A practical anatomy of incremental risk charge modeling

The Journal of Risk Model Validation (45 60) Volume 5/Number 2, Summer 2011 A practical anatomy of incremental risk charge modeling Marcus R. W. Martin University of Applied Sciences, Haardtring 100, 64295 Darmstadt, Germany; email: marcus.martin@h-da.de Helmut Lutz DekaBank, Hahnstrasse 55, 60528 Frankfurt am Main, Germany; email: helmut.lutz@deka.de Carsten S. Wehn DekaBank, Hahnstrasse 55, 60528, Frankfurt am Main, Germany; email: wehn@gmx.de This paper considers the different elements of modeling the so-called incremental risk charge. We show the increasing regulatory requirements, as well as the reasoning behind this. We start our analysis by introducing a generic multifactor multistate credit risk model (CRM). Then we compare the economic needs for credit risk modeling with the regulatory requirements for the incremental risk charge and successively reduce the complexity of the CRM to meet these regulatory requirements. We analyze in depth the different steps of this simplification, ie, factor reduction, migration matrices and valuation grids. This results in a very efficient semianalytical incremental risk charge model that is still consistent with the CRM. This analysis gives us new insight into the mechanics of both approaches and thus into the models anatomies. The efficient and consistent reduction of complexity is highly relevant to practical applications. We emphasize our analysis by applying the different approaches to a real-life example portfolio. Marcus R. W. Martin is Professor of Financial Mathematics and Stochastics at the University of Applied Sciences, Darmstadt. Helmut Lutz is senior expert in the risk modeling team at DekaBank, Frankfurt. Carsten S. Wehn is head of the risk modeling team at DekaBank, Frankfurt. The risk modeling team is responsible for developing adequate methodologies for the measurement of market risks, credit risks, liquidity risks and operational risks in the respective portfolio models. It is also responsible for validating the adequacy of the respective risk methods as well as for aspects of stress testing and economic capital models in this context. All opinions expressed herein are the authors own and should not be cited as being those of their affiliated institutions. None of the methods described herein is claimed to be in actual use at DekaBank. 45

46 M. R. W. Martin et al 1 MOTIVATION AND INTRODUCTION The measurement of so-called incremental risks, ie, default and migration risks, for those trading book positions subject to the specific regulatory risk regime in an internal market risk model is one of the most challenging exercises for both practitioners and researchers. Due to the evolving regulatory requirements (which will be described in Section 2), classical credit risk models (CRMs), as used primarily for measuring the credit risk of banking book positions so far, would have to be reasonably extended to cover (highly complex) trading book positions as well. Furthermore, these models typically suffer from the high number of simulation paths needed for generating a reliable estimate of the 99.9% quantile of the loss distribution that is necessary for the positions to be included in the incremental risk charge (IRC) as well. Hence, a semianalytical single-factor (but still multistate) approximation of the loss distribution that yields IRC estimates close to and consistent with those produced by more sophisticated but simulation-based multifactor CRMs could be of critical importance in practice. We therefore focus on the question of how to derive a semianalytical approximation model in an IRC-compliant way and also quantify and analyze the respective impacts of approximations and simplifications performed for a real-life portfolio with bond positions. This paper is organized as follows. Section 2 depicts the latest regulatory developments and new requirements as well as results from benchmark studies on their likely impact on capital requirements. We discuss diverse model alternatives in Section 3 in order to stress the necessity of obtaining a computationally efficient, yet also the most accurate possible, approximation to multifactor simulation-based credit portfolio models. This approach has been used industry-wide for measuring the credit risk capital of loan books for several years. We discuss whether a semianalytical single-factor multistate approximation of the loss distribution might serve well for measuring the IRC, in the sense that its results are close to those obtained from simulation-based multifactor multistate CRMs. We discuss these issues in Section 4 before concluding with a short summary of our results and an outlook to further research in Section 5. 2 BACKGROUND 2.1 Regulatory developments and requirements The regulatory requirements for risk measurement in the trading book have been substantially amended and changed over the last five years. The long journey began in 1996 with the Basel Market Risk Amendment (Basel Committee on Banking Supervision (BCBS) (1996)), in which the first approaches to allow internal market risk models for regulatory capital charges were made. This served to put economic and regulatory risk management in line with a risk-sensitive measure. During the next The Journal of Risk Model Validation Volume 5/Number 2, Summer 2011

