# «1 Introduction A commonly used risk metrics is the standard deviation. For examples mean-variance portfolio selection maximises the expected utility ...»

Distortion Risk Measure or the Transformation

of Unimodal Distributions into Multimodal

Functions

Dominique Guégan and Bertrand Hassani

1 Introduction

A commonly used risk metrics is the standard deviation. For examples mean-variance

portfolio selection maximises the expected utility of an investor if the utility is

quadratic or if the returns are jointly normal. Mean-variance portfolio selection using

quadratic optimisation was introduced by Markowitz (1959) and became the standard model. This approach was generalized for symmetrical and elliptical portfolio (Ingersoll 1987; Huang and Litzenberger 1988). However, the assumption of elliptically symmetric return distributions became increasingly doubtful (Bookstaber and Clarke 1984; Chamberlain 1983) to characterize the returns distributions making standard deviation an intuitively inadequate risk measure.

Recently the ﬁnancial industry has extensively used quantile-based downside risk measures based on the Value-at-Risk (V aRα for conﬁdence level α). While the V aRα measures the losses that may be expected for a given probability it does not address how large these losses can be expected when tail events occur. To address this issue the mean excess function has been introduced, (Rockafellar and Uryasev 2000; Embrechts et al. 2005; Artzner et al. 1999) and Delbaen (2000) describe the properties that risk measures should satisfy including their coherence in particular the VaR is not a coherent risk measure, failing to be sub-additive.

When we use a sub-additive measure the diversiﬁcation of the portfolio always leads to risk reduction while if we use measures violating this axiom the diversiﬁcation beneﬁt may be lost even if partial risks are triggered by mutually exclusive events.

D. Guégan ( ) University Paris 1 Panthéon-Sorbonne et New York University Polytechnic School of Engineering, Brooklyn, New York, USA CES UMR 8174, 106 boulevard de l’Hopital, 75647 Paris Cedex 13, France tel +33144078298 e-mail: dguegan@univ-paris1.fr B. Hassani University Paris 1 Panthéon-Sorbonne, CES UMR 8174, 106 boulevard de l’Hopital, 75647 Paris Cedex 13, France tel +44 (0)2070860973 e-mail: bertrand.hassani@gmail.com A. Bensoussan et al. (eds.), Future Perspectives in Risk Models and Finance, 71 International Series in Operations Research & Management Science 211, DOI 10.1007/978-3-319-07524-2_2, © Springer International Publishing Switzerland 2015 72 D. Guégan and B. Hassani The sub-additive property is required for capital adequacy purposes in banking supervision: for instance if we consider a ﬁnancial institution made of several subsidiaries or business units, if the capital requirement of each of them is dimensioned to its own risk proﬁle authorities. Consequently it has appeared relevant to construct a more ﬂexible risk measure which is sub-additive.

Nevertheless, the V aR remains preeminent even though it suffers from the theoretical deﬁciency of not being sub-additive. The problem of sub-additivity violations is not as important for assets verifying the regularity conditions1 than for those which do not and for most assets these violations are not expected. Indeed, in most practical applications the V aRα can have the property of sub-additivity. For instance, when the return of an asset is heavy tailed, the V aRα is sub-additive in the tail region for high level of conﬁdence if it is computed with the heavy tail distribution (Ingersoll 1987; Danielson et al. 2005; Embrechts et al. 2005). Non sub-additivity of the V aRα is highlighted when assets have very skewed return distributions. When the distributions are smooth and symmetric, when assets dependency is highly asymmetric, and when underlying risk factors are dependent but heavy-tailed, it is necessary to consider other risks measures.

Unfortunately, non sub-additivity is not the only problem characterizing the V aR.

First VaR only measures distribution percentiles and thus disregards any loss beyond its conﬁdence level. Due to combined effects of this limitation and the occurrence of extreme losses there is a growing interest for risk managers to focus on the tail behavior and its Expected Shortfall2 (ESα ) since it shares properties that are considered desirable and applicable in a variety of situations. Indeed, expected shortfall considers the loss beyond the V aRα conﬁdence level and is sub-additive and therefore it ensures the coherence of the risk measure (Rockafellar and Uryasev 2000).

