Suzanne Evans
Well-Known Member
Questions:
207.1. A bank's VaR, calibrated with 99% confidence and a one-day horizon, is $10.0 million. The bank conducted a backtest over the previous 500 trading days in order to ascertain, with 95% confidence, whether the 99% VaR model is good or bad ("faulty"). It is assumed losses are i.i.d. Which backtest observation would MOST LIKELY implicate the VaR model as bad (faulty) with 95% confidence?
a. Never (on no single day) did the daily loss exceed the $10.0 million VaR
b. Only once was the daily loss exactly $10.0 million
c. On exactly eight days the daily loss exceeded the $10.0 million VaR
d. The largest daily loss was over $50.0 million, which is more than 5x the daily VaR
207.2. A bank conducts a backtest of its 99% confident one-day VaR model over the previous 250 trading days (one year). The expected number of exceptions is 1% * 250 = 2.5. The null hypothesis, H(0), is that the VaR model is accurate ("good"). Each of the following is TRUE except which statement is false?
a. If eight exceptions are observed, the bank will justifiably evaluate its VaR model as bad, but will still incur the risk of a Type I error
b. If two exceptions are observed, the bank will justifiably evaluate its VaR model as good, but will still incur the risk of a Type II error if the VaR model is mis-calibrated such that exceptions occur 3% of the time rather than 1% of the time
c. If four exceptions are observed, it is possible for the bank to commit simultaneously both a Type I and Type II error
d. If the bank increases the upper cutoff, for example from six to eight exceptions such that eight exceptions does not reject the VaR model, the bank increases the probability that it will commit a Type II error (for the same T = 250 days sample size).
207.3. A bank conducts a backtest of its 95% confident one-day VaR model over the previous 700 trading days, where losses are assumed i.i.d.. As the mean of this binomial distribution is 5%* 700, the VaR model expects an average of 35 exceptions. Because the sample is large (T = 700), the binomial distribution which characterizes the backtest can be justifiably approximated by a normal distribution (as this concerns a distributional mean, please note this is entirely justified per CLT!). The bank prefers to backtest the VaR model with a two-tailed 99.0% confidence interval, such that the associated normal deviate is 2.58; i.e., one-half of 1.0% lies in each left- and right- tail at Z=-2.58 and Z=+2.58. Which number of exceptions is nearest to the cutoff point; that is, at what number of exceptions (but not lower) and above should the bank reject the VaR model and find it faulty?
a. 36
b. 39
c. 44
d. 50
Answers:
207.1. A bank's VaR, calibrated with 99% confidence and a one-day horizon, is $10.0 million. The bank conducted a backtest over the previous 500 trading days in order to ascertain, with 95% confidence, whether the 99% VaR model is good or bad ("faulty"). It is assumed losses are i.i.d. Which backtest observation would MOST LIKELY implicate the VaR model as bad (faulty) with 95% confidence?
a. Never (on no single day) did the daily loss exceed the $10.0 million VaR
b. Only once was the daily loss exactly $10.0 million
c. On exactly eight days the daily loss exceeded the $10.0 million VaR
d. The largest daily loss was over $50.0 million, which is more than 5x the daily VaR
207.2. A bank conducts a backtest of its 99% confident one-day VaR model over the previous 250 trading days (one year). The expected number of exceptions is 1% * 250 = 2.5. The null hypothesis, H(0), is that the VaR model is accurate ("good"). Each of the following is TRUE except which statement is false?
a. If eight exceptions are observed, the bank will justifiably evaluate its VaR model as bad, but will still incur the risk of a Type I error
b. If two exceptions are observed, the bank will justifiably evaluate its VaR model as good, but will still incur the risk of a Type II error if the VaR model is mis-calibrated such that exceptions occur 3% of the time rather than 1% of the time
c. If four exceptions are observed, it is possible for the bank to commit simultaneously both a Type I and Type II error
d. If the bank increases the upper cutoff, for example from six to eight exceptions such that eight exceptions does not reject the VaR model, the bank increases the probability that it will commit a Type II error (for the same T = 250 days sample size).
207.3. A bank conducts a backtest of its 95% confident one-day VaR model over the previous 700 trading days, where losses are assumed i.i.d.. As the mean of this binomial distribution is 5%* 700, the VaR model expects an average of 35 exceptions. Because the sample is large (T = 700), the binomial distribution which characterizes the backtest can be justifiably approximated by a normal distribution (as this concerns a distributional mean, please note this is entirely justified per CLT!). The bank prefers to backtest the VaR model with a two-tailed 99.0% confidence interval, such that the associated normal deviate is 2.58; i.e., one-half of 1.0% lies in each left- and right- tail at Z=-2.58 and Z=+2.58. Which number of exceptions is nearest to the cutoff point; that is, at what number of exceptions (but not lower) and above should the bank reject the VaR model and find it faulty?
a. 36
b. 39
c. 44
d. 50
Answers:
