Xavier Milhaud

Victorien Poncelet

Clément Saillard
email: clement.saillard@bnpparibas.com.

Operational choices for risk aggregation in insurance: PSDization and SCR sensitivity

Keywords: Solvency II, risk aggregation, positive semi-definite, Rebonato-Jäckel

come

Motivation

When measuring the insurer's exposure to numerous risks, in particular to determine the own funds requirement (or Solvency Capital Requirement in insurance, denoted further by SCR g , and representing the 99.5 th percentile of the aggregated loss distribution), one of the most sensitive steps is the modelling of the dependence between those risks. This question is of course important and, quite deservedly, has recently attracted attention from the scientific community, see for instance the works by [START_REF] Georgescu | Explicit Solutions to Correlation Matrix Completion Problems, with an Application to Risk Management and Insurance[END_REF], [START_REF] Cifuentes | Operational risk and the Solvency II capital aggregation formula: implications of the hidden correlation assumptions[END_REF], [START_REF] Bernard | Risk aggregation with dependence uncertainty[END_REF], [START_REF] Cheung | Bounds for Sums of Random Variables when the Marginal Distributions and the Variance of the Sum are Given[END_REF], [START_REF] Devineau | Risk aggregation in Solvency II: How to converge the approaches of the internal models and those of the standard formula?[END_REF], [START_REF] Filipovic | Multi-level Risk Aggregation[END_REF], [START_REF] Denuit | Stochastic bounds on sums of dependent risks[END_REF], and references therein. This aggregation step allows to take into account mitigation, or the potentiality of those individual risks to occur simultaneously. According to the European Directive Solvency II, there exist two main approaches to compute aggregated risk measures considering the dependence structure between risks. In the first case, this aggregation is performed through a variance-covariance approach via the "Standard Formula", that is

SCR g = i∈ 1,n j∈ 1,n ρ ij × SCR i × SCR j , (1.1)
where SCR i is the 99.5 th percentile of the loss associated to risk i, and ρ ij is the linear correlation such that ρ ij = Cov(SCR i , SCR j )/ V ar(SCR i )V ar(SCR j ). This technique was shown to be valid for elliptical loss vectors, which is not the case in full generality 1 .

Another possibility for insurers is to calculate SCR g thanks to their internal model, once the latter has been approved by supervisors. In this case, insurers usually work with copulas for the aggregation of risk factors in order to obtain the full distribution of losses.

Indeed, copulas can model the most general situation for dependence, as shown by the well known Sklar's theorem. In practice, internal models require the implementation of these successive steps:

1. calibration of marginal distributions for each risk factor (e.g. equity, interest rates);

2. aggregation of risk factors by modelling their dependence through a copula;

3. fitting of proxy functions to link the risk factors to their associated loss (e.g. through curve-fitting in ALM forecasts), which leads to get the entire distribution of the aggregated loss. Taking the 99.5 th percentile of this distribution then allows to evaluate the aggregated SCR (SCR g with our notations).

The aggregation thus requires a correlation matrix as input, whatever the technique.

This matrix, whose dimensions can be huge in practice (e.g. 1000x1000, i.e. with around 500 000 different values), is full of correlation coefficients that can result from empirical statistical measures, expert judgments or automatic formulas. By the way, it is thus rarely positive semi-definite (PSD): this is what is commonly called a pseudo-correlation matrix. Unfortunately, this matrix cannot be used directly to aggregate risks. Indeed, both variance-covariance and copula aggregation techniques require the correlation matrix to be PSD (that is with all eigenvalues ≥ 0) for the following main reasons:

• Coherence: it is a well-known property that correlation matrices are PSD. The presence of negative eigenvalues points a logical mistake made while establishing the coefficients of the matrix. For instance, consider the case of a 3x3 matrix : if the coefficients indicate a strong positive correlation between the first and second risk factors, a strong positive correlation between the second and third risk factors, but a strong negative correlation between the first and third risk factors, this will generate a negative eigenvalue corresponding to the coherence mistake made;

• Prudence: taking more risk could decrease the insurer's SCR g if there exists one negative eigenvalue associated to an eigenvector with positive coefficients. For instance, consider the loss vector (100Me, 10Me, 40Me) and the correlation matrix Using the variance-covariance approach, the riskier situation with losses (106.20Me; 16.20Me; 44.81Me) leads to a lower SCR g (70.48Me against 74.16Me)!

• Ability to perform simulations: in the copula approach, the input correlation matrix has to be PSD to apply Choleski decomposition. This is necessary for Gaussian or Student vectors, which are the most famous cases for such tasks in practice.

Using PSD correlation matrices is therefore crucial, which explains why it is explicitly required by Delegated Rules (EU) 2015/35 (Annex XVIII) of the Solvency II regulation.

In this view, insurers apply algorithms on their pseudo-correlation matrix in order to make it become PSD: this is the so-called PSDization process. Most common algorithms can be separated into three categories: Alternating Projections, Newton and Hypersphere algorithms. One focuses in this paper on the Rebonato-Jäckel algorithm, which belongs to the latter family (see Section 3 for further details about the motivation of this choice).

Given this framework, one wonders how the SCR is impacted by standard operations on the correlation matrix used for risk aggregation. Our interest lies in studying operational choices such as weighting the correlation coefficients during PSDization (to reflect the confidence experts may have on these coefficients), permutating some columns in the matrix (equivalently reorder the risks before aggregation), or adding one dimension (analogously add a business line, with the constraint that it should not impact SCR g at the end). Numerical examples support the main idea of the paper: transformations of the matrix, with low or null theoretical impact on SCR g , sometimes lead to unexpected changes of this global SCR. For large insurance companies, this is all the more important that a 1% change of SCR g can cost millions of euros in terms of capital. This would strongly affect the Return On Equity index, an essential profitability indicator for investors.