A practical anatomy of IRC modeling 47 decade the volume of traded credit products, as well as credit derivatives, in the trading book increased dramatically. This forced regulators to formulate the first proposals to strengthen the requirements (see, for example BCBS (2005, 2007)). The banking industry then suffered during the financial crisis and regulators were under political pressure to further strengthen the requirements for trading book risks and internal market risk models. The recently updated revisions to the Basel II Market Risk Framework (BCBS (2011a)) introduce many different requirements to measurement of trading book risks. Regulators see the special necessity of modeling discrete risks like migrations, defaults and event risks. Being part of the specific interest rate risk, the industry currently assumes that migrations and defaults already comprise all kind of events and this is measured by the IRC. For specific equity risk, event risks are also to be modeled, be it within the IRC model or within a different modeling framework. This comes along with the introduction of the so-called stressed value-at-risk (VaR) and special treatments for securitization and the correlation trading book. For the calculation of the IRC, the regulators require there to be a use test in place, ie, the institution has to prove that the figures serve as input for regular management decisions. The backbone of the IRC is very much in line with the requirements for the internal ratings-based approach for the banking book, especially the confidence level of 99.9% and the one-year risk horizon. The IRC model might take correlations between other defaults and migrations within the trading book into account, but not with respect to other market risks. Netting is only allowed with respect to the same financial instrument (not, for example, with respect to a certain credit unit). It is worth mentioning that the different capital charges (from VaR, stressed VaR, IRC, securitizations and comprehensive risk measure) all have to be added up to build the overall regulatory capital charge. 1 Concerning the risk horizon, the terms capital horizon and liquidity horizon have entered the proposals to reflect the observation that it does not seem to be adequate to assume, for assets from the trading book (that should by definition be tradable or at least available to hedge), that a portfolio will remain constant over a period like one year. A minimum period is set to three months for the liquidity horizon whereas the capital horizon is always one year. 2.2 Results from benchmark studies The latest available quantitative impact study comes from 2009 (BCBS (2009a)), so the latest proposals are not completely reflected in it. Nevertheless, this study provides 1 This is quite simplified since, for the final calculation, the maximum of the previous day s value and the sixty-day period multiplied by a certain factor have to be taken for VaR and stressed VaR. This factor even increases with the backtesting exceptions from the VaR part. Research Paper www.journalofriskmodelvalidation.com

48 M. R. W. Martin et al TABLE 1 Impact of the IRC under different parameterizations (different liquidity horizons and in default-only) for the BCBS (2009a) study. Incremental risk capital charge including default and migration risk for a liquidity Default-only horizon of: charge, Specific ƒ three-month risk one three six liquidity surcharge month months months horizon Mean 23 136 126 156 97 Median 17 92 84 98 66 Standard 20 131 132 159 92 deviation Minimum 1 9 5 5 7 Maximum 78 522 565 613 375 some helpful insights into the main areas of impact by the new regulations for the trading book. For the market risk capital requirements, the average increase for forty-three banks across ten countries is about 223.7%, where the IRC contributes 102.7%. 2 Table 1 shows the impact of the different parameters on the height of the IRC. The influence of the liquidity horizon is ambivalent, as the results show. On the one hand, the higher the liquidity horizon, the higher the capital charge, but on the other hand, the results vary broadly as the increasing standard deviation shows. 3 The inclusion of migrations in the IRC results in an average of a 33% higher capital requirement. From the quantitative impact study and also from other similar studies (see, for example, Algorithmics (2009)), we conclude that neither of the two approaches (the constant-level-of-risk approach (CLRA) and the constant-position approach (CPA)) is superior per se. 3 GENERAL MODELING ALTERNATIVES The regulatory requirements are quite closely related to the internal requirements for a CRM. All these requirements bring to mind the idea of applying or adopting a 2 Other main contributors are the stressed VaR with 110.8%, the changed requirements for equity specific risk under the standardized approach with 4.9% and the regulations for re-securitizations with 92.7%. 3 The overall impact of 102.7% is given by the increase of 126% at the three-month liquidity horizon and then by the capital relief of 23% due to the vanishing specific risk surcharge. The Journal of Risk Model Validation Volume 5/Number 2, Summer 2011

