How Useful are Aggregate Measures of Systemic Risk?

How Useful are Aggregate Measures of Systemic Risk? Harry Mamaysky∗ Initial draft: November 2014 Current draft: December 2015 Abstract Following the financial crisis of 2008–2009, a large literature has emerged that attempts to quantify and measure systemic risk. In this paper we focus on some of the more popular systemic risk indicators from this literature and ask how well they work, in the following sense: At the aggregate level, what information above that which is readily observable in the market do we learn from these systemic risk indicators? Several popular measures provide very little incremental information beyond that contained in implied volatilities and credit spreads – information which is commonly known to market participants. For some indicators, where neither the cross sectional mean nor standard deviation contains meaningful incremental information, we find that the tail proportion (the cross sectional proportion of indicators experiencing a tail realization) does contain information that is not already observable in the market.

∗

[email protected]. Mamaysky is a visiting reseach scholar at Columbia Business School. The author thanks Mark Flood and especially the editors (Roger Stein and Mila Sherman Getmansky) for many useful suggestions that greatly improved this manuscript.

Introduction Among the many responses to the financial crisis of 2008–2009 has been the growth of a literature focused on early warning signs for future crises. Bisias, Flood, Lo, and Valavanis [2012], in the definitive survey of the field, document 31 quantitative measures of systemic risk – and more have been created since the paper came out. Systemic risk is concerned with identifying financial imbalances that may prove large enough to cause substantial damage to the real economy. This literature has largely focused on the measurement of either realized or market implied tail behaviors of financial securities – the idea being that anomalous tail behavior may be detected earlier than changes in means or variances. Systemic risk indicators are now being used in industry and by regulators as part of their market monitoring toolkit (for example, Adrian, Covitz, and Liang [2013] and European Central Bank [2014]). Much of the empirical work on systemic risk indicators attempts to show that they worked “well” leading up to the crisis of 2008–2009. While a natural starting point, this approach is not fully satisfactory because it suffers from overfitting a unique historical event (which may never be repeated in its exact form) and because it is often silent on whether a given systemic risk indicator contains any unique information that is not already common knowledge among market participants and regulators.1 Our main point is that an effective systemic risk indicator should provide useful and unique information during periods of market stress. By focusing on market stress, rather than periods of systemic crisis, we can draw conclusions that are less sensitive to overfitting the data. And by requiring that a systemic risk indicator contain unique information we can narrow down the set of indicators that investors and regulators need to follow. Studies focusing on the forecasting ability of macro indicators often start with a set of crises, defined a priori by the researchers, and quantify the value of these indicators by their ability to predict the crises under consideration. When the number of historical crises is high (for example, Drehmann and Juselius [2013] analyze country level banking crises), this approach is particularly useful. In the context of systemic risk crises (see Lo and Zhou [2012]), the crisis-prediction approach suffers from the drawback that, by definition, systemic crises don’t happen very often. In the last 100 years we’ve had perhaps two: the Great Depression and the Great Financial Crisis. Researchers are therefore constrained to work with the small number of correlated events that occur around a particular crisis, and run the risk of finding indicators that only work well for a specific historical episode. 1

Hansen [2013] discusses other challenges and opportunities in the area of systemic risk measurement.

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We propose that an alternative way to evaluate systemic risk indicators is to focus on their ability to forecast variables which fluctuate frequently, and which we would expect to be at extreme values were a systemic crisis to occur. Such variables allow us to measure when the economy and markets are stressed, but without imposing the restriction that such stress need be a once in fifty year event. The two variables we focus on in this paper are the CBOE Volatility Index (VIX)2 and the US unemployment rate. We measure the information content of a systemic risk indicator by its ability to forecast future changes in the VIX and the US unemployment rate, once we have controlled for information that is already available to market participants.3 We now look to establish the degree of predicatability in the VIX and US unemployment rate that can be attributed to common market knowledge. For forecasting future changes in the VIX index, our set common knowledge consists of current and lagged values of the VIX and CDX indexes.4 For forecasting changes to US unemployment, we add current and lagged US unemployment to the information set. If a systemic risk indicator’s ability to forecast future stress is due predominantly to a component common to the VIX and CDX indexes, or to the US unemployment rate, then we say this indicator does not offer any unique insights. Admittedly, this requirement sets a very high bar for any risk indicator. The VIX and CDX obtain from forward looking, very liquid markets that are constantly impounding any useful information for forecasting future market stress. Were a given piece of information, such as a new systemic risk indicator, to provide very valuable forward looking information, the VIX and CDX indexes would likely impound this information and thus over time render the indicator in question not useful according to our definition. Nevertheless, the requirement that a systemic risk indicator contain information that is both orthogonal to the VIX and CDX indexes and still useful for forecasting macro outcomes is a natural benchmark.5 2

From Wikipedia: “VIX is a trademarked ticker symbol for the CBOE Volatility Index, a popular measure of the implied volatility of S&P 500 index options; the VIX is calculated by the Chicago Board Options Exchange (CBOE). Often referred to as the fear index or the fear gauge, the VIX represents one measure of the market’s expectation of stock market volatility over the next 30-day period.” 3 Giglio, Kelly, and Pruitt [2015] is a related study of the effectiveness of systemic risk indicators for forecasting macro outcomes (their main focus is on industrial production growth). However, while they control for the time series properties of the macro series they are trying to forecast, they do not control for the information content of market variables, such as the VIX and CDX indexes. 4 CDX refers to the Markit CDX North America Investment Grade Index, an extremely liquid credit derivative product that tracks 125 of the most widely traded single name credit default swaps written on credits that are rated investment grade by Moody’s, S&P, and Fitch. Our CDX series is the Bloomberg “generic” 5-year version of the index which shows at every time point the spread of the then prevailing on-the-run CDX index. The 5-year point is chosen because it is the most liquid and widely followed maturity. 5 Additionally the requirement that useful information in a systemic risk indicator be orthogonal to market risk proxies is one way to avoid the volatility paradox proposed by Brunnermeier and Sannikov [2014] and Adrian and Boyarchenko [2013]. In an economy where financial intermediaries face risk-based leverage constraints, they

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A fair criticism of the approach advocated in this paper is that a systemic risk measure may do a poor job of forecasting fluctuations in frequently observed variables, but still do a good job of forecasting infrequent systemic crises. Furthermore, the ability to forecast either market stress or economic crises should not be the sole criterion along which systemic risk measures are evaluated. For example, a systemic risk indicator may yield valuable insights for evaluating a crisis ex-post, even if it could not have helped in forecasting that the crisis would occur. Also we ignore the fact many systemic risk indicators contain rich cross-sectional information, which may be useful for understanding individual outcomes. Despite these, and other, limitations of our approach, we feel it nonetheless adds valuable insights into the systemic risk landscape.

A preview of results We focus on two prominent classes of systemic risk indicators: • Tail risk measures. These measure tail behavior either among stock prices or CDS levels. There are three directions of causality: index to single name – how does an individual stock do when the overall market experiences a tail return (marginal expected shortfall or MES); single name to index – how does the index do conditional on a single name experiencing a tail return (CoVaR); and single name to single name (CoRisk). • Covariance based estimators. Kritzman, Li, Page, and Rigobon [2010] propose to measure systemic risk by the percentage of the total variance of a set of industry returns that can be explained by a fixed and small number of principal components. Kritzman and Li [2010] use the covariance matrix of returns to generate a normalized measure of return unusualness. Exhibit 1 summarizes how systemic risk indicators fit into these classifications. Other systemic risk indicators that are similar in spirit to the ones we consider include: the distressed insurance premium (Huang, Zhou, and Zhu [2009]) which prices the risk that the entire financial system suffers a catastrophic loss in asset value; Granger causality networks (Billio, Getmansky, Lo and Pelizzon [2012]) which measure the relationship between returns will have high leverage during times when volatilities are low. As volatility spikes, intermediaries will be forced to delever as their balance sheets appear to be riskier, which in turn causes further volatility. Therefore high leverage proxies for high systemic risk, and such high leverage occurs during times of low volatility and cheap credit. Therefore systemic risk indicators which contain information different than what is contained in the VIX or CDX indexes may work better as warning signals during quiescent times.

