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A macroeconomic reverse stress test

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Abstract

Reverse stress tests are a relatively new stress test instrument that aims at finding exactly those scenarios that cause a bank to cross the frontier between survival and default. Afterward, the scenario which is most probable has to be identified. This paper sketches a framework for a quantitative reverse stress test for maturity-transforming banks that are exposed to credit and interest rate risk and demonstrates how the model can be calibrated empirically. The main features of the proposed framework are: (1) the necessary steps of a reverse stress test (solving an inversion problem and computing the scenario probabilities) can be performed within one model, (2) scenarios are characterized by realizations of macroeconomic risk factors, (3) principal component analysis helps to reduce the dimensionality of the space of systematic risk factors, (4) due to data limitations, the results of reverse stress tests are exposed to considerable model and estimation risk, which makes numerous robustness checks necessary.

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Notes

  1. Imposing a granularity assumption to eliminate idiosyncratic credit risk factors might be a way that allows these approaches to be applied to portfolios which are exposed to market as well as to credit risk.

  2. A symbol directory for Sects. 2.1 to 2.3 is given in the Appendix in the Tables 11, 12 and 13.

  3. The methodology would get more intricate when, additionally, we would model instruments that, conditional on the realization of the risk factors, have no fixed cash flows, but for which behavioral assumptions are needed to determine the involved cash flows, for example, interest rate-dependent withdrawals of saving accounts, drawings of credit lines (see, e.g., Stanhouse (2016)) or cancellations of fixed-rate loans with prepayment options. Although there are several commonalities among banks, in general, the specification of these behavioral assumptions and the extent of embedded optionalities highly depend on a bank’s business model and on its jurisdiction (see BCBS 2006, p. 212). These behavioral assumptions would add further model risk to the reverse stress test framework.

  4. In general, our proposed framework is flexible enough to consider more complex financial instruments (for example, loans structured as coupon bonds), however, at the cost of larger estimation efforts and a higher computational burden.

  5. These are reasonable numbers which can be observed from aggregated numbers for US commercial banks or for euro area banks [see monetary financial institutions (MFI) statistics by the European Central Bank (ECB)]. However, the proposed reverse stress test framework is flexible enough to consider any proportion observed in the institution carrying out the reverse stress test.

  6. These proportions are based on results in the Basel III Monitoring Report from March 2016 (see BCBS 2016a, p. 38).

  7. This maturity is obtained from banks’ unconstrained internal estimates as provided in the Basel III Monitoring Report from March 2016 (see BCBS 2016a, p.38). In the consultative document of the BCBS’s revised standards for interest rate risk in the banking book (see BCBS 2015, p. 22), the BCBS proposed a uniform slotting until six years.

  8. The risk horizon H is the point in time at which potential losses of the banking book portfolio are computed. It is assumed that the length of the time interval [0, H] is smaller or equal than \(T_n\) for all \(n \in \{1,\ldots ,12\}\).

  9. Obviously, the pricing approach could be easily modified to consider non-stochastic credit spreads that depend on the time to maturity. More data-intensive would be an approach in which the credit spreads of different times to maturity and rating grades are modeled by a multivariate distribution (under full consideration of existing dependencies).

  10. See Grundke (2011, 2012a).

  11. The mean and the standard deviation of the beta-distributed recovery rate equal Standard & Poor’s mean and standard deviation of the recovery rate of senior unsecured bonds during 1987–2011 (see Standard & Poor’s 2011b).

  12. This assumption proves to be very sensible. Due to the high volatility of the recovery rate, we can observe that the pure recovery-of-treasury assumption would lead in a surprisingly large number of cases to higher values of bonds after default. A simulation within our framework reveals that, for AA-rated obligors, the fraction of increases in value after default ranges between 13.02% (default of obligors with a maturity of \(t=1\)) and 28.98% (default of obligors with a maturity of \(t=12\)), whereas in the case of BB-rated obligors, the fraction ranges between 22.45% (default of obligors with a maturity of \(t=1\)) and 48.72% (default of obligors with a maturity of \(t=12\)). Our modified version avoids this unfavorable effect.

