Structural GARCH: The Volatility-Leverage Connection Robert Engle 1 Emil Siriwardane 1 1 NYU Stern School of Business MFM Macroeconomic Fragility Fall 2013 Meeting
Leverage and Equity Volatility I Crisis highlighted how leverage and equity volatility are tightly linked I Leverage Effect has been around - e.g. Black (1976), Christie (1982) - but... I Adynamicvolatilitymodelthatincorporatesleveragedirectly has remained elusive
Leverage and Equity Volatility I Crisis highlighted how leverage and equity volatility are tightly linked I Leverage Effect has been around - e.g. Black (1976), Christie (1982) - but... I Adynamicvolatilitymodelthatincorporatesleveragedirectly has remained elusive
Leverage and Equity Volatility I Crisis highlighted how leverage and equity volatility are tightly linked I Leverage Effect has been around - e.g. Black (1976), Christie (1982) - but... I Adynamicvolatilitymodelthatincorporatesleveragedirectly has remained elusive
BAC Leverage and Realized Volatility 1-Month Realized (Annualized) Volatility 2.4 2.2 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 100 90 80 70 60 50 40 30 20 10 Debt to Equity 1998 0 2000 2002 2004 2006 2008 2010 2012 0 Date
This Presentation I GARCH-type model where equity volatility is amplified by leverage as in structural models of credit I Statistical test of how leverage affects volatility I Heteroskedastic asset returns from observed equity returns I The Leverage Effect and Asymmetric Volatility I Capital Shortfall in a Crisis
This Presentation I GARCH-type model where equity volatility is amplified by leverage as in structural models of credit I Statistical test of how leverage affects volatility I Heteroskedastic asset returns from observed equity returns I The Leverage Effect and Asymmetric Volatility I Capital Shortfall in a Crisis
This Presentation I GARCH-type model where equity volatility is amplified by leverage as in structural models of credit I Statistical test of how leverage affects volatility I Heteroskedastic asset returns from observed equity returns I The Leverage Effect and Asymmetric Volatility I Capital Shortfall in a Crisis
This Presentation I GARCH-type model where equity volatility is amplified by leverage as in structural models of credit I Statistical test of how leverage affects volatility I Heteroskedastic asset returns from observed equity returns I The Leverage Effect and Asymmetric Volatility I Capital Shortfall in a Crisis
This Presentation I GARCH-type model where equity volatility is amplified by leverage as in structural models of credit I Statistical test of how leverage affects volatility I Heteroskedastic asset returns from observed equity returns I The Leverage Effect and Asymmetric Volatility I Capital Shortfall in a Crisis
Theoretical Foundation
Structural Models of Credit I Under relatively weak assumptions on the vol process, structural models say E t = f A t,d t,sa,t f,t I I I sa,t f At = market value of assets at time t D t = book value of debt at time t = (t-forecast) asset volatility over the life of the debt,t I Taking derivatives, de t = t At da t + f E t E t A t sa,t f ds A,t f E t t = f / A t is just our familiar in option pricing
Structural Models of Credit I Under relatively weak assumptions on the vol process, structural models say E t = f A t,d t,sa,t f,t I I I sa,t f At = market value of assets at time t D t = book value of debt at time t = (t-forecast) asset volatility over the life of the debt,t I Taking derivatives, de t = t At da t + f E t E t A t sa,t f ds A,t f E t t = f / A t is just our familiar in option pricing
Ignoring Long-Run Volatility Dynamics de t = t At da t + f E t E t A t sa,t f ds A,t f E t I For now, ignore the volatility term I Over the life of debt (e.g. 5 years), forecast long run asset volatility is virtually constant I We verify this later, and also estimate a model where we impose this Vega Analysis
Ignoring Long-Run Volatility Dynamics de t = t At da t + f E t E t A t sa,t f ds A,t f E t I For now, ignore the volatility term I Over the life of debt (e.g. 5 years), forecast long run asset volatility is virtually constant I We verify this later, and also estimate a model where we impose this Vega Analysis
