Sylvain Benoit
email: sylvain.benoit@univ-orleans.fr

Christophe Hurlin
email: christophe.hurlin@univ-orleans.fr.

Christophe Pérignon
email: perignon@hec.fr.

Yacine Ait- Sahalia

Laurent Calvet

Serge Darolles

Darrell Du¢e

Bruno Feunou

Eric Jondeau

Jean-Paul Renne

Implied Risk Exposures

Keywords: Risk Disclosure, (Stressed) Value-at-Risk, Regulatory Capital, Systemic Risk JEL Classi…cation: G21, G28, G32

We show how to reverse-engineer banks' risk disclosures, such as Value-at-Risk, to obtain an implied measure of their exposures to equity, interest rate, foreign exchange, and commodity risks. Factor Implied Risk Exposures (FIRE) are obtained by breaking down a change in risk disclosure into a market volatility component and a bank-speci…c risk exposure component. In a study of large US and international banks, we show that (1) changes in risk exposures are negatively correlated with market volatility and (2) changes in risk exposures are positively correlated across banks, which is consistent with banks exhibiting commonality in trading.

Introduction

There are many reasons for …nancial institutions to have correlated risk exposures. First, capital regulations around the world incentivize banks to over-invest in certain favorable asset classes, such as sovereign debt. Second, banks may share superior information, and as such, follow similar investment strategies [START_REF] Hirshleifer | Security analysis and trading patterns when some investors receive information before others[END_REF]. Third, banks have incentives to herd to maximize the likelihood of being bailed out (Acharya andYorulmazer, 2007, 2008; [START_REF] Farhi | Collective moral hazard, maturity mismatch and systemic bailouts[END_REF].

Correlated risks are especially problematic during …nancial crises. Indeed, as market volatility spikes, regulatory capital and collateral requirements tend to mechanically increase for …nancial institutions. In response many banks are forced to liquidate their positions, which further ampli…es market volatility [START_REF] Brunnermeier | Market liquidity and funding liquidity[END_REF][START_REF] Merrill | Why did …nancial institutions sell RMBS at …re sale prices during the …nancial crisis?[END_REF].

The resulting adverse feedback e¤ects are stronger when banks have correlated risk exposures as they tend to sell the same assets at the same time [START_REF] Morris | Risk management with interdependent choice[END_REF][START_REF] Persaud | Sending the herd o¤ the cli¤ edge: The disturbing interaction between herding and market-sensitive risk management practices[END_REF].

A traditional approach to measuring banks' risk exposures is to regress the banks' stock returns on potential risk factors [START_REF] Flannery | The e¤ect of interest rate changes on the common stock returns of …nancial institutions[END_REF][START_REF] Bhattacharyya | Risk-taking by banks: What did we know and when did we know it?[END_REF].

Alternatively, O'Brien and [START_REF] O'brien | Estimating bank trading risk: A factor model approach[END_REF] regress the daily trading revenues of six US banks on the ten-year US Treasury rate and other market risk factors. They …nd that US banks exhibit high level of heterogeneity in their risk exposures, except for interest rates. More recently, some new approaches have been proposed to infer banks' exposures to interest rate risk from accounting data. Begenau, Piazzesi and Schneider (2013) use a portfolio approach to measuring banks' exposures to interest rate risk from data on loans and interest rate swaps.

They show that derivatives increase banks' exposure to interest rate risk. [START_REF] Landier | Bank exposure to interest-rate risk and the transmission of monetary policy[END_REF] show that the interest rate sensitivity of US banks' pro…t increases with their income gap, which is de…ned as the di¤erence between assets and liabilities that mature in less than one year. This paper proposes a new and simple way to measure risk exposures. Unlike previous papers, we do not focus on interest rate risk and consider a broader spectrum of risks, namely equity risk, interest rate risk, foreign exchange (FX) risk, and commodity price risk. Furthermore, we extract implied risk exposures of banks from their public risk disclosures, with special emphasis on Value-at-Risk (VaR). 1 We exploit the fact that the level of risk disclosures depends on two main factors. It …rst re ‡ects current market conditions and as such, tends to rise with market volatility. A second driving force of a bank's risk disclosure, but one that is often hidden to the public eye, is the actual risk exposures of the bank. Indeed, taking over a major stock broker would lead to a higher equity VaR for the acquiring bank. Similarly, implementing a directional trading strategy on the commodity market would certainly in ‡ate the commodity risk …gures.

We show how to decompose a change in risk disclosure into a market volatility component and a bank-speci…c risk exposure component. The trick we use is straightforward, yet powerful.

For a broad family of distributions, the VaR is de…ned as the product of the standard deviation of the return and the dollar amount invested (up to a constant scaling factor). Consequently, the change in VaR can either be due to a change in volatility or in the amount invested, or both. As the former two pieces of information are public information, they can be used to extract an implied measure of the latter. This framework, which we dub "Factor Implied Risk Exposure" or FIRE, allows us to answer two important questions: (1) How do banks adjust their risk exposures in response to volatility shocks? [START_REF] Acharya | Cash-in-the-market pricing and optimal resolution of bank failures[END_REF] Are changes in risk exposures correlated across banks? In other words, we investigate whether banks exhibit commonality in trading and whether correlation in risk exposures strengthens when …nancial markets are under stress.

We assess the performance of the FIRE in an innovative way. For a large …nancial institution, we systematically compare the implied risk exposures given by the FIRE with statements made by the …rm about its actual risk exposures in public …lings. Using quarterly data between 2003Q1 and 2013Q3, we have not found a single occasion in which the estimated risk exposures and the stated risk exposures contradict each other. We believe that this is reassuring evidence that our method provides meaningful risk estimates. We also study by simulation the biases on the implied exposures that could be induced by model risk and estimation risk. Overall, we …nd that the bias in the exposures is relatively small whatever the experiment and the sample size considered.

To develop the intuition underlying our approach, we display in Table 1 the changes in VaR for ten large US and international banks during an episode of substantial reduction in volatility (2008Q4-2009Q4). The VaR …gures have been computed by the banks with a 99% con…dence level and a one-day horizon. One attractive feature of this dataset is that it includes risk …gures (factor VaR) that are de…ned separately for each source of risk: equity, interest rate, FX, and commodity price. During this period, volatility fell across all asset classes. The actual reduction in volatility was 46% in the equity market, 43% in the …xed-income market, 39% in the FX market, and 59% in the commodity market. 2 Despite the overall drop in volatility, we identify 17 cases, out of 40, in which the VaR increased over the same period. One potential explanation of this puzzling result is that volatility (#) and risk exposures (") moved in opposite directions and that the risk exposure e¤ect dominated the volatility e¤ect for some banks.

< Insert Table 1 >

In the empirical part of the paper, we use quarterly VaR data publicly disclosed by the same ten US and international banks between 2007 and 2013. We use separate VaR …gures for each major source of risk: equity, interest rate, FX, and commodity. To control for concurrent changes in volatility, we use several proxies including implied volatility and historical volatility.

Our empirical analysis leads to several new …ndings on the risk-taking behavior of banks. First, we …nd that VaR covaries more frequently and more strongly with risk exposures than with market volatility. This result contrasts with the abundant literature on VaR computation in which attention is made on forecasting volatility models as the portfolios' weights are assumed to be constant. We show in this paper that when we allow for time-variation in the risk exposures, we end up with a much richer VaR dynamics. Second, we show that changes in risk exposures are negatively correlated with volatility changes, which suggests that banks curb risk when …nancial markets are under stress. This …nding is consistent with the model of Adrian and Shin (2014) in which …nancial intermediaries adjust their risk exposures in reaction to changing economic conditions, in order to maintain a constant probability of default. Third, consistent with banks engaging in commonality in trading, we show that changes in risk exposures are positively correlated among banks. When contrasting periods of increasing volatility and periods of decreasing volatility, we …nd that the negative relationship between volatility and risk exposures and commonality in risk exposures is present in all market conditions. Our paper makes several contributions to the literature on …nancial risk management. First, on the methodological side, we show how to extract an implied measure of changes in banks' risk exposures from publicly available data on VaR and volatility. By doing so, we complement [START_REF] Taylor | Generating volatility forecasts from value at risk estimates[END_REF] who shows how to generate volatility forecasts from market risk disclosures. Second, we empirically document the presence of commonality in the risk exposures of large banks. Our decomposition of the changes in risk disclosure allows us to directly test for similarities in trading positions by looking at bank risk exposures and not at trading pro…t-and-loss data [START_REF] Berkowitz | Evaluating value-at-risk models with desk-level data[END_REF]. In two distinct studies of large US banks, [START_REF] Berkowitz | How accurate are value-at-risk models at commercial banks?[END_REF] and [START_REF] Jorion | Bank trading risk and systemic risk[END_REF] both report a moderate correlation between US banks' trading pro…t-and-loss, which suggests that there is signi…cant heterogeneity in banks' risk exposures. Di¤erently, our study of the joint dynamics of banks' risk exposures indicates that banks rebalance their trading portfolios in a correlated way.

