Lecture 11

Question 1

Why model behavior of tails of distribution ?

Answer 1

• Some cases, model whole distribution too complicated
• Do not need whole distribution for some applications
• Main objective = VaR computation

Question 2

What is the central parameter in modeling the behavior of tails of distribution and what does it measure ?

Answer 2

tail index and measures fatness of tails

Question 3

What is the difficulty while modeling tail distribution ?

Answer 3

Dependency between assets

Question 4

How does the dependency between assets affect the tail and extrema approaches ?

Answer 4

• Tail approach requires whole sample of returns = iid

* Extrema approach only requires subsamples = iid

Question 5

What does the extreme value distribution need ?

Answer 5

Scaled version of Mt that converges to non-degenerate distrbution

Question 6

That is the extreme value distribution theorem ?

Answer 6

Xt = sequence of iid rv with mt = max(X1,…,XT). If there exists location parameter μ, scale parameter ψ > 0 and non-degenerate distrbution function H

Question 7

What are the 3 types of distribution depending on value of tail index ξ nested by GEV ?

Answer 7

• Weibull distribution (ξ < 0) → finite support
• Gumbel distribution (ξ=0) → thin tails
• Fréchet distribution (ξ >0) → fat tails

Question 8

On what is what is based the QQ-plots ?

Answer 8

Inverse of assumed cdf F

Question 9

In the QQ-plots, what if the Gumbel distribution is used as reference function ?

Answer 9

• QQ plot = linear → limit distribution = Gumbel
• QQ plot = convex → limit distribution = Fréchet
• QQ plot = concave → limit distribtuion = Weibull

Question 10

Why are the parameters asymptotically normal in MLE of GEV ?

Answer 10

ξ >-1/2

Question 11

What does asymptotic theory requires in MLE of GEV ?

Answer 11

iidness of each subsample but not the whole sample

Question 12

How does the fact that the N-histories are chosen to ensure iidness of subsamples affect MLE ?

Answer 12

Yields consistent estimates even if raw data are not iid

Question 13

Of what consists the excess distribution and mean excess function ?

Answer 13

Selecting set of realizations over given threshold

Question 14

What does the excess distribution measure ?

Answer 14

Probability that excess realization relative to threshold (xt - u ) below certain value given u = exceeded

Question 15

What is the theorem of the Generalized Pareto distribution (GPD) ?

Answer 15

If Fx = maximum domain of attraction of GEV H, then excess distribution Fu converges, for large u, to GPD G

Question 16

Is the tail index from gdp different from gev ?

Answer 16

No

Question 17

What does H in GEV describe ?

Answer 17

Limit distribution of normalized extrema

Question 18

What does G in GDP describe ?

Answer 18

Limit distribution of scaled excesses over large thresholds

Question 19

Whatever the approach, on what depends the asymptotic behavior of extreme values ?

Answer 19

On tail index

Question 20

For which distribtuion is the Hill estimation method made ?

Answer 20

Fréchet and use ordered sample of size T

Question 21

When is ξ strongly consitent in Hill estimation method

Answer 21

If q increases s.t. 1/T → 0 as T→ ∞ and Xt = iid

Question 22

When is ξ weakly consistent and in what condition ?

Answer 22

For non-iid but stationary sequences provided not too strong dependency

Question 23

What is the Hill estimator ?

Answer 23

The MLE of ξ for tails drawn from a Pareto distribution

Question 24

What are the problems with the Hill estimator ?

Answer 24

Depends on choice of portion q/T of sample used for computing statistics

Question 25

How does the threshold impact Hill’s estimator ?

Answer 25

• Too much in tail → very inaccurate estimates because too few observations used in estimation
• Too many observations → tail observations contaminated by observations of central part

Question 26

What is the simplest approach to compute the threshold in Hill estimator ?

Answer 26

try several values of q and check if obtain sensitive value for ξ

Question 27

What does the approach of trying several values of q do for the Hill estimator ?

Answer 27

• May produce Hill “horror plots”

* Used to compute VaR directly

Question 28

How to estimate the GDP ?

Answer 28

Choose threshold u, select xt > u and fit gdp for all excess returns xt - u

Question 29

What is the advantage to estimate the GDP by using the MLE ?

Answer 29

Can use all realizations in sample exceeding u and not only maximum over N-histories

Question 30

What is the drawback to estimate the GDP by using the MLE ?

Answer 30

Have to choose threshold u

Question 31

What are the issues once the parameters of gev or gdp are estimated ?

Answer 31

• How often can one expect drop of returns beyond certain threshold ?
• Potential loss a portfolio can suffer in θ% worst cases over one month ?

Question 32

One might use historical data to address the issues once the parameters of gev or gdp are estimated. what are the problems with this approach ?

Answer 32

• Few days with exceedance of desired level, especially if large and get poor estimate of actual probability → better model entire tail for statistical stability
• Interested in variations that never occurred before

Question 33

To what does the gev correspond and what do we need to do ?

Answer 33

Corresponds to limit distribution of extrema, not returns so need to convert probability for extrema in terms of probability for returns

Question 34

What does the gev give ?

Answer 34

Probability that level x is reached for maximum/minimum of distribution

Question 35

What is the probability that level x is reached by daily return ?

Answer 35

θ* = θ^(1/N)

Question 36

What does one need to do first regarding the tail approach ?

Answer 36

Define threshold u above which have tail.

Question 37

What does one show using simulation with time dependent data ?

Answer 37

With time dependent data, precision of usual tail estimators grossly overstated by conventional significance level

Question 38

What are the practical recommendation with time dependent data ?

Answer 38

• Use extremes over subsamples → reduces dependence and is justified if subsamples long enough → still have to select size of subsamples
• Remove each realization adjacent to extreme event → not appropriate for series with large amount of time dependence
• Estimate tail of conditional rather than unconditional distribution

Question 39

What is the 2 step strategy of EVT with time dependent data ?

Answer 39

• Fit conditional mean and volatility model → account for serial correlation and heteroskedasticity = approximately iid standardized residuals
• Use EVT techniques to model tail distribution of standardized data

Question 40

What are the 2 major tasks in financial markets ?

Answer 40

• Asset mngmnt

* Risk mngmnt

Question 41

What 2 main characteristics of asset returns do Asset mngmnt and Risk mngmnt need ?

Answer 41

• Time dependency (heteroskedasticity)
• Non-normality

Incorporated in forecasting model

Question 42

Why do the 2 major tasks need these 2 main characteristics ?

Answer 42

Time dependency and non normality affect :

• Portfolio allocation strategy
• Computation VaR

Because based on unconditional distribution of returns

Question 43

To what extreme value distribution do the extremes of a Uniform distribution converge ?

Answer 43

Weibull

Question 44

To what extreme value distribution do the extremes of Normal and Lognormal distributions converge ?

Answer 44

Gumbel

Question 45

To what extreme value distribution do the extremes of a Caucy, Pareto Stuent-t distributions or Garch Processes converge ?

Answer 45

Fréchet