Mathematical Modeling of Concentration Risk under the Default Risk Charge Using Probability and Statistics Theory

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Introduction
Since 2013, the Basel Committee has led works for a new regulation to implement a more consistent regulatory market risk capital platform. Tis project is known as the Fundamental Review of the Trading Book. It replaces the Basel II International Convergence of Capital Measurement and Capital Standards text. Te complete version was prepared by the Basel Committee [1] under the title, "Minimum Capital Requirements for Market Risk." Tis regulation is summarized into four streams.
Te frst stream refers to the boundary between the trading and the banking books. Tus, this stream aims to improve the visibility of products that include market risk exposure. Te banks are brought in to list all desks of the trading book. Tey must defne the link between the positions held for the trading objective and the regulatory trading book. Te regulatory purpose is to reduce arbitrage across the trading and banking books.
Te second stream refers to rebuilding the Internal Models Approach (IMA). Hence, this stream covers all internal market risk models developed by banks. Te regulator suggests changing all market risk measurements. First, the regulator replaces the market Value at Risk (VaR) and the Stressed Value at Risk (SVaR) from two perspectives given a 10-day horizon and a 99% confdence level by the Expected Shortfall (ES) and given a 10-day horizon with diferent liquidity horizons (10,20,40, 60, 120 days) and a 97.5% confdence level. Te Basel formula is given as follows: where T � 10 days and the liquidity horizons (LH) are equal to LH j � 10, 20, 40, 60, 120 days. ES is computed sequentially by liquidity horizons and class assets. Figure 1 gives us an example of the calculation. Second, banks must distinguish between modellable and non-modellable risk factors (NMRFs). Ten, those risk factors must be quantifed using a stress scenario with zero correlation. Tird, the Comprehensive Risk Measure must be processed according to the standard approach. Finally, the Incremental Risk Charge is changed by the Default Risk Charge. We will see the detailed regulation requirement for its modeling next. At this stage, we are reminded that the migration rate risk is deleted to keep only the default one, and the equity scope is added.
Te third stream refers to improving model adequacy and backtesting. In this stream, the regulator has established two levels of VaR backtesting: 97.5% and 99%. Ten, the regulator adds the proft and loss (P&L) attribution as a new test. Such a test is based on the minimization of two ratios. On the one hand, banks must minimize the unexplained-daily P&L. It equals the difference between a risk-theoretical P&L and a hypothetical P&L over the standard deviation of hypotheticaldaily P&L. On the other hand, we fnd the ratio between the variances of unexplained-daily P&L and hypothetical-daily P&L. Te frst ratio boundary is [− 10%, +10%]. Te second ratio has to be between 0% and 20%. Tis test aims to bring the market P&L distribution closer to the theoretical P&L distribution because we know that VaR backtesting is focused on the distribution tail alone. Figure 2 shows an example of the unexplained mean and variance.
Nevertheless, one limitation of those ratios is that when the portfolio is perfectly hedged, it leads to the zero value of the hypothetical-daily P&L variance. Te two metrics converge to infnity, and we then analyze the conclusions of this test.
Te fnal stream refers to rebuilding the Standardized Approach (SA). Te new SA is based on the Sensitivities Method and covers the trading book's nonsecuritization and securitization exposures. Te regulator also defnes the SA for the DRC and the residual risk add-on that is not captured by others risk metrics. Banks will then use a linear approach based on the Delta and Vega sensitivities and a nonlinear approach for the instruments that integrate curvatures (e.g., options for ES computing). However, the DRC SA uses risk weight to weigh the Jump to Default (JTD) by obligor rating to calculate the DRC.
After that, we will focus on the FRTB guidelines for modeling the DRC on the IMA. Te regulator defnes default risk as the direct or indirect loss arising from the obligor's default. Tis risk is measured by a VaR based on a one-year horizon and a 99.9% confdence level. Te computing frequency is weekly, and the DRC capital requirement is equal to the following: We should calibrate four components and model to implement the loss function. Te frst component is the obligor's correlation. Initially, the regulator allows using the credit spread or the historical listed equity price data. Tese historical data must include at least 10 years and the stressed period, as defned in the ES model. Te chosen liquidity horizon is the one-year liquidity horizon, and the minimum for the equities is set at 60 days. Tese data must give a higher correlation for portfolios, including short and long positions. On the other hand, a low correlation is assigned to the portfolios that contain only long exposures. Next, the obligor's default must be modeled using two types of systematic factors to deduce the model correlation. Finally, the correlation measurement must be done on the one-year liquidity horizon.
Te second component is the Probability of Default (PD). Te FRTB defnes some conditions and priorities for PD estimation. Te frst two conditions are as follows: (1) the market PDs are not allowed and (2) all default probabilities are foored to 0.03%. Te Internal Ratings-Based (IRB) PDs typically become the best choice when the model is validated. Otherwise, a model must be developed respective to the IRB methodology. Terefore, historical market PDs should not be used for calibration. Institutions must base their evaluations on a historical default uploaded with a 5-year observation as a minimum calibration period. Banks could also use the external rating provided by rating agencies (e.g., S&P, Fitch, or Moody's) to estimate PDs. In this case, they must defne the priority ranking choice.
Te third component is the Loss Given Default (LGD) model. Te LGD model must catch the correlation between recovery and systematic factors. Te model must be calibrated based on IRB data if the institution already has a homologated model. Te historical data should be relevant to get accurate estimates. All LGDs must be foored to zero, and the external LGDs could be used, respective to some defned ranking choice.