The paper is organized as follows: Section 2 introduces the pseudo-correlation matrices to be considered hereafter. Section 3 describes most common PSD algorithms, and motivates our choice. Thanks to some toy examples, Section 4 illustrates to which extent PSDization leads to modify the initial pseudo-correlation coefficients. The cases of higher risk matrix dimensions, weighted correlation coefficients and risk permutations are studied. Finally, real-life sensitivities are assessed in Section 5 thanks to the use of genetic algorithms and simulations, and provides with some interesting results concerning the aforementioned operations and their impact on the global SCR of the company.

2 Pseudo-correlation matrices under study

Correlation matrices of the Standard Formula

Before studying the algorithms which enable to make a matrix be PSD when using an internal model, it is verified that the matrices defined by the standard formula are already PSD. As a reminder, the Standard Formula given by the regulation states that risks should be aggregated in a bottom-up approach, with a tree-based structure. This means that individual SCRs first have to be assessed, each one corresponding to a module (Life, Non Life, and so on). Solvency II texts then define several correlation matrices for each level of aggregation. Table 1 shows that the eigenvalues of these matrices are all positive, meaning that they are PSD. Except the global matrix which can be found in the Directive, all matrices are described in the Delegated Acts, see references in the first column of

Notations and correlation matrices under study

Throughout the paper, G xy denotes the initial pseudo-correlation matrix to be PSDized, where x refers to the matrix dimension and y is the number of the example. When using weights to apply to the correlation coefficients during PSDization, a weighting matrix H xy is defined. Then, we denote by S P A xy the PSDized matrix obtained from G xy using the Alternated Projections algorithm; S N xy the PSDized matrix with the Newton algorithm; S xy the PSDized matrix with the Hypersphere algorithm and S H xy the PSDized matrix with the Hypersphere algorithm using a weighting matrix H. The use of a weighting matrix enables to give more importance to one or several correlation coefficients with respect to other coefficients when making the correlation matrix PSD. This is very useful for insurance companies, since some coefficients can have an bigger impact on the final capital requirement, and the aim would be that PSDization modifies these coefficients as little as possible. It must be noted that some algorithms (Newton, Hypersphere) can be extended to use constraints on the correlation coefficients, such as ρ min ij ≤ ρ ij ≤ ρ max ij . However, these extensions are left for future research because of two main reasons: very few practitioners use them, and they cause non-trivial theoretical and practical issues (there could be no solution, i.e. PSDized matrix, according to the constrained set).

Our examples are built with the same simple idea: assess the impact of PSDization combined to classical operational choices on diverse situations to reflect the real world.

We thus consider correlation matrices of various dimensions, with positive and negative eigen values, and with high heterogeneity regarding their individual correlation coefficients (positive, negative, far or close to extreme values -1 or +1).

The seven examples studied all along the paper are listed in Appendix A, including the 10-dimension correlation matrix that has been created manually. This example was designed with the aim of having a certain coherence with the reality of risk aggregation in insurance.

The first risk factor, say X 1 , refers to the risk that interest rates decrease (X 2 represents the risk of interest rates rising). X 3 corresponds to unexpected high expenses, X 4 relates to the bad assessment of the level of expenses, X 5 is the risk of spreads increasing. The risk that market stocks drop is given by X 6 , X 7 accounts for the longevity risk, X 8 is the mass lapse risk, X 9 corresponds to the reserve risk in Health business line ; and X 10 is the reserve risk in short-term disability.

Correlation coefficients were determined either by statistical measures built on historical data on financial markets, or by expert opinions. To read appropriately the matrix, the linear correlation between X 1 and X 2 is the coefficient ρ 12 (or ρ 21 ), located on line 1 column 2 (of course, the coefficient equals -1 in this case). Finally, our pseudo-correlation matrix looks like

G 101 =                       1 -1 -0.28 0
                      .

Selection of an adequate algorithm for PSDization

PSDization algorithms aim to build the nearest PSD matrix to an initial non-PSD matrix, where the notion of "nearest" is detailed in the sequel. This section briefly presents the main families of PSDization algorithms, and justifies our choice to work with the Hypersphere (or Rebonato-Jäckel) algorithm. Interesting recent works on this topic and more details can be found in [START_REF] Cutajar | Actuarial Risk Matrices: The Nearest Positive Semidefinite Matrix Problem[END_REF].

3.1 Families of algorithms

Alternating Projections

The Alternating Projections (AP) algorithm, introduced by [START_REF] Higham | Computing the nearest correlation matrix -a problem from finance[END_REF], enables to find the nearest correlation matrix under the W-norm, defined by

∀A ∈ R n×n , A W = W 1 2 AW 1 2 2 ,
where

∀A ∈ R n×n , A 2 2 = (i,j)∈ 1;n 2 a 2 ij = tr(AA T
). This norm is also called the Frobenius norm, and W is a matrix with positive coefficients.

The AP algorithm corresponds to a linear optimization, projecting alternatively the matrix obtained at each step on two convex closed subsets of the matrix space R n×n . It enables in particular to show the uniqueness of the solution under this type of norm.

However, the W-norm does not in general correspond to the norm insurers may be inter-

ested in. The H-norm, defined by ∀A ∈ R n×n , A 2 H = (i,j)∈ 1;n 2 h ij a 2 ij
where H ∈ R n×n , offers more flexibility to weight coefficients according to their materiality on SCR g , or according to the confidence level one has on the coefficients. The W-norm and the H-

norm only coincide when W is diagonal (W = diag(w i ) i∈ 1;n ) and H is a rank-1 matrix (H = [(w i w j ) 1 2 ]
). Thus, the AP algorithm lacks flexibility in that it does not enable to use another general matrix to weight freely each individual correlation coefficient.

Newton algorithm

Newton algorithms with such applications were introduced by Qi and Sun (2006). They were initially designed for the traditional 2-norm, and are therefore computationally quicker than the Alternating Projections. However, their extension to the most general case (i.e. H-norm) requires optimization techniques such as the Uzawa method. Unfortunately, Uzawa method implies optimization within optimization, and makes the overall algorithm much more time consuming, as well as much more complex to interpret.