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Name Marginal Expected Shortfall (MES) CoVaR CoRisk Absorption Ratio Turbulence

Direction

Reference

index to single name Acharya et al. [2010] single name to index Adrian and Brunnermeier [2010] single name to single name International Monetary Fund [2009] covariance based Kritzman, Li, Page, and Rigobon [2010] covariance based Kritzman and Li [2010]

Exhibit 1: List of systemic risk measures that are examined in this paper. on single name A today and the lagged returns on name B (and are thus similar to CoRiskbased networks which measure contemporaneous tail dependence); the credit absorption ratio (Reyngold, Shnyra, and Stein [2015]) which works with implied asset values from a Merton debtequity model instead of with stock returns; and the performance of crowded trades (Pojarliev and Levich [2011] and Khandani and Lo [2011]) which may serve as early warning indicators of funding stress or the potential insolvency of a major financial player. We leave for future research the application of the methodology proposed in this paper to these systemic risk indicators, as well as the others described in Bisias et al. [2012]. In this paper, we abstract away from the exact implementation details of the systemic risk indicators in Exhibit 1. We try to implement the simplest possible measures – using identical data sets – that still capture the salient points of the original methodologies. This makes comparison and interpretation of our modified indicators straightforward, though at the cost of potentially missing an important element of the original implementation. The crucial differentiator of MES, CoVaR, and CoRisk is the direction of causality, whether from index to single name, single name to index, or single name to single name. The covariance based methods are primarily concerned with the factor structure of returns. Our approach accurately captures these essential elements, and we think it unlikely that the approach misses any first order effects of the original implementations. The results of our effectiveness tests are relatively poor, with most indicators from Exhibit 1 having no forecasting ability once we control for the VIX and CDX indexes, and the US unemployment rate. The one exception is CoRisk, which builds a causality network based on contemporaneous tail relationships among returns of the single name securities in our data set and then measures the connectivity of that network. A natural feature of building network links as a function of tail dependence is to choose some threshold level, such that for tail dependence above this threshold a network link is said to exist.

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It is this feature of building a network that turns out to be very important for stress forecasting. In a novel observation, we show that a systemic risk indicator defined as the proportion of single name MES and CoVaR measures that exceed some threshold at a given point in time – what we call the tail proportion – exhibits superior forecasting ability relative to the time series of cross sectional means or standard deviations of MES and CoVaR, and relative to all other indicators in Exhibit 1. We perform our tests using systemic risk indicators derived from a financials-only and a broader set of large companies. We find that the systemic risk indicators derived from the financials data perform better than the corresponding indicators derived from the broader dataset. While this conclusion may be sensitive to our choice of time period (post-2000 and including the financial crisis), it suggests that information from the financial sector may be particularly important for forecasting future economic or market stress. Our tests also allow us to ask how far into the future a given systemic risk indicator’s forecasting ability extends. We show that “useful” systemic risk indicators helps in forecasting the VIX index and the US unemployment rate one to one and a half years into the future. From a macroprudential point of view, such long lead times are helpful because they allow enough time to mount a meaningful regulatory response to a financial imbalance.

Data We use two sets of securities data. The first, called FINS, is 103 global financial firms with market capitalizations larger than US$30bln as of October 7, 2014 or larger than US$21bln6 on January 1, 2000. FINS includes securities classified as “Financials” by the Industry Classification Benchmark (ICB) in 2014, or by the Global Industry Classification Standard (GICS) in 2000. Many systemic risk analytics have focused on financials as the primary security set, and we do so here as well. As a benchmark for the results, we also use a set, called GLOBALS, of 174 large international firms with market capitalizations larger than US$75bln as of October 7, 2014 or larger than US$53bln on January 1, 2000. Exhibits 2 and 3 show the distribution by country of the market capitalizations of firms in both data sets. The US, China, and the UK are the top three countries in both data sets. Where a stock market index needs to be associated with an individual stock, we use the local 6

Deflated by 2.4% average CPI since January 1, 2000 for 14.5 years.

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country index of the exchange on which the stock trades. Exhibit 4 shows the mapping from country to stock market index. The VIX index is standard, and the CDX index in question is the Markit CDX North America Investment Grade Index with a five year maturity, expressed in basis points. Our US unemployment measure is the “US Unemployment Rate Total in Labor Force, Seasonally Adjusted” from the Bureau of Labor Statistics. All data are obtained from Bloomberg. All systemic risk indicators are computed from stock return data. The predictive regressions (discussed later) are run from November 2004 (the start of the CDX data) to December 2015.

Definitions of systemic risk indicators Our implementation of systemic risk indicators follows the spirit if not the letter of the methodologies as originally proposed. We feel that this approach simplifies the analysis, renders comparisons between efficacy of different indicators easier, and captures to first order the salient driver of the indicator as originally proposed. Exhibit 5 summarizes the key parameters of the implementation of all indicators. Let us refer to the return of security i as of time t, as ri (t). The index return of the country of listing of security i is Ii (t). All returns in the ensuing analysis are one week, overlapping returns, expressed in the local currency.

Absorption Ratio Before calculating the covariance matrix of security returns r(t), we filter the data in the following way: Any day that doesn’t have at least 25% of observations present is dropped from the sample; then in a second pass, any security with fewer than 90% of its observations present is dropped from the sample. The first step drops weekends and other days that appear to be holidays, whereas the second step only keeps securities that have enough observations present on non-weekend, non-holiday days. This insures that the composition of securities in the rolling windows over which covariances are calculated is uniform across all time periods. In a given time window, the covariance matrix is estimated using the returns only on those days for which all security return observations are present. 6

Our implementation of the absorption ratio from Kritzman, Li, Page, and Rigobon [2010] is Pn

i=1 AR(t) = PN

j=1

σi2 (t)

(1)

σj2 (t)

where σi2 is the variance of the ith principal component, n = 10, and N is the number of securities in the sample. This differs slightly from the implementation in the original paper where the denominator sums over variances of the securities, rather than the principal components. The nice feature of our version is the AR is bounded from 0 to 1. The value of AR on a given day is calculated in rolling windows of 500 business days (which matches the estimation window in in Kritzman, Li, Page, and Rigobon [2010]).

Turbulence The data filter for estimating the covariance matrix is the same as for the absorption ratio. The turbulence measure is given by turb(t) =

t 1 X (r(s) − µ(s))0 Σ(s)−1 (r(t) − µ(s)) 21 s=t−20

(2)

where the covariance matrix of returns Σ(s) and the mean return vector µ(s) are estimated over the 750 prior business days, and r(s) is understood to be the return on the final date of each successive 750 day period.7 The implementation in Kritzman and Li [2010] does not average over the prior 21 days; however without this smoothing, we find the turbulence measure too noisy to identify useful time series structure.

CoVaR and Marginal Expected Shortfall Adrian and Brunnermeier [2011] define CoVaR as “the value at risk (VaR) of the financial system conditional on institutions being under distress.” We conjecture that the high-frequency variation in their indicator comes primarily from the following measure, which we use as our 7

Kritzman and Li [2010] estimate the covariance matrix in (2) using monthly returns over their entire sample. We use a relatively long rolling window of 750 business days to estimate Σ(s) so as not to bias our forecasting tests.

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CoVaR-like indicator:8 ˆ i |ri < Qp ], CoV aRindex|i (t) = −E[I i

(3)

ˆ is the conditional sample mean over the prior 125 business days, and Qp is the sample where E i quantile for returns of security i corresponding to probability p (also computed over the prior 125 business days). This time period choice balances the desire for a timely estimate with the need to have sufficient data over which to compute tail behavior. In our analysis p = 10%. Marginal expected shortfall is calculated very similarly to the definition given in Acharya, Pedersen, Philippon, and Richardson [2010]: ˆ i |Ii < Qp ], M ESi (t) = −E[r I,i

(4)

where QpI,i is the p percent quantile of the index corresponding to security i, with the quantile and sample mean taken over the prior 125 business days, and p = 10%. Using the notation from (3), this measure is effectively CoV aRi|index though in the paper our use of CoV aR always refers to the directionality of the definition given in (3). We discuss aggregation of these measures in a later section.