Since using expected utility, the axiomatic approach to risk theory has expanded dramatically as illustrated by (Yaari 1987; Panjer et al. 1997; Artzner et al. 1999; De Giorgi 2005; Embrechts et al. 2005; Denuit et al. 2006) among others. Thus other classes of risk measures were proposed each with their own properties including convexity (Follmer and Shied 2004), spectral properties (Acerbi and Tasche 2002), notion of deviation (Rockafellar et al. 2006) or distortion (Wang et al. 1997). Acerbi and Tasche (2002) studied spectral risk measures which involve a weighted average of expected shortfalls at different levels. Then, the dual theory of choice under risk leads to the class of distortion risk measures developed by Yaari (1987) and Wang (2000), which transforms the probability distribution shifting it in order to better quantify the risk in the tails instead of modifying returns as in the expected utility framework.

Whatever the risk measures considered, the value associated to each measure is based and depends on the distribution ﬁtted on the underlying data set by risk managers strategy. Mostly of the part the distributions belong to the elliptical domain, Regularly varying (heavy tailed distributions, fat tailed) non-degenerate tails with tail index η 1 for more detail see Danielson et al. (2005).

The terminology “Expected shortfall” was proposed by Acerbi and Tasche (2002). A common alternative denotation is “Conditional Value at Risk” or CVaR that was suggested by Rockafellar and Uryasev (2002).

Distortion Risk Measure or the Transformation of Unimodal Distributions... 73 recently risk managers and researchers have focused on a class of distributions exhibiting asymmetry and producing heaving tails, All these distributions belong to the Generalized Hyperbolic class of distributions (Barndorff-Nielsen 1977), to the α-stable distributions (Samorodnitsky and Taqqu 1994) or the g- and -h distributions among others.

Nevertheless nearly all these distributions are unimodal. However, since the 2000s bubbles and ﬁnancial crises and extreme events became more and more important, restricting unimodal distributions models for risk measures. Recently debates have been opened to convince economists to consider bimodal distributions instead of unimodal distributions to explain the evolution of the economy since the 2000s (Bhansali 2012). The debate about the choice of distributions characterized by several modes is timely. We propose an approach to build and ﬁt these distributions on real data sets. An objective of this paper is to discuss this new approach and propose a theoretical framework to build multi-modal distributions to create new coherent risk measures.

The paper is organized as follows. In Section two we recall some principles and history of the risk measures: the VaR, the ES and the spectral measure. In Section three we discuss the notion of distortion to create new distributions. Section four proposes an application which illustrates the impact of the choice of unimodal or bimodal distribution associated to different risk measures to provide a value for the corresponding risk. Section ﬁve concludes.

**2 Quantile-Based and Spectral Risk Measures**

Traditional deviation risks measures such as the variance, the mean-variance analysis and the standard deviation, are not sufﬁcient within the context of capital requirements. In this section we recall the deﬁnitions of several quantile-based risk measures:3 the Value-at-Risk introduced in the 1980s, the Expected Shortfall proposed by Acerbi and Tasche (2002), the Tail Conditional Expectation suggested by Rockafellar and Uryasev (2002), and the spectral measure introduced by Acerbi and Tasche (2002).

Value at Risk initially used to measure ﬁnancial institutions market risk, was mainly popularised by J.P. Morgan’s RiskMetrics (1995). This measure indicates the maximum probable loss, given a conﬁdence level and a time horizon. The V aR is sometimes referred as the “unexpected” loss.

Deﬁnition 1 Given a conﬁdence level α ∈ (0, 1), the V aR is the relevant quantile4 of the loss distribution: V aRα (X) = inf{x | P [X x] 1 − α} = inf{x | FX (x) α} where X is a risk factor admitting a loss distribution FX.

The Expected Shortfall has a number of advantages over the V aRα. Accordingly the ES takes accounts for the tail risk and fulﬁlls the sub-additive property5 (Acerbi and Tasche 2002)6. Table 1 summarizes the link between ESα and V aRα for some distributions given α.

Expected Shortfall is the smallest coherent risk measure that dominates the V aR.

Acerbi and Tasche (2002) derived from this concept a more general class of coherent risk measures called spectral risk measures7. Spectral risk measures are a subset of coherent risk measures. Instead of averaging losses beyond the V aR, a weighted An extension can be found in Inui and Kijima (2005).

