# Journal of Risk

**ISSN:**

1465-1211 (print)

1755-2842 (online)

**Editor-in-chief:** Farid AitSahlia

# Default risk charge: modeling framework for the “Basel” risk measure

####
Need to know

- We propose a modeling framework for the Default Risk Charge (DRC) measure consistent with the requirements of the new regulatory standards for market risk for banks’ trading books.
- The properties of the proposed DRC model, the related calibration and implementation aspects and a comparison to the Standardised Approach (SA) for default risk are explored through the use of example portfolios.
- The proposals and findings are relevant for establishing industry-wide modeling standards, for benchmarking exercises between banks or for the determination of potential capital floors based on the SA.

####
Abstract

As a result of the Basel Committee on Banking Supervision’s Fundamental Review of the Trading Book, revised standards for capital requirements for market risk in banks’ trading books have been issued. Under the new standards, default risk needs to be measured and capitalized through a dedicated default risk charge (DRC). Although quantitative impact studies are ongoing and banks are preparing for these regulatory changes, this paper is the first to present a modeling framework for the DRC measure that projects losses over a one-year capital horizon at a 99.9% confidence level. We discuss selected risk factor models, which we use to derive simulation-based loss distributions and associated default risk figures. The model’s properties, aspects of its implementation and a comparison with the standardized approach for default risk are explored through the use of example portfolios.

####
Introduction

Since the mid-1990s, banks have been allowed to use internal models to calculate market risk capital requirements for activities in their trading books. These models are subject to approval by supervisory authorities. They are supposed to reflect large, adverse market moves and are usually based on a 99% ten-day value-at-risk (VaR). Owing to the fact that VaR models usually do not account for the potential illiquidity of trading positions, and since market losses can be driven by large cumulative price moves, in the aftermath of the worldwide 2007–8 financial crisis, new capital charges were proposed to complement the existing market risk capital requirements (Basel Committee on Banking Supervision 2009, 2011; European Banking Authority 2012). The concept of stressed VaR and its associated capital charges were introduced in January 2013 (in Europe); these were supplemented by the incremental risk charge (IRC), which was intended to cover market risks from credit rating migrations and defaults for flow instruments, such as bonds and credit default swaps (CDSs). More complex instruments such as collateralized debt obligations (CDOs) have been made subject to a separate, extended market risk model: the comprehensive risk measure (CRM). These changes, widely referred to as “Basel 2.5”, can be seen as rather short-term fixes, as they do not provide a consistent overall risk capital framework.

With the aim of creating a more consistent market risk capital framework, a Fundamental Review of the Trading Book (FRTB) was recently conducted by the Basel Committee on Banking Supervision (BCBS). The result of the FRTB has been the revision of the minimum capital requirements for market risk (see Basel Committee on Banking Supervision 2016). The main risk figure determining capital requirements will be changed to an expected shortfall (ES) at a 97.5% confidence level. In addition, this figure shall account for different liquidity horizons of trading positions and their associated risk factors, and it will be calibrated to a period of market stress. It is to be accompanied by a default risk charge (DRC) for nonsecuritization products, which will measure the trading portfolio’s default risk based on the 99.9% loss over a one-year capital horizon, or, at an institution’s discretion, a minimum of sixty days for equity subportfolios. In contrast to the IRC, the DRC measure does not consider rating migration risk and rules out any portfolio rebalancing assumptions. The coverage is extended to equity positions, and projected losses need to reflect stressed market conditions under the new rule set. The DRC model is to be applied in conjunction with the ES-based market risk capital charge. Should the latter fail, given quality criteria such as backtesting and profit and loss (P&L) attribution, either at the bank-wide or trading-desk level, a standardized approach (SA) will need to be applied for both ES and DRC. Otherwise, the DRC can be calculated using an institution’s own internal model, which requires validation by the relevant home supervisor and has to be computed at least on a weekly basis. For securitization products (correlation trading and noncorrelation trading), the DRC measure needs to be calculated using a dedicated SA; internal models are not allowed.

While the industry is participating in ongoing quantitative impact studies (QISs) and preparing for the implementation of the revised market risk capital standards, no literature on a suitable model for DRC has been published so far. The only notable exception is Laurent et al (2016), in which the regulation-imposed constraints on correlation matrixes and the factor structures that underpin the dependence between defaults are analyzed. In the case of IRC and CRM, Wilkens et al (2013) spearheaded the modeling discussion. By presenting a comprehensive DRC model, this paper closes a substantial gap in the research and can serve as a basis for an industry-wide discussion. Our model development is accompanied by example calculations and a discussion of implementation aspects. As in the case of IRC, high confidence levels and long projection horizons, in conjunction with limited backtesting feasibility, leave a substantial model risk.

The paper is organized as follows. Section 2 discusses the fundamental elements of a modeling framework for DRC. Section 3 focusses on the modeling of marginal, joint default and recovery rate risk. Section 4 is dedicated to the generation of P&L distributions, which are illustrated using a range of example portfolios. Aspects such as convergence and sensitivity analyses are also addressed. A quantitative comparison between the DRC based on the developed model and the SA is provided as well. Section 5 concludes. Some details on model data and calibration as well as a discussion of some model aspects and results have been relegated to an online appendix.

## 2 Fundamental elements of a modeling framework

The DRC has to capture default risk in a bank’s trading book (Basel Committee on Banking Supervision 2016, p. 60).^{1}^{1}The previously contemplated incremental default risk (IDR) was supposed to allow for the removal of any double counting between ES and default charge. This option has been removed with the modification leading to the DRC. Extending the coverage of the former IRC, the affected instruments are those which are not subject to standardized charges, and whose valuations do not depend solely on commodity prices or foreign exchange rates. Hence, bonds (including defaulted debt positions) and vanilla credit derivatives such as CDSs on single names and indexes as well as equity positions are in scope. While this is already the case for IRC, the regulation emphasizes the inclusion of sovereign exposures (including those denominated in the sovereign’s domestic currency). The risk is to be measured by means of a VaR-type measure at a 99.9% confidence level for a one-year capital horizon (ie, assuming a constant portfolio), or, at an institution’s discretion, a sixty-day horizon for designated equity subportfolios. This setup propagates the use of instantaneous shocks, ie, any time-value changes as well as cashflows are to be ignored.

With regard to the granularity of the simulation, the regulation foresees the obligor and its default risk as the primary level, while accounting for different losses from different instruments backed by the same obligor. As an example, if a parent company is the guarantor for the bond issues of two subsidiaries, the default risk for long positions amounts to the sum of the present values minus the bond-specific (simulated) recovery rates. The DRC model needs to reflect the probability of default of the obligor itself in conjunction with both (conditional) recovery rates.

## 3 Capturing default and recovery rate risk

### 3.1 Marginal default risk

As a first step, default probabilities of corporates and sovereigns need to be determined. These can be implied from market prices of bonds and credit derivatives (known as risk-neutral default probabilities) or calculated from historical default observations (known as historical or objective default probabilities). The difference between risk-neutral and historical default probabilities is discussed in Hull et al (2005), among many others. A shortcoming of market-implied probabilities is that they embed market risk premiums, which tend to bias the prediction of the actual default frequency. Regulation emphasizes that a correction of the market-implied default probabilities would be mandatory to arrive at objective probabilities of default (Basel Committee on Banking Supervision 2016, p. 61). Given the difficulties in estimating market risk premiums and the large uncertainty around the estimates, the use of historical default probabilities is usually preferable.