Third, we contribute to the debate on the procyclicality of regulatory capital. We report a negative correlation between market volatility and risk exposures, which suggests that banks actively manage their risk exposures according to market conditions. This contrarian risktaking behavior can be seen as an attempt to damper the procyclicality of bank regulatory capital. Fourth, as our methodology relies on a certain degree of commonality in volatility across the assets within a given asset class, we show that the factor structure recently documented by [START_REF] Herskovic | The common factor in idiosyncratic volatility: Quantitative asset pricing implications[END_REF] for the volatility of equity is persistent across asset classes.

Our study is also related to the theory on the propagation of …nancial shocks. In their general-equilibrium model, [START_REF] Pavlova | The role of portfolio constraints in the international propagation of schocks[END_REF] show that portfolio constraints, such as VaR constraints, can increase the comovement of the stock prices. While in their framework, only one agent is constrained in his portfolio choice, we consider a situation in which many …nancial institutions may be forced to curb their positions due to the tightening of their constraints.

Furthermore, as banks tend to rebalance their risk exposures at the same time and in the same direction after a shock, our study provides empirical evidence in support of the theoretical predictions of [START_REF] Danielsson | The impact of risk regulation on price dynamics[END_REF], which state that VaR constraints can exacerbate shocks further.

The outline of the paper is as follows. In Section 2, we present a methodology allowing us to extract information about changes in banks' risk exposures from public data. Section 3 presents the empirical analysis using actual VaR data for a sample of large US and international banks. We show in Section 4 how to extend the methodology to other types of risk disclosures and to time-varying skewness and kurtosis. Section 5 summarizes and concludes our study.

FIRE Methodology

Theory

When the distribution of the (demeaned) returns belongs to the location-scale family, the conditional VaR of an asset can be expressed as:

V aR t = t F 1 ( ) W t ( 1
)
where t is the conditional volatility of the asset return, F 1 ( ) is the -quantile of the standardized return distribution, and W is the dollar amount invested in the asset [START_REF] Jorion | Value at Risk: The New Benchmark for Managing Financial Risk[END_REF]. We see that there are two factors driving the VaR in this set-up, namely the volatility and the amount invested. 3 The change in amount invested can be due to the return of the asset or to in ‡ow/out ‡ow from the investor. The change in VaR is given by:

V aR t = V aR t+1 V aR t (2) = F 1 ( ) t+1 W t+1 t W t : (3)
While this relation only holds if F 1 ( ) remains constant over time, we relax this assumption in Section 4.2. Under this assumption, the percentage change in VaR is:

V aR t V aR t = F 1 ( ) t+1 W t+1 t W t t F 1 ( ) W t (4)
or equivalently

1 + % V aR t = 1 + % t 1 + % W t : (5)
As a result, the percentage change in the dollar amount invested in the asset is:

% W t = 1 + % V aR t 1 + % t 1: (6)
This equation is extremely useful. It allows us to infer the change in amount invested (unknown) from the change in VaR and volatility (both being observed). 4Although our methodology is very general, we focus in this paper on the actual risk disclosures of …nancial institutions. A common practice at large banks is to disclose their VaR for each risk factor, such as equity, interest rate, FX, and commodity (Pérignon and Smith, 2010; Basel Committee on Banking Supervision, 2011). Speci…cally, a factor VaR indicates the maximum loss, at the 1 con…dence level over a given horizon, that can be due to a given source of risk. For each bank i, we model the bank return on factor f , R if t , as a function of the factor return, R f t , and an idiosyncratic return, " if t :

R if t = if t R f t + " if t : (7)
For instance, for equity, this means that the return on the bank's equity portfolio can be imperfectly correlated with the US equity market, as proxied by the S&P 500 stock index. The idea behind the one-factor structure is that we focus on a subportfolio that is predominantly a¤ected by one major source of risk (e.g. equity portfolio, commodity portfolio). From Equation ( 7), we can express the variance of R if t , 2 if t , as:

2 if t = 2 if t 2 f t + 2 "t (8)
where 2 f t is the variance of the factor return and 2 "t is the variance of the idiosyncratic return. 5 In that case, VaR is de…ned as:

V aR if t = if t F 1 if ( ) W if t (9) = q 2 if t 2 f t + 2 "t F 1 if ( ) W if t (10)
and Equation (1) becomes:

V aR if t = f t F 1 if ( ) E if t (11)
where F 1 if ( ) is the -quantile of the standardized factor return and E if t is the risk exposure of …rm i with respect to factor f at time t, which is de…ned by:

E if t = W if t s 2 if t + 2 "t 2 f t (12)
' W if t if t when "t f t :

What this expression tells us is that there are two main ways for a bank to modify its risk exposure: …rst, the bank can change the dollar amount invested in the portfolio and second, it can modify the sensitivity of its portfolio with respect to a risk factor. 6 The change in VaR is given by:

V aR if t = F 1 if ( ) f t+1 E if t+1 f t E if t ( 14
)
and the percentage change in VaR is:

V aR if t V aR if t = F 1 if ( ) f t+1 E if t+1 f t E if t f t F 1 if ( ) E if t ( 15
)
1 + % V aR if t = 1 + % f t 1 + % E if t : (16)
The percentage change in risk exposure between dates t and t + 1 is given by:

% E if t = 1 + % V aR if t 1 + % f t 1: (17)
Equation ( 17) is the key result of the FIRE methodology. It gives an expression for the changes in risk exposure as a function of the changes in VaR and in the volatility of the risk factor.

It is important to notice that the FIRE methodology works with both long and short positions.

For a short position, the VaR is de…ned by:

V aR if t = if t F 1 if (1 ) E if t (18)
with E if t < 0 [START_REF] Giot | Value-at-risk for long and short trading positions[END_REF]. In that case, the percentage change is also given by Equation [START_REF] Christo¤ersen | Value-at-Risk Models, Handbook of Financial Time Series[END_REF] and the percentage change in risk exposure by Equation (17). 7

The Main Assumption under the FIRE Methodology

The main assumption in the FIRE methodology is that the quantile F 1 ( ) is constant through time. In the one-asset case (Equations ( 1)-( 4)), the quantile remains constant as long as the distribution of the asset return does not change from one date to the next. In the case of a portfolio (Equations ( 9)-( 15)), there are two sources of time variation in the quantile of the portfolio return distribution: changes in the distribution of the assets and changes in the portfolio weights. However, even when the weights are time-varying, the quantile remains constant if we consider conditional distributions for the asset returns that are closed in aggregation (e.g. normal distribution). Otherwise, the generalized FIRE presented in Section 4.2 has to be used in order to take into account the time variation in the quantile.

Conversely, when the portfolio contains a large number of assets, as it is most likely the case for the trading portfolios of the large banks studied in this paper, this distribution assumption can be relaxed. On a given date, if the number of assets tends to in…nity and the bank's portfolio is su¢ciently diversi…ed, the conditional distribution of the portfolio return tends to a normal distribution, as the Central Limit Theorem applies. As a consequence, the quantile of the standardized portfolio return converges towards

Case Study on Goldman Sachs

In order the check whether the changes in risk exposures produced by the FIRE methodology make economic sense, it would be ideal to compare the estimated risk exposure changes to the actual risk exposure changes. As the latter are typically unknown to the public, such comparison is hard to make in practice. However, we found one …rm for which the comparison is possible. Indeed, Goldman Sachs makes some statements in its quarterly public …lings about the recent changes in its trading portfolio. To our knowledge, Goldman Sachs is the only …nancial institution to make such public announcements in a systematic way over an extended period of time.