Ignoring Long-Run Volatility Dynamics de t = t At da t + f E t E t A t sa,t f ds A,t f E t I For now, ignore the volatility term I Over the life of debt (e.g. 5 years), forecast long run asset volatility is virtually constant I We verify this later, and also estimate a model where we impose this Vega Analysis
Equity Volatility as a Function of Leverage I Assets follow: da t A t = 0 z } { µ A (t)dt + p h A (t)db(t) I Straightforward to show: det vol t = t At p h A (t) E t E t I But we don t observe A t,soinvertcalloptionformula: A t g(e t,d t,s f A,t,t) = f 1 (E t,d t,s f A,t,t)
Equity Volatility as a Function of Leverage I Assets follow: da t A t = 0 z } { µ A (t)dt + p h A (t)db(t) I Straightforward to show: det vol t = t At p h A (t) E t E t I But we don t observe A t,soinvertcalloptionformula: A t g(e t,d t,s f A,t,t) = f 1 (E t,d t,s f A,t,t)
Equity Volatility as a Function of Leverage I Assets follow: da t A t = 0 z } { µ A (t)dt + p h A (t)db(t) I Straightforward to show: det vol t = t At p h A (t) E t E t I But we don t observe A t,soinvertcalloptionformula: A t g(e t,d t,s f A,t,t) = f 1 (E t,d t,s f A,t,t)
The Leverage Multiplier I Substitute in and write volatility as a function of observed leverage: I det vol t = E t LM t z apple } { 4 t g(e t /D t,1,sa,t f,t) D t dat vol t A t =LM t (D t /E t,s f A,t,t) vol t E t dat Assumes pricing function is homogenous degree one in underlying and strike. I We call LM t the leverage multiplier A t
The Leverage Multiplier I Substitute in and write volatility as a function of observed leverage: I det vol t = E t LM t z apple } { 4 t g(e t /D t,1,sa,t f,t) D t dat vol t A t =LM t (D t /E t,s f A,t,t) vol t E t dat Assumes pricing function is homogenous degree one in underlying and strike. I We call LM t the leverage multiplier A t
What Does the Leverage Multiplier Look Like? Simple Case: Black-Scholes-Merton World r = 0.03; Varying Asset s, t 9 8 Leverage Multiplier 7 6 5 4 3 σ =0.1, τ =5 σ =0.2, τ =5 2 σ =0.1, τ =10 σ =0.2, τ =10 1 0 5 10 15 20 25 30 35 40 45 50 Debt to Equity
Our Specification I The challenge is choosing the right functional form for LM t I Need a flexible function of leverage and long-run asset volatility I We modify Black-Scholes-Merton (BSM) functions: apple LM t (D t /E t,sa,t f,t)= 4 BSM t g BSM E t /D t,1,sa,t f,t D t E t f g BSM ( ) is the inverse BSM call function. delta BSM t I f 6= 1isthedeparturefromtheMertonmodel is the BSM
Our Specification I The challenge is choosing the right functional form for LM t I Need a flexible function of leverage and long-run asset volatility I We modify Black-Scholes-Merton (BSM) functions: apple LM t (D t /E t,sa,t f,t)= 4 BSM t g BSM E t /D t,1,sa,t f,t D t E t f g BSM ( ) is the inverse BSM call function. delta BSM t I f 6= 1isthedeparturefromtheMertonmodel is the BSM
Our Specification I The challenge is choosing the right functional form for LM t I Need a flexible function of leverage and long-run asset volatility I We modify Black-Scholes-Merton (BSM) functions: apple LM t (D t /E t,sa,t f,t)= 4 BSM t g BSM E t /D t,1,sa,t f,t D t E t f g BSM ( ) is the inverse BSM call function. delta BSM t I f 6= 1isthedeparturefromtheMertonmodel is the BSM
Comment Not the BSM world! Just using BSM functions... I Goal is a flexible leverage multiplier - some function of leverage I We simply use the BSM functions to build up our LM I e.g. g BSM should not be interpreted as the correct A t /D t
What Does the Leverage Multiplier Look Like? Our Specification (s A = 0.15,r = 0.03,t = 5) 11 10 9 Leverage Multiplier 8 7 6 5 4 φ =0.5 φ =1 φ =1.5 3 2 1 0 5 10 15 20 25 30 35 40 45 50 Debt to Equity
Is it Reasonable? Monte Carlo Exercise I Is our specification plausible under SV and/or non-normality? I Simulate risk-neutral asset returns as highly asymmetric GARCH (i.e. for risk-aversion) and symmetric GARCH I Try two types of shocks: conditional normal and conditional t I Calculate simulated leverage multiplier as function of leverage
Is it Reasonable? Monte Carlo Exercise I Is our specification plausible under SV and/or non-normality? I Simulate risk-neutral asset returns as highly asymmetric GARCH (i.e. for risk-aversion) and symmetric GARCH I Try two types of shocks: conditional normal and conditional t I Calculate simulated leverage multiplier as function of leverage