Journal of Probability and Statistics
Te fnal component is the Jump to Default (JTD) model. Te JTD model must catch each obligor's long and short positions. Additionally, the set assets must contain the credit (i.e., sovereign and corporate credit) and the equity exposures. Tis measure can be defned as a function of the LGD and the Exposure at the Default (EAD) for credit assets. However, it must also measure the P&L for equities when the default occurs since we know that the LGD is equal to 100% for equity assets. Te model includes equity derivatives pricing within the zero value of the stock price. Te nonlinear product JTD must integrate multidefault obligors in the case of the derivative products with a multiple underlying. A linear approach could be used for these products, such as the sensitivities approach, based on obligor default and subject to supervisor approval.
Tere are a few studies present and suggest frameworks to model the DRC. Te frst one was made by Laurent et al. [2] where they use the Hoefding decomposition to explain the loss function. Te second one was implemented by Wilkens and Predescu [3] and they propose a complete framework to build the DRC model. However, they all use the Merton model with multifactor (structural approach) and they are not studying the concentration risk issue.
In this paper, we will study the DRC modeling under the Internal Model Approach (IMA) and the regulator conditions that every DRC component must respect. Te FRTB presents the DRC measurement as a Value at Risk (VaR), over a one-year horizon, with the quantile equal to 99.9%. We will use the multifactors adjustment to measure the DRC, and we will compare it with the Monte Carlo Model to study the ftting of this approach. We will then defne the concentration in the DRC, and we will propose an approach to quantifying the concentration risk. We will fnally study the behavior of the DRC with respect to the concentration risk.  (2) corporate. Te second asset type contains regional and industry factors. We note these sets, respectively, by GA � GS, GC, R � 1 . . . r, and I � 1 . . . s as we know that two approaches allow default modeling: (1) the structural model and (2) the intensity model. In this study, we will use the Merton [4] model with multifactors and deem a portfolio with N obligor, containing credit (i.e., sovereign and corporate) and equity position. Te return variable is written for an obligor (i) as follows:

