Thomas Amorgianiotis
email: amorgianio@ceid.upatras.gr

Konstantinos Theofilatos

Sovan Mitra
email: sovan.mitra@gcu.ac.uk

Efstratios F Georgopoulos

Spiros Likothanassis

Integrating high volume financial datasets to achieve profitable and interpretable short term trading with the FTSE100 index

Keywords: Trading Strategies, Financial Forecasting, FTSE100, Genetic Algorithms, Support Vector Machines, Fuzzy logic

During the financial crisis of 2009 traditional models have failed to provide satisfactory results. Lately many techniques have been proposed to overcome the deficiencies of traditional models but most of them deal with the examined financial indices as they are cut off from the rest global market. However, many late studies are indicating that such dependencies exist. The enormous number of the potential financial time series which could be integrated to trade a single financial index enables the characterization of this problem as a "big data" problem and raises the need for advanced dimensionality reduction techniques which should additionally be interpretable in order to extract meaningful conclusions. In the present paper, ESVM-Fuzzy Inference Trader is introduced. This technique is based on the hybrid methodology ESVM Fuzzy Inference which combines genetic algorithms and some deterministic methods to extract interpretable fuzzy rules from SVM classification models. The ESVM-Fuzzy Inference Trader was applied to the task of modeling and trading the FTSE100 index using a plethora of inputs including the closing prices of various European indexes. Its experimental results were compared with a state of the art hybrid technique which combines genetic algorithm with Multilayer Perceptron Neural Networks and indicated the superiority of ESVM-Fuzzy Inference Trader. Moreover, the proposed method extracted a compact set of fuzzy trading rules which among others can be utilized to describe the dependencies between other financial indices and FTSE100 index.

Introduction

Modeling and trading financial indices remains nowadays a very challenging and demanding problem for the scientific community because of their complexity and their dynamic and noisy nature. The FTSE 100 index trades futures contracts were utilized as a case study. A future contract is a contract between two parties and its cash settlement is determined by calculating the difference between the traded price and the closing price of the index on the expiration day of the contract. Most traditional methods and more complex machine learning ones have recently failed to capture the complexity and the nonlinearities that exist in financial time series during the latest crisis period. Certain disadvantages have been identified on traditional modelling and trading methods including the difficulties in tuning the parameters of the algorithm, the disability of linear methods to provide good prediction results, the overfitting problem and the fact that modelling and trading are most of the times considered as different problems. Lately, several machine learning methods have been proposed to solve these problems [START_REF] Dunis | Computational Intelligent Techniques for Trading and Investment, Taylor and Francis[END_REF]. Despite the encouraging results of new hybrid methodologies their performance could be further enhanced.

Considering modelling and trading of the FTSE100 index, many researchers have been occupied with this problem. These methods are either based on simple traditional methods such as ARMA [START_REF] Chen | A Markov regime-switching ARMA approach for hedging stock indices[END_REF], or on machine learning techniques. The machine learning applications range from simple neural network techniques, such as Higher Order Neural Networks (HONN) [START_REF] Sermpinis | Modelling and trading the realised volatility of the FTSE100 futures with higher order neural networks[END_REF] to more elaborate techniques such as the hybrid method combining Artificial Bee Colony Algorithm with Recurrent Neural Networks [START_REF] Hsieha | Forecasting stock markets using wavelet transforms and recurrent neural networks: An integrated system based on artificial bee colony algorithm[END_REF] and the hybrid methodologies which combine Genetic Algorithms with Support Vector Machines (SVM) [START_REF] Dunis | A hybrid genetic algorithm-support vector machine approach in the task of forecasting and trading[END_REF].

Despite the promising methods of the aforementioned techniques most of them deal with the FTSE100 index as being independent and cut off from the global stock market. However, the reality is very different. As expected several articles indicated dependencies between the FTSE100 index and several other financial indices [START_REF] Doman | Dependencies between Stock Markets During the Period Including Late-2000s Financial Crisis[END_REF][START_REF] Graham | Short term and long-term dependencies of the S&P500 index and commodity prices[END_REF] and these dependencies have not yet been studied thoroughly. Thus, in order to achieve optimal prediction and trading results using the FTSE100 index, several inputs from other financial indices should be utilized alongside with the traditional autoregressive and technical indicator inputs. The size of the universe of financial indices which could possibly be dependent with FTSE100 is so high that makes this problem relevant to the "big data" [START_REF] Βoyd | CRITICAL QUESTIONS FOR BIG DATA[END_REF] topic which has gained the attention of the scientific community lately.

In the present paper, we have created an integrated dataset for modelling and daily trading with the FTSE100 index. This dataset, includes inputs and technical indicators from a variety of financial time-series including VIX, S&P500, DAXX, Euro Stoxx 50, and EURGBP exchange rate. The major problems for utilizing such integrated datasets for modelling and trading are: a) the difficulty to solve effectively the dimensionality reduction problem and b) the difficulty to achieve interpretability for extracting useful knowledge about the uncovered dependencies.

In the present study, we propose the ESVM Fuzzy Inference Trader to solve both these problems. This method is based on a Support Vector Machine methodology to achieve high performance predictions and it proposes an advanced technique to extract a meaningful compact set of fuzzy prediction rules. Moreover, a simple genetic algorithm finds the optimal feature subsets which should be used as inputs and optimizes the parameters of the overall modelling procedure. The final extracted inter-pretable fuzzy rules presents extremely high statistical and trading performance surpassing another machine learning technique (the hybrid combination of genetic algorithms with Multilayer Perceptron Neural Networks) and enabled for the extraction of important conclusions about the dependencies of FTSE100 and other examined financial indices.

Dataset description

The dataset deployed in the present study incorporates information from 5 financial indexes and one exchange rate.

The basic financial index examined in the present paper is the FTSE100 index. The FTSE 100 index is a share index of the 100 companies listed in the London Stock Exchange having highest market capitalization. It is traded by futures contracts that are cash settled upon maturity of the contract with the value of the index fluctuating on a daily basis. The cash settlement of this index is simply determined by calculating the difference between the traded price and the closing price of the index on the expiration day of the contract.

The FTSE 100 daily time series is non-normal (Jarque-Bera statistics confirms this at the 99% confidence interval), containing slight skewness and high kurtosis. In the present study arithmetic returns were used and they are estimated using the following procedure: Given the price level P 1 , P 2 ,…,P t , the arithmetic return at time t is formed by:

1 1     t t t t P P P R (1)
Forecasting and trading the FTSE100 was attempted using an extended universe of inputs (50 inputs) containing both autoregressive inputs and technical indicators, such as moving averages of the FTSE 100 index, the VIX index, the S&P500 index, the DAX index, the Euro Stoxx 50 index and the Euro-GBP exchange rate. Specifically the technical indicators which were used by our model are for the FTSE 100 index the growth rate of the previous day, the maximum reduction rate of previous day, the seven previous days, the 5,15,30,50 and 252 days moving averages, the yesterday volume of transactions, the 5,15,30,50 and 252 days moving averages of volume of transactions, the highest close price for the past 30&252 days, the lowest close price for the past 30 and 252 days. The inputs computed for the VIX index, the S&P500 index, the DAX index, the Euro Stoxx 50 index and the Euro-GBP exchange rate are the 5,15,30,50 and 252 days moving average of their closing price.

