A Multi-agent Methodology to Assess the Effectiveness of Systemic Risk-Adjusted Capital Requirements

Abstract

We propose a multi-agent approach to compare the effectiveness of macroprudential capital requirements, where banks are embedded in an artificial macroeconomy. Capital requirements are derived from alternative systemic risk metrics that reflect both the vulnerability and impact of financial institutions. Our objective is to explore how systemic risk measures could be translated into capital requirements and test them in a comprehensive framework. Based on our counterfactual scenarios, we find that macroprudential capital requirements derived from vulnerability measures of systemic risk can improve financial stability without jeopardizing output and credit supply. Moreover, macroprudential regulation applied to systemic important banks might be counterproductive for systemic groups of banks.

Appendix

1.1 Calibration of DebtRank

In general, our approach is similar to that adopted in Battiston et al. (2016), but we have adapted the algorithm to account for the structure of the underlying macro-model, as described in greater detail in section “DebtRank”. Given that the macro-environment includes firms, we first impose the shock on firms’ assets to compute the systemic vulnerability index $DR^{vul}$. Next, the induced distress transmits linearly to the assets of creditors (i.e. banks). This allows capturing the specific dynamics of the distress process.

Our calibration strategy aims to compare market- and network-based measures on a common ground. To do so, we apply to DebtRank the definition of systemic crisis employed in the SRISK framework. SRISK is computed by LRMES, which represents the expected equity loss of a bank in case of a systemic event. This is represented by a decline of market returns of $40\%$ over the next 6 months. We run 100 Monte Carlo simulations of the macro-model, record the market ROE and the firms’ losses to equity ratio. Then, we compute the change in market ROE over the past 180 periods (approximately 6 months). Finally, we construct a vector of the losses of firms to their equities in those periods where the ROE declined at least by $-40\%$.

To compute vulnerabilities by DebtRank, we randomly sample from the vector of the empirical distribution of losses/equity at each repetition of the algorithm. Finally, we obtain $DR^{vul}$ for each bank as an average of the realized values, after removing the 1st and the 99th percentiles (Fig. 11).

Fig. 11

(Top-left) rescaled market ROE from a random Monte Carlo run. (Top-right) 6-month chance of market ROE. The red dashed line represents the threshold of ${-}40\%$. (Bottom-left) Histogram of the square root of the losses/loans ratio of firms, where values equal to zero are ignored. (Bottom-right) Histogram of the losses/loans ratio of firms

1.2 DebtRank

We employ a differential version of the DebtRank algorithm in order to provide a network measure of systemic risk. Differential DebtRank (Bardoscia et al. 2015) is a generalization of the original DebtRank (Battiston et al. 2012) which improves the latter by allowing agents to transmit distress more than once. Moreover, our formulation has similarities with Battiston et al. (2016), where it is assumed a sequential process of distress propagation. In our case, we first impose an external shock on firms’ assets, and then we sequentially account for the propagation to the banking sector through insolvencies on loans, to the interbank network, and to firms’ deposits.

The relative equity loss for banks (h) and firms (f) is defined as the change in their net worth (respectively, $nw^B,$ and $nw^F$) from $\tau =0$ to $\tau$ with respect to their initial net worth. In particular, the initial relative equity loss of firms happens at $\tau =1$ due to an external shock on deposits:

$$\begin{aligned} h_i(\tau )&=\min \left[ \frac{nw^B_i(0)-nw^B_i(\tau )}{nw^B_i(0)}\right] \\ f_j(\tau )&=\min \left[ \frac{nw^F_j(0)-nw^F_j(\tau )}{nw^F_j(0)}\right] \\ \end{aligned}$$

The dynamics of the relative equity loss in firms and banks sectors is described by the sequence:

• Shock on deposits in the firms sector:

  $$f_j(1)=\min \left[ 1,\ \frac{D^F_j(0)-D^F_j(1)}{nw^F_j(0)}\right] =\min \left[ 1,\ \frac{loss_j(1)}{nw^F_j(0)}\right] $$

• Banks’ losses on firms’ loans:

  $$\begin{aligned} h_i(\tau +1)=\min \left[ 1,\ h_i(\tau )+\sum _{j\in J}\Lambda ^{fb}_{ij}(1-\varphi _j^{loan}) (p_j(\tau )-p_j(\tau -1))\right] \end{aligned}$$

• Banks’ losses on interbank loans:

  $$\begin{aligned} h_i(\tau +1)=\min \left[ 1,\ h_i(\tau )+\sum _{k\in K}\Lambda ^{bb}_{ik}(1-\varphi _k^{ib})(p_k(\tau )-p_k(\tau -1))\right] \end{aligned}$$