In this last paper, the difference between ES and T CE is conceptual and is only related to the distributions. If the distribution is continuous then the expected shortfall is equivalent to the tail conditional expectation.

If ρi is coherent risk measures for i = 1...n, then, any convex combination ρ = n βi ρi is a coherent risk measure (Acerbi and Tasche 2002).

Distortion Risk Measure or the Transformation of Unimodal Distributions... 75 Fig. 1 Spectrum of the ES for some well known distributions for several α ∈ [0.9, 0.99]. Each line corresponds to the graph of the ES as a function of α for each distribution introduced in Table 1

**average of different levels of ESα is used. These weights characterize risk aversion:**

different weights are assigned to different α levels of ESα in the left tail. The associated spectral measure could be α wα ESα, where α wα = 1. In Fig. 1 we exhibit a spectrum corresponding to the sequence of ESα for different α.

Figure 1 points out that the spectrum of the ES is an increasing function of the conﬁdence level α. It expresses the risk aversion as a weighted average for different level of ESα to generate the spectral risk measure. This is one advantage when using a spectral risk measure. Moreover a spectral risk measure being a convex combination of ESα for α ∈ [0.9, 0.99], it accounts for more information than only considering one value of α.

However the choice of weights is sensitive and need to be studied more carefully (Dowd et al. 2008). Finally, in practice the relation between spectral risk measure and risk aversion is not obvious depending on the choice of the weights.

3 Distortion Risk Measures

3.1 Notion of Distortion Risk Measures

**Deﬁnition 3 A function g : [0, 1] → [0; 1] is a distortion function if:**

1. g(0) = 0 and g(1) = 1,

2. g is a continuous increasing function.

In order to quantify the risk instead of modifying the loss distribution (as with the expected utility framework), the distortion approach modiﬁes the probability distribution. The risk measures (VaR and ES) derived from this transformation were originally applied to a wide variety of ﬁnancial problems such as the determination of insurance premiums (Wang 2000), economic capital (Hürlimann 2004), and capital allocation (Tsanakas 2004). Acerbi (2002) suggests that they can be used to set capital requirements or obtain optimal risk-expected return trade-offs and could also be used by clearing-houses to set margin requirements that reﬂect their corporate risk aversion (Cotter and Dowd 2006).

One possibility is to shift the distribution function towards the left or the right sides to account for extreme values. Wang et al. (1997) developed the concept of distortion8 risk measure by computing the expected loss from a non-linear transformation of the cumulative probability distribution of the risk factor. A formal deﬁnition of this risk measure computed from a distortion of the original distribution has been derived (Wang et al. 1997).

Deﬁnition 4 The distorted risk measure ρg (X) for a risk factor X admitting a cumulative distribution SX (x) = P(X x), with a distortion function g, is deﬁned9

**as:**

+∞ ρg (x) = [g(SX (x)) − 1]dx + g(SX (x))dx. (1) −∞ 0

Distortion functions arose from empirical10 observations that people do not evaluate risk as a linear function of the actual probabilities for different outcomes but The distortion risk measure is a special class of the so-called Choquet expected utility, i.e. the expected utility calculated under a modiﬁed probability measure.

Both integrals in (1) are well deﬁned and take a value in [0, +∞]. Provided that at least one of the two integrals is ﬁnite, the distorted expectation ρg (X) is well deﬁned and takes a value in [ − ∞, +∞].

This approach towards risk can be related to investor’s psychology as in Kahneman and Tversky (1979).

Distortion Risk Measure or the Transformation of Unimodal Distributions... 77

When g is a concave function its ﬁrst derivative g is an increasing function, g (SX (x)) is a decreasing function11 in x and g (SX (x)) represents a weighted coefﬁcient which discounts the probability of desirable events while loading the probability of adverse events. Moreover, Hardy and Wirch (2001) have shown that distorted risk measure ρg (X) introduced in (2) is sub-additive and coherent if and only if the distortion function is concave.

In his article, Wang (2000) speciﬁes that the distortion operator g can be applied to any distribution. Nevertheless in applications due to technical practical reasons he

**restricts the illustration of his methodology to a function g deﬁned as follows:**