VIX is a measure of the implied volatility of S&P 500 index options and it is used as an advanced volatility indicator which is able to measure the risk of the market. The S&P 500 consists of the 500 large companies having common stock listed on the NYSE or NASDAQ. The DAX index consists of the 30 major German companies trading on the Frankfurt stock Exchange. The Euro Stoxx 50 is made up of fifty of the largest and most liquid stocks of Eurozone. It was first introduced on February 26, 1988. Historical data of the five examined indexes were downloaded from the finance yahoo site (http://finance.yahoo.com/) and the Euro-GBP exchange rate was down-loaded from the global-view site (http://www.global-view.com/forex-tradingtools/forex-history/).

Our dataset consists of a three-year in-sample dataset (02/01/2009 -31/12/2011) and a two-year out-of-sample dataset (02/01/2012 -31/12/2013). Specifically the insample dataset consists of 757 rows and 50 columns and the out-of-sample dataset consists of 502 rows and 50 columns. Each row represents a day and each column the 49 input features and the output. This dataset includes inputs and technical indicators from a variety of financial time-series including VIX, S&P500, DAXX, Euro Stoxx 50, and EUR-GBP exchange rate. The ESVM fuzzy inference trader tries to find the dependencies between the above financial indices and the FTSE100. FTSE100 index can also be dependent on an enormous number of indices. From this point of view, the problem of modeling and forecasting the FTSE100 can be seen as a big data problem. The ESVM fuzzy inference trader can find these dependencies and eliminate the financial time series that seem to have no dependencies with the FTSE100 index and this is how we deal with the big data problem. This is done by the feature selection technique of the proposed evolutionary algorithm which is explained at the next section. Due to the market's non stationary nature and its continuing, we deployed a sliding window approach. In particular, the out of sample dataset (testing dataset) was divided in three-month sub-datasets and in order to forecast and trade each individual sub-dataset the algorithms were trained using as in-sample dataset (training dataset) the past three years.

3

Method description

GAMLP methodology

The first machine learning methodology which was applied in the present study is a hybrid combination of Genetic Algorithms and Multi-Layer Perceptron Neural Networks. It has been successfully applied in forecasting and trading the DJIE index [START_REF] Theofilatos | Modelling and Trading the DJIA Financial Index Using Neural Networks Optimized with Adaptive Evolutionary Algorithms[END_REF] and the FTSE100 index [START_REF] Theofilatos | Advanced short-term forecasting and trading deploying neural networks optimized with an adaptive evolutionary algorithm[END_REF]. This technique deploys an adaptive evolutionary algorithm to optimize the inputs, the structure and the parameters of MLP NNs. At this work three layered MLP NNs were utilized. The first is the input layer, the second is the hidden layer and the third is the output layer. The fitness function which was utilized by the evolutionary algorithm is described below:

where #selected_inputs is the number of inputs selected by every iteration of the evolutionary algorithm. This fitness function enables the extraction of profitable trading strategies while at the same time selecting only the most compact feature subset to be used as inputs.

The deployed evolutionary algorithm utilized an adaptive mutation rate operator which decreases the mutation probability as the evolutionary process progresses. Moreover, when stagnation effect is diagnosed the mutation probability is increased to escape local optimal solutions. The adaptive mutation rate, enables the overall al-gorithm to present advanced exploration properties in its initial steps and gradually reinforcing its exploitation properties.

ESVM Fuzzy Inference Trader

Support Vector Machines (SVM) is a method which belongs to the category of supervised learning models and it is nowadays considered as state-of-the-art in the field of machine learning. SVM attempts to locate the optimal hyperplane which can classify the points at categories and presents the largest distance between the points of different categories. SVMs do not require prior knowledge of the problem and they are general purpose classifiers. Non linear SVMs use a kernel function to project the input space to a space of higher dimensions in hope of transforming non-linear separable problems to linear separable ones. Through the years, there have been proposed several kernel functions. At this work we used the radial basis function because it is the most commonly used kernel function in non-linear classification problems [START_REF] Theofilatos | A Hybrid Sup-port Vector Fuzzy Inference System for the Classification of Leakage Current Waveforms Portraying Discharges[END_REF]. The main limitations of the SVM models are their black-box nature which restricts its interpretability, and the dependence of their performance to the selection of their inputs and parameters.

To overcome the disadvantages of the SVMs we combined them with a Genetic algorithm and a fuzzy rule based system. The genetic algorithm is used for the feature selection problem in an embedded framework and for the optimization of the SVM's parameters. Fuzzy logic is more close to the linguistic language and domain experts can interpret fuzzy rule-based classifiers more easily. Fuzzy logic has the inherit advantage of hiding the imprecise knowledge through fuzziness. Their main disadvantage is overfitting when the dataset includes non linear data.

The ESVM Fuzzy Inference Trader approach deploys an advanced technique to extract interpretable fuzzy rules from SVM classification models.

The problem of extracting fuzzy rules from trained classification SVMs models has been of interest to many researchers through the last decade. Our model is based at the work of [START_REF] Papadimitriou | Efficient and interpretable fuzzy classifiers from data with support vector learning[END_REF]. The proposed methodology initially deploys the technique which was proposed by [START_REF] Chen | Support vector learning for fuzzy rule based Classification[END_REF]. This is an algorithm to implicitly transform the SVM classification model to a set of Support Vector Fuzzy Inference (SVFI) rules which are proved to be equivalent to the initial classification model. SVFI rules are in the following general form: Rule k: if and and… then where are fuzzy clauses, having the form where . These fuzzy clauses examine the membership of the i-th input value in the i-th fuzzy set of the k-th support vector. The support vectors are the training samples that are selected by the SVM algorithm to define the final classification hyperplane. A Gaussian function of the form , estimates the membership function by quantifying the distance of the inputs component x i from the value x i k of the ith component of SV k . The parameters σ k are real constant numbers (σ k ∈ R).

Because SVFI rules are large and do not include linguistic clauses they are hard to be interpreted by domain experts. The ESVM Fuzzy Inference Trader applies a series of filtering and merging rules to obtain a more compact set of rules. In particular, fuzzy clauses with membership values less than a predefined threshold (β) and even whole fuzzy rules whose total membership value is less than another predefined threshold (δ) are discarded. As a second step, in the produced more compact set of rules CloseToSV(k, i) clauses are replaced with linguistic clauses (low, medium, high) which are represented as Gaussian functions with centers placed in the minimum, median and maximum value of each feature.

To fine tune the whole procedure of training SVMs and extracting interpretable fuzzy rules from them, the ESVM Fuzzy Inference Trader algorithm deploys a simple Genetic Algorithm. The GA is used to select the optimal feature subset, and find the optimal values for the regularization parameter C of the SVM, the parameter gamma for the RBF Kernel function, the threshold β for discarding a fuzzy clause, the threshold δ for discarding a fuzzy rule and the variations of the linguistic clauses' Gaussian function (low, medium, high).