Rebonato-Jäckel algorithm

Belonging to the family of Hypersphere algorithms, Rebonato-Jäckel method was initially introduced in Rebonato and Jäckel (1999). Since then, it has been extensively studied in the literature, and many papers have proved its efficiency in reaching a robust solution.

Its large success follows from the following theorem (see the proof in Jäckel ( 2002)): any correlation matrix ρ can be written as

ρ = BB T
where the coefficients of the matrix B ∈ R n×n can be written as:

∀(i, j) ∈ 1; n × 1; n -1 , B ij = cos(θ ij ) j-1 k=1 sin(θ ik ); ∀i ∈ 1; n , B in = n-1 k=1 sin(θ ik ).
In addition, the angular vector θ is unique if :

∀(i, j) ∈ 1; n × 1; n -1 , θ ij ∈ [0, π]; ∀i ≤ j, θ ij = 0.
The Hypersphere algorithm thus consists in looking for the solution matrix under the above-mentioned form. It offers several advantages; in particular it is simple to use, easily understandable, and is the most widely used algorithm in the bank and insurance sectors.

Moreover, it allows the use of the H-norm and converges fairly quickly. However, its main weakness lies in that it sometimes converges to local minima, and therefore does not guarantee that the output is the nearest PSDized correlation matrix. This drawback has to be kept in mind, since it may cause other side effects. Let us mention for instance the fact that the order in which risk factors are considered in the correlation matrix matters, although it should not (see Sections 4 and 5).

Choice of the algorithm for the rest of the paper

To check for the robustness when performing PSDization with these algorithms, one first compares the distances between the initial pseudo-correlation matrix and its PSDized version in the three different cases. Of course, the lower this distance the better the algorithm. We use the Frobenius-norm (see 3.1.1), since it is common for all algorithms.

Results about PSDized versions for each example are detailed in Appendix B : notice that the PSDized correlation matrices are the same when the dimension remains low, whatever the algorithm under consideration. More precisely, the three algorithms give very similar results with PSDized versions of G 31 , G 32 , G 41 , G 42 , G 51 , and G 52 (same coefficients, up to 10 -4 ). On the contrary, the PSDized versions of G 101 are slightly different depending on the algorithm used. As an illustration, we get the following distances:

||S P A 101 -G 101 || 2 = ||S N 101 -G 101 || 2 = 1.211 ≤ ||S 101 -G 101 || 2 = 1.213.
In this example, it seems that the two first algorithms give better results. The Rebonato-Jäckel algorithm is likely to have selected a locally-optimal solution. Despite not being the best technique in this particular case, we will use the latter for three main reasons in coming analyses : i) distances do not seem to be significantly different from one method to another, ii) the Rebonato-Jäckel algorithm enables to easily integrate confidence weights to the individual correlation coefficients in practice (through the H-norm), which is a key point, iii) the Rebonato-Jäckel algorithm is fast and easy to interpret. Indeed, experts know that some initial coefficient values can be particularly reliable, or they can anticipate that some of them will have a significant impact on the global SCR (SCR g ).

One

PSDization example: the 10-dimensioned matrix The three last values equal 10 -5 , which is the lower bound defined in the algorithm. This means that the three negative eigen values of G 101 (see Table 2) have been replaced by the lowest possible value. One then measures the standardized distance between the initial pseudo-correlation matrix G 101 and its PSDized version S H 101 :

D H 101 = ||S H 101 -G 101 || H ||G 101 || H = 19, 7%.
It thus seems that PSDization globally had a great impact on the pseudo-correlation matrix. Some coefficients were strongly modified, see for instance ρ 65 (fictional correlation between equity and spreads). Indeed, ρ 65 equals 0.83 at the beginning in G 101 , but is close to 0.54 after PSDization in S H 101 . Such an example highlights the need for the insurer to check all the modifications, to gain control and monitor her internal model.

Sensitivity of the matrices to PSDization

In this section, one would like to illustrate how the correlation matrix can be modified when performing PSDization by the Rebonato-Jackël algorithm, with toy examples. In particular, we investigate how the coefficients of G 31 (Appendix A) change during the sole PSDization. We also investigate the evolution of individual correlation coefficients when considering other classical operations for practitioners: permutations of some coefficients before PSDization, change of matrix dimension (before PSDization), or weights given to the correlation coefficients during PSDization. In this view, we consider the following initial weighting matrix

H init 31 =     1 0.1 0.9 0.1 1 0.5 0.9 0.5 1     .
These operations mainly correspond to the decisions actuaries have to make when developing internal models for risk aggregation. Note that the impact on the capital requirement can be substantially different from the impact in terms of matrix norm, according to the respective importance of the loss marginals. This impact will be studied in Section 5.

Impact of permutations

To study how permutations of risks defining the correlation matrix impacts the standard PSDization process, one first considers the permutation σ such that

    1 2 3 ↓ ↓ ↓ 2 3 1     .
As a result, one obtains the following modifications on the correlation coefficients: Examining the coefficients, we notice that they are significantly modified: first by the PS-Dization process itself, but also by the permutation of risks. This last result is surprising, since there should be no theoretical impact with this operation. However, because our algorithm presents local minima issues, the choice of the order of risk factors (arbitrarily made by the insurer) matters when performing PSDization.

To figure out more comprehensively the impact of permutations, the best would be to look at the exhaustive list of permutations for a given pseudo-correlation matrix. Recall that a D-dimensioned matrix admits D! permutations, and let us consider the example G 51 . Figure 1 shows the Frobenius distance to the initial matrix for the 5! permutations of G 51 , knowing that this distance between G 51 and S H 51 initially equals 1.68 without any permutation. The Frobenius norm is clearly not the norm that the algorithm optimizes, but it simply illustrates to which extent the solution matrix S H 51 is modified. Two remarks can be made here. First, the distances follow a block pattern which is due to the order of the permutations. Second, the permutations do not always lead to increase the distance to the initial pseudo-correlation matrix.