To be able to extract the implied risk exposures, we collect quarterly equity VaR …gures from all Goldman Sachs 10-Q forms between 2003Q1 and 2013Q3. These …gures are one-day 95%

VaRs averaged over a given quarter. Furthermore, we control for contemporaneous changes in volatility in the stock market using the VIX index. Figure 1 displays the quarterly values of the equity VaR along with the VIX index (both are average measures over the quarter).

Eyeballing the …gure shows little covariation between the VaR and the market volatility. In fact, if anything, the correlation is negative. 8 For instance, the sharp increase in volatility between 2007 and 2008 corresponds to a period of massive reduction in equity VaR for the …rm. The negative relationship between equity VaR and VIX may seem surprising at …rst sight, and especially if we refer to the abundant literature on tail risk in which the positive relationship between tail risk and volatility is crucial (see for instance the excellent survey by [START_REF] Christo¤ersen | Value-at-Risk Models, Handbook of Financial Time Series[END_REF]. The fundamental positive relationship between VaR and volatility is of course true if the risk exposure remains constant through time. However, in practice, this condition is violated as trading positions can signi…cantly vary from one quarter to the next.

< Insert Figure 1 and Table 2 >

For each quarter Q in year Y, we extract the change in equity risk exposure between quarter Q in year Y and quarter Q in year Y-1 using the FIRE methodology. We display the changes in equity VaR, volatility, and risk exposure in Table 2. We notice that the VaR increased steadily between 2003 and 2007 whereas the volatility decreased over the same period. This preliminary piece of evidence con…rms that VaR is not only driven by the volatility and that changes in risk exposure are likely to play an important role in the dynamics of the risk disclosure. The relationship between the VaR and the market volatility remains negative over the entire sample period. Di¤erently, the changes in VaR and in risk exposures are positively correlated.

As a cross-validation exercise, we contrast the risk exposure estimates with statements made by the …rm about its current equity risk exposure. In each quarterly report, Goldman Sachs complements the VaR …gures with information about any substantive changes in its investment strategy over the past year. For instance, in its 10-Q form dated May 2008, Goldman Sachs We conduct a similar analysis for all 30 quarterly reports between 2004Q1 and 2013Q3.

For each quarter, we compare the change in equity risk exposure provided by the FIRE methodology with the information disclosed by the …rm in its 10-Q report. As shown in Table 2 and Figure 2, we have not found a single case in which the FIRE estimate and the 10-Q form contradict each other. Note that this result is not due to any major trend in risk exposures as reductions in risk exposures are almost as frequent as increases in risk disclosure in our sample (nine decreases and eleven increases). Furthermore, there are another ten quarters for which Goldman Sachs made no particular comments. Interestingly, we notice that these quarters correspond to periods during which the equity risk exposure revealed by the FIRE was more stable. We …nd that during high VaR change quarters (j V aR=V aRj > 30%), the …rm makes comments in 94.1% of the cases (16 out of 17 quarters), whereas during low VaR change quarters (j V aR=V aRj < 30%), the …rm makes comments in only 30.8% of the cases (4 out of 13 quarters).

We consider a series of robustness checks. First, we replace average VaR and VIX values by their end-of-quarter values. We, again, systematically compare the estimated change in risk exposures given by the FIRE methodology to actual statements made by the …rm for the 30 di¤erent quarters. Second, we conduct a similar analysis using annual 10-K forms between 2004 and 2013, which leads to another 20 comparisons. In annual reports, the company compares its average (respectively year-end) equity-risk exposures in year Y to its average (respectively year-end) equity-risk exposures in year Y-1. For these 50 comparisons, there are speci…c comments from the …rm in 27 cases. In two cases only the sign of the change in implied risk exposure does not match the company's report. However, in both cases, the implied changes in risk exposures is small (-2% and 6%), which makes misclassi…cation more likely.

Overall, the results in this case study are encouraging. Despite the assumptions we made about the distribution and the factor structure of the return, the FIRE methodology seems to produce some risk estimates that …t well with reality. In Section 3, we expand the analysis to more banks and factors and investigate the comovements in risk exposures across banks.

Monte Carlo Simulations

In practice, both the VaR and the volatility estimates can be a¤ected by estimation risk or model risk. For instance, banks may not correctly and promptly incorporate dynamic volatility in their VaR models. This is for instance the case when the VaR is computed by historical simulation. In this section, we study by simulation the potential biases on the FIRE that come from estimation and model risks.

To better understand the problem, we need to distinguish three elements: model whereas the bank uses historical simulation. We will see below that only the latter type of model risk can be problematic to extract risk exposures.

Moreover, estimation risk is also at play as soon as the parameters of the bank VaR model and/or of the FIRE model have to be estimated [START_REF] Gourieroux | Estimation adjusted VaR[END_REF]. When the parameters are estimated with errors, the resulting implied exposure may also be biased. It is well known that this bias tends to disappear as the sample size increases.

The basic idea of these simulations is to assume a particular process for the changes in the bank's risk exposure and to check whether the FIRE methodology correctly estimates them.

For simplicity, we consider only one asset and some discrete exposures to ease the comparison between the true and estimated exposures. On each date, the bank's exposure in the asset is assumed to change by W t %, where W t % is drawn from a multinomial distribution on {-20%, -15%, -10%, -5%, -2%, 2%, 5%, 10%, 15%, 20%} with equal probabilities.

In all experiments, the DGP of the asset return R t is a GARCH(1,1) process. 9 Moreover, given its exposure and a simulated sample of the returns, denoted fR s t g T t=1 ; the bank computes its VaR using its internal risk model. We consider three types of internal models: a parametric model with estimated parameters (GARCH), a parametric model with …xed parameters (RiskMetrics) and a non-parametric method (historical simulation). Finally, bank

VaRs are used to estimate the implied exposure of the bank with the FIRE methodology. In our simulations, we consider three types of FIRE volatility models: GARCH, RiskMetrics, and historical volatility based on a rolling window of 250 days.

We present our four experiments in Panel A of Table 3

< Insert Table 3 >
In order to quantify the relative importance of model and estimation risks, we need to compare the true and implied risk exposures obtained for each simulation. This comparison is based on three statistical criteria: the percentage of matching signs, the Mean Absolute Error (MAE), and the Mean Absolute Percentage Error (MAPE). We also report the average R 2 statistics obtained by regressing the true position change on a constant and the exposure extracted with the FIRE methodology. The sample size T ranges from 250 to 2,000 observations and we run 100,000 simulations for each experiment.

The results are reported in Panels B-E of Table 3. Overall, we observe that the percentage of positive matching signs between the true and implied changes in exposure is always greater than 92% (91% for the percentage of negative matching signs). This result indicates that the FIRE methodology accurately predicts the direction of the change in the true risk exposure.

Moreover, the bias in the exposures is relatively small whatever the experiment and the sample size considered since the R 2 is always larger than 89%.

Several other conclusions can be drawn from this series of experiments. First, risk model does not a¤ect the performance of the FIRE except if it stems from a mismatch between the bank VaR model and the FIRE model (see experiments 1 and 2). Second, according to all evaluation criteria and sample sizes, the bias is the largest in the fourth experiment. For instance, for a sample size of 250 observations, the MAE is about 0.5% in experiments 2 and 3 whereas it is equal to 3% in the fourth experiment. This result clearly indicates that estimation risk as modeled in experiment 2, or moderate model risk as modeled in experiment 3, have limited impact on the FIRE methodology. In particular, the in ‡uence of estimation risk is very limited even in small samples (T = 250). Third, the magnitude of the bias decreases with sample size when there is estimation risk. For instance in the second experiment, the MAE drops from 0.65% to 0.48% when the sample size goes from 250 to 2,000 observations. When only model risk is present, the MAE does not change with sample size. In experiment 3, it is constant and equal to 0.58%.

