Extreme Financial Risks: From Dependence to Risk Management (Springer Finance) by Yannick Malevergne

1

 On the Origin of Risks and Extremes
 1.1 The Multidimensional Nature of Risk
 and Dependence
 In finance, the fundamental variable is the return that an investor accrues from
 his investment in a basket of assets over a certain time period. In general, an
 investor is interested in maximizing his gains while minimizing uncertainties
 (“risks”) on the expected value of the returns on his investment, at possibly
 multiple time scales– depending upon the frequency with which the manager
 monitors the portfolio– and time periods– depending upon the investment
 horizon. From a general standpoint, the return-risk pair is the unavoidable du-
ality underlying all human activities. The relationship between return and risk
 constitutes one of the most important unresolved questions in finance. This
 question permeates practically all financial engineering applications, and in
 particular the selection of investment portfolios. There is a general consensus
 among academic researchers that risk and return should be related, but the
 exact quantitative specification is still beyond our comprehension [414].
 Uncertainties come in several forms, which we cite in the order of increasing
 aversion for most human beings:
 (i) stochastic occurrences of events quantified by known probabilities;
 (ii) stochastic occurrences of events with poorly quantified or unknown prob-
abilities;
 (iii) random events that are “surprises,” i.e., that were previously thought
 to be impossible or unthinkable until they happened and revealed their
 existence.
 Here we address the first form, using the mathematical tools of probability
 theory.
 Within this class of uncertainties, one must still distinguish several
 branches. In the simplest traditional theory exemplified by Markowitz [347],
 the uncertainties underlying a given set of positions (portfolio) result from
 the interplay of two components: risk and dependence.

 (a) Risk is embedded in the amplitude of the fluctuations of the returns. its
 simplest traditional measure is the standard deviation (square-root of the
 variance).
 (b) The dependence between the different assets of a portfolio of positions
 is traditionally quantified by the correlations between the returns of all
 pairs of assets.
 Thus, in their most basic incarnations, both risk and dependence are thought
 of, respectively, as one-dimensional quantities: the standard deviation of
 the distribution of returns of a given asset and the correlation coefficient
 of these returns with those of another asset of reference (the “market” for
 instance). The standard deviation (or volatility) of portfolio returns provides
 the simplest way to quantify its fluctuations and is at the basis of Markowitz’s
 portfolio selection theory [347]. However, the standard deviation of a portfolio
 offers only a limited quantification of incurred risks (seen as the statistical fluc-
tuations of the realized return around its expected– or anticipated– value).
 This is because the empirical distributions of returns have “fat tails” (see
 Chap. 2 and references therein), a phenomenon associated with the occur-
rence of non-typical realizations of the returns. In addition, the dependences
 between assets are only imperfectly accounted for by the covariance matrix
 [309].
 The last few decades have seen two important extensions.
 • First, it has become clear, as synthesized in Chap. 2, that the standard
 deviation offers only a reductive view of the genuine full set of risks em-
bedded in the distribution of returns of a given asset. As distributions of
 returns are in general far from Gaussian laws, one needs more than one
 centered moment (the variance) to characterize them. In principle, an in-
finite set of centered moments is required to faithfully characterize the
 potential for small all the way to extreme risks because, in general, large
 risks cannot be predicted from the knowledge of small risks quantified by
 the standard deviation. Alternatively, the full space of risks needs to be
 characterized by the full distribution function. It may also be that the dis-
tributions are so heavy-tailed that moments do not exist beyond a finite
 order, which is the realm of asymptotic power law tails, of which the stable
 L´ evy laws constitute an extreme class. The Value-at-Risk (VaR) [257] and
 many other measures of risks [19, 20, 73, 447, 453] have been developed to
 account for the larger moves allowed by non-Gaussian distributions and
 non-linear correlations.
 • Second and more recently, the correlation coefficient (and its associated
 covariance) has been shown to only be a partial measure of the full de-
pendence structure between assets. Similarly to risks, a full understanding
 of the dependence between two or more assets requires, in principle, an
 infinite number of quantifiers or a complete dependence function such as
 the copulas, defined in Chap. 3

Extreme Financial Risks: From Dependence to Risk Management (Springer Finance) by Yannick Malevergne