**1. Introduction**

Systemic risk is the risk that the distress of one or more institutions triggers a collapse of the entire financial system. The CoVaR measure for systemic risk contributions was first proposed by Adrian and Brunnermeier (2008). This measure is the value-at-risk (VaR) of the financial system conditional on an institution (bank) being in financial distress. The systemic risk contribution of an institution is defined as a difference of VaR conditioning on the institution being under distress and being in its normal state. Huang and Uryasev (2017) replaced VaR by conditional value-at-risk (CVaR) and proposed the CoCVaR measure. CVaR has superior mathematical properties as compared to VaR; see, for instance, Rockafellar and Uryasev (2002). CVaR takes into account losses in the distribution tail, while VaR is not sensitive to outcomes in the tail.

Similar to Huang and Uryasev (2017), this paper is based on CVaR, but returns are replaced by drawdowns. The relevant risk measure is called conditional drawdown-at-risk (CDaR). By applying the CoCVaR approach to drawdowns, we defined CoCDaR. Therefore, CoCDaR is CDaR of the financial system conditioned on an institution in distress measured by drawdown. The intuition behind CDaR instead of VaR or CVaR is that these two measures do not take into account consecutive losses. As a result, small consecutive losses resulting in a large cumulative loss are not picked up by VaR or CVaR. Drawdown, which is capturing cumulative losses, is popular in active portfolio management. The idea behind CoCDaR is that large drawdowns of financial institutions have a strong effect on the system as a whole. Hence, by conditioning on large drawdowns of institutions we can analyze systemic risk contributions (compared to effect of one-period negative returns).

We further extended CoCDaR with multiple regression framework and developed so-called mCoCDaR. This measure allows for multiple institutions being in distress, while CoVaR, CoCVaR and CoCDaR assume that only one institution is in distress and others are in normal states. Similar to mCoCDaR, we considered a multiple regression version of CoCVaR, called mCoCVaR. Therefore, mCoCVaR and mCoCDaR account for multiple marginal risk contributions of institutions and are well-defined Shapley values. This approach is motivated by the idea of identifying a risk contribution of each institution that is independent of contributions of other institutions. The estimation of CoCDaR and mCoCDaR was performed with CVaR regression developed in Rockafellar et al. (2014) and Golodnikov et al. (2019). The CVaR regression in CoCDaR uses drawdowns, while CoCVaR uses returns.

The mCoCDaR framework was also illustrated with fund style classification by using drawdowns instead of returns. This approach extends Bassett and Chen (2001), which used quantile regressions of fund returns depending on returns of indices. In addition, we have considered portfolio optimization formulations with CoCVaR and CoCDaR objectives and risk constraints.

CoCDaR and mCoCDaR approaches were demonstrated with a case study for the 10 largest USA banks. Furthermore, we have performed drawdown style classification of the Magellan fund using four stock indices. CVaR regression was implemented with Portfolio Safeguard (PSG) developed by AORDA (http://aorda.com). Case studies results and codes are posted on the web for verification purposes.

#### **2. Methodology**

#### *2.1. Drawdown Definition*

Suppose *r*1, ... ,*rT* are the rates of return of a risky instrument coming from a distribution of return random variable *X*. Let *ξ<sup>t</sup>* be the cumulative rate of return of the instrument for time *t*, which can be either uncompounded and defined by *ξ<sup>t</sup>* = ∑*<sup>t</sup> <sup>k</sup>*=<sup>1</sup> *rk* or compounded and defined by *ξ<sup>t</sup>* = ∏*<sup>t</sup> <sup>k</sup>*=1(1 + *rk*) − 1. Further analysis in this section holds for either definition of the cumulative return, however, for the sake of tractability of optimization problems, *ξ<sup>t</sup>* is defined as uncompounded cumulative rate of return.

The drawdown of the instrument at time *t* with *τ*-window is defined as follows (see Chekhlov et al. (2005); Zabarankin et al. (2014)),

$$y\_t = \max\_{t\_\tau \le k \le t} \mathbb{Z}\_k - \mathbb{Z}\_{t\prime} \qquad t\_\tau = \max\{t - \tau, 1\}, \qquad t = 1, \dots, T, \quad \tau = 1, \dots, T. \tag{1}$$

At time *t*, the drawdown is the loss of the instrument, since a peak of *ξ<sup>t</sup>* that occurs within the *<sup>τ</sup>*-window [*tτ*, *<sup>t</sup>*] (*t<sup>τ</sup>* <sup>=</sup> 1 for *<sup>t</sup>* <sup>≤</sup> *<sup>τ</sup>* and *<sup>t</sup><sup>τ</sup>* <sup>=</sup> *<sup>t</sup>* <sup>−</sup> *<sup>τ</sup>* for *<sup>t</sup>* <sup>&</sup>gt; *<sup>τ</sup>*). If at time *<sup>t</sup>*, the cumulative rate of return *ξ<sup>t</sup>* is the highest on [*tτ*, *t*], then *yt* = 0. The drawdown is always nonnegative and is often referred to as underwater curve. It is zero for all time moments only if returns are nonnegative for all period. See Figure 1 for the illustration of the drawdown definition (the figure is borrowed from Zabarankin et al. (2014)).