Is it Reasonable? Monte Carlo Exercise I Is our specification plausible under SV and/or non-normality? I Simulate risk-neutral asset returns as highly asymmetric GARCH (i.e. for risk-aversion) and symmetric GARCH I Try two types of shocks: conditional normal and conditional t I Calculate simulated leverage multiplier as function of leverage
Is it Reasonable? Monte Carlo Exercise I Is our specification plausible under SV and/or non-normality? I Simulate risk-neutral asset returns as highly asymmetric GARCH (i.e. for risk-aversion) and symmetric GARCH I Try two types of shocks: conditional normal and conditional t I Calculate simulated leverage multiplier as function of leverage
Leverage Multiplier with Stochastic Vol/Non-Normality SV Parameters s.t. Unconditional Asset Volatility = 0.15. t = 2,r = 0 15 Leverage Multiplier 10 5 BSM GARCH-N GARCH-t GJR-N GJR-t φ =1. 21 φ =0. 97 0 0 5 10 15 20 25 30 35 40 45 50 Debt to Equity
Comments I In symmetric setting, making the tails longer via GARCH decreases LM for larger levels of debt I Moving from normal to t-dist. errors amplifies this effect I Volatility asymmetry makes asset returns negatively skewed ) shorter right tails ) increases LM I t-dist. errors shortens right tail further, so increases LM I Letting f vary in our model captures all of these cases well ) Our LM specification is reasonable in SV/non-normal environment
Comments I In symmetric setting, making the tails longer via GARCH decreases LM for larger levels of debt I Moving from normal to t-dist. errors amplifies this effect I Volatility asymmetry makes asset returns negatively skewed ) shorter right tails ) increases LM I t-dist. errors shortens right tail further, so increases LM I Letting f vary in our model captures all of these cases well ) Our LM specification is reasonable in SV/non-normal environment
Comments I In symmetric setting, making the tails longer via GARCH decreases LM for larger levels of debt I Moving from normal to t-dist. errors amplifies this effect I Volatility asymmetry makes asset returns negatively skewed ) shorter right tails ) increases LM I t-dist. errors shortens right tail further, so increases LM I Letting f vary in our model captures all of these cases well ) Our LM specification is reasonable in SV/non-normal environment
Comments I In symmetric setting, making the tails longer via GARCH decreases LM for larger levels of debt I Moving from normal to t-dist. errors amplifies this effect I Volatility asymmetry makes asset returns negatively skewed ) shorter right tails ) increases LM I t-dist. errors shortens right tail further, so increases LM I Letting f vary in our model captures all of these cases well ) Our LM specification is reasonable in SV/non-normal environment
Structural GARCH
The Full Recursive Model Structural GARCH r E,t = LM t 1 p h A,t e A,t h A,t GJR(w,a,g,b) apple LM t 1 = 4 BS t 1 g BS E t 1 /D t 1,1,sA,t f 1,t D t 1 E t 1 v " # sa,t f 1 = u t+t t Et 1 Â h A,s s=t f So parameter set is =(w,a,g,b,f)
Observations I f tunes our leverage multplier I I f = 0 is a vanilla GARCH. Leverage doesn t affect equity vol f = 1 is the classic Merton model I Half-life of GARCH process means s f A,t I We re-estimate the model using a constant sa,t f essentially unchanged 1 is basically constant 1... results I Dividing equity returns by LM t 1 gives daily asset returns I Actually asset returns? Schaefer and Strebulaev (2008)
Observations I f tunes our leverage multplier I I f = 0 is a vanilla GARCH. Leverage doesn t affect equity vol f = 1 is the classic Merton model I Half-life of GARCH process means s f A,t I We re-estimate the model using a constant sa,t f essentially unchanged 1 is basically constant 1... results I Dividing equity returns by LM t 1 gives daily asset returns I Actually asset returns? Schaefer and Strebulaev (2008)
Observations I f tunes our leverage multplier I I f = 0 is a vanilla GARCH. Leverage doesn t affect equity vol f = 1 is the classic Merton model I Half-life of GARCH process means s f A,t I We re-estimate the model using a constant sa,t f essentially unchanged 1 is basically constant 1... results I Dividing equity returns by LM t 1 gives daily asset returns I Actually asset returns? Schaefer and Strebulaev (2008)