  13. This assumption corresponds to an accounting standard under which firms are not allowed to consider value variations of their equity caused by changes in their own credit quality in a way that affects their net income. For an alternative modeling with time-varying bank rating, see Grundke (2012a). With a time-varying bank rating, care has to be taken to avoid circularity problems.

  14. Note that when a scenario would be modeled as a point in \({\mathbb {R}}^n\) and the systematic risk factors would be continuous random variables, the probability of occurrence would be zero for each scenario. In this case, the plausibility of a scenario could, for example, be measured by its Mahalanobis distance (see Grundke 2012a, p. 95).

  15. The setting \(B=V_E(0)\) corresponds to a gone concern perspective that the bank uses for its risk coverage calculations. Of course, other settings are possible.

  16. The idiosyncratic risk is the only source of uncertainty in the case of the conditional distribution. Therefore, the small number of Monte-Carlo simulation runs is sufficient.

  17. The interpretation as asset returns results from the seminal paper Merton (1974). More generally, the credit quality of an obligor is assumed to be driven by some creditworthiness index (see, for example, Dorfleitner et al. 2012). The lower the index is, the worse is the rating grade of the obligor. When the index is below a given threshold, this event is set equal to a default of the obligor.

  18. A more sophisticated approach would be a dynamic credit portfolio model such as the one in Rösch and Scheule (2007) where the default threshold is explained by time-lagged systematic risk factors or CreditPortfolioView\(^{TM}\) (see Wilson 1997a, b) where the probabilities of the unconditional rating transition matrices are shifted depending on forecasted default rates.

  19. Estimating factor loadings in linear factor models for asset returns by maximum likelihood (based on default data) is a frequently employed approach in the credit portfolio risk literature (see, for example, Gordy and Heitfield 2002; Frey and McNeil 2003; Hamerle and Rösch 2006; Rösch and Scheule 2007).

  20. An additional constraint \(\rho _{i,Z} \in (0, 1)\) ensures that we do not divide by zero or compute the square root of a negative value.

  21. See McNeil et al. (2005, p. 275).

  22. The normal copula does not exhibit tail dependence.

  23. A detailed introduction to copula functions is given, for example, in McNeil et al. (2005) and Nelson (2006).

  24. See Genest et al. (2009, p. 201).

  25. See Hill et al. (2011, p. 238).

  26. The internal codes in Datastream are USGDP…D and S&PCOMP.

  27. The internal codes are FRTCM3M, FRTCM6M, FRTCM1Y, USBDS2Y, USBDS3Y, USBDS5Y, USBDS7Y, USBD10Y and USBD30Y.

  28. Otherwise, the null hypothesis that the time series contain unit roots cannot be rejected at reasonable significance levels by the ADF test.

  29. An earlier report of Standard & Poor’s (2003, p. 8) made of breakdown according to various regions and shows that most defaults are caused by U.S. companies. A quite similar dataset from Moody’s (2011) shows that 84% of defaults are triggered by North American companies for the period from 1986 to 2010. Furthermore, worldwide and U.S. default rates are highly correlated. For the period 1983–2010, we calculated a correlation of 97.41% when using data from Standard & Poor’s.

  30. Reilly et al. (2010) examine statistical properties of credit spreads for speculative grade obligors.

  31. Data was adjusted for rating withdrawals.

  32. Datastream uses a linear combination of sovereign bonds in order to match the maturities of corporate bonds precisely.

  33. A negative spread can be explained by low liquidity shortly before the maturity date. If a bond is not traded on a day, the last observed price is taken as the current price. Therefore, the bond price does not converge against the face value, and, for bonds priced above their face value, a negative yield (and a negative credit spread) can be calculated.

  34. See Kaiser (1960).

  35. In focusing on GDP and the interest rates (principal components of the risk-free interest rates) as stressed macroeconomic systematic risk factors, we follow Virolainen (2004) and Sorge and Virolainen (2006). Other studies add further risk factors like commodity prices (see, for example, Misina et al. 2006) or credit spreads (see, for example, Avouyi-Dovi et al. 2009). Of course, many other macroeconomic risk factors might also be relevant to explaining defaults (such as industry production or money supply indicators; see, for example, Dorfleitner et al. 2012). However, any additional macroeconomic risk factor that we add to the linear factor model explaining the obligors’ asset returns complicates the reverse stress test due to computational issues. Thus, we face the classical conflict between accuracy and practicability for the desired purpose.

