Dominique Guegan
email: dguegan@univ-paris1.fr

Bertrand K Hassani
email: bertrand.hassani@malix.univ-paris1.fr.

Combining risk measures to overcome their limitations -spectrum representation of the sub-additivity issue, distortion requirement and added-value of the Spatial VaR solution: An application to Regulatory Requirement for Financial Institutions

Keywords:

To measure the major risks experienced by financial institutions, for instance Market, Credit and Operational, regarding the risk measures, the distributions used to model them and the level of confidence, the regulation either offers a limited choice or demands the implementation of a particular approach. In this paper, we highlight and illustrate the paradoxes and issues observed when implementing an approach over another, the inconsistencies between the methodologies suggested and the problems related to their interpretation. Starting from the notion of coherence, we discuss their properties, we propose alternative solutions, new risk measures like spectrum and spatial approaches, and we provide practitioners and supervisor with some recommendations to assess, manage and control risks in a financial institution 3 .

Introduction

The ECB-SSM 4 , the EBA 5 and the Basel Committee have been constantly reviewing the methodological framework of risk modelling for the past 20 years. In this paper, we analyse some of the issues observed when measuring the risks that would be worth addressing in future regulatory documents.

Problematic

During the 2007/08' crisis 6 , the failure of models and the lack of capture of extreme exposures have led regulators to change the way risks were measured, either by requiring financial institutions to use particular families of distributions (Gaussian (BCBS ( 2005)), sub-exponential (EBA (2014b))), or by changing the way dependencies are captured (EBA (2014b)) or by suggesting a shift from the Value-at-Risk (VaR) 7 to sub-additive risk measures like the Expected Shortfall (ES) 8 (BCBS (2013)). Indeed, inappropriate risk modelling had played a major role during the crisis which began in 2008 either as a catalyst or trigger. The latest changes proposed by the authorities have been motivated by the will to come closer to the reality of financial markets. 4 European Central Bank -Single Supervisor Mechanism 5 European Banking Authority 6 Known as the subprime crisis.

7 Given a confidence level p ∈ [0, 1], the VaR associated to a random variable X is given by the smallest number

x such that the probability that X exceeds x is not larger than (1 -p) V aRp = inf(x ∈ R : P (X > x) ≤ (1 -p)).

(1.1)

8 For a given p in [0, 1], η the V aRp, and X a random variable which represents losses during a pre-specified period (such as a day, a week, or some other chosen time period) then,

ESp = E(X|X > η). (1.2)
In a recent paper we have discussed the importance of the choice of the distributions in measuring the risks (Guégan and Hassani (2016)). In this paper, in a first step, we go beyond the findings of our previous paper, and discuss the concept of coherence 9 and in particular the impact of the notion of sub-additivity, which interested many researchers in the past 20 years [START_REF] Artzner | Coherent measures of risk[END_REF], [START_REF] Dhaene | Can a coherent risk measure be too subadditive?[END_REF] and [START_REF] Jorion | Value at Risk: The New Benchmark for Managing Financial Risk[END_REF]), complying with regulatory requirements (BCBS (2011a), [START_REF] Bcbs | Interpretative issues with respect to the revisions to the market risk framework[END_REF], [START_REF] Bcbs | Fundamental review of the trading book: A revised market risk framework[END_REF], EBA (2014a)). Then in a second step, we discuss the spectral approach, the spectrum representation of a risk measure (i.e. the value of the risk measure for each and every percentile of a given distribution), the distribution of risk measures themselves, and the combination of risk measures to create ranges allowing to capture the "true" risk instead of an approximation of this one by a point.

Thus, the purpose of this paper is to address the methodological aspects of the regulatory framework related to risk modelling and its evolution since 1995, focusing on supervisors' strong incentive to use: (i) specific distributions to characterise the risks, (ii) specific risk measures, (iii) specific associated confidence level, and to apply these strategies independently from each other.

We argue that approaches proposed by the regulator engender a bias (positive or negative) in the assessment of the risks, and consequently a distortion in both the corresponding capital requirements and the management decision taken since the problem of the measurement is not dealt with in its entirety, and as such we question the motivation of the regulator.

Some of the following points are addressed in this paper: (i) Is the choice of a particular risk measure ensuring conservativeness? (ii) When moving from a V aR p to sub-additive risk measures such as the ES p , for which distributions is the sub-additivity 10 property fulfilled given that we consider several risk factors? (iii) Given that each risk type is modelled based on different distributions and using different p-s, how can the sub-additivity criterion be fulfilled? Is that really important in practice? (iv) Should we combine risk measures? (v) Should we focus on ranges of risk values rather than unique value? These different points are linked to the choice of a particular distribution, to the choice of the confidence level p and to the risk measure itself.

The regulatory documents state with respect to market risk -since 1995 (BCBS for instance)that "the VaR risk measure is inadequate for measuring the risks because it does not take into account the extreme events" and also "one of the problems of recognising banks"' value-at-risk measures as an appropriate capital charge is that the assessments are based on historical data and that, even under a 99% confidence interval, extreme market conditions are excluded"11 . To confirm this fact, in the Consultative Document concerning the Fundamental review of the trading book (BCBS ( 2013)), the Basel Committee proposes "to move from Value-at-Risk (VaR) to

Expected Shortfall (ES) as a number of weaknesses have been identified using VaR for determining regulatory capital requirements, including its inability to capture tail risk". The Committee has agreed "to use a 97.5th ES for the internal models-based approach and to use it to calibrate capital requirements under the revised market risk standardised approach". We may argue that this modification has been decided with the Gaussian distribution in mind as the values of the 97.5th ES and the 99th VaR are very close to each other.

In these documents the regulator states that the choice of the VaR as a risk measure does not take into account extreme values. This statement is not correct as the choice of the VaR is not the issue; it is the choice of the underlying distribution with which the associated quantile is evaluated that determines if the extreme events are captured or not. This point actually implies a second question about what an extreme event is and answering it would suppose a complete information set. Then in 2013, it seems that the regulator thought that the use of the ES instead of the VaR would be more effective to capture the most relevant information to measure the risks. This is not necessarily true as once again, it depends on the choice of the distributions used for the computation of this ES. Nevertheless, we know that this last measure is more interesting than the VaR when considering the same distribution because it provides better information concerning the amplitude of the risk, but if the fitted distribution is inappro-priate 12 the problem of capturing extreme events remains the same. Besides, the choice of the level of confidence, for instance 97.5 is also arbitrary (this point will be illustrated in the next section). Indeed, why did the regulator move from 99% (in 1995) to 97.5 % (in 2013)? -Why did they not suggest 95% or another value p?

Another point is considered by regulators for modelling operational risk (EBA (2014b) 13 ). Indeed, they consider that a "risk measure means a single statistic extracted from the aggregated loss distribution at the desired confidence level, such as Value-at-Risk (VaR), or shortfall measures (e.g. Expected Shortfall, Median Shortfall)". This definition is particularly, reductive, limiting and dangerous. How can the risk measures computed for different factors with different levels be aggregated? If we use the ES measure, it loses its sub-additivity property in that latter case. Thus, other approaches could be more robust and realistic, for instance the use of spectral measure, or a spectrum of the previous risk measures.

It appears that some documents are too prescriptive, preventing banks from going beyond the proposals and focusing more on the capital calculations than on the risk measurement and management itself. Regarding the calculation of the capital requirement from the knowledge of the risk factors, the main points concern the choice of the distribution, the choice of the risk measure and the choice of p: these choices are not studied in a uniform manner and the approach proposed by regulators does not constitute a robust approach for measuring the risks of financial institutions.

In Section Two, we investigate the notion of sub-additivity for a risk measure showing that this property do not only depends on the choice of the risk measure but also depends on the choice of the distribution. We illustrate the point that the restriction imposed by regulators prevents a reliable approach to measure the risk. We illustrate our statements with examples. In Section Three we show that it is the choice of the distributions which is definitively the key point in risk modelling. Then in Section Four, we propose an innovative way to look at risk measures,

12 Goodness-of-fit test are only relevant with respect to the information considered. Besides, though a distribution might be appropriate given a data set, if the underlying information evolves (in other words if the sample changes), the distribution might not be valid anymore according to that test. 13 The discussed philosophy is also implied in the final version of the document.

working in several directions; discussing the spectral measure, introducing a new one, the spatial risk measure which relies on a confidence region, or using a measure based on the distortion of the distribution in order to capture the multi-modal aspect of some risk factors. Section Five concludes.

2 Sub-additivity property: a real added-value for risk management?

The concept of sub-additivity which has been largely studied in the past 20 years appears interesting if we consider that the measure of the risk of a portfolio14 is obtained when we calculate the risks of each factor of this portfolio. This is a very restrictive approach for measuring these exposures. An appropriate solution would be to use a multivariate quantile approach based on copula or vines (Guégan and Maugis (2010a), Guégan and Hassani (2013), etc). Nevertheless, if

we maintain this method of assessing the risks, the idea of the sub additive risk measure is that the risk measure of the sum has to be smaller than the sum of the risk measures obtained for each factor taken individually, and following the works of [START_REF] Artzner | Coherent measures of risk[END_REF], it seems that this property is only verified by the ES risk measure in all cases. In fact, this property is also verified by the VaR measure, only now it depends on the distribution used [START_REF] Degen | Evt-based estimation of risk capital and convergence of high quantiles[END_REF]). Indeed, the VaR is known to be sub-additive (i) for stable distribution, (ii) for all log-concave distributions, (iii) for the infinite variance stable distributions with finite mean, (iv) and for distribution with Generalised Pareto Distribution type tails when the variance is finite (i.e. when the shape (usually denoted ξ) is inferior to 0.5). The non sub-additivity of the VaR can occur (i) when P&L distributions are greatly skewed; (ii) when the dependency between factors is highly asymmetric, and (iii) when underlying risk factors are independent but highly heavy-tailed. This data set is characterised by a right skewed (positive skewness) and leptokurtic distribution.

