Book 1. Market Risk

FRM Part 2

MR 3. Parametric Approaches (II)- Extreme Value

Presented by: Sudhanshu

Module 1. Extreme Values

Topic 1. Managing Extreme Values

Topic 2. Extreme Value Theory

Topic 3. Peaks-Over-Threshold

Topic 4. Generalized Pareto Distribution

Topic 5. VaR and Expected Shortfall

Topic 6. Generalized Extreme Values and Peaks-Over-Threshold

Topic 7. Multivariate EVT

Topic 1. Managing Extreme Values

Importance:
- Extreme events, though rare, can be extremely costly.
- They result from large market declines, institutional failures, financial/political crises, or natural catastrophes.
Challenges in Modeling:
- Limited observations of extreme values make model building difficult.
- There are ranges of extreme values that have not yet occurred.
- Researchers must assume a distribution, which introduces error as it may not perfectly match the true distribution.
- Traditional distribution choices based on central tendency often miss the specific needs of extreme value estimation.
- Similar challenges are faced in other fields, like modeling flood lines for dam construction.

Topic 2. Extreme Value Theory

Definition: EVT is a branch of applied statistics focused on problems related to extreme outcomes.
Focus: Unlike "central tendency" statistics, EVT specifically addresses the unique aspects of extreme values.
Application: EVT provides a framework for estimating parameters that describe extreme movements.
Fisher-Tippett Theorem: As sample size (n) increases, the distribution of extremes ( $M_{n}$ ) converges to the Generalized Extreme Value (GEV) distribution.
- if
- Parameters:
  - $μ$ : Location parameter
  - $σ$ : Scale parameter
  - $ξ$ : Tail index (shape/heaviness of the tail)

F(X|\xi, \mu, \sigma)=exp[-\left(1+\xi\times \frac{x-\mu}{\sigma}\right)^{-1/\xi}]

\xi \neq 0

F(X|\xi, \mu, \sigma)=exp[-exp\left( \frac{x-\mu}{\sigma}\right)] \text{ if }\xi=0

Three General Cases based on $ξ$ (Tail Index):
1. $ξ > 0$ (Frechet distribution): "Heavy" tails, common in t-distribution and Pareto distributions.
2. $ξ = 0$ (Gumbel distribution): "Light" tails, characteristic of normal and lognormal distributions.
3. $ξ < 0$ (Weibull distribution): Tails are "lighter" than a normal distribution.
Financial Risk Management Focus: Analysis typically concentrates on $ξ > 0$ and $ξ = 0$ as $ξ < 0$ distributions are less common in financial models.
Choosing $ξ$ : Researchers can choose $ξ$ by:
1. Confidence in the parent distribution (e.g., if t-distribution, assume $ξ > 0$ ).
2. Statistical testing where the hypothesis $ξ = 0$ cannot be rejected.
3. Being conservative and assuming $ξ > 0$ to avoid model risk.
Parameter Estimation: Parameters ( $μ$ , $σ$ , $ξ$ ) can be estimated using maximum likelihood (ML), regression, or semi-parametric methods (e.g., Hill estimator for tail index).

Practice Questions: Q1

Q1. According to the Fisher-Tippett theorem, as the sample size $n$ gets large, the distribution of extremes converges to a:

A. normal distribution.

B. uniform distribution.

C. generalized Pareto distribution.

D. generalized extreme value distribution.

Practice Questions: Q1 Answer

Explanation: D is correct.

The Fisher-Tippett theorem says that as the sample size n gets large, the distribution of extremes, denoted , converges to a generalized extreme value (GEV) distribution.

M_n

Topic 3. Peaks-Over-Threshold

Application of EVT: POT models the distribution of excess losses over a high threshold.
Parameter Efficiency: Generally requires fewer parameters than approaches based on extreme value theorems.
Modeling Excesses: Provides a natural way to model values exceeding a high threshold, aligning with GEV theory's focus on maxima/minima of large samples.
Definition of Excess Losses:
- Let $X$ be the loss and $u$ be the threshold.
- The conditional distribution of excess losses is:
- This represents the distribution of X given that the threshold is exceeded by no more than x.
Parent Distribution: The parent distribution of X is usually unknown, even if it could be normal or lognormal.

F_u(x)=P{X-u \leq x|X >u}=\frac{F(x+u)-F(u)}{1-F(u)}

Practice Questions: Q2

Q2. The peaks-over-threshold approach generally requires:

A. more estimated parameters than the GEV approach and shares one parameter with the GEV.

B. fewer estimated parameters than the GEV approach and shares one parameter with the GEV.

C. more estimated parameters than the GEV approach and does not share any parameters with the GEV approach.

D. fewer estimated parameters than the GEV approach and does not share any parameters with the GEV approach.

Practice Questions: Q2 Answer

Explanation: B is correct.

The POT approach generally has fewer parameters, but both POT and GEV approaches share the tail parameter ξ.

