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)

CoVar and Systemic Risk 2

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


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)


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


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


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


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


( ) 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


( ) 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


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


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


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


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


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


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


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


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


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


Time-Invariant CoVaR

•  Then the estimate of CoVaRqj|i is given by


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)


[ ] [ ]( ) ( ) ( )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


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


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)


Step 3: Form the CoVaRs For Each Institution

•  Estimate the VaRqi via historical simulation

(or other method) and find the median of each Xi


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


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


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) :


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:


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:


1||| )()( −++= tisysi

tisys

qisysi

t MqVaRqCoVaR q γβα

…and Conditional ΔCoVaR

•  Again, ΔCoVaR is the difference between the “distress” CoVaR and the median CoVaR:


( ))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


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


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)


Example: Adrian and Brunnermeier (2010)

•  System: 1269 U.S. financial firms (banks, broker-dealers, insurance companies, real estate companies), weekly 1986-2010


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


Results: Time-Varying VaR


Results: Conditional ΔCoVaR


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


Scatter Plots—US Institutions


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


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


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


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


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


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


Adrian & Brunnermeier, Table 3


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)


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


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


Adrian & Brunnermeier, Table 4


Further Variables to Include

•  Derivatives positions •  Off-balance-sheet exposures •  Interdependence measures •  Supervisory information


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


Adrian & Brunnermeier, Fig. 5


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


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


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


A&B, Figure 2


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


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)


covar and systemic risk - cemla and systemic risk 6 pr (x ≤ var q) = q what is x? • since we are...

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