Incremental Default Risk (IDR): Modeling Framework for the “Basel 4” Risk Measure Sascha Wilkens*, Mirela Predescu** Initial version: July 31, 2015
Abstract. As part of the upcoming comprehensive overhaul of the capital requirements for market risks in a bank’s trading book, the Fundamental Review of the Trading Book (“Basel 4”), default events need to be modeled to produce a dedicated Incremental Default Risk (IDR) charge. In the absence of standards and while industry-wide quantitative impact studies are still being conducted, this paper is the first to present a modeling framework for this Value-at-Risk-type measure that projects losses over a one-year capital horizon at a 99.9% confidence level. The article discusses selected risk factor models to derive simulation-based loss distributions and the associated risk figures. Example calculations and implementation aspects complement the discussion. JEL classification: G13; G18. Keywords: Fundamental Review of the Trading Book; Incremental Default Risk; IDR; Basel 2.5; Basel 4; Banking Regulation.
*
BNP Paribas, Risk Analytics & Modelling, London. E-Mail:
[email protected]. Corresponding author. Address: BNP Paribas, Risk, 10 Harewood Avenue, London NW1 6AA, United Kingdom.
**
BNP Paribas, Risk Analytics & Modelling, London. E-Mail:
[email protected].
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, Lee Moran and Bruno Thiery. The views expressed in this paper are those of the authors and do not necessarily reflect the views and policies of BNP Paribas.
1
Introduction and regulatory background
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 the respective supervisory authorities and are supposed to reflect large adverse market moves and are usually based on a 99% ten-day Value-atRisk (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 world-wide 2007/2008 financial crisis, new capital charges to complement the existing market risk capital requirements were proposed (Basel Committee on Banking Supervision (2009, 2011), European Banking Authority (2012)). The concept of Stressed VaR and associated capital charges were introduced in January 2013 (in Europe), supplemented by the Incremental Risk Charge (IRC) that is intended to cover market risks from credit rating migrations and defaults for flow instruments such as bonds and credit default swaps (CDS). More complex instruments such as collateralized debt obligations (CDO) 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.
A comprehensive overhaul of the capital requirements for market risks has been put forward as the Fundamental Review of the Trading Book (FRTB) (see Basel Committee on Banking Supervision (2013), with some subsequent updates), among industry practitioners also referred to as “Basel 4”.1 The main risk figure determining the capital requirements will be changed to an Expected Shortfall (ES) at a 97.5% confi-
1
The “Basel 3” framework focused on counterparty credit risk modeling.
1
dence level, shall account for different liquidity horizons of trading positions and their associated risk factors and shall be calibrated to a period of market stress. It is to be accompanied by an Incremental Default Risk (IDR) charge, which is to measure the trading portfolio’s default risk based on the 99.9% loss percentile over a one-year capital horizon. In contrast to the IRC it does not consider rating migration risk and also 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 ruleset. The IDR is based on an institution’s own model, which requires validation by the relevant home supervisor, and has to be computed at least on a weekly basis. A designated IDR model is a pre-requisite for a bank to use an ES-based market risk capital charge, otherwise a standardized approach that usually attracts much higher capital charges will need to be applied.
Given the FRTB regulation is not even fully finalized regarding all components (Basel Committee on Banking Supervision (2014, 2015)), no literature on a suitable IDR model is available as yet. The only notable exception is the work by Laurent et al. (2015), where the regulation-imposed constraints on correlation matrices and factor structure 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 IDR model this paper therefore closes a substantial research gap and can serve as a basis for an industry-wide discussion. The model development is accompanied by example calculations and the 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 fundamental elements of an IDR modeling framework. Section 3 focusses on the modeling of marginal and joint
2
default and recovery rate risk. In Section 4, the question of capturing only incremental losses (vis-à-vis VaR and ES) in IDR is discussed briefly. Section 5 is dedicated to the generation of profit and loss (P&L) distributions, illustrated with a range of example portfolios. Aspects such as convergence and sensitivity analyses are addressed. Section 6 concludes.
2
Fundamental Elements of a Modeling Framework
The IDR charge has to capture default risk in a bank’s trading book (“direct loss due to an obligor’s default as well as the potential for indirect losses that may arise from a default event”, Basel Committee on Banking Supervision (2013), p. 93). Extending the coverage of the former IRC, affected instruments are all 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 CDS on single names and indices as well as equity positions are in scope. While it 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 for a one-year liquidity horizon (i.e., assuming a constant portfolio) at a one-tail, 99.9% confidence level. This setup propagates the use of instantaneous shocks, i.e., any time value changes as well as cash flows 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 (simulat-
3
ed) recovery rates. The IDR model needs to reflect the probability of default of the obligor itself, in conjunction with both (conditional) recovery rates.
3 3.1
Capturing Default and Recovery Rate Risk 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 premia, 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. Given the difficulties in estimating market risk premia 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 (e.g., 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 (2011b)). Additionally, certain highly rated types of obligors that did not experience defaults in the past (e.g., AAA-rated sovereigns) will lead to a default probability of zero unless additional assumptions are imposed.
4
Generally, default rates tend to vary over business cycles, with more defaults observed during recessions (see Altman (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 (2011)). One can generally distinguish between throughthe-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 pro-cyclical risk measures and as such can lead to pro-cyclical capital requirements. For the application in the context of IDR, through-the-cycle probabilities hence seem preferable.
Based on these considerations, in the following, through-the-cycle default probabilities differentiated by corporates and sovereigns as provided by Standard and Poor’s are used.2 Table 1 provides the average one-year default probabilities over 1981-2012 for corporates and 1975-2012 for sovereigns. For corporates with ratings better than AA- and for sovereigns with ratings better than BB+, default probabilities are floored at 3bps, which is the minimum value prescribed by regulation.3
[Insert Table 1 here]
2
According to the latest regulatory guidelines (Basel Committee on Banking Supervision (2015)) banks with an Internal Ratings-Based (IRB) Approach must use probabilities of default from their own framework.
3
See also the critical assessment by Chourdakis and Jena (2013) on levels of default probability that can be inferred for events with few or even no occurrences in history, such as sovereign defaults.
5
For simulation purposes, asset returns
are generated for each obligor . In a one-
factor setup (relaxed later in Section 3.2), the asset return is driven by one systematic factor ( ) and an idiosyncratic factor (
):
(1)
acts as a weight and controls the asset return correlation between obligors. Assuming that
and all the
follow independent standard normal distributions, the
resulting asset return is again standard normally distributed. The deterministic vector codifies the default probabilities for corporates and sovereigns across the spectrum of
ratings according to Table 1.
This is achieved by setting
as the inverse of the cumula-
with
tive standard normal distribution function. The
indicate the
type (corporate or sovereign) and rating of obligor . In this setup, the overall asset return
3.2
has a default threshold equal to zero, i.e., a default is triggered when
.
Correlation of defaults across obligors
The IDR model needs to reproduce the dependence between defaults of different obligors. Regulation prescribes the use of a two-factor default correlation model, based on listed equity prices,4 and a calibration period of at least ten years reflecting a period of stress and a one-year liquidity horizon (Basel Committee on Banking Supervision (2013), p. 97).
4
The latest iteration of the ruleset (Basel Committee on Banking Supervision (2015)) allows the use of equity prices or CDS spreads. In the following, the use of equity prices is illustrated; the application of CDS spreads would be analogous.
6
These 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) and 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 (e.g., company size). In general, equity (and CDS) prices embed much more co-dependence than the simple “default correlation”. With default being an absorbing state market prices of non-defaulted 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; they rely, however, on the correlations implied from 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 towards the use of a direct measure of correlation between them. This requirement rules out alternatives such as structural type models or other approaches where additional information (e.g. debt information, asset values, actual default correlations) is used. Using directly observed equity (or CDS spread) correlations has a significant advantage for portfolio risk measurement, management and analysis, as it allows one to clearly explain the drivers and the magnitude of the co-movements between the names in the portfolio or sub-portfolios, perform stress-tests and sensitivity analyses with respect to the market level of correlations.
7
From a very fundamental perspective, “correlation” can be defined and measured in many different ways, posing a series of challenges and pitfalls especially in the context of tail events (see, for instance, Embrechts et al. (1999)). Regulation does not provide any guidance on the intended interpretation; given the simple implementation and measurement it can be assumed that the framework refers to classical linear (Pearson) correlation, in spite of its known shortcomings.