In an attempt to increase the accuracy of future predictions, historical default rates, by rating, can be differentiated by attributes such as type of obligor (eg, corporate, sovereign), region and industry. A high granularity of default probabilities needs to be balanced against the data availability of historical default observations and whether differences are statistically significant (see, for example, Moody’s 2011). In addition, certain highly rated types of obligors that did not experience defaults in the past (eg, AAA-rated sovereigns) will lead to a default probability of zero unless additional assumptions are imposed.

In general, default rates tend to vary over business cycles, with more defaults observed during recessions (see Altman and Kalotay (2014), among many others). Changes in default risk can be traced further, for example, to account for global and industry effects (see Aretz and Pope 2013). One can generally distinguish between through-the-cycle and point-in-time default probabilities. While point-in-time estimates tend to be more risk sensitive, as they better reflect current economic conditions, they can imply instability in the forecast as well as potentially procyclical risk measures. As such, they can lead to procyclical capital requirements. For the application in the context of DRC, through-the-cycle probabilities therefore seem preferable. Based on these considerations, in the following, through-the-cycle default probabilities differentiated by corporates and sovereigns as provided by Standard & Poor’s (S&P) are used.

In practice, regulation requires that banks with an internal ratings-based approach (IRBA) use probabilities of default from their own internal framework (see Basel Committee on Banking Supervision 2016, p. 62).^{2}^{2}This requirement is similar in nature to the rules in the United States, where the Dodd–Frank Wall Street Reform and Consumer Protection Act (2010) requires federal agencies to remove any references to external credit ratings from federal regulations, and to replace such ratings with different standards of credit quality; see Section 939A, paragraph (b). The spirit of the regulation is therefore to encourage banks not to rely on external ratings and corresponding default probabilities for the determination of capital, but rather to have their own assessment of the credit quality of different obligors or types of securities.

Table 1 provides the average one-year default probabilities between 1981 and 2012 for corporates and between 1975 and 2012 for sovereigns. For corporates with ratings better than AA$-$, and for sovereigns with ratings better than BB$+$, default probabilities are floored at 3 basis points (bps), which is the minimum value prescribed by regulation.^{3}^{3}See the critical assessment by Chourdakis and Jena (2013) on the levels of default probability that can be inferred for events with few or even no occurrences in history, such as sovereign defaults. In general, the regulatory requirement of a floor can be incorporated into more advanced methods to calibrate default probability curves (see, for example, Tasche 2013). The sensitivity of the DRC measure to the floor is likely material, given that banks’ portfolios tend to comprise significant positions in securities of well-rated sovereigns and corporates.

Rating | Corporates | Sovereigns |
---|---|---|

(S&P) | 1981–2012 | 1975–2012 |

AAA | 0.00* | 0.00* |

AA$+$ | 0.00* | 0.00* |

AA | 0.02* | 0.00* |

AA$-$ | 0.03* | 0.00* |

A$+$ | 0.06 | 0.00* |

A | 0.07 | 0.00* |

A$-$ | 0.07 | 0.00* |

BBB$+$ | 0.14 | 0.00* |

BBB | 0.20 | 0.00* |

BBB$-$ | 0.35 | 0.00* |

BB$+$ | 0.47 | 0.10 |

BB | 0.71 | 0.41${}^{\&}$ |

BB$-$ | 1.21 | 1.70 |

B$+$ | 2.40 | 2.06${}^{\&}$ |

B | 5.10 | 2.50 |

B$-$ | 8.17 | 6.30 |

CCC/C | 26.85 | 34.00 |

The default probabilities in Table 1 determine the asset return thresholds that trigger defaults within the simulation model.

### 3.2 Correlation of defaults across obligors

#### 3.2.1 Overview

The DRC model needs to reproduce the dependence between defaults of different obligors. Regulation prescribes the use of a default correlation model with two types of systematic factors, based on listed equity prices or CDS spreads (Basel Committee on Banking Supervision 2016, p. 60). The correlations must be based on “a calibration period of at least ten years that includes a period of stress” as well as “measured over a one-year liquidity horizon” (Basel Committee on Banking Supervision 2016, p. 61).

These regulatory requirements provide only loose guidance on the correlation estimates to be used in the default model. Different proxies have been considered in the literature for the objective of measuring default correlations. Moody’s (2008) makes the case for using asset correlations (estimated via a structural Merton-type model (Merton 1974)) as predictors for subsequently realized default correlations. Alternatively, equity correlations have been shown to considerably overestimate asset correlations (see, for example, Düllmann et al 2008) and to be, at best, noisy indicators of default correlations (see, for example, De Servigny and Renault 2002; Qi et al 2015). Vassalou and Xing (2004) analyze the intimate relationship between default risk and equity returns, also providing a risk-based interpretation of certain stylized effects (eg, company size). In general, equity (and CDS) prices embed much more codependence than the simple default correlation. With default being an absorbing state, market prices of nondefaulted names by nature cannot provide a straightforward measure of default correlation. Some recent approaches, such as Liu et al (2012), allow the direct use of equity correlations in the prediction of joint defaults; however, they rely on the correlations implied by actual defaults as an additional calibration source.

While one should take into account all the previous findings on default correlation estimates, the explicit reference in the regulation to listed equity prices (or CDS spreads) points toward the use of a direct measure of correlation between them. This requirement rules out alternatives, such as structural-type models or other approaches, which use additional information (eg, debt information, asset values, actual default correlations). Using directly observed equity (or CDS spread) correlations has a significant advantage for portfolio risk measurement, management and analysis, as it allows us to clearly explain the drivers and the magnitude of the co-movements between the names in the portfolio or subportfolios, and perform stress tests and sensitivity analyses with respect to the market level of correlations.

#### 3.2.2 Data

In this paper, equity prices are used in the estimation of the correlation model. The application to CDS spreads would be analogous. In general, equity data offers a significantly wider coverage than CDS spreads for corporate obligors; hence, it is preferable, given the wide scope of the DRC, which encompasses both equity and credit products.

From a very fundamental perspective, “correlation” can be defined and measured in many different ways. This poses a series of challenges and pitfalls, especially in the context of tail events (see, for example, Embrechts et al 1999). Regulation does not provide any guidance on the intended interpretation; given the simple implementation and measurement, we can assume that the framework refers to a classical linear (Pearson) correlation, in spite of its known shortcomings. For the actual parameter determination, return windows and the measurement interval need to be chosen. While the targeted forecasting horizon refers to a one-year period, it is not viable to measure correlations based on (nonoverlapping) one-year equity returns, since the great uncertainty in the measurement prevents this. Along the same lines, the measurement interval needs to have a minimum length in order to allow for a meaningful estimation of return correlations. As an illustration, in the case of a measured (Pearson) correlation between two equities of $+$10%, the 95% confidence interval around a correlation estimate amounts to approximately [$-$86%, $+$90%] for a sample of five observations, and shrinks to about [$-$36%, $+$52%] for twenty observations and [$-$24%, $+$42%] for thirty-five observations. Hence, the combination of length of return window and measurement interval needs to allow for a certain minimum number of observations in order to render the exercise reasonable.