**Figure 1.** Drawdown example: the solid line is the uncompounded cumulative rate of return, which at time *t* is the sum of rates of return over periods 1, ... , *t*. Here, *τ* = 6. For *t* = 5, *ξ*<sup>5</sup> = 0.5%, whereas the maximum of *ξ<sup>t</sup>* over time moments preceding *t* = 5 occurs at *t* = 2 with *ξ*<sup>2</sup> = 1.5%. Consequently, *y*<sup>5</sup> = 1.5% − 0.5% = 1%. The instrument maximum drawdown over time period [0, 6] occurs at *t* = 5.

#### *2.2. CoCDaR Definitions*

Conditional value-at-risk (CVaR) of a random variable *X* (see Rockafellar and Uryasev (2000, 2002)), can be defined as follows,

$$\text{CVaR}\_{\mathfrak{A}}(X) = \min\_{\mathbb{C}} \left\{ \mathbb{C} + \frac{1}{1 - a} \, E[(X - \mathbb{C})^{+}] \right\},$$

where *<sup>A</sup>*<sup>+</sup> <sup>=</sup> max{0, *<sup>A</sup>*}. The *<sup>α</sup>*-conditional drawdown is an expectation over the worst <sup>1</sup> <sup>−</sup> *<sup>α</sup>* drawdowns occurring in the considered horizon. We can look at {*yt*}1≤*t*≤*<sup>T</sup>* as a nonlinear transformation of observations from the random return variable *X* and denote the random variable for drawdowns by *Y*. The conditional drawdown-at-risk (CDaR) for *X* is defined as CVaR of *Y*:

$$\operatorname{CDaR}\_{\mathfrak{A}}(X) = \operatorname{CVaR}\_{\mathfrak{A}}(Y) \; .$$

Let *Xsys* denote return of a financial system and let returns of financial institutions *i* = 1, . . . , *I* be denoted as *X<sup>i</sup>* . Given a sample path of data {*x sys <sup>t</sup>* , *<sup>x</sup>*<sup>1</sup> *<sup>t</sup>* , ... , *x<sup>I</sup> <sup>t</sup>* }1≤*t*≤*<sup>T</sup>* , we can obtain the drawdown observations for the financial system as well as all the institutions and denote them by {*y sys <sup>t</sup>* , *<sup>d</sup>*<sup>1</sup> *<sup>t</sup>* , ... , *d<sup>I</sup> <sup>t</sup>* }1≤*t*≤*<sup>T</sup>* . Let *<sup>Y</sup>sys*, *<sup>D</sup>*1, ..., *<sup>D</sup><sup>I</sup>* denote random variables associated with these observations.

Similar to Huang and Uryasev (2017), we can define CoCDaR as:

$$\text{Cov}\text{D}aR\_{\mathfrak{A}}^{\text{sys}|i} = \text{C}\text{D}aR\_{\mathfrak{A}}(X^{\text{sys}}|X^{i}, M\_{1}, \dots, M\_{\mathfrak{n}}) = \text{C}VaR\_{\mathfrak{A}}(Y^{\text{sys}}|D^{i}, M\_{1}, \dots, M\_{\mathfrak{n}}) \ .$$

Here *M*1, ..., *Mn* are state factor variables which we define in the next section. They are the lagged system variables used in Huang and Uryasev (2017), but transformed to provide more explanatory powers in drawdown regression (considered later on).

*CoCDaRsys*|*<sup>i</sup> <sup>α</sup>* gives a measure of CDaR of a system (index) conditioning on the drawdown level of an individual institution (stock) *i*, along with some other state variables. By using drawdown instead of return, we are looking specifically at the impact of individual institution drawdown to the entire financial system drawdown as a measure of systemic risk contribution (that takes into account consecutive distress periods). This intuition will be further developed in Section 2.6.