The chromosome of the deployed GA consisted of input genes and parameter genes. The input genes take values 0 or 1 and force the classifier to use a specific feature as input if the feature value for this input is 1. The parameter genes are represented as double variables. The basic variation operators which were applied are the one point crossover and mutation operators. The crossover probability was set to 90% as this is the most important operator in GAs. For the mutation operator we used the binary mutation operator for the feature genes and the Gaussian mutation for the other genes as they are real valued genes. The binary mutation randomly alters a gene value from 0 to 1 and opposite. The Gaussian mutation operator adds a random number in a randomly selected gene. This random number is taken from the Gaussian distribution using as center the zero value and as width the interval of allowed values for this gene divided by 10.

The fitness function which guides the evolutionary process of ESVM Fuzzy Inference System is described below:

(3) where is the annualized return of the extracted trading strategy, geometric mean is the geometric mean of sensitivity and specificity metrics, = and = .

The parameters g1, g2, g3 and g4 were set to 2, 1, 0.5 and 0.25 respectively to reflect the importance of each goal. In particular the most important goal is to extract profitable trading strategies and the second most important goal is to achieve high classification performance. Secondary goals are the reduced number of fuzzy rules and selected inputs in order to increase the interpretability of the final set of fuzzy rules. The population of the genetic algorithm was set to 30 and the generations to 500.

The ESVM Fuzzy Inference Trading System also calculates a regression prediction for every data sample by calculating its distance from the classification hyperplane.

Experimental Results

In this section we present the results of the studied models in the problem of trading the FTSE100 index. The trading strategy for the examined models is simple and identical for both of them: go or stay long when the forecast return is above zero and go or stay short when the forecast return is below zero. Additionally, a confirmation filter was used which disables changing positions when the forecasted value is below the threshold 0.001.

Because of the stochastic nature of the proposed methodologies a simple run is not enough to measure their performance. This is the reason why ten runs where executed and the mean results are presented in the next table. When using the proposed trading strategy traders could utilize the execution which provided the highest in sample performance or use a more elaborate technique to combine the independent traders as proposed in [START_REF] Pendaraki | Constructing Portfolios Using Argumentation Based Decision Making and Performance Persistence of Mutual Funds[END_REF].

Table 1. Trading Results

GA-MLP ESVM Fuzzy Inference Trader

Annualized Return (excluding costs) [START_REF] Lindemann | Probability distributions, trading strategies and leverage: an application of Gaussian mixture models[END_REF] From Table 3 it is easily observed that both methodologies presented highly profitable trading results even when transaction costs were considered. This is mainly attributed to the extended universe of candidate inputs which was utilized and the sliding window approach which enables the model to be recalculated every three months. Furthermore, both deployed machine learning methods utilize an advanced feature selection mechanism which enables them to ignore redundant feature and extract more compact and simple prediction models.

The ESVM Fuzzy Inference Trader, clearly outperformed GAMLP approach in all examined metrics [START_REF] Lindemann | Probability distributions, trading strategies and leverage: an application of Gaussian mixture models[END_REF]. The main reason for this fact is the advanced generalization properties that the SVM provides to the overall methodology of ESVM Fuzzy Inference Trader. Moreover, the prediction model of ESVM Fuzzy Inference Trader could be further analyzed by studying the extracted interpretable fuzzy rules sets. The ten most highly rated, in terms of aggregated weight, rules which came up during the sliding window runs of ESVM Fuzzy Inference Trader are provided in Table 2. From this table it is shown that traditional autoregressive inputs where not so significant in the prediction models which were extracted from the ESVM Fuzzy Inference Trader. Important role seem to play as expected the different moving averages technical indicators. Moreover, other not traditional inputs such as past volumes of transactions, highest value of the indexes, lowest values of the indexes and external inputs such as VIX, DAX, S&P500 and EURGBP related inputs seems to be present in the final prediction models. With a closer examination of the extracted fuzzy rules low values of the moving averages of VIX past values seem to be an indicator for positive predictions. This is explained from the fact that low volatility in the market which is indicated by low values of the VIX index may have positive effects on the FTSE100 index. It is noteworthy that both DAX and S&P500 present negative effects on the FTSE100 outcome. In particular the extracted fuzzy rules predict positive values when negative values for the moving averages of these indexes are observed. EURGBP has similar action on the extracted fuzzy rules as when high values are observed for its moving averages the fuzzy rules decide to predict negative values.

Conclusions

In the present paper, the problem of modelling and daily trading of the FTSE100 index was approached for the first time with as a big data problem. In specific an extended universe of 50 potential inputs were integrated to a single dataset. These inputs were related to simple autoregressive inputs, and technical indicators of the FTSE100 index and several other financial time-series.

The proposed novel machine learning algorithm for solving the aforementioned problem was the ESVM Fuzzy Inference Trader. It is a hybrid methodology which combines genetic algorithms, Support Vector Machines, Fuzzy systems and some deterministic techniques to extract a compact and interpretable set of linguistic fuzzy prediction rules. The extracted fuzzy rules presented extremely high statistical and trading performance and outperformed one other machine learning approach which was utilized even when transactions costs were taken into account. Further analysis of the extracted interpretable fuzzy rules revealed that traditional autoregressive inputs are not significant for trading the FTSE100 index. Moreover, other technical indicators, such as moving averages, of FTSE100 and of other examined time series are the most important inputs for forecasting and trading the FTSE100 on a daily bases. Considering the role of the other financial time series which were studies, both the moving averages of VIX, S&P500, DAX and EURGBP were found to be inversely related with the FTSE100 index.

The present study could be further extended using additional time series for generating even more potential inputs. The ESVM Fuzzy Inference system is selecting only the more relevant inputs and its performance is not affected by the number of the initial potential features used. For this reason, the incorporation of even more inputs will only affect positively the performance of the overall method and will additionally reveal even more dependencies of the FTSE100 with the rest of the global market. Moreover, The proposed algorithmic technique can be applied to extract intraday trading strategies which are even closer to the big data problem definition.

Table 2 .

 2 Ten most significant extracted fuzzy rules


Gareth W Peters
email: gareth.peters@ucl.ac.uk

Pavel V Shevchenko
email: pavel.shevchenko@csiro.au

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

Ariane Chapelle
email: a.chapelle@ucl.ac.uk

Standardized Measurement Approach for Operational risk: Pros and Cons

This response has been put together by academics and in total independence of any corporate or individual interests. Our results are solely driven by scientific analysis and presented in the interest of the

The structure of this response involves a collection of summary results and comments for studies performed on the proposed SMA model which include:

 Capital instability;  Capital sensitivity;  Reduction of risk responsivity and interpretability;  Incentivized risk taking;  Discarding key sources of Operational risk data;  Possibility of super additive capital under SMA. The detailed analysis of these points is developed in the manuscript [START_REF] Peters | Should AMA be Replaced with SMA for Operational Risk[END_REF].

The response then concludes with suggestions relating to maintaining the AMA internal model framework with standardization recommendations that could be considered to unify internal modelling of Operational risk.