  36. The maximization was performed using the function constrOptim in R and is regarded as numerically stable. Calculations were performed using the Nelder-Mead method (see Nelder and Mead 1965) with different initial values. The improper integral was solved using the function int of the program R, which is based on the Gauss-Kronrod quadrature (see Kronrod 1965). Numerical issues due to the improper integral were considered, too. For the integral \(\int _{-\infty }^{+\infty } \left( {\begin{array}{c}N_{i}(t)\\ d_{i}(t)\end{array}}\right) q_i (z,x(t),c_1(t),c_2(t))^{d_{i}(t)} (1-q_i(z,x(t),c_1(t),c_2(t)) )^{N_{i}(t)-d_{i}(t)} \phi (z) dz\), we substituted \(y={\varPhi }(z)\) and \(\frac{dy}{dz}=\phi (z)\), respectively. This leads to the expression \(\int _{0}^{1} \left( {\begin{array}{c}N_{i}(t)\\ d_{i}(t)\end{array}}\right) q_i ({\varPhi }^{-1}(y),x(t),c_1(t),c_2(t))^{d_{i}(t)} (1-q_i({\varPhi }^{-1}(y),x(t),c_1(t),c_2(t)) )^{N_{i}(t)-d_{i}(t)} dy\). The following optimization delivered the same result as that one using the improper integral.

  37. A possible explanation is that the parameter’s variance is increased due to several observations without defaults. However, it turns out that the coefficients are numerically stable (see Footnote 36) and have the correct sign.

  38. The latent systematic credit risk factor Z is assumed to be standard normally distributed.

  39. Figure 6 shows that the data for U.S. GDP log-returns includes exactly one outlier (realization in 2009). The same is true for the data of the S&P 500 log-returns (realization in 2008) and the second principal component (realization in 2009) that also includes one outlier. When omitting these outliers, we could not reject normality for all risk factors at reasonable significance levels.

  40. Modeling the right tail of the distribution of the U.S. GDP log-return and the S&P 500 log-return, respectively, is not necessary because we are interested in scenarios generating a sufficiently large loss. Thus, due to the positive sign of the asset return sensitivity with respect to the U.S. GDP log-return and the S&P 500 log-return, large U.S. GDP or S&P 500 log-return increases are less relevant. The second principal component, in contrast, has an ambiguous effect on losses because it weights interest rate changes with a short time to maturity with a positive sign and interest rate changes with a long time to maturity with a negative sign. The net effect depends on the portfolio sensitivities towards interest rates for different times to maturity and, therefore, both tails should be modeled by the GPD.

  41. These are graphs that map for every u a mean excess function \({\mathbb {E}}[X-u|X>u]\) (see, for example, Ghosha and Resnick 2010). For an application, see, for example, Gourier et al. (2009).

  42. This is required due to the linearity of the mean excess function of the GPD.

  43. The data was transformed by multiplication with −1.

  44. Three observations ensure that we have more observations than parameters to estimate.

  45. The parameters were estimated by maximum likelihood and by the probability-weighted moment method, respectively. To take estimation risk into account, the most conservative estimates were used (those with the highest shape parameter \(\xi\) indicating a fat tail).

  46. For a detailed description, see Genest and Rémillard (2008).

  47. The Clayton copula benefits of its sparse parametrization and the comparatively good fit, while the elliptical copulas are punished due to their high number of parameters and the other Archimedean copulas possess a much worse fit.

  48. The copula parameters and their significance were estimated by maximum pseudo-likelihood and by inverting Kendall’s Tau (see, for example, McNeil et al. 2005, pp. 228–237) using the functions gofCopula and fitCopula of the package copula in R. Since the estimators deviated less than the standard error of each other, we employed only the maximum pseudo-likelihood estimators. The use of different estimation techniques takes account of the estimation uncertainty and serves as an internal robustness check.

  49. Due to the construction principle of principle components (that is based on orthogonalization), the theoretically true value for \(\rho _{C_1,C_2}\) is zero.

  50. For the reverse stress test, we use the default thresholds as specified in Table 8 instead of the estimated ones in (2.11) in conjunction with (2.12).

