Abstract:
In this dissertation we take a closer look at how copulas can be used to improve the risk
measurement at a financial institution. The focus is on market risk in a trading environment.
In practice risk numbers are calculated with very basic measures that are easy to explain to
senior management and to traders. It is important that traders understand the risk measure
as that helps them to understand the risk inherent in any deal and may assist them in
deciding on the optimal hedge. The purpose of a hedge is to reduce the risk in a portfolio.
As senior management is responsible for deciding on the optimal risk limits and risk appetite
of the financial institution, it is important for them to understand what the risks are and how
to measure these.
The simplicity of the risk measures leads to certain inadequacies that can have very negative
consequences for a financial institution. If the risk measure does not adequately capture the
risk of a deal, the financial institution may suffer big losses when there are stress events in
the market. Alternatively, when the risk measure overestimates the risk of a deal, too much
economic capital is tied up in the deal. This inhibits the trader from adding more deals to a
portfolio that may potentially lead to big profits. Economic capital is the capital that has to
be held against positions to protect the financial institution if and when extreme market
moves occur.
In this dissertation the focus is on how copulas can be used to improve current risk
measures. We focus on bivariate copulas. Bivariate copulas are easier to depict graphically
than multivariate copulas with more than two dimensions. It is also easier to prove that the
fitted bivariate copulae do adequately describe the underlying dependence structure
between risk factors. Even though the focus is on the bivariate case, all methodologies can
easily be extended to higher dimensions.
In Chapter 1 copulas are defined and some basic copula properties are shown. We consider
the definition of elliptical copulas and discuss some drawbacks to using them in a financial
application. Some useful Archimedean copula properties are discussed and it is shown how
to generate the copula function for n 2 dimensions. The various ways in which to estimate
the parameters of a copula are also discussed as well as goodness-of-fit tests that are used
to test whether the copula fits the underlying data adequately. Finally the chapter ends with
an example that illustrates the theory. A back-test is done to establish whether the copula
adequately describes the dependence structure over time. It is also shown how the fitted copula can be used to generate stress scenarios that are used as an alternative to historical
scenarios when calculating a value-at-risk (VaR) number.
In chapter 2 the properties of a dependence measure are discussed and it is argued that
linear correlation does not conform to these desired properties. Rank correlation measures
have some additional properties that make them more efficient than linear correlation
measures in certain instances. We also consider their relationship to copulas. Finally it is
shown how copulas can be used in practice to get another view on the dependence
structure between risk factors.
In risk measurement we are mainly concerned with extreme moves that market variables
may show. In chapter 3 some of the techniques used in risk management are discussed as
well as some of their shortcomings. The shortcomings are addressed by applying extreme
value theory to calculate stress factors and using copulas to model the dependence structure
between risk factors. The theory underlying bivariate extreme copulas is discussed and
illustrated with a practical example.