*3.3. CoCDaR Calculation Results*

The drawdown-at-risk values of each institution at two different risk levels, *α* <sup>1</sup> = 0.9 for distress level and *α* <sup>2</sup> = 0.5 for normal level, are computed using the quantile regression defined in Section 2.5. The CoCDaR values of the system at a specific risk level *α* = 0.9 conditioned on each institution's DaR being at a distress level and a normal level are respectively computed based on the CVaR regression in Section 2.4. Following Section 2.6, the difference in CoCDaR values is taken and this results in a time series of Δ*CoCDaRsys*|*<sup>i</sup> <sup>t</sup>*,*<sup>α</sup>* for each institution *i* and for each observation time *t*.

We observe that the quantile regression for DaR using the state variables as regressors yields different behaviors for different institutions. Responses are different for the state variables: some are positive while others are negative. This is true for both the distress level and the normal level. The pseudo *R*<sup>2</sup> metrics for these quantile regressions are generally between 0.5 to 0.7, which indicates a descent level of explanatory power as compared with using just the original state factors. The observation is consistent with that made in Huang and Uryasev (2017). The CoCDaR and DaR calculation results are posted in the CoCDaR case study2, see Problems 1 and 3. For the ten CVaR regressions of index drawdowns on the state variables and respective institution drawdowns, we observe that the coefficients for each factor typically have the same sign (with a few exceptions). The pseudo *R*<sup>2</sup> for CoCDaR regressions are all above 0.8.

We averaged each bank's contributions to CoCDaR across time and ranked the ten banks accordingly, where larger values correspond to stronger contributions to system's CoCDaR:


The number in brackets is the ranking based on Δ*CoCVaR* in Section 3.2. Results show that only Citigroup Inc. has negative CoCDaR contribution to the index on average, hinting that its drawdowns could have a negative correlation with index drawdowns. All other institutions are contributing positively to the system's conditional drawdown-at-risk.

In particular, PNC Financial Services Group Inc (PNC) was ranked third by Δ*CoCVaR* but ranked seventh by Δ*CoCDaR*. On the other hand, Bank of America (BAC) was ranked fifth by Δ*CoCVaR* but ranked second by Δ*CoCDaR*. Clearly, these two approaches provide different perspectives.

#### *3.4. mCoCVaR Calculation Results*

This section demonstrates the performance of suggested mCoCVaR, which is the multiple version of the CoCVaR approach developed in Huang and Uryasev (2017). We begin by performing the mCoCVaR analysis of the ten financial institutions in one CVaR regression. The pseudo *R*<sup>2</sup> for mCoCVaR regression is 0.76. The value-at-risk for normal and distress states are calculated for every institution respectively using quantile regressions on the original state variables. The procedure for VaR calculation is described in Huang and Uryasev (2017), Sections 2.3 and 3.3.2. By holding all other institutions' return values to their VaR values in a normal state (which corresponds to the median) and looking at the differences resulting from changing one particular institution's return value to its VaR value in a distress state, we obtain a time series of Δ*mCoCVaRsys*|*<sup>i</sup> <sup>t</sup>*,*<sup>α</sup>* for each institution and for each observation time *t*. We averaged each bank's contributions to mCoCVaR across time and ranked the ten banks accordingly, where larger values correspond to stronger contributions to system's mCoCVaR:


The number in the bracket is the ranking according to Δ*CoCVaR* in Section 3.2. The results based on Δ*mCoCVaR* are similar to those based on Δ*CoCVaR*, but there are some significant differences. For instance, WFC, originally ranked the highest, dropped to the seventh place in this new ranking. This might have been caused by its returns having a high correlation to returns of other institutions, for example The BB&T Corporation (BBT). This effect is neglected in the previous CoCVaR method, but in our multiple regression setting, by explicitly fixing the other institutions' returns to their respective normal states, we are analyzing the marginal impact of WFC's distress. Hence, the drop in ranking may indicate that WFC is not a key systemic risk contributor in the sense that its risk contributions are dependent on the high risk contributions of other institutions. Clearly, CoCVaR and mCoCVaR provide different perspectives regarding the ranking of financial institutions' risk contributions.

### *3.5. mCoCDaR Results*

The drawdown-at-risk of each institution at two different risk levels, *α* <sup>1</sup> = 0.9 for distress level and *α* <sup>2</sup> = 0.5 for normal level, are computed using the quantile regression defined in Section 2.5; this step is identical to the first step performed in Section 3.3. The mCoCDaR values of the system at a specific risk level *α* = 0.9 conditioned on each institution's DaR being at a distress level and a normal level are computed respectively based on the CVaR regression with multiple institutions as specified in Section 2.8. Following Section 2.9, the difference in mCoCDaR values is taken and this results in a time series of Δ*mCoCDaRsys*|*<sup>i</sup> <sup>t</sup>*,*<sup>α</sup>* for each institution *i* and for each observation time *t*.