As a first step in 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 (non-overlapping) one-year equity returns since the large uncertainty in the measurement prevents this. Along the same lines, the measurement interval needs to have a minimum length in order to allow 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 only to about [-45%, +60%] for 20 and [-24%, +42%] for 35 observations. Hence, the combination of length of return window and measurement interval needs to allow for a certain minimum number of observations to render the exercise reasonable. The practical interpretation chosen here refers to monthly (non-overlapping) returns over three-year periods, which leads to 35 returns for each measured pairwise correlation. This can be seen as a reasonable compromise between 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 correla-
8
tions).5 This hypothesis can be challenged (confer studies such as Erb et al. (1994), 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.
Based on this fundamental setup, the prescribed calibration period of at least ten years is interpreted as the task to measure correlations over rolling time windows covering at least ten years and identifying a period of stress. The latter is thereby not defined further; it is likely supposed to point to a period for which the IDR model provides the comparatively highest loss estimate for a bank. This leaves two routes of investigation. As a first approach, the period can be identified as the one with the highest (pairwise) correlation. This is a suitable route in case the IDR is – ceteris paribus – higher the higher the correlation. For a directional portfolio (for example, holding bonds) this assumption is valid for reasonable marginal distributions. As it 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.
For the subsequent illustrations, historical equity prices of the constituents of a broad set of world-wide equity indices are used. Using the index compositions as of end of
5
Note the guidance that “[t]hese 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 (2013), p. 93)
9
December 2013, the time series between 1990 and 2013 of members of Eurostoxx 50, SMI, FTSE-100, S&P500, ASX-200, HSI and Nikkei-225 are selected (source: Bloomberg).6, 7 Table 2 provides an overview of the breakdown of the 1,145 equities by country and industry (Panel (a)) as well as statistics for monthly and annual (log-)returns (Panel (b)). The sample consists of about 40% of U.S.-based and Asian-Pacific firms each, while European names make up about 15%. The industry split is reasonably balanced across the different categories. From the return statistics it is evident that the constituents of all equity indices exhibit, on average, a positive return since 1990 (between 4% and 10% p.a.), with comparable levels of volatility (of the order of magnitude of 50% p.a.).8
[Insert Table 2 here]
Pairwise equity correlations are calculated over a rolling window spanning three years and using monthly returns. Figure 1 shows different percentiles of the distribu-
6
Since the IDR covers both equity- and credit-sensitive trading book positions, one might enlarge the equity dataset to cover explicitly the constituents from the main credit indices such as iTraxx and CDX. Many of the names, however, are already covered by the equity index selection used here – for example, the iTraxx Europe is well represented by the Eurostoxx 50 and FTSE-100.
7
The selection from the main equity indices puts an inherent focus on larger companies. In case 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.
8
Note that the theoretical relationship between monthly and annual returns in terms of number of observations (approximately 12:1) and corresponding averages, for example, is not fully preserved because of missing or invalid data points in the time series – especially for the individual equity time series.
10
tion of correlations, as well as the average width of the 95% confidence level around the individual correlation estimates.9 The peak of the correlation is found around the 2008/09 time period, as expected, with a median correlation of around 45%; the corresponding surrounding three-year period is marked in the figure and chosen as September 2007 through September 2010 in the following. While the estimation uncertainty is high, as indicated by the confidence intervals, the correlation pattern over time clearly points towards this crisis period. Notably, this holds as well when conducting the analysis with different return windows (e.g., quarterly) and measurement periods (e.g., two years) – not shown here.
[Insert Figure 1 here]
As a second approach, the IDR can be run multiple times for a series of correlation measurement periods and the stress period can be identified as the one leading to the highest loss. While this approach does not require an ex-ante assumption on the IDR as a function of correlation levels, it is much more involved from a computational perspective and likely less stable since it is portfolio-dependent; especially the latter is not a desirable feature for a risk measure with a long projection horizon. The example portfolios in Section 5.2 re-address the topic and study the influence of correlation assumptions on the IDR by means of sensitivity analyses.
Continuing with the interpretation of a two-factor model – this would usually refer to a setup with two systematic factors and one obligor-specific idiosyncratic factor.10
9
The figure 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.
10
The latest ruleset speaks more explicitly of “two systemic risk factors” (Basel Committee on Banking Supervision (2015), p. 218).
11
Laurent et al. (2015) propose to calibrate 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, stability of such fitted matrices over time might not be sufficient. In the following, an alternative strategy is pursued: in order to reflect the overall state of the economy and especially potential periods of stress, one would usually assume a global factor common to all obligors. As for the second systematic factor, a country- and an industry-specific factor are considered here.11 Within the strict framework of a two-factor model, either country or industry only could 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. As an example: Ford Motor Company would be associated with United States and/or Consumer-Cyclical, respectively. In this regard, Aretz and Pope (2011), for example, argue that changes in default risk depend most strongly on global and industry effects, with country effects usually 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 non-corporate obligors), one can express returns as function of the global (
11
) and a country-specific return factor (
).
Related to the comment in footnote 7, other factors such as company size could be of relevance; see Fama and French (1993), among many others. 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 (2013), p. 94) could be interpreted and accommodated.
12
The first practical step in the model building and calibration consists in standardizing each of the individual time series of corporate returns (
) to a mean of zero and
standard deviation of one. At each time point, global (
), country (
industry returns ( returns.12 Let
) and
) are derived from the relevant cross-section of the corporate and
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 between global returns on the one side and country and industry returns on the other side, one can express the latter as
(2)
with
and
as weights given to the global factor and
the country- and industry-specific residuals. Let and
and
and
as
denote the standard deviation of
reflect the standard deviation of the residuals
and
. Table 3 shows the results of a regression analysis run on the basis of (2) for the identified stress period. Most of the country and industry returns are moving in line with the global returns (
and
not statistically different from one) and
the explanatory content reflected by the
is high (between 65 and 95%), indicating
the dominant role of the global factor. The corresponding
coefficients can be used
readily in the simulation.
[Insert Table 3 here]
12
At least five observations in a cross-sectional return set are required here for a valid systematic return.
13
In order to simulate country returns factor
, one would use the realization of a global
and that of a country-specific factor
. Assuming that both follow in-
dependent standard normal distributions one obtains
(3)
with
as the sign function. The following straightforward expression can be
used to calculate the country/country correlation implied by the model in (3):
(4)
with as the indicator function.
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 (2), only the residuals
and
are considered
as additional explanatory factors for the corporate returns. It is worth noting that reusing the coefficients
and
from (2) as the weights for the global factor is
not possible without introducing assumptions on the relationship of country and industry returns vis-a-vis corporates. Therefore, in the following, the sensitivity to the global factor itself is expressed via a separate coefficient
:
(5)
Table 4 summarizes the results of a multi-linear panel regression analysis run on the basis of (5) for the stress period. In order to address the problem of multicollinearity
14
in the model setup – each corporate belongs to one country and one industry –, one country (here: ‘United States’) and one industry (here: ‘Utilities’) are omitted from (5); this means that excess returns stemming from the country and industry factors are expressed relative to this base case. From (5) one obtains estimates for and
,
as well as an overall
as a goodness-of-fit measure. Individual
can be obtained by calculating an aggre-
gated systematic return per corporate, based on the estimation in (5), and measuring the correlation between the time series of the systematic and corporate returns.
[Insert Table 4 here]
Using only the global factor as explanatory variable accounts for about 43% of the variation in equity returns. Unsurprisingly, the corresponding best fitting return model equals
, i.e., on average, the global return is the best predictor for the
individual corporate return. Adding country factors increases the explanatory power to about 48%. The coefficients
are nearly all equal to one, i.e., the best predic-
tor for an individual corporate return is, on average, the sum of global and countryspecific excess return.13 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 as reflected by the set of coefficients
of about 51% and the factor structure and
becomes a bit more diverse. In a
second step, the returns of each obligor are regressed against the corresponding aggregated systematic factors, which leads to name-specific values for
. The result-
ing statistics are shown at the bottom of Table 4. The corresponding values range from 0% to about 90%, with an average corresponding approximately to the overall
13
Given that the ‘United States’ are treated as base case here, the excess return of a country
is
expressed relative to the global return that encompasses the US-specific excess country return.