#### *2.3. State Variables*

Lagged state variables *M*1,*t*−1, ..., *Mn*,*t*−<sup>1</sup> used in the regression in the following section were introduced by Adrian and Brunnermeier (2008):

(1) VIX = The Chicago Board Options Exchange Volatility Index;

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

(2) Liquidity Spread = A short-term liquidity spread, defined as the difference between the three-month repurchase agreement rate and the three-month treasury bill rate;

(3) 3M Treasury Change = The change in the three-month T-bill rate;

(4) Term Spread Change = The change in the slope of the yield curve, measured by the yield spread between the ten-year treasury rate and the three-month bill rate;

(5) Credit Spread Change = The change in the credit spread between Baa-rated bonds and the treasury rate;

(6) Equity Returns = The equity market return from S&P 500 Index;

(7) Real Estate Excess Return = The real estate sector return in excess of the market return.

#### *2.4. Estimation of CoCDaR*

Consider the following regression similar to Adrian and Brunnermeier (2008) and Huang and Uryasev (2017),

$$Y\_t^{sys} \sim \beta\_0 + \beta\_1 D\_t^i + \omega\_1 M\_{t-1,1} + \dots + \omega\_n M\_{t-1,n} \dots$$

We define the residual random variable as:

$$L = Y^{sys} - \left(\beta\_0 + \beta\_1 D^i + \omega\_1 M\_1 + \dots + \omega\_n M\_n\right).$$

This regression problem uses a single institution's drawdown and lagged state variables as factors to model the drawdown of the financial system. We have *T* observations of the system drawdowns, drawdowns of institution *i*, and the state factors. We next perform a CVaR regression with the above model and find CVaR of the system's drawdown conditioned on drawdowns of institution *i*. Here, the state factors are cumulative changes of each fundamental factor in the period of the current drawdown of the financial system. In particular, VIX and liquidity spread are given in numbers so we calculate the time lagged difference of them in the current period of system drawdown. The other state factors are given in percentage changes, so we calculate their cumulative changes in the current period of system drawdown.

For each time step *t*, we consider the system cumulative returns *ξ sys <sup>t</sup>* and find the historic peak time (used in drawdown definition), denoted by

$$\nu(t) = \underset{t\_{\mathbb{T}} \le s \le t}{\text{arg}\max} \ \xi\_s^{sys} \ . \tag{2}$$

Let the original state variable values (numeric or cumulative changes in percentage) be denoted by *mt*. The transformed state variables for the CoCDaR regression are hence defined for each *j* = 1, . . . , *n* :

$$\mathcal{M}\_{t-1,j} = m\_{t-1,j} - m\_{\nu(t),j} \ :$$

The estimate of the *α*-CVaR of *Ysys* can be obtained by minimizing the CVaR (superquantile) error from Rockafellar et al. (2014):

$$\mathcal{E}\_{\mathfrak{a}}^{\text{CVaR}}(L) = \frac{1}{1 - \mathfrak{a}} \int\_0^1 \mathbb{C}V a R\_{\gamma}^+(L) \, d\gamma - E[L] \,. \tag{3}$$

Golodnikov et al. (2019) proved that minimization of error (3) for CVaR regression can be reduced to the minimization of the Rockafellar error (convex and liner programming formulations are in Appendix A, Golodnikov et al. (2019)). The Rockafellar error belongs to the mixed quantile quadrangle, as defined by Rockafellar and Uryasev (2013). For given confidence levels *α<sup>k</sup>* ∈ (0, 1) and weights *λ<sup>k</sup>* > 0, *k* = 1, . . . , *K* such that *K* ∑ *k*=1 *λk*=1, the Rockafellar error equals:

$$\mathcal{E}^{\rm ROC}(L) = \min\_{\mathbb{C}\_1, \dots, \mathbb{C}\_K} \left\{ \sum\_{k=1}^K \lambda\_k \mathcal{E}\_{a\_k}^{KB} (L - \mathbb{C}\_k) \mid \sum\_{k=1}^K \lambda\_k \mathbb{C}\_k = 0 \right\},\tag{4}$$

where the rescaled Koenker–Bassett (KB) error equals:

$$\mathcal{E}\_{\mathfrak{a}}^{KB}(L) = E\left[\frac{\mathfrak{a}}{1-\mathfrak{a}}L^{+} + (-L)^{+}\right].\tag{5}$$

Koenker and Bassett (1978) suggested estimating a conditional quantile by minimizing error (5). Since CVaR is an integral of quantile (VaR), then it is not surprising that CVaR can be estimated with Rockafellar error (4) which is a weighted average of KB-errors. The Rockafellar error is quite a complicated function: it is a minimum of a convex nonsmooth function with respect to variables *C*1, ... , *CK* with a linear constraint. However, since this error is a convex piece-wise linear function, it can be minimized very efficiently; see for instance results of numerical experiments in Golodnikov et al. (2019). The resulting coefficients will provide an estimate of the *α*-CVaR of the dependent variable conditioned on the independent variables.