One possible interpretation of the prescribed calibration period is to measure correlations over rolling time windows covering at least ten years, thereby identifying a period of stress. The latter is not defined further; it is likely supposed to point to a period for which the DRC model provides the comparatively highest loss estimate for a bank. This leaves two routes for investigation. As a first approach, the period can be identified as that with the highest (pairwise) correlation. This is a suitable route in case the DRC is ceteris paribus higher the higher the correlation. For a directional portfolio (for example, holding bonds), this assumption is valid for reasonable marginal distributions.^{4}^{4}As is well known from VaR-based models, a high correlation reduces the diversification benefit within a portfolio and allows for more extreme outcomes. This is especially true if the marginal distributions are fat-tailed, as in the digital case of default/nondefault. In order to put this approach into practice, the running pairwise correlation over a period of at least ten years can serve for the stress identification. As a second approach, the DRC can be run multiple times for a series of correlation measurement periods, and the stress period can be identified as that leading to the highest loss. While this approach does not require an ex ante assumption on the DRC as a function of correlation levels, it is much more involved from a computational perspective, and likely less stable, since it is portfolio dependent. The latter, in particular, is not a desirable feature for a risk measure with a long projection horizon. The example portfolios in Section 4.2 readdress this topic and study the influence of correlation assumptions on the DRC by means of sensitivity analyses.

Another possible interpretation of the prescribed calibration period is the direct use of at least ten years of data to measure correlations, justifying that it covers a period of stress. Under this interpretation, the correlation estimates would exhibit more of a through-the-cycle character. Nevertheless, one would likely need to revert to the tools described for the first interpretation.

In the following, the first approach is adopted, targeting the identification and use of a period of stress for the correlation input data used to calibrate the model. The practical interpretation chosen here refers to monthly (nonoverlapping) returns over three-year periods, which leads to thirty-five returns for each measured pairwise correlation. This can be seen as a reasonable compromise between the forecasting horizon of the model and the parameter estimation from historical data; robustness checks with regard to the choice are generally recommended. The embedded assumption is that correlations measured over monthly and annual intervals are identical (and a good predictor for future one-year correlations).^{5}^{5}Note the guidance that “these correlations should be based on objective data and not chosen in an opportunistic way where a higher correlation is used for portfolios with a mix of long and short positions and a low correlation used for portfolios with long only exposures” (Basel Committee on Banking Supervision 2016, p. 61). This hypothesis can be challenged (see studies such as Erb et al (1994) and Turley (2012), among many others), but, given the uncertainty of the correlation measurement itself (confer the width of the confidence intervals), it is hard to reject from a statistical perspective.

For the subsequent illustrations, historical equity prices of the constituents of a broad set of worldwide equity indexes are used. Using the index compositions at end-December 2013, the time series for 1990–2013 of members from EURO STOXX 50, Swiss Market Index (SMI), Financial Times Stock Exchange (FTSE) 100, S&P 500, Australian Securities Exchange (ASX) 200, Hang Seng index (HSI) and Nikkei 225 are selected.^{6}^{6}Since the DRC covers both equity- and credit-sensitive trading book positions, one might enlarge the equity data set to explicitly cover the constituents from the main credit indexes, such as iTraxx and the credit default swap index (CDX). Many of the names, however, are already covered by the equity index selection used here, for example, iTraxx Europe is well represented by EURO STOXX 50 and FTSE 100. The selection from the main equity indexes puts an inherent focus on larger companies. If a bank’s portfolio were mainly concentrated on smaller companies with listed equities, the selection could be amended as required in order to capture potentially different correlation structures among names. Section A of the online appendix provides further details on the data.

Pairwise equity correlations are calculated over a rolling window spanning three years and using monthly returns. Figure 1 shows different percentiles of the distribution of correlations as well as the average width of the 95% confidence level around the individual correlation estimates.^{7}^{7}Figure 1 shows correlations only between mid-1991 and mid-2012, since the three-year window requires one-and-a-half years of return data before and after the reference date. The peak of the correlation is found around the 2008–9 time period, as expected, with a median correlation of about 45%; the corresponding surrounding three-year period is marked in the figure and chosen as September 2007 through to September 2010 in the following. While the estimation uncertainty is high, as indicated by the confidence intervals, the correlation pattern over time clearly points toward this crisis period. Notably, this also holds when conducting the analysis with different return windows (eg, quarterly) and measurement periods (eg, two years; not shown here).

#### 3.2.3 Model description

One way of addressing the requirement that factor correlation models have two types of systematic factors, proposed by Laurent et al (2016), consists of calibrating empirically observed correlations to a “nearest” correlation matrix with a given number of factors. While this approach is a best fit in mathematical terms, it will usually be difficult to attribute an economic meaning to the resulting factors. Furthermore, the stability of such fitted matrixes over time might not be sufficient.

An alternative approach pursued here is to define the economic factors driving equity returns. The first systematic factor is a global factor that is common to all obligors, reflecting the overall state of the economy and, in particular, potential periods of stress. For the second systematic factor, we consider both a country- and an industry-specific factor.^{8}^{8}As noted in footnote 6, other factors such as company size could be relevant (see, for example, Fama and French 1993). With regard to the prescribed regulation, it is unclear how the requirement to “reflect all significant basis risks in recognising these correlations, including, for example, maturity mismatches, internal or external ratings, vintage, etc” (Basel Committee on Banking Supervision 2016, p. 62) could be interpreted and accommodated. While this in essence reflects a full set of country and industry factors, it is in line with the rule set, which refers to two types of systematic factors. Nevertheless, with these constraints, only the country or the industry would be amenable as explanatory factors. Subsequently, and in accordance with industry practice and the established modeling technique adopted for the IRC charge, the most general form with a global factor and country and industry factors is explored, in addition to the simplified variants. For example, Ford Motor Company would be associated with the United States and/or consumer cyclicals, respectively. In this regard, Aretz and Pope (2013), for example, argue that changes in default risk depend most strongly on global and industry effects, with country effects usually being more dependent on the sample period. The last return model component is a corporate-specific, idiosyncratic factor. As for countries themselves (or, say, municipals and similar noncorporate obligors), one can express returns as a function of a global and a country-specific return factor.