Commonality in Volatility Within an Asset Class

The FIRE methodology relies on a certain degree of commonality in volatility across the assets that belong to the same asset class, as shown in Equation [START_REF] Benoit | A theoretical and empirical comparison of systemic risk measures[END_REF]. While [START_REF] Herskovic | The common factor in idiosyncratic volatility: Quantitative asset pricing implications[END_REF] have recently documented that equity volatility exhibit a strong factor structure, we test in

The volatility measures are the historical standard-deviations of the daily returns, which are available for the period January 1, 1999 to June 20, 2014 (respectively, end of 2013 for equities). For equity, we extract from CRSP the daily returns of the 500 constituents of the S&P 500 stock index at the end of 2013. For …xed income, we extract from the FRED database the daily yields of all (148) securities within four categories: commercial papers [START_REF] Landier | Bank exposure to interest-rate risk and the transmission of monetary policy[END_REF], corporate bonds (98), Treasury bills (4), and Treasury constant maturity [START_REF] Danielsson | The impact of risk regulation on price dynamics[END_REF]. For FX, we select the ten largest currencies based on the percentage shares of average daily turnover in April 2013 (BIS, 2013). 11 Then, we extract from Bloomberg the daily exchange rates for the 45 pairs of currencies and compute their daily returns. For commodities, we consider the constituents of the Dow Jones-UBS Commodity Index. To avoid issues due to expiration dates, we extract from Bloomberg the price of the Generic 1st month Futures for 20 components of the commodity index, as well as the S&P GSCI Kansas Wheat Index. (56.3%-66.8%), interest rate (45.1%-50.9%), foreign exchange (52.2%-59.2%), and slightly lower for commodities (29.5%-34.1%). Note that for some asset classes, the average intercept and slope coe¢cients di¤er from zero and one, respectively, because of the unbalanced panel structure of the data. We also notice that for equity, the R 2 reported in Table 4 with a yearly frequency tend to be higher than those in [START_REF] Herskovic | The common factor in idiosyncratic volatility: Quantitative asset pricing implications[END_REF], which are around 0.35.

This di¤erence is likely due to the much longer sample period (1926-2010) and much broader cross-section of assets (20,000 stocks) considered in their original study. We complement the univariate analysis by displaying the R 2 of a pooled regression obtained from a panel regression model with securities' …xed-e¤ects (within estimator). The results indicate that our …ndings are robust in a panel model that imposes common loadings for all the assets.

The degree of commonality in volatility remains particularly strong within equities and …xed income securities.

< Insert Table 4 >

3 Changes in Risk Exposures at Large Banks

First Input: VaR

In this section, we study the actual changes in risk exposures at large banks before, during, and after the 2008 crisis. These risk exposure changes are extracted from the VaR of ten large US and international banks between 2007Q3 and 2013Q3 (see Appendix for a list of the sample banks). VaR …gures are publically disclosed in the quarterly and annual reports of the …rms. These reports have been retrieved from the EDGAR database for US banks and from the …rms' websites for international banks. The VaR …gures typically have a one-day horizon and a 99% con…dence level and are available on four di¤erent risk factors: equity, interest rate, FX, and commodity. In our tests, we use end-of-quarter VaRs for all banks, except for Bank of America and BNP Paribas for which we use average VaRs over the quarter. 13 We …rst show in Figure 3 and Table 5 that the factor VaRs only exhibit some weak positive covariation across banks. Figure 3 displays the Value-at-Risk of four sample banks (Citigroup, Credit Agricole, Credit Suisse, and Deutsche Bank). We notice in this graph that the evolution of the VaR is quite erratic, with large changes from one quarter to the next. It is indeed not uncommon to see a VaR changing by a factor of 3 or 5 within a given year. For some risk types, there is a common trend over the sample period. For instance, the interest-rate VaR of all banks increased over 2007-08 and decreased afterwards. Similarly, there is a clear negative trend for equity risk starting at the end of 2008. Di¤erently, there is much less comovement in the FX and commodity VaRs for these banks. We extend the analysis to all sample banks in Table 5 and report the average correlation between the quarterly VaR of a bank, V aR if t , and the quarterly VaR of all other sample banks for each risk factor, V aR jf t , j 6 = i (upper panel). We report a positive average correlation for all four risk factors, which re ‡ects the fact that VaR numbers are a¤ected by some common volatility shocks. However, the magnitude of these correlations is not very high: in the 40%-45% range for equity and interest rate and less than 20% fo FX and commodity. Furthermore, we measure in the lower panel of Table 5 the frequency with which the VaRs of banks i and j move in the same direction. The percentage of matching signs between V aR if t and V aR jf t is rather low, between 44% and 57%.

< Insert Figure 3 and Table 5 > 1 3 Our initial sample was the largest 25 banks in the world according to their total assets as of June 2012. We then selected all banks disclosing end-of-quarter or average VaRs for the four main risk factors (equity, interest rate, foreign exchange, and commodity). Then, we selected the longest possible sample period allowing us to get a balanced panel. See the Appendix for more details about the VaR …gures.

Second Input: Volatility

In order to control for concurrent changes in volatility, we use some factor volatility indices. These indices are extracted from options written on the di¤erent underlying factors and with maturities between one and three months. Speci…cally, we use the CBOE VIX index to proxy the volatility of the equity market. The volatility on the …xed income market is measured by the Merrill Lynch Move index, which tracks the implied volatility of Treasury bond prices.

The volatility on the FX market is measured by the CVIX, a measure of implied volatility of major currency exchange rates. Finally, the volatility on the commodity market is measured by the OVX, a measure of implied volatility in West Texas Intermediate crude oil prices. 14 We display the evolution of the volatility of each risk factor in Figure 4. As expected, there is strong commonality in the volatility of these risk factors, with spikes after the Lehman collapse in October 2008 and the European sovereign-debt crisis during the summer 2011.

< Insert Figure 4 >

We show in Table 6 that the VaR and the factor volatility tend to be positively correlated for all risk factors. 15 The average correlation is lowest for commodity (16%) and highest for interest rate (53%). We also notice that this correlation is not positive for all banks. In fact, there are only …ve banks in our sample for which the correlation is positive for all four factors. Furthermore, when we compute the percentage of matching signs between the changes in VaR and in volatility, we …nd a frequency in the 40%-55% range. This …nding suggests that in many occasions, the evolutions of the bank risk disclosures and market volatility diverge. 1 4 We use the same volatility indices as in the Risk (2010) annual VaR survey. We obtain daily data on the factor volatility indices from Bloomberg and Datastream. 1 5 For banks that disclose end-of-quarter VaRs, the correlation is computed using end-of-quarter volatility. Similarly, for banks that disclose average VaRs, the correlation is computed using average volatility.

Another implication of our preliminary set of results is that market volatility does not seem to be a dominant driving force for factor VaR.

< Insert Table 6 >

Implied Risk Exposures

To formally gauge the impact of volatility and risk exposure changes on VaR, we implement the FIRE methodology that was presented in Section 2. For each bank/quarter, we plug the percentage change in VaR and the percentage change in volatility into Equation [START_REF] Ernst | Progress in …nancial services risk management: A survey of major …nancial institutions[END_REF] to get the implied risk exposure variation for each risk factor. To have a …rst look at the results, we superimpose the evolution of the VaR, volatility, and implied risk exposure for equity in Figure 5. The message we obtain is unambiguous: the change in risk exposures is the main driving force for equity VaR.

Another important …nding is that changes in risk exposure and volatility tend to move in opposite directions. We analyze the relationship between risk exposure and volatility for all factors and all banks in Table 7. In the upper panel of the table, we show that the percentage changes in risk exposure and volatility are negative for virtually all …rms and all factors. On average, this correlation is -53% for equity, -56% for interest rate, -25% for FX, and -56% for commodity. Moreover, as shown in the lower panel of Table 7, rarely do the changes in risk exposure and volatility move in the same direction.

Our conclusion on the negative relationship between risk exposures and market volatility is consistent with the model and empirical …ndings of Adrian and Shin (2014). They claim that …nancial …rms cut back their asset exposure when the environment becomes more risky in order to maintain a constant probability of default. They show that large US banks reacted to the volatility spike in 2008 by sharply reducing their leverage. At the same time, the VaR to equity ratio barely changed.

< Insert Figure 5 and Table 7 >

We then move to the cross-sectional analysis of banks' risk exposures. To test whether risk exposures are correlated across banks, we report in the upper panel of Table 8 the average correlation between the percentage change in risk exposure of a bank, % E if t , and the percentage change in risk exposure of all other sample banks, % E jf t , j 6 = i. The lower panel of this table displays the frequency with which changes in risk exposure of banks i and j move in the same direction. The main takeaway from this table is that there is some strong commonality in bank risk exposures. Indeed, 39 out of the 40 average correlation coe¢cients among the changes in risk exposures are positive. Moreover, risk adjustments at two random sample banks go in the same direction between 58% and 66% of the time, which is between 5 and 22 percentage points higher than the values for the VaR in Table 5.