Topic 4. Generalized Pareto Distrobution

GPBdH Theorem: As the threshold (u) gets large, the distribution of excess losses ( $F_{u} (x)$ ) converges to a Generalized Pareto (GP) distribution.
GP Distribution Formulas:
Parameters:
- $ξ$ : Tail (or shape) index parameter, same as in GEV theory (positive, zero, or negative; mainly interested in $ξ \geq 0$ ).
- $β$ : Scale parameter (positive value).
Behavior: The GP distribution dips below the normal distribution before the tail, then rises above it in the extreme tail, providing a linear approximation that better matches empirical data.
Natural Model: Because all distributions of excess losses converge to the GP distribution, it is the natural model for excess losses.
Threshold Selection Tradeoff:
- Must be high enough for the GPBdH theorem to apply.
- Must be low enough to ensure sufficient observations ( $N_{u}$ ) above the threshold for parameter estimation.

1-[1+\frac{\xi x}{\beta}]^{-1/\xi} \text{ if }\xi \neq 0

1-exp[-\frac{x}{\beta}] \text{ if }\xi=0

Practice Questions: Q3

Q3. In setting the threshold in the POT approach, which of the following statements is the most accurate? Setting the threshold relatively high makes the model:

A. more applicable but decreases the number of observations in the modeling procedure.

B. less applicable and decreases the number of observations in the modeling procedure.

C. more applicable but increases the number of observations in the modeling procedure.

D. less applicable but increases the number of observations in the modeling procedure.

Practice Questions: Q3 Answer

Explanation: A is correct.

There is a tradeoff in setting the threshold. It must be high enough for the
appropriate theorems to hold, but if set too high, there will not be enough
observations to estimate the parameters.

Topic 5. VaR and Expected Shortfall

Goal of POT: To compute Value at Risk (VaR).
Expected Shortfall (ES): Can be derived from VaR estimates.
- ES is the average or expected value of all losses greater than the VaR.
- Expression: $E [L_{P} ∣ L_{P} > Va R]$
- Provides insight into the distribution of the size of losses exceeding VaR, making it a popular measure to report.
VaR Formula (using POT parameters):

- Where:
  
  $u =$ threshold, $n =$ total observations, $N_{v} =$ observations exceeding threshold.
Expected Shortfall Formula (using POT parameters):

V a R=u+\frac{\beta}{\xi}\left\{\left[\frac{n}{N_v}(1-\text { confidencelevel })\right]^{-\xi}-1\right\}

E S=\frac{V a R}{1-\xi}+\frac{\beta-\xi u}{1-\xi}

Practice Questions: Q4

Q4. A researcher using the POT approach observes the following parameter values: , and . The 5% VaR in percentage terms is:

A. 1.034 .

B. 1.802 .

C. 2.204 .

D. 16.559 .

\beta=0.9, \xi=0.15, \mathrm{u}=2 \%

N_u / n=4 \%

Practice Questions: Q4 Answer

Explanation: B is correct.

\begin{aligned} & \mathrm{VaR}=2+\frac{0.9}{0.15}\left\{\left\lfloor\frac{1}{0.04}(1-0.95)\right\rfloor^{-0.15}-1\right\} \\ & \mathrm{VaR}=1.802 \end{aligned}

Practice Questions: Q5

Q5. Given a VaR equal to 2.56 , a threshold of 1 %, a shape parameter equal to 0.2 , and a scale parameter equal to 0.3 , what is the expected shortfall?

A. 3.325 .

B. 3.526 .

C. 3.777 .

D. 4.086 .

Practice Questions: Q5 Answer

Explanation: A is correct.

\mathrm{ES}=\frac{\mathrm{VaR}}{1-\xi}+\frac{\beta-\xi \mathrm{u}}{1-\xi}=\frac{2.560}{1-0.2}+\frac{0.3-0.2 \times 1}{1-0.2}=3.325

Topic 6. Generalized Extreme Value vs. Peaks-Over-Threshold

Shared Origin: Both GEV and POT approaches originate from Extreme Value Theory.
Shared Parameter: Both approaches utilize the same tail parameter, $ξ$ .
Subtle Difference in Focus:
- GEV theory concentrates on the distributions of extremes (e.g., block maxima).
- POT focuses on the distribution of values that exceed a specific high threshold.
Considerations for Choice:
1. Parameter Estimation: GEV typically requires estimating one more parameter than POT. Popular GEV approaches can also lead to more data loss compared to POT.
2. Threshold Choice: The POT approach necessitates the selection of a threshold, which can introduce additional uncertainty.
3. Data Nature: The characteristics of the data might make one approach more suitable than the other.

Topic 7. Multivariate Extreme Value Theory

Importance: Extreme values are often dependent across different assets or markets (e.g., a terrorist attack affecting oil companies and broader financial assets).
Goal: Similar to univariate EVT, the objective is to move from central-value distributions to methods that estimate multivariate extreme events.
Tail Dependence: Multivariate EVT's central focus is on tail dependence, which captures how extreme values in different variables move together.
Limitations of Traditional Methods: Assumptions of elliptical distributions and the use of covariance matrices are of limited use for multivariate EVT.
Modeling with Copulas:
- Modeling multivariate extremes requires the use of copulas.
- Multivariate EVT states that the limiting distribution of multivariate extreme values will belong to the family of EV copulas.
- EV copulas can model multivariate EV dependence.
- Copulas can have as many dimensions as there are random variables.

Challenges with Increasing Dimensions:
- Increased dimensionality leads to a drastic reduction in the number of multivariate extreme observations available for analysis.
- For instance, with two independent variables, if univariate extreme events occur 1 in 100 times, a multivariate extreme event (both variables extreme) occurs 1 in 10,000 times.
- This also increases the number of parameters to estimate.