SMA Introduces Capital Instability

Our analysis of the SMA and OpCar model shows that SMA fails to achieve the objective of capital stability. Consider a simple representative model for a bank's annual Operational risk loss process comprised of the aggregation of two generic loss processes, one high frequency with low severity loss amounts and the other corresponding to low frequency and high severity loss amounts given by Poisson-Gamma and Poisson-Lognormal models respectively. We set the business indicator (BI) constant to Euro 2 billion at half way within the interval for bucket 2 of the SMA, we kept the model parameters static over time and simulated a history of 1,000 years of loss data for three differently sized banks (small, medium and large) using different parameter settings for the loss models to characterize such banks. For a simple analysis we set a small bank corresponding to capital in the order of Euro 10's of million average annual loss, a medium bank in the order of Euro 100's of million average annual loss and a large bank was set to have in the order of Euro 1billion average annual loss. We then studied the variability that may arise in the capital under the SMA formulation, under the optimal scenario that models did not change, model parameters were not recalibrated and the business environment did not change significantly, in the sense that BI was kept constant. In this case we observe the core variation that arise just from loss history experience of many banks of the three different sizes over time.

Our analysis shows that, a given institution can experience the situation in which its capital can more than double from one year to the next, without any changes to the parameters, the model or the BI structure. Annual variation can be as large as 2 times the long-term average capital (Figure 1).

It follows from the results in this sections analysis that two banks, with the same risk profile, can produce SMA capital numbers differing by a factor of more than 2.

Details of the analysis are available on the complete paper posted on ssrn.com (see [START_REF] Peters | Should AMA be Replaced with SMA for Operational Risk[END_REF]) and the code used is available upon request. In summary, the simulation takes the case of BI fixed over time, the loss model for the institution is fixed according to two loss processes given by Poisson(λ)-Gamma(α,β) and Poisson(λ)-Lognormal(µ,σ). In this example, Gamma(α,β) is the distribution of the loss severities with the mean equal to αβ and the variance αβ 2 , and the Lognormal(µ,σ) is the distribution of severities with the mean of the log severity equal to µ and the variance of the log severity equal to σ 2 .

The total institutions losses are set to be on average around 1,000 per year with 1% coming from heavy tailed loss process Poisson-Lognormal component. We perform two case studies, one in which the shape parameter of the heavy tailed loss process component, LogNormal is with σ = 2.5 and the other with σ = 2.8. We summarize the settings for the two cases below in Table 1 andTable 2. The ideal situation that would indicate that SMA was not producing capital figures which were too volatile would be if each of the sub-figures below in Figure 1 and Figure 2 were very closely constrained around 1. However, as we see from the analysis, the variability in capital from year to year, in all size institutions can be significant. In particular medium to large size institutions both demonstrated that in any given year under SMA, the capital required to be held could double. Understanding Variability in BI when SMA matches AMA As a second study of the SMA capital instability we again consider a Poisson-Lognormal loss process model Poisson(λ)-Lognormal(µ,σ). Except in this case, instead of fixing the BI to the midpoint of Bucket 2 of the SMA formulation, we instead numerically solve for the BI that would match the SMA capital to the Value-at-Risk for a Poisson-LogNormal Loss Distributional Approach (LDA) model at the annual 99.9% quantile level.

In other words, for these simulations we obtain the BI such that the LDA capital will match SMA capital in the long term. This is achieved by solving the following non-linear equation numerically via root search for the BI. That is we solve for the BI such that SMA(BI) = VaR(0.999).

For this experiment we took 0.999 VaR under Poisson-Lognormal model according to the single loss approximation given by

                              2 1 2 1 exp 999 . 0 1 1 exp ) 999 . 0 (       VaR where ) ( 1   
is the standard Normal cdf.

The results of this analysis are presented in Table 3 for λ = 10 and the results for the LogNormal σ and µ parameters are varied in the table to present different BI values.

Table 3. In addition, we consider a second capital instability study where we use the BI obtained from matching the long term average SMA capital with the long term LDA capital, as described above for an example generated by Poisson(10)-Lognormal(µ =12, σ =2.5) and corresponding solved BI=14.714 bln (bucket 4). In this case SMA capital (ideal where averaging over many years rather than 10 years) is 1.87 bln is about the same as LDA 0.999 quantile = 1.87 (calculated through Monte Carlo). Then the year on year variability in the capital with this combination of implied BI and Poisson-LogNormal loss model is given in Figure 3. It shows that again we get capital instability with capital doubling from year to year compared to the long term average SMA capital.

Implied BI in billions

Figure 3. Ratio of the SMA capital to the long term average.

SMA is Excessively Sensitive to Behaviour of Dominant Loss Process

In this section we consider an institution with a wide range of different types of Operational risk loss processes present in each of its business units and risk types. As in our first study above, we again consider in a stylized manner to split these loss processes into two categories, high frequency and low severity, and low frequency high severity types. As in the previous illustration we consider a simplified banking structure with two loss processes given by Poisson(λ)-Gamma(α,β) and Poisson(λ)-Lognormal (µ,σ). In this study we consider the sensitivity of SMA capital to the dominant loss process.

More precisely, we study the sensitivity of SMA capital to the parameter that dictates how heavy the tail of the most extreme loss process will be. This analysis is based on the simulation setup presented in Table 4 with simulations performed over 1,000 years. These results in Figure 4 can be interpreted to mean that banks with more extreme loss experiences as indicated by heavier tailed dominant loss processes (increasing σ) tend to have significantly greater capital instability compared to banks with less extreme loss experiences. Importantly, these findings demonstrate how non-linear this increase in SMA capital can be as the heaviness of the dominant loss process tail increases. For instance, banks with relatively less heavy tailed dominant loss processes (σ=2) tend to have capital variability year on year of between 1.1 to 1.4 multipliers of long term average SMA capital. Already this is not a good outcome. However, banks with relatively heavy tailed dominant loss processes (σ = 2.5, 2.75 or 3) tend to have excessively unstable year on year capital figures, with variation in capital being as bad as 3 to 6 times multipliers of the long term average SMA capital. Furthermore, it is clear that when one considers each boxplot as representing a population of banks with similar dominant loss process characteristics, then as the tail-heaviness of the dominant loss process in each population increases the population distribution of capital becomes increasingly skewed and demonstrates increasing kurtosis in the right tail, with our findings demonstrating that this can result in excessive variability in capital year on year for banks with heavy tailed dominant loss processes.

Therefore, SMA fails to achieve the claimed objective of robust capital estimation. Capital produced by the proposed SMA approach will be neither stable nor robust with worsening robustness as the severity of Operational risk increases. In other words, banks with higher severity Operational risk exposures will be substantially worse of in SMA approach with regard to capital sensitivity.

SMA Reduces Risk Responsivity and Interpretability

One can consider that the SMA capital is less responsive to risk drivers and the variation in loss experience that is observed in a bank at granularity of the Basel II 56 Business Unit Risk Type units of measure. This is due to the naive approach of modeling at the level of granularity assumed by the SMA which is only capturing variability at the institution level and not the intra-variability within the institution at division or business unit levels explicitly. Choosing to model at institution level, rather than the units of measure or granularity of the 56 Basel categories reduces model interpretability and reduces risk responsivity of the capital.

