ERM Chapter 20

Question 1

Outline the significance of low frequency/high severity events.

Answer 1

• Can have a devastating effect on companies and investment funds. Their low frequency means little data exists to model their effects accurately. Examples include:
  > Global financial crisis in 2007
  > Black Monday in 2987
  > 9/11 attacks
• Financial data is more narrowly peaked and has fatter tails than the normal distribution. As such, when equity values are modelled, extreme events occur more frequently than predicted by the normal distribution. The normal distribution may be of limited use for modelling low frequency/high severity events.
• Fat tails are observed in financial data as the result of:
  > returns are heteroskedastic
  > innovations in a heteroskedastic model are best modelled using a fat-tailed distribution
• Fitting a fat-tailed distribution on the whole dataset does a poor job of fitting the tails, as parameter estimates are heavily influenced by the main bulk of the data in the middle of the distribution.

Question 2

What are the two main results of extreme value theory?

Answer 2

1. The distribution of the standardised block maxima Xm = max(X1, X2, …, Xn) is approximately described by the Generalised Extreme Value (GEV) family of distributions if n is sufficiently large.
2. The tail of the distribution above a threshold, P(X>x+u| x>u), can be approximated, for large values of u, by the Generalised Pareto Distribution (GPD).

Question 3

Outline the Extreme Value Theory (EVT).

Answer 3

• We are interested in the distribution of Xm = max(X1, X2, …, Xn), where each Xi is an observed loss.
• Where losses are iid, P(Xm<=x) = P(X10 and a1, …, an:
  H(x) = lim[Fn(Bnx + an)]

Question 4

Outline GEV distribution.

Answer 4

• The generalised extreme value family of distributions has three parameters:
  > a is the location parameter
  > B>0 is the scale parameter
  > y is the shape parameter
• The standard GEV distribution is given by a=0 and B=1.
• The value of y leads to three distributions:
  > if y=0, the distribution is a Gumbel distribution. This has a tail that falls exponentially.
  > if y<0, the distribution is a Weibull GEV distribution. This has a finite upper-bound e.g. temperature, wind-speed, ages
  > if y>0 the distribution is a Frechet-type GEV distribution. The tails become heavier and follow a power law. e.g. extreme financial (loss) events.

Question 5

What are the underlying distributions of the Weibull GEV distribution?

Answer 5

• Beta
• Uniform
• Triangular
• underlying distributions that have finite upper limits

Question 6

What are the underlying distributions of the Gumbell GEV distribution?

Answer 6

• Chi-square
• Exponential
• Gamma
• Log-normal
• Normal
• Weibull
• light-tail distributions that have finite moments

Question 7

What are the underlying distributions of the Frechet GEV distribution?

Answer 7

• Burr
• F
• Log-gamma
• Pareto
• t
• heavy tail distributions whose higher moments can be infinite

Question 8

What are the advantages and disadvantages of the GEV distributions?

Answer 8

A: - can be used to investigate the limiting distributions for the minimum values of a distribution.

D: - a lot of data is lost (as everything apart from the maxima in each block is ignored)
- the choice of block size can be subjective, and represent a compromise between granularity and parameter uncertainty

Question 9

Outline the Generalised Pareto Distribution (GPD).

Answer 9

• Considers all the claim values that exceed some threshold as extreme values
• Fu(x) = P(X-uu)
  = [F(x+u) - F(u)]/[1 - F(u)]
• GPD is a two parameter distribution
• CDF for the standardised GPD is given by setting B=1
• The distribution:
  > has a lower bound when y>0
  > has an upper bound when y<0
  > if y>0 then the GPD becomes the Pareto distribution

Question 10

What is the asymptotic property?

Answer 10

• When the standardised maxima of a distribution converges to a GEV distribution, the excess distribution converges to a GPD distribution with an equivalent shape parameter y.

Question 11

What is the mean excess function?

Answer 11

e(u) = E(X-u|X>u)

Question 12

How should the subjective threshold u be chosen when fitting a GPD?

Answer 12

• it should reflect the context e.g. the excess over which a reinsurer might be liable to cover
• typically around the 90-95th percentile of the distribution, as lower thresholds would bring into consideration some values that are not in the tail
• there will be a trade-off between the quality of the approximation (good for high u) against the level of bias the bit will achieve (good for low u)
• the mean excess function e(u) will become a linear function of u for higher values of u, so it can be shown that the slope of this line is related to the shape parameter y, allowing the fitting of a GPD distribution to the empirical excess distribution.