15
for the four models. When studying the correlations (not shown here) it is noteworthy that there are a few names with negative values, i.e., an anti-cyclical equity performance.
In order to simulate the – standardized – return and industry
of a corporate from country
, one would use the realizations of a global factor
and that of the
relevant country-specific, industry-specific and idiosyncratic factors, :14
and
(6)
where
represents the coefficient for the systematic factor in the regression of the
return
of
obligor
on
the
aggregated
systematic
return.
is a normalization coefficient and all factors are assumed to follow independent standard normal distributions. Based on (6), the pairwise correlation between any two corporates can be expressed in closed form as:
(7)
14
Note that this expression assumes that all country and industry residual factors are independent from each other, which is a simplification, albeit backed up empirically: the average correlation among the residual country (industry) factors in the setup presented here amounts to about 2% (-8%), the average correlation between residual country and industry factors to ly -1%.
16
The integration with the marginal default probabilities and associated recovery rates as discussed in Section 3.1 is straightforward: using the factor realizations leads to asset returns for all obligors that determine the default event while reflecting the correct joint behavior across obligors.
The empirical distribution that the factor model is supposed to reproduce is illustrated in Figure 2. It shows the distribution of the corporate/corporate correlations during the identified stress period. Additionally, based on the country returns defined as cross-sectional averages through time, the distribution of country/country correlations is displayed.15 The comparatively high country/country correlations point towards a dominant common (global) driver of returns.
[Insert Figure 2 here]
The final judgment as to whether the proposed model is suitable is provided by the differences between model-implied and empirically measured (pairwise) correlations.16 Figure 3 shows the corresponding distributions, differentiated by corporate/corporate and the associated four model variants and the country/country case. To a varying degree, all four model flavors for corporate names result, on average, in an adequate estimation of the correlations (average difference of about 3%).17 Adding country and/or industry factors to the global factor does not change the picture
15
For a qualifying country return time series, at least five valid returns (out of 35) are required.
16
It is worth pointing out that “[a] bank must validate that its modelling approach for these correlations is appropriate for its portfolio, including the choice and weights of its systematic risk factors.” (Basel Committee on Banking Supervision (2013), p. 93)
17
Recall that all correlation pairs are equally weighted in this analysis (see also footnotes 7 and 11).
17
too much, at least on average, as expected from the results in Table 4. The country/country correlation tends to be well estimated by the factor model as well (average difference of about 2%).
[Insert Figure 3 here]
3.3
Integrating Recovery Rate Risk
The IDR model has to reflect the “dependence of the recovery [rates] on the systemic risk factors” (Basel Committee on Banking Supervision (2013), p. 94). This renders recovery rates dependent on the economic cycle and, during economic downturns, their simulated values will tend to be comparatively lower.
To capture dependence between recovery rates and the economic cycle different approaches have been considered in the literature. 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, proxying for the economic cycle. A range of recovery rate distribution functions are considered in the literature, such as a beta (see Moody’s (2005), RiskMetrics Group (2007)), log-normal (see Bade et al. (2011a, 2011b)), logit (see Wilkens et al. (2013)) or Vasicek-type (see Frye (2014)).
In addition to the economic cycle, Schürmann (2004) finds other factors driving the differences between historical recovery rate distributions. These include seniority in the capital structure (senior vs. subordinated debt), debt collateralization (secured vs. unsecured debt) and industry conditions. Similar factors are confirmed by Alt18
man (2014) who takes into account the observed differences in recovery rate distributions by means of a mixture distribution model for recovery rates.
Here a log-normal recovery rate model similar to the one 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. Specifically, the following linear factor model is assumed for the log-recovery rate of obligor :18
(8)
The drivers of the log-recovery rate are the systematic asset return ( ) and an idiosyncratic recovery factor (
) – all assumed to follow independent standard normal
distributions. The systematic factor is the corresponding factor from the process in (1). Within the multi-factor setup developed in Section 3.2, the systematic factor in (8) represents the aggregated asset return driven by the corresponding global/country/industry returns. The parameter
controls the extent to which the recov-
ery rate is influenced by the systematic asset return. The deterministic vectors
and
reflect specific calibration parameters for corporates and sovereigns across the spectrum of ratings, similar to is given by
in the case of asset returns. The recovery rate itself
.
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 (e.g., by
18
The setup in (8) differs from that in Bade et al. (2011) in that here no correlation between the idiosyncratic asset return and the idiosyncratic recovery rate components is assumed.
19
industry, debt seniority) 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 recovered fully 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.19
As for the calibration, Table 5 summarizes the main parameters. In Altman (2014) the marginal distribution of the recovery rates conditional on default is derived from a large set of defaulted bonds over the period 1988 through 2011. The data provides the empirical mean (
in Table 5) and standard deviation (
in Table 5) 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, i.e., the empirical distribution needs to be matched given
. Differentiating 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. Probability mass attributed to recovery rates larger than one is limited to one percent as part of the fitting process in order to avoid an ill-fitted model.20
19
Notably, there is an established empirical literature, starting with Franks and Torous (1989), documenting violations from the strict APR. This is due to the fact that junior creditors have the ability to delay bankruptcy resolutions and senior creditors may be willing to accept less and thus reduce additional costs from lengthy resolutions.
20
Given the log-normality assumption for the unconditional recovery rate the fitting of mean, 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.
20
[Insert Table 5 here]
Given limited data availability and to avoid calibration instability one parameter from the comprehensive study in Bade et al. (2011a) is re-used: the correlation between log-recovery rates, which implies that
in (8) is set equal to
. Bade et
al. (2011a) use Maximum Likelihood Estimation to jointly fit asset and recovery rate process, based on a data set of approximately 188,000 annual observations for nonfinancial bonds spanning the period 1982 through 2009, with a default rate of about 1%. Their derived value for
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. Note that the correlation between asset return and recovery rate model is a function of the asset return correlation; therefore the correlation parameters
and
from the
equity correlation model in Section 3.2 influence the model behavior.
In order to study the asset-recovery rate model more closely, Figure 4 provides an insight into the properties for the one- and two-obligor case. Panel (a) shows the marginal distribution of recovery rates that the model produces conditional on default. This largely preserves a log-normal distributional shape. Panel (b) illustrates the asset-recovery rate dependence conditional on default. In line with the model philosophy a more negative asset return tends to be associated with a lower recovery rate. The strength of the correlation depends on the assumed level of assumed level of
. Given the
of about 4%, the unconditional correlation between the asset
return and recovery rate is about 13% while conditional on default it amounts to about 5%. In order to analyze the model behavior across obligors Panel (c) shows the asset correlation between two names, conditional on a joint default. The dependence
21
is less pronounced than the unconditional parameterization (
of about 43%)
suggests. Panel (d) provides an illustration of the joint recovery rates conditional on two obligors defaulting. In line with the assumed low value for
of about 4%, the
correlation between recovery rates conditional on joint defaults is close to zero.
[Insert Figure 4 here] For sovereigns, in the absence of sufficient empirical observations, the model parameterization is inherited from the corporate case. The different asset correlations and default probabilities of sovereigns feed into the otherwise unchanged model. Similar qualitative effects with respect to recovery-default correlation as shown in Figure 4 for corporates are also observed for sovereign obligors. The correlation between the asset return and (log-)recovery rates for sovereign obligors ( comparable to the corporate case (
around 18%) is
around 13%).
Concerning the modeling and estimation of the recovery rate and the default probabilities for sovereign debt, one would likely want to differentiate between local currency and foreign currency issued debt since, from a foreign investor’s point of view, the former carries both default and foreign exchange rate risk (not to be reflected in the IDR) due to the expected devaluation of the local currency in the event of default. While, for example, Du and Schreger (2013) examine the properties and driving factors in a comprehensive way, very limited historical data for defaults and recovery rates on local currency debt (confer Moody’s (2011a)) renders largely qualitative or relative assessments in the calibration of recovery rates for local versus foreign currency debt necessary. One possibility would be to use the different ratings for local currency and foreign currency debt issued by sovereign entities (as provided by Standard & Poor’s) to derive expected recovery rates for such debt.