Conceptually, it relates to the simplification of the Advanced Measurement Approach (AMA) to instead the SMA adopting a top down formulation that reduces Operational risk modelling to a single unit of measure, as if all operational losses were following a single generating mechanism. This is equivalent to considering that earthquakes, cyber-attacks and human errors are all generated by the same drivers and manifest in the loss model and loss history in the same manner as other losses that are much more frequent and have lower consequence, such as credit card fraud, when viewed from the institution level loss experience. It follows quite obviously that the radical simplification and aggregation of such heterogeneous risks in such a simplified model cannot claim the benefit of risk-sensitivity, even remotely.

Therefore, SMA fails to achieve the claimed objective of capital risk sensitivity. Capital produced by the proposed SMA approach will be neither stable nor related to the risk profile of an institution.

SMA Incentivizes Enhanced Risk-Taking

Besides extreme conceptual flaws, the SMA induces risk-taking behaviors, failing to achieve the Basle committee objectives of stability and soundness of the financial institutions.

Moral hazard and other unintended consequences are:

-More risk-taking: without the possibility of capital reduction for better risk management, in the face of increased funding costs due to the rise in capital, it is predictable that financial institutions will raise their risk-taking to a level sufficient enough to pay for the increased cost of the new fixed capital. The risk appetite a financial institution would mechanically increase. This effect goes against the Basel Committee objective of a safe and secured financial system.

-Denying loss events: whilst incident data collection is a constant effort for over a decade in every institutions, large or small, the SMA is the most formidable disincentive to report losses. There are many opportunities to compress historical losses such as ignoring, slicing, transferring to other risk categories. The wish expressed in the consultation that "Banks should use 10 years of good-quality loss data" is actually meaningless if the collection can be gamed. Besides, what about new banks or BIA banks which do not have any loss data collection process as of today?

-Hazard of reduced provisioning activity: provisions, which should be a substitution for capital, are vastly discouraged by the SMA, as they are penalized twice, counted both in the BI and in the losses, and not accounted for as a capital reduction. The SMA captures both the expected loss and the unexpected loss, when the regulatory capital should only reflect the unexpected loss. We believe that this confusion might come from the use of the OpCar model as a benchmark because the OpCar captures both equally.

The SMA states in the definition of "Gross loss, net loss, and recovery definitions" on page 10 Section 6.2 of [BCBSd355,2016] under item (c) that the loss data set gross loss and net loss should Documents de travail du Centre d'Economie de la Sorbonne -2016.64

include "Provisions or reserves accounted for in the P&L against the potential operational loss impact". This clearly indicates the nature of the double counting of this component since they will enter in the BI through the PnL and in the loss data component of the SMA capital.

-Ambiguity in provisioning and resulting capital variability: the new guidelines on provisioning under SMA framework follow similar general concept as those that recently came into effect in credit risk with the International Financial Reporting Standard (IFRS9) set forward by the International Accounting Standards Board (IASB), who completed the final element of its comprehensive response to the financial crisis with the publication of IFRS 9 Financial Instruments in July 2014. The IFRS9 guidelines explicitly outline in Phase 2 an impairment framework which specifies in a standardized manner how to deal with delayed recognition of (in this case) credit losses on loans (and other financial instruments).

IFRS9 achieves this through a new "…expected loss impairment model that will require more timely recognition of expected credit losses. Specifically, the new Standard requires entities to account for expected credit losses from when financial instruments are first recognised and it lowers the threshold for recognition of full lifetime expected losses."

However the SMA Operational risk version of such provisioning concept for Operational risk losses fails to provide such a standardized and rigorous approach. Instead the SMA framework simply states that loss data bases should now include "Losses stemming from operational risk events with a definitive financial impact, which are temporarily booked in transitory and/or suspense accounts and are not yet reflected in the P&L ("pending losses"). Material pending losses should be included in the SMA loss data set within a time period commensurate with the size and age of the pending item."

However, unlike the more specific IFRS9 accounting standards, under the SMA there is a level of ambiguity. Furthermore, this ambiguity can propagate now directly into the SMA capital calculation causing potential for capital variability and instability.

For instance, there is no specific guidance or regulation requirements to standardize the manner in which a financial institution decides what is to be considered as "definitive financial impact" and what they should consider as a threshold for deciding on existence of a "material pending loss" nor what is specifically the guidance or rules relating to time periods related to inclusion of such pending losses in an SMA loss data set and therefore into the capital. The current guidance simply states "Material pending losses should be included in the SMA loss data set within a time period commensurate with the size and age of the pending item". This is too imprecise and may lead to manipulation of provisions reporting and categorization that will directly reduce the SMA capital over the averaged time periods in which the loss data component is considered.

Furthermore, if different financial institutions adopt differing provisioning rules, the capital obtained for two banks with identical risk appetites and similar loss experience could differ substantially just as a result of their provisioning practices.

-Imprecise Guidance on Timing Loss Provisioning:

The SMA guidelines also introduce an aspect of "Timing Loss Provisioning" in which they state:

"Negative economic impacts booked in a financial accounting period, due to operational risk events impacting the cash flows or financial statements of previous financial accounting periods (timing losses"). Material "timing losses" should be included in the SMA loss data set when they are due to operational risk events that span more than one financial accounting period and give rise to legal risk."

However, we would argue that for standardization of a framework there needs to be more explicit guidance as to what constitutes a "Material timing loss". Otherwise, different timing loss provisioning approaches will result in different loss databases and consequently can result in differing SMA capital just as a consequence of the provisioning practice adopted. In addition, the Documents de travail du Centre d'Economie de la Sorbonne -2016.64

ambiguity of this statement doesn't make it clear as to whether such losses may be accounted for twice.

-Grouping of Losses: Under previous AMA internal modelling approaches the unit of measurement or granularity of the loss modelling was reported according to the 56 Business Unit and Risk Type categories specified in Basel II. However, under the SMA the unit of measure is just at the institution level so the granularity of the loss processes modelling and interpretation is lost. This has consequences when it is considered in light of the new SMA requirement that "Losses caused by a common operational risk event or by related operational risk events over time must be grouped and entered into the SMA loss data set as a single loss."

Previously, in internal modelling losses within a given Business Unit Risk Type would be recorded as a random number (frequency model) of individual independent loss amounts (severity model). Then, for instance under an LDA model such losses would be aggregated only as a compound process and the individual losses would not be "grouped" except on the annual basis and not on the per-event basis. However, there seems to be a marked difference in the SMA loss data reporting on this point, under the SMA it is proposed to aggregate the individual losses and report them in the loss database as a "single grouped" loss amount. This is not advisable from a modelling or an interpretation and practical risk management perspective.

Furthermore, the SMA guidance states "The bank's internal loss data policy should establish criteria for deciding the circumstances, types of data and methodology for grouping data as appropriate for its business, risk management and SMA regulatory capital calculation needs."

One could argue that if the aim of SMA was to "Standardize" Operational risk loss modelling in order to make capital less variable due to internal modelling decisions, then one can fail to see how this will be achieved with imprecise guidance such as the one provided above. One could argue that the above generic statement on criteria establishment basically removes the internal modelling framework of AMA and replaces it with internal heuristic (non-model based, nonscientifically verifiable) rules to "group" data. This has the potential to result in even greater variability in capital than was experienced with non-standardized AMA internal models. At least under AMA internal modelling, in principle the statistical models could be scientifically criticized.