Denote by *β*ˆ*<sup>α</sup>* <sup>0</sup>, *<sup>β</sup>*ˆ*<sup>α</sup>* <sup>1</sup>, *<sup>ω</sup>*<sup>ˆ</sup> *<sup>α</sup>* <sup>1</sup> , ..., *<sup>ω</sup>*<sup>ˆ</sup> *<sup>α</sup> <sup>n</sup>* the regression coefficients obtained by minimizing the Rockafellar error (4). CoCVaR of the system's drawdown, which is CoCDaR of the system, is estimated by:

$$\text{CoCDaR}\_{t,\alpha}^{\text{sys}} = \hat{\beta}\_0^a + \hat{\beta}\_1^a D\_t^i + \hat{\omega}\_1^a M\_{t-1,1} + \dots + \hat{\omega}\_n^a M\_{t-1,n} \dots$$

This regression estimation is done for every institution, *i* = 1, . . . , *I*.

#### *2.5. Institutional Drawdown-at-Risk*

To calculate system CoCDaR at some risk level conditioned on institution *i* being in drawdown distress, we need to set an institutional distress level *D<sup>i</sup> t*.

*α*-value-at-risk (VaR), which is also *α*-quantile, of a random loss variable *L* is defined as:

$$V a R\_{\alpha}(L) = \inf \{ \alpha : F\_L(\alpha) \ge \alpha \} \dots$$

We define *α*-drawdown-at-risk (*α*-DaR) of an institution *i* as the *α*-quantile (VaR) of the drawdown loss random variable *D<sup>i</sup>* corresponding to its return random variable *X<sup>i</sup>* , where *α* ∈ [0, 1],

$$
\operatorname{DaR}\_{\mathfrak{a}}(X^{i}) = \operatorname{VaR}\_{\mathfrak{a}}(D^{i})\ .
$$

Similar to Huang and Uryasev (2017), we can use quantile regression for estimation of *α*-DaR:

$$D\_t^i \sim \gamma\_0^i + \gamma\_1^i M\_{t-1,1}^i + \dots + \gamma\_n^i M\_{t-1,n}^i \ .$$

Here, the state factors *M<sup>i</sup>* <sup>1</sup>, ..., *<sup>M</sup><sup>i</sup> <sup>n</sup>* are defined differently compared to the CoCDaR regression. They are the same fundamental factor changes but calculated in the current period of each institution drawdown. Define *<sup>ν</sup>i*(*t*) = *argmaxtτ*≤*s*≤*<sup>t</sup> <sup>ξ</sup><sup>i</sup> <sup>s</sup>* for each institution *i* = 1, ... , *I*, where *ξ<sup>i</sup> <sup>t</sup>* are the cumulative returns observations. The transformed state variables for the DaR regression are hence defined for each *j* = 1, . . . , *n*:

$$M\_{t-1,j}^{i} = m\_{t-1,j} - m\_{\nu\_i(t),j} \cdot$$

Let the residual term be denoted as:

$$G^i = D^i - \left(\gamma^i\_0 + \gamma^i\_1 M^i\_1 + \dots + \gamma^i\_n M^i\_n\right) \ .$$

By minimizing KB-error, <sup>E</sup>*KB <sup>α</sup>* (*G<sup>i</sup>* ), we find coefficients *γ*ˆ*<sup>i</sup>* <sup>0</sup>, ..., *<sup>γ</sup>*ˆ*<sup>i</sup> <sup>n</sup>* and estimate the *α*-quantile of *D<sup>i</sup> t*:

$$DaR^i\_{t, \alpha} = \hat{\gamma}^i\_0 + \hat{\gamma}^i\_1 M^i\_{t-1, 1} + \dots + \hat{\gamma}^i\_n M^i\_{t-1, n} \dots$$

#### *2.6. Sytemic Risk Contribution*

We have defined the CoCDaR measure and suggested an estimation procedure with CVaR regression. Next we show how to use this measure for systemic risk contribution measurement. We follow definitions from Huang and Uryasev (2017) and define:

$$X\_t^{sys} = 100 \ln \frac{I\_t}{I\_{t-1}} \text{ } \epsilon$$

as the system's return variable which is the log return of the index value, such as the Dow Jones Index. Similarly, the *i*-th financial institution log return *X<sup>i</sup> <sup>t</sup>* is defined as:

$$X\_t^i = 100 \ln \frac{P\_t^i}{P\_{t-1}^i} \text{ / }$$

where *P<sup>i</sup> <sup>t</sup>* is the closing price of institution *i* at time *t*.