22
Equity positions are assumed to have a zero recovery rate. Defaulted debt positions need to also be captured in the IDR measure (see also the requirements in Basel Committee on Banking Supervision (2015), p. 218). 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 (see Figure 4) and then sample from it to generate recovery rate scenarios.
4
Capturing Only Incremental Losses in IDR
Obviously addressing the industry critique around the IRC with regard to double counting mark-to-market losses in VaR and IRC, the regulation around IDR allows to reflect only the non-VaR part of the potential loss (“incremental loss from default in excess of the mark-to-market losses already taken at the time of default”, Basel Committee on Banking Supervision (2013), p. 94).21 As an example, if a long bond position with a (nominal and) market value of 100 is assumed to lose 10 in a certain credit spread-widening simulation scenario in the VaR, the corresponding IDR scenario – should it lead to a default of the obligor with an assumed bond recovery of 40% of the nominal – would only reflect a loss 100-40-10 = 50 instead of 60. Essentially, the IDR is expected to reflect a sudden jump-to-default, without a prior deterioration of the credit quality.
While conceptually a sound idea, such an approach poses a series of further challenges for the model. In order to allow a meaningful “aggregation”, the scenarios between ES and IDR need to be somehow aligned. But losses (or, more generally, P&L distributions) are simulated over different time horizons and with likely differ-
21
Similarly for the ES as the new main market risk (capital) measure.
23
ent distributional assumptions (e.g., historical vs. variance-covariance or MonteCarlo VaR/ES vis-à-vis IDR) and furthermore calculated at different confidence levels. The topic might be suitable for future research. The expectation at this point is that the option to avoid double-counting (which should be less pronounced than in the case of IRC) will hardly be used in practice.22
5 5.1
P&L Generation and Distributions Overview
In order to generate P&L distributions and derive associated tail measures, joint realizations of the risk factors are to be applied as instantaneous shocks to the deals in the IDR coverage. The risk factors in scope are the obligors’ defaults and the associated recovery rates.
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. Using a discretization of the recovery rate – a grid between 0% and 100% with steps of, say, 5% – the run time can be improved. 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 whose P&L cannot be decomposed in terms of that of the constituents. Another challenge arises for pathdependent derivatives (e.g., variance swaps): without simulating an actual path to a potential default the associated P&L is ill-defined. One possible approach – although in contradiction with the paradigm of instantaneous shocks – would consist in simu-
22
Regulation is currently contemplating a change towards a Default Risk Charge (DRC), which would remove the option of addressing double-counting.
24
lating an actual random default time within the one-year period and complementing this by a Brownian bridge to define a path to default.
5.2
Example portfolios
Model properties are best explored with example portfolios. Table 6 illustrates the P&L distributions by means of selected percentiles (including 0.1th for the IDR) for a set of bond and equity portfolios. The results are based on calibrations according to Section 3 and simulations with one million scenarios each.23 Relative estimation errors at a 95% confidence level are shown in parentheses.
[Insert Table 6 here]
For the long investment grade bond portfolio (Portfolio A), the IDR is around 663k EUR, representing about 7% of the total absolute notional (10m EUR). Changing the setup to a long/short portfolio (Portfolio B), the IDR reduces to about 196k EUR, or 2% of the total absolute notional. The equivalent high-yield portfolios (Portfolios C and D) yield larger IDR figures of around 3.1m EUR and 750k EUR, respectively, reflecting higher default risk in these portfolios compared to the investment grade cases. Applying the model to long (Portfolio E) and long/short positions (Portfolio F) in sovereign bonds results in IDR figures of around 899k EUR and 466k EUR, respectively. The IDR for the selected sovereign portfolios lie between those for investment grade portfolios and high-yield corporate portfolios. This is 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
23
Recovery rates are capped at 100% in the simulation. Given the fitting procedure (see Section 3.3) this capping would affect not more than one percent of the default scenarios.
25
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 IDR for the aggregate portfolio G is around 1m EUR, reflecting a diversification benefit of about 25% when compared to the sum of the IDR 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 non-negative P&Ls and thus render the IDR figures equal to zero. For a long equities portfolio (Portfolio H), the IDR measure amounts to 1m EUR (10% of the total absolute notional). This corresponds to a scenario where 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 (Portfolio I) has an IDR of 400k EUR. 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 IDR for the aggregate portfolio J is around 1.1m EUR, reflecting a diversification benefit of around 40%, when compared to the sum of IDR figures for portfolios B, D, F and I.
Established tools for VaR-type risk measures such as marginal contributions can be applied to the IDR as well. For Portfolio J, for example, one finds Ukraine as the biggest contributor with a marginal IDR of about 187k EUR. Another way of investigating the figures consists in analyzing the simulated scenarios close to the IDR 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 the one from the marginal IDR, again with Ukraine as the biggest single contributor. Laurent et al. (2015) suggest a factor decomposition of the IDR. Applying this technique to Portfolio J (in a simplified setup with a one-factor model and constant recovery rates)
26
yields a decomposition of the IDR into approximately 5% coming from the expected loss and 25% stemming from the systematic factor.
Analyzing the relationship between the different tail measures for the test portfolios, one can observe that the P&L distributions have generally fatter tails than a standard normal distribution. For example, for portfolios A and B, the ratio between the 0.1th and 1st percentile is around 3.6 and 3.0, respectively, compared to 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.
5.3 5.3.1
Model properties Convergence
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 IDR stretching over many business lines, the overall P&L distribution can be asymmetric, fat-tailed and non-smooth, thus requiring a high number of simulations. On the example of Portfolio J from Section 5.2 Figure 5 shows the simulated P&L distribution (Panel (a)) and the IDR estimate as a function of the number of simulations, in conjunction with the corresponding 95% confidence interval (Panel (b)). The pattern reveals that the confidence interval around the IDR 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.
27
[Insert Figure 5 here]
Standard variance reduction techniques can be useful in the Monte-Carlo setup. Antithetic variates are one standard means in this regard and can be applied by re-using the standard normal random draws with a reversed sign; in this way, a more equal “spanning” of the scenarios can be achieved. However, for several random factors as in the case of IDR, efficient application and benefit are less obvious. More targeted for the risk factor modelling in a portfolio context is importance sampling, as put forward, for example, in Kalkbrener et al. (2004) and Glasserman and Li (2005), and further refined in Reitan and Aas (2010). With the main idea to ensure that the simulated scenarios sufficiently cover the relevant tail area of the distribution by means of a shift vector for the systematic factors, the uncertainty of the estimated percentile can potentially be reduced. In the concrete case of IDR, as evident from the examples in Table 6 and Figure 5, an acceptable accuracy can be achieved with pure MonteCarlo simulation and a limited number of scenarios already.
The P&L distribution tail is usually not smooth, which tends to coincide with large estimation uncertainty and missing robustness of the IDR – small changes in portfolio compositions or model parameters such as default probabilities leading to significant changes in the risk figure (“cliff effects”). Extreme Value Theory (EVT) is an available tool to address this problem. The technique essentially consists in making use of a universally valid approximation of the extreme tail of the distribution with a generalized Pareto distribution (see McNeil et al. (2005), pp. 264-326 for an overview). Some of the main challenges in practice consist in controlling the approximation error and the difficulties in identifying the actual IDR scenario in a smoothed, parametrized distribution tail.
28
5.3.2
Parameter sensitivity
Besides the inherent model risk, the parameterization itself poses an important challenge. Amongst the model parameters discussed in Section 3, Table 7 shows selected sensitivity analyses of the IDR figures with respect to changes in (a) obligor credit quality, (b) average recovery rates and (c) average default correlation.
[Insert Table 7 here]
In particular, in case (a) the obligor ratings are each decreased and increased by one notch, to attract a different default probability according to Table 1.24 For Portfolios A, C, E and H, as expected, the IDR is monotonically increasing (decreasing) with the rating quality worsening (improving). The IDR increases to 35% for these portfolios if all the obligors are downgraded by one notch. In the opposite case where all obligors are upgraded by one notch, the decrease in IDR varies between 15% and 23%, hence generally smaller in absolute terms than in the case of rating downgrades. For the long/short Portfolios B, D, F and I, the influence of changes to the initial ratings is qualitatively similar although theoretically less obvious. For the equity portfolio I the IDR does not change for a one-notch downgrade or upgrade. For portfolios B, D, F the effects are similar to the long portfolios.