A practical anatomy of IRC modeling 49 multifactor CRM, whenever this already exists in an institution and covers the fundamentals like migration risk and defaults. In this section, we discuss the advantages and disadvantages of applying a multifactor CRM already implemented for credit risk calculations in a financial institution. 3.1 Pros and cons of credit portfolio based multifactor incremental risk charge models Most financial institutions already use multifactor CRMs for credit risk measurement and management purposes, with the CRMs either provided by a vendor or developed internally. Using these potentially available CRMs for calculating the IRC as well provides several advantages. In this case, for example, the CRM might become the starting point for a unified and consistent framework for measuring default and migration risks across market and credit risks in the long run. At first sight, one may levy on the fact that the model is well-known to all parties involved (at least for the calculation of credit risk economic capital), who should in this case have a working experience and knowledge of the model and its results. With regard to an internal capital adequacy process, one could expect to have all necessary processes already in place for the CRM. These points are also of great importance for internal acceptance of results and from the point of view of the so-called use-test. Nonetheless, there are some parameterizational issues that have to be considered when modifying any internally applied CRM. The CRM primarily serves the internal needs, which may or may not be compliant with the regulatory minimum requirements. It should be emphasized that such an internally applied CRM is especially designed to serve the particular needs of modeling the loan book. Hence, it might reflect portfolio-specific needs in depth by, for example, certain factor loadings, or other parameters, or by treating certain derivatives differently from regulatory requirements. Therefore, by adopting any internally applied CRM to measure IRC, one might expect to simply (and canonically) extend the existing infrastructure to be applicable across all asset classes or across banking and trading books. Unfortunately, reality is less comfortable. Trading book products do not typically enter credit risk capital calculations at the level of rigidity (in terms of single product evaluation, in particular with respect to full valuation requirements) or granularity required for regulatory IRC purposes. Sometimes methodological simplifications are in place for measuring the credit risk of certain product classes where economic needs are different from regulatory requirements for calculating IRC. Finally, IRC has to be calculated at least on a weekly basis, which usually implies the need to reassess the performance of data quality maintenance processes as well as the computational performance of the underlying credit portfolio model itself, Research Paper www.journalofriskmodelvalidation.com

50 M. R. W. Martin et al since economic capital for credit risk is usually calculated less frequently (say, on a quarterly or monthly basis). These facts motivated us to simplify the generic CRM to fulfil regulatory requirements for the IRC. Hence, despite the evident advantages, the drawbacks mentioned above lead us to question whether it is possible to find an adequate or at least reasonable approximation to the results of a multifactor CRM that is computationally efficient and easier to adapt to the requirements for calculating IRC. We intend to show that a semianalytical IRC implementation (as is well-known in credit derivatives pricing) will provide such a solution in a CPA for calculating IRC. 3.2 The constant-position approach versus the constant-level-of-risk approach Whilst our approach is to simplify a standard multifactor CRM with regard to the regulatory requirements, we mention one additional point for the sake of completeness: the difference between the CPA and the CLRA (see also Section 2.1). While the capital horizon of one year, as for credit risk, is fixed for any regulatory capital calculation, a bank can either choose to apply a one-year CPA for all portfolios or to model its incremental risks based on the (usually) shorter liquidity horizons under the assumption of a constant level of risk over the one-year holding period. The CLRA requires (see BCBS (2009b, Guidelines, Section 16)) that a bank rebalances, or rolls over, its trading positions over the one-year capital horizon in a manner that maintains the initial risk level, as indicated by a metric such as VaR or the profile of exposure by credit rating and concentration. This means incorporating the effect of replacing positions whose credit characteristics have improved or deteriorated over the liquidity horizon with positions that have risk characteristics equivalent to those that the original position had at the start of the liquidity horizon. The frequency of the assumed rebalancing must be governed by the liquidity horizon for a given position. In practice, implementing a CLRA therefore means implementing a multistep model for calculating the cumulated one-year profits and losses (P&L) based on the P&L generated for each time step according to the applicable liquidity horizon (which is subject to a regulatory three-month floor). In this paper we attempt to simplify the CRM to meet IRC requirements. We will therefore stick to the CPA. The reasons for this are twofold. First, the data needed to calibrate a CLRA calculation of IRC, in particular short-term migration and default probabilities and short-term asset correlations as well as short-term losses given default and exposures at default, is mostly neither readily available nor marketimplied from any liquidly traded financial products nor internally available (since typically credit risk is measured on the one-year time horizon). No best-practice methods to infer it from market data have been developed so far. Furthermore, the The Journal of Risk Model Validation Volume 5/Number 2, Summer 2011

