# Credit Risk Meets Random Matrices: Coping with Non-Stationary Asset Correlations

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Random Matrix Theory for Non-Stationary Asset Correlations

#### 2.1. Wishart Model for Correlation and Covariance Matrices

#### 2.2. New Interpretation and Application of the Wishart Model

## 3. Modeling Fluctuating Asset Correlations in Credit Risk

#### 3.1. Random Matrix Approach

#### 3.2. Average Loss Distribution

#### 3.3. Adjusting to Different Market Situations

#### 3.4. Value at Risk and Expected Tail Loss

## 4. Concurrent Credit Portfolio Losses

#### 4.1. Simulation Setup

#### 4.2. Empirical Credit Portfolios

## 5. Discussion

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Standard deviation time series for Goodyear from 1992–2018. The return interval is $\Delta t=1$ trading day, and the time window has a length of $T=60$ trading days.

**Figure 2.**Correlation matrices of $K=306$ companies for the fourth quarter of 2005 and the first quarter of 2006; the darker, the stronger the correlation. The companies are sorted according to industrial sectors. Reproduced with permission from (Schmitt et al. 2013), EPLA.

**Figure 3.**Aggregated distribution of normalized returns $\tilde{r}$ for fixed covariances from the S&P 500 dataset, $\Delta t=1$ trading day and window length $T=25$ trading days. The circles show a normal distribution. Reproduced with permission from (Schmitt et al. 2013), EPLA.

**Figure 4.**Aggregated distribution of the rotated and scaled returns $\tilde{r}$ for $\Delta t=1$ (

**top**) and $\Delta t=20$ (

**bottom**) trading days. The circles correspond to the aggregation of the distribution (13). Reproduced with permission from (Schmitt et al. 2013), EPLA.

**Figure 5.**Aggregated distribution for the normalized monthly returns with the empirical covariance matrix on a logarithmic scale. The black line shows the empirical distribution; the red dotted line shows the theoretical results. The insets show the corresponding linear plots.

**Top**left/right: S&P 500 (1992–2012)/(2002–2012);

**bottom**left/right: NASDAQ (1992–2012)/(2002–2012). Reproduced with permission from (Schmitt et al. 2015), Infopro Digital.

**Figure 6.**Aggregated distribution for the normalized monthly returns with the effective correlation matrix on a logarithmic scale. The black line shows the empirical distribution; the red dotted line shows the theoretical results. The insets show the corresponding linear plots.

**Top**left/right: S&P 500 (1992–2012)/(2002–2012);

**bottom**left/right: NASDAQ (1992–2012)/(2002–2012). The average correlation coefficients are $c=0.26$, 0.35, 0.21 and $0.25$, respectively. Reproduced with permission from (Schmitt et al. 2015), Infopro Digital.

**Figure 7.**Average portfolio loss distribution for different portfolio sizes of $K=10$, $K=100$ and the limiting case $K\to \infty $. At the

**top**, the maturity time is one month; at the

**bottom**, it is one year. Reproduced with permission from (Schmitt et al. 2015), Infopro Digital.

**Figure 8.**Average loss distribution for different parameters taken from Table 1. The dashed line corresponds to the calm period 2002–2004; the solid line corresponds to the global financial crisis 2008–2010.

**Figure 9.**Underestimation of the VaR if fluctuating asset correlations are not taken into account. The empirical covariance matrix is used and compared for different values of N. Reproduced with permission from (Schmitt et al. 2015), Infopro Digital.

**Figure 10.**Heterogeneous correlation matrix illustrating a financial market. The two rimmed squares correspond to two non-overlapping credit portfolios. Taken from (Sicking et al. 2018).

**Figure 11.**Average loss copula histograms for homogeneous portfolios with vanishing average asset correlations $c=0$. The asset values are multivariate log-normal ($N\to \infty $) in the top figure and multivariate heavy-tailed (${N}_{\mathrm{eff}}=5$) in the bottom figure. The color bar indicates the local deviations from the corresponding Gaussian copula. Taken from (Sicking et al. 2018).

**Figure 12.**Average loss copula histograms for homogeneous portfolios with asset correlations $c=0.3$. The asset values are multivariate log-normal ($N\to \infty $). The drifts are $\mu ={10}^{-3}\phantom{\rule{0.222222em}{0ex}}{\mathrm{day}}^{-1}$ (

**top**), $3\times {10}^{-4}\phantom{\rule{0.222222em}{0ex}}{\mathrm{day}}^{-1}$ (

**middle**) and $-3\times {10}^{-3}\phantom{\rule{0.222222em}{0ex}}{\mathrm{day}}^{-1}$ (

**bottom**). The color bar indicates the local deviations from the corresponding Gaussian copula. Taken from (Sicking et al. 2018).

**Figure 13.**Average loss copula histograms for two portfolios with heterogeneous volatilities drawn from a uniform distribution in the interval $(0,0.25)$. The color bar indicates the local deviations from the corresponding Gaussian copula. Taken from (Sicking et al. 2018).

**Figure 14.**Time averaged loss copula histograms for two empirical copulas of size $K=50$. The asset values are multivariate log-normal ($N\to \infty $).

**Top**: Portfolio 1 is always drawn from S&P 500 and Portfolio 2 from Nikkei 225;

**middle**: both portfolios are drawn from S&P 500;

**bottom**: both portfolios are drawn from Nikkei 225. The color bar indicates the local deviations from the corresponding Gaussian copula. Taken from (Sicking et al. 2018).

**Table 1.**Average parameters used for two different time horizons. Taken from (Schmitt et al. 2015).

Time Horizon for Estimation | K | ${\mathit{N}}_{\mathbf{eff}}$ | $\mathit{\rho}$ in Month${}^{-1/2}$ | $\mathit{\mu}$ in Month${}^{-1}$ | c |
---|---|---|---|---|---|

2002–2004 | 436 | 5 | 0.10 | 0.015 | 0.30 |

2008–2010 | 478 | 5 | 0.12 | 0.01 | 0.46 |

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**MDPI and ACS Style**

Mühlbacher, A.; Guhr, T.
Credit Risk Meets Random Matrices: Coping with Non-Stationary Asset Correlations. *Risks* **2018**, *6*, 42.
https://doi.org/10.3390/risks6020042

**AMA Style**

Mühlbacher A, Guhr T.
Credit Risk Meets Random Matrices: Coping with Non-Stationary Asset Correlations. *Risks*. 2018; 6(2):42.
https://doi.org/10.3390/risks6020042

**Chicago/Turabian Style**

Mühlbacher, Andreas, and Thomas Guhr.
2018. "Credit Risk Meets Random Matrices: Coping with Non-Stationary Asset Correlations" *Risks* 6, no. 2: 42.
https://doi.org/10.3390/risks6020042