CoRisk The CoRisk measures proposed in International Monetary Fund [2009] calculates the percent increase in the 95th percentile of the CDS level of security i conditional on security j being in its 95th percentile. This measures the contribution to the tail risk of security i that arises from i’s dependence on security j. Figure 2.6 in their paper shows a network graph of the the CoRisk estimates among the set of financial institutions that they analyze. The authors note that “only co-risk estimates above or equal to 90 percent are depicted.” This highlights the need for a threshold level of dependence when constructing tail linkage networks. Our measure of CoRisk captures the flavor, though not the exact implementation, of the 8

It is somewhat misleading to refer to this measure as “VaR” because it is a conditional expectation not a quantile. But we’ll continue with this nomenclature for ease of exposition. As discussed later in the paper, the choice of conditional expectation vs conditional quantiles does not effect the results.

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IMF measure.9 We define CoRisk between institutions i and j as follows ˆ j |ri < Qp ], CoRisk j|i (t) = −E[r i

(5)

where Qpi is, as before, the p percent quantile of the return distribution of security i estimated over the prior 125 days, and p = 10%. The conditional mean is calculcated over the same 125-day time period. One advantage of using stock data instead of CDS is that stock data is more widely available, and is observable at a higher frequency (especially during calm markets) than CDS data, though CDS levels are arguably more sensitive to credit-specific information. Also note that i and j may not trade in the same time zone, which leads to an asynchronicity issue that should largely be mitigated by our use of weekly returns. Our codependence network is constructed by having a link go from i to j if CoRisk j|i (t) is above 3%. This is an arbitrary choice, though as Exhibits 12 and 13 show it leads to an indicator with appropriate time series properties. The choice of optimal threshold level is an interesting area for future work. Our measure of network connectivity simply counts the connections that exist, as a fraction of all possible connections,10 or N X X 1 1[CoRisk j|i (t) > θ] CoRisk(t) = 2 N − N j=1 i6=j

(6)

where N is the number of securities in the sample and θ = 3%. As will be seen shortly, we use the idea that a connection doesn’t exist unless the value of a systemic risk indicator exceeds some threshold to define a proportion-in-tail indicator for MES and CoVaR.

A comment on the use of conditional expectations Our CoVaR, MES, and CoRisk measures are all calculated as conditional expectations (see equations (3), (4), and (5)). An alternative calculation, more in keeping with the original definition of CoVaR and CoRisk, would be to compute conditional quantiles. For example, in 9

As section A comment on the use of conditional expectations points out, computing conditional quantiles instead of conditional expectations does not affect our results in any meaningful way. 10 Using eigenvector centrality (the eigenvalue of the first eigenvector of the network matrix) gives almost identical results.

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the case of CoRisk, equation (5) would be replaced by CoRisk j|i (t) = −(ˆ ap + ˆbp · Qpi − Qpj ),

(7)

where a ˆp and ˆbp are coefficient estimates from a quantile regression for the pth quantile of the returns of security j conditional on the returns of security i. For p = 10%, (7) tells us how much lower j’s tenth percentile return would be if i experienced a return equal to its tenth percentile relative to j’s unconditional tenth percentile return. Using this and the analogous definitions for CoVaR and MES leaves all of our empirical results largely unchanged. For parsimony, we use the simpler conditional expectation measure in the remainder of the paper. The quantile regression based results are available upon request.

Aggregation of CoVaR and Marginal Expected Shortfall One natural way to aggregate single name CoVaR and MES is to look at the time series of the cross-sectional average. This is the aggregation approach most often followed in the literature. For MES, we have N 1 X M ESi (t), M ES(t) = N i=1

(8)

with the analogous definition for CoVaR. Other authors propose looking at the second moment. For example, Lo and Zhou [2012] report that the cross-sectional standard deviation of MES, i.e. M ESSD (t) = StdDev({M ES1 (t), . . . , M ESN (t)}), is the best performing of their set of indicators based on the signal-to-noise ratio. We also will study M ESSD and CoV aRSD in our set of systemic risk indicators.

Tail proportion Borrowing an idea from the CoRisk network definition, where an edge is drawn from i to j if CoRisk j|i exceeds some threshold, we define threshold versions of M ES and CoV aR as the

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fraction of single name measures in a given time period that exceed some threshold: N 1 X M ESP T (t) = 1[M ESi (t) > θ], N i=1

(9)

with the analogous definition for CoV aR. We refer to this measure as the tail proportion, and use the subscript P T (proportion-in-tail) to indicate this version of a given indicator. Intuitively, the tail proportion will be very sensitive to increases in the cross-sectional dispersion of systemic risk indicators at a given time. A well behaved cross sectional mean of MES, for example, may still exhibit a high tail proportion. It is this sensitivity to cross sectional outliers that makes the tail proportion indicators particularly effective based on our tests, as will be seen. In our analysis, θ = 3% for both MES and CoVaR, though an optimal choice of θ is an interesting area for future research.11

Properties of systemic risk indicators Exhibit 5 summarizes the parameters of the indicators described in the previous section. We have nine systemic risk indicators of interest, three flavors each of CoVaR and MES, CoRisk, absorption ratio and turbulence. We also have two high frequency market stress measures, in the VIX and the CDX investment grade indexes. Exhibits 12 and 13 show the systemic risk indicators for the FINS and GLOBALS data sets respectively (as described in the Data section). A scaled picture (in grey) of the VIX index is juxtaposed with each systemic risk indicator. Exhibit 6 (7) show the correlation of these time series for the FINS (GLOBALS) data set. For FINS and GLOBALS, MES, CoVaR, and CoRisk are highly correlated with the VIX index – the cross-sectional means of both indicators seem to be good contemporaneous indicators for the state of volatility. The SD and P T versions of MES and CoVaR are also highly correlated. CoRisk has a similar behavior to the P T indicators, as may be expected given that all three are threshold based measures. Turbulence and especially the absorption ratio have lower correlations with the other systemic risk measures and with the VIX and CDX indexes. The high correlations that some systemic risk indicators have with the VIX and CDX indexes calls into question how much information these indicators may contain for future market stress that is not already 11

One simple method would be to choose θ so that a given tail proportion indicator is above 0.25 a quarter of the time, or some other such criterion which would adjust for the volatility of the underlying systemic risk measure.

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contained in the VIX and CDX indexes. Formal test of this are discussed in the next section. From Exhibits 12 and 13, we see that for both the FINS and GLOBALS data sets, the tail proportion versions of MES and CoVaR are much more sensitive to small VIX and CDX fluctuations than the cross sectional mean and standard deviation versions. CoRisk – also a threshold based measure – shares some features of this sensitivity with the tail proportion measures. The absorption ratio (i.e. the percentage of total variance of the FINS and GLOBALS data sets that is captured by the top ten principal components) proxies for very long lasting factor regimes, rather than the higher frequency volatility tracked by the VIX index. Both FINS and GLOBALS show three absorption ratio regimes in our data, a medium pre-2005 regime, a low regime heading into the crisis, and a high absorption ratio regime post crisis. Interestingly, though financial market volatility has been rather benign in recent experience, the absorption ratio has stayed elevated post-crisis. Finally, the turbulence measure appears very VIX-like in the part of the sample prior to and including the financial crisis, but becomes much less VIX-like in the later part of the sample. In fact, the turbulence measure for GLOBALS shows large spikes in the post-crisis sample at times when the VIX index remains relatively benign.

A test of efficacy A systemic risk indicator is useful to the extent that it contains previously unknown information that is useful for forecasting future episodes of economic stress. We proxy for such episodes by identifying time periods during which the VIX index and the US unemployment rate rise. Exhibit 14 shows a time series of the VIX and US unemployment rate, with US recessions represented as grey bars. Both series show pronounced countercyclical variation. Therefore incremental ability to forecast rises in either variable should prove useful for forecasting future periods of economic stress.

