CoVaR and Systemic Risk - CEMLA and Systemic Risk 6 Pr (X ≤ VaR q) = q What is X? • Since we are...
Transcript of CoVaR and Systemic Risk - CEMLA and Systemic Risk 6 Pr (X ≤ VaR q) = q What is X? • Since we are...
CoVaR and Systemic Risk
Professor Connel Fullenkamp Duke University
IMF Institute -CEMLA Course on Macro-
Prudential Policies
October 2013
What is CoVaR?
• A technique for capturing a financial institution’s contribution to systemic risk based on market data and the value-at-risk (VaR) methodology
• Adrian and Brunnermeier, NY Fed Staff Reports, 2008 (revised 2009 and 2010)
• The “Co” in CoVaR stands for Conditional, Co-movement, Contagion, or Contributing (authors’ own suggestions)
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CoVaR and VaR • CoVaR uses Value at Risk (VaR) as the basic
measure of risk and systemic risk • Systemic Risk is measured by the VaR of the
financial system (or a subset of it) • CoVaR measures what happens to the
system’s VaR when one particular institution is under financial stress, as measured by its own individual VaR
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Questions Answered by CoVaR
• What is the VaR of the financial system if a particular institution is under financial stress? (CoVaR)
• How does the VaR of the system change when a particular institution becomes financially stressed? (ΔCoVaR)
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Idea Behind CoVaR • The distribution of asset values of the financial
system depends on the financial health of individual institutions and their effects on each other
• When a financial institution undergoes stress, this will change the distribution of asset values of the system
• CoVaR estimates the size of the tail of the distribution of asset values in the system—and how it changes
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Review of VaR
• The q% VaR is the “minimum large loss” that occurs only q% of the time, or the loss that is not exceeded (1- q%) of the time
• For some random variable X, we can define the q-percent VaR, denoted VaRq or VaR(q), as the number that satisfies
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qX =≤ )VaR(Pr q
What is X?
• Since we are thinking about the financial distress of banks or other institutions, X should be a function of the market value of the institution’s assets – Market Value of Assets (MVA) – Change in MVA – Growth rate of MVA
• When MVA falls below the value of liabilities, the institution is insolvent
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General Definition of CoVaR
• Let CoVaRqj|i denote the VaR of institution
(or set of institutions) j, conditional on some event C(Xi) occurring to institution i
• CoVaRqj|i is a number such that
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( ) qXCX ij =≤ )(|CoVaRPr )|C(Xjq
i
Specific Definition of CoVaR
• The conditioning event C(Xi) is usually chosen to be that institution i is under stress, so that Xi = VaRq
i . • So CoVaRq
j|i is a number such that
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( ) qXX ij ==≤ iq
i|jq VaR|CoVaRPr
CoVaR in Words
• CoVaRqj|i tells us the q-percent VaR value for
institution j when institution i is at its q-percent VaR value
• Example: What is Citibank’s 5% VaR when JP Morgan Chase is at its 5% VaR?
• If we let institution “j” be the financial system, then CoVaRq
j|i tells the system’s q-percent VaR when institution i is at its q-percent VaR
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What Does CoVaR Tell Us?