Observations I f tunes our leverage multplier I I f = 0 is a vanilla GARCH. Leverage doesn t affect equity vol f = 1 is the classic Merton model I Half-life of GARCH process means s f A,t I We re-estimate the model using a constant sa,t f essentially unchanged 1 is basically constant 1... results I Dividing equity returns by LM t 1 gives daily asset returns I Actually asset returns? Schaefer and Strebulaev (2008)
Observations I f tunes our leverage multplier I I f = 0 is a vanilla GARCH. Leverage doesn t affect equity vol f = 1 is the classic Merton model I Half-life of GARCH process means s f A,t I We re-estimate the model using a constant sa,t f essentially unchanged 1 is basically constant 1... results I Dividing equity returns by LM t 1 gives daily asset returns I Actually asset returns? Schaefer and Strebulaev (2008)
Observations I f tunes our leverage multplier I I f = 0 is a vanilla GARCH. Leverage doesn t affect equity vol f = 1 is the classic Merton model I Half-life of GARCH process means s f A,t I We re-estimate the model using a constant sa,t f essentially unchanged 1 is basically constant 1... results I Dividing equity returns by LM t 1 gives daily asset returns I Actually asset returns? Schaefer and Strebulaev (2008)
Estimation Details I QMLE, iterate over t 2 [1,30] I Estimate both dynamic forecast and constant forecast, take best LL I 88 financial firms I D t is exponentially smoothed book value of debt I smoothing parameter = 0.01, so half-life of weights 70 days
Estimation Results
Parameter Values Cross-Sectional Summary of Estimated Parameters Parameter Median Median t-stat % with t > 1.64 w 1.0e-06 1.43 30.9 a 0.0442 3.16 85.2 g 0.0674 2.50 72.8 b 0.9094 71.21 98.8 f 0.9876 2.87 75.3 I Average t = 8.28 I Leverage matters I BSM leverage multiplier does well I Schaefer and Strebulaev (2008)
Application: The Leverage Effect
Restating the Leverage Effect I Equity volatility is negatively correlated with equity returns (i.e. volatility asymmetry) I One explanation: financial leverage, e.g. Black (1976), Christie (1982) I Second explanation: risk-premium effect, e.g. Schwert (1989) I Which one is it? e.g. Bekaert and Wu (2000)
Structural GARCH and the Leverage Effect I g parameter in GJR model is a measure of volatility asymmetry I Structural GARCH models asset returns as GJR - effectively unlevers the firm I Median g for asset returns is 0.0674 I Median g for equity returns is 0.0846 23% of so-called leverage effect comes from leverage
Structural GARCH and the Leverage Effect I g parameter in GJR model is a measure of volatility asymmetry I Structural GARCH models asset returns as GJR - effectively unlevers the firm I Median g for asset returns is 0.0674 I Median g for equity returns is 0.0846 23% of so-called leverage effect comes from leverage
Structural GARCH and the Leverage Effect I g parameter in GJR model is a measure of volatility asymmetry I Structural GARCH models asset returns as GJR - effectively unlevers the firm I Median g for asset returns is 0.0674 I Median g for equity returns is 0.0846 23% of so-called leverage effect comes from leverage
Structural GARCH and the Leverage Effect I g parameter in GJR model is a measure of volatility asymmetry I Structural GARCH models asset returns as GJR - effectively unlevers the firm I Median g for asset returns is 0.0674 I Median g for equity returns is 0.0846 23% of so-called leverage effect comes from leverage
Structural GARCH and the Leverage Effect I g parameter in GJR model is a measure of volatility asymmetry I Structural GARCH models asset returns as GJR - effectively unlevers the firm I Median g for asset returns is 0.0674 I Median g for equity returns is 0.0846 23% of so-called leverage effect comes from leverage
More Tests Higher Leverage and Higher Asymmetry Gap? I Firms with more leverage should have larger (g E,i g A,i ) I Run regression: g E,i g A,i = a + b D/E i + error i Variable Value t-stat R 2 b 0.0029 4.0471 17.8% Asset Asymmetry and Risk-Premia? I Higher market betas should mean higher asset asymmetry I Run two-stage regression: Stage 1: r A i,t = c + b A mkt,i r E mkt,t + e i,t Stage 2: g A,i = e + f b A mkt,i + e i Variable Value t-stat R 2 f 0.0287 1.98 4.95%