Using the definitions in previous sections, we get the drawdown observations *Ysys <sup>t</sup>* , *<sup>D</sup><sup>i</sup> <sup>t</sup>* for the financial system and an institution *i*. We also have state factors, *Mt*−1,1, ..., *Mt*−1,*n*, for every time moment *t* is in the considered horizon.

We first perform the quantile regression in Section 2.5 to estimate *DaR<sup>i</sup> <sup>t</sup>*,*α* for all *t* for two particular levels: *α* <sup>1</sup> = 0.9 and *α* <sup>2</sup> = 0.5. The level *α* <sup>1</sup> = 0.9 corresponds to the distress level of the institution in terms of its drawdown and *α* <sup>2</sup> = 0.5 corresponds to the median (normal) state of the institution.

Next we perform the CVaR regression from Section 2.4 and obtain an estimate of the *α*-CoCDaR of the financial system conditioned on the drawdown level of institution *i* and state factors. Here *α* is different from *α* used in the previous quantile regression. For every time step *t*, we calculate:

$$\text{CoCDaR}\_{t,\mathfrak{a}}^{\text{sys}|D\_t^i = DaR\_{t,\mathfrak{a}'}^i} = \hat{\beta}\_0^a + \hat{\beta}\_1^a DaR\_{t,\mathfrak{a}'}^i + \hat{\omega}\_1^a M\_{t-1,1} + \dots + \hat{\omega}\_n^a M\_{t-1,n} \dots$$

By choosing *α* <sup>1</sup> = 0.9 and *α* <sup>2</sup> = 0.5 for the DaR level for an individual institution and selecting a separate risk level *α* for system CoCDaR, we obtain:

$$\Delta CoCDaR\_{t,\alpha}^{sys|i} = CoCDaR\_{t,\alpha}^{sys|D\_t^i = DaR\_{t,0.9}^i} - CoCDaR\_{t,\alpha}^{sys|D\_t^i = DaR\_{t,0.5}^i}.$$

This difference is defined as the systemic drawdown risk contribution of institution *i* to the financial system at the selected risk level *α*. More concretely, it calculates the difference in conditional drawdown-at-risk values of the financial system given that the drawdown level of institution *i* is at its distress level or its normal level as a measure of systemic drawdown risk contribution.

#### *2.7. mCoCDaR Definition*

Using the same set of state factors and extending the idea of CoCDaR as a measure of systemic risk contribution, we propose a more comprehensive measure called multiple-CoCDaR, which measures the conditional drawdown-at-risk of the financial system conditioned on the distress levels of all *I* institutions being considered. The idea is an extension of the CoCDaR approach defined above by combining it with a generalization of the multiple-CoVaR method defined in Bernardi et al. (2013) and Bernardi and Petrella (2014). In their paper, a similar approach was developed that defines conditional tail risk of a system/institution conditioned on the distress level of multiple institutions at the same time. A similar approach was also seen in Cao (2013). Different from their methods, our approach uses a simple multiple regression formulation. In the multiple regression framework, we can measure risk contribution of an institution by taking the difference between CoCDaR values of the system under different drawdown levels of that institution alone, while holding other institutions' drawdown values fixed at their normal levels. We define mCoCDaR as:

$$m\text{CoCD}aR\_{\mathfrak{a}}^{\text{sys}|1,\dots,I} = \text{CDaR}\_{\mathfrak{a}}(X^{\text{sys}}|X^1,\dots,X^I,M\_1,\dots,M\_{\mathfrak{n}}) = \text{CVaR}\_{\mathfrak{a}}(Y^{\text{sys}}|D^1,\dots,D^I,M\_1,\dots,M\_{\mathfrak{n}}) \ . \ .$$

#### *2.8. Estimation of mCoCDaR*

Consider the following regression using the same set of state factors as in CoCDaR regression,

$$Y\_t^{sys} \sim \beta\_0 + \beta\_1 D\_t^1 + \dots + \beta\_I D\_t^I + \omega\_1 M\_{t-1,1} + \dots + \omega\_n M\_{t-1,n} \ \lambda$$

with residual:

$$L = Y^{sys} - \left(\beta\_0 + \beta\_1 D^1 + \dots + \beta\_I D^I + \omega\_1 M\_1 + \dots + \omega\_n M\_n\right).$$

This regression problem uses *I* institutions drawdowns and lagged state variables as factors to model the drawdown of the financial system. We have *T* observations for the system drawdown random variable, the *I* institutions' drawdown random variables, and the state factors' random variables. We next perform a CVaR regression of the above model to find CVaR of the system drawdown conditioned on all *I* institution drawdown. Denote by *β*ˆ*<sup>α</sup>* <sup>0</sup>, *<sup>β</sup>*ˆ*<sup>α</sup>* <sup>1</sup>, ..., *<sup>β</sup>*ˆ*<sup>α</sup> <sup>I</sup>* , *<sup>ω</sup>*<sup>ˆ</sup> *<sup>α</sup>* <sup>1</sup> , ..., *<sup>ω</sup>*<sup>ˆ</sup> *<sup>α</sup> <sup>n</sup>* coefficients obtained by minimizing Rockafellar error (4). These coefficients allow one to compute the *α*-CDaR of the financial system conditioned on drawdowns of all the institutions and state factors.