-Ignoring the future: all forward-looking aspects of risk identification, assessment and mitigation such as scenarios and emerging risks have disappeared in the new consultation. This in effect introduces the risk of setting back the banking institutions in their progress towards a better understanding of the threats, even though such threats may be increasing in frequency and severity and the bank exposure to such threats may be increasing due to business practices, this cannot be reflected in the SMA framework capital. In that sense, the SMA is only backward looking.

SMA Fails to Utilize Range of Data Sources and Fails to Provide Risk Management Insight

Both Basel II and Basel III regulations emphasize the significance of incorporating a variety of loss data into Operational risk modelling and therefore ultimately into capital calculations. The four primary data sources to be included are Internal Loss Data, External Loss Data, Scenario Analysis and Business Environment and Internal Control Factors (BEICF), where only the first data source is in SMA and the others are not.

To understand the importance of BEICF data in the form of Key Risk Indicators (KRIs), Key Performance Indicators (KPIs) and Key Control Indicators (KCIs) we first briefly recall their properties.

A KRI is a metric of a risk factor. It provides information on the level of exposure to a given operational risk of the organization at a particular point in time. KRIs are useful tools for business lines managers, senior management and Boards to help monitor the level of risk taking in an activity or an organization, with regard to their risk appetite.

Performance indicators, usually referred to as KPIs, measure performance or the achievement of targets. Control effectiveness indicators, usually referred to as KCIs, are metrics that provide information on the extent to which a given control is meeting its intended objectives. Failed tests on key controls are natural examples of effective KCIs.

KPIs, KRIs and KCIs overlap in many instances, especially when they signal breaches of thresholds: a poor performance often becomes a source of risk: poor technological performance such as system downtime for instance becomes a KRI for errors and data integrity. KPIs of failed performance provide a good source of potential risk indicators. Failed KCIs are even more obvious candidates for preventive KRIs: a key control failure always constitutes a source of risk.

Indicators can be used by organizations as a means of control to track changes in their exposure to operational risk. When selected appropriately, indicators ought to flag any change in the likelihood or the impact of a risk occurring.

For financial institutions that calculate and hold operational risk capital under more advanced approaches such as the previous AMA internal model approaches, KPIs, KRIs and KCIs are advisable metrics to capture BEICF. While the definition of BEICF differs from one jurisdiction to another and in many cases is specific to individual organizations, these factors must:  be risk sensitive (here the notion of risk goes beyond incidents and losses);  provide management with information on the risk profile of the organization;  represent meaningful drivers of exposure which can be quantified; and  should be used across the entire organization.

While some organizations include the outputs of their risk and control self-assessment programs under their internal definition of BEICF's, indicators are an appropriate mechanism to satisfy these requirements, implying that there is an indirect regulatory requirement to implement and maintain an active indicator program, see discussion in [START_REF] Chapelle | The Importance Preventive KRIs[END_REF].

For instance, incorporating BEICF's into Operational risk modelling is a reflection of the modelling assumption that one can see Operational risk as a function of the control environment. If the control environment is fair and under control, large operational losses are less likely to occur and Operational risk can be seen as under control. Therefore, understanding the firm's business processes, mapping the risks on these processes and assessing how the controls implemented behave is the fundamental role of the Operational risk manager. However, the SMA does not provide any real incentive mechanism firstly for undertaking such a process and secondly for incorporating this valuable information into the capital calculation.

In terms of using pieces of information such as BEICF's and Scenario data, since under the SMA framework the level of model granularity is only at institution level, it does not easily lend itself to incorporation of these key Operational risk data sources.

To business lines managers, KRIs help to signal a change in the level of risk exposure associated with specific processes and activities. For quantitative modellers, key risk indicators are a way of including BEICF into operational risk capital. However, since BEICF data does not form a component of required data for the SMA model, there is no longer a regulatory requirement or incentive under the proposed SMA framework to make efforts to develop such BEICF data sources. Therefore, this not only reduces the effectiveness of the risk models through the loss of a key source of information, but in addition the Documents de travail du Centre d'Economie de la Sorbonne -2016.64

utility of such data for risk management practitioners and managers is reduced as this data is no longer collected with the same required scrutiny, including validation, data integrity and maintenance and reporting that was previously required for AMA internal models using such data.

These key sources of Operational risk data are not included in the SMA and furthermore cannot easily be incorporated into an SMA framework even if there were a desire to do so due to the level of granularity implied by the SMA. This makes capital calculations less risk sensitive. Furthermore, the lack of scenario based data incorporated into the SMA model makes it less forward looking and anticipatory as an internal model based capital calculation framework.

SMA can be a super additive capital calculation

The SMA seems to have the unfortunate feature that it may produce capital at a group level compared to the institutional level in a range of jurisdictions which has the property that it is super additive. To understand this, the following two examples should be helpful.

In each case, consider two banks with identical BI and identical Loss Component (LC). However, the first bank has only one entity where as the second has two entities. The two entities of the second bank have the same BI and the same LC, and those are equal to both half the BI and half the LC of the first joint bank.

In case one (Table 1) we consider the situation of a bucket shift, where the SMA capital obtained for the joint bank is 5,771 million while the sum of the SMA capital obtained for the two entities of the second bank is only 5,387 million. In this example, the SMA does not capture a diversification benefit, on the contrary, it assumes that the global impact of an incident is larger than the sum of the parts.

In the second case (Table 2) we consider no bucket shift between the joint bank and the two entity bank.

Again in this case we see that the joint bank has an SMA capital of 11,937mil, whereas the two entity bank has an SMA capital of 10,674mil capital. Again there is a super additive property. To conclude this section we state a mathematical expression that a bank could utilize in business structure planning to decide in the long term if it will be advantageous under the new SMA framework to split into two entities (or more) or remain in a given joint structure, according to the cost of funding Tier I SMA capital.

For illustration we can assume the joint institution is simply modelled by a Poisson-Lognormal model Poisson(λJ)-Lognormal(µJ,σJ) with parameters sub-indexed by J for the joint institution and a BIc for the joint institution denoted by BIcJ. Furthermore, we assume that if the institution had split into two separate entities for Tier I capital reporting purposes then each would have its own stylized annual loss modelled by two independent Poisson-Lognormal models: Entity 1 denoted Poisson(λ1)-Lognormal(µ1,σ1); and Entity 2 denoted by Poisson(λ2)-Lognormal(µ2,σ2) with BIc1 and BIc2, respectively.

Consider the long term SMA capital behavior averaged over the long term history of the SMA capital for each case, joint and disaggregated business models. Then, according to a long term analysis of these stylized Poisson-Lognormal models, the following expressions can be used to determine the point at which the SMA capital would be super additive. If it is super additive in the long term, it would indicated that there is therefore an advantage to split the institution in the long-run into disaggregated separate components. Furthermore, the expression provided allows one to maximize the long term SMA capital reduction that can be obtained under such a partitioning of the institution.

Note: the following calculations are based on truncated Poisson expected loss expressions for Lognormal models, see details in [START_REF] Peters | Advances in Heavy Tailed Risk Modeling: A Handbook of Operational Risk[END_REF].