• CoVaR simply tells us the boundary on a large loss for some institution(s), given that a particular institution is stressed to a certain degree
• We need to compare the CoVaR measure to another “reference” measure in order to see the change in the boundary caused by institution i’s financial stress
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Delta CoVaR
• One way to measure the contribution to systemic risk is to show what happens when an institution changes from “normal” to “stressed”
• “Normal” means that its asset values are at their median, while “stressed” means that its asset values are at the q-percent VaR level
• Compare the CoVaRs for “stressed” and “normal” realizations of for institution i
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Formal Definition of ΔCoVaR
• Change in the boundary of large-loss region for institution j when institution i goes from a “normal” to a “stressed” realization of X
• When j is a financial system, ΔCoVaRqj|i gives
an estimate of institution i’s contribution to systemic risk—how much the system’s large loss increases because of firm i’s stress
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iiiq
i MedianX|jq
VaRX|jq
i|jq CoVaRCoVaRCoVaR == −=Δ
Using CoVaR and ΔCoVaR
• CoVaR estimates can be used to infer the size of the losses to the system caused by financial distress of one institution
• ΔCoVaR estimates can be used as a measure of contribution to systemic risk that in turn can be the basis of policy – Base of a systemic risk surcharge or tax – Trigger for regulatory intervention
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Estimating CoVaR and ΔCoVaR
• Many methods can potentially be used – Bootstrapping past returns – Extreme value theory
• Adrian and Brunnermeier (2008) use quantile regression, since this is both convenient and it imposes relatively little structure on the distribution
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Quantile Regression
• Standard (OLS) regressions estimate the mean of the distribution of the dependent variable Y, given the explanatory variables Z
• Quantile regression is a technique to estimate the location of the percentiles of this conditional distribution
• In other words, quantile regression estimates the qth percentile of the distribution of Y, given Z
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Output from Quantile Regression
• Given the model Y = α + β Z + ε, the quantile regression estimates a different set of coefficients associated with each percentile of interest
• Therefore, the estimate of the qth percentile of Y, given the value of Zi, is given by
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iqqq ZY βα ˆˆˆ +=
Quantile Regression and CoVaR
• Estimate αq and βq for some lower-tail value of q, and then choose the q-percent VaR values for Z
• Then the fitted Y variable is the estimate of the q-th quantile of Y, given that Z = VaRq
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iq
iq
i|jq VaRˆˆ)VaR(|ˆ CoVaR qq
iq ZY βα +===
Unconditional CoVaR Estimates
• Let Xti = growth rate of the market value of
the total assets of financial institution i • Let Xt
sys = growth rate of the market value of the total assets of all financial institutions
• Estimate Xtsys = α + β Xt
i + ε via quantile regression
• Estimate VaRqi via historical simulation or
other method, and find the median Xi also
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Time-Invariant CoVaR
• Then the estimate of CoVaRqj|i is given by
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iq
iq
i|jq VaRˆˆ)VaR(|ˆ CoVaR qq
isysq XX βα +===
Time-Invariant ΔCoVaR
• Recall that our working definition of ΔCoVaRq
j|i is given by
• Note that VaR.50i = median(Xi)
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[ ] [ ]( ) ( ) ( )i
.50iq
i.50
iq
i.50
iq
i|jq
VaRVaRˆVaRˆˆVaRˆˆ
)VaR(|ˆ)VaR(|ˆ CoVaR
−=+−+=
=−==Δ
qqqqq
isysq
isysq XXXX
ββαβα
Example
• Suppose that we wish to find the unconditional CoVaR and ΔCoVaR for the financial system of our country
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Step 1: Estimate Asset Values
• Gather data on banks: – Stock prices and shares outstanding – Balance sheet equity (BVE) and total assets (BVA)
• Form market value of equity (MVE) = stock price * shares outstanding
• Let market value of assets (MVA) = book value of assets (BVA) * (MVE / BVE)
• This assumes that market-to-book ratios for equity and assets are equal
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Step 2: Quantile Regressions
• Form MVAtsys = Σ MVAt
i • Form Xt
i = (MVAti – MVAt-1
i ) / MVAt-1i for
each bank and for the system • Perform quantile regressions of the form
Xtsys = α + β Xt
i + εt
(regress the system’s change in asset value on each individual institution’s asset value)