  51. This setting reduces the vulnerability to interest rate risk. For AA (BB)-rated obligors and without interest rate swaps, the Basel II 200 bps parallel upward shift would lead to a shrinking of the bank’s equity value to 186.66 (24.33) monetary units, whereas the equity would increase to 308.55 (85.64) monetary units after a 200 bps parallel downward shift. The swap contract, however, reduces the bank’s equity sensitivity to interest rate changes. In this case, the bank’s equity value amounts to 195.17 (32.84) after an upward shift and to 278.09 (55.17) after a downward shift.

  52. The computation of the probabilities was done using the function pCopula of the package copula in the program R. In order to calculate probabilities, pCopula refers to the function pmvt of the package mvtnorm which uses randomized Quasi-Monte-Carlo methods (see, for example, Genz and Bretz 1999, 2002). As the assigned probabilities on the edge of the considered part of the support are very low, numerical issues may lead to us obtaining implausible results, especially negative probabilities. To solve this problem, we calculate probabilities in the case of the t-copula as the mean over several repetitions. However, this works only for initially BB-rated obligors.

  53. It should be noted that the latent systematic risk factor takes a slightly positive value in the case of the specification using the GDP as the economic indicator in combination with the t-copula with normally distributed margins.

  54. In total, 255 scenarios are classified as reverse stress test scenarios in the case of the t-copula with GPD tails and U.S. GDP as the economic indicator. For the t-copula with normally distributed margins, we got 262 reverse stress test scenarios. In the case of the Clayton-copula, 265 (normal, GDP), 250 (GPD, GDP), 337 (normal, S&P 500) and 331 (GPD, S&P 500) scenarios exhausted the initial capital buffer B for initially BB-rated obligors.

  55. Note that we do not need this multivariate distribution for identifying the set \(\varvec{{\varOmega }}^*\) of all reverse stress scenarios. Thus, estimation risk with respect to the multivariate distribution of the systematic risk factors does not influence this identification process.

  56. Using an Intel Core i7-3970X Extreme Edition, 32 GB RAM and Revolution R 8.0.

  57. Effectively, this is also true for explaining the indirect effect on the value of the zero-coupon bonds that yield curve movements have: as interest rate factors drive the obligors’ asset returns [see (2.7)], the obligors’ ratings (up to a default) and, hence, their credit spreads depend on them. In (2.7), we also employ only the first two principal components instead of the yield-to-maturities of all relevant times to maturity.

  58. A further portfolio-based dimension reduction technique is proposed by Skoglund and Chen (2009). Their approach is based on the Kullback–Leibler divergence.

  59. See BCBS (2006, pp. 212–213) and BCBS (2016b, p. 15).

  60. The BCBS suggests to consider both perspectives (see BCBS 2016b). The European Banking Authority’s (EBA) guidelines on management of interest rate risk arising from non-trading activities (IRRBB) propose using mandatory both methods (see EBA 2015).

  61. See Schmidt (1981).

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Authors and Affiliations

  1. Osnabrueck University, Katharinenstraße 7, 49069, Osnabrück, Germany

    Peter Grundke

  2. Deutsche Bundesbank, Wilhelm-Epstein-Straße 14, 60431, Frankfurt am Main, Germany

    Kamil Pliszka

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  1. Peter Grundke
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  2. Kamil Pliszka
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Correspondence to Kamil Pliszka.

Additional information

Previously, the paper circulated under the title ‘Empirical implementation of a quantitative reverse stress test for defaultable fixed-income instruments with macroeconomic factors and principal components’. We wish to thank an anonymous referee who helped to improve the paper significantly and the participants of the Bundesbank Seminar. This paper represents the authors’ personal opinions and does not necessarily reflect the views of the Deutsche Bundesbank or its staff.

Appendix

Appendix

1.1 Symbol directory for Sect. 2

See Tables 11, 12 and 13

Table 11 Overview of the symbols in Sect. 2.1
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Table 12 Overview of the symbols in Sect. 2.2
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Table 13 Overview of the symbols in Sect. 2.3
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Grundke, P., Pliszka, K. A macroeconomic reverse stress test. Rev Quant Finan Acc 50, 1093–1130 (2018). https://doi.org/10.1007/s11156-017-0655-8

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