The first practical step in model building and calibration consists of standardizing each of the individual time series of corporate returns (${r}_{i,t}$) to a mean of zero and a standard deviation of one. At each time point, global (${r}_{\mathrm{G},t}$), country (${r}_{\mathrm{C}(j),t}$) and industry returns (${r}_{\mathrm{I}(k),t}$) are derived from the relevant cross-section of the corporate returns.^{9}^{9}At least five observations in a cross-sectional return set are required here for a valid systematic return. Let ${N}_{\mathrm{C}}$ and ${N}_{\mathrm{I}}$ denote the number of countries and industries, respectively. The resulting factor time series, then, all have a mean of zero. In order to capture the dependence of country and industry returns on global returns, the following linear regressions are run:

${r}_{\mathrm{C}(j),t}$ | $={\beta}_{\mathrm{C}(j)}{r}_{\mathrm{G},t}+{\epsilon}_{\mathrm{C}(j),t},$ | |||

${r}_{\mathrm{I}(k),t}$ | $={\beta}_{\mathrm{I}(k)}{r}_{\mathrm{G},t}+{\epsilon}_{\mathrm{I}(k),t},$ | (3.1) |

where ${\beta}_{\mathrm{C}(j)}$ and ${\beta}_{\mathrm{I}(k)}$ are weights given to the global factor, and ${\epsilon}_{\mathrm{C}(j),t}$ and ${\epsilon}_{\mathrm{I}(k),t}$ are the country- and industry-specific residuals. Let ${\sigma}_{\mathrm{G}}$ denote the standard deviation of ${r}_{\mathrm{G},t}$, and let ${\sigma}_{\mathrm{C}(j)}$ and ${\sigma}_{\mathrm{I}(k)}$ denote that of the residuals ${\epsilon}_{\mathrm{C}(j),t}$ and ${\epsilon}_{\mathrm{I}(k),t}$, respectively.

The details of the regression analysis run on the basis of (3.1) for the identified stress period are provided in Section B of the online appendix. Most of the country and industry returns move in line with the global returns (${\beta}_{\mathrm{C}(j)}$ and ${\beta}_{\mathrm{I}(k)}$ are not statistically different from one), and the explanatory content reflected by ${R}^{2}$ is high (between 65% and 95%), indicating the dominant role of the global factor.

Moving on to the case of single corporates and their returns, one can postulate that these are, in the most general case, a function of the respective global, country and industry returns. Since the country and industry returns have already been expressed via the global factor in (3.1), only the residuals ${\epsilon}_{\mathrm{C}(j)}$ and ${\epsilon}_{\mathrm{I}(k)}$ are considered as additional explanatory factors for the corporate returns. It is worth noting that reusing the coefficients ${\beta}_{\mathrm{C}(j)}$ and ${\beta}_{\mathrm{I}(k)}$ from (3.1) as the weights for the global factor is not possible without introducing assumptions on the relationship of country and industry returns vis-à-vis corporates. Therefore, in the following, the sensitivity to the global factor itself is expressed via a separate coefficient. The sensitivities to the country-specific and industry-specific factors are ${\gamma}_{\mathrm{C}(1)},{\gamma}_{\mathrm{C}(2)},\mathrm{\dots},{\gamma}_{\mathrm{C}({N}_{\mathrm{C}}-1)}$ and ${\gamma}_{\mathrm{I}(1)},{\gamma}_{\mathrm{I}(2)},\mathrm{\dots},{\gamma}_{\mathrm{I}({N}_{\mathrm{I}}-1)}$, respectively:

${r}_{i,t}={\gamma}_{\mathrm{G}}{r}_{\mathrm{G},t}$ | $+{\displaystyle \sum _{j=1}^{{N}_{\mathrm{C}}-1}}{\gamma}_{\mathrm{C}(j)}{\epsilon}_{\mathrm{C}(j),t}{\mathrm{?}}_{\{\mathrm{C}(i)=\mathrm{C}(j)\}}$ | |||

$+{\displaystyle \sum _{k=1}^{{N}_{\mathrm{I}}-1}}{\gamma}_{\mathrm{I}(k)}{\epsilon}_{\mathrm{I}(k),t}{\mathrm{?}}_{\{\mathrm{I}(i)=\mathrm{I}(k)\}}+{\epsilon}_{i,t},$ | (3.2) |

where $\mathrm{?}$ is the indicator function. In order to address the problem of multicollinearity in the model setup – each corporate belongs to one country and one industry – one country (here, United States) and one industry (here, utilities) is omitted from (3.2); this means that excess returns stemming from the country and industry factors are expressed relative to this base case.

The panel regression in (3.2) uses the full sample information in order to identify the systemic factor structure explaining the cross-sectional and time series covariation in corporate returns. The goodness-of-fit for (3.2) is assessed via an overall ${R}^{2}$ coefficient. The parameter estimates from (3.2), ${\widehat{\gamma}}_{\mathrm{G}},{\widehat{\gamma}}_{\mathrm{C}(1)},\mathrm{\dots},{\widehat{\gamma}}_{\mathrm{C}({N}_{\mathrm{C}}-1)},{\widehat{\gamma}}_{\mathrm{I}(1)},{\widehat{\gamma}}_{\mathrm{I}({N}_{\mathrm{I}}-1)}$, are then used to calculate an aggregated systematic return for each obligor. All names in a specific country $\mathrm{C}(i)$ and industry $\mathrm{I}(i)$ have a common aggregated systematic return component, as given by ${\widehat{\gamma}}_{\mathrm{G}}{r}_{\mathrm{G},t}+{\widehat{\gamma}}_{\mathrm{C}(i)}{\epsilon}_{\mathrm{C}(i),t}+{\widehat{\gamma}}_{\mathrm{I}(i)}{\epsilon}_{\mathrm{I}(i),t}$. The sensitivity of each obligor to this systematic return component as well as name-specific values of ${R}^{2}$ are obtained in a second estimation step by regressing the returns of each obligor against the corresponding systematic component:

$${r}_{i,t}={\beta}_{i}({\widehat{\gamma}}_{\mathrm{G}}{r}_{\mathrm{G},t}+{\widehat{\gamma}}_{\mathrm{C}(i)}{\epsilon}_{\mathrm{C}(i),t}+{\widehat{\gamma}}_{\mathrm{I}(i)}{\epsilon}_{\mathrm{I}(i),t})+{\epsilon}_{i,t},$$ | (3.3) |

where ${\widehat{\gamma}}_{\mathrm{G}}$, ${\widehat{\gamma}}_{\mathrm{C}(i)}$ and ${\widehat{\gamma}}_{\mathrm{I}(i)}$ represent the parameter estimates from (3.2).

The model proposed here and described in (3.1)–(3.3) bears some similarity to the Moody’s KMV (GCorr) model, to the extent that it captures global, country and industry factors as the common drivers of asset returns (see Bohn and Stein (2009, Chapter 8) for a description of the Moody’s KMV model). However, there are some important differences between the two models. First, the correlation model in this paper is based on directly observable equity returns, while the GCorr model is based on (unobserved) asset returns estimated from equity prices and balance sheet information. As discussed above, the observability is a feature that the regulatory text emphasizes. Second, the GCorr model is a multifactor model with a number of systematic factors that exceeds the maximum according to the regulation.^{10}^{10}For instance, there are fourteen common systematic factors in addition to industry- and country-specific factors.