Adding a risk: higher the matrix dimension

Another arbitrary element chosen by the actuaries of the company is the number of risk factors to be aggregated. In some cases, it might be necessary to model many risk factors, according to the use of the internal model that is made by the business units (need to model many lines of business when modelling the reserving loss factor for instance). These choices have to be made for risk factors which are not material at the Group level, even if they can be important for the concerned subsidiaries. Nevertheless, these choices will impact the final SCR (SCR g ) through the modification brought to the overall correlation matrix during PSDization. Still based on the same example, let us consider the following case:

    1 -0.9 -0.5 -0.9 1 -0.5 -0.5 -0.5 1     →       1 -0.9 -0.5 0.1 -0.9 1 -0.5 0.1 -0.5 -0.5 1 0.1 0.1 0.1 0.1 1      
As can be noticed, the correlation is low between the added risk and others.

Coef. Before PSDization Direct PSDization PSDization with higher dimension ρ 12 -0.9 -0.585 -0.577 ρ 13 -0.5 -0.473 -0.471

ρ 23 -0.5 -0.437 -0.435
Direct PSDization obviously gives the same results, whereas correlation coefficients are slightly modified after PSDization when introducing the new risk. Changes on these coefficients are hard to anticipate, but it seems that the impact is lower in this case than with permutations. A natural question would be to understand whether the value of the correlation coefficient that was added is key to explain the modifications obtained in the PSDized correlation matrix. Figure 2 shows this impact on the Frobenius norm, with a new risk whose correlation coefficients vary from 0.1 to 1. Results are intuitive: the bigger the coefficients added, the larger the Frobenius norm. Indeed, PSDization is a whole process that takes into account every coefficient, including the one that was added.

Impact of confidence weights

Finally, the choice of the weights associated to the terms of the correlation matrix, which somewhat represents the confidence level given by experts to the individual correlation coefficients, can also have a significant impact on the PSDized matrix. To illustrate this, still keeping in mind the example G 31 with the initial weights listed in H init 31 , we consider the new following weights: As it can be seen, correlation coefficients significantly vary. Notice that the modifications are all the more far from the standard PSDized coefficient that the weight is low, which is in line with intuition. Indeed, more weight on some correlation coefficients means more importance in the PSDization process, and thus less modification for them (so as to minimize the Frobenius norm). To generalize and understand deeper to which extent the weights could impact the correlation coefficients after PSDization, Figure 3 shows the Frobenius norm between the solution S H 51 and the initial matrix G 51 , with weights varying from H 51 to a limit weighting matrix given by

    ω 11 ω 12 ω 13 ω 21 ω 22 ω 23 ω 31 ω 32 ω 33     =     1 0.2 0.8 0.
H limit 51 =         
1 0.5 0.5 0.5 0.5 0.5 1 0.2 0.5 0.5 0.5 0.2 1 0.6 0.6 0.5 0.5 0.6 1 0.1 0.5 0.5 0.6 0.5 1

         .
This limit weighting matrix is used for illustration purposes only. It corresponds to increase linearly by +0.4 the lowest weights of the matrix H 51 , while decreasing highest weights by the same factor. Figure 3 shows that the Frobenius norm decreases as the weighting matrix is distorted towards the limit H limit 51 , and a closer analysis reveals that this phenomenon is mainly due to the correlation coefficients ρ 24 , ρ 25 , ρ 34 and ρ 35 of G 51 (and their transposed coefficients) which are much less modified than with the initial weighting matrix since their weights are significantly increased: from 0.1 to 0.5 and respectively from 0.2 to 0.6.

Summary

To put it in a nutschell, Figure 4 shows the impact of permutations and weights (the two most prominent operations) on the Frobenius norm, in the conditions stated above. It shows that the impact on the Frobenius norm is more important when weights vary than when the order of risk factors is changed. After these primary illustrations, we now move to the analysis of such transformations on the capital requirement, SCR g .

Analysis of SCR sensitivity

One has just highlighted the importance of PSDization on the final correlation matrix to be used to assess the insurer's own funds requirement. The correlation coefficients chosen by the experts, or even those defined by statistical means can be significantly modified. The aim of this section is to provide with some real-life sensitivities concerning the computation of the global SCR thanks to internal models. One would like to see SCR g as a function of the main parameters in the actuary's hand. To make it, one applies genetic algorithms to find a range [min, max]2 of values to which the SCR belongs; given a copula, realizations of risk factors, and proxy functions (more details further). The implemented algorithms are disclosed in Appendices D.1 and D.2, respectively for the case of permutations and weights. They correspond to an adaptation of the Rebonato-Jackël algorithm that incorporate these operations. At the end, the range is obtained for a given pseudo-correlation matrix G, a given weighting matrix H, and possibly a given permutation σ. Hence we want to evaluate the function g such that g : (G, H, σ) → g(G, H, σ) = SCR(P SD H (σ.G)), where σ.G stands for the effect of σ on the risk factors represented in G, and P SD H (G) represents the nearest (from G) PSD matrix obtained using the Rebonato-Jackël algorithm with weights H.

The considered permutation σ is the one presented at the beginning of Section 4.1.

Loss factors or risk factors?

In full generality, it is a very hard task to estimate the loss generated by the occurrence of some given risk. Much easier is to describe the behavior of risk factors, through marginal distributions. For instance if the interest rates rise, the potential loss for the insurer depends on impacts on both assets (e.g. value of obligations drops) and liabilities (contract credited rates may vary, which should modify expected lapse rates). To compute the loss associated to the variation of some risk factors, one thus needs a (very) complex transformation. Practically speaking, and to save computation time, simple functions (polynomial form) approximate these losses. However, the insurer can sometimes directly evaluate the loss related to one given loss factor: this is the case for example when considering the reserve risk, which can be modeled by classical statistical methods (bootstrap). The insurer's total loss, P , thus reads

P = 1 1 R =∅ f ((X i ) i∈R ) + 1 1 P =∅ i∈P X j ,
where f is a given (proxy) function, X i and X j are random variables, P is the set of loss factors and R is the set of risk factors.