More Tests Higher Leverage and Higher Asymmetry Gap? I Firms with more leverage should have larger (g E,i g A,i ) I Run regression: g E,i g A,i = a + b D/E i + error i Variable Value t-stat R 2 b 0.0029 4.0471 17.8% Asset Asymmetry and Risk-Premia? I Higher market betas should mean higher asset asymmetry I Run two-stage regression: Stage 1: r A i,t = c + b A mkt,i r E mkt,t + e i,t Stage 2: g A,i = e + f b A mkt,i + e i Variable Value t-stat R 2 f 0.0287 1.98 4.95%
Application: SRISK with Leverage Amplification
SRISK I Acharya et. al (2012) and Brownlees and Engle (2012) I Three steps 1. GJR-DCC model using firm equity and market index returns 2. Expected firm equity return if market falls by 40% over 6 months LRMES 3. Combine LRMES with book value of debt to determine capital shortfall in a crisis I The crisis in this case is a 40% drop in the stock market index over 6 months
SRISK I Acharya et. al (2012) and Brownlees and Engle (2012) I Three steps 1. GJR-DCC model using firm equity and market index returns 2. Expected firm equity return if market falls by 40% over 6 months LRMES 3. Combine LRMES with book value of debt to determine capital shortfall in a crisis I The crisis in this case is a 40% drop in the stock market index over 6 months
SRISK I Acharya et. al (2012) and Brownlees and Engle (2012) I Three steps 1. GJR-DCC model using firm equity and market index returns 2. Expected firm equity return if market falls by 40% over 6 months LRMES 3. Combine LRMES with book value of debt to determine capital shortfall in a crisis I The crisis in this case is a 40% drop in the stock market index over 6 months
The Role of Leverage? Thought Experiment with Structural GARCH I Firm experiences sequence of negative equity (asset) shocks I Level of leverage goes up rapidly I Leverage multiplier increases, equity vol amplification higher I Painfully obvious in the crisis, so build into SRISK
The Role of Leverage? Thought Experiment with Structural GARCH I Firm experiences sequence of negative equity (asset) shocks I Level of leverage goes up rapidly I Leverage multiplier increases, equity vol amplification higher I Painfully obvious in the crisis, so build into SRISK
The Role of Leverage? Thought Experiment with Structural GARCH I Firm experiences sequence of negative equity (asset) shocks I Level of leverage goes up rapidly I Leverage multiplier increases, equity vol amplification higher I Painfully obvious in the crisis, so build into SRISK
The Role of Leverage? Thought Experiment with Structural GARCH I Firm experiences sequence of negative equity (asset) shocks I Level of leverage goes up rapidly I Leverage multiplier increases, equity vol amplification higher I Painfully obvious in the crisis, so build into SRISK
Asset Volatility or Leverage? The Financial Crisis Annualized Volatility Agg. Leverage Multiplier 2 1.5 1 0.5 EVW Equity Vol Index EVW Asset Vol Index 0 2007 2008 2009 2010 Date 4 3.5 3 2.5 2 2007 2008 2009 2010 Date
Asset Based Systemic Risk: Preliminary Numbers
Bank of America LRMES: Full Sample
Bank of America LRMES: 2006-2011
Bank of America Capital Shortfall: 2006-2011
Citigroup LRMES: 2006-2011
Citigroup Capital Shortfall: 2006-2011
What s Next
More Granular Debt Measurement I Right now we use book value of debt I We can decompose debt further. For example, short term vs long term: D = q 1 LT Debt + q 2 ST Debt + q 3 Non-Debt Liabilities I q 1,q 2,andq 3 are now estimated parameters
Other Applications I Endogenous Crisis Probability with Structural GARCH I Estimation of Distance to Crisis I Endogenous Capital Structure and Leverage Cycles I Counter-cyclical Capital Regulation
Appendix
Ignoring the Vega Term I Can we ignore the vega term? de t = t At da t + f E t E t A t sa,t f ds A,t f E t I Without ignoring it, the volatility of equity is: det var t E t = A 2 t dat t var t E t A t s A t nt dat +2 t var t E t E t A t 2 nt + var t dsa,t f E t var t dsa,t f dat r t,dsa,t f A t I We investigate the rough magnitudes of the additional terms
Magnitude of Volatility Terms I Use Black-Scholes vega I Use estimated asset volatility series to compute ds f A,t I Assume vol of vol and correlation with asset returns is constant I Use in-sample moments I Vol of vol = 4.9737e 4 I Plot all terms that contribute to equity variance
Decomposition of Equity Variance: JPM On average, LM term 12 times the size of vega terms 12 x 10 3 10 LM Term Cov Term Vega Term Contribution to Equity Variance 8 6 4 2 0 2 1998 2000 2002 2004 2006 2008 2010 2012 Date Back