LO 45.6: Explain the importance of multivariate EVT for risk management.

Multivariate EVT is important because we can easily see how extreme values can be dependent on each other. A terrorist attack on oil fields will produce losses for oil companies, but it is likely that the value of most financial assets will also be affected. We can imagine similar relationships between the occurrence of a natural disaster and a decline in financial markets as well as markets for real goods and services.

Multivariate EVT has the same goal as univariate EVT in that the objective is to move from the familiar central-value distributions to methods that estimate extreme events. The added feature is to apply the EVT to more than one random variable at the same time. This introduces the concept of tail dependence, which is the central focus of multivariate EVT. Assumptions of an elliptical distribution and the use of a covariance matrix are of limited use for multivariate EVT.

Modeling multivariate extremes requires the use of copulas. Multivariate EVT says that the limiting distribution of multivariate extreme values will be a member of the family of EV copulas, and we can model multivariate EV dependence by assuming one of these EV copulas. The copulas can also have as many dimensions as appropriate and congruous with the number of random variables under consideration. However, the increase in the dimensions will present problems. If a researcher has two independent variables and classifies univariate extreme events as those that occur one time in a 100, this means that the researcher should expect to see one multivariate extreme event (i.e., both variables taking extreme values) only one time in 100 x 100 = 10,000 observations. For a trinomial distribution, that number increases to 1,000,000. This reduces drastically the number of multivariate extreme observations to work with, and increases the number of parameters to estimate.

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Topic 45 Cross Reference to GARP Assigned Reading – Dowd, Chapter 7

Ke y C o n c e pt s

LO 45.1 Estimating extreme values is important since they can be very costly. The challenge is that since they are rare, many have not even been observed. Thus, it is difficult to model them.

LO 45.2 Extreme value theory (EVT) can be used to model extreme events in financial markets and to compute VaR, as well as expected shortfall.

LO 45.3 The peaks-over-threshold (POT) approach is an application of extreme value theory. It models the values that occur over a given threshold. It assumes that observations beyond the threshold follow a generalized Pareto distribution whose parameters can be estimated.

LO 45.4 The GEV and POT approach have the same goal and are built on the same general principles of extreme value theory. They even share the same shape parameter:

LO 45.5 The parameters of a generalized Pareto distribution (GPD) are the scale parameter (3 and the shape parameter Both of these can be estimated using maximum-likelihood technique

When applying the generalized Pareto distribution, the researcher must choose a threshold. There is a tradeoff because the threshold must be high enough so that the GPD applies, but it must be low enough so that there are sufficient observations above the threshold to estimate the parameters.

LO 45.6 Multivariate EVT is important because many extreme values are dependent on each other, and elliptical distribution analysis and correlations are not useful in the modeling of extreme values for multivariate distributions. Modeling multivariate extremes requires the use of copulas. Given that more than one random variable is involved, modeling these extremes can be even more challenging because of the rarity of multiple extreme values occurring at the same time.

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Topic 45 Cross Reference to GARP Assigned Reading – Dowd, Chapter 7

C o n c e pt Ch e c k e r s

1.

2.

3.

4.

5.

According to the Fisher-Tippett theorem, as the sample size n gets large, the distribution of extremes converges to: A. a normal distribution. B. a uniform distribution. C. a generalized Pareto distribution. D. a generalized extreme value distribution.

The peaks-over-threshold approach generally requires: A. more estimated parameters than the GEV approach and shares one parameter

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

with the GEV.

with the GEV.

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

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

parameters with the GEV approach.

parameters with the GEV approach.

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

B.

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

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

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

procedure.

procedure.

A researcher using the POT approach observes the following parameter values: (3 = 0.9, = 0.15, u = 2% and Nu/n = 4%. The 5% VaR in percentage terms is: A. 1.034. B. 1.802. C. 2.204. D. 16.559.

Given a VaR equal to 2.56, a threshold of 1%, a shape parameter equal to 0.2, and scale parameter equal to 0.3, what is the expected shortfall? A. 3.325. B. 3.526. C. 3.777. D. 4.086.

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Topic 45 Cross Reference to GARP Assigned Reading – Dowd, Chapter 7

C o n c e pt C h e c k e r An s w e r s

1. D The Fisher-Tippett theorem says that as the sample size n gets large, the distribution of

extremes, denoted M , converges to a generalized extreme value (GEV) distribution.

2. B The POT approach generally has fewer parameters, but both POT and GEV approaches

share the tail parameter t

3. A There is a trade-off 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.

,

0.9 [ 1 0.15 [0.04

– a -0.95)

J – 0.15 J

-1

4. B V aR – 2 +

VaR = 1.802

5. A ES =

5

1-t

1 – t

0.3 0.2×1 2.560 ——- + ————– = 3.325 1- 0.2

1- 0.2

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The following is a review of the Operational and Integrated Risk Management principles designed to address the learning objectives set forth by GARP. This topic is also covered in:

Va l i d a t i n g Ra t i n g M o d e l s

Topic 46

Ex a m Fo c u s

This is a specialized and rather detailed topic that deals with rating system validation. There is broad coverage of both qualitative and quantitative validation concepts with greater importance being assigned to qualitative validation. For the exam, focus on best practices as well as the specific elements of qualitative and quantitative validation. Within the realm of quantitative validation, focus specifically on the concepts of calibration and discriminatory power. Note that this material is an extension of the Rating Assignment Methodologies topic from Book 2 (Topic 19).

M o d e l V a l i d a t i o n