< Insert Table 8>

We also model changes in risk exposures using a multivariate panel regression. Our baseline speci…cation is:

% E if t = i + 1 % E jf t + 2 % f t + 3 R f t + 4 CDS it + 5 RoE it + e if t ( 20
)
where i is a bank-speci…c intercept, % E jf t = P i6 =j % E jf t =(N 1) denotes the average percentage change in banks' risk exposures, % f t is the percentage change in factor volatility, R f t denotes the quarterly return of the risk factor, CDS it is the senior 5-year CDS of the bank, and RoE it is the bank quarterly return on equity. 16 The 1 parameter aims to capture any commonality in risk exposures among banks whereas the 2 parameter measures the relationship between risk exposure and market volatility.

In our tests, we use the following indices for the four risk factors: S&P500 Index (equity), We present the estimation results in Table 9 for each risk factor (columns 1-8) and then for all risk factors stacked together (columns 9-10). We …nd evidence of strong commonality in risk exposures as the OLS estimated coe¢cients associated with other banks' risk exposures are positive and signi…cant ( ^ 1 > 0). This …nding holds true for all factors but the e¤ect is particularly strong for equity and interest rate. This result is suggestive of commonality in risk exposures due to similar investment or hedging policies across banks. We also report a negative and signi…cant relationship between risk exposure and factor volatility ( ^ 2 < 0), which is consistent with the univariate results in Table 7. We however …nd no evidence that banks with particularly poor performance or higher probability of default tend to take on more risk (risk shifting). We also notice that the inclusion of the control variables (R f t ,

CDS it , RoE it ) does not alter our conclusions on commonality in risk exposures and on the relationship between risk exposure and market volatility, with the exception of FX.

< Insert Table 9 >

Robustness Checks

We consider a series of robustness checks. First, we use alternative proxies for the average change in banks' risk exposures. We replace the equally-weighted commonality proxy by a value-weighted commonality proxy (Table 10, Panel A) and by the …rst principal component of the covariance matrix of the percentage changes in risk exposures (Table 10, Panel B).

Overall we …nd that our result on commonality in risk exposures remains strong and signi…cant with all commonality proxies. Second, we change the volatility proxy for the risk factors.

Instead of using implied volatility indices, we compute historical volatility measures within a given quarter. Speci…cally, we compute the 3-month historical standard deviation of the return of the risk factor (Table 10, Panel C). Using these new proxies for volatility changes, we recompute the implied change in risk exposures using the FIRE method and re-run our regression. Overall, we see that our main …ndings are robust to this change of volatility proxy.

< Insert Table 10 >

Several reasons can explain the commonality in risk exposures documented in Tables 9 and10. First, banks can rebalance their trading portfolios in a correlated way because of common information. Second, they may have to curb risk at the same time because they face similar regulatory constraints. For instance, when several banks operate at their VaR limit, even a small increase in volatility would force them to unwind their positions in a correlated way.

Third, the exposure of two banks with respect to a given factor can also increase because the return of this factor was positive. In order to control for the latter e¤ect, we estimate the following panel regression:

% E if t R f t = i + 1 % E jf t R f t + 2 % f t + e if t (21)
where

% E jf t R f t is P i6 =j (% E jf t R f t )=(N 1)
. In this speci…cation, we systematically remove the return on the factor from the change in risk exposure. Results in Panel D of Table 10 clearly indicate that commonality in risk exposures is not mainly due to factor returns.

Indeed, the coe¢cient associated with other banks' changes in risk exposures ( 1 ) remains positive and signi…cant for all factors, at least at the 10% con…dence level. We also notice that the strong negative relationship between volatility and risk exposure is preserved ( 2 ).

Finally, in order to test whether our conclusions remain valid in di¤erent market conditions, we split the sample into two subperiods. The …rst one covers 2007Q3-2008Q4 and corresponds to a period of sharp increase in market volatility (see Figure 4). The second subperiod, 2009Q1-2010Q1, corresponds to a period of massive reduction in market volatility. We show in Table 11 that the quarterly average change in factor volatility ranges between 22% and 27% in the …rst period and between -11% and -18% in the second period. Overall, we …nd that our conclusions about the dynamics of the risk exposures are persistent through the di¤erent phases of the volatility cycle. In particular, we …nd that the negative relationship between changes in volatility and risk exposure is a robust feature of the data. Furthermore, we report evidence of commonality in risk exposures across banks in both volatility regimes.

< Insert Table 11 > 4 Extensions

Other Types of Risk Disclosures

So far in this study, we have only focused on one type of bank risk disclosure, namely the VaR.

We now show how to infer information about risk exposures from other types of banks' risk disclosures. Under Basel III, all …nancial institutions with material trading activities must compute both their VaR using recent data and their stressed VaR (sVaR) using data from a particularly volatile period (Basel Committee on Banking Supervision, 2011; Rossignolo, Fethi and Shaban, 2013). This measure is intended to replicate a VaR calculation that would be generated on the bank's current portfolio if the relevant market factors were experiencing a period of stress. As an example, for many portfolios, a 12-month period relating to signi…cant

c t = max fV aR t ; m V aR avg g + max fsV aR t ; m s sV aR avg g (22)
where m and m s are two positive multiplicative factors set by the regulators and subject to an absolute minimum of 3, and the avg subscript stands for an average computed over sixty business days.

We show in this section that it is possible to use the FIRE methodology with stressed, instead of standard, VaR …gures. In fact, it turns out that it is much easier to learn about changes in risk exposures from stressed VaRs than it is from standard VaRs. The reason being that changes in stressed VaR are only due to changes in risk exposures, and not to changes in volatility (recall that, with stressed VaR, volatility is always measured during the same highvolatility period). We make this point formally by de…ning the stressed VaR as:

sV aR t = F 1 ( ) E t (23)
where denotes the conditional variance of the return measured over a particularly volatile period. We note that the variance parameter is not changing from one day to the next as it refers to a given high-volatility episode in the past. As a result, the change in stressed VaR is given by: sV aR t = sV aR t+1 sV aR t

= F 1 ( ) E t+1 E t : (24)
The percentage change in VaR is:

sV aR t sV aR t = F 1 ( ) E t+1 E t F 1 ( ) E t (26) = E t+1 E t E t : (27)
Then, we conclude that:

% sV aR t = % E t : (28)
This equation shows that changes in stressed VaR only re ‡ect changes in risk exposures.

Unlike with standard VaR, changes in stressed VaR are completely immunized from volatility shocks, which greatly simpli…es the analysis.

Generalized FIRE with Time-Varying Skewness and Kurtosis

It was shown in Section 2 that the -quantile, F 1 ( ), of the standardized return distribution must be constant for the FIRE to work. Obviously, if the skewness and/or the kurtosis of the conditional distribution of the returns are/is dynamic, the -quantile may not be constant anymore and the implied exposure given by FIRE can be biased. To illustrate this, we consider a simple model in which the return is given by R t = t " t where " t is i:i:d: with E (" t ) = 0 and V (" t ) = 1: Denote by F t (:) the cumulative density function, s t the skewness and k t the kurtosis of the distribution of " t : Using the Cornish-Fisher expansion, we know that for any 2 [0; 1]:

F 1 t ( ) = z + s t 6 z 2 1 + k t 3 24 z 3 3z s 2 t 36 2z 3 5z (29)
where z = 1 ( ) denotes the th quantile of the standard normal distribution. Then, if s t or k t is dynamic, F 1 t ( ) is not constant over time. As a consequence, a generalized version of the implied exposure becomes:

% W t = 1 + % V aR t (1 + % t ) 1 + % F 1 t ( ) 1 (30)
with

1 + % F 1 t ( ) = F 1 t+1 ( ) =F 1 t ( ).
In this case, we need to make an assumption on the dynamics of s t and k t . For instance, we can use the generalized skewed Student's t distribution of [START_REF] Hansen | Autoregressive conditional density estimation[END_REF] with ARCH-type models for the skewness and kurtosis, or the extension of Harvey andSiddique (1999, 2000).