Note that Operational Risk losses are defined on ] -∞, 0[, as a result, in the following com-putations and discussions, the sign will not be considered and the data treated as positive values.

In order to follow the regulatory guidelines, we choose to fit on this data set some of the distributions prescribed and also others which seem more appropriate considering the properties of the data set. We retained eight distributions. They are estimated (i) on the whole sample: the empirical distribution, the lognormal distribution (asymmetric and medium tailed), the Weibull distribution (asymmetric and thin tailed), a Generalised Hyperbolic (GH) distribution (symmetric or asymmetric, fat tailed on an infinite support), an α-Stable distribution (symmetric, fat tailed on an infinite support), a Generalised Extreme value (GEV) distribution (asymmetric and fat tail), (ii) on an adequate subset: the Generalised Pareto (GPD) distribution (asymmetric, fat tailed) calibrated on a set built over a threshold, a Generalised Extreme Value (GEVbm) distribution (asymmetric and fat tailed ) fitted using maxima coming from the original set. The whole data set contains 98082 data points, the sub-sample used to fit the GPD contains 2943 data points and the sub-sample used to fit the GEV using the block maxima approach contains 3924 data points. The objective of these choices is to evaluate the impact of the selected distributions on the risk representation, i.e. how the initial empirical exposures are captured and transformed by the model. It is interesting to note that using empirical distributions instead of fitted analytical distributions could be of interest as the former one captures multi-modality by construction. Unfortunately, this solution was initially rejected by regulators who consider this non-parametric approach not able to capture tails properly: looking at table 11 this idea might be a false statement. However, recently the American supervisor seems to be re-introducing empirical strategies in practice for CCAR 16 purposes.

Table 1 exhibits parameter estimates for each distribution selected 17 . The parameters are estimated by maximum likelihood, except for the GPD which implied a POT [START_REF] Guégan | An efficient threshold choice for the computation of operational risk capital[END_REF] approach and the GEV fitted on the maxima of the data set (maxima obtained using a block maxima method [START_REF] Fisher | Limiting forms of frequency distributions of the largest or smallest member of a sample[END_REF], [START_REF] Gnedenko | Sur la distribution limite du terme d'une série aléatoire[END_REF])). The quality of the adjustment is measured using both the Kolmogorov-Smirnov and the Anderson-Darling tests. The results presented in Table 1 show that none of the distributions is adequate. This is usually the case when fitting uni-modal distributions onto a multi-modal data set. Indeed, the multi-modality of distributions is a frequent issue when modelling risks, in particular operational risks, as the units of measures usually combine multiple kinds of incidents; for instance a category combining external frauds will contain the fraud card on the body, commercial paper fraud in the middle, cyber attack and Ponzi scheme in the tail, but we have also observed a similar pattern using market data.18

In the introduction we indicated that the regulator recommends (for market risk in particular) the use of the ES instead of the VaR because the former is sub-additive, property unverified by the VaR. In the following, we question these assertions showing that, even if it is true that the ES is sub-additive, (i) the VaR also has this property for a lot of distributions as we have mentioned before; (ii) the sub-additivity property can be verified for some values of p, and not verified for others; (iii) the sub additivity of the VaR is very often verified for fat-tailed distributions (note that the problem is addressed here in a context different from [START_REF] Daníelsson | Fat tails, var and subadditivity[END_REF]; (iv) the sub-additivity is not verified anymore for the ES when we aggregate them. We illustrate these different facts making some simulations computing VaR p (X + Y ) and VaR p (X) + VaR p (Y ) for X and Y two risk factors. We proceed in the following way (we will proceed similarly for the ES):

• As VaR p (X) is a quantile, p ∈ [0, 1], the entire spectrum of the VaR has been built, considering the inverse of the cumulative distribution function, sometimes referred to as the quantile function. Summing VaR p (X) and VaR p (Y ) for each value of p provides us with VaR p (X) + VaR p (Y ).

• To obtain VaR p (X +Y ), another approach is adopted. In a first step we randomly generate X and Y using specific distributions. Then X and Y are aggregated. The resulting cumulative distribution function is built and its inverse provides the spectrum of VaR p (X + Y ) 19 .

In Table 2 we provide the values obtained for both the VaR and the ES for fully correlated random variables. It is interesting to note that the risk measures obtained on fully correlated random variables and the sum of the risk measures obtained univariately are really similar. This means that as soon as we sum the VaR obtained on two variables we mechanically assume an upper tailed correlation of the random variables. Therefore, as well as being conservative, the sum of univariate VaRs taken at the same level prevents the capture of any diversification benefit. Fully correlated random variables do not embed any diversification benefit by definition.

Consequently, are we really comparing what is comparable i.e. mechanically highly correlated factors with potentially independent or negatively correlated ones? One may argue that the sub-additivity inequality compares results obtained from a multivariate distribution on one side with highly correlated variables on the other, and this may bias our understanding of the phenomenon in practice.

Then we work with the data sets we have previously introduced. We randomly generated values from the distribution fitted before and combined them two by two. By carrying out this process we generated some random correlations and incidentally some diversification. Then, we compared the risk measures obtained from the combination of random variables and the sum of the risk measures computed on the random variables taken independently. Tables 3,4, 5 and 6 are exhibiting the differences between VaR(X) + VaR(Y ) and VaR(X + Y ) where X and Y are random variables successively following each of the distributions presented earlier. These tables are now analysed.

From Table 3, for fixed p, we observe that the VaR is never sub-additive if the lognormal distribution is associated with a GPD; while if the lognormal distribution is associated with any of the others, the VaR is usually sub-additive in the tails but not at the end of the body part.

Note that if the lognormal is associated with an identical lognormal, the differences we have observed are only due to numerical errors related to sampling. We expect the two values to be absolutely identical. However, it is interesting to note that the random generation of numbers can be at the origin of non sub-additive results. An identical analysis can be done on other combinations such as those containing the Weibull distribution as observed in Table 4. From

Table 5 it appears that when the GPD has a positive location parameter, this prevents any combination from being sub-additive, because by construction the 0th percentile of the GPD is equal to the location parameter which should, according to Pickand's theorem [START_REF] Pickands | Statistical inference using extreme order statistics[END_REF]), be sufficiently high. At the 95th percentile, the VaR is always sub-additive whenever a lognormal distribution is involved, except if it is combined with a GPD. For the other distributions, it is not always true. For example, the VaR obtained after combining a Weibull and a GEV fitted on the whole sample is not sub-additive. Table 6 shows that the use of an Alpha-Stable combined with any other distribution, except for the GPD, provides sub-additive risk measures at the 99% level. Other examples are provided in Table 6 with the Weibull distribution.

Building the ES always respects the sub-additivity property (see Tables 7,8, 9 and 10), contrary to the VaR for which this property is not always verified and depends on the underlying distribution as discussed previously: the results for the ES can be compared to those obtained with the VaR looking at Tables 3 and7, Tables 4 and8, then Tables 5 and9 and finally Tables 6 and10. It is interesting to note that if we combine two ES measures taken at two different levels of confidence p, the ES may not be sub-additive anymore. This is a point that the regulators do not discuss when they ask to aggregate the results provided by each risk measure. We fail to understand the consistency of their mindful process. What is the point of demanding a coherent risk measure when this property fails when aggregating the risks measured as required by the regulation (see first section)? This issue is particularly important for risk managers, since the level of confidence prescribed in the regulation guidelines is different from one risk factor to another and seems totally arbitrary though some rational may support them.

In parallel, Figures 1 to 5 allow a more discriminating analysis of the behaviour of the component

V aR p (X +Y ) versus V aR p (X)+V aR p (Y ).
In Figure 1, we show that the sub-additivity property is only verified for high percentiles when we combine a Weibull and a GH distribution, i.e. for p > 90%. In addition, the gap tends to widen as the percentiles increase. Figure 2 exhibits a non sub-additive VaR from the 95th percentile, when we use the combination of an α-Stable distribution and a GEV fitted with the block maxima method, but the differences are not as great as in Figure 1. Figure 3 shows that combining two identical distributions does not always produce sub-additive risk measures though it should always be the case: this can be due to numerical errors caused by the random generation of data points and the discretisation of the distribution. In Figures 4 and5 we observe that the VaRs obtained from the combination of an Alpha-Stable distribution and a GH distribution or an Alpha-stable distribution and a GEV distribution calibrated on maxima are never sub-additive below 70%. For comparison purposes, Figures 6 and 7 illustrate the fact that the combination of two elliptical distributions (respectively the Gaussian and the Student-t distributions) always leads to sub-additive VaRs (given p).

Role of the distributions in the computation of VaR and ES measures

In the previous Section analysing the role of the sub-additivity concept, we point out the importance of the choice of the distributions. In this section we illustrate this influence on the univariate risk measure computation. Table 11 provides the values obtained for the VaR p and the ES p computed from the eight distributions either fitted on the whole data set or some subsamples. We also illustrate the fact that an a priori on the choice of a distribution provides arbitrary results which can be disconnected from reality, and also the importance to use an appropriate approach for estimation.

From Table 11 we note that the values provided by VaR p can be larger than the values derived from an ES p and conversely. We observe that the results obtained from the GPD and the αstable distributions are of the same order. Second, the differences between the GPD and the GEV fitted on the block maxima are huge, illustrating the fact that, despite being two extreme value distributions, the information captured is quite different. The ES calculations are also linked to the distribution used to model the underlying risks. Looking at Table 11, at 95%, we observe that the ES goes from 1979 for the Weibull to 224 872 for the GPD. Therefore, depending on the distribution used to model the same risk, at the same p level, the ES obtained is completely different. The corollary of that issue is that the ES obtained for a given distribution at a lower percentile will be higher than the ES computed with another distribution at a higher percentile. For example, Table 11 shows that the 90% ES obtained from an Alpha-Stable distribution is much higher than the 99.9% ES computed on a lognormal distribution.