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Step 3: Form the CoVaRs For Each Institution
• Estimate the VaRqi via historical simulation
(or other method) and find the median of each Xi
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iq
isys|q VaRˆˆ CoVaR i
qiq βα +=
( )i.50
iq
isys|q VaRVaRˆ CoVaR −=Δ i
qβ
Adding Time Variation
• Unconditional CoVaR and ΔCoVaR give the average contribution of risk, but we know that this contribution will change over time
• We want to make CoVaR and ΔCoVaR dynamic, so we can estimate how these measures increase during times of stress
• In order to do this, we need to add another layer of assumptions…that X depends on a set of state variables
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Time Variation in Asset Returns
• Assume an underlying factor model for asset returns, where the return on each asset depends linearly on these factors: – A set of lagged state variables Mt-1 (to be defined
shortly) – The system-wide growth in assets, Xsys
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Implications of Factor Model
• The asset growth of each bank depends on lagged state variables, while the growth rate of system assets depends on individual bank asset growth and lagged state variables (see Adrian & Brunnermeier Appendix A) :
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itt
iiit MX εγα ++= −1
isystt
isysit
isysisyssyst MXX |
1||| εγβα +++= −
Conditional VaR
• If we take the individual bank asset growth equation and estimate a quantile regression, we can form the q-level VaR for bank i, conditional on the state variables at time t-1:
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1)( −+= tiq
iq
it MqVaR γα
Conditional CoVaR
• Substitute the VaR(q) from the previous slide into the equation for the growth of the system’s assets to obtain the conditional CoVaR:
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1||| )()( −++= tisysi
tisys
qisysi
t MqVaRqCoVaR q γβα
…and Conditional ΔCoVaR
• Again, ΔCoVaR is the difference between the “distress” CoVaR and the median CoVaR:
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( ))50(.)(
)50(.)()(| i
tit
isysq
it
it
it
VaRqVaRCoVaRqCoVaRqCoVaR−=
−=Δ
β
Choosing State Variables
• Variables that capture the time variation in the conditional moments of returns
• Variables that capture the time variation in the tails of asset returns
• The variables should come from assets that are liquid and easily tradable
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State Variable Choices
• Implied stock market volatility (VIX or equivalent)
• Liquidity spread, such as the difference between 3-month repo and 3-month tbills
• Change in 3-month tbill rate (seems to capture tail variation)
• Change in the slope of the yield curve, where slope is the difference between long (10-yr) and short (3-mo) government bond
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State Variables, Continued
• Change in credit spread, where credit spread is difference between long (10-yr BAA) corporate bonds and long (10-yr) govt bonds
• Weekly equity market return • One-year cumulative equity return in real
estate companies (to capture property market influence on banks)
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Example: Adrian and Brunnermeier (2010)
• System: 1269 U.S. financial firms (banks, broker-dealers, insurance companies, real estate companies), weekly 1986-2010
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Results of Conditional Estimation Variable Mean Standard Dev.
Xi (asset growth, percent) 0.378 11.875 1% VaRi -11.745 8.251 1% VaRsys -6.267 3.477 ΔCoVaRi (.01) -1.217 1.235
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CoVaR and VaR
• Adrian and Brunnermeier (2010) find that there is only a loose link between the VaR of an individual institution and its CoVaR with the system
• They made scatter plots of ΔCoVaR versus VaR for groups of institutions, in terms of weekly returns
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Implications
• VaR may be acceptable as a micro-prudential risk management tool, but it does not appear to capture an institution’s contribution to systemic risk
• Basing macro-prudential regulation on VaR or individual institution risk measurements may not be sufficient
• Note that VaR and ΔCoVaR do seem to be strongly correlated across time, however
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Should We Use ΔCoVaR as the Base of Regulation?
• Conditional ΔCoVaR is a high-frequency measure of tail risk, and by its nature is imprecise
• Rebonato’s “accuracy versus relevance” tradeoff at work
• ΔCoVaR is backward-looking (or at best, contemporaneous)
• Market-based risk measures like VaR and CoVaR have procyclicality problems
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A Better Approach?