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Baseline for predictability To establish the baseline level of predictability exhibited by the VIX and US unemployment rate, we run the following regressions: V IX(t + ∆d ) − V (t) = L X ` a+ bV IX V IX(t − ` · 21) + b`CDX CDX(t − ` · 21) + (t),

(10)

`=0

U nemp(t + ∆m ) − U nemp(t) = L X ` α+ cU U nemp(t − `) + c`V IX V IX(t − `) + c`CDX CDX(t − `) + η(t), `=0

(11) where U nemp(t) is the US unemployment rate, L is the number of monthly lags, and ∆d (∆m ) is the forecasting horizon in business days (calendar months). The VIX (unemployment) equation is estimated at a daily (monthly) frequency. The results reported in this paper use L = 2 (note that this implies regressors at times t, and one and two months prior to t), though the qualitative nature of the results is not sensitive to this choice of L. Note that all coefficients in the above regressions depend on the forecasting horizon ∆ – there is a new set of coefficients for every ∆ for which these regression are evaluated – though this dependency is suppressed to avoid cluttered notation. The forecasting regressions in (10) and (11) are simple, easily to implement, and rely only on easily obtainable data. They should serve, therefore, as an appropriate benchmark for what is readily knowable about the future evolution of the VIX and the unemployment rate. There are undoubtedly better forecasting models, but if we set the bar too high the possibility of any systemic risk variable adding value will be minimal. That a systemic risk indicator adds incremental forecasting power to these regressions is therefore a necessary but not sufficient condition for said risk indicator to be useful in practice. Note that the explanatory variables in (10) and (11) may suffer from collinearity. While this affects the interpretation of the coefficient estimates from these regressions, it does not affect the model’s R2 – and in particular, collinearity of the explanatory variables will not affect our measures of the incremental contribution of a systemic risk indicator to this forecasting regression.

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Exhibits 15 (VIX) and 16 (US unemployment) show the coefficients and R2 ’s from the above forecasting regressions, as a function of the forecasting horizon ∆. The VIX series is persistent, and in the short-term the best predictor of future VIX changes is the current VIX. Over a medium-term horizon, the level of CDX becomes an important predictor for the VIX as well. The R2 of the regression increases steadily as the forecasting horizon ∆ increases, going from under 10% to close to 60%, indicating there is more predictability in long-term than short-term changes in the VIX. The unemployment series is also very persistent, though collinearity in lagged values of unemployment make interpretation of cˆ0U , cˆ1U , and cˆ2U difficult since they tend to offset each other. Over the medium term there is evidence that the CDX index is useful for forecasting changes in the unemployment rate – perhaps via a drying up of credit effect – and over the long run the VIX starts to have some effect as well. The R2 of forecasts for unemployment rate changes increases steeply with ∆ but then levels off around 70% after a forecasting horizon of 100 or so business days. Regressions (10) and (11) establish that we have identified important sources of predictability in the VIX and the US unemployment series. This sets the bar for our systemic risk indicators: They have to provide incremental information that is not already captured in our simple forecasting model.

Usefulness of systemic risk indicators We now introduce each of the nine systemic risk indicators under consideration as explanatory variables in regressions (10) and (11). We refer to these as the augmented regressions. For example, the MES-augmented version of the VIX regression from (10) is V IX(t + ∆d ) − V (t) = a +

L X

b`V IX V IX(t − ` · 21) + b`CDX CDX(t − ` · 21)

`=0

+ bSRI M ES(t) + (t),

(12)

where we suppress the implicit dependence of all regression coefficients on M ES and ∆d to avoid clutter. Because we have two data sets (FINS and GLOBALS) for calculating the systemic risk indicators, we have two sets of results for each indicator – though the results are broadly similar. The above regression is estimated with L = 2. The in-sample contribution of a systemic risk indicator to the explanatory power of this regression clearly decreases with L, but importantly 14

the qualitative nature of the results and the inter-indicator comparisons are not sensitive to the choice of L. For the FINS data set, Exhibits 17 and 18 summarize the results for the VIX index and the US unemployment rate respectively.12 Each row of the figures corresponds to three summary statistics for each systemic risk indicator (there are nine rows). The first chart in each row shows the unadjusted R2 from the forecasting regression without the systemic risk indicator (in red) and with the systemic risk indicator (in blue). The next chart shows the difference in the two series, which is non-negative by construction. This difference in R2 ’s between the augmented (with the systemic risk indicator) and the baseline forecasting regressions, which we call the information value (IV ) of a systemic risk indicator, is our key metric of usefulness. For example, for MES we define IVM ES as IVM ES (∆) ≡ R2 (MES-augmented regression, ∆) − R2 (baseline regression, ∆),

(13)

with the analogous definitions for the other indicators. Note that IV is a function of the forecasting horizon ∆, and is defined for both the VIX and the unemployment forecasting regressions. The introduction of a systemic risk indicator helps us in forecasting future risk episodes only to the extent that it increases the proportion of the variance in the macro state variables that is explainable by the model, and this is what IV captures. A corroborating piece of information is the sign and significance of the loading on the systemic risk indicator for each forecasting horizon of the augmented regression (i.e. bSRI from equation (12)) – as all systemic risk indicators are expected to have higher values in stressed times, our prior is that these coefficients should be positive. A plot of bSRI is shown in the third chart in each row. The x-axis for all plots in Exhibits 17 and 18 is the forecasting horizon, ∆, in business days. Exhibit 8 shows summary statistics for each indicator, including the maximum value of IV (labeled “Max R2 Diff”), the time horizon ∆ (in business days) at which this maximum is achieved, and the percentage of forecasting horizons for which coefficient estimates bSRI for a given systemic risk indicator in the augmented regression are significant and positive at the 10% and 5% levels. 12

Results for the GLOBALS dataset are qualitatively similar, and are not shown to conserve space.

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VIX tests with the FINS data set Let us first examine the results for forecasting the VIX (equations (10) and (12)) using MES derived from the FINS data set. These are shown in the second row of Exhibit 17. First we see that for horizons of less than 250 days, there is zero incremental value to including the systemic risk indicator as IVM ES (∆) ≈ 0 for ∆ < 250. There is some incremental improvement in forecasting ability, especially as we approach the one and a half year forecasting horizon, with IVM ES peaking at 4.6%.13 Supporting evidence that MES becomes useful for longer forecasting horizons can be seen in the third chart of the MES row which shows that most coefficient estimates bSRI for MES in the augmented regression are significant at the 5% level for ∆ > 300. However, comparing the R2 ’s of the baseline and augmented regressions shows that the economic benefit of MES is rather marginal – the augmented R2 (blue line) doesn’t ever get very far from the baseline R2 (red line). Our tests for the predictive power for the VIX of M ES, M ESSD , CoV aR, AbsRatio, and T urbulence suggest that these five indicators either yield incremental or negligible economic benefits in forecasting future risk events, once we control for the current level of the V IX and CDX indexes. Note that this does not imply that some of these indicators do not contain important cross-sectional information. Of the original indicators summarized in Exhibit 1, CoRisk and CoV aRSD are best performers in our VIX tests using the FINS data set. As can be seen from the CoRisk row in Exhibit 17, IVCoRisk is above zero for all forecasting horizons between 100 and 400 days, peaking at an incremental benefit of 11.3% around day 200. Furthermore, as Exhibit 8 shows, 65% of the CoRisk coefficient estimates (as a function of ∆) are significant at the 10% level. CoV aRSD performs comparably on the IVCoV aR SD measure, and better in terms of significance of bSRI from (12) as all (94%) of coefficient estimates are positive and significant at the 10% (5%) level. For the remaining indicators from Exhibit 1 we find the negative result that, at the aggregate level, they are not useful for forecasting future increases in the VIX once we control for current and lagged values of the VIX and CDX indexes. Interestingly, the two best performing indicators in this set of tests are the tail proportion versions of MES and CoVaR. As can be seen from Exhibit 17 and Exhibit 8, both M ESP T and CoV aRP T yield meaningful improvements in the R2 ’s of the forecasting regressions at most time horizons, with peak improvements of 13.5% and 11.3% respectively, and the associated 13

We do not perform formal statistical tests on IV , though it is an interesting area for future work.