Consequently, one question arises. What should the regulator ask to use: the VaR or the Expected Shortfall? To answer this question, we can consider several points: (i) Conservativeness:

Regarding that point, the choice of the risk measure is only relevant for a given distribution, i.e.

for any given distribution the VaR p will always be inferior to the ES p (assuming only positive values) for a given p. But, if we consider two distributions to characterise a risk it may happen that for a given level p, the VaR p obtained from a distribution is superior to the ES p for another distribution. For example, Table 11 shows that the 99.9% VaR obtained using the GH distribution is superior to the ES obtained for the Weibull or the lognormal distributions at the same level p; (ii) Distribution and confidence level p impacts: Table 11 shows that potentially a 90% level ES obtained from a given distribution is larger than a 99.9% VaR obtained with another distribution, e.g. the ES obtained from a GH distribution at 90% is higher than the VaR obtained from a lognormal distribution at 97.5%. Thus is it always pertinent to use a high value for p? (iii) Parameterisation and estimation: the impact of the calibration of the estimates of the parameters is not negligible [START_REF] Guégan | An efficient threshold choice for the computation of operational risk capital[END_REF]), for instance when we fit a GPD. Indeed, in that latter case, due to the instability of the estimates of the threshold, the practitioners can largely over-represent the risks. The fitting, for instance, of a Weibull distribution using a whole sample in place of a set of maxima (build for example using block maxima method) is totally counterproductive (if we want to model correctly the tails) because the values obtained for the VaR with this distribution are difficult to interpret.

4 Extending traditional approaches: Spectral measure, Spectrum, Spatial risk measure and Distortion

Spectral Risk Measure vs Spectrum

Discussing the importance of the choice of the distributions and also the choice of the level p in the previous sections we point the fact that it is nearly impossible to decide the appropriate level p to perfectly capture the risks associated to any factor, suggesting the idea that these two notions are intrinsically linked. Indeed in Tables 2 and11, we exhibit for different p the values of both VaR and ES for various distributions and it appears that using multiple levels could be informative in terms of real exposures. This is the objective of two other concepts: the spectral risk measures and the spectrum associated to each risk measure. We specify these two notions and show how these can be built.

A spectral risk measure is obtained considering a weighted average of outcomes. By construction, it is a coherent risk measure. Spectral measures found their usefulness in the fact that they can be related to risk aversion through the weights chosen to reflect the possible risk exposures. A formal representation of spectral risk measure ρ (Acerbi, 2002) is a function defined from L → R such that:

ρ(X) = - 1 0 φ(p)F -1 (x)(p)dp (4.1)
where φ is positive or null, non-increasing, right-continuous, integrable function defined on [0, 1] such that 1 0 φ(p)dp = 1 and F (x) is the cumulative distribution function for x. Any spectral risk measure satisfies the conditions for coherence making them useful in practice as well as law-invariance and comonotonic additivity 20 [START_REF] Adam | Evt-based estimation of risk capital and convergence of high quantiles[END_REF]).

The ES is a spectral risk measure for which φ(p) = 1, ∀p in (4.1). The limitation of the spectral ES is that all the weights are uniform whatever p by definition, however, this insures its coherence. When financial institutions provide the ES, i.e., a unique value to the regulator to represent its risks, it provides a spectral risk measure. Consequently, up to certain extent, the aggregation of the risks computed for each department within a financial institution can be considered as a spectral risk measure. Unfortunately, as introduced, the regulation requires these risks to be assessed for different p levels, set by the regulator: 99,9% for operational risk, 95% for market risks, etc. In that case the property of coherence of the risk measure cannot be ensured anymore, however the coherence of the risk measure might still be achieved, as presented in Guégan et al. (2016), depending on the distributions selected to represent the risks.

Note that as a quantile, and not a combination of them, the VaR is not a spectral risk measure but may have a spectral representation. The general representation of spectral risk measures is more flexible than the particular case of the ES. Indeed, for a given risk factor, we can associate different weights to different levels depending on if we want to privilege the body or the tail of the distribution. This can be equivalent to fitting a multi-modal distribution to the data characterising the risk factor, and then to recalculate the risk measure (see below the section on the distortion). In Tables 2 and11 the VaR for different p. The spectral approach is supposed to provide a more complete measure of risk as soon as we choose to associate to each quantile a weight. In fine, we get an unique value which is different of the concept of the spectrum that we are going to describe now.

We call spectrum of a risk measure, the set of all the values we get for this risk measure given a set of levels p i . We exhibit in Figure 8 an illustration of a spectrum of the ES for a given risk factor for different distributions and for p i ∈ [0.70, 0.999]. This graph provides several information: the role of the distribution, the influence of the choice of p, the large difference depends on if we are close to the body of the distributions or in the tails (Note that the differences can be extremely significant depending on the distribution chosen). Indeed, while the use of several levels p i , i = 1, • • • , k allows a spectrum representation of the risk measures (VaR or ES) and could be more interesting for risk management than the approach proposed by regulators using the idea of spectral measure which combines distribution and confidence level. Indeed, the 70% ES of some combinations may lead to a much higher value than the 99.9th (Table 8, WE-GPD vs WE-GH); on the contrary we provide in Tables 12,13 and 14, the differences V aR(X) + V aR(Y ) -V aR(X + Y ) for several distributions. In Table 12 we use Weibull and a lognormal distributions, in Table 13 Weibull and α-stable distributions and in Table 14 two GEV distributions. We do this exercise for 90% < p < 99.9% in Tables 12 and13, and for 1% < p < 99.9% with a step of 1% in Table 14. In that last table when the values are positive, the VaR is sub-additive, when the values are negative it is not sub-additive.

The turning points are highlighted in bold. This provides an interesting picture of the property of these distributions. The spectrum of the VaR given in these tables provides good information in terms of risk management; indeed, it shows that relying directly on risk measures to evaluate a capital requirement may not be representative of the risk profile of the target entity.

In fact, it can even be misleading, as from one p i to another because the risk measures may have dramatically different orders. The spectrum of the VaR approach shows a risk measure obtained at a particular level cannot be representative of the whole risk profile, and assuming the contrary could lead to dreadful failures and inappropriate management decision. Thus we can incite risk managers to construct the spectrum to have a better understanding of these risks.

A Spatial Risk Measure: the SVaR

In this section, following an idea developed in a paper of Guégan et al. (2016), we present a way of measuring risks spatially, capturing the intrinsic uncertainty of the VaR both vertically and horizontally. By vertically, we consider the confidence interval (CI) of the VaR at a given percentile and horizontally by assuming that the level p is not properly evaluated (usually, supposed to represent the 95th, the 97.5th or the 99.9th -see section 1 for the regulatory requirements). The construction of the CI is specifically described in the following paragraphs. Therefore, instead of using a value for p, a range, in which p lies, is selected. For example for the 95th, we may want to select a range from the 92nd to the 98th. The combination of the vertical CI and the horizontal one leads to a quadrilateral figure giving the area in which the true VaR is supposed to lie. This area -delimited by V aR p i and the upper bound of CI q i -corresponds to a Spatial risk measure we propose to use as an alert indicator. Indeed, having the VaR for different p provides us with the spectrum of the VaR. Then using a set of confidence intervals around this spectrum provides us with an acceptable range of variation for the V aR p -s. Only considering the upper bounds on the confidence intervals gives this spatial VaR. The upper envelop (or the lower depending on the sign) allows us to use multiple values at a particular percentile level. To build the confidence interval we use a recent result introduced by Guégan et al. (2016).

In their paper, Guégan et al. (2016) studied the asymptotic Gaussian property of the distribution of the estimator V aR p of V aR p , which allows to build a confidence interval CI q around V aR p .

Since the convergence speed of this estimate depends on the underlying unknown distribution f , the sample size n of the data set and the confidence level p of V aR p , the paper provides a comprehensive analysis of application for finite samples with a panel of distributions on risk data. The confidence interval they propose is the following:

V aR p -Z 1-q 2 , V aR p -Z q 2 (4.2)
where Z q = Ψ -1 q ( √ n ω ). Ψ and ω are provided in expressions below.

Ψ(

√ nω ) = 1 -Φ( √ nω ) ω = ω + 1 nω log 1 ψ(-ω) (4.3) ψ(-ω) = ω(t -1) t -p ( r 0 1 -p ) 1 2 ω = -2h(t)sign(t -p) (4.4) h(t) = p log p t + (1 -p) log 1 -p t (4.5)
where sign(x) = 1 if x ≥ 0 and sign(x) = -1 otherwise. Besides, φ represent the Gaussian distribution.

In their article Guégan et al. (2016) call this approach a saddle point approximation for the confidence interval. It is based on a result of [START_REF] Zhu | Saddlepoint approximation for sample quantiles with some applications[END_REF] which provides an asymptotic convergence of the estimate of the V aR p towards its true value with a relatively fast rate of convergence, which is maintained in case of finite samples. 21 .

In order to show the power of this new spatial risk measure, we illustrate its use on an example.

Here, we are interested in the lower envelop as considering a distribution defined on an infinite support (-∞, +∞), the losses are represented by negative values. Figure 9 exhibits the construction of the Spatial VaR using S&P 500 data from 01/01/2008 to 31/12/2008. The abscissa provides the "p"-s at which the VaR estimate (in ordinate) has been calculated. On the left, we present an truncated axis presenting the "q"-s. Here the Spatial VaR tells us in which range the 97 th percentile of the log returns of the S&P500 is located. For an intuitive understanding of our approach, note that the 98 th percentile of the distribution considered is included in the CI obtained for the estimate of the VaR at 96%.

Distorted distributions

To assess an exposure, the first point to consider is to fit an appropriate distribution on the data.