• Adrian and Brunnermeier suggest that we estimate the relationship between ΔCoVaR and lower-frequency, easily observable, institution-specific variables: – Size – Leverage – Maturity Mismatch
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Gains from Indirect Estimation of ΔCoVaR
• More robust inference about the size and direction of ΔCoVaR (assuming that there are stable relationships between firm characteristics and ΔCoVaR)
• Ability to forecast ΔCoVaR and make it (somewhat) more forward-looking
• Chance to reduce some of the procyclicality of using a market-based measure of tail risk
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Indirect Estimation of ΔCoVaR
• Panel regressions of institutions’ ΔCoVaRs on lagged characteristics of the firms (separate estimations using 1 quarter, 1 year, 2 year lags)
• Firm characteristics measured quarterly • ΔCoVaR measured quarterly as the sum of
weekly conditional ΔCoVaR estimates during the quarter
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Data: Firm Characteristics
• Leverage: total assets / total equity (BVA/BVE)
• Maturity Mismatch: (short-term debt – cash) / total liabilities
• Market/Book: MVE/BVE • Size: total assets (MVA) • Daily equity return volatility during quarter • Equity beta calculated from daily data
during the quarter CoVar and Systemic Risk 46
And One More Characteristic
• Although Adrian and Brunnermeier do not discuss it explicitly in their paper, (quarterly) conditional VaR estimates are also included as one of the main explanatory variables
• Strong time-series correlation between VaR and ΔCoVaR
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Interpreting the Table
• The coefficients measure sensitivities of ΔCoVaR to changes in the various variables
• Example: the -.164 variable on Leverage in Regression 1, Panel A, implies that an increase in leverage ratio of 1 (from 2 to 3) in one quarter would increase the systemic risk contribution by 16.4 basis points, two years later (smaller ΔCoVaR means a more extreme tail observation and hence more risk)
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Note on Table 3
• Instead of absolute size, Adrian & Brunnermeier apparently use relative size = market share, in terms of market value of total assets in institution i, divided by total market value of assets in the system
• Thus, the “Relative Size” coefficient gives the impact on ΔCoVaR of a change in the market share—a .1% increase in market share increases systemic risk by about 40 basis points
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Improving the Estimation
• More detailed firm characteristics should help improve the fit and accuracy of the ΔCoVaR forecasts
• For banks, use asset categories such as loans, loan-loss allowances, intangibles, and trading assets, as a share of total book assets
• Use liability categories such as interest-bearing core deposits, non-interest-bearing deposits, large time deposits and demand deposits, as a share of total book assets
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Further Variables to Include
• Derivatives positions • Off-balance-sheet exposures • Interdependence measures • Supervisory information
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Forward ΔCoVaR
• This is the set of predicted values from the forecasting equations
• Adrian and Brunnermeier find that the 2-year Forward ΔCoVaR is strongly negatively correlated with the contemporaneous ΔCoVaR, suggesting that the Forward ΔCoVaR offsets the procyclicality of the contemporaneous ΔCoVaR
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Using ΔCoVaR Estimations and Forward ΔCoVaR
• A&B suggest that coefficients from the forecasting equations can be used as the basis for systemic risk regulation and possibly as the base of a systemic risk tax
• The Forward ΔCoVaR itself could also be used as a regulatory measure or the base of a regulatory tax
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Using ΔCoVaR Forecasts
• A&B point out that the coefficients from the forecasting regressions show the tradeoffs between different firm characteristics that may be manipulated to alter the bank’s overall systemic risk contribution
• Idea: set a Forward ΔCoVaR target that an institution must meet, but allow it to meet it (and demonstrate compliance) via changes in firm characteristics of the bank’s choice
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Extension: Co-Expected Shortfall
• CoESqi = Expected shortfall of the financial
system, given that Xi ≤ VaRqi
• Can this really be estimated precisely? CoVar and Systemic Risk 58
)|( iq
syssysiq CoVaRXXECoES ≤=
)|(
)|(
50.isyssys
iq
syssysiq
CoVaRXXE
CoVaRXXECoES
≤−
≤=Δ
Extension: CoVaR Networks
• CoVaR is directional, so that CoVaRj|i is not necessarily equal to CoVaRi|j
• This raises the possibility of mapping the magnitude of spillovers as different institutions go into financial distress
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A&B, Figure 2
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Top number is the CoVaR of the bank at the point of the arrow, in US$ billions, conditional on distress of bank at the origin of arrow. Bottom number is CoVaR in opposite direction.
Extension: Exposure CoVaR
• A&B call CoVaRqi|sys the “exposure CoVaR” of
institution i—it is the impact of distress in the financial system on institution i
• Complement to stress testing on individual institutions
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Summary • CoVaR and ΔCoVaR extend the VaR
framework to measuring systemic risk rather than individual institution risk
• Quantile regression makes CoVaR easy to implement
• Forward ΔCoVaR estimations based on firm characteristics may offer antidote to procyclicality of CoVaR as well as best practical regulatory implementation (such as systemic risk taxes)
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