###### Systemic Risk

The Systemic Risk measures described below are based on the seminalpapers by Acharya, Pedersen, Philippon, and Richardson (2010) as well as Brownlees and Engle (2010). The paper by Engle, Jondeau, and Rockinger (2012) provides an extension handling European particularities.

### 1 Capital Requirements

To understand the methodology behind systemic risk measures, consider an elementary balance sheet of bank i at time t, with the book value of assets (BA_{i,t}) on the left hand side and the book value of equity (BW_{i,t}) and debt (D_{i,t}) on the right hand side.

One defines the ‘quasi-market value’ of the assets as the sum of the book value of debt (D_{i,t}) plus the market value of equity (W_{i,t}, also called market capitalization). Formally, A_{i,t} = D_{i,t} + W_{i,t}, therefore A_{i,t} = BA_{i,t} - BW_{i,t} + W_{i,t}.

What is Systemic Risk? It is the propensity of a firm to be under-capitalized when the financial system as a whole is under-capitalized, i.e., in case of a new financial crisis. A bank is said to be under-capitalized (or in financial stress) if its equity falls below a given fraction θ of its assets, i.e., if θA_{i,t} -W_{i,t} becomes positive. The parameter θ is defined below as a prudential ratio, typically set by the regulator.

Example: Assume θ = 8%, the assets of the bank are A_{i,t} = 1000, and its market capitalization is W_{i,t} = 80. Then the bank is just capitalized since 8% of 1000 is 80. Its debt is D_{i,t} = 920 and financial leverage is A_{i,t}∕W_{i,t} = 12.5 (because 12.5 = 1∕8%). Now assume a financial crisis. The financial market breaks down and the credit market dries up. The market capitalization of the bank falls below 80, say 50 (its leverage is now 20). To satisfy the regulatory ratio (W_{i,t} = 0.08 × A_{i,t}), the bank should raise 30 of equity capital from other financial institutions or from the market. However, as the market has dried up, the bank cannot refinance itself and may default, as it happened to Bear Stearns, Lehman Brothers, and others.

In this case, without the massive intervention of central banks and governments, one would probably see a cascade of bank defaults. It is exactly to be prepared for such situations that one needs a measure of the potential level of under-capitalization that a given bank would face in a crisis.

### 2 Capital Shortfall

Define Crisis_{t:t+T} an indicator variable that defines if there is a financial crisis between dates t and t + T. The expected capital shortfall of bank i in case of such a crisis is defined as:

It measures how much capital would be needed for that bank as to be correctly capitalized after the crisis. It should be noticed that if the resulting capital shortfall is very large, one obtains a link with the notion of too big to fail, which remains a source of concern for governments and regulators. The expected capital shortfall can be rewritten in terms of parameters that are measured from the balance sheet or estimated econometrically:

In this expression L_{i,t} = A_{i,t}∕W_{i,t} is the financial leverage and LRMES_{i,t:t+T} is the long-run marginal expected shortfall of thebank, i.e., the sensitivity of its equity return to the evolution of the world market in case of a financial crash. The market capitalization (W_{i,t}) and the financial leverage (L_{i,t}) are readily available from market and accounting data, respectively. What remains to be estimated is LRMES.

### 3 Long-Run Marginal Expected Shortfall

LRMES is defined as the sensitivity to a (hypothetical) 40% semiannual market decline:

where T = 6 months and cumulative returns are defined as:

and

with r_{i,t} and r_{M,t} the daily log-return of firm i and the daily log-return of the market at date t, respectively.

LRMES is particularly difficult to estimate because it corresponds to an extremely rare event. We had only three 40% market crashes over the last century (1929, 2000, and 2008). Brownlees and Engle (2010) advocated for two complementary approaches to estimate the LRMES:

The first (direct) approach consists of estimating LRMES as the expected return of the firm in case of a 40% semiannual decline in the market return. Directly estimating the LRMES relies on the simulation of the model over T periods using all information available at date t. It is estimated by:

where R_{i,t:t+T}^{(s)} and R_{M,t:t+T}^{(s)} are simulated by the model described below, and (x) = 1 if x is true and 0 otherwise.This approach will provide accurate estimates of the true expectation provided the number of simulated data is sufficiently large. We use S = 50000 draws.

- In the second (indirect) approach, LRMES is based on the expected return of the firm in case of a (relatively modest) 2% decline in the daily market return, which is then extrapolated to match a ‘once-per-decade” crisis. The sensitivity to a 2% daily world market decline, called Short-run marginal expected shortfall (SRMES) is defined as:

Then under some (not too straightforward) assumptions, the LRMES can be approximated by:

The parameter k has been estimated via extreme value theory. For T = 6 months, it was found to be k = 18. This approximation allows a much faster estimation of the risk measures, but does not allow a multifactor approach.

The European risk measures we report are based on the direct approach in a multifactor setting (See Section 5).

The Worldwide systemic risk measures we report from NYU Stern’s Volatility Lab are based on the indirect approach. A single factor (the World market) is used to estimate the LRMES. (Volatility Lab also reports risk measures based on the direct approach for U.S. financial institutions, assuming a single factor approach.)

### 4 Systemic Risk of Financial Institutions

We define the systemic risk of bank i as the expected capital shortfall when it is positive:

Having a negative capital shortfall means that the firm has more equity than required by the prudential ratio θ, so that the firm is no at risk.