A practical anatomy of IRC modeling 51 results from benchmark studies do not necessitate the application of the CLRA either (see Table 1 on page 48 and Algorithmics (2009)). The second reason refers to the easiest way of implementing an IRC model based on the CLRA. Such an implementation relies, loosely speaking, on just repeating a CPA-type simulation sequentially over the minimal liquidity horizon (of three months according to the regulatory floor) four times to generate the quarterly P&L. Assuming that the algorithms for replacing matured or defaulted positions are independent of the approach that is used for generating the quarterly P&L, there is, in principle, no particular difference between comparing the results of different alternative models on each liquidity horizon with the results over the full time horizon. For our purposes, the CPA means that the risk is fully represented in the valuation grids for the portfolio as it is today. This assumes that the composition of the portfolio remains constant over the capital horizon, that maturing products will be replaced identically at the capital horizon, and that no rebalancing is necessary. 3.3 Prerequisites for applying a semianalytical incremental risk charge model and building blocks In order to perform a reliable computation of IRC, we have to ensure that the semianalytical IRC model approximates the full multifactor credit portfolio model within an adequate error bound. In doing so, the first step is to decide whether we are able to calibrate the semianalytical model to the CRM in such a way that it reproduces the same risk characteristics without missing any important available information (see Section 4.2). Thereafter, the different components necessary for generating an IRC estimate are conceptually identical for semianalytical approximation and the CRM. (1) We have to ensure by the calibration that the asset values generated by our firstorder semianalytical approximation to the multifactor model are adequate. (2) Therefore, the remaining fundamental building block of the model is the migration matrix used to generate future rating transitions for every position (over the liquidity horizon of one year in the CPA). Consequently, the granularity of the rating grid on the one hand, and the dispersion of migration probabilities across the migration matrix on the other also influence the IRC results. (3) In the next step, a so-called valuation grid has to be provided for each position in the portfolio. It contains the full revaluation, ie, the present value, of the position under consideration for each future rating state generated before. The particular shape of these valuation grids and the granularity of the rating classes have an important impact on the hypothetical P&L generated by the model. We consider these points more closely below. Research Paper www.journalofriskmodelvalidation.com

