Modelling Extreme Events

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Modelling Extreme Events

• David Sanders

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What is an extreme event ?

• Relative to the entity
• House fire
• Florida Windstorm
• Asteroid collision
• Not generally covered by insurance process
• Losses will occur which do have a detrimental
  effect on a number of players. These are
  generally termed extreme events.

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Overview

• Assessment of Reserves and pricing usually based
  on law of large numbers and one distribution
• But it should be considered as at least two
• the central part of the distribution, which deals
  with the normal claims and
• the extreme end of the distribution

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Type of Risks

• Catastrophe Risks
• Mortality Risks
• Asset Value Risks
• Operational Risks
• Other examples at end

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Types of Models

• Statistical/actuarial models, where past
  experience is used to estimate the consequence of
  future events
• Physical models, where, for example, the
  consequence of a landslide or submarine
  earthquake is estimated from a scaled down model
  in a laboratory and
• Simulation or catastrophe models depending on
  computer simulations of events which include
  pre-determined parameters and physical
  constraints, for example in weather forecasts.
  Catastrophe models tend to make use of
  statistical and formulated physical models

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Structure of Models

• Diagnostic, where, for example post event
  hurricane loss assessment is made using current
  and historic observed data, with, perhaps
  physical constraints such as topography, combined
  with extrapolating and interpolating the
  estimates for known locations to those where
  there has been no historic data
• Investigative where an explanation is formulated
  as to why hurricanes occur and the relationship
  between hurricane intensity and, for example,
  ocean temperature, and the conditions in the
  western Sahara Desert and
• Predictive which, for example, attempts to
  forecast the number of hurricanes in a season in
  certain category or higher.

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Issues in determining model

• Parameter estimation
• How do you estimate 1 in 200 year storm on 25
  years data
• Outliers are fundamental
• Need to consider range of predictions and not
  fixed point
• Data

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Generalised Poisson Distribution

• In this talk ignore Frechet. Weibull, Gumbell
• Over some threshold the Generalised Pareto
  Distribution (GPD) approximates most
  distributions. Therefore to fit the tail of a
  distribution (for example, to price a high XL
  layer) one needs to select a threshold and then
  fit the GPD to the tail
• This approach is sometimes known as Peaks over
  Threshold or POT. The main issue is the
  selection of the threshold, as the distribution
  is conditional on that amount.

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Modelling

• Pr(YgtyuYgtu) G(y u, ? , s ) 1 ?y/s
  -1/? , for ygt0.
• A simple approach is to fit a GPD to a random
  variable (say a claim amount) by plotting the
  mean exceedance of the random variable,
  E(Y-uYgtu) against u.
• After some threshold point, the fit will approach
  linearity.
• Above this point fit a GPD to the data by
  calculating ?u Pr(Ygtu) and then using standard
  techniques (MLE, Method of Moments) to calculate
  ? and s.

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Mean Excess Plot Taken from Excel Spreadsheet
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Issues

• At the chosen threshold of 406,000 there are 123
  data points
• At a threshold of 1,000,000 this reduces to 28.
• At a threshold of 1,500,000 there would be just 7
  observations used to fit the GPD.
• There is a trade-off between approximation to
  the underlying distribution (good for high
  threshold) and bias (good for low threshold)
• This brutal in the world of extreme values.

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The Power Function

• A Power Law is a function f(x) where the value y
  is proportional to some power of the input x
• f(x) y x-a
• Power law models tend to relate to geophysical
  events.
• An example is the Gutenberg-Richter law. Data
  indicates that the number of earthquakes of
  magnitude M is proportional to 10-bM

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Earthquake Numbers in 1995 and Gutenberg-Richter
Prediction
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Predicted Power Law Exponents for Atmospheric
Phenomena
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The Connection

• Power Laws are related to Pareto Law
• How many people have an income greater than a
  specific amount ?
• Equivalent in EVT formulation
• How many aggregate claims are greater than a
  specific amount ?
• The GPD is an extension of Paretos Law
• Self Organising Criticality and Complexity

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Other considerations

• The theorems depend on variables being iid
• There should not be any trend lines, cycles etc
  in the data
• These need to be removed for the theory to work

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Example Hurricane Data

• Remove trends
• Increase in value of property due to inflation
• Changes of population
• Changes in construction and construction type
• Remove cycles
• Seasonal fluctuations
• El Nino

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Example from modelling storm 90 A
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Hurricane Data from Hogg and Klugman
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Hurricane Data from Hogg and Klugman
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Observations

• Used 5000 (low) threshold
• Highest 1638000(Andrew)- next 863881

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Removing large value
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Observations

• Used 5000 (low) threshold
• Highest 1638000(Andrew)- next 863881
• Adjust Excess to 50000

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Mean Excess at 50000
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PP plot with 50000 threshold
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Parameters
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Observations

• Used 5000 (low) threshold
• Highest 1638000(Andrew)- next 863881
• Adjust Excess to 50000
• Hogg Klugman suggested Lognormal
• GDP gives better fit
• Range of outcomes is calculated

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Range PP Plot
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CDF with 5,000 POT
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CDF with 50,000 POT
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Other Applications

• Operational Loss (Basel II)
• PML for Cargo
• Excess Mortality Bonds
• Terrorism Cover

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Other Applications

• Casulty Excess of Loss
• Patrik found that the right tail of the ISO US
  liability claim severity model fitted to claims
  generally up to a limit of 1,000,000 is a US
  Pareto (or Burr distribution with x-exponent 1)
  
• The shape parameter of the GPD was approximately
  - 1, and that this applied over several lines of
  business.
• Patrik defined ? -1/a, and estimated the
  parameter a.

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Patrik a parameters
PC actuaries should use the GPD to model excess
losses, or make some lame excuse. Extreme
Value Distributions are a necessary part of the
actuarial pricing toolbox, and the parameter -1
is a good initial estimate based on Patriks work.