16

bM ES estimates are almost all positive and significant for M ESP T (though less impressive for CoV aRP T ). Why the proportion-in-tail estimators perform so well is an interesting question for future research.

US Unemployment tests with the FINS data set The results for the augmented version of equation (11) are even more stark. Of the original five indicators, except CoRisk, none have any incremental forecasting ability for future stress as proxied by change in the US unemployment rate (Exhibit 18). CoV aRSD and M ESSD do not fare any better. The baseline and augmented R2 ’s are almost completely identical. CoRisk and CoV aRP T have incremental value (IV peaks at around 8.5% one and a half years out). By far the best performing indicator is (again) M ESP T , with a peak level of IVM ES of 17.8%, with the majority of bM ES coefficient positive and significant at the 5% level. Exhibit 9 summarizes these results. One possible reason for the outperformance of M ESP T relative to CoV aRP T is that the 3% threshold (see Exhibit 5) is a more stringent requirement for CoVaR (conditional index returns) than for MES (conditional single name returns) since single names are more volatile than indexes – thus CoV aRP T only spikes in response to relatively larger shocks than M ESP T . Perhaps a more clever choice of θ would render these two indicators more comparable in their effectiveness (see footnote 11).

Results with the GLOBALS data set Exhibits 10 and 11 summarize the same tests (the analogous figures are not shown to conserve space), but using systemic risk indicators derived from the GLOBALS data set. With this broader data set, all five (including CoRisk) indicators from Exhibit 1, as well as M ESSD , do not contain meaningful economic value for forecasting future risk episodes, once we control for market observables. CoV aRP T and CoV aRSD show some forecasting ability, but are a clear second behind M ESP T which is once again the best performing indicator, and by a large margin, showing maximal IM ES values of 12.2% and 9.1% in the VIX and unemployment regression respectively.

17

These results are broadly consistent with those obtained using the FINS dataset, though the results are stronger for systemic risk indicators derived using the FINS dataset (compare the improvement in R2 and the fraction of significant coefficients in Exhibits 8 and 9 versus the comparable results in Exhibits 10 and 11). This suggests that looking at the financial sector for information about future market stress is more useful than looking at a broader set of companies. However, this conclusion may be particularly sensitive to our choice of a time period (post-2000) which contained the financial crisis of 2008–2009. Using a longer time period, Giglio et al. [2015] find a similar result that financial sector volatility is a better predictor of macroeconomic outcomes than non-financial sector volatility.

Summary of VIX and US unemployment tests The unemployment and VIX tests with both the FINS and GLOBALS datasets all point to the same set of conclusions: • Except CoRisk, the original five systemic risk measures from Exhibit 1 are either marginal or ineffective at forecasting future stress once we control for known market variables; • CoRisk is effective in both the VIX and unemployment tests, possibly due to its threshold rule for establishing a network connection; • Our newly proposed tail proportion versions of the MES and CoVaR indicators add incremental value for forecasting changes in the VIX and US unemployment; • The MES proportion-in-tail indicator is the best performer across all tests, suggesting this direction of conditionality (i.e. single name conditional on an index tail return) is perhaps the most important tail risk measure.

Time horizon of efficacy By looking at the forecasting horizon ∆∗ at which IV (i.e. maximal difference in R2 ’s) is maximized for the augmented versions of regressions in (10) and (11), we can see the horizon at which effective systemic risk indicators are most valuable. Using the FINS data set, the best performing indicators for VIX forecasting are M ESP T , CoV aRP T and CoRisk. The IV measure achieves its maximum value at forecasting horizons of 235 business days for all three

18

(see Exhibit 8). For unemployment, these optimal forecasting horizons are either 378 or 420 business days (Exhibit 9). Using the GLOBALS data set, the best performing indicator is M ESP T which achieves its maximum IV value in the VIX forecasting regressions at 305 business days (Exhibit 10) and at 399 business days in the unemployment regressions (Exhibit 11). The long lead times provided by these indicators suggest that they may be useful tools not only for market participants but for the regulatory community as well.

Conclusion In this paper we have argued that the usefulness of systemic risk indicators should be measured by their forecasting ability once we control for easily observable, market primitives such as the VIX volatility and CDX credit spread indexes. Rather than using an event study methodology, we use state variables (the VIX and the US unemployment rate) that vary frequently, and whose values are likely to be high in a systemic crisis. We then show that many of the systemic risk indicators proposed in the literature do not have any incremental forecasting power for future levels of the VIX or US unemployment, once we control for current and lagged values of the VIX and CDX indexes, and of the US unemployment rate. This calls into question just how useful these systemic risk indicators will prove in practice. On a more constructive note, we show that CoRisk and a tail proportion indicator for MES and CoVaR (which measures the fraction of single name indicators at a given time point that experience tail realization) do have incremental forecasting power for future changes in VIX and US unemployment. The effectiveness of proportion-in-tail systemic risk indicators is a new result.

19

References Acharya, V., L. Pedersen, T. Philippon, and M. Richardson, 2010, “Measuring systemic risk,” working paper, New York University. Adrian, T. and M. Brunnermeier, 2010, “CoVaR,” FRBNY Staff Report No. 348. Adrian, T., D. Covitz, and N. Liang, 2013, “Financial stability monitoring,” FRBNY Staff Report No. 601. Adrian, T. and N Boyarchenko, 2013, “Intermediary leverage cycles and financial stability,” FRBNY Staff Report no. 567. Billio, M., M. Getmansky, A. Lo and L. Pelizzon, 2012, “Econometric measures of connectedness and systemic risk in the finance and insurance sectors,” Journal of Financial Economics, 104, 535–559. Bisias, D., M. Flood, A.W. Lo, and S. Valavanis, 2012, “A survey of systemic risk analytics,” Office of Financial Research Working Paper no. 1. Brunnermeier, M.K., and Y. Sannikov, 2014, “A macroeconomic model with a financial sector,” American Economic Review 104(2), 379–421. Drehmann, M. and M. Juselius, 2013, “Evaluating early warning indicators of banking crises: Satisfying policy requirements,” BIS Working Papers No. 421 . European Central Bank, 2014, Financial Stability Review, May 2014, https://www.ecb.europa.eu/pub/pdf/other/financialstabilityreview201405en.pdf. Giglio, S., B. Kelly, and S. Pruitt, 2015, “Systemic risk and the macroeconomy: An empirical evaluation,” NBER Working Paper 20963. Hansen, L.P., 2013, “Challenges in identifying and measuring systemic risk,” working paper. Huang, X., H. Zhou, and H. Zhu, 2009, “Assessing the systemic risk of a heterogeneous portfolio of banks during the recent financial crisis,” Board of Governors of the Federal Reserve, Finance and Economics Discussion Series 2009-44. International Monetary Fund, 2009, “Assessing the systemic implications of financial linkages,” Global Financial Stability Review, April 2009, 73–110.

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Khandani, A. and A. Lo, 2011, “What happened to the quants in August 2007? Evidence from factors and transactions data,” Journal of Financial Markets, 14 (1), 1–46. Kritzman, M., and Y. Li, 2010, “Skulls, financial turbulence, and risk management,” Financial Analysts Journal, 66(5), 30–41. Kritzman, M., Y. Li, S. Page, and R. Rigobon, 2010, “Principal components as a measure of systemic risk,” Revere Street Working Paper Series: Financial Economics 272-28. Lo, A. and A. Zhou, 2012, “A comparison of systemic risk indicators”, unpublished working paper, MIT Laboratory for Financial Engineering. Pojarliev, M. and R. M. Levich, 2011, “Detecting crowded trades in currency funds,” Financial Analysts Journal, 67(1), 26–39. Reyngold, A., K. Shnyra, and R.M. Stein, 2015, “Aggregate and firm-level measures of systemic risk from a structural model of default,” Journal of Alternative Investments, 17 (4), 58–78.