Mathematical Modeling of the DRC
where Z G , Z g , Z R j , and Z I l are independent of set and follow N(0, 1), with g ∈ GA, j ∈ R, l ∈ I. β gives the correlation between obligors and systematic factors, whereas ε i ∼ N(0, 1) represents the specifc risk, and they are independent and identically distributed for i ∈ 1 . . . N { } and independent of all systematic factors. Also, the following formula is used to keep X i ∼ N(0, 1): Te initial choice of a systematic factor does not allow an independent set structure. However, we can run the Gram-Schmidt algorithm to get the orthogonal sets before calibrating the model correlations. We then fx the global systematic factor and orthogonalize each axis in GA set with the global one. Te new axes are defned as follows: We do the same thing for regions and add the orthogonal projection under Z G and Z ⊥ g to get Z R ⊥ j . Finally, we proceed for industries by adding the projection on Z R ⊥ j to build Z I ⊥ j . We center and reduce at each orthogonalization step to keep a centered and reduced variable.
We will use the new systematic factors in the following calculation. Terefore, the implied correlation between obligors can be deduced by where ρ I represents the obligor implied correlation matrix, N × N; ρ F is the systematic factor intracorrelation matrix, K × K and K � (3 + r + s); β represents the correlation factors between the obligors matrix, N × (3 + r + s), and the systematic factors; β ′ represents the transposed matrix; σ 2 is the vector of σ 2 i ; and I is the identity matrix. We deem a set of 1,481 issuers within a 10-year historical spread. Our population contains 69 sovereigns in six regions and 11 industries. Terefore, we build a set of systematic factors orthogonal by subsets, which gives us the block of zeros in building a ρ F with 20 × 20 as the dimensions.
Te relevance of our model could be measured by comparing the implied correlation with the historical correlation. We assume that we use the log return of historical data and note ρ H , the Pearson historical correlation between obligors. First, we propose plotting the density function of ρ H and ρ I to see if the implied distribution fts with the historical one. Figure 3 shows that the implied density is close to the historical density, and both look very similar. However, the plot of densities is not enough to measure ft. We can also use the same ratios defned on the P&L attribution to compare the mean and standard deviation of the two distributions. Hence, these ratios equal 2.26% and 12.14%, respectively. We suggest building the confdence interval on ρ H using the following Fisher transformation to complete this analysis: We directly conclude the confdence interval, with α, the confdence level. We then compute the percent numbers of this confdence interval pair-wise to conclude the model accuracy. Given that α � 10%, the result tells us that 91.9% of the population is inside the confdence interval. We observe that the quality of our model correlation is sensitive to how we build the systematic factor, and we can use this to rebuild the set that gives the closest implied correlation to the observed one. However, the drawback of this approach is that it is very time-expensive.
After this calibration, the obligor default is defned under the Merton model as follows: In other words, we can write which represents the probability of default for the obligor (i). We will use the Standard & Poor's (S&P) PDs with a foor of 0.03% as specifed in the FRTB. Table 1 shows the one-year probability of default by rating and category.
Hence, the conditional default probability for the systematic factors under this model is equal to Given that β i are the obligor lines of the β matrix, Z ′ is the systematic vectors transpose and is defned as

LGD Model. Te
LGD computation depends on the recovery rate. However, the FRTB guidelines require the dependency between recovery rate and systematic factors ("Te model must incorporate the dependence of the recovery on the systemic risk factors" [1], p. 62). Hence, we should resort to the models that allow this condition. Tere are many models developed in this optic. For example, Michael [5] proposed an exponential function between the recovery rate and the systematic factors. In another approach, Hull and White [6] suggested an exponential function between recovery and default rates. Tis model indirectly links the LGD to systematic factors because the default rate is a function of them. In this paper, we opt for a similar model to the one based on the default rate. We deem the following relation between the LGD and the conditional default probability of the systematic factor: LGD where We use the IRB data for the calibration of LGD min and LGD max to conform to the FRTB regulation. Te asset class can make this calibration, so we have to defne a sov , b sov for sovereign and a corp , b corp for corporate obligors. It could also be done by seniority. However, we keep the sovereign and corporate subdivisions in our case, taking the following values for calibration: LGD min (SOV) � 0.0, LGD max (SOV) � 0.8, LGD min (CORP) � 0.6, LGD max (CORP) � 0.99.
Given these values, we fnd our parameters as follows: Tus, we deem the following transformation: where We can calculate the distribution and the density of Y. Hence, the calculations lead to the following results: Terefore, we can deduce the recovery rate distribution since it is a function of Y: Ten, We then conclude the expectation and the variance of the recovery: where For the small values of Y, we can approximate e − aY ≈ 1 − aY. We then have a close formula for the recovery expectation and variance:

JTD Model.
Te standardized approach defnes a long and short JTD for the same obligor. We then aggregate to get the gross JTD. In this approach, JTD is a function of LGD, the notional amount, and the P&L. Terefore, we have the following equations: JTD(long) � max(LGD × notional + P&L, 0),

Journal of Probability and Statistics
where P&L � market value-notional and LGD � 25%, 75%, 100%, respectively, for covered bonds, senior debts, and nonsenior debts. We could keep the same formula by replacing it with the LGD model. However, the other option is to compute the EAD as we did in the banking book. Terefore, it will be explained by asset type (i.e., credit and equity) and in mono and multi-underlying contexts. We then describe the formula for each asset type as follows: where T � 1 year; Y i is the obligor underlying price (i.e., the stock price in the case of equity); V is the function value of the obligor (i) (i.e., the aggregate position of mono and multi-underlying); w i � |s i |/ K 1 |s i |, s i � zv((x 1 , . . . , X i , . . . , X K ))/zx i represents the sensitivity weight; and A is Mono (24) Figure 4 gives the exposure density of the portfolio used in this paper.

