Measuring and managing market, credit and Op risk [106]

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Bullet points include: To relate LR test to VaR back testing, suppose have time series sample of n portfolio losses xt and corresponding VaR estimates, VaRt (for given alpha), and define If model correct then yt should be a sample of independent binomial distributed random variables with probability alpha equal to VaR confidence level. Likelihood of j exceptions (so j = sum of yt) is then proportional to: ML estimator of alpha is then (using ‘first order condition’, i.e. taking partial differentials):

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