Joint Institution (Long term behavior):

Long Term Average Loss Component (LTALC) is given by Therefore the Long term ILM (LTILM) is given by

                                                J J J J J J J J J J J J J J J J J J H L LTALC                   ln 2 1 exp 5 ln 2 1 exp 7 2 1 exp 7 ) , , ( 2
           J J J J J J J BIC LTALC LTIM ) , , ( 1 ) 1 exp( ln ) , , (      
These are then used to calculate the Long Term SMA (LTSMA) which will be an explicit function of LDA model parameters (λ J ,µ J ,σ J ) according to Long Term Average Loss Component (LTALC) is given by Therefore the Long term ILM (LTILM) is given by

       5 - 2 Buckets if ), , , ( Mln) 110 (
                                                i i i i i i i i i i i i i i i i i i H L LTALC                  
           i i i i i i i BIC LTALC LTIM ) , , ( 1 ) 1 exp( ln ) , , (  
   
These are then used to calculate the Long Term SMA (LTSMA) which will be an explicit function of LDA model parameters (λ i ,µ i ,σ i ) according to

       5 - 2 Buckets if ), , , ( Mln) 110 (
2 1 1 1             LTSMA LTSMA LTSMA J J J ) , , ( ) , , ( 2 2
Using this stylized condition, banks may be able to determine for instance if in the long term it would be economically efficient to split their institution into 2 or more separate entities. Furthermore, they can use this expression to optimize the capital reduction for each of the individual entities, relative to the combined entities SMA capital. Hence, what we show here is the long term average behavior which will be the long run optimal conditions for split or merge.

We also note that due to the volatility of the loss experience, and the BI figures over time it could also be the case that a given institution could have an SMA capital which would be switching between super vs sub-additivity of the capital over time. This would imply that SMA model could provide a time varying incentive and disincentive in merge and split depending on the current environment for capital funding. Proposition: a Standardization of AMA In the following section the recommendations made are based on detailed discussions proposed in [START_REF] Cruz | Fundamental Aspects of Operational Risk and Insurance Analytics: A Handbook of Operational Risk[END_REF], [START_REF] Peters | Advances in Heavy Tailed Risk Modeling: A Handbook of Operational Risk[END_REF] and the preprint [START_REF] Peters | Should AMA be Replaced with SMA for Operational Risk[END_REF].

SMA cannot be considered as an alternative to AMA models. We suggest that AMA should not be discarded, but instead could be improved by addressing its current weaknesses. It should be standardized! Details of how a rigorous and statistically robust standardization can start to be considered, with practical considerations, are provided below.

Rather than discarding all Operational risk modelling as allowed under the AMA, instead the regulator could make a proposal to standardize the approaches to modelling based on the accumulated knowledge to date of Operational risk modelling practice.

We propose one class of models that can act in this manner and allow one to incorporate the key features offered by AMA LDA type models which involve internal data, external data, BEICF's and Scenarios, with other important information on factors that the SMA method and OpCar approaches have tried to achieve but failed. As noted in this response, one issue with the SMA and OpCar approaches is that they try to model all Operational risk processes at the institution or group level with a single LDA model and simplistic regression structure, this is bound to be problematic due to the very nature and heterogeneity of Operational risk loss processes. In addition it fails to allow for incorporation of many important Operational risk loss process explanatory information sources such as BEICF's which are often no longer informative or appropriate to incorporate at institution level, compared to individual Business Line/Event Type (BL/ET) level.

We propose a standardization of the AMA internal models to remove the wide range of heterogeneity in model type. Our recommendation involves a bottom up modeling approach where for each BL/ET Operational risk loss process we model the severity and frequency components in an LDA structure which is comprised of a hybrid LDA model with factor regression components, allowing to include the factors driving operational risks in the financial industry, at a sufficient level of granularity, while utilizing a class of models known as the Generalized Additive Models for Location, Shape and Scale (GAMLSS) in the severity and frequency aspects of the LDA framework. The class of GAMLSS models can be specified to make sure that the severity and frequency families are comparable across institutions, allowing both risk-sensitivity and capital comparability. We recommend in this regard Poisson and Generalized Gamma classes for the family of frequency and severity model as these capture all typical ranges of loss model used in practice over the last 15 years in Operational risk, including Gamma, Weibull, Lognormal, Pareto type severities.

Standardizing Recommendation 1:

This leads us to the first standardizing recommendation relating to the level of granularity of modelling in Operational risk. The level of granularity of the modelling procedure is important to consider when incorporating different sources of Operational risk data such as BEICF's and scenarios and this debate has been going for the last 10 years, with discussion on bottom-up versus top-down based Operational risk modelling, see overview in [START_REF] Cruz | Fundamental Aspects of Operational Risk and Insurance Analytics: A Handbook of Operational Risk[END_REF] and [START_REF] Peters | Advances in Heavy Tailed Risk Modeling: A Handbook of Operational Risk[END_REF]. We advocate that a bottom-up based approach be recommended as the standard modelling structure as it will allow for greater understanding and more appropriate model development of the actual loss processes under study. Therefore, we argue that sticking with the 56 BL/ET structure of Basel II is in our opinion best for a standardizing framework with a standard aggregation procedure to institution level / group level. We argue that alternatives such as the SMA and OpCar approaches that are trying to model multiple different featured loss processes combined into one loss process at the institution level will be bound to fail as they need to capture high frequency events, as well as high severity events, this in principle is very difficult if not impossible to capture with a single LDA model at institution level and Documents de travail du Centre d'Economie de la Sorbonne -2016.64 should be avoided. Furthermore, such a bottom-up approach allows for greater model interpretation and incorporation of Operational risk loss data such as BEICF's.

Standardizing Recommendation 2:

This brings us to our second recommendation for standardization in Operational risk modelling. Namely, we propose to standardize the modelling class to remove the wide range of heterogeneity in model type. We propose a standardization that involves a bottom up modelling approach where each BL/ET level of Operational risk loss process we model the severity and frequency components in an LDA structure which is comprised of a hybrid LDA model with factor regression components. The way to achieve this is to utilise a class of GAMLSS regression models for the severity and frequency model calibrations.

That is two GAMLSS regression models are developed, one for the severity fitting and the other for the frequency fitting. This family of models is flexible enough in our opinion to capture any type of frequency or severity model that may be observed in practice in Operational risk data whilst incorporating factors such as BEICF's (Key Risk Indicators, Key Performance Indicators, Key Control Indicators) naturally into the regression structure. This produces a class of hybrid factor regression models in an Operational risk LDA family of models that can easily be fit, simulated from and utilised in Operational risk modelling to aggregate to the institution level. Furthermore, as more data years of history become available, the incorporation of time-series structure in the severity and frequency aspects of each loss process modelling can be naturally incorporated in a GAMLSS regression LDA framework.