• Firms’ losses on deposits:

  $$\begin{aligned} f_{j}(\tau +1)=\min \left[ 1, f_j(\tau )+\Lambda ^{fb}_{jk}(1-\varphi _k^{dep})(p_k(\tau )-p_k(\tau -1))\right] \end{aligned}$$

where $p_j$ is the default probability of debtor j and $\varphi ^{i},\ i=\{loan, ib, dep\}$ is the recovery rate on loans, interbank loans, and deposits. Recovery rates on each kind of asset are randomly extracted from a vector of observations generated by the benchmark model.

For the sake of simplicity, we can define it as linear in $f_j$ ($h_k$ for banks), so that $p_j(\tau )=h(\tau )$.¹¹ $\Lambda$ is the exposure matrix that represents credit/debt relationships in the firms–banks network. It is written as a block matrix, where $\Lambda ^{bb}$ refers to the interbank market, $\Lambda ^{bf}$ refers to deposits, $\Lambda ^{fb}$ refers to firm loans and $\Lambda ^{ff}$ is a matrix of zeros.

$${\Lambda }= \begin{bmatrix} {\Lambda }^{bb} &{} {\Lambda }^{bf}\\ {\Lambda }^{fb} &{} {\Lambda }^{ff}\\ \end{bmatrix}$$

The exposure matrix $\Lambda$ represents potential losses over equity related to each asset at the beginning of the cycle, where each element has the value of assets at the numerator and the denominator is the net worth of the related creditor in our specification firms have no intra-sector links, hence $\Lambda ^{ff}=0$. In case there are $N^b=2$ banks and $N^f=3$ firms, the matrix $\Lambda$ looks like

$$ {\Lambda }= \begin{bmatrix} 0 &{} \frac{Ib_{12}}{nw^B_2} &{} \frac{D_{13}}{nw^F_1} &{} \frac{D_{12}}{nw^F_2} &{} \frac{D_{15}}{nw^F_3}\\ \frac{Ib_{21}}{nw^B_1} &{} 0 &{} \frac{D_{23}}{nw^F_1} &{} \frac{D_{24}}{nw^F_2} &{} \frac{D_{25}}{nw^F_3}\\ \frac{L^f_{31}}{nw^B_1} &{} \frac{L^f_{32}}{nw^B_2} &{} 0 &{} 0 &{} 0\\ \frac{L^f_{41}}{nw^B_1} &{} \frac{L^f_{42}}{nw^B_2} &{} 0 &{} 0 &{} 0\\ \frac{L^f_{51}}{nw^B_1} &{} \frac{L^f_{52}}{nw^B_2} &{} 0 &{} 0 &{} 0\\ \end{bmatrix} $$

1.3 SRISK

SRISK (Brownlees and Engle 2012) is a widespread measure of systemic risk based on the idea that the latter arises when the financial system as a whole is under-capitalized, leading to externalities for the real sector. To apply the measure to our model, we follow the approach of Brownlees and Engle (2012). The SRISK of a financial firm i is defined as the quantity of capital needed to re-capitalize a bank conditional to a systemic crisis

$$ SRISK_{i,t}=\min \left[ 0,\ \frac{1}{\lambda }\mathcal {L}_i-\left( 1-\frac{1}{\lambda }\right) nw^B_{i,t}(1-MES^{Sys}_{i,t+h|t})\right] $$

where $MES^{Sys}_{i,t+h|t}=E\left( r_{i,t+h|t}|r<\Omega \right)$ is the tail expectation of the firm equity returns conditional on a systemic event, which happens when i’s equity returns r from $t-h$ to t are less than a threshold value $\Omega$.

Acharya et al. (2012) propose to approximate $MES^{Sys}$ with its Long Run Marginal Expected Shortfall (LRMES), defined as a

$$LRMES_{i,t}=1-\exp \{-18MES^{2\%}_{i,t}\}$$

LRMES represents the expected loss on equity value in case the market return drops by $40\%$ over the next 6 months. Such approximation is obtained through extreme value theory, by means of the value of MES that would be if the daily market return drops by ${-}2\%$.