Indeed, looking at Figure 10, we observe that the "natural" distribution fitted on the underlying market data set 22 is multi-modal. In this section, we use distributions defined on an infinite support (-∞, +∞), and the losses are represented by negative values. In practice, we often observe this kind of pattern for financial or economic data sets. We observe from this graph that we can separate large losses from the other ones and then obtain a better understanding of the probability of these outcomes. In this section we propose an alternative to our proposal in paragraph 3 for the fit of the distributions characterising the risk factors, and by doing so we introduce an alternative risk measure approach.

Given a risk factor X characterised by a distribution function F X , fitting a multi-modal distributions can be done in several manners. The most natural way of representing multi-modal data is to implement a kernel density strategy, however, if we are interested in capturing other aspects of exposure not captured by data, another possibility is to transform a uni-modal distribution fitted to the data into another one, as illustrated in Figure 10.

Multi-modal distribution construction has been investigated by many statisticians considering mainly multimodal distributions inside the exponential family [START_REF] Fisher | On the mathematical foundations of theoretical statistics[END_REF]) and more recently by economists within the dual theory of choice [START_REF] Yaari | The dual theory of choice under risk[END_REF]). Both approaches extend the notion of multimodality appearing as a mixture of normal or possibly other unimodal densities and suggest transforming the original distribution into a new one using a distortion function g(.) with appropriate properties. A function g : [0, 1] → [0; 1] is a distortion function if (i) g(0) = 0 and g(1) = 1, (ii) g is a continuous increasing function. Different distortion functions have been proposed in the literature. A wide range of parametric families of distortion functions are mentioned in [START_REF] Wang | A class of distortion operators for pricing financial and insurance risks[END_REF] or [START_REF] Hardy | Distortion risk measures: coherence and stochastic dominance[END_REF]. [START_REF] Cobb | Estimation and momet recursion relations for multimodal distributions of the exponential family[END_REF] also proposes an interesting approach which is more general and whose applicability is based on robust statistical techniques.

We introduce some functions g resulting in a bimodal distribution23 . To create multi-modality we need to use a function g which possesses saddle points. The saddle point generates a second mode in the new distribution which allows us to take into account different patterns located in the tails. The distortion function g fulfilling this objective can be an inverse S-shaped polynomial function of degree 3 -for instance given by the following equation and characterised by two parameters δ and β:

g δ (x) = a x 3 6 - δ 2 x 2 + δ 2 2 + β x . (4.6)
We note that g δ (0) = 0, and to get g δ (1) = 1 this implies that the coefficient of normalisation is

equal to a = 1 6 -δ 2 + δ 2 2 + β -1
. The function g δ will increase if g δ > 0 requiring 0 < δ < 1.

The parameter δ ∈ [0, 1] allows us to locate the saddle point. The curve exhibits a concave part and a convex part. The parameter β ∈ R controls the information under each mode in the distorted distribution. Illustration of the role of δ on the location of the saddle points and of β for the shape of this bimodal distribution can be found in Guégan and Hassani (2015a) (other functions can be found also in this article). We provide a graph (Figures 11) below which shows the creation of a bi-modal distribution using the transformation g(F X ).

As soon as we have obtained a bi-modal (or multi-modal) distribution, it is interesting to compute a risk using this distribution. In Guégan and Hassani (2015a) we suggest to consider the following new risk measure ρ g (X) (associated with a risk factor X admitting a cumulative distribution S X (x) = P(X > x), transformed by a distortion function g), defined by

ρ g (X) = 0 -∞ [g(S X (x)) -1]dx + +∞ 0 g(S X (x))dx, (4.7)
as soon as the two integrals are finite. Assuming g is differentiable on [0, 1] and F X (x) is continuous, then a new risk measure -the distortion risk measure -can be re-written as:

ρ g (X) = E[Xg (S X (x))] = 1 0 F -1 X (1 -p)dg(p) = E g [F -1 X ]. (4.8)
Distortion functions arose from empirical observations that people do not evaluate risk as a linear function of the actual probabilities for different outcomes but rather as a non-linear distortion function. It is used to transform the probabilities of the loss distribution to another probability distribution by re-weighting the original distribution. This transformation increases the weight given to desirable events and deflates others. Different distortions g have been proposed in the literature. A wide range of parametric families of distortion functions have been mentioned in [START_REF] Fisher | On the mathematical foundations of theoretical statistics[END_REF][START_REF] Wang | A class of distortion operators for pricing financial and insurance risks[END_REF]. We can also use specific distortion functions if we want to increase the influence of asymmetry in the transformation of the original distribution and work for instance as follows [START_REF] Sereda | Distortion risk measures in portfolio optimisation[END_REF]):

ρ g i (X) = 0 -∞ [g 1 (S X (x)) -1]dx + +∞ 0 g 2 (S X (x))dx, (4.9) with g i (u) = u + k i (u -u 2 ) for k ∈]0, 1] et ∀i ∈ {1, 2}
. With this approach one models loss and gains differently, relative to the values of the parameters k i , i = 1, 2. Thus upside and downside risks are modelled in different ways. Nevertheless the calibration of the parameters With this approach we obtain another way to use the concept of spectrum risk measure discussed previously. Here the weights are g 1 and g 2 : they permit to separate the risks associated for instance to the center from the risks associated to the tail. Thus the corresponding coherent risk measure will be defined as:

k i , i = 1,
ρ(X) = E g [F -1 X (x)|F -1 X (x) > F -1 X (δ)]. (4.10)
While Figure 10 illustrates the distortion, we provide the quantiles computed on a uni-modal distribution and the VaR computed on the multi-modal distribution. For instance, the empirical 95 th percentile equals 0.0316, the Gaussian 95 th percentile equals 0.0272 and the distorted 95 th percentile equals 0.0433. However, here we are interested in the negative part of the distribution for risk management purposes, and the results are for closer, indeed, the empirical 5 th percentile equals -0.0276, the Gaussian 5 th percentile equals -0.0286 and the distorted 5 th percentile equals -0.027.

The distortion of the distribution induces a mass transfer from one part of the distribution to another, here as we capture the hump in the right tail, the body and the left tail are mechanically impacted. As a result, though the 95 th percentile of the distorted distribution is much higher than the others, the risk measure applied to the left tail (the negative part of the distribution) is the lowest. Consequently, we observe that with a better capture of the exposure the risk measure decreases and therefore the capital requirement impact might be non negligible for financial institutions.

Conclusion and Recommendations

In the introduction, which analysed several guidelines issued by the EBA and the Basel Committee, we pointed out the fact that the regulators impose specific distributions, risk measures and confidence levels to analyse the risk factors in order to evaluate capital requirements of financial institutions. It appears that their approach is non holistic and their analysis of the risks relies on a disconnection between the components outlined in the previous sentence, i.e. the tools necessary to assess the risks.

In this paper we show that risk measurement in financial institutions depends intrinsically on how the tools are chosen, i.e. the distribution, the combinations of these distributions, the type of risk measure and the level of confidence. Therefore, the existence of a risk measure as discussed in the regulation is questionable, as for example modifying the level of confidence by a few percent would result in completely different interpretations. The regulators fail to propose an appropriate approach to measure these risks in financial institutions as soon as they do not take into account the problem of risk modelling in its globality.

Regulators are far too prescriptive and their choices questionable:

• Imposing distributions does not really make sense whatever the risks to be modelled as these may change quite quickly. We may wonder where these a priori are coming from.

• The regulation reflects some misunderstanding regarding distribution properties (probabilistic approach) and of the particular properties surrounding their fittings (statistical approach).

• The levels of confidence p seem rather arbitrary. They neither take into account the flexibility of risk measures nor the impact of the underlying distribution, misleading risk managers.

While these fundamental problems are not addressed, others are completely ignored such as the concept of spectral analysis, spectrum or distortion risk measures [START_REF] Sereda | Distortion risk measures in portfolio optimisation[END_REF], Guégan and Hassani (2015a)). Despite the cosmetic changes included in Basel II and III, the propositions do not enable a better risk management, and the response of banks to regulatory points 20 Documents de travail du Centre d'Economie de la Sorbonne -2016.66

is not appropriate as they do not correspond to the reality. It is therefore not surprising that capital calculations and stress testing are still unclear, and that these are not able to capture asymmetric shocks corresponding to extreme incidents.

Some other questions should also be addressed:

• Is it more efficient in terms of risk management to measure the risk and then build a capital buffer or to adjust the risk taken, considering the capital we have? In other words, maybe banks should start optimising their income generation with respect to the capital they already have.

• The previous points are all based on uni-modal parametric distributions to characterise each risk factor. What is the impact of using multi-modal distributions in terms of risk measurement and management? We believe that an empirical evaluation of the risks provides bank with a reliable benchmark and a starting point in terms of what would be an acceptable capital charge or risk assessment.

• One of the biggest issues lies in the fact that we do not know how to combine or aggregate V aR(X) p 1 , V aR(Y ) p 2 and V aR(Z) p 3 evaluated on three different kinds of risks at three different confidence levels p 1 , p 2 , p 3 . This mechanically prevents banks from building a holistic approach from a capital point of view. How should we proceed to solve the problem, should we use p = max(p 1 , p 2 , p 3 ), or the minimum or the average?

• Although in this paper we have focused on each factor taken independently, the question of dependence is quite important too. Maybe not as important as the impact of the distribution selected for the risk factor [START_REF] Guégan | Multivariate vars for operational risk capital computation : a vine structure approach[END_REF]) but not addressing this issue properly could lead to a mis-interpretation of the results. The choice of the copula has a direct impact on the dependence structure we would like to apply and the capture of shocks. For instance, a Gaussian or Student t-copula is symmetric, despite the fact that a t-copula with a low number of degrees of freedom could capture tail dependencies; these would not capture asymmetric shocks. Archimedean or extreme value copulas associated with a vine strategy would be more appropriate (Guégan and Maugis (2010b)).

