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

−1 gα (u) = (u) + α, (3) where is the Gaussian cumulative distribution. In other words he applies the same perspective of preference to quantify the risk associated to gain and risk. Thus, a risk manager evaluates the risk associated to the upside and downside risks with the same function g implying a symmetric consideration for the two effects due to the distortion. Moreover it induces the same conﬁdence level for the losses and the gain which implies the same level of risk aversion associated to the losses and the gains.

In Fig. 2 we illustrate the impact of the Wang (2000) distortion function introduced in Eq. (3) on the logistic distribution provided in Table 1. We can remark that the distorted distribution is always symmetrical under this kind of distortion function, and we observe a shift of the mode of the initial distribution towards the left.

To avoid the problem of symmetry in the previous distorsion, Sereda et al. (2010) propose to use two different functions issued from the same polynomial with different

**coefﬁcients, say:**

Fig. 2 Distortion of logistic distribution with mean 0 using a Wang distortion function with conﬁdence level 0.65. It illustrates the effect of distortion with gi (u) = u + ki u − u2 for ki ∈ ]0, 1] et ∀i ∈ {1, 2}. With this approach one models loss and gains differently relatively to the values of the parameters ki, i = 1, 2.

Thus upside and downside risks are modeled in different ways. Nevertheless the calibration of the parameters ki, i = 1, 2 remains an open problem.

To create bimodal or multi-modal distributions we have to impose other properties to the distortion function g. Indeed, transforming an unimodal distribution into a bimodal one provides different approaches to the risk aversion of losses and gains.

This will allow us to introduce a new coherent risk measure in that latter case.

**3.2 A New Coherent Risk Measure**

We begin to discuss the choice of the function g to obtain a bimodal distribution. To do so we need to use a function g which creates saddle points. The saddle point generates a second hump in the new distribution which allows us to take into account different patterns located in the tails. The distortion function g fulﬁlling this objective is an inverse S-shaped polynomial function of degree 3 given by the following equation

**and characterized by two parameters δ and β:**

Fig. 3 Curves of the distortion function gδ introduced in Eq. (5) for several value of δ and ﬁxed values of β = 0.001 In Fig. 3, the value of the level of the discrimination of an event is given by β = 0.001 then we plot the function gδ for different values of δ. This parameter β illustrates the fact that some events are discriminating more than others. Figure 3 shows the location of the saddle point creating convex and concave parts inside the domain [0, 1]. The convex part can be associated to the negative values of the returns associated to the losses and the concave part is associated to positive returns. We observe in this picture that for high values of δ the concave part diminishes and then the effect of saddle point decreases.

Variations in β in Fig. 4 exhibit different patterns for a ﬁxed value of δ.

To understand the inﬂuence of the parameter β on the shape of the distortion function we use three graphs in Fig. 4. The two left graphs correspond to the same value of the parameters. The middle ﬁgure zooms on the x-axis from [0, 1] to [−4, 4].

We show that the function g may not have a saddle point on ]0, 1[ depending on the values of β. The right graph provides different representations of the distorsion function for several values of β. We observe that if β tends to 1 then the distortion function g tends to the identity mapping and when β tends to 0 the curve is more important and the effect of g on the distribution will be more important.

Figure 5 illustrates the effect of distortion of the Gaussian distribution for several values of β and ﬁxed δ = 0.50. We observe the same effects as in Fig. 4. For small values of the parameter β (0.00005 or 0.005) the distortion function has two distinct parts, one convex part for x ∈ ]0, 0.5[ and one concave part for x ∈ ]0.5, 1[.

Moreover when β is close to 1 then the distorted cumulative distribution tends to the initial Gaussian variable.

80 D. Guégan and B. Hassani Fig. 4 The effect of β on the distortion function for a level of security δ = 0.75 showing that if β tends to 1, the distortion function tends to the identity function Fig. 5 The effect of β on the cumulative Gaussian distribution for δ = 0.50 Figure 6 points out the effect of distortion on the density of the Gaussian distribution using the same values of the parameters than those used in Fig. 5. Again we generate a new distribution with two humps. Making both parameters varying permits to solve one of our objective: to create a asymmetrical distribution with more than one hump.

It is important to notice that the function gδ creates a distorted density function which associates a small probability in the centre of the distribution and put greater weight in the tails. This phenomenon is illustrated in Fig. 7 where the derivative of g (density) indicates how weights on the tails can be increased.

Such discrimination is also illustrated in Fig. 8 which exhibits the particular effect of parameter β when δ is ﬁxed to 0.75 for the creation of humps. From a Gaussian distribution, applying gδ deﬁned in (5), with δ = 0.75 and β = 0.48 we create a distribution for which the probability of occurrences of the extremes in the right part is bigger than the probability of occurrence of the extremes in the left part which can be counter-intuitive for risk management but interesting from a theoretical point of view.

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