21

Exhibits Country UNITED STATES CHINA BRITAIN CANADA AUSTRALIA HONG KONG JAPAN FRANCE SWITZERLAND GERMANY SPAIN ITALY BRAZIL NETHERLANDS SWEDEN INDIA QATAR RUSSIA SINGAPORE SAUDI ARABIA IRELAND

Mkt Cap (USD,bn) Mkt Cap (%) 2893.2 42.9 1239.2 18.4 420.8 6.2 293.3 4.3 286.8 4.3 227.0 3.4 213.1 3.2 206.3 3.1 191.7 2.8 146.2 2.2 144.5 2.1 117.9 1.7 80.5 1.2 63.6 0.9 43.4 0.6 40.9 0.6 31.7 0.5 30.2 0.4 28.9 0.4 23.2 0.3 22.2 0.3

Exhibit 2: Market capitalizations in US dollars aggregated by country of incorporation for firms in the FINS data set, as of 2015-12-18.

22

Country UNITED STATES CHINA BRITAIN SWITZERLAND JAPAN GERMANY FRANCE AUSTRALIA SPAIN HONG KONG NETHERLANDS BELGIUM CANADA SOUTH KOREA DENMARK BRAZIL TAIWAN INDIA SAUDI ARABIA ITALY MEXICO RUSSIA NORWAY

Mkt Cap (USD,bn) Mkt Cap (%) 10039.2 56.7 1565.1 8.8 1203.4 6.8 814.8 4.6 667.7 3.8 599.9 3.4 514.3 2.9 355.1 2.0 238.2 1.3 232.7 1.3 205.4 1.2 200.2 1.1 185.9 1.0 159.8 0.9 147.6 0.8 140.3 0.8 112.1 0.6 71.9 0.4 70.5 0.4 53.6 0.3 50.3 0.3 44.4 0.3 44.0 0.2

Exhibit 3: Market capitalizations in US dollars aggregated by country of incorporation for firms in the GLOBALS data set, as of 2015-12-18.

23

Country USA Russia China Japan Mexico Brazil India Spain France Germany Swiss UK Italy Hong Kong Australia Canada Belgium Saudi Arabia Singapore Netherlands Denmark Sweden Norway Qatar Ireland South Korea Taiwan

Index Name S&P 500 INDEX MICEX INDEX SHANGHAI SE COMPOSITE NIKKEI 225 MEXICO IPC INDEX BRAZIL IBOVESPA INDEX Nifty 50 IBEX 35 INDEX CAC 40 INDEX DAX INDEX SWISS MARKET INDEX FTSE 100 INDEX FTSE MIB INDEX HANG SENG INDEX S&P/ASX 200 INDEX S&P/TSX COMPOSITE INDEX BEL 20 INDEX TADAWUL ALL SHARE INDEX Straits Times Index STI MSCI NETHERLANDS OMX COPENHAGEN 20 INDEX OMX STOCKHOLM 30 INDEX OBX STOCK INDEX QE Index IRISH OVERALL INDEX KRX 100 INDEX TAIWAN TAIEX INDEX

Exhibit 4: Stock market indexes associated with country of listing for stocks in the FINS and GLOBALS data sets.

24

Class Return Window Calculation Window Tail Prob Threshold Number PC Average Window

CoRisk MES 5 5 125 125 0.1 0.1 0.03 0.03

CoVaR AbsRatio Turbulence 5 5 5 125 500 750 0.1 0.03 10 21

Exhibit 5: Parameters for systemic risk runs. Return, calculation, and average windows are reported in business days. PC stands for principal components. For details on the implementation of each systemic risk indicator, see discussion in section Definitions of systemic risk indicators.

25

CoRisk MES MES SD MES PT CoVaR CoVaR SD CoVaR PT AbsRatio Turbulence VIX CDX IG

CoRisk MES 1.00 0.89 1.00 0.65 0.75 0.91 0.77 0.92 0.97 0.61 0.65 0.96 0.83 0.31 0.38 0.57 0.69 0.74 0.81 0.78 0.88

MES SD

MES PT

CoVaR

CoVaR SD

CoVaR PT

1.00 0.53 0.69 0.71 0.65 0.33 0.53 0.68 0.72

1.00 0.78 0.53 0.89 0.41 0.42 0.63 0.69

1.00 0.68 0.89 0.31 0.71 0.83 0.85

1.00 0.62 0.07 0.57 0.62 0.66

1.00 0.26 0.53 0.72 0.71

AbsRatio Turbulence VIX

1.00 0.07 0.33 0.57

1.00 0.69 0.61

CDX IG

1.00 0.86

1.00

AbsRatio Turbulence VIX

CDX IG

Exhibit 6: Correlation of systemic risk indicators using the FINS dataset.

CoRisk MES MES SD MES PT CoVaR CoVaR SD CoVaR PT AbsRatio Turbulence VIX CDX IG

CoRisk MES 1.00 0.93 1.00 0.62 0.75 0.90 0.85 0.95 0.97 0.68 0.79 0.97 0.88 0.30 0.32 0.52 0.56 0.79 0.84 0.76 0.86

MES SD

MES PT

CoVaR

CoVaR SD

CoVaR PT

1.00 0.58 0.61 0.75 0.57 0.30 0.43 0.69 0.76

1.00 0.86 0.69 0.93 0.36 0.48 0.72 0.73

1.00 0.72 0.91 0.32 0.55 0.82 0.83

1.00 0.66 0.17 0.47 0.65 0.68

1.00 0.32 0.49 0.74 0.71

1.00 -0.03 0.35 0.52

Exhibit 7: Correlation of systemic risk indicators using the GLOBALS dataset.

26

1.00 0.53 0.41

1.00 0.86

1.00

CoRisk MES MES SD MES PT CoVaR CoVaR SD CoVaR PT AbsRatio Turbulence

Max R2 Diff Max R2 Days % pos and sig at 10 % pos and sig at 5 11.3 235 65 53 4.6 345 47 41 1.2 435 14 8 13.5 235 96 96 2.5 345 31 20 10.5 135 100 94 11.3 235 59 51 4.6 135 0 0 3.9 365 71 55

Exhibit 8: Summary of prediction results using the FINS dataset for the VIX index. Data starts on Nov 2004 and ends on Dec 2015. “Max R2 Diff” refers to the maximum difference in R2 ’s between the augmented and baseline regressions with L = 2 (IV from the text). Max R2 Days refers to business days. “% pos and sig” refers to the percentage of coefficient estimates for the systemic risk indicator that are significant and positive in the augmented regression, as a function of the forecasting horizon ∆. Standard errors are obtained using Newey-West with 10 lags.



Exhibit 9: Summary of prediction results using the FINS dataset for US unemployment. Data starts on Nov 2004 and ends on Dec 2015. “Max R2 Diff” refers to the maximum difference in R2 ’s between the augmented and baseline regressions with L = 2 (IV from the text). Max R2 Days refers to business days. “% pos and sig” refers to the percentage of coefficient estimates for the systemic risk indicator that are significant and positive in the augmented regression, as a function of the forecasting horizon ∆. Standard errors are obtained using Newey-West with 10 lags.

27



Exhibit 10: Summary of prediction results using the GLOBALS dataset for the VIX index. Data starts on Nov 2004 and ends on Dec 2015. “Max R2 Diff” refers to the maximum difference in R2 ’s between the augmented and baseline regressions with L = 2 (IV from the text). Max R2 Days refers to business days. “% pos and sig” refers to the percentage of coefficient estimates for the systemic risk indicator that are significant and positive in the augmented regression, as a function of the forecasting horizon ∆. Standard errors are obtained using Newey-West with 10 lags.