52 M. R. W. Martin et al 4 MECHANICS OF INCREMENTAL RISK CHARGE MODELS AND IMPORTANT PARAMETERS 4.1 A multifactor credit risk model As mentioned above, in this section we start with a multifactor CRM approach to identify the main building blocks and to get the areas of potential approximations for IRC uses. Hence, we start with a multifactor Merton-based approach (see Bluhm et al (2008)) that can be used to model firms joint migrations and defaults, generically referred to as credit rating changes. Given the set of possible rating grades f1;:::;rg, the firm u with current rating r u.0/ is assigned a rating r u.t/ 2f1;:::;Rgat horizon t if and only if the firm s asset value return a u.t/ falls between well defined thresholds: a u.t/ 2.s ru.0/;r u.t/;s ru.0/;r u.t/c1 with s ru.0/;1 D 1;s ru.0/;rc1 DC1 These thresholds are calibrated in such a way that the possibility of a credit rating change from r u.0/ to r u.t/ is equal to the migration probability: P.a u.t/ 2.s ru.0/;r u.t/;s ru.0/;r u.t/c1 / To incorporate economic dependencies between firms we will use a factor model. An asset value return s stochastic process is driven by a set of systematic and idiosyncratic risk factors. Correlations between systematic risk factors induce economic dependencies between firms asset value returns. In this model, a firm s asset value return is given by: a u.t/ D NX nd1 ˇcu ;i u InX n.t/ C ˇidio;u " u.t/ The asset value returns a u.t/, systematic risk factors X n.t/, n D 1;:::;N; and idiosyncratic risk factors " u.t/ are modeled as standard normally distributed random variables with correlation matrix W X.t/ 0;. Idiosyncratic risk factors are assumed to be independent with respect to all other risk factors. Factor loadings ˇcu ;i u In with respect to the systematic risk factors are estimated to be identical for firms with the same country industry assignment c u, i u. The variance of the systematic part is given by: X N 2 Rc 2 u ;i u D E ˇcu ;i u In X n.t/ nd1 hence the variance of the idiosyncratic part follows as: q ˇidio;u D 1 Rc 2 u ;i u The Journal of Risk Model Validation Volume 5/Number 2, Summer 2011

4.2 The semianalytical incremental risk charge model 4.2.1 Factor reduction for incremental risk charge purposes A practical anatomy of IRC modeling 53 Due to the differences between the internal and regulatory requirements, the use of the CRM in the above-mentioned sense poses several challenges that together lead to it being necessary to simplify the CRM. 4 We therefore focus on the potential simplifications of the CRM, as introduced in Section 3.3, to serve our needs. This is done by implementing a semianalytical approximation and reducing complexity to a one-factor model, further referred to as the IRC model: q a u.t/ D Oˇcu ;i u X.t/ C 1 Oˇ2c u ;i u " u.t/ (4.1) We do not want to lose the advantages of an internally used CRM and its established risk management processes and thus, to ensure consistency, we calibrate our parameters to the internal model s parameters. Factor loadings with respect to the systematic risk factors are again estimated to be identical for firms with the same country industry assignment. We consider a portfolio consisting of M positions. Conditional on the factor realization X.T / D x at the risk horizon T, the credit rating changes from r u.0/ to r u.t / D r with probability: P.a u.t/ 2.s ru.0/;r;s ru.0/;rc1 j X.t/ D x/ sru.0/;rc1 D Oˇcu ;i u x sru.0/;r 0;1 q Oˇcu ;i u x 0;1 q 1 Oˇcu ;i u 1 Oˇcu ;i u (4.2) Let V m D v m;r be the loss of the mth position due to the above credit rating change and let u.m/ be the firm corresponding to this position. Conditional on the above factor realization and credit rating change we find: fv m D v m;r j X.T / D x ^ r u.m/.t / D rg () fa u.m/.t / 2.s ru.m/.0/;r;s ru.m/.0/;rc1 j X.T / D xg (4.3) 4 On the one hand, the internal treatment of positions might be different from the regulatory minimum requirements. On the other hand, there is a need for a higher computation frequency and hence for saving computational effort. Research Paper www.journalofriskmodelvalidation.com