• In a situation such as one depicted by the stress-testing process with a forward looking perspective, if the risks are not correctly measured then the foundations will be very fragile and the outcome of the exercise not reliable. Indeed, stressing a situation requires an appropriate initial assessment of the real exposure, otherwise the stress would merely model what should have been captured originally and therefore be useless (Bensoussan et al. (2015), Guégan and Hassani (2015b), [START_REF] Hassani | Risk appetite in practice: Vulgaris mathematica[END_REF]).

We came up to the conclusion that the debate related to the selection of a risk measure over another is not really relevant, and considering issues raised in the previous sections our main recommendation would be to leave as much flexibility as possible to the modellers to build the most appropriate models for risk management purposes initially and then extend with conservative buffers for capital purposes. The objective would be to suggest that good risk management would mechanically limit the exposures and the losses and therefore ultimately reduce the regulatory capital burden. Models should only be a reflection of the underlying risk framework and not a tool to justify a reduced capital charge. We would like to see the supervisory face of the authorities more and their regulatory face less; in other words we would like them to stop focusing so much on a bank's risk measurement comparability and more on financial institutions risk understanding. It would probably be wise if both regulators and risk managers worked together (e.g., academic formation open to both corpus, regular workshops, etc., [START_REF] Guégan | Former les analystes et les opérateurs financiers[END_REF])) rather than as opponents, in order to reach their objective of stability of the financial system first and profitability second.

Finally, we believe that the implementation of combinations of risk measures such as the spectral risk measures, spatial VaR, risk measure spectrum or the distortion of the risk measure may help addressing the limitations, the inefficiencies or blind spots of the more traditional risk measures for instance the VaR or the ES. Indeed, the combinations help capturing a more diffuse risk, and not a specific value in a spot, providing a better representation of the exposure, incorporating the uncertainty related to the selection of the distribution used to assess the risks and the fittings. Furthermore, they allow capturing the multi-modality of some distributions.

Besides, the combination also smooths the risk measurement reducing the volatility of these over time. Consequently, we would suggest financial institutions to start implementing the methodology to measure their risk more accurately, and regulators to start considering them for regulatory capital calculations. As presented in the previous section, it is really important to understand that capturing the exposure more accurately does not necessarily lead to larger regulatory capital, but mechanically to better risk management.

Parameters

Table 1 -Parameter estimates

This table provides the estimated parameters for the seven parametric distributions fitted on the operational risk data set. If for the α stable distribution α < 1, for the GEV, GEVbm and GPD ξ > 1 we are in the presence of an infinite mean model. The p-values of both Kolmogorov-Smirnov and Anderson-Darling tests are also provided for the fit of each distribution. The sum of VaR(X) and VaR(Y) (line 1) versus VaR(X + Y) (line 2) for couples of distributions: LN = lognormal, WE = Weibull, GPD = Generalised Pareto, GH = Generalised Hyperbolic, AS = Alpha-Stable, GEV = Generalised Extreme Value, GEVbm = Generalised Extreme

LN-LN

Value calibrated on maxima. The percentiles represented are the 70th, 80th, 90th, 95th, 99th, 99.5th and 99.9th. We use the distributions fitted on the data set representing "Execution, Delivery, and process Management". The sum of VaR(X) and VaR(Y) (line 1) versus VaR(X + Y) (line 2) for couples of distributions: LN = lognormal, WE = Weibull, GPD = Generalised Pareto, GH = Generalised Hyperbolic, = Alpha-Stable, GEV = Generalised Extreme Value, GEVbm Generalised Extreme

WE-WE

Value calibrated on maxima. The percentiles are the 70th, 80th, 90th, 95th, 99th, 99.5th and 99.9th. We use the distributions fitted on the data set representing "Execution, Delivery, and process Management". This plot represents the sum of VaR(X) and VaR(Y) (red) versus VaR(X + Y) (black). The random variable X has been generated using an Alpha-stable distribution and Y has been obtained from a GEV distribution calibrated on maxima. The percentiles represented are the 70th, 80th, 90th, 95th, 99th, 99.5th and 99.9th. For high percentiles, the VaR is not sub-additive. This plot represents the sum of VaR(X) and VaR(Y) (red) versus VaR(X + Y) (black). The random variable X has been generated using a Alpha-stable distribution and Y has been obtained from a GEV distribution calibrated on maxima. The percentiles represented are sequentially going from the 10th to the 70th with a step of 1% between two points. The VaR represented are never sub-additive.

GPD-GPD

38

Documents de travail du Centre d'Economie de la Sorbonne -2016.66 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0 20 40 This plot represents the sum of VaR(X) and VaR(Y) (doted line) versus VaR(X + Y) (solid line). The random variable X has been generated using a Gaussian distribution (0, 1) and Y has been obtained from a Gaussian distribution (2, 1). The VaRs represented are always sub-additive. q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0 20 40 60 80 100 40 Documents de travail du Centre d'Economie de la Sorbonne -2016.66 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0.00 0.01 0.02 0.03 0.04 0.05

  , we have shown the great variability of the values obtained for 20 Law-Invariance: for any combination of risks X and Y with respective cumulative distribution functions F (x) and F (y), if F (x) = F (y) then ρ(X) = ρ(Y ); comonotonic additivity: for every comonotonic random variables X and Y , ρ(X + Y ) = ρ(X) + ρ(Y ).

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  2 remains an open problem. Estimation procedures are provided in a companion paper.

.

Figure 1 -

 1 Figure 1 -VaR(X) + VaR(Y) vs VaR(X + Y) Plot -Weibull & GHThis plot represents the sum of VaR(X) and VaR(Y) (red) versus VaR(X + Y) (black). The random variable X has been generated using a Weibull distribution and Y has been obtained from a Generalised Hyberbolic distribution. The percentiles represented are the 70th, 80th, 90th, 95th, 99th, 99.5th and 99.9th. For high percentiles, the VaR seems to be sub-additive.

Figure 2 -

 2 Figure 2 -VaR(X) + VaR(Y) vs VaR(X + Y) Plot -Alpha & GEV max

Figure 3 -

 3 Figure 3 -VaR(X) + VaR(Y) vs VaR(X + Y) Plot -GEV & GEVThis plot represents the sum of VaR(X) and VaR(Y) (red) versus VaR(X + Y) (black). The random variables X and Y have been obtained from two identical GEV distributions. The percentiles represented are the 70th, 80th, 90th, 95th, 99th, 99.5th and 99.9th. For high percentiles, the VaR is sub-additive.

Figure 4 -

 4 Figure 4 -VaR(X) + VaR(Y) vs VaR(X + Y) Plot -Alpha & GH enlargedThis plot represents the sum of VaR(X) and VaR(Y) (red) versus VaR(X + Y) (black). The random variable X has been generated using an Alpha-stable distribution and Y has been obtained from a Generalised Hyperbolic distribution. The percentiles represented are sequentially going from the 10th to the 70th with a step of 1% between two points. The VaR represented are never sub-additive.

Figure

  Figure 5 -VaR(X) + VaR(Y) vs VaR(X + Y) Plot -Alpha & GEV max enlarged

Figure 6 -

 6 Figure 6 -VaR(X) + VaR(Y) vs VaR(X + Y) Plot -Gaussian case

  Figure 7 -VaR(X) + VaR(Y) vs VaR(X + Y) Plot -Student case This plot represents the sum of VaR(X) and VaR(Y) (doted line) versus VaR(X + Y) (solid line). The random variable X has been generated using a Student-t distribution (3 df) and Y has been obtained from a Student-t distribution (4 df). The VaRs represented are always subadditive.

Figure 8 -

 8 Figure 8 -This figure illustrates how the ES evolves depending on the distribution used.

  Figure 9 -Spatial VaRWe provide the area corresponding to the SVaR, the lower bound corresponds to V aRp i and the upper bound to the upper bound of CIq i ,p i , i = 1, ..., k.. The area in blue represent the Spatial VaR capturing simultaneously the uncertainty related to the choice of the distribution and the uncertainty related to the estimation of the parameters. The VaR has been evaluated on the distribution of the return of S&P500 from 01/01/2008 to 31/01/2008.

Table 3 -

 3

VaR(X) + VaR(Y) vs VaR(X + Y) part 1

Table 4

 4

	-VaR(X) + VaR(Y) vs VaR(X + Y) part 2

Table 5 -VaR(X) + VaR(Y) vs VaR(X + Y) part 3

 5

	The sum of VaR(X) and VaR(Y) (line 1) versus VaR(X + Y) (line 2) for couples of distributions: LN = lognormal, WE = Weibull, GPD =
	Generalised Pareto, GH = Generalised Hyperbolic, AS = Alpha-Stable, GEV = Generalised Extreme Value, GEVbm = Generalised Extreme
	Value calibrated on maxima. The percentiles represented are the 70th, 80th, 90th, 95th, 99th, 99.5th and 99.9th. We use the distributions
	fitted on the data set representing "Execution, Delivery, and process Management".
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Table 6 -VaR(X) + VaR(Y) vs VaR(X + Y) part 4

 6 The sum of VaR(X) and VaR(Y) (line 1) versus VaR(X + Y) (line 2) for couples of distributions: LN = lognormal, WE = Weibull, GPD = Generalised Pareto, GH = Generalised Hyperbolic, AS = Alpha-Stable, GEV = Generalised Extreme Value, GEVbm = Generalised Extreme Value calibrated on maxima. The percentiles represented are the 70th, 80th, 90th, 95th, 99th, 99.5th and 99.9th. We use the distributions fitted on the data set representing "Execution, Delivery, and process Management".