At this point, a natural question arises. What is the cost of neglecting the dynamics of the skewness and kurtosis when extracting risk exposures? One way to answer this question is to compare the exposures given by the FIRE and a generalized version of the FIRE allowing for time-varying skewness and kurtosis. The di¤erence in exposure depends on the value of 1 + % F 1 t ( ). From Equation ( 29), we get:

1 + % F 1 t ( ) = 1 + z 2 1 6 % s t + z 3 3z 24 % k t 2z 3 5z 18 s t % s t : (31)
For = 0:01, s t = 0:2, and a range of [ 10%,10%] for both % s t and % k t , the value of 1 + % F 1 t ( ) remains between 0:9181 and 1:0819. This means that the size of the bias of the exposure induced by neglecting the dynamics of the skewness and/or kurtosis ranges from 8:92% to 7:57%. Moreover, we notice in Equation ( 31) that % F 1 t ( ) is more sensitive with respect to the skewness than to the kurtosis. Indeed, the partial derivative with respect to the change in skewness is z 2 1 =6 2z 3 5z s t =18 = 0:5848 and the partial derivative with respect to the change in kurtosis is 3 3z =24 = 0:2338.

Conclusion

Because of the G20 Data Gap Initiative, more data will have to be disclosed by …nancial institutions to allow policy makers and supervisors to better assess the evolution of the …nancial system, as well as the intervention required [START_REF] Cerutti | Systemic risk in global banking: What can available data tell us and what more data are needed?[END_REF]. However, opportunities to observe actual positions or risk exposures of banks remain extremely rare in practice (e.g. European Banking Authority's 2011 stress tests). In this paper, we present FIRE, a new technique to infer banks' risk exposures from current public disclosures; very much in the spirit of implied volatility extracted from option prices.

The performance of the FIRE turns out to be quite good in practice, despite the assumptions made to derive our key result. In the case study on Goldman Sachs, we show that the implied risk exposures are systematically in line with the statements made by the …rm about its risk taking in public …lings. We believe that this is reassuring evidence that our method provides meaningful estimates. In addition, we assess the performance of the FIRE by simulation by considering several situations in which model risk and estimation risk could arise. Overall, we show that, in most situations, the bias induced by model and estimation risks remains moderate.

Using a sample of large US and international banks, we …nd that the main driving force of bank risk disclosures is the shifts in risk exposures and not market volatility. Furthermore, we show that changes in risk exposures are negatively correlated with volatility changes, which suggests that banks aim to reduce the variability of their VaR and regulatory capital. Most importantly, we provide empirical evidence of commonality in risk exposures across banks, which supports the view that banks exhibit quite similar behavior in trading. Our empirical conclusions have some important implications for the dynamics of banks' regulatory capital. Notes: The upper panel of this table presents the correlation between the percentage change in the quarterly risk exposure of a bank and the percentage change in quarterly factor volatility between 2007Q3 and 2013Q3 (25 observations per bank). The lower panel reports the frequency with which the quarterly risk exposure of a given bank move in the same direction as the factor volatility (+/+ or -/-). For each bank, we compute the percentage of matching signs between its E if t and the f t . Values are expressed in percentage points. Notes: This table presents the estimated coe¢cients and robust standard errors (in parentheses) for several regressions of the percentage changes in risk exposures for the 10 sample banks using an OLS panel regression with bank …xed e¤ects in single factor regressions, columns (1)-( 8), and with bank and factor …xed e¤ects in the regressions aggregating all factors, columns ( 9)- [START_REF] Begenau | Banks' risk exposures[END_REF]. The dependent variable is the percentage change in risk exposure (% E if t ). ***, **, * indicate that the coe¢cient is statistically signi…cant at the 1%, 5% and 10% con…dence level, respectively. R it denotes the factor return over the quarter, CDS it denotes the CDS of bank i at the beginning of the quarter, and RoE it denotes the return on equity of bank i over the quarter. Each regression is run separately over an estimation period covering 2007Q3-2013Q3. Notes: In table, we contrast two subsamples. The upper (lower) panel presents the results for an episode of increase (decrease) in market volatility. In each panel, we present the average quarterly percentage change in the factor volatility index (% V olatility), the average quarterly percentage change in factor VaR (% V aR), the average quarterly percentage change in risk exposure (% E), and the average correlation between the percentage change in risk exposures of a bank, % E if t , and quarterly changes risk exposure of the nine other banks, % E jf t , j 6 = i: Values are expressed in percentage points. For each quarter Q in year Y, we extract the change in equity risk exposure between quarter Q in year Y and quarter Q in year Y-1 using the FIRE methodology. The 30 quarters have been divided into three subsamples according to statements made by the …rm in its 10-Q reports regarding its actual change in risk exposures. There are nine quarters during which the …rm stated that its equity risk exposure did go down (Panel A), ten quarters during which the …rm made no statements about its change in equity risk exposure (Panel B), and eleven quarters during which the …rm stated that its equity risk exposure did go up (Panel C). In each panel, the quarters are ranked chronologically. See Table 2 for a list of the quarters in each panel.

  mentions that "Our average daily VaR increased to $184 million for the second quarter of 2008 from $133 million for the second quarter of 2007. The increase was primarily due to higher levels of volatility [...]. These increases were partially o¤set by a decrease in exposures to equity prices". Over this particular period (2008Q2 vs. 2007Q2), the FIRE methodology successfully indicates the direction of the change in risk exposures. It generates a 51% decrease in implied risk exposure for equity while, at the same time, the VIX index increased by 61%.

  [START_REF] Acharya | Too many to fail -An analysis of timeinconsistency in bank closure policies[END_REF] the true data generating process (DGP) of the return, (2) the internal model used by the bank to compute its VaR, and (3) the volatility model used by the econometrician to implement the FIRE method. For simplicity, we call the latter model the FIRE model.In our context, there are two sources of model risk. First the bank VaR model may not match with the DGP[START_REF] Escanciano | Robust backtesting tests for value-at-risk models[END_REF]. For instance, the bank computes historical simulation VaRs whereas the DGP is a GARCH[START_REF] Acharya | Too many to fail -An analysis of timeinconsistency in bank closure policies[END_REF][START_REF] Acharya | Too many to fail -An analysis of timeinconsistency in bank closure policies[END_REF]. Second, the FIRE model may not match with the bank VaR model. For instance, the econometrician uses a GARCH[START_REF] Acharya | Too many to fail -An analysis of timeinconsistency in bank closure policies[END_REF][START_REF] Acharya | Too many to fail -An analysis of timeinconsistency in bank closure policies[END_REF]

3 -

 3 Year Treasury Constant Maturity Rate (interest rate), Trade Weighted U.S. Dollar Index (FX), and Dow Jones-UBS Commodity Index (commodity). The data have been retrieved from Datastream and the Federal Reserve Economic Database (FRED) and cover the period 2007Q3-2013Q3.

  losses in 2007/2008 would adequately re ‡ect a period of such stress. The stressed VaR is an important innovation in …nancial risk management. The Ernst & Young (2012) survey of …nancial services risk management reveals that stress testing and stressed VaR have been the top two areas of improvement in 2012: 55% of the respondents identify stressed VaR as the top area of improvement in transparency. Moreover, under Basel III, stressed VaR is included in the computation of the capital requirements for market risk, c t :

  Indeed, our paper documents two sources of procyclicality in bank capital. The …rst one is due to the original increase in volatility while the second one arises from further volatility increases triggered by correlated risk exposures across banks, through a feedback e¤ect. This new framework could lead a variety of applications in the future. Implied risk exposures could, for instance, be used to study the empirical performance of the trading strategies of banks, in the spirit of the study of Agarwal et al. (2013) on hedge funds. One could also test whether some …nancial institutions lead their peers in terms of investment behavior. FIRE could also be used in banking supervision by complementing existing systemic risk measures (see[START_REF] Benoit | A theoretical and empirical comparison of systemic risk measures[END_REF] for a survey). Indeed, a situation in which a pool of large banks have a growing, common exposure to an asset class can become a serious source of concerns for banking regulators.

