EVT

Question 1

What are some financial applications of EVT

Answer 1

Distribution of income has fat tails
Pricing puroses
Reinsurance pricing and modelling
CAT Models
Captal management or reservings

Question 2

What an example oF A non financial application of EVT?

Answer 2

Meteorology for example - not interested in focusing on the average events want to do a risk assessment for extreme conditions to power plants, flooding, heat waves, hurricanes, droughts

Question 3

What is the difference in focus between classical stats and extreme value theory

Answer 3

Classical stats - focus is on average behaviour of stochastic process: CLT
In EVT focus is on rare events: Fisher tippett theorem

Question 4

What does the fisher tipett theorem state

Answer 4

Tells us the cdf for the block maximum follows approximately a generalised extreme value distribution with three parameters determining the model flexibility
Mu- location
Sigma - scale
Zeta - shape parameter

Question 5

Define Mn

Answer 5

Block maximum Mn= Max(x1,….xn)

Question 6

For GEV what are the three types of distributions

Answer 6

Determined by the value of shape parameter zeta
Gumbel: exponential tail with zeta=0
Frechet: fat tail with zeta equal to 1
Weibull : upper finite end point with zeta equal to -1

Question 7

Name the two methods of EVT

Answer 7

Block maximum and peak over threshold

Question 8

What is the asymptotic distributions int he case of peak over threshold

Answer 8

Yi=Xi-u|Xi>u is asymptotically distributed as a generalized pareto distribution with two parameter and one indirect parameter (threshold):
Sigma is scale parameter
Zeta is shape parameter

Question 9

What are the three variations of GPD

Answer 9

Gumbell gives exponential tail : Zeta=0
Pareto/Frechet: gives polynomial tail behaviour Zeta>0
Weibull gives upper tail end point and has zetaa<0

Question 10

Why is EVT important in particular in finance

Answer 10

More extreme losses are the thing determining the reserves for the company - they are most worrying. Isolating justt he extreme values we can plot a distribution to more accurately model these ont heir own

Question 11

What length tails do the three generalized EVT distributions have

Answer 11

Heavy tailed - Frechet
Medium tailed - Gumbell
Short tailed - Weibull

Question 12

What does Picklands balkema de Haan Theorem say?

Answer 12

For many loss distribution the distribution fo losses above a high threshold is a generalized Pareto distribution

Question 13

What is GPD most efficient at modelling

Answer 13

Modelling values beyond a very high threshold. GPD would not describe the full data well more so the bigger losses.

Question 14

What is a spline and how are they used in EVT

Answer 14

A spline function f is a piecewise polynomial function defined on an interval [xmin, xmax] with specified continuity constraints

Often we will model separate chunks of data using different distributions to get a more accurate fit of extreme values - may need a few GPD distributions to model the few extreme values

Question 15

What is the effect of a higher shape parameter in EVT distribution

Answer 15

Thicker tails implying more extreme losses

Question 16

What effect does reducing threshold u have

Answer 16

Extreme right hand tail is reached sooner
More data available to base distribution fitting on int he tail
Contains smaller parameter error as a consequence
BUT larger model error as not as focused on the right tail

Question 17

What effect does increasing threshold u have

Answer 17

Extreme values are reached later
High u gives small model error
Less data in tail so there is larger parameter error

Question 18

What trade off is present when choosing a threshold for extreme values

Answer 18

Trade off to accurately describe extremes vs model being robustly fitted to the region you’re interested in.