Mod 20: Extreme value theory

Question 1

Outline the main disadvantages of quantifying low frequency / high severity events by fitting distributions to the whole (past) dataset
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Answer 1

Modelling extreme events by fitting to the whole dataset
1. fitting a distribution to such a dataset will likely lead to a distribution that underestimates extreme events as financial data is typically leptokurtic
2. fitting the entire dataset may result in the parameters being heavily influenced by the main bulk of the data
3. fitting the entire dataset may ignore features that change over time, eg financial returns are often heteroskedastic, structural breaks etc
4. extreme value theory cannot be applied because it focuses on modelling the tails, ignoring the main body of the distribution.
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Question 2

Outline the block maxima approach to fitting an extreme value distribution

Answer 2

See 652/1136

Question 3

Outline the threshold exceedances approach to fitting and extreme value distribution, including how it is linked to the block maxima approach
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Answer 3

Threshold exceedances approach

1. Let X be a random variable with the distribution function F(x) . If the losses are iid then, as the threshold, u, increases, the distribution of the conditional losses (the exceedances), X -u |X>u , will converge to a generalised Pareto distribution (GPD).
2. When the maxima of a distribution converge to a GEV distribution (as is true for all commonly used statistical distributions), the exceedances converge to a GPD distribution with an equivalent shape parameter gamma .
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Question 4

Outline:
1. the two different ways that observed losses can be analysed using the GEV distribution
2. the two main disadvantages of the GEV approach ©

Answer 4

Methods of analysis using the GEV distribution
The GEV distribution can be used to analyse a set of observed losses in two different ways:
1. select the maximum observation in each block (the return-level approach)
2. count the observations in each block that exceed some set level (the return-period approach).

Under both methods the periods over which we are taking maxima must be selected.
The main disadvantages of the GEV approach are:
1. a lot of data (everything apart from the maxima in each block) is effectively ignored
2. the choice of block size is subjective and represents a compromise between granularity and parameter uncertainty.
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Question 5

See 657

Answer 5

see 658

Question 6

Describe the choices involved in fitting a GPD to threshold exceedances

Answer 6

Fitting a GPD to threshold exceedances

1. A suitably high threshold must be chosen:
   - to reflect the context (eg reinsurance excess)
   - to focus on the tail (eg  90 th percentile)
   - above which the mean excess is linear
   - there is a trade off between quality of approximation to the GPD (good for high thresholds) and level of bias in the fit (lower for lower thresholds).
2. The GPD must be parameterised to best fit the distribution of threshold exceedances:
   - using standard techniques (eg maximum likelihood estimation, method of moments).
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Question 7

State formulae for the mean excess and empirical mean excess functions

Answer 7

see 662

Question 8

see 663

Answer 8

see 664