  ) the FIRE model used to extract the conditional volatility. HV denotes historical volatility and HS historical simulation. For each experiment, we report the percentage of positive and negative (in parentheses) matching signs between the true change in risk exposure and the implied change in risk exposure extracted with the FIRE methodology. We also display the average of the Moving Absolute Error (MAE) and the Moving Absolute Percentage Error (MAPE) between the changes in the true risk exposure and the implied risk exposure extracted with the FIRE methodology. Finally, we report the average R 2 statistic obtained by regressing the true position changes on a constant and the implied risk exposure. In each experiment, we vary the sample size from 250 to 2,000 observations and we use 100,000 simulations.

Figure 1 :

 1 Figure 1: FIRE Analysis of Goldman Sachs' Equity VaR

Figure 2 :

 2 Figure 2: Empirical Performance of the FIRE Methodology

  . In the …rst experiment, we consider the …rst type of model risk in which both the bank VaR model and the FIRE model are assumed to be RiskMetrics whereas the DGP is a GARCH[START_REF] Acharya | Too many to fail -An analysis of timeinconsistency in bank closure policies[END_REF][START_REF] Acharya | Too many to fail -An analysis of timeinconsistency in bank closure policies[END_REF]. Note that there is no estimation risk in this case. In the second experiment, the bank VaR model is RiskMetrics and the FIRE model is a GARCH. Since the GARCH model nests RiskMetrics, there is no model risk in this case. However, since the GARCH parameters have to be estimated, estimation risk is present. In the third experiment, there is model risk (second type) but no estimation risk. The bank VaRs are produced by historical simulation and the FIRE is based on a historical volatility obtained with the same rolling estimation window. Finally in the fourth experiment, the bank uses historical simulation to produce its VaR whereas the FIRE is based on a GARCH model, inducing both model risk and estimation risk.

Table 4

 4 whether this holds true for other asset classes, such as …xed income, foreign exchange, and commodity. We follow Herskovic et al. (2014) and regress, for each asset, the asset-level volatility, if t , on the equally-weighted average of volatility within the asset class f , f t : 10 if t = intercept i + loading i f t + e if t :

  12

Table 4

 4 reports the cross-sectional averages of the intercept and loading coe¢cient estimates and of the R 2 for each asset class. In this table, we consider three frequencies: yearly in Panel A (like in[START_REF] Herskovic | The common factor in idiosyncratic volatility: Quantitative asset pricing implications[END_REF], quarterly in Panel B (like in the rest of this study), and monthly in Panel C. The main conclusion from all three panels is that the high degree of commonality in volatility discovered by[START_REF] Herskovic | The common factor in idiosyncratic volatility: Quantitative asset pricing implications[END_REF] for equities, is persistent across all main asset classes. The cross-sectional average R 2 is particularly high for equity1 1 The list of currencies, sorted in decreasing order of importance, includes the US Dollar, Euro, Japanese Yen, British Pound, Australian Dollar, Swiss Franc, Canadian Dollar, Mexican Peso, Chinese Renminbi, and New Zealand Dollar.1 2 The 20 Generic 1st month Futures are Natural Gas, WTI Crude Oil, Brent Crude Oil, Heating Oil, Live Cattle, Lean Hogs, Wheat, Corn, Soybeans, Soybean Oil, Soybean Meal, Aluminum, Copper, Zinc, Nickel, Gold, Silver, Sugar, Cotton, and Co¤ee.

Table 1 :

 1 Change in Factor VaR and Factor Volatility between 2008 and 2009The source for the VaR …gures are the EDGAR database for US banks and …rms' websites for international banks. We use a speci…c implied volatility index for each risk factor. The volatility on the equity market is measured by the Chicago Board Options Exchange VIX index. The volatility on the …xed income market is measured by the Merrill Lynch MOVE index, which tracks the volatility of Treasury bond prices using implied volatility from 30-day options. The volatility on the foreign exchange market is measured by the Deutsche Bank CVIX index, an average 3-month implied volatility for all the major currency pairs. The volatility on the commodity market is measured by the Chicago Board Options Exchange OVX index, a measure of 30-day implied volatility in West Texas Intermediate crude oil prices. Bold …gures denote positive percentage changes. All sample banks report end-ofquarter daily VaR except Bank of America and BNP Paribas that report average daily VaR for each quarter. Values are expressed in percentage points.

	% V aR	Equity	Interest Rate Foreign Exchange	Commodity
	Bank of America	54	137	355	442
	BNP Paribas	-30	-35	-48	25
	Citigroup	-18	-40	-62	20
	Credit Agricole	-56	-73	-57	200
	Credit Suisse	33	36	-56	17
	Deutsche Bank	7	-15	-37	10
	Goldman Sachs	161	-46	-42	0
	JPMorgan Chase	-7	-51	-74	-12
	Morgan Stanley	0	45	86	4
	UBS	11	-26	-56	-40
	% V olatility	-46	-43	-39	-59
	Notes:

Table 3 :

 3 Monte Carlo ExperimentsPanel A: Design of the Monte Carlo Experiments

		Exp. 1	Exp. 2	Exp. 3	Exp. 4
	DGP of the return	Garch(1,1)	Garch(1,1)	Garch(1,1)	Garch(1,1)
	Bank VaR Model	RiskMetrics	RiskMetrics	HS	HS
	FIRE Model	RiskMetrics	Garch(1,1)	HV	Garch(1,1)
		Panel B: Experiment 1 -Model Risk
	Sample Size	Matching Signs (%)	MAE	MAPE (%)	R 2
	250	100 (100)	0.0000	0.0000	1.000
	1,000	100 (100)	0.0000	0.0000	1.000
	2,000	100 (100)	0.0000	0.0000	1.000
		Panel C: Experiment 2 -Estimation Risk
	Sample Size	Matching Signs (%)	MAE	MAPE (%)	R 2
	250	99.18 (99.63)	0.0065	11.8779	0.9924
	1,000	99.38 (99.83)	0.0054	9.9131	0.9948
	2,000	99.51 (99.91)	0.0048	8.7629	0.9960
		Panel D: Experiment 3 -Model Risk
	Sample Size	Matching Signs (%)	MAE	MAPE (%)	R 2
	250	99.29 (99.33)	0.0058	10.6340	0.9925
	1,000	99.29 (99.33)	0.0058	10.6336	0.9923
	2,000	99.29 (99.33)	0.0058	10.6332	0.9923
	Panel E: Experiment 4 -Model and Estimation Risks
	Sample Size	Matching Signs (%)	MAE	MAPE (%)	R 2
	250	92.01 (91.71)	0.0306	56.1063	0.8978
	1,000	92.03 (91.57)	0.0305	55.8513	0.8987
	2,000	92.05 (91.49)	0.0304	55.6789	0.8995

Notes: This table presents the design and the results of the Monte Carlo simulations. In the four experiments, we vary (1) the data generating process (DGP) of the return, (2) the bank VaR model, and

Table 7 :

 7 Bank Risk Exposures and Volatility

	Correlation	Equity Interest Rate Foreign Exchange Commodity
	Bank of America	-59	-59	-33	-34
	BNP Paribas	-50	-35	-27	-54
	Citigroup	-34	-46	-13	-55
	Credit Agricole	-35	-50	-3	-41
	Credit Suisse	-59	-70	-26	-67
	Deutsche Bank	-58	-78	-28	-61
	Goldman Sachs	-69	-59	-22	-47
	JPMorgan Chase	-21	-44	8	-79
	Morgan Stanley	-77	-68	-70	-82
	UBS	-65	-47	-39	-43
	Sample Average	-53	-56	-25	-56
	% of Matching Signs	Equity	Interest Rate Foreign Exchange Commodity
	Bank of America	12	28	20	24
	BNP Paribas	32	36	36	28
	Citigroup	56	16	20	36
	Credit Agricole	32	36	36	20
	Credit Suisse	24	32	32	20
	Deutsche Bank	24	24	32	24
	Goldman Sachs	24	40	28	28
	JPMorgan Chase	40	44	40	20
	Morgan Stanley	12	28	20	16
	UBS	28	24	24	20
	Sample Average	28	31	29	24

Table 8 :

 8 Commonality in Bank Risk Exposures

	Average Correlation	Equity Interest Rate Foreign Exchange Commodity
	Bank of America	30	35	-2	26
	BNP Paribas	42	19	7	23
	Citigroup	15	19	8	43
	Credit Agricole	21	40	5	29
	Credit Suisse	38	42	13

Table 9 :