The multiple-CoCVaR of the system drawdown, which is equivalent to the multiple-CoCDaR of the financial system, is estimated by:

$$m\text{CoC}\text{D}a\text{R}^{\text{sys}}\_{t,\mathfrak{a}} = \hat{\beta}^a\_0 + \hat{\beta}^a\_1 D^1\_t + \dots + \hat{\beta}^a\_I D^I\_t + \hat{\omega}^a\_1 M\_{t-1,1} + \dots + \hat{\omega}^a\_n M\_{t-1,n} \text{ .} $$

This procedure applies one regression problem using all institutions' drawdown observations to obtain coefficient estimates. The institutional DaRs are calculated exactly the same way as in Section 2.5.

#### *2.9. Sytemic Risk Contribution using mCoCDaR*

We have defined mCoCDaR measure and the estimation procedure with CVaR regression. We use this measure for systemic risk contribution measurement, following the definitions in Section 2.4. The drawdown observations are denoted by *Ysys <sup>t</sup>* , *<sup>D</sup>*<sup>1</sup> *<sup>t</sup>* , ..., *D<sup>I</sup> <sup>t</sup>* for the financial system and all *I* institutions respectively. We also have lagged state variables *Mt*−1,1, ..., *Mt*−1,*<sup>n</sup>* for every time moment *t*.

We first perform the quantile regression in Section 2.5 to estimate *DaR<sup>i</sup> <sup>t</sup>*,*α* for all *t* and for all *i* for two particular levels: *α <sup>i</sup>*,1 = 0.9 and *α <sup>i</sup>*,2 = 0.5. Level *α <sup>i</sup>*,1 = 0.9 corresponds to the distress level of the *i*-th institution in terms of its drawdowns and *α <sup>i</sup>*,2 = 0.5 corresponds to its median (normal).

Next we perform the CVaR regression from Section 2.8 and estimate the financial system's conditional drawdown-at-risk conditioned on the drawdown levels of all *I* institutions and state factors. For every time step *t*, we calculate,

$$\begin{aligned} \text{mCoCDaR}^{\text{sys}[D] = DaR}\_{t,\mu\_1^1} \dots \alpha\_1^{I\_l} &= \beta\_0^a + \beta\_1^a DaR^1\_{t,\mu\_1^l} + \dots + \beta\_1^a DaR^1\_{t,\mu\_1^l} + \widehat{\alpha}\_1^a M\_{t-1,1} + \dots + \widehat{\alpha}\_n^a M\_{t-1,n} \dots \alpha\_1^{I\_l} \end{aligned}$$

Now, to analyze the effect of a single institution *i* on the financial system, we compute the mCoCDaR values based on *α <sup>i</sup>*,1 = 0.9 and *α <sup>i</sup>*,2 = 0.5, while holding *α* <sup>−</sup>*<sup>i</sup>* <sup>=</sup> 0.5 fixed where <sup>−</sup>*<sup>i</sup>* means all the institutions other than *i*, and calculate the difference in mCoCDaR,

$$\Delta m \text{CoCDaR}\_{t,\mu}^{\text{sys}|i} = m \text{CoCDaR}\_{t,\mu}^{\text{sys}|D\_t^i = \text{Da}R\_{t,0.9}^i, D\_t^{-i} = \text{Da}R\_{t,0.5}^{-i}} - m \text{CoCDaR}\_{t,\mu}^{\text{sys}|D\_t^i = \text{Da}R\_{t,0.5}^i, D\_t^{-i} = \text{Da}R\_{t,0.5}^{-i}} - 1$$

This difference is the incremental/marginal systemic drawdown risk contribution of the distress of institution *i* to the financial system, while other institutions are at their normal states.

We can switch back to the original return observations instead of the drawdown observations and perform the regression procedure in Sections 2.7 and 2.8. This way we get another measure for systemic risk contribution which we call mCoCVaR. It measures the incremental/marginal conditional value-at-risk of the financial system's returns conditioned on one institution's return being in distress while all the other institutions are in their normal states.

