Market risks are the prospect of financial losses - or gains - due to unexpected changes in market prices and rates. Evaluating the exposure to such risks is nowadays of primary concern to risk managers in financial and non-financial institutions alike. Until late 1980s market risks were estimated through gap and duration analysis (interest rates), portfolio theory (securities), sensitivity analysis (derivatives) or ''what-if'' scenarios. However, all these methods either could be applied only to very specific assets or relied on subjective reasoning.
Since the early 1990s a commonly used market risk estimation
methodology has been the Value at Risk (VaR).
A VaR measure is the highest possible loss incurred from holding
the current portfolio over a certain period of time at a given
confidence level ([27,40,51]):
![]() ![]() |
The VaR figure has two important characteristics: (1) it provides a common consistent measure of risk across different positions and risk factors and (2) it takes into account the correlations or dependencies between different risk factors. Because of its intuitive appeal and simplicity, it is no surprise that in a few years Value at Risk has become the standard risk measure used around the world. However, VaR has a number deficiencies, among them the non-subadditivity - a sum of VaR's of two portfolios can be smaller than the VaR of the combined portfolio. To cope with these shortcomings, [2] proposed an alternative measure that satisfies the assumptions of a coherent, i.e. an adequate, risk measure. The Expected Shortfall (ES), also called Expected Tail Loss or Conditional VaR, is the expected value of the losses in excess of VaR:
ES![]() ![]() |
The essence of the VaR and ES computations is estimation of low quantiles in the portfolio return distributions. Hence, the performance of market risk measurement methods depends on the quality of distributional assumptions on the underlying risk factors. Many of the concepts in theoretical and empirical finance developed over the past decades - including the classical portfolio theory, the Black-Scholes-Merton option pricing model and even the RiskMetrics variance-covariance approach to VaR - rest upon the assumption that asset returns follow a normal distribution. But is this assumption justified by empirical data?
No, it is not! It has been long known that asset returns are not
normally distributed. Rather, the empirical observations exhibit
excess kurtosis (fat tails). The Dow Jones Industrial Average (DJIA)
index
is a prominent example, see Fig. 1.1 where the index
itself and its returns (or log-returns) are depicted. In
Fig. 1.2 we plotted the empirical distribution of the
DJIA index. The contrast with the Gaussian law is striking. This heavy
tailed or leptokurtic character of the distribution of price changes
has been repeatedly observed in various markets and may be
quantitatively measured by the kurtosis in excess of , a value
obtained for the normal distribution
([13,20,42,67,87]). In
Fig. 1.2 we also plotted vertical lines representing
the Gaussian and empirical daily VaR estimates at the
and
confidence levels. They depict
a typical situation encountered in financial markets. The Gaussian
model overestimates the VaR number at the
confidence
level and underestimates it at the
confidence level.
These VaR estimates are used here only for illustrative purposes and
correspond to a one day VaR of a virtual portfolio consisting of one
long position in the DJIA index. Note, that they are equal to the
absolute value of the and
quantiles,
respectively. Hence, calculating the VaR number reduces to finding the
quantile. The empirical
quantile is obtained by taking
the
th smallest value of the sample, where
is the smallest
integer greater or equal to the lenght of the sample times
. The Gaussian
quantile is equal to
,
where
is the normal distribution function. Since algorithms for
evaluating the inverse of the Gaussian distribution function are
implemented in practically any computing environment, calculating the
quantile is straightforward.
Interestingly, the problem of the underestimation of risk by the
Gaussian distribution has been dealt with by the regulators in an ad
hoc way. The [7] suggested that for the purpose of
determining minimum capital reserves financial institutions use a ten
day VaR at the
confidence level multiplied by
a safety factor
, with the exact value of
depending on
the past performance of the model. It has been argued by [93]
and [26] that the range of the safety factor comes
from the heavy-tailed nature of the returns distribution. Indeed, if
we assume that the asset returns distribution is symmetric and has
finite variance
then from Chebyshev's inequality
([58]) we obtain
, where
represents the random loss over the
specified time horizon. So if we want to calculate the upper bound for
a
VaR, setting
yields
, which in turn implies that
VaR
. However, if we
assumed a Gaussian distribution of returns then we would have
VaR
, which is roughly
three times lower than the bound obtained for a heavy-tailed, finite
variance distribution.
Having said this much about the inadequacy of the Gaussian distribution for financial modeling and risk management we have no other choice but offer some heavy-tailed alternatives. We have to mention, though, that all distributional classes described in this chapter present computational challenge. Large parts of the text are thus devoted to numerical issues. In Sect. 1.2 we deal with the historically earliest alternative - the stable laws and briefly characterize their recent generalizations - the so-called truncated stable distributions. In Sect. 1.3 we study the class of generalized hyperbolic laws. Finally, in Sect. 1.4 we introduce the notion of copulas and discuss the relation between VaR, asset portfolios and heavy tails.
All theoretical results are illustrated by empirical examples which utilize the quantlets (i.e. functions) of the XploRe computing environment ([43]). For reference, figure captions include names of the corresponding quantlets (Q). The reader of this chapter may therefore repeat and modify at will all the presented examples via the local XploRe Quantlet Server (XQS) without having to buy additional software. Such XQ Servers may be downloaded freely from the XploRe website http://www.xplore-stat.de. Currently, no other statistical computing environment offers a complete coverage of the issues discussed in this chapter. However, when available links to third-party libraries and specialized software are also provided.