Since we are using the same quantile estimates for DaR, we obtained the same observations as that in Section 3.3. For the CVaR regression of the drawdowns of the index on the state variables and the institution drawdowns, we observe that some institutions' regression coefficients are positive in the CVaR regression, while others are negative. The pseudo *R*<sup>2</sup> for mCoCDaR regression is 0.9. The mCoCDaR and DaR results are posted in the CoCDaR case study2, see Problems 2 and 3.

We averaged each bank's contributions to mCoCDaR across time and ranked the ten banks accordingly, where larger values correspond to stronger contributions to system's mCoCDaR:


The number in the bracket is the ranking according to Δ*CoCDaR*. Δ*mCoCDaR* and Δ*CoCDaR* rankings are mostly similar, yet have some interesting differences as well. While WFC is ranked highest by Δ*CoCDaR*, it is ranked last by Δ*mCoCDaR*. This observation coincides with what we saw in Section 3.4, indicating the high correlation that WFC might have with other top risk contributors such as BB&T and BAC. Furthermore, while STT is ranked second last by Δ*CoCDaR*, it is ranked fourth by Δ*mCoCDaR*.

#### *3.6. Comparative Summary of the Proposed Methods*

Table 1 provides a complete summary of the rankings of the ten banks with the four risk measures. Compared with CoCVaR, CoCDaR takes into account drawdowns and focuses on consecutive losses. Using drawdowns is particularly insightful because drawdowns identify cumulative losses (negative cumulative returns), hence the dependencies between institutions and the system in "good" times are ignored. Dependencies in "bad" times are captured, which is important for risk analysis. We observe that CoCVaR and CoCDaR may provide very different rankings. For instance, USB with mCoCDaR and CoCDaR are ranked 3 and 5, accordingly (i.e., BAC is a top contributor), but with mCoCVaR and CoCVaR it is ranked 9 (i.e., close to bottom contributor). Even more surprisingly, JPM is ranked 9 and 8 with mCoCDaR and CoCDaR, but ranked 4 with mCoCVaR and CoCVaR.

mCoCVaR and mCoCDaR approaches add further insights to CoCVaR and CoCDaR, since they employ a multiple regression that marginalizes the systemic risk contributions of individual institutions. Running the multiple regression instead of individual ones enables us to look at institutions' contributions in a unified way, since their fraction contributions sum up to one.

Risk contributions based on CoVaR and CoCVaR measures, as a function of time, demonstrate a similar pattern for different institutions, see Huang and Uryasev (2017). This is probably because the methodology is based on separate regression for each institution. On the other hand, mCoCDaR results (plotted below) show that the time series of mCoCDaR risk contributions exhibit quite different patterns compared to CoVaR and CoCVaR, and compared across different institutions. With multiple regression, marginal risk contributions of each institution change significantly over time.


**Table 1.** Systemic Risk Contribution Ranking Summary.

Furthermore, we plot time dependent drawdowns and mCoCDaR contributions; see Figures 2–11. Each institution graph on the left plots its drawdown curve in blue versus the orange curve showing drawdowns of the Dow Jones index in the same time period, both based on cumulative uncompounded returns on a weekly basis. Every graph on the right plots fraction contribution to the total systemic risk from an individual bank. This fraction is obtained by normalizing individual contributions measured by Δ*mCoCDaR* described in Section 3.5. Normalization is done by dividing individual contributions by the total contribution from the ten banks. By construction, the normalized contributions sum up to one for each time step. As a result of applying the mCoCDaR regression setting, we observe that individual contributions significantly vary over time as well as across institutions. Moreover, risk contributions may have different signs. For instance, JPM and WFC always have negative contributions (see, Figures 2 and 5). Citigroup starts with negative contributions and moves to contributing positively (see, Figure 4), while the others always have positive contributions.

**Figure 2.** JP Morgan Chase & Company.

**Figure 3.** Bank of America.

**Figure 4.** Citigroup Inc.

**Figure 5.** Wells Fargo & Company.

**Figure 6.** The Bank of New York Mellon Corporation.

**Figure 7.** US Bancorp.

**Figure 8.** Capital One Financial Corporation.

*J. Risk Financial Manag.* **2020**, *13*, 270

**Figure 9.** PNC Financial Services Group Inc.

**Figure 10.** State Street Corporation.

**Figure 11.** The BB&T Corporation.

#### **4. mCoCDaR Application to Style Classification**

This section extends the approach to hedge fund style classification. We show how to estimate CDaR as a function of drawdowns of several market indices. Style classification is a well studied topic approached by Sharpe (1992) and Carhart (1997) with a standard regression (for returns of instruments). Furthermore, it was extended by Bassett and Chen (2001) using quantile regression. Here, we demonstrate results with the mCoCDaR method. This classification explains fund drawdowns, as a function of drawdowns of several market indices (as factors). Codes, data, and results for this case study are posted at this link4.