When stressing the (average) recovery rates, case (b), the picture is as follows. The recovery rate plays a role only in case of a default; therefore, there is an obvious link between the effects from recovery rates and ratings (default probabilities). A change in the average recovery rate of 25% has a significant effect on IDR. The change in IDR due to an increase or decrease in average recovery rate is quite symmetric and
24
The worst attainable rating equals CCC/C, reflecting the last entry in Table 1.
29
overall the effect of a 25% change in the average recovery rate is comparable to a one-notch change in the ratings (see for example portfolio J). Similar observations hold for the long-short and overall portfolios.
The default correlation assumption and its sensitivity to changes are analyzed as case (c). Two changes in correlation are considered: one where the average correlation is decreased by 10% and another one where the average correlation is decreased by 25%.25 This case also addresses the question raised in Section 3.2 with regard to the behavior of IDR in periods of stress. In all long-only portfolios – Portfolios A, C, E and H – the IDR is monotonically increasing in correlation, i.e., the higher the correlation the less diversification benefit the portfolio offers through its constituents. Moving on to the long/short Portfolios B, D, F, G, I and J, the picture is qualitatively similar, however, the magnitude of the change in IDR is significantly reduced compared to the corresponding long-only portfolios. For example, for the high-yield corporate portfolio C, the two changes in correlation are associated with decreases in IDR of around 8% and 19%, respectively, compared to 3% and 6% for the long-short high-yield portfolio D. For portfolios F and I, a decrease in correlation has (almost) no effect on the IDR.
Notably, in general, the correlation influence on the IDR is not that obvious to predict. Starting with a theoretical assumption of a 0% pair-wise correlation and ignoring the uncertainty of the recovery rates: all obligors would default purely according to their marginal probability and the joint distribution of defaults can be derived from the convolution of the individual density functions. The other extreme case of a theoretical 100% pair-wise correlation means that the same default factor realization is
25
Only downward changes in correlation are considered given that the average correlation in the model is already at the highest level over the considered time period.
30
applied to all obligors. Hence all obligors with a certain probability or higher will default and none of the others will. The P&L distribution in the 100% correlation case reflects the sum of all obligors’ PVs for a given default probability. For example, the 0.1th (99.9th) percentile of the P&L (loss) distribution is equal to sum of P&Ls from defaults for all obligors with an individual default probability of 0.1% or higher. In a long/short portfolio, it is possible that this value is zero because of a “dominance” of short exposures to the defaulting names. The IDR could hence be lower (in absolute terms) for a higher pair-wise correlation. Particularly, also monotonicity of the IDR for in-between cases with correlations between 0% and 100% correlation is not ensured. As pointed out in Laurent et al. (2015) the required number of factors in the correlation model is usually portfolio-dependent: in a directional portfolio, a one-factor approximation will tend to perform quite well, while for more realistic long/short portfolios even two factors are hardly able to capture the dependence structure.
As an example of a more structural model parameterization risk the sensitivity of the IDR to the systematic factor weight in the recovery rate model
is analyzed as
case (d). A change in this parameter from 4% to 25% increases the IDR for all portfolios. This is expected given that an increase in
has the effect of decreasing the
average conditional recovery rate for extreme scenarios. In these scenarios, which are driven by very negative global asset returns, the average recovery rate across the defaulted names will be monotonically lower for higher values of
. For example,
taking the most extreme simulated scenario for the long-only investment-grade portfolio A – corresponding to 75 defaults out of 125 names – the average conditional recovery rate amounts to about 34% for
= 4% and to about 23% for
= 25%.
These values are significantly lower than the marginal average conditional recovery rate of 45%, reflecting the depressed recoveries in such an extreme scenario. For long-only portfolios A, C and E, the IDR increase due to the increase in
31
is in the
range of 3% to 11%. For long-short portfolios B, D and F, the increase is around 2% to 3% while for the aggregate portfolios G and J, the increase is around 5%.
The sensitivity analyses illustrate a reasonable degree of parameterization risk for the suggested IDR model; notably, the presented analyses are conducted ceteris paribus and certain “cross effects”, for example, between ratings (default probabilities) and correlations, might be worth monitoring in a practical application.
6
Conclusion and Outlook
The Fundamental Review of the Trading Book (“Basel 4”) aims at consolidating several building blocks from the existing rules for internal market risk model. Intensive discussions between the industry and regulators as well as work-intense quantitative impact studies over the past years have helped to achieve more coherent risk and capital measures. While this comprehensive regulation will become binding only in 2018 (as the discussion stands now), the banking industry needs to prepare for this fundamental step, not least against the background of ongoing and continuing QIS exercises and potential revisions to the rules. This paper is the first to present a comprehensive IDR model that is compliant with the regulatory framework in its current form. The IDR 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 two-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 by Glasserman and
32
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. As far as the model setup is concerned the decomposition of individual contributions of the factors in the correlation model to the IDR as proposed in Laurent et al. (2015) could be another angle of future investigation.
From a practitioner’s point of view, the IDR might actually serve as a useful risk measure instead of reflecting only a capital charge. This, however, crucially depends on the final regulatory framework and the imposed model assumptions and restrictions.
References Altman, Edward I. (2014), Ultimate Recovery Mixtures, Journal of Banking and Finance, Vol. 40, pp. 116-129. Altman, Edward I.; Brady, Brooks; Resti, Andrea; Sironi, Andrea (2005), The Link between Default and Recovery Rates. Theory, Empirical Evidence, and Implications, The Journal of Business, Vol. 78, pp. 2203-2228. Aretz, Kevin; Pope, Peter F. (2011), Common Factors in Default Risk Across Countries and Industries, European Financial Management, Vol. 17, pp. 1-45. Bade, Benjamin; Rösch, Daniel; Scheule, Harald (2011a), Default and Recovery Risk Dependencies in a Simple Credit Risk Model, European Financial Management, Vol. 17, pp. 120-144. Bade, Benjamin; Rösch, Daniel; Scheule, Harald (2011b), Empirical Performance of Loss Given Default Prediction Models, The Journal of Risk Model Validation, Vol. 5, Summer, pp. 25-44. Basel Committee on Banking Supervision (2009), Guidelines for Computing Capital for Incremental Risk in the Trading Book, July. Basel Committee on Banking Supervision (2011), Revisions to the Basel II Market Risk Framework, February. Basel Committee on Banking Supervision (2013), Fundamental Review of the Trading Book. A Revised Market Risk Framework, Consultative Document, October.