DRC Model.
We now have all components to defne the loss function, given by the following equation: Tis quantity can be computed using the Monte Carlo simulation to generate the loss distribution. Indeed, we note M, the number of simulations, and L m , the sampled path for m � 1. . .M. As we know, the DRC is a VaR at 99.9% for a one-year horizon; thus, it can be estimated as follows: Knowing that m ′ represents the path that gives the 99.9% × M loss, the Monte Carlo simulation gives the following DRC value with M � 1, 000, 000 after decreasingly ordering all simulated paths: (27) Figure 5 shows the loss distribution density of the portfolio.
Tis approach is straightforward and gives a good result with a large enough M. However, it takes much time for large portfolios and does not support quantifying concentration risk cost and defning whether it has been captured or not. First, we describe the loss induced by the systematic risk factor as follows: Tis equality comes where the JTD is Z-measurable. Terefore, we deem the following transformation as . By substituting the loss function, we get the following result: We applied a Monte Carlo simulation to this expression to compute the distribution of L Z and the VaR because the model contains more than one factor. However, we must relate our model to the one-factor model to get a direct computation of the VaR. Michael [5] defned this relation using an aggregate systematic factor Z as follows: where c k is chosen to maximize the correlation between elements of Z and Z. Furthermore, we can rewrite the obligor default variable as follows: where ρ i � cor(Z i , Z) � K k�1 c k α i,k . For the rest of this study, we redefne the recovery rate as a function of Z. Given these results, the loss function under the one-factor model becomes . We use the following problem optimization to fnd the appropriate c k . Tis method allows minimizing the norm one between the DRC value under the Monte Carlo approach and VaR 99.9% (L Z ): In our case, the quantile of the systematic loss for the optimized c k is equal to Hence, the remaining part of the DRC is the diference between the Monte Carlo DRC and the systematic loss, which equals 1,465,919. Tis quantity represents 23.9%, and it will be approximated using the adjustment.
It remains to compute the correlation and concentration efects since the FRTB guideline specifes that the model must refect the name concentration risk and the sectorial one by asset class ("Te model must refect the efect of issuer and market concentrations, as well as concentrations that can arise within and across product classes during stressed conditions" [1], p. 62).
Terefore, the frst and second derivatives of μ(z) according to the systematic factor are given by with We use the variance decomposition to compute σ 2 (z) as follows: We can explain that the frst term gives correlation effects between issuers and sectors. Hence, it indirectly gives the sector correlation since implied correlation depends on intrasectorial correlation. Te second one integrates the name concentration (i.e., specifc) risk and is known as the Granularity Adjustment (GA). We then get We compute the covariance between two individual loss issuers as follows: Terefore, the frst term is equal to  Journal of Probability and Statistics where Φ 2 is the bivariate normal cumulative distribution function and ρ Z ij represents the implied correlation between two issuers conditional to the systematic factors. Te derivative with respect to z is equal to Te second term gives the name concentration part of the adjustment, and we can compute it knowing that the individual losses are independent conditional to the systematic factors: By computing the individual variance of loss, we get By substitution, the result is Te derivative respect to Z then is equal to Given these results, we can rewrite Δ q (L ε ) as the sum of the two quantities. Te frst one will represent the efect correlation and the sectorial concentration, while the second will represent the name concentration. We then have where Journal of Probability and Statistics Te DRC approximation is calculated by the following formula: However, L Z is a monotonically decreasing function of Z. Tis property leads to Tus, the calculations give the following results: Tis approximation explains that the DRC is a sum of the systematic, specifc, and correlation contribution losses. Te relative error with the Monte Carlo approach is 1.6%. We then have a granularity contribution to make on the concentration risk efects. In the next section, we propose two approaches for determining concentration risk. Te frst one uses the concentration ratio (Ad Hoc), and the second employs granularity adjustment (Add-On).

