These Lecture series are relating the use R language software, its interface and functions required to evaluate financial risk models. Furthermore, R software applications relating financial market data, measuring risk, modern portfolio theory, risk modeling relating returns generalized hyperbolic and lambda distributions, Value at Risk (VaR) modelling, extreme value methods and models, the class of ARCH models, GARCH risk models and portfolio optimization approaches.

1. FINANCIAL RISK MGT – FRM
Lecture by;
Dr. Syed
Muhammad Ali
Tirmizi
1

2. 2
TOPICS OF CHAPTER NO. 9
 In this lecture, we will cover the following topics:
9. Modelling Dependence
i. Overview
ii. Correlation, Dependence, and Distributions
iii. Copulae
a. Motivation
b. Correlations and Dependence Revisited
c. Classification of Copulae
iv. Synopsis of R packages
a. The package BLCOP
b. The package copula
c. The package fCopulae
d. The package gumbel
e. The package QRM

3. 3
TOPICS OF CHAPTER NO. 9
v. Empirical Applications of Copulae
a. GARCH – copula model
b. Motivation
c. Description
d. Application (R Codes)
vi. Mixed Copula Approaches
e. Application (R Codes)
f. Results and Discussion

4. 4
CHAPTER OVERVIEW
 In this chapter the topic of financial risk modelling
in the context of multiple financial instruments is
addressed.
 Discussion is on the correlation coefficient between
two assets and investigates its appropriateness as a
measure of dependence between two assets.
 We further discusses alternative measures of
dependence, namely the use of rank correlations and
the concept of the copula.

5. 5
CHAPTER OVERVIEW
 Discussion is further extended to the synopsis of the
R packages that specifically include copula
modelling.
 Finally, we will learn how copula models can be
fruitfully combined with the techniques outlined in
Chapters 6, 7 & 8.
 In particular, a copula-GARCH model is proposed
for measuring the market risk of a portfolio.

6. 6
CORRELATION, DEPENDENCE,
AND DISTRIBUTIONS
 The computation and usage of Pearson’s correlation
coefficient is quite pervasive in the quantitative
analysis of financial markets.
 However, applied quantitative researchers are often
unaware of the pitfalls involved in careless application
and usage of correlations as a measure of risk.
 It is therefore appropriate to investigate this
dependence concept in some detail and point out the
shortcomings of this measure.

7. 7
CORRELATION, DEPENDENCE,
AND DISTRIBUTIONS

8. 8
CORRELATION, DEPENDENCE,
AND DISTRIBUTIONS
 The implication for multivariate risk modelling is that only
in the case of jointly elliptically distributed risk factors can
the dependence between these be captured adequately by
the linear correlation coefficient.
 Given the stylized facts of financial market returns this
assumption is barely met.
 It should be pointed out at this point that with respect to
risk modelling one is usually more concerned with the
dependence structure in the tail of a multivariate loss
distribution than with an assessment of the overall
dependence.

9. 9
CORRELATION, DEPENDENCE,
AND DISTRIBUTIONS
Ellipses are common
in physics, astronomy and engineering.
For example, the orbit of each planet in
the solar system is approximately an
ellipse with the Sun at one focus point
(more precisely, the focus is
the barycenter of the Sun planet pair).
The same is true for moons orbiting
planets and all other systems of two
astronomical bodies.

10. 10
CORRELATION, DEPENDENCE,
AND DISTRIBUTIONS
 To highlight the fact that the correlation coefficient
depends on the marginal distributions of the random
variables in question and the possibility that the
correlation coefficient cannot take all values in the
interval −1 ≤ 𝜌 ≤ 1 if one views the multivariate
distribution of these random variables.

11. 11
CORRELATION, DEPENDENCE,
AND DISTRIBUTIONS

12. 12
CORRELATION, DEPENDENCE,
AND DISTRIBUTIONS
 The dependence between financial instruments can
only be depicted correctly with the linear correlation
coefficient if these are jointly elliptically distributed.
 It was also shown that the value of the correlation
coefficient depends on the marginal distributions
and that not all values in the range [−1, 1] are
attainable.