54 M. R. W. Martin et al With V.m/ D P m id1 V i being the loss of a portfolio containing the first m positions, we can compute the portfolio s conditional loss distribution recursively: ( 0; v < 0 P.V.0/ 6 v j X.T / D x/ D 1; v > 0 P.V.m/ 6 v j X.T / D x/ RX D P.V.m 1/ 6 v v m;r j X.T / D x/ rd1 P.V m D v m;r j X.T / D x ^ r u.m/.t / D r/ RX D P.V.m 1/ 6 v v m;r j X.T / D x/ rd1 P.a u.m/.t / 2.s ru.m/.0/;r;s ru.m/.0/;rc1 j X.T / D x/; 1 6 m 6 M Finally, we find the portfolio s unconditional loss distribution by numerical integration: Z C1 P.V.M / 6 v/ D P.V.M / 6 v j X.T / D x/ ' 0;1.x/ dx (4.4) 1 The semianalytical approach for generating the loss distribution was first proposed by Andersen et al (2003) in a collateralized debt obligation pricing context. 5 4.2.2 Calibration to the multifactor case To calibrate the parameters of IRC models to those of the CRM, we assume that the variance of the systematic part: X N Rc 2 u ;i u D E nd1 2 ˇcu ;i u InX n.t/ is explained by only one systematic risk factor. Hence Oˇcu ;i u WD R cu ;i u is a meaningful choice concerning factor loadings in the one-factor case. The correlation between asset value returns is induced by: O uv D ( 1; u D v Oˇcu ;i u Oˇcv ;i v ; u v 5 The idea for applying this technique to IRC calculations as presented here was inspired by a talk given by Gerstein (2010), who focused on the use of different types of copulas. The Journal of Risk Model Validation Volume 5/Number 2, Summer 2011

A practical anatomy of IRC modeling 55 Compared with the correlation: ( 1; u D v Q uv D ˇTc u ;i u ˇcv ;i v ; u v (4.5) induced in the multifactor case we can try to minimize the term P u;v. Q uv O uv / 2 to improve our first guess concerning the factor loadings. At a slightly more advanced level we can perform an eigenvalue decomposition. We find the following factorization of the correlation matrix: 2 3 v 11 v 1N D A T A; A T D V T 1=2 6 ; V D : 4 : :: 7 : 5 v N1 v NN as the eigenvector matrix (ie, columns of this matrix are the eigenvectors) and D diag. 1 ;:::; n / as the diagonal matrix with the respective eigenvalues. Without loss of generality, let 1 > 2 > > N. Given a vector of standard normally distributed random variables F.t/ 0;I, the following fundamental relationship holds: A T F.t/ 0;. For this reason we can formulate the following factor model for the systematic risk factors: X n.t/ D NX j D1 q j v jn F j.t/; n D 1;:::;N with independent standard normally distributed random variables F j.t/, j D 1;:::;N. Restricting ourselves to the first factor X.t/ D F 1.t/, corresponding to the largest eigenvalue 1, we can formulate the following one-factor model: X n.t/ D p 1 v 1n X.t/; n D 1;:::;N In this model, firms asset value returns are given by: a u.t/ D Oˇcu ;i u X.t/ C q1 Oˇ2cu;iu " u.t/ with Oˇcu ;iu D p X N 1 nd1 ˇcu ;i u Inv 1n Example 4.1 We analyze a real-life portfolio with bond positions. With respect to the market value, 95% of the positions can be explained by only three country industry combinations. The following correlations O uv are induced by the CRM: Research Paper www.journalofriskmodelvalidation.com

56 M. R. W. Martin et al Correlation matrix by CRM 1.0000 0.2203 0.3356 0.2203 1.0000 0.2183 0.3356 0.2183 1.0000 Using the heuristic approach, we find the following correlations O uv in the IRC model with only one factor: Correlation matrix by IRC 1.0000 0.2552 0.3306 0.2552 1.0000 0.2508 0.3306 0.2508 1.0000 Since the two correlation matrices are close to each other, we can expect only a small effect during a change from a multifactor model to a one-factor model. To check this, we run one million simulations in both multifactor and one-factor implementations. To check the semianalytical implementation, we finally compare both simulationbased and semianalytical one-factor implementations. The multifactor model with one million simulation runs serves as the benchmark (100.0%). Quantile Quantile Quantile 0.1% 1.0% 5.0% Multifactor, one million simulations 100.0% 100.0% 100.0% One-factor, one million simulations 102.2% 103.1% 100.7% One-factor, semianalytical 99.4% 101.7% 100.6% The results are within the interval for simulation errors, that is, about 1.5% for the 0.1% quantile. The example also confirms the intuitive impression that the simplified approaches, especially the reduction of the number of factors made by the IRC model, are quite adequate for our purposes. 4.2.3 Migration matrices Let p ij D P.r.t/ D i j r.0/ D j/, i;j D 1;:::;R, be the migration probabilities and let n j be the sample size of the j th initial rating class serving as the basis for the The Journal of Risk Model Validation Volume 5/Number 2, Summer 2011

