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Answer: As the threshold value is increased, the distribution of losses over a fixed threshold value converges to a generalized Pareto distribution.
## Explanation **Correct Answer: A** In Extreme Value Theory (EVT), when examining distributions of losses exceeding a threshold value: - **Option A is correct**: As the threshold value increases, the distribution of losses over a fixed threshold value converges to a generalized Pareto distribution (GPD). This is a fundamental result in EVT known as the Pickands-Balkema-de Haan theorem, which states that for sufficiently high thresholds, the excess distribution converges to the GPD. **Why other options are incorrect:** - **Option B**: If the tail parameter (shape parameter ξ) of the GEV distribution goes to infinity, it does not become a normal distribution. When ξ → 0, the GEV becomes the Gumbel distribution, not the normal distribution. - **Option C**: EVT can be applied to any underlying distribution, not just normal or lognormal distributions. In fact, EVT is particularly useful for modeling extreme events in distributions with heavy tails that are not well-described by normal or lognormal distributions. - **Option D**: The number of exceedances actually increases as the threshold value decreases (not decreases), which provides more data points and improves the reliability of parameter estimates, not decreases reliability. **Key EVT Concepts:** - **Peaks Over Threshold (POT) method**: Uses GPD to model exceedances over high thresholds - **Block Maxima method**: Uses GEV distribution to model maxima over fixed time periods - EVT is particularly valuable for modeling tail risk and extreme events that traditional risk models may underestimate.
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A CRO is concerned that existing internal risk models of a firm, which are governed mainly by the central limit theorem, are not adequate in addressing potential random extreme losses of the firm. The CRO then recommends the use of extreme value theory (EVT). When applying EVT and examining distributions of losses exceeding a threshold value, which of the following is correct?
A
As the threshold value is increased, the distribution of losses over a fixed threshold value converges to a generalized Pareto distribution.
B
If the tail parameter value of the generalized extreme-value (GEV) distribution goes to infinity, then the GEV essentially becomes a normal distribution.
C
To apply EVT, the underlying loss distribution must be either normal or lognormal.
D
The number of exceedances decreases as the threshold value decreases, which causes the reliability of the parameter estimates to increase.