#### *2.10. Advantages of mCoCDaR and mCoCVaR*

As we have seen in the previous section, the multiple version of the systemic risk conditional estimation provides a more general framework to analyze the effect on the financial system posed by a particular institution's distress, or perhaps multiple financial institutions' joint distress. It is based on the idea that during periods of financial instability, several institutions may experience financial distress at the same time, so their risk contributions can be highly correlated. Switching from the CoCVaR and CoCDaR to their multiple regression versions helps to mitigate these dependencies on risk contribution measures.

With mCoCDaR, we can measure the contribution to the financial system's conditional drawdown-at-risk conditioned on the drawdown levels of two institutions *i*, *j* as follows,

$$\Delta m \text{CoCDa} R\_{t,\alpha}^{\text{sys}|i,j} = m \text{CoCDa} R\_{t,\alpha}^{\text{sys}|D\_t^i = DaR\_{t,0.9}^i, D\_t^j = DaR\_{t,0.9}^j, D\_t^{-i,j} = DaR\_{t,0.5}^{-i,j}}$$

$$-m \text{CoCDa} R\_{t,\alpha}^{\text{sys}|D\_t^j = DaR\_{t,0.5}^j, D\_t^j = DaR\_{t,0.5}^j, D\_t^{-i,j} = DaR\_{t,0.5}^{-i,j}}$$

There is a lot of flexibility on the risk levels to choose for this type of analysis, which means the DaR level for the two institutions in distress can be set differently. This approach considers the joint impact of two institutions without distinguishing their respective contributions, which is not included in the original framework without using multiple regression. The flexibility given by mCoCDaR, and similarly mCoCVaR, does not come at additional computation costs. In fact, by combining all institutions in one regression problem, we save computational time.

Another advantage of the multiple-CoCVaR and multiple-CoCDaR are their consistency as risk distribution measures. Bernardi et al. (2013) noticed that the original Δ*CoVaRsys*|*<sup>i</sup>* is not a desirable risk distribution measure, because summing up Δ*CoVaRsys*|*<sup>i</sup>* for all institutions *i* does not generally equal their overall effect on the system. This issue is addressed in Bernardi et al. (2013) and Bernardi and Petrella (2014) via the Shapley value, which transforms the calculated contribution using <sup>Δ</sup>*Multiple* − *CoVaR* to a Shapley value for each institution so that their contribution adds up to the joint contribution of all institutions together on the system. The Shapley value methodology was originally proposed to measure shared utility or cost among participants of a cooperative game.

We observe that the individual risk contribution calculated with Δ*mCoCVaR* or Δ*mCoCDaR* does not have this drawback. For instance, for mCoCDaR:

$$\begin{aligned} \text{mCoCDaR}^{\text{sys}|D\_l^1 = \text{DaR}^1\_{t,\mu'\_1}, \dots, \text{D}\_l^1 = \text{DaR}^1\_{t,\mu'\_l}} \ &= \hat{\rho}\_0^a + \hat{\rho}\_1^a \text{DaR}^1\_{t,\mu'\_l} + \dots + \hat{\rho}\_1^a \text{DaR}^1\_{t,\mu'\_l} + \hat{\omega}\_1^a M\_{t-1,1} + \dots + \hat{\omega}\_n^a M\_{t-1,n} \dots \text{N}\_l \end{aligned}$$

Once we have estimated the coefficients via CVaR regression, we can calculate the individual contribution of institution *i* entering stress level 0.9 as:

$$\Delta m \text{CoCDaR}\_{t,\mu}^{\text{sys}|i} = m \text{CoCDaR}\_{t,\mu}^{\text{sys}|D\_t^i = DaR\_{t,0:9}^i D\_t^{-i} = DaR\_{t,0:5}^{-i}} - m \text{CoCDaR}\_{t,\mu}^{\text{sys}|D\_t^i = DaR\_{t,0:5}^i D\_t^{-i} = DaR\_{t,0:5}^{-i}}$$

$$= \beta\_i^{\text{it}} (DaR\_{t,0:9}^i - DaR\_{t,0:5}^i) \equiv V\_{\text{sys}}(i) \; . $$