13. 13
COPULAE
 Motivation: The copula approach was introduced by
Sklar (1959). Detailed textbook expositions can be
found in Schweizer and Sklar (1983), Nelsen
(2006), Joe (1997), and McNeil et al. (2005).
 However, only since the mid-1990s have copulae
been used as a tool for modelling dependencies
between assets in empirical finance.
 The word “copula” derives from the Latin verb
copulare and means to “bond” or “tie.”

14. 14
COPULAE
 The marginal distributions of jointly distributed random
variables as well as their dependence are contained in
their joint distribution function, which is as follows;
 This measure may take values less than 1 even if there is
perfect dependence between the two random variables.
 Hence, it is necessary to separate the marginal
distributions from the dependence structure between the
random variables.
 This separation can be achieved by means of a copula.

15. 15
COPULAE

16. 16
COPULAE
 Hence, a copula is the distribution function in ℝ𝑑
space of a d-element random vector with standard
uniform marginal distributions U(0, 1).
 Alternatively, a copula can be interpreted as a function
that maps from the d-dimensional space [0, 1]d into
the unit interval:
 In this interpretation further conditions must be met
by the function C in order to qualify as a copula.
 The dependence structure of the random vector X is
embodied in th copula.

17. 17
CORRELATIONS AND DEPENDENCE REVISITED
 The linear correlation coefficient does not capture
dependencies well in the case of non-elliptically
distributed random variables.
 Furthermore, while this measure depicts the overall
dependence, in the assessment of the riskiness of a
portfolio what matters most is the dependence in the
tail of the joint distribution.
 Hence, in this subsection two further concepts for
capturing the dependence between risk factors are
introduced, namely, concordance and tail dependence.

18. 18
CORRELATIONS AND DEPENDENCE REVISITED

19. 19
CLASSIFICATION OF COPULAE

20. 20
CORRELATIONS AND DEPENDENCE REVISITED
 Two uniformly distributed random variables U1 and
U2 are assumed. Between these two variables three
extreme cases of dependence can be distinguished:
concordance, independence, and discordance.
 First, the case of perfect positive dependence will be
investigated (e.g., U1 = U2). The copula for this case
could then be written as;

21. 21
CORRELATIONS AND DEPENDENCE REVISITED
 The copula would also be valid if a monotone
transformation is applied to the random variables.
 As an intermediate step between this case and that of
discordant random variables, independence will be
investigated next.
 The copula is then the product of the two random
variables:

22. 22
CORRELATIONS AND DEPENDENCE REVISITED
 The joint density function of independent random
variables is the product of the respective marginal
distributions, which equals the independence copula:

23. 23
CORRELATIONS AND DEPENDENCE REVISITED
 We will now classify copulae into two broad
categories, namely Archimedean copulae on the one
hand and distribution-based copulae on the other.
 Within the latter group of copulae the dependence
between the random variables will be captured
implicitly by a distribution parameter.
 For instance, the bivariate case of a Gauss copula is
defined and expressed in the next slide.

24. 24
CORRELATIONS AND DEPENDENCE REVISITED

25. 25
CORRELATIONS AND DEPENDENCE REVISITED
 We will now focus on the Student’s t copula, which is
defined as;

26. 26
CORRELATIONS AND DEPENDENCE REVISITED

27. 27
CORRELATIONS AND DEPENDENCE REVISITED

28. 28
CORRELATIONS AND DEPENDENCE REVISITED

29. 29
CORRELATIONS AND DEPENDENCE REVISITED
 In principle, the unknown parameters of a copula can be
estimated in two ways.
 The first procedure is fully parametric. Because for
multivariate distribution models derived from a copula of
higher dimensions this procedure can be quite burdensome,
Joe and Xu (1996) and Shih and Louis (1995) proposed a
two-step estimation.
 Here, the unknown parameters for the assumed models of
the marginal distributions are estimated first. Based upon
these fitted models, the pseudo-uniform variables are
retrieved from the inverse distribution functions.
 These can then be used for maximizing the likelihood
(copula).
 This approach is, for obvious reasons, often termed
inference functions for margins.