An important property of the SRISK measure is that it allows aggregation. We define the marginal expected shortfall of a given country or the entire financial system as:

where R_{F,t:t+T} denotes the cumulative return of the financial industry between t and t + T. As the return of the industry is just the value-weighted sum of the return of the N financial institutions (R_{F,t:t+T} = ∑ _{i=1}^{N}w_{i,t}R_{i,t:t+T}, with w_{i,t} = W_{i,t}∕∑ _{i=1}^{N}W_{i,t}), we obtain that the marginal contribution of a given institution to the overall LRMES is simply the LRMES of the institution. Theaggregate marginal expected shortfall is therefore obtained by aggregation:

Similarly, the systemic risk of the entire financial system is just:

### 5 Econometric Methodology

There are substantial differences across European countries in terms of macroeconomic dynamics, fiscal and monetary policies, and regulation. For this reason, we need a fine description of what drives the risk of a financial firm. Our stratification allows for three drivers of a firm’s return:

the country-wide index (r_{C,t}),

the European index (r_{E,t}),

the World index (r_{W,t}).

A further complication stems from the asynchronicity of time zones. The stock market in a given country may be affected by a shock on the world index one day later, if the shock is initiated late in the U.S. or overnight in Asia. For these reasons, our system includes five series, r_{t} = .

The objective of the model is to capture the dependence of the return of firm i with respect to the drivers. Our econometricapproach aims at capturing this dependence by designing a factor model with time-varying parameters, time-varying volatilities and correlations, and a general, non-normal dependence structure for the innovations.

We use the following recursive multifactor model with time-varying parameters, after having preliminarily demeaned all return series:

where the L superscript corresponds to the lagged world index. The parameters of the model are estimated using the Dynamic Conditional Beta approach proposed by Engle (2012). See Engle, Jondeau, and Rockinger (2012) for additional details.

The error terms ε_{t} =

are uncorrelated across time and across series, but may be non-linearly dependent both in the time series (such as heteroskedasticity) and in the cross-section (such as tail dependence). To deal with heteroskedasticity, we assume a univariate asymmetric GARCH model:

where

for k ∈{i,C,E,W}. Innovations z_{t} = {z_{i,t},z_{C,t},z_{E,t},z_{W,t}} are such that E[z_{t}] = 0 and V [z_{t}] = I_{4}. It is commonly accepted that theconditional distribution of stock market returns is fat-tailed and asymmetric. To capture these features, the innovations are assumed to have a univariate skewed t distribution, z_{k,t} ~ f(z_{k,t}; ν_{k},λ_{k}), where f denotes the pdf of the skewed t distribution, with degree of freedom ν_{k} and asymmetry parameter λ_{k}.

Although innovations z_{t} have been preliminarily orthogonalized, they are not a priori independent. Their joint distribution should allow for possible non-linear dependencies. A convenient modeling approach is to use copula. We define u_{t} = as the margin of z_{t} with u_{k,t} = F(z_{k,t}; ν_{k},λ_{k}), where F is the cdf of the skewed t distribution. The copula is then the joint distribution of u_{t}, denoted by C(u_{t}).

After investigating several alternative copulas, we selected the t copula, which has been found to capture the dependence structure of the data very well. It accommodates tail dependence and its elliptical structure provides a convenient way to deal with large-dimensional systems. The t copula is defined as:

where t_{ν} is the cdf of the univariate t distribution with degree of freedom ν and t_{Γ,ν} is the cdf of the multivariate t distribution with correlation matrix Γ and degree of freedom ν.

To summarize, our model combines a Dynamic Conditional Beta model for the returns’ dynamic, univariate GARCH models for the dynamic of the volatility of the error terms, and a t copula for the dependence structure between the innovations. To deal with the possible time variability of some of the model parameters, we estimate the model over a rolling window of 10 years of data as soon as a new observation is made available.

### 6 A Comment on Units

For European SRISK measures, all data are expressed in billion euros.

For Worldwide SRISK measures (from NYU Stern’s Volatility Lab), all data are expressed in billion U.S. dollars.

One key ingredient of systemic risk measures is firm’s financial leverage. An important issue in measuring leverage is that the firms in the U.S. and Europe are currently under two different accounting standards: Generally Accepted Accounting Principles (GAAP) in the U.S. and International Financial Reporting Standards (IFRS) in Europe. The balance sheet of U.S. banks presents derivatives on a net basis, meaning that derivatives represent a small part of the assets, whereas the balance sheet of European banks reports derivatives on a gross basis. Some crude estimates suggest that the total assets (and therefore the leverage) of large U.S. banks (which are highly active in derivatives markets) would be 40-60% larger under IFRS than under U.S. GAAP.

To deal with this important source of bias, we use a different prudential ratio θ in Europe: we use θ = 5.5% for Europe institutions and θ = 8% for the U.S. and the rest of the world.

### References

Acharya, V.V. , L.H. Pedersen, T. Philippon, and M. Richardson (2010), Measuring Systemic Risk. Available at SSRN: http:// ssrn.com/abstract=1573171.

Brownlees, C.T., and R.F. Engle (2010), Volatility, Correlation and Tails for Systemic Risk Measurement. Available at SSRN: http://ssrn.com/abstract=1611229.

Engle, R.F., E. Jondeau, and M. Rockinger (2012), Conditional Beta and Systemic Risk in Europe. HEC Lausanne workingpaper.