33
Basel Committee on Banking Supervision (2014), Analysis of the Trading Book Hypothetical Portfolio Exercise, September. Basel Committee on Banking Supervision (2015), Instructions for Basel III Monitoring. Version for Banks providing Data for the Trading Book Part of the Exercise, January. Chourdakis, Kyriakos; Jena, Rudra P. (2013), Bayesian Inference of Default Probabilities, Working Paper, Nomura. De Servigny, Arnaud; Renault, Olivier (2002), Default Correlation. Empirical Evidence, Presentation, Standard & Poor’s, November. Du, Wenxin; Schreger, Jesse (2013), Local Currency Sovereign Risk, Working Paper, Board of Governors of the Federal Reserve System, December. Düllmann, Klaus; Küll, Jonathan; Kunisch, Michael (2008), Estimating Asset Correlations from Stock Prices or Default Rates. Which Method is Superior?, Working Paper, Deutsche Bundesbank, April. Embrechts, Paul; McNeil, Alexander; Straumann, Daniel (1999), Correlation: Pitfalls and Alternatives, Working Paper, ETH Zurich, March. Erb, Claude B.; Harvey, Campbell R.; Viskanta, Tadas. E. (1994), Forecasting International Equity Correlations, Financial Analysts Journal, Vol. 50, November/December, pp. 32-45. European Banking Authority (2012), EBA Guidelines on the Incremental Default and Migration Risk Charge (IRC), May. Fama, Eugene F.; French, Kenneth R. (1993), Common Risk Factors in the Returns on Stocks and Bonds, Journal of Financial Economics, Vol. 33, pp. 3-56. Franks, Julian R.; Torous, Walter N. (1989), An Empirical Investigation of U.S. Firms in Renegotiation, The Journal of Finance, Vol. 44, pp. 747-769. Frye, Jon (2014), The Simple Link from Default to LGD, Risk, Vol. 27, March, pp. 64-69. Glasserman, Paul; Li, Jingyi (2005), Importance Sampling for Portfolio Credit Risk Management Science, Vol. 51, pp. 1643-1656. Glasserman, Paul; Xu, Xingbo (2014), Robust Risk Measurement and Model Risk, Quantitative Finance, Vol. 14, pp. 29-58. Hull, John C.; Predescu, Mirela; White, Alan (2005), Bond Prices, Default Probabilities and Risk Premiums, Journal of Credit Risk, Vol. 1, pp. 53-60. Kalkbrener, Michael; Lotter, Hans; Overbeck, Ludger (2004), Sensible and Efficient Capital Allocation for Credit Portfolios, Risk, Vol. 17, January, pp. S19-S24. Laurent, Jean-Paul; Sestier, Michael; Thomas, Stéphane (2015), Trading Book and Credit Risk: How Fundamental is the Basel Review?, Working Paper, University Paris-1 Panthéon-Sorbonne, April. 34
Liu, Sheen; Qi, Howard; Shi, Jian; Xie, Yan Alice (2012), A Structural Approach for Predicting Default Correlation, Working Paper, Washington University, August. McNeil, Alexander J.; Frey, Rüdiger; Embrechts, Paul (2005), Quantitative Risk Management. Concepts, Techniques and Tools, Princeton (NJ). Merton, Robert C. (1974), On the Pricing of Corporate Debt. The Risk Structure of Interest Rates, The Journal of Finance, Vol. 29, pp. 449-470. Moody's (2005), LossCalc V2: Dynamic Prediction of LGD. Modeling Methodology, January. Moody's (2007), Moody's Ultimate Recovery Database, Special Comment, April. Moody's (2008), Asset Correlation, Realized Default Correlation, and Portfolio Credit Risk. Modeling Methodology, March. Moody's (2011a), Sovereign Default and Recovery Rates, 1983-2010, Special Comment, May. Moody's (2011b), Statistical Testing of Differences in Default, Special Comment, May. Pykhtin, Michael (2003), Unexpected Recovery Risk, Risk, Vol. 16, August, pp. 74-78. Qi, Howard; Xie, Yan Alice; Liu, Sheen; Wu, Chun (2015), Inferring Default Correlation from Equity Return Correlation, European Financial Management, Vol. 21, pp. 333-359. Reitan, Trond; Aas, Kjersti (2010), A New Robust Importance Sampling Method for Measuring Value-at-Risk and Expected Shortfall Allocations for Credit Portfolios, The Journal of Credit Risk, Vol. 6, Winter, pp. 113-149. RiskMetrics Group (2007), CreditMetrics. Technical Document, September. Schürmann, Til (2004), What Do We Know About Loss Given Default?, in: Shimko, David (ed.), Credit Risk Models and Management, 2nd edition, London, pp. 249-274. Standard & Poor's (2013a), Annual Global Corporate Default Study and Rating Transitions, Report, March. Standard & Poor's (2013b), Sovereign Defaults and Transitions Data. 2012 Update, Report, March. Turley, Robert (2012), Price Comovement and Time Horizon: Fads and Fundamentals, Working Paper, Harvard University, January. Vassalou, Maria; Xing, Yuhang (2004), Default Risk in Equity Returns, The Journal of Finance, Vol. 59, April, pp. 831-868. Wilkens, Sascha; Brunac, Jean-Baptiste; Chorniy, Vladimir (2013), IRC and CRM. Modelling Framework for the 'Basel 2.5' Risk Measures, European Financial Management, Vol. 19, pp. 801-829.
35
Table 1: Default probabilities by rating This table shows the average historical one-year default rates (in %) by rating for corporate and sovereign issuers, to serve as probabilities for future defaults. The rates are calculated as weighted averages of the one-year default rates for each year in the sample, with weights based on the number of issuers rated at the beginning of each year. Source: Standard & Poor's (2013a, 2013b). Regulation prescribes the use of a floor of 3bps, therefore the default rates marked with “*” are to be floored accordingly. In order to ensure monotonicity of default rates by rating, the historical values for BB and B+ sovereigns, 0.00% and 0.60%, respectively, have been fitted from the adjacent rates – indicated by “&”.
Rating (Standard and Poor's)
Corporates
Sovereigns
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
i
&
&
Table 2: Equity data for correlation modeling This table provides statistics of the equity data used as a basis for the return correlation modeling. The index constituents of Eurostoxx 50, SMI, FTSE-100, S&P500, ASX-200, HSI and Nikkei-225 reflect the index compositions as of end of December 2013. Source: Bloomberg (the country reflects the 'country of risk', the industry the 'industry sector'). Panel (a) shows the breakdown by country and industry, Panel (b) provides statistics of the (log-)returns for monthly and annual returns during the period 1990 through 2013.
Constituents from selected equity indices FTSES&P500 ASXHSI 100* 200**
Eurostoxx 50
SMI
50
20
101
500
199
Country Australia Belgium China France Germany Hong Kong Ireland Italy Japan Jersey Mexico Netherlands New Zealand Singapore Spain Switzerland United Kingdom United States
1 19 14 1 5 4 6 -
19 1
2 1 1 2 2 92 1
1 1 2 496
Industry Basic Materials Communications Consumer Cyclical Consumer Non-cyclical Diversified Energy Financial Industrial Technology Utilities
2 4 5 8 3 13 8 2 5
2 1 2 6 1 5 3 -
9 7 17 24 7 20 10 2 5
25 39 69 100 1 45 80 65 46 30
Total
Nikkei225
All
50
225
1145
186 1 6 1 1 4
27 22 1 -
225 -
186 1 27 19 14 22 5 5 225 1 1 6 6 1 6 22 96 502
34 21 23 30 1 14 40 28 1 7
1 3 6 3 4 6 21 1 1 4
32 11 44 30 3 26 66 8 5
105 86 166 201 6 79 205 181 60 56
(a) Breakdown by country and industry
(b) Return statistics Monthly
N Average Std. dev. Minimum Maximum
10,527 0.4% 9.4% -87.6% 73.7%
4,125 0.5% 9.4% -105.8% 99.3%
20,468 0.8% 9.9% -407.6% 96.7%
100,245 0.8% 10.8% -299.6% 149.5%
29,539 0.7% 12.3% -140.5% 180.8%
8,853 0.9% 12.8% -131.8% 190.6%
50,966 0.0% 10.8% -132.7% 100.7%
224,723 0.6% 10.9% -407.6% 190.6%
Annual
N Average Std. dev. Minimum Maximum
289 7.3% 54.7% -211.1% 268.3%
111 3.5% 78.2% -281.4% 287.2%
540 9.9% 56.3% -303.6% 260.4%
14,427 7.6% 38.0% -422.0% 281.8%
915 6.2% 44.9% -227.2% 195.4%
260 4.5% 48.7% -193.3% 371.4%
1,257 9.7% 42.5% -196.3% 267.0%
17,799 7.7% 40.2% -422.0% 371.4%
* The FTSE-100 consists of 100 companies, but there are 101 listings since Royal Dutch Shell has both A and B class shares listed. ** With only quarterly rebalancing, the number of names in the index can slightly diverge from 200.
ii
Table 3: Equity return factor model – Country and industry factors This table provides the results of the country and industry factor analysis derived from the equity returns of the index constituents of Eurostoxx 50, SMI, FTSE-100, S&P500, ASX-200, HSI and Nikkei-225. The equity index compositions reflect the status as of end of December 2013. The analysis is based on non-overlapping monthly (log-)returns and conducted over the period September 2007 through September 2010 that has been identified as the one with the most "stressed" correlation (see Figure 1; 35 return periods). After standardization the equity returns are used to derive global, country and industry factors as cross-sectional averages at each time point. The country and industry factors are 2 regressed onto the global factor; coefficients (C and I, respectively) as well as R are provided in the table,
complemented by the standard deviation of the residual returns (see also (2)). t-tests are conducted to evaluate whether the coefficients differ from one; *** (**, *) indicates significance [two-sided test] at the 1% (5%, 10%) level.
