Extreme value theory

Quantitative Risk Management in Python

Jamsheed Shorish

Computational Economist

Extreme values

  • Portfolio losses: extreme values
  • image of extreme events exceeding VaR threshold
  • Extreme values: from tail of distribution
    • Tail losses: losses exceeding some value
    • Model tail losses => better risk management
  • image of tail risk and VaR
Quantitative Risk Management in Python

Extreme value theory

  • Extreme value theory: statistical distribution of extreme values
  • Block maxima

line interval from 0 to T

Quantitative Risk Management in Python

Extreme value theory

  • Extreme value theory: statistical distribution of extreme values
  • Block maxima:
    • Break period into sub-periods

line from 0 to T divided into subintervals

Quantitative Risk Management in Python

Extreme value theory

  • Extreme value theory: statistical distribution of extreme values
  • Block maxima:
    • Break period into sub-periods
    • Form block from each sub-period

line interval showing blocks

Quantitative Risk Management in Python

Extreme value theory

  • Extreme value theory: statistical distribution of extreme values
  • Block maxima:
    • Break period into sub-periods
    • Form blocks from each sub-period
    • Set of block maxima = dataset
  • Peak over threshold (POT):
    • Find all losses over given level
    • Set of such losses = dataset

interval with blocks and maximum loss per block

Quantitative Risk Management in Python

Generalized Extreme Value Distribution

  • Example: Block maxima for 2007 - 2009

    • Resample losses with desired period (e.g. weekly)
      maxima = losses.resample("W").max()

  • Generalized Extreme Value Distribution (GEV)

    • Distribution of maxima of data
    • Example: parametric estimation using scipy.stats.genextreme
      from scipy.stats import genextreme
params = genextreme.fit(maxima)
Quantitative Risk Management in Python

VaR and CVaR from GEV distribution

  • 99% VaR from GEV distribution
    • Use .ppf() percent point function to find 99% VaR
    • Requires params from fitted GEV distribution
    • Finds maximum loss over one week period at 99% confidence
  • 99% CVaR from GEV distribution
    • CVaR is conditional expectation of loss given VaR as minimum loss
    • Use .expect() method to find expected value
VaR_99 = genextreme.ppf(0.99, *params)

CVar_99 = ( 1 / (1 - 0.99) ) * genextreme.expect(lambda x: x, *params, lb = VaR_99)
Quantitative Risk Management in Python

Covering losses

  • Risk management: covering losses
    • Regulatory requirement (banks, insurance)
    • Reserves must be available to cover losses
      • For a specified period (e.g. one week)
      • At a specified confidence level (e.g. 99%)
  • VaR from GEV distribution:
    • estimates maximum loss
      • given period
      • given confidence level
Quantitative Risk Management in Python

Covering losses

  • Example: Initial portfolio value = $1,000,000
  • One week reserve requirement at 99% confidence
    • $\text{VaR}_{99}$ from GEV distribution: maximum loss over one week at 99% confidence
  • Reserve requirement: Portfolio value x $\text{VaR}_{99}$
    • Suppose $\text{VaR}_{99}$ = 0.10, i.e. 10% maximum loss
    • Reserve requirement = $100,000
  • Portfolio value changes => reserve requirement changes
  • Regulation sets frequency of reserve requirement updating
Quantitative Risk Management in Python

Let's practice!

Quantitative Risk Management in Python

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