	LN-LN 1	1, 895	2, 587	4, 226	6, 616	16, 572	23, 652	50, 725
	LN-LN 2	1, 895	2, 587	4, 226	6, 616	16, 572	23, 652	50, 725
	LN-WE 1	1, 727	2, 302	3, 574	5, 293	11, 739	16, 021	31, 500
	LN-WE 2	1, 541	1, 970	2, 882	4, 092	8, 841	12, 269	25, 675
	LN-GPD 1	87, 496	101, 478	140, 329	211, 059	682, 080	1, 191, 608	4, 513, 150
	LN-GPD 2	87, 065	100, 726	138, 767	208, 277	674, 213	1, 180, 157	4, 488, 157
	LN-GH 1	2, 146	2, 984	5, 081	8, 347	23, 114	33, 777	71, 406
	LN-GH 2	1, 996	2, 698	4, 383	6, 898	17, 681	25, 214	50, 397
	LN-AS 1	16, 694	24, 801	48, 732	95, 726	459, 981	905, 044	4, 350, 967
	LN-AS 2	16, 545	24, 525	48, 067	94, 322	454, 147	895, 398	4, 326, 497
	LN-GEV 1	8e18	12e18	24e18	48e18	244e18	489e18	2, 447e18
	LN-GEV 2	8e18	12e18	24e18	48e18	244e18	489e18	2, 447e18
	LN-GEVbm 1	5, 608	8, 174	15, 460	29, 105	126, 073	237, 314	1, 032, 332
	LN-GEVbm 2	5, 457	7, 888	14, 762	27, 640	120, 229	227, 765	1, 008, 148

Table 7

 7

	-ES(X) + ES(Y) vs ES(X + Y) part 1
	The sum of ES(X) and ES(Y) (line 1) versus ES(X + Y) (line 2) for couples of distributions: LN = lognormal, WE = Weibull, GPD =
	Generalised Pareto, GH = Generalised Hyperbolic, AS = Alpha-Stable, GEV = Generalised Extreme Value, GEVbm = Generalised Extreme
	Value calibrated on maxima. The percentiles represented are the 70th, 80th, 90th, 95th, 99th, 99.5th and 99.9th. We use the distributions
	fitted on the data set representing "Execution, Delivery, and process Management".

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Table 8 -ES(X) + ES(Y) vs ES(X + Y) part 2

 8

	The sum of ES(X) and ES(Y) (line 1) versus ES(X + Y) (line 2) for couples of distributions: LN = lognormal, WE = Weibull, GPD =
	Generalised Pareto, GH = Generalised Hyperbolic, AS = Alpha-Stable, GEV = Generalised Extreme Value, GEVbm = Generalised Extreme
	Value calibrated on maxima. The percentiles represented are the 70th, 80th, 90th, 95th, 99th, 99.5th and 99.9th. We use the distributions
	fitted on the data set representing "Execution, Delivery, and process Management".

Table 9 -ES(X) + ES(Y) vs ES(X + Y) part 3

 9

	The sum of ES(X) and ES(Y) (line 1) versus ES(X + Y) (line 2) for couples of distributions: LN = lognormal, WE = Weibull, GPD =
	Generalised Pareto, GH = Generalised Hyperbolic, AS = Alpha-Stable, GEV = Generalised Extreme Value, GEVbm = Generalised Extreme
	Value calibrated on maxima. The percentiles represented are the 70th, 80th, 90th, 95th, 99th, 99.5th and 99.9th. We use the distributions
	fitted on the data set representing "Execution, Delivery, and process Management".

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Table 10 -ES(X) + ES(Y) vs ES(X + Y) part 4

 10

	The sum of ES(X) and ES(Y) (line 1) versus ES(X + Y) (line 2) for couples of distributions: LN = lognormal, WE = Weibull, GPD =
	Generalised Pareto, GH = Generalised Hyperbolic, AS = Alpha-Stable, GEV = Generalised Extreme Value, GEVbm = Generalised Extreme
	Value calibrated on maxima. The percentiles represented are the 70th, 80th, 90th, 95th, 99th, 99.5th and 99.9th. We use the distributions
	fitted on the data set representing "Execution, Delivery, and process Management".

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Table 11 -Univariate Risk Measures This

 11 table exhibits the VaRs and ESs for the height types of distributions considered -for instance empirical, lognormal, Weibull, GPD, GH, α-stable, GEV and GEV fitted on a series of maximafor five confidence level (for instance, 90%, 95%, 97.5%, 99% and 99.9%) evaluated on the period 2009-2014. Note that the parameters obtained for the α-stable and the GEV fitted on the entire data set lead to infinite mean model and therefore, the ES are hardly applicable.

	80.00%	81.00%	82.00%	83.00%	84.00%	85.00%	86.00%	87.00%
	-81.843	-74.943	-66.539	-57.410	-47.461	-35.212	-20.496	-3.984
	88.00%	89.00%	90.00%	91.00%	92.00%	93.00%	94.00%	95.00%
	16.247	40.129	67.997	102.443	144.756	196.882	266.676	360.135
	96.00%	97.00%	98.00%	99.00%	99.50%	99.90%	99.95%	99.99%
	489.356	677.618	1, 011.196	1, 696.400	2, 581.672	4, 858.396	5, 761.766	10, 964.930

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Table 12 -VaR(X) + VaR(Y) vs VaR(X + Y) valuation -Weibull and lognormal

 12 This table shows the differences between the sum VaR(X) and the VaR(Y) and the VaR(X + Y). The random variable X has been generated using a Weibull and Y has been obtained from a lognormal distribution. When the values are positive, the VaR is sub-additive, when the values are negative the VaR is not. The turning points are highlighted in bold.

	80.00%	81.00%	82.00%	83.00%	84.00%	85.00%	86.00%	87.00%
	-86.104	-82.891	-80.004	-75.764	-69.887	-63.385	-55.082	-45.380
	88.00%	89.00%	90.00%	91.00%	92.00%	93.00%	94.00%	95.00%
	-34.810	-21.030	-2.510	23.340	54.970	99.660	159.200	249.830
	96.00%	97.00%	98.00%	99.00%	99.50%	99.90%	99.95%	99.99%
	393.730	632.630	1, 098.500	2, 170.800	3, 052.900	4, 784.190	17, 905.440	-633, 422.500

Table 13 -VaR(X) + VaR(Y) vs VaR(X + Y) valuation -Weibull & Alpha This

 13 table shows the differences between the sum VaR(X) and the VaR(Y) and the VaR(X + Y). The random variable X has been generated using a Weibull and Y has been obtained from an Alpha-stable distribution. When the values are positive, the VaR is sub-additive, when the values are negative the VaR is not. The turning points are highlighted in bold.

	-0.012	0.022	-0.013	-0.018	-0.015	-0.031	-0.020	-0.026
	-0.038	0.011	0.028	0.023	0.022	0.024	0.044	0.073
	0.074	0.080	0.139	0.144	0.194	0.171	0.167	0.163
	0.142	0.141	0.134	0.150	0.179	0.175	0.105	0.107
	0.016	-0.001	-0.002	-0.003	0.013	-0.021	-0.048	-0.011
	-0.016	0.045	0.074	0.032	0.074	0.166	0.124	0.104
	0.098	0.019	-0.037	-0.079	-0.100	-0.120	-0.144	-0.047
	-0.070	-0.086	-0.136	-0.234	-0.291	-0.352	-0.272	-0.197
	-0.098	0.038	0.121	-0.313	-0.299	-0.483	-0.621	-0.422
	-0.457	0.099	0.272	0.381	0.430	0.656	0.754	0.533
	0.693	1.035	0.715	1.087	0.778	-0.167	-0.479	-0.522
	-0.759	-3.391	-2.265	-4.190	-3.137	-6.484	-1.975	9.502
	6.873	16.636	69.495	50.091	7, 118.689	8, 798.144	-148, 979.500	NA

35

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Sorbonne -2016.66

Table 14 -

 14 This table shows the differences between the sum VaR(X) and the VaR(Y) and the VaR(X + Y). The random variable X and Y have been obtained on 2 identical GEV distributions. When the values are positive, the VaR is sub-additive, when the values are negative the VaR is not. The turning points are highlighted in bold. The percentiles represented are sequentially going from 1% to 99% by 1%, and to capture the tail, the 99.95th, 99.9th, 99.95th and 99.99th percentiles are added.

VaR(X) + VaR(Y) vs VaR(X + Y) valuation -GEV & GEV

There is no mean to know for sure that the most extreme market conditions have been met.4Documents de travail du Centre d'Economie de laSorbonne -2016.66

Documents de travail du Centre d'Economie de laSorbonne -2016.66

The term portfolio is here used stricto sensus and not necessarily as a combination of assets.

In our demonstration, the data set which has been sanitised here is not of particular importance since the same data set has been used for each and every distribution tested.6Documents de travail du Centre d'Economie de laSorbonne -2016.66

It could be more appropriate to consider empirical distributions than fitted analytical distributions as it may help to capture multi-modality. We will come back to this last point in Section 4.

We acknowledge the numerical error that this process may engender, though this one appears negligible here.8Documents de travail du Centre d'Economie de laSorbonne -2016.66

For more details, we suggest the previous cited articles

Daily returns Hang Seng Index from 24/07/2006 to 24/07/2008

Here our approach is mainly descriptive, in another paper we provide robust estimation from original data sets using maximum likelihood and the weighted moment method.

Santander's position. As a result, Santander cannot be held responsible for them. 3 This work was achieved through the Laboratory of Excellence on Financial Regulation (Labex ReFi) supported by PRES heSam under the reference ANR10LABX0095. It benefited from a French government support managed by the National Research Agency (ANR) within the project Investissements d'Avenir Paris Nouveaux Mondes (investments for the future Paris New Worlds) under the reference ANR11IDEX00060 1 Documents de travail du Centre d'Economie de la Sorbonne -2016.Documents de travail du Centre d'Economie de la

Table 2 -Correlated Risk Measures

This table presents the VaRs and the ESs obtained on fully correlated random variables simulated with seven distributions.

Hang Seng Index Log Return Distribution


Bertrand K Hassani
email: bertrand.hassani@malix.univ-paris1.fr.