Max R2 Diff 2.2 2.2 1.3 9.1 4.1 1.5 2.6 0.9 0.4

Max R2 Days % pos and sig at 10 % pos and sig at 5 63 0 0 105 0 0 21 17 12 399 62 58 126 0 0 42 17 8 441 0 0 210 0 0 483 0 0

Exhibit 11: Summary of prediction results using the GLOBALS dataset for US unemployment. Data starts on Nov 2004 and ends on Dec 2015. “Max R2 Diff” refers to the maximum difference in R2 ’s between the augmented and baseline regressions with L = 2 (IV from the text). Max R2 Days refers to business days. “% pos and sig” refers to the percentage of coefficient estimates for the systemic risk indicator that are significant and positive in the augmented regression, as a function of the forecasting horizon ∆. Standard errors are obtained using Newey-West with 10 lags.

28

Systemic risk indicators for FINS 2015−12−17 VIX (scaled)

0.7 0.5 0.3 0.1

0.7 0.5 0.3 0.1

VIX (scaled)

VIX (scaled)

0.9

AbsRatio

0.16

MES

VIX (scaled)

0.9

CoVaR_PT

CoRisk

0.9

0.12 0.09 0.06

0.8

0.7

0.03 VIX (scaled)

0.06 0.04 0.03 0.02 VIX (scaled)

1.4

60

VIX

MES_PT

2.0

74

0.7 0.6

45

0.4

31

0.2

17 VIX (scaled)

VIX (scaled)

255

CDX IG

0.09

CoVaR

2.6

0.9

0.9

0.07 0.06 0.04 0.02

205 155 104 54

0.03

CoVaR_SD

VIX (scaled)

3.1

Turbulence

MES_SD

0.07

VIX (scaled)

2000

2005

2010

0.02

0.01

2000

2005

2010

2015

Exhibit 12: Behavior of systemic risk indicators, and the VIX and CDX investment grade indexes. Systemic risk indicators are calculated using the FINS data set. Parameters for the calculation of different indicators are given in Exhibit 5.

29

2015

Systemic risk indicators for GLOBALS 2015−12−17 VIX (scaled)

0.6 0.5 0.3

VIX (scaled)

0.12

AbsRatio

MES

0.07 0.05

0.06

VIX (scaled)

VIX (scaled)

0.7

VIX (scaled)

2.7

Turbulence

MES_SD

0.3

0.6

0.03

0.05 0.04 0.03 0.02

2.2 1.8 1.4 0.9

VIX (scaled)

0.9

74 60

0.7 0.6

VIX

MES_PT

0.5

0.8

0.10

45 31

0.4 0.3

17 VIX (scaled)

VIX (scaled)

255

CDX IG

0.09

CoVaR

0.7

0.1

0.1

0.07 0.05 0.03 0.02

CoVaR_SD

VIX (scaled)

0.9

CoVaR_PT

CoRisk

0.8

205 155 104 54

VIX (scaled)

0.026

2000

2005

2010

0.022 0.018 0.014 0.009 2000

2005

2010

2015

Exhibit 13: Behavior of systemic risk indicators, and the VIX and CDX investment grade indexes. Systemic risk indicators are calculated using the GLOBALS data set. Parameters for the calculation of different indicators are given in Exhibit 5.

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2015

10

20

30

VIX index 40 50 60

70

80

Dependent variables and the business cycle

1990

2000

2010

1980

1990

2000

2010

4

5

US unemployment 6 7 8 9

10

11

1980

Exhibit 14: The VIX index (top) and US unemployment (bottom), with NBER recessions shown in grey.

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No SRI predictive test for VIX ●

100

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400

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Exhibit 15: Coefficients from regression (10) plotted as a function of the forecasting horizon ∆ (shown in business days). The notation “ l[`]” indicates a lag of ` months. Coefficients significant at the 5% (10%) level are shown with a large (small) square. We use Newey-West standard errors with a lag of 10 days.

32

No SRI predictive test for US Unemployment ●

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●

● ● ●

● ● ●

●

●

● ● ●

100

200

300

∆ (Days forward)

400

500

●

●

●

●

●

● ●

●

●

●

●

−0.3

0.01

0.4

unemp_l0

●

0.2

●

−0.1

cdx_l2

−0.015 −0.010 −0.005

●

●

100

200

300

∆ (Days forward)

400

500

Exhibit 16: Coefficients from regression (11) plotted as a function of the forecasting horizon ∆ (shown in business days). The notation “ l[`]” indicates a lag of ` months. Coefficients significant at the 5% (10%) level are shown with a large (small) square. We use Newey-West standard errors with a lag of 10 months.

33

Predictive tests for VIX using FINS with 2 lags

0.00

Diff in R2 Diff in R2 0.02 0.06 0.10 0.000 0.015 Diff in R2 Diff in R2 0.02 0.04 0.00 0.06

100

200

300

400

500

0.00 0.005

Diff in R2 0.025

0

5 t−val sri 2 3 4

●

●

● ● ● ● ● ● ●●●●

● ● ● ● ●●●●●● ● ●● ● ●● ● ●●●

1 4

t−val sri 1 2 3 0

●● ●●●

● ● ● ● ● ● ● ●● ●●●●●●●●●●●●●

●● ● ● ● ●● ● ●

●●●●●●●● ●●● ● ●● ● ● ● ●● ●●●●● ● ● ● ●● ●●● ● ●●●● ● ● ●● ●● ●●●● ●

t−val sri 3 4 5

Max value = 0.135 ● ● ● ●● ● ● ● ●● ●●●●●●

● ● ●●● ● ●● ● ●●●● ● ● ● ●● ●●●●● ● ●

●

6

●●●●● ●●● ●● ● ● ● ●●

●●● ● ● ●●●●● ● ●● ● ●●

2 t−val sri 0 1

● ●

●● ●● ● ●● ●● ●● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ●●● ●● ● ● ●●●●● ● ●● ●●● ● ● ● ● ● ● ●

−1

●

●

●

● ●●

● ● ●

Max value = 0.025

● ● ●●

●

● ●● ●●● ●●●● ●●●●●● ●●●

●

● ●●●●●●

● ● ●●●●●●●

●●● ●● ●● ● ●● ● ● ● ● ● ●● ● ●●● ● ●●●● ● ●● ● ● ●●●●●●●● ●

●

●

●

● ●

● ● ●

● ● ●●●

● ●

●

● ● ●● ● ● ●●●● ●●●●●● ●● ●●● ●●●●●●●●●●

● ● ● ● ●● ●●●

● ●● ●●● ●● ● ●

Max value = 0.039 ●●● ● ●

●● ● ●● ● ●● ●

200

300

400

● ●● ● ● ● ● ●● ●●●● ●● ● ● ●

500

● ●●● ● ●● ●●● ●● ●● ● ●●●● ● ●● ●●● ●● ●●● ●●

●

0

100

200

300

400

Exhibit 17: Each row corresponds to regression (10) augmented with a systemic risk indicator. The first chart in each row shows the R2 from the unaugmented regression (red hollow circles) and the R2 from the augmented regression (blue solid squares). The next chart shows the difference in the R2 ’s. The final chart shows the coefficient on the systemic risk indicator estimated in the augmented regression. Large (small) squares indicate significance at the 5% (10%) level using Newey-West standard errors with 10 lags. The forecasting horizon ∆ is on the x-axis, in business days. 34

● ● ● ● ● ● ●● ●

●●●● ●● ●● ● ● ●● ● ●● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ●●●●

●●●● ●●

● ●●

●

● ● ● ● ● ● ● ● ● ●●● ● ● ●● ●

●

●● ●

● ●● ● ● ● ● ● ●

Max value = 0.046

●● ●

● ●●●●●●●●● ●● ● ● ●● ● ●● ● ● ● ●●● ●● ● ● ● ● ●●

● ●● ● ●● ● ●● ● ●● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●●●●● ● ● ● ●●● ●● ●● ● ● ●●● ●●●

