Chapter 20: Extreme value theory

Question 1

Why tail events may be incorrectly modelled STICO

Answer 1

• Statistical Distribution of extreme data may differ from smaller loss data
• True data is more skewed and leptokurtic
• Influence of bulk of data on parameter estimates
• Changes in data features over time (heteroskedasticity and structural breaks)
• Origin of the event differs from standard event

Question 2

Description of GEV BITSS

Answer 2

Block maxima sample X_m=max⁡(X_1,X_2,…,X_n)
IID assumed
Tend to a particular distribution with an increase in observations
Standardized CDFs exist to solve for
Shape parameter is gamma

Question 3

Weibull, γ<0 RUN

Answer 3

o Reinsured losses
o Upper bound
o Natural events may it

Question 4

EVT Gumbel, γ=0 FEN

Answer 4

o Fat tails
o Exponentially falling
o No Bounds

Question 5

Fréchet γ>0 HPF

Answer 5

o Heavy tails with increased variance
o Power laws in the tails
o Financial loss events

Question 6

Fitting a GEV distribution SMDC MET

Answer 6

1. Subdivide data into blocks
2. Maximum for each block calculated
   a. Max observation in each block (return level)
   b. Exceeding some set level (return period)
   c. Trade off between information and variance
3. Distribution selection based on data behaviour
4. Calculate parameters of GEV distribution with method of moments or MLE

Question 7

Character of GPD AM PULE

Answer 7

Asymptotic property – excess distribution of standardised maxima converges to GPD
Mean excess loss function used for fitting
Pareto GPD: If gamma > 0
Upper bound when γ<0
Lower bound γ>= 0
Exponential GPD: If gamma = 0

Question 8

Choosing a threshold for the distribution LASS BRS

Answer 8

Linearity between mean excess loss
function and threshold – lower point chosen
Asymptotic properties applied to increase accuracy
Slope of mean excess loss functions determines value of the shape parameter γ
Subjectivity introduced in selecting the slope level

Bias to quality trade-off

Reliability of the data
Scanty data

Question 9

Parameterisation of the GDP function MIRS

Answer 9

• MLE of MOM can be used
• IID assumed for all observations
• Random amount of variables exceed threshold
• Shortcomings of the approach should be considered