A practical anatomy of IRC modeling 57 estimation of migration matrices, ie, the number of firms with initial rating j within the sample. Hence, we can expect that m ij D p ij n j firms will migrate from initial rating j to rating i within one year. If we aggregate rating classes of both disjoint subsets I;J f1;:::;rg to two larger rating classes, then P P i2i j 2J m ij is the number of firms for which we can expect a migration from J to I. Hence: Op IJ D X X X m ij n j i2i j 2J j 2J is the corresponding migration probability. So we have reduced the rating granularity by calibration to expected migrations. Alternatively, we can reduce the granularity by calibration to expected losses. Given the potential losses L ij with respect to rating migrations from j to i, we can calculate the expected losses by EL ij D L ij p ij. P j 2J EL ij. A mean- Because expected losses are additive, we find EL IJ D P i2i ingful definition of L IJ is as follows. Within the final rating class I we calculate the mean potential loss with respect to the initial rating classes and then we sum up over all initial rating classes belonging to J : L IJ D X j 2J 1 ji j X L ij ; i2i I J Now we find the corresponding migration probability by Op IJ D EL IJ =L IJ ;I J ; Op II is easily found to be the difference between the value and one. Example 4.2 To check this we reduce the highly granular R R migration matrix to a less granular 7 7 matrix by calibration to expected losses and compare it with both the CRM and the IRC model. The multifactor model with R R matrix serves as the benchmark (100.0%). Quantile Quantile Quantile 0.1% 1.0% 5.0% Multifactor R R matrix, 100.0% 100.0% 100.0% one million simulations Multifactor 7 7 matrix, 95.5% 100.8% 100.7% one million simulations One-factor 7 7 matrix, 94.7% 102.1% 101.7% semianalytical We find that there is only a small effect due to migration matrix reduction and find similar results with both reduced multifactor and semianalytic implementation. Research Paper www.journalofriskmodelvalidation.com

58 M. R. W. Martin et al 4.2.4 Valuation grids and pricing impact To obtain the position loss at the risk horizon T given a credit rating change to r u.m/.t / D r we have to revalue the position with its specific forward zero curves for the changed rating F cu.m/ ;i u.m/ ;r.t / and the current rating F cu.m/ ;i u.m/ ;r u.m/.0/.t / both as of the risk horizon T : v m;r D V T.F cu.m/ ;i u.m/ ;r.t // V T.F cu.m/ ;i u.m/ ;r u.m/.0/.t //; r < R In case of a default, the loss is simply given by the specific recovery applied to the nominal: v m;r D.1 u.m/ /Nom. To further improve the ease and speed of the IRC, the idea is to use valuation grids with present values typically used in the market risk environment instead of forward values. Hence we can try to approximate forward values at the risk horizon by present values as of today: v m;r V 0.F cu.m/ ;i u.m/ ;r.0// V 0.F cu.m/ ;i u.m/ ;r u.m/.0/.0//; r < R It should be mentioned that the positions on shortened maturities remain unconsidered, but incurring payments during the risk horizon remain considered as opposed to the pure forward-value approach. Example 4.3 We compared both approaches using forward values and present values, both with a less granular migration matrix and within the semianalytic implementation: Quantile Quantile Quantile 0.1% 1.0% 5.0% Forward-value approach 100.0% 100.0% 100.0% Present-value approach 108.2% 138.6% 150.5% We find that the effect is not negligible. The overestimation within the present-value approach is caused especially by the pull-to-par effect for bonds maturing close to the risk horizon. The example underlines that using present values instead of forward values results in overestimation of the IRC. Hence, this approximation seems not to be appropriate. 5 CONCLUSION AND OUTLOOK Summarizing the results of Section 4, we were able to show that a carefully calibrated semianalytical single-factor approximation of the portfolio s loss distribution can be The Journal of Risk Model Validation Volume 5/Number 2, Summer 2011