#### 3.2.4 Estimation results

Table 2 summarizes the results of the multilinear panel regression analysis run on the basis of (3.2) for the stress period. Using only the global factor as an explanatory variable accounts for about 43% of the variation in equity returns. Unsurprisingly, the corresponding best-fitting return model equals the global return (${r}_{i,t}\approx {r}_{\mathrm{G},t}$), ie, on average, the global return is the best predictor for an individual corporate return. Adding country factors increases the explanatory power to about 48%. The estimates for ${\gamma}_{\mathrm{C}(j)}$ are nearly all equal to one, ie, the best predictor for an individual corporate return is, on average, the sum of global and country-specific excess returns.^{11}^{11}Given that “United States” is treated as a base case here, the excess return of a country is expressed relative to the global return that encompasses the US-specific excess country return. Using industry returns in conjunction with the global one results in a very similar picture, and an explained variance of about 47%. The joint use of all contemplated factors leads to an ${R}^{2}$ of about 51%; the factor structure as reflected by the set of coefficients ${\gamma}_{\mathrm{C}(j)}$ and ${\gamma}_{\mathrm{I}(k)}$ becomes a bit more diverse.

The descriptive statistics for the name-specific ${R}^{2}$, estimated on the basis of (3.3), are shown at the bottom of Table 2. The corresponding values range from 0% to about 90%, with an average that corresponds approximately to the overall ${R}^{2}$ for the four models. When studying the correlations (not shown here), it is noteworthy that there are a few names with negative values, ie, an anticyclical equity performance.

More details on the empirical correlations that the factor model is supposed to reproduce and the resulting differences between model-implied and empirically measured correlations are provided in Section B of the online appendix. In conclusion, to a varying degree, all four model flavors for corporate names result, on average, in an adequate estimation of the correlations (with an average difference of about 3%). Adding country and/or industry factors to the global factor does not change the picture too much, at least on average, as is expected from the results in Table 2. The country/country correlation tends to be well estimated by the factor model as well (with an average difference of about 2%).

#### 3.2.5 Model simulation

Section C of the online appendix outlines the way the calibrated model can be applied in a simulation context.

### 3.3 Integrating recovery rate risk

The DRC model has to reflect the “dependence of the recovery [rates] on the systemic risk factors” (Basel Committee on Banking Supervision 2016, p. 62). This renders recovery rates dependent on the economic cycle. So, during economic downturns, their simulated values will tend to be comparatively lower.

(1) | (2) | (3) | (4) | |

Global | Global and | Global and | Global, country | |

factor | country | industry | and industry | |

Model | only | factors | factors | factors |

N | 36 124 | 36 124 | 36 124 | 36 124 |

Coefficients | ||||

Global | 1.0021 | 1.0025 | 1.0022 | 1.0031 |

Australia | 1.0017 | 0.9498 | ||

Belgium | 0.0000 | 0.0000 | ||

China | 1.0000 | 0.9728 | ||

France | 1.0000 | 0.9301 | ||

Germany | 1.0000 | 1.0442 | ||

Hong Kong | 0.9998 | 0.8970 | ||

Ireland | 0.9943 | 0.9537 | ||

Italy | 1.0000 | 0.9091 | ||

Japan | 0.9998 | 0.9696 | ||

Jersey | 0.0000 | 0.0000 | ||

Mexico | 0.0000 | 0.0000 | ||

Netherlands | 1.0000 | 0.9038 | ||

New Zealand | 0.2867 | 0.3374 | ||

Singapore | 0.0000 | 0.0000 | ||

Spain | 1.0000 | 0.9822 | ||

Switzerland | 0.9995 | 0.9380 | ||

United Kingdom | 1.0002 | 0.9546 | ||

United States | — | — | ||

Basic materials | 0.9877 | 0.8178 | ||

Communications | 0.9901 | 0.9667 | ||

Consumer cyclical | 0.9998 | 0.8617 | ||

Consumer noncyclical | 0.9990 | 1.0437 | ||

Diversified | 1.0000 | 0.5506 | ||

Energy | 0.9996 | 1.0166 | ||

Financial | 0.9987 | 0.9084 | ||

Industrial | 1.0458 | 0.4605 | ||

Technology | 0.9987 | 0.9968 | ||

Utilities | — | — | ||

${R}^{\text{2}}$ | 42.8% | 47.5% | 46.6% | 50.8% |

Individual R${}^{\mathit{\text{2}}}$ versus systematic factor(s) | ||||

Average | 45.5% | 50.2% | 49.1% | 53.2% |

SD | 19.2% | 19.8% | 19.4% | 19.4% |

Minimum | 0.0% | 0.0% | 0.0% | 0.0% |

Maximum | 87.8% | 88.3% | 88.5% | 88.6% |

In the literature, different approaches have been considered to capture the dependence between recovery rates and the economic cycle. Altman et al (2005) estimate a linear inverse relation between historical observed recoveries and annual default rates, signaling the indirect dependence of both recovery rates and default rates on the economic cycle, which also drives defaults. Other papers take a different approach and calibrate a recovery rate distribution, assuming that the recovery rate is linked to one unobservable global risk factor and proxying for the economic cycle. A range of recovery rate distribution functions is considered in the literature, such as the beta (see Moody’s 2005; RiskMetrics Group 2007), lognormal (see Bade et al 2011a, 2011b), logit (see Wilkens et al 2013) and Vasicek-type (see Frye 2014).

In addition to the economic cycle, Schürmann (2004) finds other factors that drive the differences between historical recovery rate distributions. These include seniority in the capital structure (senior versus subordinated debt), debt collateralization (secured versus unsecured debt) and industry conditions. Similar factors are confirmed by Altman and Kalotay (2014), who take into account the observed differences in recovery rate distributions by means of a mixture-distribution model for recovery rates.

Here, a lognormal recovery rate model, similar to that proposed in Bade et al (2011a, 2011b) and based on previous work by Pykhtin (2003), is adopted. This model captures the correlation between the default process and recovery rate given default via a common systematic factor. It allows for the simultaneous estimation of the default and recovery rate model parameters by taking into account the fact that realized recoveries are observed only when defaults occur. In a separate paper, Bade et al (2011b) conclude that their model has better predictive power for future recoveries than alternative models, in which (log) recovery rates are estimated separately from the default model as linear functions of observable variables.

Adopting the setup in Bade et al (2011a), the following linear factor model is assumed for the log recovery rate of obligor $i$:^{12}^{12}The setup in (3.4) differs from that in Bade et al (2011a) in that, here, no correlation between the idiosyncratic asset return and the idiosyncratic recovery rate components is assumed.

$${Y}_{i}={\gamma}^{\prime}{x}_{i}+{\sigma}^{\prime}{x}_{i}(\sqrt{{\rho}^{Y}}{Z}_{\mathrm{G}}+\sqrt{1-{\rho}^{Y}}{\epsilon}_{i}^{Y}).$$ | (3.4) |

The drivers of the log recovery rate are the systematic return (${Z}_{\mathrm{G}}$) and an idiosyncratic recovery factor (${\epsilon}_{i}^{Y}$), both of which are assumed to follow independent standard normal distributions. The systematic factor is the same global return as in the asset return model for sovereigns and corporates (see Section C of the online appendix). The parameter ${\rho}^{Y}$ controls the extent to which the recovery rate is influenced by the global asset return. The deterministic vectors

$$\gamma =({\gamma}_{1}^{\mathrm{Corp}},{\gamma}_{2}^{\mathrm{Corp}},\mathrm{\dots},{\gamma}_{L}^{\mathrm{Corp}},{\gamma}_{1}^{\mathrm{Sov}},{\gamma}_{2}^{\mathrm{Sov}},\mathrm{\dots},{\gamma}_{L}^{\mathrm{Sov}})$$ |

and

$$\sigma =({\sigma}_{1}^{\mathrm{Corp}},{\sigma}_{2}^{\mathrm{Corp}},\mathrm{\dots},{\sigma}_{L}^{\mathrm{Corp}},{\sigma}_{1}^{\mathrm{Sov}},{\sigma}_{2}^{\mathrm{Sov}},\mathrm{\dots},{\sigma}_{L}^{\mathrm{Sov}})$$ |

reflect specific calibration parameters for corporates and sovereigns across the spectrum of ratings. ${x}_{i}=(\mathrm{\dots},0,1,0,\mathrm{\dots})$ indicates the type (corporate or sovereign) and rating of obligor $i$. The recovery rate itself is given by ${\mathrm{RR}}_{i}=\mathrm{exp}({Y}_{i})$.

