Lecture 11 Flashcards
Why model behavior of tails of distribution ?
- Some cases, model whole distribution too complicated
- Do not need whole distribution for some applications
- Main objective = VaR computation
What is the central parameter in modeling the behavior of tails of distribution and what does it measure ?
tail index and measures fatness of tails
What is the difficulty while modeling tail distribution ?
Dependency between assets
How does the dependency between assets affect the tail and extrema approaches ?
- Tail approach requires whole sample of returns = iid
* Extrema approach only requires subsamples = iid
What does the extreme value distribution need ?
Scaled version of Mt that converges to non-degenerate distrbution
That is the extreme value distribution theorem ?
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
What are the 3 types of distribution depending on value of tail index ξ nested by GEV ?
- Weibull distribution (ξ < 0) → finite support
- Gumbel distribution (ξ=0) → thin tails
- Fréchet distribution (ξ >0) → fat tails
On what is what is based the QQ-plots ?
Inverse of assumed cdf F
In the QQ-plots, what if the Gumbel distribution is used as reference function ?
- QQ plot = linear → limit distribution = Gumbel
- QQ plot = convex → limit distribution = Fréchet
- QQ plot = concave → limit distribtuion = Weibull
Why are the parameters asymptotically normal in MLE of GEV ?
ξ >-1/2
What does asymptotic theory requires in MLE of GEV ?
iidness of each subsample but not the whole sample
How does the fact that the N-histories are chosen to ensure iidness of subsamples affect MLE ?
Yields consistent estimates even if raw data are not iid
Of what consists the excess distribution and mean excess function ?
Selecting set of realizations over given threshold
What does the excess distribution measure ?
Probability that excess realization relative to threshold (xt - u ) below certain value given u = exceeded
What is the theorem of the Generalized Pareto distribution (GPD) ?
If Fx = maximum domain of attraction of GEV H, then excess distribution Fu converges, for large u, to GPD G
Is the tail index from gdp different from gev ?
No
What does H in GEV describe ?
Limit distribution of normalized extrema
What does G in GDP describe ?
Limit distribution of scaled excesses over large thresholds
Whatever the approach, on what depends the asymptotic behavior of extreme values ?
On tail index
For which distribtuion is the Hill estimation method made ?
Fréchet and use ordered sample of size T
When is ξ strongly consitent in Hill estimation method
If q increases s.t. 1/T → 0 as T→ ∞ and Xt = iid
When is ξ weakly consistent and in what condition ?
For non-iid but stationary sequences provided not too strong dependency
What is the Hill estimator ?
The MLE of ξ for tails drawn from a Pareto distribution
What are the problems with the Hill estimator ?
Depends on choice of portion q/T of sample used for computing statistics
How does the threshold impact Hill’s estimator ?
- 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
What is the simplest approach to compute the threshold in Hill estimator ?
try several values of q and check if obtain sensitive value for ξ
What does the approach of trying several values of q do for the Hill estimator ?
- May produce Hill “horror plots”
* Used to compute VaR directly
How to estimate the GDP ?
Choose threshold u, select xt > u and fit gdp for all excess returns xt - u
What is the advantage to estimate the GDP by using the MLE ?
Can use all realizations in sample exceeding u and not only maximum over N-histories
What is the drawback to estimate the GDP by using the MLE ?
Have to choose threshold u
What are the issues once the parameters of gev or gdp are estimated ?
- How often can one expect drop of returns beyond certain threshold ?
- Potential loss a portfolio can suffer in θ% worst cases over one month ?
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 ?
- 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
To what does the gev correspond and what do we need to do ?
Corresponds to limit distribution of extrema, not returns so need to convert probability for extrema in terms of probability for returns
What does the gev give ?
Probability that level x is reached for maximum/minimum of distribution
What is the probability that level x is reached by daily return ?
θ* = θ^(1/N)
What does one need to do first regarding the tail approach ?
Define threshold u above which have tail.
What does one show using simulation with time dependent data ?
With time dependent data, precision of usual tail estimators grossly overstated by conventional significance level
What are the practical recommendation with time dependent data ?
- 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
What is the 2 step strategy of EVT with time dependent data ?
- 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
What are the 2 major tasks in financial markets ?
- Asset mngmnt
* Risk mngmnt
What 2 main characteristics of asset returns do Asset mngmnt and Risk mngmnt need ?
- Time dependency (heteroskedasticity)
- Non-normality
Incorporated in forecasting model
Why do the 2 major tasks need these 2 main characteristics ?
Time dependency and non normality affect :
- Portfolio allocation strategy
- Computation VaR
Because based on unconditional distribution of returns
To what extreme value distribution do the extremes of a Uniform distribution converge ?
Weibull
To what extreme value distribution do the extremes of Normal and Lognormal distributions converge ?
Gumbel
To what extreme value distribution do the extremes of a Caucy, Pareto Stuent-t distributions or Garch Processes converge ?
Fréchet