 9 Panel Regression Analysis of Changes in Risk Exposures

		Equity	Interest Rate	Foreign Exchange	Commodity	All Factors
		% E if t	% E if t	% E if t	% E if t	% E if t	% E if t	% E if t	% E if t	% E it	% E it
	% E jf t	0.450***	0.323**	0.475*** 0.479***	0.367*	0.360*	0.375*	0.388*	0.438*** 0.435***
		(0.122)	(0.130)	(0.086)	(0.085)	(0.171)	(0.178)	(0.173)	(0.207)	(0.064)	(0.065)
	% f t	-0.370*** -0.304*** -0.479*** -0.490*** -0.420**	-0.310	-0.652*** -0.661*** -0.437*** -0.433***
		(0.061)	(0.056)	(0.075)	(0.090)	(0.184)	(0.356)	(0.153)	(0.146)	(0.060)	(0.061)
	R f t		0.702		-0.013		-0.659		0.095		0.027
			(0.390)		(0.079)		(1.889)		(0.274)		(0.075)
	CDS it		0.0391		-0.0232		0.0513		-0.0510		0.0001
			(0.0005)		(0.0002)		(0.0008)		(0.0003)		(0.0002)
	RoE it		-0.0016		0.0049		-0.0019		0.0010		0.0011
			(0.0028)		(0.0040)		(0.0027)		(0.0021)		(0.0024)
	Observations	250	250	250	250	250	250	250	250	1,000	1,000
	R 2	0.284	0.293	0.339	0.348	0.081	0.088	0.259	0.266	0.225	0.226

Table 10 :

 10 Robustness CheckPanel D: Controlling for Factor Returns% E if t R f t % E if t R f t % E if t R f t % E if t R f t % E if t R f tThis table presents the estimated coe¢cients and robust standard errors (in parentheses) for several regressions of the percentage changes in risk exposures for the 10 sample banks using an OLS panel regression with bank …xed e¤ects in single factor regressions, columns (1)-(8), and with bank and factor …xed e¤ects in the regressions aggregating all factors, columns (9)-[START_REF] Begenau | Banks' risk exposures[END_REF]. The dependent variable is the percentage change in risk exposure (% E if t or % E if t R f t ). ***, **, * indicate that the coe¢cient is statistically signi…cant at the 1%, 5% and 10% con…dence level, respectively. Each regression is run separately over an estimation period covering 2007Q3-2013Q3.

		Equity	Interest Rate Foreign Exchange Commodity	All Factors
		% E if t	% E if t	% E if t	% E if t	% E if t
			Panel A: Value-Weighted Commonality Proxy
	% E jf t	0.275**	0.389***	0.321**	0.383*	0.349***
		(0.089)	(0.079)	(0.128)	(0.177)	(0.059)
	% f t	-0.472***	-0.508***	-0.507**	-0.652***	-0.501***
		(0.068)	(0.078)	(0.159)	(0.166)	(0.057)
		Panel B: First Principal Component as Commonality Proxy
	% E jf t	0.234***	0.225***	0.204**	0.181*	0.211***
		(0.067)	(0.039)	(0.080)	(0.099)	(0.016)
	% f t	-0.197	-0.332***	-0.321	-0.503*	-0.300***
		(0.127)	(0.072)	(0.184)	(0.255)	(0.053)
			Panel C: Historical Volatility
	% E jf t	0.749***	0.630***	0.567***	0.313	0.619***
		(0.113)	(0.172)	(0.016)	(0.181)	(0.056)
	% f t	-0.206**	-0.292**	-0.292*	-0.638***	-0.300***
		(0.070)	(0.109)	(0.142)	(0.111)	(0.053)
	% E jf t R f t	0.384**	0.885***	0.378**	0.400*	0.663***
		(0.142)	(0.067)	(0.164)	(0.200)	(0.052)
	% f t	-0.303***	-0.284**	-0.467**	-0.481***	-0.261***
		(0.055)	(0.091)	(0.192)	(0.130)	(0.053)
	Notes:

Table 11 :

 11 Subsample Analysis Equity Interest Rate Foreign Exchange Commodity Episode of Increase in Volatility (2007Q3-2008Q4)

	% V olatility	26	23	27	22
	% V aR	1	26	24	3
	% E	-15	9	-0.1	-13
	Corr(% E if t ; % E jf t )	19	28	5	46
	% of Matching Signs	70	52	70	69
	Episode of Reduction in Volatility (2009Q1-2010Q1)
	% V olatility	-15	-12	-11	-18
	% V aR	10	-5	14	13
	% E	31	16	30	41
	Corr(% E if t ; % E jf t )	3	58	24	23
	% of Matching Signs	49	73	63	62

The VaR corresponds to a loss that should only be exceeded with a given target probability over a given time horizon(Jorion,

2007). We show in Section 4.1 that our methodology can also be implemented with other types of risk disclosures.

We use a speci…c volatility index for each risk factor (see caption of Table1for more details).

The two-dimensional nature of VaR is made clear in Goldman Sachs' 2013 10-K report (page 103): "even if our positions included in VaR were unchanged, our VaR would increase with increasing market volatility and vice versa".

With non-zero mean processes, the conditional mean of the return needs to be subtracted in Equation[START_REF] Acharya | Too many to fail -An analysis of timeinconsistency in bank closure policies[END_REF]. However, given the short horizon considered, the variance term is much larger than the mean so that the mean can safely be ignored.

Consistent with Equation (8),[START_REF] Herskovic | The common factor in idiosyncratic volatility: Quantitative asset pricing implications[END_REF] show that there exists a strong factor structure for the volatility of equities. We show in Section 2.5 that other asset classes also exhibit strong volatility factor structures.

If no single exposure in the portfolio accounts for more than an arbitrarily small share of the portfolio, then the variance of the portfolio return obtained when the portfolio size tends to in…nity is fully determined by the variance of the common factor and "t f t (see[START_REF] Gordy | A risk-factor model foundation for rating-based bank capital rules[END_REF].

If we further assume that the marginal distribution F is symmetric, then the VaR becomes V aR if t = if t F 1 f ( ) jE if t j for both long and short positions. Under the symmetry assumption, the FIRE methodology is robust to a change in position from a long position to a short position, and vice versa. In the symmetric case, the percentage change in risk exposure is still given by Equation[START_REF] Ernst | Progress in …nancial services risk management: A survey of major …nancial institutions[END_REF].

We obtain similar pattern when we replace the VIX by the standard-deviation of daily returns on the S&P500 stock index using a three-month estimation window.

The parameters are the following: constant = 8:5965e[START_REF]Revisions to the Basel II market risk framework[END_REF] , ARCH parameter = 0:0692, and GARCH parameter = 0:9242.

0 While[START_REF] Herskovic | The common factor in idiosyncratic volatility: Quantitative asset pricing implications[END_REF] also document commonality in idiosyncratic volatility, we only test for commonality in total volatility since the FIRE methodology is based on total volatility. In each regression, we require a minimum of 10 observations.

[START_REF]Triennial central bank survey -Foreign exchange turnover in April 2013: preliminary global results[END_REF] The CDS data were retrieved from Datastream and the RoE from the banks' quarterly and annual reports.

thank Bertrand Maillet support from the Deloitte -Société Générale Chair in Energy and Finance at HEC Paris

Notes: This …gure displays the daily factor volatility for each risk factor (equity, interest rate, foreign exchange, and commodity) from 2007Q2 to 2013Q3. The volatility on the equity market is measured by the Chicago Board Options Exchange VIX index. The volatility on the …xed income market is measured by the Merrill Lynch MOVE index, which tracks the volatility of Treasury bond prices using implied volatility from 30-day options. The volatility on the foreign exchange market is measured by the Deutsche Bank CVIX index, an average 3-month implied volatility for all the major currency pairs. The volatility on the commodity market is measured by the Chicago Board Options Exchange OVX index, a measure of 30-day implied volatility in West Texas Intermediate crude oil prices. Notes: We transform 10-day VaRs into 1-day VaRs by dividing the former by p 10 (square root of time rule). We also transform 95% and 98% VaRs into 99% VaRs by multiplying the former VaRs respectively by 1.4143 and 1.1327, which are equal to 1 (0:99)= 1 (0:95) and 1 (0:99)= 1 (0:98) (normal distribution).