Here, two main debt categories are considered for the estimation of the recovery rate model parameters: corporate senior and sovereign. This is in line with the categories for the default probabilities. More granular recovery rate model calibrations (eg, by industry, debt seniority, etc) can be carried out if necessary, given sufficient data availability. If separate parameters were estimated for different debt seniorities, one might need to additionally enhance the recovery rate model to enforce the absolute priority rule (APR), which states that the junior creditors would only recover something if the more-senior creditors fully recovered their claims. In other words, the recovery rate for junior debt should be greater than zero only if the senior debt has a 100% recovery rate.^{13}^{13}Notably, there is an established empirical literature, starting with Franks and Torous (1989), documenting violations of the strict APR. This is due to the fact that junior creditors have the ability to delay bankruptcy resolutions, while senior creditors may be willing to accept less and thus reduce additional costs from lengthy resolutions.

As for the calibration, Table 3 summarizes the main parameters. In Altman and Kalotay (2014), the marginal distribution of the recovery rates conditional on default is derived from a large set of defaulted bonds over the period 1988–2011. The data provides the empirical mean (${\mu}^{\mathrm{RR}}$ in Table 3) and standard deviation (${\sigma}^{\mathrm{RR}}$ in Table 3) that the model should reflect. The objective is to fit the empirical recovery rate distribution. Importantly, the fitting needs to be carried out conditional on default, ie, the empirical distribution needs to be matched given $$. When we differentiate by corporates and sovereigns as well as rating-implied default probabilities (as given in Table 1), separate tuples are estimated in an iterative numerical procedure designed to fit the empirical conditional recovery rate distribution. Specifically, the objective function that is minimized in order to obtain the parameters for the corporate recovery rate model reads as follows:

$$ |

such that

$$ | (3.5) |

where ${\widehat{\mu}}^{\text{RR}}$ and ${\widehat{\sigma}}^{\text{RR}}$ are the empirical estimates for ${\mu}^{\text{RR}}$ and ${\sigma}^{\text{RR}}$ (shown in part (a) of Table 3). The probability mass attributed to recovery rates larger than one is limited to 1% ($\varphi =0.01$) in order to avoid an ill-fitted model.^{14}^{14}Given the lognormality assumption for the unconditional recovery rate, the fitting of the mean and standard deviation as well as fulfilling the constraint on the probability mass beyond one tends to result in a lower fitted conditional standard deviation than the target, but a well-fitted conditional mean. The estimates $$, $$ and $$ are determined from the joint simulation of the default process (see Section C of the online appendix) and the recovery rate process (as per (3.4)). The sovereign parameters are estimated similarly to the corporate case using a fitting procedure, as in (3.5). The differences in the calibration procedure are the estimates $$, $$ and $$, which depend on the sovereign return correlations (${\rho}^{V,\mathrm{Sov}}$).

(a) External parameters | ||

Parameter | Value (%) | Source |

${\rho}^{V,\text{Corp}}$ | 42.82 | Equity correlation model, Section 3.2 |

${\rho}^{V,\text{Sov}}$ | 81.88 | Equity correlation model, Section 3.2 |

${\mu}^{\text{RR}}$ | 44.90 | Altman and Kalotay (2014, Table 1) [used in fitting] |

${\sigma}^{\text{RR}}$ | 37.90 | Altman and Kalotay (2014, Table 1) [used in fitting] |

${\rho}^{\text{ln}(\text{RR})}$ | 4.11 | Bade et al (2011a, Table 5) |

(b) Resulting parameters | |
---|---|

Parameter | Value (%) |

${\rho}^{Y}$ | 4.11 |

${\rho}^{VY,\text{Corp}}$ | 13.27 |

${\rho}^{VY,\text{Sov}}$ | 18.35 |

(c) Calibrated parameters | ||||
---|---|---|---|---|

Parameter/value | ||||

Rating | ${?}^{\text{????}}$ | ${?}^{\text{????}}$ | ${?}^{\text{???}}$ | ${\sigma}^{\text{???}}$ |

AAA | $-$0.7131 | 0.4301 | $-$0.6275 | 0.4192 |

AA$+$ | $-$0.7131 | 0.4301 | $-$0.6275 | 0.4192 |

AA | $-$0.7131 | 0.4301 | $-$0.6275 | 0.4192 |

AA$-$ | $-$0.7131 | 0.4301 | $-$0.6275 | 0.4192 |

A$+$ | $-$0.7461 | 0.4824 | $-$0.6275 | 0.4192 |

A | $-$0.7476 | 0.4834 | $-$0.6275 | 0.4192 |

A$-$ | $-$0.7476 | 0.4834 | $-$0.6275 | 0.4192 |

BBB$+$ | $-$0.7517 | 0.4534 | $-$0.6275 | 0.4192 |

BBB | $-$0.7615 | 0.4361 | $-$0.6275 | 0.4192 |

BBB$-$ | $-$0.7764 | 0.4108 | $-$0.6275 | 0.4192 |

BB$+$ | $-$0.7876 | 0.4117 | $-$0.6819 | 0.4555 |

BB | $-$0.7964 | 0.4081 | $-$0.7080 | 0.4271 |

BB$-$ | $-$0.8114 | 0.4129 | $-$0.7634 | 0.4173 |

B$+$ | $-$0.8324 | 0.4182 | $-$0.7690 | 0.4151 |

B | $-$0.8474 | 0.4158 | $-$0.7778 | 0.4160 |

B$-$ | $-$0.8577 | 0.4165 | $-$0.8099 | 0.4229 |

CCC/C | $-$0.8927 | 0.4165 | $-$0.8769 | 0.4188 |

Note that our calibration of the marginal distribution of recovery rates conditional on default differs from the approach in Bade et al (2011a); there, all default and recovery rate process parameters are determined jointly via a maximum likelihood estimation. Due to the joint fitting of all parameters, the moments of the marginal distribution of the recovery rate conditional on default in Bade et al (2011a) may differ from the empirical estimates to a greater extent than those in this paper, as our approach is designed to fit the two moments directly. In addition, the fitting approach used here can be easily applied in practice to estimate recovery rate parameters for more granular debt categories.