Country
C
t-value
R2
C
G
Australia Belgium Switzerland China Germany Spain France United Kingdom Hong Kong Ireland Italy Jersey Japan Mexico Netherlands New Zealand Singapore United States
0.8573 1.0267 0.9945 1.0968 1.1881 1.0384 0.9485 1.0380 0.8838 1.2470 1.0399 0.9970 1.0424
-2.268** 0.466 -0.054 1.580 1.768* 0.480 -0.764 0.419 -1.065 2.597** 0.437 -0.027 1.120
84.5% 90.9% 74.2% 90.4% 78.6% 83.2% 85.3% 79.4% 67.2% 83.5% 81.2% 70.4% 95.7%
24.2% 14.0% 15.2% 18.7% 32.8% 37.5% 20.9% 10.6% 20.3% 40.3% 31.1% 42.6% 14.6%
65.9%
Industry
I
t-value
R2
I
G
Basic Materials Communications Consumer Cyclical Consumer Non-cyclical Diversified Energy Financial Industrial Technology Utilities
1.0111 0.9796 1.0370 0.8668 1.1938 0.9620 1.0540 1.0965 1.0794 0.8345
0.188 -0.562 0.934 -2.739*** 2.268** -0.390 0.995 3.492*** 1.506 -1.578
89.7% 95.5% 95.3% 90.3% 85.2% 74.1% 91.7% 97.9% 92.5% 65.1%
22.6% 14.0% 15.2% 18.7% 32.8% 37.5% 20.9% 10.6% 20.3% 40.3%
65.9%
iii
Table 4: Equity return factor model – Corporates This table provides the results of the panel regression for the equity returns of the index constituents of Eurostoxx 50, SMI, FTSE-100, S&P500, ASX-200, HSI and Nikkei-225. The equity index compositions reflect the status as of end of December 2013. The model estimation is based on the following settings: (1) only global factor; (2) global and country factors; (3) global and industry factors and (4) global, country and industry factors. Country and industry factors are defined as cross-sectional averages at each time point and have already been decomposed into global factor and residuals (see Table 3). Country 'United States' and industry 'Utilities' serve as a base case for the (dummy-based) regression. The estimation is based on non-overlapping monthly (log-)returns and conducted over the period September 2007 through September 2010 that has been identified as the one with the most "stressed" correlation (see Figure 1; 35 return periods). Explanatory factors with too few valid (less than five) observations are greyed out; all other factors, are statistically different from zero at 95% confidence level. The bottom part of the table provides the statistics for 2 the R between the individual names and the systematic factor(s).
Model
(1) Global factor only
(2) Global and country factors
(3) Global and industry factors
(4) Global, country and industry factors
N
36,124
36,124
36,124
36,124
1.0021
1.0025
1.0022
1.0031
Coefficients Global Australia Belgium China France Germany Hong Kong Ireland Italy Japan Jersey Mexico Netherlands New Zealand Singapore Spain Switzerland United Kingdom United States
1.0017 0.0000 1.0000 1.0000 1.0000 0.9998 0.9943 1.0000 0.9998 0.0000 0.0000 1.0000 0.2867 0.0000 1.0000 0.9995 1.0002 -
Basic Materials Communications Consumer Cyclical Consumer Non-cyclical Diversified Energy Financial Industrial Technology Utilities R2
0.9498 0.0000 0.9728 0.9301 1.0442 0.8970 0.9537 0.9091 0.9696 0.0000 0.0000 0.9038 0.3374 0.0000 0.9822 0.9380 0.9546 0.9877 0.9901 0.9998 0.9990 1.0000 0.9996 0.9987 1.0458 0.9987 -
0.8178 0.9667 0.8617 1.0437 0.5506 1.0166 0.9084 0.4605 0.9968 -
42.8%
47.5%
46.6%
50.8%
45.5% 19.2% 0.0% 87.8%
50.2% 19.8% 0.0% 88.3%
49.1% 19.4% 0.0% 88.5%
53.2% 19.4% 0.0% 88.6%
Individual R2 vs. systematic factor(s) Average Std. dev. Minimum Maximum
iv
Table 5: Recovery rate model – Calibration This table summarizes the parameterization of the recovery rate model according to (8). The asset correlation parameters
and
stem from the equity correlation model in Section 3.2. In particular, they are set to the R
2
2 for corporates [Table 4, with Model (1) as a proxy] and the average R across countries for sovereigns [Table 3]. The
calibration aims at matching the empirically observed mean (
) and standard deviation (
) of recovery rates of a
set of 2,828 corporate bonds over the period 1988 through 2011 according to data in Altman (2014). These moments of the recovery rate distribution need to be matched conditional on default; with probabilities of default varying across the rating spectrum separate tuples (
) are calibrated each in an iterative procedure. In order to avoid an ill-fitting with is added as a constraint to
significant probability mass attributed to recovery rates greater than one,
the fitting procedure. Given limited data availability and to avoid calibration instability one parameter from the comprehensive study in Bade et al. (2011a) is re-used, namely the correlation between log-recovery rates ( log-recovery rate correlation implies the weight of the systematic asset return in the recovery rate model (
The correlation between the asset return and recovery model (
). The
):
) is given by
and is hence different for corporates and sovereigns, given the different asset (equity) correlations.
A. External parameters Parameter
V, Corp V, Sov RR RR ln(RR)
B. Resulting parameters
Value
Source
Parameter
42.82%
Equity correlation model, Section 3.2
81.88%
Equity correlation model, Section 3.2
44.90%
Altman (2014), Table C.1 [used in fitting]
VY, Corp VY, Sov
37.90% 4.11%
Y
Altman (2014), Table C.1 [used in fitting] Bade et al. (2011a), Table 5
C. Calibrated parameters Parameter/Value Rating
Corp
Corp
Sov
Sov
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
v
Value 4.11% 13.27% 18.35%
Table 6: P&L distributions and IDR figures for selected example portfolios This table presents calculation results for a set of IDR example portfolios. Portfolio A consists of equally weighted investment-grade corporate bonds with issuers from the iTraxx (125 names, Series 20) and assumes an overall notional of 10 million EUR. Portfolio B represents a long/short corporate bond portfolio where half of the bond positions (63) are long, the others are short (62). Portfolios C and D represent the equivalent setup, but with high-yield bonds and issuers from the iTraxx CrossOver (100 names, Series 20). Long and long/short portfolios in sovereign bonds are reflected in portfolios E and F; the issuers are from the SovX Western Europe (Series 8), SovX CEEMEA ex-EU (Series 8) and SovX AsiaPacific (Series 8), for a total of 34 names. Portfolio G is a combination of portfolios B, D and F. A long position in constituents of the Eurostoxx 50 is reflected in portfolio H and portfolio I represents a long/short equity portfolio where half of the positions are long, the others are short (25 each). Finally, portfolio J aggregates portfolios B, D, F and I to a mixture of long/short positions in bond and equity positions across a wide spectrum of issuers. Note that short portfolios in either bonds or equities are not considered since they only allow non-negative P&Ls and thus render the IDR figures equal to 0 EUR. All positions are assumed to be equally weighted with respect to the notional; bonds are considered as trading at par, which implicitly puts slightly higher weight on lesser rated bonds whose PVs – ceteris paribus – would be lower than those of better rated ones. Probabilities of default use the data from Table 1 and reflect the S&P rating of the issuers as of December 2014; Moody’s ratings serve as primary and the index average ratings as secondary fall-back. Initial recovery rates (subject to simulation) reflect
from Table 5; equity positions are assumed to have a 0%
recovery rate. With not all issuers’ countries being part of the correlation model calibration, fall-backs based on geographical location and type of country (developed vs. emerging) are used. Data source for index members and ratings: Bloomberg. The P&L figures (in EUR) are derived by means of one million simulations each. The estimation error in parentheses is calculated as
with
as the percentile estimate and
as the upper
and lower bounds representing a 95% confidence level.