Similar to Bassett and Chen (2001), we investigated dependence of drawdowns of the Magellan fund (fund) from four indices: Russell 1000 value index (rlv), Russell 1000 growth index (rlg), Russell 2000 value index(ruj), and Russell 2000 growth index (ruo). These indices correspond to four equity classes: large value stocks, large growth stocks, small value stocks, and small growth stocks.

<sup>4</sup> http://uryasev.ams.stonybrook.edu/index.php/research/testproblems/financial\_engineering/case-study-styleclassification-with-mcocdar-regression/

We used a dataset from a quantile regression style classification case study posted at5. Golodnikov et al. (2019) considered the same dataset for testing CVaR regression, which is posted at this link6. The dataset contains 1264 weekly return observations for the fund and indices.

We calculated drawdowns for the fund and the four indices using weekly returns in the considered time period. CVaR regression of drawdowns is done by minimizing CVaR2 error in PSG3, as follows,

$$D\_{fund,t} = \beta\_0 + \beta\_1 D\_{rlv,t} + \beta\_2 D\_{rl\lg,t} + \beta\_3 D\_{ruj,t} + \beta\_4 D\_{ruo,t} \ \ \ \ \ \ \ \beta\_6$$

where *Di*,*<sup>t</sup>* for *i* = *rlv*, *rlg*, *ruj*, *ruo* are (uncompounded) drawdowns of index *i* at time *t* and *Df und*,*<sup>t</sup>* are (uncompounded) drawdowns of the fund. See definition of drawdowns of a financial instrument in Section 2.1.

For 0.9-CVaR regression the pseudo-R square equals 0.91 and the estimated coefficients are:

*β*ˆ = 0.3713, *β*ˆ = 0.4621, *β*ˆ = 0.5493, *β*ˆ <sup>=</sup> <sup>−</sup>0.0171, *<sup>β</sup>*<sup>ˆ</sup> = −0.0591 .

We considered also 0.0-CVaR regression, which estimates mean and corresponds to an ordinary least squares regression. Pseudo-R square equals 0.91 and estimated coefficients are:

$$
\vec{\beta}\_0 = -0.2733, \ \vec{\beta}\_1 = 0.4891, \ \vec{\beta}\_2 = 0.5150, \ \vec{\beta}\_3 = -0.0618, \ \vec{\beta}\_4 = -0.0003\ .
$$

Regression coefficients show that both large and average drawdowns of the Magellan fund are mostly explained by drawdowns in large value stocks index (coefficient *β*ˆ 1) and large growth stocks index (coefficient *β*ˆ 2). The fund exhibits roughly 50–50% mix of these two classes of stocks in the sense of drawdown behavior.

Furthermore, we compared these results with previous studies, which used CoVaR- and CoCVaR-based measures. The CoVaR approach based on quantile regression<sup>5</sup> (see Problem 1 in the link) gives the following coefficient estimate:

$$\hat{\beta}\_0 = -0.0089, \; \hat{\beta}\_1 = 0.4602, \; \hat{\beta}\_2 = 0.5176, \; \hat{\beta}\_3 = -0.0156, \; \hat{\beta}\_4 = 0.0001\,\, \text{A}$$

and the CoCVaR approach based on CVaR regression6 (see Problem 1, *α*=0.9 in the link) gives the following estimate:

$$\hat{\beta}\_0 = 0.0105, \; \hat{\beta}\_1 = 0.6058, \; \hat{\beta}\_2 = 0.4721, \; \hat{\beta}\_3 = -0.0778, \; \hat{\beta}\_4 = -0.0071\,\, .$$

We observe that this particular dataset considered regressions of a similar style with around a 50–50% mix of two stock indices.