● ●● ●● ●● ●●●●●●●●●

● ● ● ●●

100

●

● ●●●● ● ● ● ● ●●● ● ● ●●

● ● ●

● ● ●● ●

Max value = 0.113

● ● ● ●●●●● ● ● ●●●●●●●●●

0

●●●● ●●● ● ● ● ●●

● ● ●● ● ● ● ●●●

●

Max value = 0.105

● ●

●

●

● ● ● ●●●

●

● ●● ● ● ●●

● ● ● ● ●●●● ● ●●

CoVaR_PT

Turbulence

●

● ●

2

●●●●●●●●● ●●● ●●● ● ●● ●● ●●●● ●●● ●●●●●●●●●●●● ●●●● ●●● ● ● ● ● ●

● ●

Max value● = 0.012 ●

CoVaR_SD

AbsRatio

● ● ● ● ● ● ●●●

● ● ●● ● ● ● ● ● ●●●● ●●● ● ●●●●●● ●●●●●●●●

t−val sri 0 1 2

R2 from Model 0.1 0.3 0.5

●●●●●●●●● ●●● ●●● ● ●● ●●●● ●●● ●● ●●●●●●●● ●●● ● ●●●● ●●● ● ● ● ● ●

CoVaR

● ● ● ● ● ● ● ● ● ●●

Max value = 0.046

●● ●●●● ● ●● ● ●● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ●●●● ●● ●●●●● ● ●●●●

−1


●●●●●● ●●●● ●●● ●● ● ● ● ●● ● ●● ● ● ●● ●●●●●●●● ●●● ● ●●●● ●●● ● ● ● ● ●

MES_PT

● ● ● ● ●● ●● ●●● ● ● ● ●●●●●●

t−val sri 3.0


●●●●●●●●● ●●● ●●● ● ●● ●● ●●●● ●●● ●●●●●●●●●●●● ●●●● ●●● ● ● ● ● ●

MES_SD

● ● ● ● ● ●● ●● ●● ●● ●●●●●●●

2.0


●●●●●● ●●●● ●●● ●● ● ●● ●● ● ●● ● ● ●● ●●●●●●●● ●●● ● ●●●● ●●● ● ● ● ● ●

MES

●

t−val sri 1 2 3 4 5


●●●●●●●●● ●●● ●●● ● ●● ●● ●● ●●● ●● ●●●●●●●● ●●● ● ●●●● ●●● ● ● ● ● ●

CoRisk

Max value = 0.113 ●●●●●

t−val sri −1.5 −0.5


●●●●●●●●● ●●● ●●● ● ●● ●● ● ●● ● ● ●● ●●●●●●●● ●●● ● ●●●● ●●● ● ● ● ● ●

t−val sri

● ●● ●●

t−val sri 1.0 2.0 3.0 4.0−2.5


●●●●●●●●● ●●● ●●● ● ●● ●● ●● ●●● ●● ●●●●●●●● ●●● ● ●●●● ●●● ● ● ● ● ●

difference in R2

Diff in R2 Diff in R2 Diff in R2 Diff in R2 0.06 0.120.000 0.006 0.012 0.00 0.02 0.04 0.00 0.06


●●●●●● ●●●● ●●● ●● ● ●● ●● ● ●● ● ● ●● ●●●●●●●● ●●● ● ●●●● ●●● ● ● ● ● ●


R2 from predictive regs

500

Predictive tests for US Unemployment using FINS with 2 lags

● ● ● ● ● ● ● ● ●

CoVaR_SD ●

●

AbsRatio ● ●

● ● ● ● ● ● ● ● ●

● ● ● ● ● ● ● ● ● ● ●

●

Turbulence ●

100

200

300

400

500

t−val sri 0 1 2

● ● ● ●

t−val sri 0.0 0.4

●

●

● ● ●

● ●

●

●

● ● ● ● ●

●

●

● ● ●

● ●

●

●

●

● ● ●

● ● ● ●

● ● ● ●

● ●

● ● ● ●

● ● ● ● ● ● ● ● ● ● ●

●

● ● ● ● ●

● ● ●

−0.4

● ●

● ● ● ●

●

● ●

●

●

●

● ●

●

● ● ● ● ● ● ●

● ● ● ● ●

●

● ●

● ●

● ●

● ●

●

● ●

●

● Max value = 0.018 ● ●

●

●

●

●

●

●

●

● ● ● ● ●

● ● ● ● ● ●

Max value = 0.084

● ●

● ●

●

● ● ● ● ● ● ● ● ● ● ● ●

●

●

● ● ● ●

● ● ● ● ● ●

●

●

● = 0.009 Max value ● ● ● ●

● ●

●

● ●

● ●

● ● ● ●

100

●

● ● ●

●

● ●

●

●

● ● ● ● ●

●

● ●

●

●

● ● ●

●

● ● ● ● ●

●

●

● ● ● ● ●

200

● ●

●

● ● ●

● ● ●

300

400

●

●

●

● ● ●

● ● ●

● ●

500

● ● ● ● ●

●

●

●

●

● ● ●

●

●

● ● ●

●

●

●

● ●

●

● ● ● ● ● ● ●

● ● ●

● ● ● ● ●

100

200

300

400

Exhibit 18: Each row corresponds to regression (11) augmented with a systemic risk indicator. The first chart in each row shows the R2 from the unaugmented regression (red hollow circles) and the R2 from the augmented regression (blue solid squares). The next chart shows the difference in the R2 ’s. The final chart shows the coefficient on the systemic risk indicator estimated in the augmented regression. Large (small) squares indicate significance at the 5% (10%) level using Newey-West standard errors with 10 lags. The forecasting horizon ∆ is on the x-axis, in business days. 35

●

● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

Max value = 0.007

●

● ●

●

●

● ● ●

●

● ●

● ● ● ●

● ● ●

●

●

●

● ●

● ● ● ●

●

● ●

●

●

● ● ● ● ● ● ● ● ●

●

● ●

●

●

Max value = 0.023 ●

●

● ●

●

●

●

●

●

●

● ●

Max value = 0.178

● ● ● ● ● ● ● ● ● ● ●

●

●

●

●

Diff in R2 0.004

●

0.00

CoVaR_PT ● ● ● ● ● ● ● ● ●

●

● ●

●

● ● ● ● ● ● ● ● ● ● ●

●

●

●

● ●

0.000

●

● ● ● ● ● ● ● ● ● ●

● ● ●

●

●

● ● ● ● ● ● ● ● ● ● ●

●

● ● ●

● ● ●

●

●

● ●

●

t−val sri −1.5 −0.5

CoVaR ● ●

● ● ●

Max value = 0.001

● ● ● ● ● ● ● ● ● ● ●

●

●

●

●

Diff in R2 Diff in R2 Diff in R2 0.04 0.08 0.004 0.010 0.0160.000 0.015

●

● ● ● ● ● ● ● ● ●

●

●

t−val sri 0 1 2 3 4 5

●

●

●

Diff in R2 0.10

MES_PT

●

●

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

●

●

●

●

Max value = 0.004

● ●

t−val sri 1.2 1.8 −2.5

●

●

●

● ●

0.6

MES_SD ●

● ●

●

● ● ● ● ● ● ● ● ● ● ●

●

●

● ● ● ● ● ●

● ● ●

●

●

t−val sri −1 0 1 2

● ● ● ● ● ● ● ● ●

Diff in R2 6e−04

● ●

● ●

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

●

t−val sri t−val sri 0.5 1.0 1.5 2.0 −1.0 0.0

Diff in R2 0.002

MES

t−val sri

●

●

● ● ● ● ● ● ● ● ● ● ●

●

● ● ● ● ● ● ●

●

●

●

● ●

●

0.000

●

●

t−val sri −0.5 0.5 −1

Diff in R2 0.04 0.08

●

0.00

CoRisk ● ● ● ● ● ● ● ● ●

Max value = 0.085

● ● ● ● ● ● ● ● ● ● ●

●

●

difference in R2

0e+00

●

● ● ● ● ● ● ● ● ●

0.00

●

Diff in R2 0.002 0.006

R2 from Model R2 from Model R2 from Model R2 from Model R2 from Model R2 from Model R2 from Model R2 from Model R2 from Model 0.4 0.6 0.4 0.6 0.40 0.55 0.70 0.4 0.6 0.4 0.6 0.4 0.6 0.8 0.40 0.55 0.70 0.4 0.6 0.40 0.55 0.70

R2 from predictive regs

500