Standardizing Recommendation 3:

The class of models considered for the conditional response in the GAMLSS severity model can be standardized. There are several possible examples of such models that may be appropriate [START_REF] Cruz | Fundamental Aspects of Operational Risk and Insurance Analytics: A Handbook of Operational Risk[END_REF] and [START_REF] Ganegoda | A scaling model for severity of operational losses using generalized additive models for location scale and shape (GAMLSS)[END_REF], however we advocate for the severity models that the class of models be restricted in regulation to one family, the Generalized Gamma family of models, see details in [START_REF] Peters | Should AMA be Replaced with SMA for Operational Risk[END_REF] where these models are developed in an LDA hybrid factor GAMLSS model. This work shows that such models are appropriate for Operational risk as they admit special members which correspond to the LogNormal, Pareto, Weibull and Gamma. All of these models are popular Operational risk severity models used in practice and represent the range of best practice by AMA banks as observed in the recent survey [BCBS160b]. Since the Generalized Gamma family contains all these models as special sub-cases it means that banks would only have to ever fit one class of severity model to each BL/ET LDA severity profile, and the most appropriate family member would be resolved in the fitting through the estimation of the shape and scale parameters, in such a manner that if a LogNormal model was appropriate it would be selected, where as if a Gamma model were more appropriate it would also be selected from one single fitting procedure.

Furthermore, the frequency model could be standardized as a Poisson GAMLSS regression structure as the addition of explanatory covariates and time varying and possible stochastic intensity allow for a flexible enough frequency model for all types of Operational risk loss process.

Standardizing Recommendation 4:

The fitting of these models should be performed in a regression based manner in the GAMLSS framework, which incorporates truncation and censoring in a penalized maximum likelihood framework, see [START_REF] Stasinopoulos | Generalized additive models for location scale and shape (GAMLSS) in R." Stasinopoulos DM, Rigby RA[END_REF]. We believe by standardizing the fitting procedure to one that is statistically rigorous, well understood in terms of the estimator properties and robust when incorporating a censored likelihood appropriately, will remove the range of heuristic practices that has arisen in fitting models in Operational risk. The penalized regression framework, based on L1 parameter penalty will also allow for shrinkage methods to be used to select most appropriate Documents de travail du Centre d'Economie de la Sorbonne -2016.64 explanatory variables in the GAMLSS severity and frequency regression structures.

Standardizing Recommendation 5:

The standardization in form of Bayesian versus Frequentist type models be left to the discretion of the bank to decide which version is best for their practice. However, we note that under a Bayesian formulation, one can adequately incorporate multiple sources of information including expert opinion and scenario based data, see discussions in [START_REF] Cruz | Fundamental Aspects of Operational Risk and Insurance Analytics: A Handbook of Operational Risk[END_REF] and [START_REF] Peters | Dynamic operational risk: modeling dependence and combining different sources of information[END_REF] and [START_REF] Shevchenko | Modelling operational risk using Bayesian inference[END_REF].

Standardizing Recommendation 6:

The sets of BEICF's and factors to be incorporated into each BL/ET LDA factor regression model for severity and frequency should be specified by the regulator. There should be a core set of factors to be incorporated by all banks which include BEICF's and other factors to be selected. The following types of KRI's categories can be considered in developing the core family of factors (see [START_REF] Chapelle | The Importance Preventive KRIs[END_REF]):

Exposure Indicators: any significant change in the nature of the business environment and in its exposure to critical stakeholders or critical resources. Flag any change in the risk exposure.

Stress Indicators: any significant rise in the use of resources by the business, whether human or material. Flag any risk rising from overloaded humans or machines.

Causal Indicators: metrics capturing the drivers of key risks to the business. The core of preventive KRIs.

Failure Indicators: poor performance and failing controls are strong risk drivers. Failed KPIs and KCIs.

In this approach, a key difference is that instead of fixing the regression coefficients for all banks (as is the case for SMA and OpCar) pretending that all banks have the same regression relationship as the entire banking population, instead one should standardize the class of factors. Specify explicitly how they should be collected, the frequency and then specify that they should be incorporated in the GAMLSS regression. This will allow each bank to then calibrate the regression model to their loss experience through a rigorous penalized Maximum Likelihood procedure. With strict criterion on cross validation based testing on the amount of penalization admitted in the regression when shrinking factors out of the model. This approach has the advantage that banks will not only start to better incorporate in a structured and statistically rigorous manner the BEICF information into Operational risk models, but they will be forced to better collect and consider such factors in a principled manner.

Figure 2 .

 2 Figure 2. Ratio of the SMA capital to the long term average (Case 2)

  du Centre d'Economie de la Sorbonne -2016.64

Figure 4 .

 4 Figure 4. Ratio of the SMA capital to the long term average (Case 3 -SMA sensitivity)

  denotes the Normal CDF, L is Euro 10 million and H is Euro 100 million. Documents de travail du Centre d'Economie de la Sorbonne -2016.64

  .) denotes the Normal CDF, L is Euro 10 million and H is Euro 100 million.

  Documents de travail du Centre d'Economie de la Sorbonne -2016.64

Table 1 .

 1 Case 1: σ = 2.5.

	MODEL	Parameters	Mean Annual Loss	Annual Loss Capital
			(Euro Million)	(99.9% VaR)
				(Euro Million)
	Poisson-Lognormal	(λ, σ , µ) =	15 (small)	260 (small)
	Loss Component	(10, 2.5, {10; 12; 14})	136 (medium)	1,841 (medium)
	Poisson-Gamma	(λ, α, β) =	769 (Large)	14,610 (Large)
	Loss Component	(990, 1, {10,000;100,000; 500,000})

Table 2 .

 2 Case 2: σ = 2.8.

	MODEL	Parameters	Mean Annual	Annual Loss Capital
			Loss	(99.9% VaR)
			(Euro Million)	(Euro Million)
	Poisson-Lognormal	(λ, σ , µ) =	21 (small)	772 (small)
	Loss Component	(10, 2.8, {10; 12; 14})	181(medium)	5,457 (medium)
	Poisson-Gamma	(λ, α, β) =	1,101(Large)	41,975 (Large)
	Loss Component	(990, 1, {10,000;100,000; 500,000})

Documents de travail du Centre d'Economie de la Sorbonne -2016.64

Table 4

 4

	. Case 3: Varying σ.
	MODEL	Parameters
	Poisson-Lognormal	(λ, σ , µ) = (10, {2; 2.25; 2.5; 2.75; 3} , 14)
	Loss Component
	Poisson-Gamma	(λ, α, β) = (990, 1, 500,000)
	Loss Component

Table 5 :

 5 Super-additivity issue illustrated (in million)

	Bucket Shift
		Component Group
	Bank 1	BIC LC	6920 BI=32,000 4000
		SMA	5771
		Component Entity 1	Entity 2
	Bank 2	BIC LC	3120 2000	3120 BI=32,000 2000
		SMA	2694	2694
		Sum SMA	5387

Documents de travail du Centre d'Economie de la Sorbonne -2016.64

Table 6 :

 6 Super-additivity issue illustrated (in million)

	Same Bucket
		Component Group
				BI =
	Bank 1	BIC	17940	70,000
		LC	4000
		SMA	11937
		Component Entity 1	Entity 2
	Bank 2	BIC LC	7790 2000	7790 BI=70,000 2000
		SMA	5337	5337
		Sum SMA	10674

Figure 1. Ratio of the SMA capital to the long term average (Case 1) Documents de travail du Centre d'Economie de la Sorbonne -2016.64

Documents de travail du Centre d'Economie de la Sorbonne -2016.64