and allocation of capital.

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a The null hypothesis to test would be The VaR model is accurately calibrated with a 99 confidence l... View the full answer