For our coming analysis, Table 3 gives the different functional forms depending on the risk dimension and the risk factors X i . For the sake of simplicity, one considers that all our marginals (X i and X j ) follow the same distribution, but with different parameters. This common distribution is lognormal LN (µ, λ), since it is widely used in insurance for prudential reasons. Table 4 sums up the parameters involved in the eight different cases under study: vectors X a k (where k refers to the dimension of the vector) will be used to compute the global loss in the case of loss factors aggregation (meaning that R = ∅), whereas vectors X b k will be the input of proxy functions defined in Table 3 for risk factors aggregation (P = ∅). We distinguish these two configurations to see whether taking into account proxy functions gives very different SCR sensibilities as compared to only aggregate risk factors.

Variance-covariance or copula approach, pros and cons

Except PSDization step itself that generates different results, another important choice is the aggregation approach. Here, one would like to detail the reasons for choosing one of them (i.e. copula or variance-covariance). Let us consider the simplest framework: the loss P only depends on loss factors (no need to apply proxy functions that link risk factors to loss factors). It is then possible to model this loss as following: P = i∈P X i .

Dimension Functional form under consideration

3 P = f ((X i ) i∈[1,3] ) = 0.5X 2 1 + 2X 4 2 + 0.3X 3 + 10X 1 X 2 4 P = f ((X i ) i∈[1,4] ) = 5X 1 + 0.02X 2 2 + 2000X 3 + 5X 3/2 4 5 P = f ((X i ) i∈[1,5] ) = X 1 + 0.1X 2 2 + 50X 3 + X 1/2 4 -1.2X 5 10 P = f ((X i ) i∈[1,10] ) = 0.5X 3 1 + 0.4X 3 2 + 3X 2 3 + 2X 2 4 + 7 i=5 X 2 i + 0.5X 2 8 + 10 i=9 X i
Table 3: Polynomials used as proxy functions to get the overall insurer's loss.

Individual loss factors have to be modeled and estimated by the actuaries for internal models, or come from standardized shocks if using the Standard Formula. Fortunately, it is likely that extreme events corresponding to the 99.5 th percentile of every loss factors

do not occur at the same time: there is thus a mitigation effect, that generally yields to q 99.5% (P ) ≤ i∈P q 99.5% (X i ).

As already mentioned in Section 1, the regulation states that the variance-covariance approach can be used to aggregate risks, with the given correlation matrices. This method has some advantages, but also some drawbacks [START_REF] Embrechts | Model uncertainty and VaR aggregation[END_REF]). Of course, it is the easiest way to aggregate risks: the formula is quickly implemented (which allows to compute sensitivities without too much effort), and easy to understand. However it does not provide with the entire distribution of the aggregated loss, knowing that the insurer is sometimes interested in other risk measures than the unique 99.5 th percentile.

Name Dim. Parameters Moreover, this approach is not adequate for modeling non-linear correlations, which is often the case when considering the tails of loss distributions. It means that it is very tricky to calibrate the correlation matrix so as to ensure that we can effectively estimate the 99.5 th percentile of the aggregated loss. Finally, the variance-covariance approach is too restrictive since it does not allow to correlate risk factors, but only correlate losses.

X 1 X 2 X 3 X 4 X 5 X 6 X 7 X 8 X 9 X 10 X a 3 3 (µ, λ) (6
This makes the interpretation of scenarios generating a huge aggregated loss very difficult.

For all these reasons, internal models are generally developed using copulas: they enable to simulate a large number of joint replications of the risk factors, before applying proxy functions (most of time). By this technique, insurers obtain the full distribution of P , and thus a richer information among which the quantile of interest [START_REF] Embrechts | Copula Theory and Its Applications[END_REF], Lescourret and Robert (2006)). Very familiar copulas in the insurance industry (Gaussian and Student copulas) are parameterized by the linear correlation matrix, whose marginals are the risk and loss factors. This is linked to the main property of copulas: they allow to define the correlation structure and the marginals separately.

For example, aggregation with the Gaussian copula can be simulated with the following steps:

• simulation of the marginals stand-alone (stored in the vector X ∈ R n×B , where n stands for the number of risk factors and B the number of random samples);

• simulation of a Gaussian vector Y through the expression Y = T Z, where T represents the Choleski decomposition of the correlation matrix ρ = (ρ ij ) (i,j)∈ 1,n and Z is an independent Gaussian vector of size n;

• ordering X in the same order as Y to ensure that ∀j ∈ {1, ..., B}, ∀i ∈ {1, ..., n}, q(X ij ) = q(Y ij ) (q(x) stands for the quantile corresponding to x).

Results using a simplified internal model

Applications presented hereafter were designed to consider a wide range of operational situations in which the insurer aims to estimate its global SCR, i.e. SCR g .

For a given PSD correlation matrix S H xy , and given values for the vector of risk factors (simulated with lognormal distributions), one performs Q = 131072 = 2 17 simulations for the aggregation of risk factors (dependence structure). As a matter of fact, there are two sources of uncertainty explaining the variation of SCR g values (SCR g = (SCR g q ) q=1,...,Q ). First, the genetic algorithm itself is likely to have reached different local solutions depending on the simulation. Second, the simulation of the Gaussian or Student vectors to model the correlation through copulas may change. In order to focus the study on correlation, it must be noted that marginals were simulated initially and then kept fixed.

Roughly, it can be assumed that the confidence interval of the global SCR is similar to that of a Gaussian distribution (SCR g q ∼ SCR g ∼ N (m, σ)) because of the Central Limit Theorem and of the independence of the simulations of each SCR, i.e.

P(|SCR

g -m| < 1.65 σ) = 90%,
where σ stands for the standard deviation of SCR g , and m its mean. Of course, the estimation of these parameters is made simple using their empirical counterparts, denoted by m = (1/Q) q SCR g q and σ2 = (1/(Q -1)) q (SCR g q -m) 2 . Table 5 summarizes the estimated quantities for each case in our framework.