#### **5. On Portfolio Optimization with mCoCDaR and mCoCVaR**

Previous sections defined and tested mCoCDaR and mCoCVaR multiple regression versions for systemic risk measurement. It should be considered that risk measures can be used for other purposes. For instance, we can build a portfolio minimizing CoCVaR or CoCDaR, conditioned on the distress level of several market indices (or factors), under the constraint that the expected return meets some target. Similar problems were studied in Kurosaki and Kim (2013a, 2013b) with CoAVaR and CoVaR measures for conditional risk. Here, we present portfolio optimization problems using mCoCVaR and mCoCDaR risk measures:

$$\min\_{\vec{w}\_{l}} \quad m\text{CoCVaR}\_{a,t}^{\vec{w}\_{l}|f\_{l}^{1},\ldots,f\_{l}^{K}} \quad \text{s.t.} \quad \sum\_{i=1}^{I} w\_{l}^{i}r\_{l}^{i} = r^{\star} \quad \sum\_{i=1}^{I} w\_{l}^{i} = 1$$

<sup>5</sup> http://uryasev.ams.stonybrook.edu/index.php/research/testproblems/financial\_engineering/style-classification-withquantile-regression/

<sup>6</sup> http://uryasev.ams.stonybrook.edu/index.php/research/testproblems/financial\_engineering/on-implementation-ofcvar-regression/

$$\min\_{\vec{w}\_l} \quad m\\ \text{CoCDaR}\_{\alpha, l}^{\vec{w}\_l | f\_l^1, \dots, f\_l^K} \quad \text{s.t.} \quad \sum\_{i=1}^I w\_l^i r\_l^i = r^\star \quad \sum\_{i=1}^I w\_t^i = 1$$

where *K* is the number of market index factors, *f* <sup>1</sup> *<sup>t</sup>* , ..., *f <sup>K</sup> <sup>t</sup>* are risk levels at time *t* of *K* factors (market indices), *w <sup>t</sup>* is vector of portfolio weights for *I* stocks, *r<sup>i</sup> <sup>t</sup>* is return of stock *i* at time *t*, and *r* is a target return. Systemic risk-driven portfolio selection problems were also studied in Capponi and Rubtsov (2019), where they considered portfolio optimization given a systemic event. Detailed analysis of these portfolio optimization problems is beyond the scope of this paper. We have included a short description to show that considered risk measures can be used in various areas of finance.

#### **6. Conclusions**

This paper proposed a new systemic risk measure, CoCDaR, which is based on conditional drawdown-at-risk and inspired by the CoCVaR approach from Huang and Uryasev (2017). We further extended the approach to mCoCDaR, which calculates conditional drawdown-at-risk of the financial system conditioned on all the institutions' drawdown distress levels. These measures can rank institutions according to their incremental (marginal) contributions to the systemic risk of the system, conditional on other institutions' distress levels. The multiple regression setting is applied to the CoCVaR measure from Huang and Uryasev (2017) and resulted in so-called mCoCVaR. Since mCoCDaR and mCoCVaR are based on multiple regression, they have the flexibility to measure joint contributions of multiple institutions. These measures are also well-defined Shapley value functions with desirable mathematical properties for a risk contribution measure. After normalization, individual risk contributions sum up to one. These advantages do not come at any additional computational cost.

CoCDaR and mCoCDaR measures are based on drawdowns (path dependent cumulative losses). These two measures capture the impact of an institution's drawdowns on the financial system's drawdowns, which is particularly suitable for market crash situations. They are useful for determining which institution may lead to a bigger crash in the market in terms of large drawdown events.

We performed a case study for the three proposed methods, CoCDaR, mCoCVaR, and mCoCDaR, using data from the ten largest banks and the Dow Jones Index, along with some state factors. The case study with codes and data are posted on the web. We have also reproduced the case study for CoCVaR measure from Huang and Uryasev (2017), with corrected signs in the returns data. We compared the ranking of institutions based on contributions to system's CoCDaR, mCoCDaR, mCoCVaR, and CoCVaR. The difference in applying CVaR- and CDaR-based measures is observed from quite different rankings of institutions. Multiple regression identifies key drivers in systemic risk because effects are marginalized. We compared time dependent curves of risk contributions for mCoCDaR and CoCVaR. Risk contributions based on CoCVaR are quite similar across institutions, while those based on mCoCDaR have very different patterns. These different patterns are implied by both the use of drawdowns and the use of multiple regressions.

Other applications of the proposed method include fund style classifications based on mCoCDaR regression. We have conducted a case study analyzing drawdowns of the Magellan fund as a function of drawdowns of four market indices. We have posted this case study to the web. The suggested methodology may also be used in various other areas of finance. In particular, we have stated portfolio selection problems with mCoCVaR or mCoCDaR objectives and constraints on expected returns.

**Funding:** This research received no external funding.

**Author Contributions:** R.D. and S.U. stated the problem; S.U. obtained the data and provided the software; R.D. processed the data, wrote the programs and obtained the results; R.D. and S.U. analyzed the results; R.D. prepared the first draft manuscript; R.D. and S.U. revised the text and the conclusions. All authors have read and agreed to the published version of the manuscript.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


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