Given limited data availability, and to avoid calibration instability, one parameter from the comprehensive study in Bade et al (2011a) is reused: the correlation between log recovery rates, which implies that ${\rho}^{Y}$ in (3.4) is set equal to ${\rho}^{\mathrm{ln}(\mathrm{RR})}$. Bade et al (2011a) use a data set of approximately 188 000 annual observations for nonfinancial bonds, spanning the period 1982–2009, with a default rate of about 1%. Their derived value for ${\rho}^{\mathrm{ln}(\mathrm{RR})}$ of about 4% suggests a low correlation between log recovery rates, implying a near-zero correlation between individual recoveries conditional on default. In other words, their estimates imply that, on average, recovery rates conditional on default are driven mostly by idiosyncratic factors. The correlation parameter (${\rho}^{Y}$) in this paper, estimated from the corporate data in Bade et al (2011a), is assumed to have the same value for both corporates and sovereigns. A separate estimation of this parameter for sovereigns would likely be difficult or result in significant measurement uncertainty, given the very small number of historical sovereign default events. Note that the correlation between the asset return and recovery rate model is a function of the asset return correlation; therefore, the correlation parameters ${\rho}^{V,\text{Corp}}$ and ${\rho}^{V,\text{Sov}}$ from the equity correlation model in Section 3.2 influence the model behavior. The properties of the asset-recovery model are explored in further detail in Section D of the online appendix.

Equity positions are assumed to have a zero recovery rate. Defaulted debt positions need also be captured in the DRC measure (Basel Committee on Banking Supervision 2016, p. 61). The recovery rate model can be used to model changes in values of defaulted debt. Specifically, one can assume that the recovery rate distribution is equal to the fitted conditional one, and then sample from it to generate recovery rate scenarios.

## 4 P&L generation and distributions

### 4.1 Overview

In order to generate P&L distributions and derive associated tail measures, joint realizations of the risk factors are applied as instantaneous shocks to the deals in the DRC coverage. The risk factors in scope are the obligors’ defaults and the associated recovery rates.

In practice, for large portfolios, the computational requirements for the DRC calculation can pose a concern. In order to ease the computational burden, one could potentially pre-generate the P&L per obligor and only read the corresponding values in each Monte Carlo scenario. Given that the DRC captures losses from defaults, only the P&L needs to be generated, and only for the default scenarios, since it is zero in all nondefault scenarios. For the default scenarios, a P&L grid can be pre-generated using a discretization of the recovery rate: a grid between 0% and 100% with steps of, say, 5%. Notably, this technique is not applicable in case the P&L is not separable by obligor, for example, for CDS index option positions, or certain multi-underlying equity derivatives, for which the P&L cannot be decomposed into that of the constituents.^{15}^{15}For multi-underlying equity derivatives only, subject to supervisory approval, the rule set allows “simplified modelling approaches (for example […] that rely solely on individual jump-to-default sensitivities to estimate losses)” (Basel Committee on Banking Supervision 2016, p. 62). Another challenge arises for path-dependent derivatives (eg, variance swaps): without simulating an actual path to a potential default, the associated P&L is ill-defined.^{16}^{16}One possible approach, although it contradicts the paradigm of instantaneous shocks, consists of simulating an actual random default time within the one-year period and complementing this with a Brownian bridge to define a path to default.

In spite of the one-year projection horizon with constant positions, it may be necessary to simulate (joint) defaults at shorter horizons as well, for example, to reflect potential mismatches between the maturity of a position and its hedge. Section E of the online appendix discusses this aspect further.

### 4.2 Example portfolios

The properties of the DRC model are explored using example portfolios. For these portfolios, the DRC model results are also compared with those based on the SA for default risk.^{17}^{17}The SA is expected to serve as “a credible fallback for, as well as a floor to, the IMA [internal model approach]” (Basel Committee on Banking Supervision 2016, p. 1). The SA for the default risk charge is described in Basel Committee on Banking Supervision (2016, pp. 43–46). Note that the DRC is a capital measure and, as such, a (nonnegative) loss figure. For ease of illustration and discussion, the tables and figures in this paper show the actual (signed) P&L corresponding to the 0.1th (for DRC) or other percentiles. Table 4 illustrates the P&L distributions by means of selected percentiles (including the 0.1th for DRC) for a set of bond and equity portfolios. The results are based on the calibrations according to Section 3 and simulations with one million scenarios each.^{18}^{18}Recovery rates are capped at 100% in the simulation. Given the fitting procedure (see Section 3.3), this capping would affect not more than 1% of the default scenarios. For a straightforward comparison, the liquidity horizon is assumed to be one year for all portfolios. Relative estimation errors at a 95% confidence level for the P&L percentiles are shown in parentheses.