A practical anatomy of IRC modeling 59 achieved, one that yields values for the incremental risk charge consistent with and close to a full-blown simulation-based multifactor model. Thus, the use of the risk management processes already in place is possible. In particular, we were able to provide estimates of the capital impact of our simplifying assumptions for a reallife portfolio, implying the adequacy of the semianalytical approximation while still fulfilling the regulatory requirements for calculating the capital charge. Hence, we provide some potential reasons for choosing a flexible and computationally efficient alternative to simulation-based models. Future research might include the following. Alternative recursion formulas to the one implemented above are provided, for example, by Hull and White (2004). These could be studied in more detail since they allow for reflection of stochastic recovery rates within the semianalytical framework as well, which is a topic of growing importance in credit portfolio modeling. The semianalytical approximation can also be extended to copulas other than the Gaussian as used above and should be subject to further investigation (see also Gerstein (2010)). Furthermore, the comparison with other approaches for efficiently generating the conditional loss distribution, such as fast Fourier transform, could also be discussed. One might expect similar results, namely that the recursive method is still faster than a fast Fourier transform-based approach, as is already known in the literature (see Andersen et al (2003) and Jackson et al (2007)). REFERENCES Algorithmics (2009). Comments on Revisions to the Basel II market risk framework and Guidelines for computing capital for incremental risk in the trading book. URL: www. bis.org/publ/bcbs14849/ca/algorithmics.pdf. Annexes: URL: www.algorithmics.com/en/ media/pdfs/algo-gc1008-irc-baselcommitteeannexes.pdf. Andersen, L., Sidenius, J., and Basu, S. (2003). All your hedges in one basket. Risk 16(11), 67 72. Basel Committee on Banking Supervision (1996). Amendment to the capital accord to incorporate market risks. URL: www.bis.org/publ/bcbs24.pdf. Basel Committee on Banking Supervision (2005). The application of Basel II to trading activities and the treatment of double default effects. URL: www.bis.org/publ/ bcbs116.pdf. Basel Committee on Banking Supervision (2007). Guidelines for computing capital for incremental default risk in the trading book. Consultative Document. URL: www.bis.org/ publ/bcbs134.pdf. Basel Committee on Banking Supervision (2009a). Analysis of the trading book quantitative impact study. URL: www.bis.org/publ/bcbs163.pdf. Basel Committee on Banking Supervision (2009b). Guidelines for computing capital for incremental risk in the trading book. URL: www.bis.org/publ/bcbs159.pdf. Research Paper www.journalofriskmodelvalidation.com

60 M. R. W. Martin et al Basel Committee on Banking Supervision (2011a). Revisions to the Basel II market risk framework updated as of February 2011. URL: www.bis.org/publ/bcbs193.pdf. Basel Committee on Banking Supervision (2011b). Interpretive issues with respect to the revisions to the market risk framework. URL: www.bis.org/publ/bcbs193a.pdf. Bluhm, C., Overbeck, L., and Wagner, C. (2008). An Introduction to Credit Risk Modeling, 2nd edition. Taylor & Francis. Gerstein, M. (2010). Modelling IRC: a semianalytical Gaussian copula implementation. Presentation at Understanding Developments to Market Risk Frameworks in the Trading Book Conference, June 17 18, 2010, London. Hull, J. C., and White, A. (2004). Valuation of a CDO and an nth to default CDS without Monte Carlo simulation. Journal of Derivatives 12(2), 8 23. Jackson, K., Kreinin, A., and Ma, X. (2007). Loss distribution evaluation for synthetic CDOs. Working Paper, University of Toronto. The Journal of Risk Model Validation Volume 5/Number 2, Summer 2011