Then, we study the impact of our transformations (permutation, weights varying, and higher dimension) as compared to this standard deviation σ. More precisely, we consider one operation, perform the same number of simulations, and store the minimum and maximum values of the vector (SCR g q ) q=1,...,Q . This way, it is possible to define a normalized range (NR) for these values, whose expression follows

N R = max(SCR g ) -min(SCR g ) m .
If NR is lower than (2 × 1.65 × σ), the transformation is said to have a limited impact on SCR g . Otherwise, it is considered as a significant impact. The worst cases correspond to situations where NR is greater than (2 × 2.89 × σ). The multiplier 2 enables to take into account the fact that there are two sources of uncertainty (genetic algorithm and simulated dependence structure). All the results are stored in Table 6, where column 'Imp.' describes the strength of the impact of the transformation under study : '0' refers to a limited impact

Gaussian copula Student copula (3 d.f.) Loss factors X a k Risk factors X b k Loss factors X a k Risk factors X b k Example Dim. k m σ m σ m σ m σ S H
(Q = 2 17 simulations). (N R < 2 × 1.65σ), '+' means a significant impact (N R ∈ [2 × 1.65σ, 2 × 2.89σ]), and
'++' a very strong impact (N R > 2 × 2.89σ).

Impact of permutations on the global SCR

On the 14 examples under study (7 pseudo-correlation matrices G xy , times Gaussian or Student copula), the permutation systematically causes a very strong impact on the global SCR. This change can represent up to 6.7% in practice, although it should have no theoretical impact. This is mainly due to the PSDization process that leads to select different local minima after permutation is made. This highlights two phenomenons: the need to control the bias coming from the initial choice of the insurer concerning the order of risk factors, and the need to initially define PSD correlation matrices (revisiting the experts' opinions, and identifying incoherent correlation submatrices).

To have a more comprehensive view of this impact, Figure 5 illustrates it on the total loss distribution (rather than the sole 99.

Impact of the modification of weights on SCR

Our sensitivities relate to weighting coefficients varying in a given range. This range is defined by weights between 6 ) shows that on 28 examples, 7 cases lead to a very significant impact on the final SCR, with a range likely to represent more than 4.7% of the SCR! Once more, this shows the necessity to properly define correlation coefficients at the very beginning.

H min = [0] (i,j)∈ 1,n 2 and H max = [1] (i,j)∈ 1,n

Impact of adding a dimension to the correlation matrix

In practice, the insurer's global loss often incorporates some negligible loss factor. In the simple case where there are only loss factors affecting the global loss, it means that

P = j∈P,j =n+1 X j + X n+1 ,
where thus tends to 0. The limit case would be = 0, which means that the (n+1) th risk factor would have no impact on the insurer's loss, but still plays a role through its presence in matrix and its impact in the PSDization process. The correlation between this risk factor and others is fixed to 0.1 (as in Section 4.2). We measure SCR g value before and after adding this dimension.

On the 28 examples analyzed, almost one third (9 cases exactly) leads to a statistically significant impact on the final SCR (strong or very strong impact on SCR g ). However, except one particular case involving an impact value around 6%, most of the impacts seem to be lower than with other operations. Once again, it is important to realize that this transformation should have no theoretical impact. Of course, it suggests that it would be worth conducting deeper analyses on this aspect, especially about adding more than one dimension and changing the correlation coefficients of the added risk factor.

Concluding remarks and discussion

A Pseudo-correlation matrices under study

Let us present the pseudo-correlation matrices, but also the weighting matrices coming from expert judgments to be taking into account during PSDization.

Example #1: Example #3:

G 31 =     1 -0.9 -0.5 -0.9 1 -0.5 -0.5 -0.5 1     H 31 =     1
G 41 =       1
G 51 =         
1 -0.8 -0.9 -0.9 0.2 -0.8 1 0.9 -0.7 0.6 -0.9 0.9 1 0.2 -0.6 -0.9 -0.7 0.2 1 -0.1

B PSDized matrices, without weighting coefficients

We recall here that except for the highest dimension under consideration (dimension 10), the three algorithms give the same solution.

Examples #1 and #2: .

Example #7: on this example, the PSDized versions of the initial pseudo-correlation matrix differ depending on the algorithm used.

S P A 101 =                      

Figure 1 :

 1 Figure 1: Impact of permutations on the Frobenius norm, in the case of G 51 and H 51 .

Figure 2 :

 2 Figure 2: Impact of adding a new dimension on the Frobenius norm (through modified PSDized coefficients), with various correlation coefficients corresponding to the new risk.

Figure 3 :

 3 Figure 3: Impact of weights on the Frobenius norm, in the case of the example G 51 .

Figure 4 :

 4 Figure 4: 3D plot showing the impact of permutations and weights on the Frobenius norm, with G 51 and H 51 .

  5 th percentile), with G 52 , H 52 and considering the aggregation of loss factors. The red curve corresponds to the loss distribution after applying the permutation (1 → 5; 4 → 1; 5 → 4) to G 52 in the case of a Gaussian copula, whereas the blue one corresponds to the permutation (2 → 3; 3 → 4; 4 → 5; 5 → 2).