The total contribution of all financial institutions distress on the systemic risk is:

$$\Delta mCoCDaR\_{t,a}^{sys|1,...,I}$$

$$\mathcal{B} = \begin{array}{c} m \text{CoC} \text{Da} \text{R} \text{R}\_{t,a}^{\text{sys}} = \text{Da} \text{R}\_{t,0.9}^{1}...\text{D}\_{t}^{I} = \text{Da} \text{R}\_{t,0.9}^{I} \\\\ \text{T} = m \text{CoC} \text{Da} \text{R}\_{t,a}^{\text{sys}} \end{array} \\ \text{sys} \\ \begin{array}{c} \text{ys} | D\_{t}^{1} = \text{Da} \text{R}\_{t,0.9}^{1}...\text{D}\_{t}^{I} = \text{Da} \text{R}\_{t,0.9}^{I} \\\\ \text{T} = m \text{CoC} \text{R}\_{t,0.9}^{1} = \text{Da} \text{R}\_{t,0.9}^{1} \end{array}$$

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

$$=\sum\_{i=1}^{I} \beta\_i^{\alpha} (DaR\_{t,0.9}^i - DaR\_{t,0.5}^i) \ = \sum\_{i=1}^{I} V\_{sys}(i) \ .$$

A similar statement is valid for *mCoCVaR*. The entire systemic risk is exactly distributed to its institutional components. Δ*mCoCVaR* and Δ*mCoCDaR* are both Shapley value functions, denoted by *Vsys*(*i*) for contributor *i*, such that they satisfy the following desirable mathematical properties as outlined in Bernardi et al. (2013). Let S be a set of *I* institutions:

(1) Efficiency: ∑*<sup>I</sup> <sup>i</sup>*=<sup>1</sup> *Vsys*(*i*) = *Vsys*(S). This axiom states that the total risk is distributed to participants.

(2) Symmetry: For *i* = *j* such that *Vsys*(*H* ∪ *i*) = *Vsys*(*H* ∪ *j*), ∀*H* such that *i*, *j* ∈/ *H*, then *Vsys*(*i*) = *Vsys*(*j*). This axiom states that the contribution measure is permutation invariant and fair for all contributors.

(3) Dummy axiom: *Vsys*(*H* ∪ *i*) = *Vsys*(*i*), ∀*i* ∈ *H* and *H* ⊇ S. This means, if the risk of institution *j* is independent of all other institutions, then its risk contribution to the system should be its own risk. Generally, in CoCVaR and CoCDaR approaches (also CoVaR), the risks are not orthogonal among institutions. Hence, their ranking should differ from those provided by a Shapley value measure such as *mCoCVaR* and *mCoCDaR*.

(4) Linearity (additivity): If *i*, *j* ∈ *H*, *i* = *j* are two institutions, where *Vsys*(*i*) = *Vsys*(*j*), let *wi* > 0, *wj* > 0, *k* = *wii* + *wj j*, then new combined risk contributions equal the weighted average of individual risk contributions: *Vsys*(*k*) = *wiVsys*(*i*) + *wjVsys*(*j*).

(5) Zero player: If *i* ∈/ *S*, *Vsys*(*i*) = 0. A null player receives zero risk contribution.

#### *2.11. mCoCDaR Versus mCoCVaR*

We do not claim that one of the considered risk measures is better for analyzing systemic risk contributions than the other one. CoCVaR is concerned with the conditional risk in terms of the returns' tail behavior, while CoCDaR is concerned with the conditional risk in terms of the drawdowns. These measures have a nonlinear relationship embedded in their definitions.

When a financial system's large drawdowns are significantly correlated with large drawdowns of some particular institutions, it can be hypothesized that the CoCDaR measure will provide a more reasonable estimate of the risk contributions and therefore give a more reasonable ranking of the systemic risk contribution of each institution. This can be generalized to comparing mCoCVaR and mCoCDaR measures which are proposed in this work. Another intuition for using drawdown based approaches is that drawdown measures a psychological effect from a consistent distress in stock returns.

#### **3. Case Studies**

This case study uses data from CoCVaR paper Huang and Uryasev (2017), which is posted at this link 1. Codes, data, calculation results for this case study are posted at this link 2.

We first computed the drawdowns from the return data and transformed the state factors corresponding to each regression problem. Next, we proceeded to the quantile regressions on institutional drawdowns and the CVaR regression on the system's drawdowns. The CVaR regression is implemented using Portfolio Safeguard (PSG)3 in MATLAB environment. PSG includes efficiently implemented (precoded) Koenker–Bassett, Rockafellar and CVaR errors.

<sup>1</sup> http://uryasev.ams.stonybrook.edu/index.php/research/testproblems/financial\_engineering/case-study-cocvarapproach-risk-contribution-measurement

<sup>2</sup> http://uryasev.ams.stonybrook.edu/index.php/research/testproblems/financial\_engineering/case-study-cocdarapproach-systemic-risk-contribution-measurement

<sup>3</sup> Portfolio Safeguard (PSG) is a product of American Optimal Decisions: http://aorda.com

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