Ch5 Parametric Approach: Extreme Value (1)

1. GEV Theory
*video clip: http://www.youtube.com/watch?v=o-cpu1IH3tM

1-1. Extreme Events: events that are unlikely to occur, but can be very costly when they do (low-probability, high-impact events)

1-2. GEV (Generalized Extreme Value)
- Fisher-Tippett theorem (1928)
- as the sample size n gets large, the distribution of extremes, Mn, converges to the following distribution: where:
1 + ksi(x - mu)/delta > 0
mu = location parameter of the limiting distribution (a measure of the central tendency of Mn)
delta = scale parameter of the limiting distribution (a measure of the dispersion of Mn)
ksi = tail index, an indication of the shape (or heaviness) of the tail of the limiting distribution

- if ksi > 0, Frechet distribution, heavy-tailed, estimate of ksi is positive but less than 0.35
- if ksi = 0, Gumbel distribution, exponential tails (relatively light tail)
- if ksi < 0, Weibull distribution, lighter than normal tails



- both are skewed to the right, but the Frechet is more skewed than the Gumbel and has a longer right-hand tail
- probability mass will lie between x values of -2 and +6

where:
alpha = VaR confidence level associated with the threshold Mn*

1-3. Choice of distribution
- if the researcher is confident the parent distribution is a t-distribution -> ksi > 0
- if the researcher applies a statistical test and cannot reject the H0: ksi = 0 -> ksi = 0
- if the researcher may wish to be conservertive & to avoid model risk -> ksi > 0