Figure 5 :

 5 Figure 5: Illustration of the impact of the permutation on the loss distribution, with a focus on the right on the area around the capital requirement (99.5 quantile)

  Insurers using internal models, as well as supervisors, are legitimate to ask themselves about the robustness of their PSDization algorithm. Our study shows and highlights the importance of PSDization through quantified answers to very practical questions on a series of real-life examples. On the 98 (3*28+14) examples based on various configurations (different copulas and ways to consider risks, see

  S P A 31 = S N 31 = S 31 , and S P A 32 = S N 32 = S 32 , where

Table

Table 1 :

 1 Eigen values of correlation matrices in Solvency II regulation.

	Module	Dimension λ 1	λ 2	λ 3	λ 4	λ 5	λ 6	λ 7	λ 8	λ 9	λ 10 λ 11 λ 12 PSD
	Global (Ann. IV)	5	1.92 1.16 0.75 0.75 0.40					Yes
	Market up (Art. 164)	6	2.47 1.18 1.00 0.68 0.50 0.15				Yes
	Market down (Art. 164)	6	2.89 1.00 0.87 0.57 0.50 0.15				Yes
	Life (Art. 136)	7	2.18 1.51 1.07 0.81 0.70 0.57 0.12			Yes
	Health SLT (Art. 151)	6	2.04 1.43 1.00 0.81 0.58 0.12				Yes
	Health non SLT (Ann. XV)	4	3	0.5 0.5 0.5						Yes
	Health (Art. 144)	3	1.68 0.81 0.5							Yes
	Non Life (Art. 114)	3	1.25	1	0.75							Yes
	Prem. Reserve (Ann. IV)	12	4.91 1.45 1.09 0.97 0.73 0.68 0.61 0.48 0.38 0.33 0.20 0.12 Yes

Table 2

 2

		sums up the eigen values of our seven pseudo-correlation matrices: notice that
	none of the considered matrices is PSD. However, some of them are not very far from
	having this property.
	Example # Dim. Notation λ 1	λ 2	λ 3	λ 4	λ 5	λ 6	λ 7	λ 8	λ 9	λ 10 PSD
	1	3	G 31	1.90 1.39 -0.29					No
	2	3	G 32	1.91 1.44 -0.35					No
	3	4	G 41	2.17 1.27 0.88 -0.32				No
	4	4	G 42	1.95 1.74 0.96 -0.64				No
	5	5	G 51	2.80 1.98 1.16 -0.17 -0.77			No
	6	5	G 52	2.44 1.97 1.67 -0.19 -0.88			No
	7	10	G 101	3.93 2.72 2.00 1.14 0.74 0.57 0.25 -0.03 -0.55 -0.77 No

Table 2 :

 2 Eigen values in our examples, before PSDization.

Table 4 :

 4 Marginals for each risk factor and corresponding 99.5 th percentile.

Table 5 :

 5 Mean and standard deviation of the global SCR, SCR g

Table 6 :

 6 Sensitivities of the global SCR to transformations on the correlation matrix (permutation, two different cases for weighting the correlation coefficients, and addition of a new risk); depending on copula, and risk factors / loss factors aggregation. 21 represent more than 5.4% of the SCR, which is really huge in practice.The same analysis with stronger constraints (weights belonging to an interval of width 0.2 around the initial weights, i.e. H min = H -[0.1] (i,j)∈ 1,n 2 and H max = H + [0.1] (i,j)∈ 1,n 2 , see 'W Sensi2' in Table

	Gaussian copula

2

. This sensitivity is denoted by 'W Sensi1' in Table

6

. On the 28 examples analyzed here (7 pseudocorrelation matrices G xy , times (Gaussian or Student copula), times (risk or loss factors)),

Table 6 )

 6 , approximately one half (exactly 47) generates significant impacts on the global SCR (up to 6%) when studying sensitivities to our three tuning parameters (weights given to individual correlation coefficients, permutations, and addition of a fictive business line). Knowing that these transformations are either theoretically neutral, or should not lead to big effects on the global capital requirement, this underlines that practicioners' choices (risk managers, actuaries) are fundamental when performing risk aggregation in internal models. A strong control of PSDization by supervisors thus makes sense. Proxy functions do not seem to change conclusions here: SCR g sensitivity is similar when considering only loss factors.The following best practices were identified: i) develop a sound internal control framework on both the triggers generating negative eigenvalues (e.g. expert judgments) and the PSDization step itself, ii) assess the need for adding a new risk (e.g. new business line) in terms of its impact on the correlation matrix and thus on the global SCR. On the former point, independent validations and systematic reviews of the modifications brought to the correlation matrix by the algorithm should be analyzed, and a wide number of sensitivities has to be implemented to challenge the results. Concerning the dimension of the risk matrix, there seems to be a trade-off to find: adding business lines allows to increase granularity when describing the correlation between risks, but tend to cause more disturbance on the individual correlation coefficients during PSDization. As usual, the best choice lies in an intermediate dimension for the risk matrix.

	Finally, this work could be extended in several ways, among which the definition of
	algebraic tests to detect in advance inconsistencies in the experts' choices; and a deeper
	understanding of the permutations leading to the minimum or maximum values of the
	SCR. In particular, if these permutations show some similar features, it would be possible
	to define best practices when ordering risk factors.

  Since the only method that allows to integrate weights when looking for the closest PSDized matrix is the Rebonato-Jäckel algorithm, we have here seven results coming from our seven examples.