Percentile of P&L distribution A. Investment-grade bonds – long position B. Investment-grade bonds – long/short position C. High-yield bonds – long position D. High-yield bonds – long/short position E. Sovereign bonds – long position F. Sovereign bonds – long/short position G. Portfolios B, D and F together H. Equity index constituents – long position I. Equity index constituents – long/short position J. Portfolios B, D, F and I together
vi
10%
1%
0.1%
0
-184,696
-662,808
(0.0%)
(1.6%)
(3.3%)
0
-66,048
-196,464
(0.0%)
(1.0%)
(3.1%)
-629,037
-1,768,448
-3,116,519
(0.4%)
(0.6%)
(1.3%)
-137,433
-421,220
-749,059
(0.3%)
(0.6%)
(1.2%)
-208,180
-386,529
-899,212
(0.1%)
(0.5%)
(3.6%)
-207,003
-351,149
-466,263
(0.1%)
(0.6%)
(0.8%)
-264,601
-646,645
-1,064,022
(0.4%)
(0.6%)
(1.0%)
0
-200,000
-1,000,000
(0.0%)
(0.0%)
(0.0%)
0
-200,000
-400,000
(0.0%)
(0.0%)
(0.0%)
-263,129
-651,687
-1,105,615
(0.4%)
(0.6%)
(1.6%)
Table 7: Selected sensitivity analyses for IDR figures This table presents the results of selected sensitivity analysis of the IDR figures relative to the base cases in Table 6; a positive number indicates a more negative P&L (i.e. an increase in IDR). The stresses to the average recovery rate are achieved by adjusting
from Table 5. The stresses in the average default correlation are achieved by means of adjusting the R2 of
the names in the portfolio while leaving the systematic risk factor structure unchanged. In both cases, the recovery rate model is recalibrated to ensure that the recovery rate distribution conditional on default across all ratings matches the targeted moments (see Table 5).
(a) Obligor ratings
(b) Average recovery rate
(c) Average default correlation
(d) Recovery rate: weight of systematic factor
Downgrade by
Upgrade by
Decrease by
Increase by
Decrease by
Decrease by
one notch
one notch
25%
25%
25%
10%
A. Investment-grade bonds – long position
33.7%
-23.2%
16.4%
-16.2%
-29.0%
-12.2%
7.7%
B. Investment-grade bonds – long/short position
16.0%
-19.4%
15.4%
-16.1%
-11.8%
-5.1%
3.2%
C. High-yield bonds – long position
23.4%
-21.1%
16.0%
-15.7%
-18.8%
-7.6%
10.8%
D. High-yield bonds – long/short position
20.4%
-12.8%
15.0%
-15.8%
-6.0%
-2.7%
3.6%
E. Sovereign bonds – long position
34.2%
-15.8%
16.7%
-16.4%
-11.8%
-1.7%
2.9%
F. Sovereign bonds – long/short position
12.4%
-2.7%
11.7%
-13.6%
2.0%
1.0%
1.7%
G. Portfolios B, D and F together
13.4%
-10.9%
15.4%
-15.7%
-5.7%
-2.6%
4.4%
H. Equity index constituents – long position
20.0%
-20.0%
N/A
N/A
-40.0%
-20.0%
N/A
0.0%
0.0%
N/A
N/A
0.0%
0.0%
N/A
14.4%
-11.0%
14.0%
-13.4%
-6.9%
-2.7%
4.6%
I. Equity index constituents – long/short position J. Portfolios B, D, F and I together
vii
= 0.25
100%
300%
80% 250%
200%
20% 0% ‐20%
150%
1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013
Pairwise equity‐equity correlation
40%
100%
‐40% ‐60%
Average width of 95% confidence interval
60%
50%
‐80% ‐100% 10th percentile
0%
25th percentile
50th percentile
75th percentile
90th percentile
The figure shows the evolution of the equity correlation over time, based on the index constituents of Eurostoxx 50, SMI, FTSE-100, S&P500, ASX-200, HSI and Nikkei-225. The equity index compositions reflect the status as of end of December 2013. The return correlation is calculated from non-overlapping monthly (log-)returns and conducted on a rolling three-year window. The distribution of the pairwise equity correlations is illustrated by means of the 10th, 25th, 50th, 75th and 90th percentile. The shaded area reflects the three-year window (September 2007 through September 2010) surrounding the peak in correlation (median > 40%). The bars on the bottom of the graph (related to the secondary axis) provide the width of average 95% confidence interval from the correlation estimation.
Figure 1: Evolution of equity correlation over time
viii
40% 35%
Observed frequency
30% 25% 20% 15% 10% 5%
1.00
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
‐0.10
‐0.20
‐0.30
‐0.40
‐0.50
‐0.60
‐0.70
‐0.80
‐0.90
‐1.00
0%
Pairwise correlation Corporate/Corporate
Country/Country
The figure shows the distribution of the equity correlation during the identified stress period (September 2007 through September 2010), based on the index constituents of Eurostoxx 50, SMI, FTSE-100, S&P500, ASX-200, HSI and Nikkei-225. The equity index compositions reflect the status as of end of December 2013. The return correlation is calculated from non-overlapping monthly (log-)returns. Besides the corporate/corporate correlations, country/country correlations are derived from the corresponding cross-sectional averages of corporate returns.
Figure 2: Distribution of correlations during the stress period
ix
40%
Observed frequency
35% 30% 25% 20% 15% 10% 5%
1.00
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
‐0.10
‐0.20
‐0.30
‐0.40
‐0.50
‐0.60
‐0.70
‐0.80
‐0.90
‐1.00
0%
Difference between model and empirical correlation Model 1: global factor only
Model 2: global and country factors
Model 3: global and industry factors
Model 4: global, country and industry factors
Model [countries]
The figure shows the distribution of the differences between the model-implied and empirical (pairwise) correlations. As for the model, the four variants (global factor only, global/country, global/industry, global/country/industry factors) are explored for the corporate/corporate correlations. The differences between the model-implied and empirical country/country correlations are illustrated by means of a separate histogram.
Figure 3: Distribution of differences between model and empirical correlations
x
(a)
(b) 100%
7%
90%
6% Recovery rate conditional on default
80%
5% 4% 3% 2% 1%
70% 60% 50% 40% 30% 20% 10%
95%
100%
90%
85%
80%
75%
70%
65%
60%
55%
50%
45%
40%
35%
30%
25%
20%
15%
5%
10%
0%
0%
0% ‐2.0
Recovery rate (RRi) conditional on default
‐1.8
‐1.6
‐1.4
‐1.2
‐1.0
‐0.8
‐0.6
‐0.4
‐0.2
0.0
Asset return (Vi)
(c)
(d)
0.0 ‐2.0
‐1.8
‐1.6
‐1.4
‐1.2
‐1.0
‐0.8
‐0.6
‐0.4
‐0.2
0.0
Asset return obligor 2 (V2) conditional on joint default
‐0.2 ‐0.4 ‐0.6 ‐0.8 ‐1.0 ‐1.2 ‐1.4 ‐1.6 ‐1.8 ‐2.0
Asset return obligor 1 (V1) conditional on joint default
The figure provides an illustration of the calibrated asset-recovery rate model on the example of one and two BBB-rated corporate obligors. Panel (a) shows the marginal distribution of recovery rates that the model produces conditional on default. Panel (b) illustrates the asset-recovery rate dependence conditional on default. In Panel (c) the asset correlation between two obligors is shown, conditional on a joint default. Panel (d) provides an illustration of the joint recovery rates conditional on both obligors defaulting.
Figure 4: Analysis of asset-recovery rate model xi
(a)
(b) ‐800,000
12%
‐900,000
10%
‐1,000,000
8%
‐1,100,000
P&L
14% 50%
10,000 40,000 70,000 100,000 130,000 160,000 190,000 220,000 250,000 280,000 310,000 340,000 370,000 400,000 430,000 460,000 490,000 520,000 550,000 580,000 610,000 640,000 670,000 700,000 730,000 760,000 790,000 820,000 850,000 880,000 910,000 940,000 970,000 1,000,000
1,000,000
875,000
750,000
625,000
500,000
375,000
250,000
125,000
0
‐125,000
‐250,000
‐375,000
‐500,000
‐625,000
‐1,500,000
‐750,000
0%
‐875,000
‐1,400,000
‐1,000,000
2%
‐1,125,000
‐1,300,000
‐1,250,000
4%
‐1,375,000
‐1,200,000
‐1,500,000
6%
Number of simulations
P&L Estimate
95% Confidence interval
The figure shows in Panel (a) the simulated P&L distribution on the example of portfolio J (see Table 6), with the vertical dotted line indicating the 0.1th percentile (i.e., the IDR figure). Panel (b) illustrates the estimated 0.1th percentile as a function of the number of Monte-Carlo simulations, together with the corresponding 95% confidence interval.
Figure 5: P&L distribution and model convergence
xii