30. 30
CORRELATIONS AND DEPENDENCE REVISITED
 The second procedure is based upon a semi-parametric
approach.
 In contrast to the first procedure, no models are assumed for
the marginals, but rather the empirical distribution functions
are employed to retrieve the pseudo-uniform variables.
 These are then used for maximizing the pseudo-likelihood.
 The parameters of the copula are determined by means of
numerical optimization techniques.
 For the case of a Student’s t copula a simplification results if
Kendall’s tau is calculated first and then the pseudo-
likelihood has to be maximized only with respect to the
degrees of freedom parameter 𝜈.

31. 31
SYNOPSIS FOR R PACKAGES
 The package BLCOP: The package BLCOP (see
Gochez et al. 2015) implements the Black–Litterman
approach (see Black and Litterman 1990) to portfolio
optimization and the framework of copula opinion
pooling (see Meucci 2006a,b, 2010).
 The package depends on the packages methods, MASS
(see Venables and Ripley 2002), and quadprog the latter
is a non-standard package and will be installed
automatically if not found in the search path.

32. 32
SYNOPSIS FOR R PACKAGES
 The package copula: In the package copula the copula
concept is implemented in a broad and self-contained
manner (see Hofert and Mächler 2011; Hofert et al.
2015; Kojadinovic and Yan 2010; Yan 2007).
 The package is considered to be a core package of the
CRAN “Distributions” Task View and is also listed in
the “Finance” and “Multivariate” Task Views. S4 classes
and methods not only for elliptical and Archimedean but
also for extreme value copulae are defined, enabling the
user to estimate copulae and conduct statistical tests with
respect to the appropriateness of a chosen copula.

33. 33
SYNOPSIS FOR R PACKAGES
 The package fCopulae: Similar to the package
copula, the package fCopulae offers a unified
treatment of copulae (see Würtz and Setz 2014).
 The package is part of the Rmetrics suite of
packages and is considered a core package in the
CRAN “Distributions” and “Finance” Task Views.
 It employs S4 as well as S3 classes and methods.

34. 34
SYNOPSIS FOR R PACKAGES
 The package gumbel: The package gumbel is
dedicated solely to the Gumbel copula (see Dutang
2015).
 It is contained in the CRAN “Distribution” Task
View.
 In addition to the density and distribution function
and the generation of random variates, the
generating function of the Gumbel copula and its
inverse are available.

35. 35
SYNOPSIS FOR R PACKAGES
 The package QRM: The package QRM contains
functions for the Frank, Gumbel, Clayton, normal,
and Student’s t copulae.
 Here, the densities and the generation of random
variates are implemented.
 The fitting of Archimedean copulae is conducted
with the function fit.AC().
 Numerical maximization of the likelihood is
accomplished by nlminb().

36. 36
EMPIRICALAPPLICATIONS OF COPULAE
 GARCH–copula model – Motivation
 In particular, it was stated that the assumptions of an iid
and normal return process are generally not met in
practice.
 The stylized facts regarding “fat tails” and “volatility
clustering” can be viewed as the flip side of these
violated assumptions.
 As should be evident from this chapter the appropriate
determination of the dependence between the financial
instruments in a portfolio is of pivotal importance.

37. 37
EMPIRICALAPPLICATIONS OF COPULAE
 GARCH–copula model – Motivation
 The question of the kind of model to employ in
simultaneously addressing the univariate and
multivariate stylized facts for market returns in the
portfolio context arises.
 We are now in a position to put together the pieces
from the previous chapters of this part of the book,
that is, to combine GARCH and copula models.

38. 38
EMPIRICALAPPLICATIONS OF COPULAE
 GARCH–copula model – Motivation
1. GARCH models possess a higher kurtosis than the normal
distribution, hence the higher probability of witnessing
extreme returns can be captured with these volatility
models.
2. Risk measures based on the variance-covariance approach
are unconditional in nature and therefore imply an iid
process. It is therefore impossible to model volatility
clusters. The GARCH models specifically capture this
empirical artifact and allow the computation of
conditional risk measures.

39. END OF LECTURE NO. 10
39