				1	-0.671 -0.313 -0.018 0.563 -0.243 0.159	0.396	0.010	0.010	
			-0.671 -0.313 0.592 1 -0.018 0.422 0.563 -0.110 0.146 0.592 1 0.296 -0.243 0.676 0.259 0.159 0.233 -0.223 0.025 0.422 -0.110 0.676 0.296 0.146 0.259 -0.223 0.233 1 0.217 0.281 0.025 0.217 1 0.521 0.310 0.281 0.521 1 0.625 0.310 0.625 1 0.396 0.392 0.294 0.507 0.704 0.669 0.535 0.010 0.262 0.479 0.486 -0.000 0.018 0.224	0.392 0.294 0.507 0.704 0.669 0.5350.224 0.224 0.262 0.479 0.486 -0.000 -0.000 0.262 0.479 0.486 0.018 0.018 1 0.249 0.249 0.249 1 0.761	                    	.
			0.010	0.262	0.479	0.486 -0.000 0.018	0.224	0.249	0.761	1
	Examples #1 and #2:
							1	-0.848 -0.505			1	-0.278 0.187	
				S H 31 =	  	-0.848 -0.505 -0.029 1	-0.029 1	   ,	S H 32	  	-0.278 0.187	1 0.892	0.892 1	   .
	Examples #3 and #4: S P A 41 = S N 41 = S 41 , and S P A 42 = S N 42 = S 42 , where
					1		0.844 -0.401 -0.707			1	0.041 -0.614 0.041	
	S H 41 =	1 -0.707 -0.329 0.216 -0.681 -0.310 -0.008 0.574 -0.241 0.158 0.390 0.017 1 0.041 0.144 0.763      0.844 1 -0.029 -0.329 -0.401 -0.029 1 0.216      , S H 42 =      0.041 1 0.129 0.144 1 0.017 -0.614 0.129 1 0.763      .
				-0.681	1	0.596		0.434 -0.010 0.684	0.233 0.390 0.269	0.269
				-0.310 0.596	1		0.295		0.144	0.256 -0.222 0.293 0.480	0.480
	-0.008 0.434 Examples #5 and #6: S P A 0.295 51 = S N 51 = S 51 , and S P A 1 0.210 52 = S N 0.276 52 = S 52 , where 0.024 0.508 0.487	0.487
				0.574 -0.010 0.144 1 -0.822 -0.795 -0.811 0.132 0.210 1	0.520	0.308 0.710 -0.001 -0.001
	S H 51 =	       	-0.241 0.684 0.158 0.233 -0.222 0.024 0.256 0.276 0.390 0.390 0.293 0.508 0.017 0.269 0.480 0.487 -0.001 0.019 0.520 1 0.308 0.625 0.710 0.674 -0.822 1 0.834 0.353 0.155 -0.795 0.834 1 0.371 -0.412 -0.811 0.353 0.371 1 -0.193	0.625 0.674 0.019 1 0.537 0.226 0.537 1 0.245 0.226 0.245 1	0.019 0.226 0.245 0.760
				0.017	0.269	0.480		0.487 -0.001 0.019	0.226 0.245 0.760	1

C PSDized matrices with weighted coefficients

As the CEIOPS (ex-EIOPA) admitted in its Solvency II calibration paper of April

.

Recall that the genetic algorithms must be seen as "clever" sensitivities rather than fully convergent optimization algorithms. There is no guarantee that the convergence to a minimum is obtained.

Acknowledgment

This work is partially supported by BNP Paribas Cardif, through the Research Chair DAMI. We also would like to thank two researchers for useful and constructive comments, Hansjörg Albrecher and Léonard Vincent.


Tao Guang

Michel Xu

Adrien Visonneau

Francisco Chinesta Leygue

Chinesta

Space-time separated representation for solving Navier-Stokes equations

come

The Proper Generalized Decomposition (PGD) method has been used to solve many kinds of problem in multi-dimensional spaces: quantum chemistry, kinetic theory description of complex fluid and chemical master equations, etc...The interested reader can refer to [START_REF] Chinesta | An overview of the proper generalized decomposition with applications in computational rheology[END_REF] for a recent review in the context of computational rheology. The present work describes a first implementation of the PGD formulation into a general unstructured finite-volume unsteady Navier-Stokes solver dedicated to viscous ship hydrodynamics (ISIS-CFD [START_REF] Queutey | An interface capturing method for free-surface hydrodynamic flows[END_REF]).

PGD decomposition for the time-space domain

The following separated decomposition for the pressure and velocity fields is injected into the unsteady Navier-Stokes equations:

u(x, t) ≈ N i=1 X u i (x)T i (t) + X u T v(x, t) ≈ N i=1 X v i (x)T i (t) + X v T p(x, t) ≈ N i=1 X p i (x)T i (t) + X p T (1)
which leads to coupled non-linear steady spatial and temporal equations replacing the traditional incremental approach. The spatial modes are determined by using the fully coupled approach implemented in ISIS-CFD [START_REF] Deng | A New Fully Coupled Method for Computing Turbulent Flows[END_REF]. This steady formulation is based on the iterative solution of a saddle point linear system coupling together momentum and mass conservation equation transformed into a pressure equation. Steady spatial 2D or 3D partial differential equations and temporal ordinary differential equation are coupled with the help of a fixed-point algorithm. The discretisation based on an unstructured finite-volume formulation is described in [START_REF] Queutey | An interface capturing method for free-surface hydrodynamic flows[END_REF]. The result shows that the new computational algorithm reduces drastically the computational time needed for the simulation of complex unsteady flows. This non-incremental solution strategy opens a new perspective towards radical evolutions of the computational methods for the simulation of unsteady viscous flows.

Results -CPU time comparison

The laminar viscous flow inside a lid-driven cavity with an oscillating upper wall is used to compare the performances of the PGD formulation with respect to the traditional incremental algorithm. The PGD formulation applied to the unstructured finite-volume industrial Navier-Stokes solver ISIS-CFD, has been assessed successfully on a first academic test case. An impressive reduction of the computational time has been observed on this first application, illustrating promising perspectives.

Fig.( 1 )

 1 shows the CPU time used by both methods, illustrating the strong reduction implied by the PGD formulation.

Figure 1 :

 1 Figure 1: CPU time comparisons for the lid-driven cavity

  11th. World Congress on Computational Mechanics (WCCM XI) 5th. European Conference on Computational Mechanics (ECCM V) 6th. European Conference on Computational Fluid Dynamics (ECFD VI) July 20 -25, 2014, Barcelona, Spain -T. Xu ,1 , M. Visonneau 2 , A. Leygue 3 and F. Chinesta 4 1 LHEEA & GeM, Ecole Centrale Nantes 44321 France, guangtao.xu@ec-nantes.fr 2 LHEEA UMR CNRS-Ecole Centrale Nantes 44321 France, michel.visonneau@ec-nantes.fr 3 GeM, Ecole Centrale Nantes 44321 France, adrien.leygue@ec-nantes.fr 4 ESI International Chair & Institut Universitaire de France & GeM, UMR CNRS-Centrale Nantes 44321 Nantes France, francisco.chinesta@ec-nantes.fr

	SPACE-TIME SEPARATED REPRESENTATION FOR
	SOLVING NAVIER-STOKES EQUATIONS
	Key words: PGD, space-time separation, Navier-Stokes, incompressible, unstructured
	finite volume

G