DRC and Concentration.
Te IMA text's guidelines impose that the model must catch concentration risk efects. Since we can have two types of concentrations, we will ensure that the DRC increases with the name and sector concentrations. For this, we will defne a concentration ratio that provides this property for the name concentration.
However, building this ratio is not straightforward, like in the case of the loan book that defnes only a positive exposure. We have long and short EADs in the DRC. Te frst one increases the concentration, whereas the second should decrease it. Tus, the frst step is to defne two subsets by EAD issuers. Te frst one contains long exposures, and the second is built with short exposures: We now defne the share for each subset as follows: Te concentration ratio is a function of these shares, and there are many of these ratios. However, we will use the Herfndahl-Hirschman Index (HHI) [9], getting one for the long positions and another for the short positions: Terefore, the concentration ratio of the global portfolio is defned as follows: HHI � max HHI long − HHI short , 0 . (61) Tis ratio is verifed by constructing the concentration properties [10], and we then compute it directly for our portfolio: HHI long � 0.210%, HHI short � 0.208%, We conclude that the portfolio concentration under the HHI measure is very small. Hence, we can increase the concentration by increasing the long EADs and decreasing the short EADs to study the concentration efect. However, the impact cannot be signifcant since the contribution is minimal in both the Monte Carlo and the GA approaches.
Terefore, the second step is ordering the EAD issuers by the distance to the default Φ − 1 (PD). We defne a decreasing order for the long and short EADs. Tis approach allows us to see most EADs that contribute to the DRC in the case of the Monte Carlo approach and the contribution weight in the case of the GA model. We now have all the tools to verify that the DRC model has caught the name concentration. We then stress the portfolio by augmenting the frst long EADs and decreasing the fnal long EADs. Te frst impact arises on the HHI because it automatically increases the concentration under this measure. It remains to verify that the same efect appears in the Monte Carlo DRC and the GA (Δ 2 q (L ε )). For that, we sort the EADs by Φ − 1 (PD) decreasingly, and we use the transfer principal property to increase the concentration. We then compute the Monte Carlo DRC and Δ 2 q (L ε ) to study the behavior of the concentration efect. Figure 6 shows that the DRC also increases with HHI, which proves that the model has captured the name concentration.
We also conclude the same behavior for Δ 2 q (L ε ) in Figure 7.
Te DRC behavior, respective to the sector concentration, can be studied using the intrasectorial correlation. Terefore, we can increase these correlations and recompute the Monte Carlo DRC to verify whether it augments or not. We can also use Δ 1 q (L ε ) to see if it increases. Figures 8 and 9 show that sectorial concentration increases with intrasectorial correlation.

Conclusion
In this paper, we attempt to implement an approach that allows DRC modeling respective to the FRTB guidelines. First, we describe the regulatory requirement to build the conformance model. Te DRC model needs four components: (1) PD, (2) recovery, (3) JTD, and (4) loss function. We propose the model and calibration issues for each of these situations. We also describe the Monte Carlo approach to compute the DRC VaR. Nevertheless, this approach cannot give the concentration risk contribution. Additionally, it does not provide its impact on the DRC model. We suggest multiadjustment to fx this issue since the model must include multisystematic factors. Furthermore, we propose an adaptable HHI ratio to measure the portfolio name concentration since we have long and short positions. We then compute the evolution of DRC and Δ 2 q (L ε ) with respect to the HHI measure. We conclude that the model captured the concentration risk since the DRC increased with concentration. Regarding the sector concentration risk, we conclude that it increased respective to intrasectorial correlation. Terefore, all of these results prove that the model includes this component and verifes the regulatory requirement.
However, this approach is based on assumptions that may carry risks. Te frst assumption supposes that we are in the Merton environment, and the second assumption uses the Gaussian copula. Hence, we suggest using other copulas, like the Student or Gumbel copula, to study the impact of the   second assumption on the obtained results. For the frst assumption, we suggest to replace the structural approach with the intensity approach and remaking the study to see if these results remain the same.

Data Availability
Te input.zip data used to support the fndings of this study are included within the supplementary information fle(s) in the following google drive link: https://drive.google.com/fle/d/ 1O2e74mFDlPiJyNphRZlqXjIxCdG8yNVz/view?usp�sharing. All data are in csv fle and only who has this link could access these data.

Conflicts of Interest
Te author declares that there are no conficts of interest.