Chapter 13. Extreme Value Theory
Natural Catastrophes
Mad-made Catastrophes
Financial Events (Market Crashes)
Uses:
Pricing
Reserving
Capital Modelling
Model Types:
Statistical/Actuarial:
Using past data, most relevant, can adjust past data for trends, closed form simple solutions
Based on past data, current prejudice without much critical thinking, data may not be complete or sufficient
Physical Tests in a Lab:
Most objective, actual situation, carried out by experts
Only natural catastrophes, results need to be translated to insurance contracts, difficult and expensive to commission
Simulation/Cat Models:
Combined stats and physical, can build up models of extreme events
Requires computing power and is expensive, not closed form (black box), difficult to explain
EVT has two main results:
Distribution of a maxima to determine MPL
Distribution above a set value to price XOL contracts
Normally want to ignore the outliers, but here we want to ignore the main losses and target the tail
Generalised Extreme Value (GEV):
Suitable for finding a maxima and the distribution around it
Adjust for exposure
Possibly non-stationary (does not revert to mean over time)
Paramterise using method of moments (MoM) or max likelihood estimator (MLE)
Generalised Pareto Distribution (GPD):
Used more often than GEV as we are more interested in the distribution than predicting a maxima
Used to model claims over some threshold
Better approximation for high threshold, but less data as it increases
Usually between 90-95 percentile of losses
Scale is like your mean, shape is like the variance
Paramaterise:
Need the probability of exceeding the threshold
Determining threshold requires judgment
Use MLE or MoM to determine shape and scale
Can also use Probability-Weighted Moments
Use the Hill Estimator to determine the threshold u
Use Goodness of Fit to determine if the distribution is correct
Frequency:
Poisson:
Frequency of events per unit time - using data per unit time (normally per unit exposure and predicted that way)
Good where threshold is high and probability is low
Uses:
Catastrophe Modelling - alongside traditional models for a broader spectrum
Reinsurance High Layer Pricing
Capital Modelling - to generate large and cat losses in our DFA
IBNR Reserving
IBNER Reserving
Considerations:
Allow for IBNER in losses
Allow for IBNR in loss counts
Claims Inflation - increases size and frequency
Adjust for Exposure (like premium) - depending on how the exposure measure changes - eg. double the policies or just double the risk of the same policies
Pros:
Mathematical view on quant measure of risk
Describes extremes
Tail is the point of interest, not the centre
Reasonable formula for tail from a priori effects
View to deal with skewness, fat tails, rare events and stress scenarios
Cons:
Difficult to make assumptions (scale and shape, threshold)
Setting threshold is difficult, not easy choice
Assume Independent, Identical
Difficult to explain
Limitations of EVT in General Insurance:
Frequency not Poisson (if variance > mean then using negative binomial)
Fitting to extrapolated data:
Severity - should account for uncertainty and volatility in our IBNER's
Frequency - same as above but for IBNR
Can model for both the IBNER and IBNR
No cap on claims size in EVT
Parameter Error - can increase volatility
Uncertain future inflation - can model inflation
GPD not always best there are other severity models. Use Goodness of Fit
Losses are not Independent and Identically Distributed
Different policy characteristics, details and claims change over time
Severity Correlations (eg. inflation)
Frequency Correlations (eg. legislation changes, economic conditions)