Percentile of P&L distribution | 10% | 1% | 0.1% | SA |
---|---|---|---|---|

A. Investment-grade bonds: | 0 | $-$184 696 | $\mathbf{-}$662 808 | $-$347 400 |

long position | (0.0%) | (1.6%) | (3.3%) | |

B. Investment-grade bonds: | 0 | $-$66 048 | $\mathbf{-}$196 464 | $-$91 584 |

long/short position | (0.0%) | (1.0%) | (3.1%) | |

C. High-yield bonds: | $-$629 037 | $-$1 768 448 | $\mathbf{-}$ 3 116 519 | $-$1 669 500 |

long position | (0.4%) | (0.6%) | (1.3%) | |

D. High-yield bonds: | $-$137 433 | $-$421 220 | $\mathbf{-}$749 059 | $-$452 250 |

long/short position | (0.3%) | (0.6%) | (1.2%) | |

E. Sovereign bonds: | $-$208 180 | $-$386 529 | $\mathbf{-}$899 212 | $-$465 441 |

long position | (0.1%) | (0.5%) | (3.6%) | |

F. Sovereign bonds: | $-$207 003 | $-$351 149 | $\mathbf{-}$466 263 | $-$248 713 |

long/short position | (0.1%) | (0.6%) | (0.8%) | |

G. Portfolios B, D | $-$ 264 601 | $-$ 646 645 | $\mathbf{-}$ 1 064 022 | $\mathrm{-}$793 828 |

and F together | (0.4%) | (0.6%) | (1.0%) | |

H. Equity index constituents: | 0 | $-$200 000 | $\mathbf{-}$ 1 000 000 | $-$396 000 |

long position | (0.0%) | (0.0%) | (0.0%) | |

I. Equity index constituents: | 0 | $-$200 000 | $\mathbf{-}$400 000 | $-$84 000 |

long/short position | (0.0%) | (0.0%) | (0.0%) | |

J. Portfolios B, D, F | $-$ 263 129 | $-$ 651 687 | $\mathbf{-}$ 1 105 615 | $\mathrm{-}$877 292 |

and I together | (0.4%) | (0.6%) | (1.6%) |

For the long investment grade bond portfolio (A), the DRC is around €663 000, representing about 7% of the total absolute notional (€10 million). If we change the setup to a long/short portfolio (B), the DRC reduces to about €196 000, or 2% of the total absolute notional. The equivalent high-yield portfolios (C and D) yield larger DRC figures of around €3.1 million and €750 000, respectively, reflecting higher default risk in these portfolios compared with the investment grade cases. Applying the model to long (portfolio E) and long/short positions (portfolio F) in sovereign bonds results in DRC figures of around €899 000 and €466 000, respectively. The DRCs for the selected sovereign portfolios lie between those for investment grade portfolios and high-yield corporate portfolios. This is to be expected, given that the sovereign portfolio has a mix of investment grade and subinvestment grade (high-yield) issuers, with examples such as Ukraine (CCC) and Vietnam (BB$+$) among the high-yield ones. In order to reflect a balanced long/short portfolio across investment grade and high-yield corporate as well as sovereign bonds, portfolio G represents the aggregation of portfolios B, D and F. One can observe that the DRC for the aggregate portfolio G is around €1 million, reflecting a diversification benefit of about 25% when compared with the sum of the DRC figures for portfolios B, D and F. It is worth noting that pure short portfolios in bonds and/or equities are not considered, since they only allow for nonnegative P&Ls and thus render the DRC figures equal to zero. For a long equities portfolio (H),^{19}^{19}If one were to use a liquidity horizon of sixty days, the DRC figures for portfolios H and J would change to €400 000 and €200 000, a reduction of 60% and 50% compared with the DRC for a one-year horizon. the DRC measure amounts to €1 million (10% of the total absolute notional). This corresponds to a scenario in which five names default, given that the recovery rate is 0% for equities and that each name in portfolio H has an equal weight. The corresponding long/short equities portfolio (I) has a DRC of €400 000. Finally, reflecting a reasonably well-diversified long/short portfolio of bonds and equities with obligors of different credit quality, portfolio J represents the aggregation of portfolios B, D, F and I. The DRC for the aggregate portfolio J is around €1.1 million, reflecting a diversification benefit of around 40% when compared with the sum of DRC figures for portfolios B, D, F and I.

A comparison between the DRC results based on the model in this paper and the corresponding SA yields interesting results. The SA-based DRC is smaller than the model-based one for all example portfolios. For portfolio J, the SA-based DRC amounts to about 80% of the model-based one. For some portfolios, the ratio between the SA-based and the model-based DRC can be as low as 20% (portfolio I). These differences suggest that the SA-based DRC corresponds to a less extreme scenario than the 0.1th percentile from the P&L distribution generated by the model. Indeed, the SA-based DRC reflects the 0.3th and third percentile of the P&L distribution for portfolios J and I, respectively. Section F of the online appendix sheds further light on the results for the SA.

Focusing on the model-based DRC, established tools for VaR-type risk measures such as marginal contributions can be applied as well. For portfolio J, for example, one finds Ukraine is the biggest contributor, with a marginal DRC of about €187 000. Another way of investigating the figures consists of analyzing the simulated scenarios close to the DRC one. In the case of portfolio J, one finds that the average P&L across the ten next-worst and ten next-best scenarios provides a very similar picture to that from the marginal DRC, again with Ukraine as the biggest single contributor. Laurent et al (2016) suggest a factor decomposition of the DRC. Applying this technique to portfolio J (in a simplified setup, with a one-factor model and constant recovery rates) yields a decomposition of the DRC into approximately 5% coming from the expected loss and 25% stemming from the systematic factor. Note that, in this second approach, a single scenario (corresponding to the 0.1th P&L percentile) is used to determine the marginal contributions to the DRC.

If we analyze the relationship between the different tail measures for the test portfolios, one can observe that the P&L distributions generally have fatter tails than a standard normal distribution. For example, for portfolios A and B, the ratio between the 0.1th and first percentile is around 3.6 and 3.0, respectively, compared with 1.33 for the standard normal distribution. Similar characteristics can be observed across the P&L distributions for all portfolios, with the exception of the sovereign bond portfolios E and F, where the tail is slightly thinner than the corresponding one for a standard normal distribution.

### 4.3 Model properties

#### 4.3.1 Convergence

The convergence of a simulation model is an important property to investigate. It allows the user to gauge an acceptable relationship between computational burden and accuracy of the model estimate. For typical large-scale bank portfolios, with the DRC stretching over many business lines, the overall P&L distribution can be asymmetric, fat-tailed and nonsmooth, thus requiring a high number of simulations. Using the example of portfolio J from Section 4.2, Figure 2 shows the simulated P&L distribution (part (a)) and the DRC estimate as a function of the number of simulations, in conjunction with the corresponding 95% confidence interval (part (b)). The pattern reveals that the confidence interval around the DRC decreases with the number of simulations, as expected. Using 10 000 simulations results in an uncertainty, measured as the relative width of the confidence interval, of around 27%. This decreases to around 2% when using one million simulations.

In order to improve convergence, standard variance reduction techniques such as importance sampling (see Glasserman and Li (2005), among many others) can be used in the Monte Carlo setup. Alternatively, or in conjunction with this, extreme value theory (EVT) is a technique that could be used to address the problem of usually nonsmooth P&L distribution tails, which tend to coincide with large estimation uncertainty and missing robustness of extreme quantiles (see McNeil et al (2005, pp. 264–326) for an overview of EVT). In the concrete case of DRC, as evidenced by the examples in Table 4 and Figure 2, an acceptable accuracy can already be achieved with pure Monte Carlo simulation and a limited number of scenarios.

#### 4.3.2 Parameter sensitivity

Beside the inherent model risk, the parameterization itself poses an important challenge. Section G of the online appendix provides a series of sensitivity studies for the example portfolios and discusses the resulting effects. The analyses therein illustrate a reasonable degree of parameterization risk for the suggested DRC model.

## 5 Conclusion and outlook

The revised standards for capital requirements for market risk in the trading book issued as a result of the FRTB aim to consolidate several building blocks from the existing rules for market risk capital. Intensive discussions between the industry and regulators as well as work-intense QISs over the past few years have helped to achieve more coherent risk and capital measures. Although this comprehensive regulation will only become binding in 2019 at the earliest (as the discussion now stands), the banking industry needs to prepare for this fundamental step, not least due to the current background of ongoing QIS exercises and potential revisions to the rules. This paper is the first to present a comprehensive model framework for DRC that is compliant with the revised regulatory framework. The DRC is supposed to complement short-term (continuous) risk measures such as VaR and ES by incorporating event risk in the form of defaults. It requires model components for marginal default and recovery rate risk as well as a factor correlation model to link them together.

As a general conclusion, an extreme tail measure and a long projection horizon, in conjunction with very limited backtesting feasibility, leave a substantial model and parameterization risk, as in the case of the Basel 2.5 risk (capital) measures. Following industry trends (see, for example, the approach of Glasserman and Xu 2014), analyzing the model risk further might be a point for future research. The same applies to aspects such as variance reduction techniques and EVT. From a practitioner’s point of view, the DRC, or a DRC-like quantity, might actually serve as a useful risk measure instead of reflecting only a capital charge. In light of the imposed model assumptions and restrictions, however, it is not obvious that this goal can be achieved.

## Declaration of interest

The views expressed in this paper are those of the authors and do not necessarily reflect the views and policies of BNP Paribas.

## Acknowledgements

We are very thankful to Hamid Skoutti, who provided valuable ideas for the model design. We also appreciate helpful input and suggestions from Andrei Greenberg, Jean-Paul Laurent, Lee Moran and Bruno Thiery as well as an anonymous referee.

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