The bivariate process driving firms’ ($r_i$) and market ($r_m$) returns is

$$\begin{aligned}&r_{m,t}=\sigma _{m,t}\epsilon _{m,t}\\&r_{i,t}=\sigma _{i,t}\rho _{i,t}\epsilon _{m,t}+\sigma _{i,t}\sqrt{1-\rho _{i,t}^2}\xi {i,t}\\&(\xi _{i,t},\epsilon _{m,t})\sim F \end{aligned}$$

where $\sigma _{m,t}$ is the conditional standard deviation of market returns, $\sigma _{i,t}$ is the conditional standard deviation of firms’ returns, $\rho _{i,t}$ is the conditional market/firm correlation and $\epsilon$ and $\xi$ are i.i.d. shocks with unit variance and zero covariance.

$MES^{2\%}$ is expressed setting $\Omega =-2\%$:

$$MES^{\Omega }_{i,t-1}=\sigma _{i,t}\rho _{i,t}E_{t-1}\left( \epsilon _{m,t}|\epsilon _{m,t}<\frac{\Omega }{\sigma _{m,t}}\right) +\sigma _{it}\sqrt{1-\rho ^2_{i,t}}E_{t-1}\left( \xi _{i,t}|\epsilon _{m,t}<\frac{\Omega }{\sigma _{m,t}}\right) $$

Conditional variances $\sigma _{m,t}^2,\ \sigma _{i,t}^2$ are modelled with a TGARCH model from the GARCH family (Rabemananjara and Zakoian 1993). Such specification captures the tendency of volatility to increase more when there are bad news:

$$\begin{aligned} \sigma ^2_{m,t}&= \omega _m+\alpha _m r^2_{m,t-1}+\gamma _{m}r^2_{m,t-1}I^-_{m,t-1}+\beta _{m}\sigma ^2_{m,t-1}\\ \sigma ^2_{i,t}&= \omega _i+\alpha _i r^2_{i,t-1}+\gamma _{i}r^2_{i,t-1}I^-_{i,t-1}+\beta _{i}\sigma ^2_{i,t-1} \end{aligned}$$

$I^-_{m,t}=1$ if $r_{m,t}<0$ and $I^-_{i,t}=1$ when $r_{i,t}<0$, 0 otherwise.

Conditional correlation $\rho$ is estimated by means of a symmetric DCC model (Engle 2002). Moreover, to obtain the MES it is necessary to estimate tail expectations. This is performed with a non-parametric kernel estimation method (see Brownlees and Engle 2012).

Open-source Matlab code is available, thanks to Sylvain Benoit, and Gilbert Colletaz, Christophe Hurlin, who developed it in Benoit et al. (2013).

1.4 $\mathbf {\Delta CoVaR}$

Following Adrian and Brunnermeier (2016) $\Delta CoVaR$ is estimated through a quantile regression (Koenker and Bassett Jr 1978) on the $\alpha {th}$ quantile, where $r_{sys}$ and $r_i$ are, respectively, market-wide returns on equity and bank i’s returns. Quantile regression estimates the $\alpha {th}$ percentile of the distribution of the dependent variable given the regressors, rather than the mean of the distribution of the dependent variable as in standard OLS regressions. This allows comparing how different quantiles of the regress and are affected by the regressors; hence, it is suitable to analyse tail events. While Adrian and Brunnermeier (2016) employ an estimator based on an augmented regression, we further simplify the estimation of $\Delta CoVaR$ following the approach in Benoit et al. (2013), which is consistent with the original formulation.

First we regress individual returns on market returns:

$$\begin{aligned}&r_{sys,t}=\gamma _1+\gamma _2 r_{i,t}+\varepsilon _{\alpha ,t}^{sys|i} \end{aligned}$$

The estimated coefficients (denoted by $\widehat{}$) are employed to build CoVaR. The conditional VaR of bank i ($Var^{i}_{\alpha ,t}$) is obtained from the quasi-maximum likelihood estimates of conditional variance generated by the same TGARCH model described above (see Benoit et al. 2013, p. 38).

$$\begin{aligned}&CoVar^{sys|i}_{\alpha ,t}= \widehat{\gamma }_1 + \widehat{\gamma }_2 Var^{i}_{\alpha ,t} \end{aligned}$$

Finally $\Delta CoVar$ is obtained from the difference between the $\alpha {th}$ and the median quantile of CoVar.

$$\begin{aligned}&\Delta CoVar^{sys|i}_{\alpha ,t} = CoVar^{sys|i}_{\alpha ,t}-CoVar^{sys|i}_{0.5,t}\\&\Delta CoVar^{sys|i}_{\alpha ,t} =\hat{\gamma }_2\left( VaR^{i}_{\alpha ,t}-VaR^{i}_{0.5,t} \right) \end{aligned}$$