Bringing the Customer Back to the Foreground: The End of Conduct Risk?

Keywords:

In this chapter we argue that conduct risk arising from the way nancial institutions are conducting business with respect to their customers might be prevented, mitigated and potentially annihilated. Indeed, we believe that data science, proper segmentation, product design and control will lead to a tremendous reduction of conduct risk exposure and as such these topics are addressed here.

1 Introduction Financial institutions mis-conduct or perception of mis-conduct leads to conduct risk. Indeed, the terminology "conduct risk" gathers various processes and behaviours which fall into operational risk Basel category 4 (Clients, Products and Business Practices) [START_REF] Bcbs | International convergence of capital measurement and capital standards[END_REF]) in terms of event, but goes beyond as it generally implies a non-negligible reputational risk. Indeed, nancial institutions might be perceived as being unprofessional or not knowing what they are doing. Through the operational risk angle, conduct risk can lead to huge losses, usually resulting from compensations, nes or remediation costs. But from a reputational point of view, it may result in reduced revenues as customers may think that banks does not treat them fairly or worse, are unprofessional as they have been caught cheating. Contrary to other operational risks, conduct risk is connected to the activity of the nancial institution (up to a certain extent), i.e. how they generate their income, how they conduct their business with respect to sta, customers, institutions, regulations etc. (e.g. money laundering and non-compliance are issues falling into the lack of integrity matter and consequently are part of conduct risk.)

According to the Financial Stability Board (FSB), "One of the key lessons from the crisis was that reputational risk was severely underestimated; hence, there is more focus on business conduct and the suitability of products, e.g., the type of products sold and to whom they are sold. As the crisis showed, consumer products such as residential mortgage loans could become a source of nancial instability" (FSB (2013)).

Here the FSB conrmed our statements in the rst paragraph, as they identied conduct risk as the main source of reputational risk. In that case we are talking about products sold to consumers. Rightfully, consumers are expecting the products they buy to perform as they have been led to believe, and that these are suitable for them. Again we see here the connection, the relationship operational risk and reputational risk.

Regulatory bodies aim at protecting investors and customers, ensuring fair, orderly and eciently functioning markets; and facilitating capital formation and transmission. The objective is to promote a trustable market environment, i.e. an environment founded on transparency and integrity. Indeed, institutions such as the SEC, the UK's Financial Conduct Authority (FCA) or the Australian Securities and Investments Commission (ASIC) are focusing on ensuring public trust and consumer and investor protection. Activities that may arm these objectives, such as selling products that are not suitable for their customers, or operating outside a culture that respects consumers interests. In otherwords, these institutions are targeting nancial institutions mis-conducts. Most nancial institution are customer oriented businesses, in particular retail banks. When a nancial institution is accused of money laundering, customers are always on one side of the issue, and when anancial institution produces a awed product, the customer is the person impacted.

By bringing the customer back in the foreground and thinking about doing things right before acting on sole potential prot generation basis, nancial institutions will generate more ecient and more sustainable revenues. By dealing with conduct risk, nancial institutions are changing the way od conducting business.

To achieve their regulatory prerogatives, authorities are interested in: how rms comply with the regulation, customers experience, product approval processes, decision making process, the rm's behaviour and the remuneration structures. These are indeed inherent factors leading to mis-conduct.

As our goal is to bring customers back in the foreground, in this chapter, we will address the question of how we can increase the knowledge of our customer better, using particular segmentation strategies relying on big data. Then, we will focus on the product design and more specically on how to address potential aws and mis-conception with respect to the target market. Indeed, many techniques coming from other industries can be applied to measure conduct risk and address it while designing new products. We believe that a scenario analysis type of process should be implemented every time a new product is designed. Here the term design comprise the set of strategic and tactical tasks and activities, starting from the initial idea, the concept, the rst model to the commercialisation of the products, in other words the design phase for us nishes when the rst product is sold to a rst customer.

The products will be designed with respect to the market segment targeted. Indeed, we will argue that most conduct risk issue can be addressed through the process of designing a products, as a product is usually design for a particular market segment, it has to comply with a set of rules and regulations, sales people remuneration is usually linked to product sales, and the intrinsic characteristic of the product have to be properly set so the product behave as expected. Issues have to be proactively tackled by understanding customers, improving our knowledge of them and analysisng how the product may fail them and consequently the bank. Scenario analysis and bid data are likely to be the tools to deal with conduct risk and hopefully annihilate it. The sole focus on prot generation seems to lead to an over conduct risk exposure and to an less ecient generation of income.

Therefore, in this chapter we will address issues leading to conduct risk i.e. the lack of integrity of the nancial institutions due to dealing with unknown customers, aws in the product, or the sales process, e.g. how customers (or any third party) are treated and how product are designed. These matters are clearly linked to the risk "culture" of the nancial institutions.

Remark 1.1. Note that the issue might be related to the customer perception and not an actual issue with the product created by the nancial institution, and this might be quite complicated to address as sometimes doing the right thing might not be perceived as such.

2

Know your customer -Data Mining for Segmentation

In a rst stage, it is important for nancial institutions to understand (now more than ever) who they are dealing with, i.e. what business they are accepting and to whom they are selling their products. Thought the statement "know your customer"

is simple, what it implies is rather complicated as it requires a dynamic capture of customer evolution and behaviours. Indeed, the same person taken at two dierent point in time of her life should be considered and treated as two dierent customers (up to a certain extent).

Dynamic segmentation

To ensure that products are sold to the right people, it is necessary to change the way the market is segmented.

Segmentation is a marketing strategy involving splitting the target market into subsets that have common characteristics and designing product adapted to the segment.

Segmentation allows identifying and further dening the target customers, providing data to elaborate the marketing plan. The product created are then adapted to the dierent market segment, in other words, it would not be possible to oer a mortgage to a rst year student. Financial institutions traditionally develop dierent strategies involving specic products or product lines depending on the attributes of the targeted segment.

Though practitioners usually (or initially) segment with respect to geographic criterianations, states, regions, countries, cities, neighborhoods, or postal codes, in a rst stage and with respect to demography i.e. considering age, sex, generation, religion, occupation and education level [START_REF] Riley | Market segmentation[END_REF]) or according to perceived benets which a product or service may provide in a second stage, we would argue that behavioral segmentation divides consumers into groups according to their knowledge, attitude, usage of something would be the most appropriate way to dene market segments [START_REF] Kotler | Principles of Marketing[END_REF]) in particular considering our objective to nip conduct risk in the bud.

Indeed, this allows a dynamic adaptation of the segment and regular switch from one segment to another according to the segment characteristic considered. However, on the more complex side, it is rather complicated to obtain all the data necessary to achieve such a dynamic approach, to make sure that we know our customer all the time and not point in time. Once again, a customer one day cannot be considered identical the next and the product sold might not be appropriate anymore. This is where data sciences introduced in the next section might be useful.

Data science support

The purpose of data science and here in particular data mining [START_REF] Hastie | The Elements of Statistical Learning: Data mining,Inference,and Prediction[END_REF])

is to extract information from data sets. The analysis of large quantities of data allows detecting interesting patterns such as clusters [START_REF] Everitt | Cluster Analysis[END_REF]), anomalies, and dependencies, and the outcome can then be used for further analysis in machine learning, predictive analytics or more traditional modelling and in our case to achieve the dynamic segmentation in question.

Data mining is widely used by companies with a strong consumer focus, in other words, retail, nancial, communication, and marketing organisations (Palace (n.d.)).

These types of companies were mining data to analyse relationships between endogenous and exogenous factors such as respectively price, product positioning, and economic indicators, competition, or customer demographics. as long as their in-uence or impacts on sales, reputation, corporate prots etc. Besides, it permitted summarising the information analysed. Generally, any of four types of relationships are sought: classes, clusters [START_REF] Everitt | Cluster Analysis[END_REF]), associations (Piatetsky-Shapiro (1991)) and sequential patterns.

To summarise the data mining process, this one consists in the following major elements: capture, pre-processing -data selection, cleansing and anomalies detection and transformation -, processing -outlier detection, relationship analysis, pattern recognition and summarisation -and integration in the decisions making process.

Product creation

Once customers are understood, the segmentation properly achieved it can be used as a base to proper product design. As suggested in the introduction, product issued by nancial institutions should be subject to a drastic quality control, not only once the product is nished, but all along the product creation, design, management, etc.

The quality control dened by the norm ISO 9000 -"A part of quality management focused on fullling quality requirements" [START_REF] Poksinska | The state of iso 9000 certication: A study of swedish organizations[END_REF]) -should be transformed into a quality defect prevention process, and here by defects we will obviously focus on conduct risk.

Consequently here we are interested in analysing and testing the products to uncover defects or potential risk of failures leading to conduct risk and reporting to the senior management who can decide to prevent the product from being released or sponsor corrective actions, i.e. this is an ex ante process by opposition to an ex post quality control. We believe that prevention may help nancial institutions reducing their exposure to conduct risk related losses and help them mitigating their potential remediation costs.

Many approaches have been proposed over time by both practitioners and academics to address the issue, such as Ishikawa Fishbone diagram, or root cause analysis among others (Hassani (2016(Hassani ( -2017))).

Ishikawa diagrams

Ishikawa diagrams are causal diagrams created for quality management processes purposes [START_REF] Ishikawa | Guide to Quality Control[END_REF]). Indeed they are used for product design and quality defect prevention. Ishikawa diagram are also refred to as shbone diagram because of its gure similar to the side view of a sh skeleton.Each cause of potential imperfection should be considered as a source of product design variation. As illustrated in Figure 1 potential fault sources are identied and gathered into various categories as follows:

1. People: Anyone involved with the process 2. Process: Goverance, policies, committees, procedures, etc.

3. Equipment: Any equipment, computers, tools, etc. required to achieve a task 4. Materials: Raw materials (in a large sense)

5. Management and Measurements: Data generated from the process used to evelaute the quality 6. Environment: Culture, law, regulation etc.

Each and every source of variation should be considered to avoid defect and faults at any stage of the product, for example this allows analysing how adapted the product is to a specic market segment and make sure that it has been designed consistently.

Fault Tree

The FTA is a top down, deductive failure analysis in which an undesired state of a system is analysed using boolean logic to combine a series of lower-level events 8 Documents de travail du Centre d'Economie de la Sorbonne -2016.67 [START_REF] Martensen | The fault-tree compiler[END_REF], [START_REF] Delong | Ta fault tree manual[END_REF], [START_REF] Larsen | Fault tree analysis[END_REF], FAA (1998)).

To be more specic regarding how the FTA can be used to, the following enumeration should be enlightening 1. understand the logic, the chain of events and the conditions leading to an undesired event.

2. show compliance with the regulation and the commercial principles.

3. prioritise the contributors leading to the top event -4. optimise resources.

5. assist in designing a system. Indeed, the FTA may be used as a design tool, helping avoiding common failures and measuring eect of design modications.

In this section, we will focus on this aspect of the methodology.

6. function as a diagnostic tool to identify and correct causes of the undesired event. This will alos be useful to design the product as it may help identifying sources of variations 7. quantify the exposure by calculating the probability of the undesired event.

To illustrate the process, the undesired outcome is taken as the top of a tree such as: the insurance product has been mis-sold. Working backward it is possible to determine what led to the failure, i.e. a fault in the product intrinsic characteristics or the wrong target segment targeted. This condition is a logical OR. Considering the branch analysing when the wrong market segment is targeted, we are in the presence of another logical OR, i.e. the population targeted is not in capacity to understand the product characteristics or the customers targeted are only those who have been customers for at least ten years and trust the bank no matter what and are easier targets for the sales people in the branch. On the other side of the tree, regarding the fault in the product characteristics, a logical AND relationship appears, as we may assume that the term sheet of the product is misleading and the product does not behave as expected due to a high volatility in the markets. Fault trees events can be associated with failure probabilities, and it is possible to calculate the likelihood of the undesired event to occur.The diagram is usually drawn using conventional logic gate symbols. The route between an event and an initiator in the tree is called a cut set. The shortest credible way through the tree from fault to initiating event is usually referred as a minimal cut set.

Figure 2 illustrates the Fault tree described above. If the fault tree is used to analyse the potential causes of failures, it necessary means that these have been identied

and therefore these can be dealt with, i.e. the awareness of the potential causes of failure lead to the implementation of new controls and a better attentions to some particular aspect of the product design.

Indeed, conduct risk related to product aws have to be prevented as the remediation costs experienced are far too large to be acceptable, besides the loss of prot is not acceptable for any company.

The construction of a fault tree is an iterative process, which has 6 clearly dened steps, for instance (Fault Tree Analysis (1999)):

1. Review the gate events 2. Identify all the possible causes of this event and ensure that none are missed 3. Identify the cause-impact relationships at each node 4. Structure the tree consistently 5. Ensure events are unique at each gate. While informing each gate node involves a 3 steps:

• Step 1 -Immediate, Necessary and Sucient (INS)

•

Step 2 -Primary, Secondary and Command (PSC)

•

Step 3 -State of the System or Component

Analysing this rst step in detail, the question to be answered is are the factors immediate, necessary and sucient (INS)to cause the intermediate event?

The we need to consider the fault path for each enabling event and identify each causing event identifying if they are primary fault, secondary faults or command faults. Then, it is possible to structure the sub events and gate logic from the path type. The last step requires addressing the question is the intermediate event a state of the system or a state of the component. "State of the component" implies that we are at the lowest level for that issue, while "state of the system" implies subsequent issues.

Failure mode and eects analysis (FMEA)

The FMEA might be used to systematically analyse product component potential failures [START_REF] Koch | Jet propulsion laboratory reliability analysis handbook[END_REF]). All signicant potential failure should be considered to design the product. The primary benet of the methodology is the early identication of issues to be addressed during the design of the product (recall that we considered the governance is considered part of the design phase).

The main benets of the FMEA process are the following:

• It helps maximising the chance of designing an appropriate product.

• It supports the assessment of potential failure as well as their their impacts, allowing to rank them and prioritise them.

• It permits the early identication of single points of failure.

13 Documents de travail du Centre d'Economie de la Sorbonne -2016.67

• It allows troubleshooting and product variation at any point in the design process.

• Easy to implement.

In that case, I would not consider a particular template as in that case it is necessary to adapt it to the design of nancial products.

Root Cause Analysis

Root cause analysis (RCA) aims at solving problem by dealing with their origination [START_REF] Wilson | Root cause analysis: A tool for total quality management[END_REF][START_REF] Vanden | Root Cause Analysis Handbook: A Guide to Ecient and Eective Incident Management[END_REF][START_REF] Horev | Root Cause Analysis in Process-Based Industries[END_REF] and [START_REF] Barsalou | Root Cause Analysis: A Step-By-Step Guide to Using the Right Tool at the Right Time[END_REF]). RCA allows identifying and correcting the root causes of events before they materialise, rather than dealing with the symptoms. This methodology aims for prevention while it acknowledges the fact that it is not always possible to create a failure proof product.

Though the analysis is usually done after an event has occurred, post-portem, this one could be considered to feed future product design or to analyse a scenario of failure or fault to manage them eectively and eciently.

RCA allows identifying the factors leading to a failure, preventing recurrence of issues, focusing on the lesson learnt and documenting the sequence of events leading to the failures supports controls positioning. The methodology helps transforming a reactive culture into a forward-looking culture. Note that absolute transparency is required and that RCA might be highly data consuming [START_REF] Shaqdan | Root-cause analysis and health failure mode and eect analysis: two leading techniques in health care quality assessment[END_REF]).

Why-Because Strategy

The Why-Because analysis has originally been developed to analyse accidents (Ladkin and Loer (1998)). A why-because graph presents causal relationships between factors of a failure easily applicable to nancial products, as a directed acyclic graph in which the factors are represented by nodes and relationships between factors by directed edges.

"What?" is always the rst question to ask, though as a forensic analysis it is a simple analysis as consequences are understood, for conduct risk it might slightly be more complicated as sometimes potential issues are not necessarily straight forward. We need to remember that a nancial institution may think they have done something appropriate but customer may believe the contrary, or may perceive dierently. The methodology relies on an iterative process to determine causes of failure.

Each contributing cause must be a necessary and all causes must be sucient to cause it [START_REF] Ladkin | The glenbrook why-because graphs, causal graphs, and accimap[END_REF]).

Bayesian Network

A Bayesian network is a probabilistic graphical model representing random variables and their conditional dependencies using a directed acyclic graph (DAG). Formally, the nodes represent random variables in the Bayesian sense, in our case product specicities. Edges exhibit conditional dependencies, if the nodes are not connected, they are assumed independent. Each node has a probability function which from the node's parent information, returns the probability of the variable represented by the node.

Bayesian Belief Networks are very valuable to design nancial products as they combine of data and expertise. They can be constructed using few data points, with or without knowledge of the underlying failure process, besides Bayesian networks are scalable. [START_REF] Corney | Designing food with bayesian belief networks[END_REF] shows that they are as accurate as neural networks, but with the advantage of being reversible, and therefore permits assessing the likelihood of a design to fail and on the contrary the likelihood of a design to succeed.

Modelling failure factors help understanding the impact and consequently support the control implementation. Besides the quality of the control implemented can be evaluated and considered in the analysis. For proper calculation we will refer to [START_REF] Ha | An identity for the wishart distribution with applications[END_REF] analysing and representing each every factory potentially leading to a failure may be more appropriate, requires less quantitative background (except for the Bayesian networks) and are easier to present to the senior management ultimately responsible for the approval of a product.

However, it is important to bear in mind that conduct risk modelling implies another kind of risk: model risk, and this one must be addressed too. Indeed, if the methodology used to analysed and design the product is itself awed, the product will not be properly designed and the conduct risk unmitigated [START_REF] Hassani | Model risk -from epistemology to management[END_REF]). However, it is interesting to note that the scenario methodologies presented such as Bayesian networks can be extended to other risk issues.

Figure 1 :

 1 Figure 1: Hishikawa diagram illustration

Figure 2 :

 2 Figure 2: Simple Fault Tree: This fault tree gives a simplied representation of the process leading to the materialisation of a conduct risk due to a product mis-sold.

Figure 3

 3 Figure 3 illustrates the construction of a simple Bayesian network. The node from which the arrow (arc) starts is refered to as the parent, and the node where the arc ends as the the child. In our example, Aa is a parent of Dd, and Dd is the child of Aa. Nodes that reachable from other nodes are called descendants. Nodes on the path to a specic node are called ancestors. Here, Ee and Ff are descendents of Bb, and Bb and Dd are ancestors of Ff.

  requires changing the way the business is conducted and the income generated. For instance, nancial institutions may need to close some revenue channel because it is the right thing to do as the products sold are inappropriate, or they should change the way sales people are remunerated if it appears they were forced to deliver a certain number of sales per day no matter what otherwise they might end up in a position of selling quanto products to grand-mothers, or accepting transactions from unknown people, assuming that these are genuine without any other form of verications.

Documents de travail du Centre d'Economie de laSorbonne -2016.67

Documents de travail du Centre d'Economie de laSorbonne -2016.67

Aa

In this chapter we actually addressed conduct risk awareness and management. To summarise, it is necessary to make sure that we know our customer and that the products sold are appropriately designed to make sure that these will not lead to the materialisation of conduct risk. After introducing the requirements of a dynamic segmentation strategy, we introduced methodologies traditionally used for product design in other industries which should be adapted systematically before any product is released. Other alternatives (Hassani (2016(Hassani ( -2017))) are available such as Neural networks or assessment through workshops, though we believe in that case that carefully