Structural GARCH: The Volatility-Leverage Connection

Transcription

1 Structural GARCH: The Volatility-Leverage Connection Robert Engle 1 Emil Siriwardane 1 1 NYU Stern School of Business MFM Macroeconomic Fragility Fall 2013 Meeting

2 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

3 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

4 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

5 BAC Leverage and Realized Volatility 1-Month Realized (Annualized) Volatility Debt to Equity Date

6 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

7 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

8 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

9 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

10 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

11 Theoretical Foundation

12 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

13 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

14 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

15 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

16 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

17 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)

18 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)

19 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)

20 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

21 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

22 What Does the Leverage Multiplier Look Like? Simple Case: Black-Scholes-Merton World r = 0.03; Varying Asset s, t 9 8 Leverage Multiplier σ =0.1, τ =5 σ =0.2, τ =5 2 σ =0.1, τ =10 σ =0.2, τ = Debt to Equity

23 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

24 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

25 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

26 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

27 What Does the Leverage Multiplier Look Like? Our Specification (s A = 0.15,r = 0.03,t = 5) Leverage Multiplier φ =0.5 φ =1 φ = Debt to Equity

28 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

29 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

30 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

31 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

32 Leverage Multiplier with Stochastic Vol/Non-Normality SV Parameters s.t. Unconditional Asset Volatility = t = 2,r = 0 15 Leverage Multiplier 10 5 BSM GARCH-N GARCH-t GJR-N GJR-t φ =1. 21 φ = Debt to Equity

33 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

34 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

35 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

36 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

37 Structural GARCH

38 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)

39 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)

40 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)

41 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)

42 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)

43 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)

44 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)

45 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

46 Estimation Results

47 Parameter Values Cross-Sectional Summary of Estimated Parameters Parameter Median Median t-stat % with t > 1.64 w 1.0e a g b f I Average t = 8.28 I Leverage matters I BSM leverage multiplier does well I Schaefer and Strebulaev (2008)

48 Application: The Leverage Effect

49 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)

50 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 I Median g for equity returns is % of so-called leverage effect comes from leverage

51 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 I Median g for equity returns is % of so-called leverage effect comes from leverage

52 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 I Median g for equity returns is % of so-called leverage effect comes from leverage

53 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 I Median g for equity returns is % of so-called leverage effect comes from leverage

54 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 I Median g for equity returns is % of so-called leverage effect comes from leverage

55 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 % 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 %

56 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 % 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 %

57 Application: SRISK with Leverage Amplification

58 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

59 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

60 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

61 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

62 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

63 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

64 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

65 Asset Volatility or Leverage? The Financial Crisis Annualized Volatility Agg. Leverage Multiplier EVW Equity Vol Index EVW Asset Vol Index Date Date

66 Asset Based Systemic Risk: Preliminary Numbers

67 Bank of America LRMES: Full Sample

68 Bank of America LRMES:

69 Bank of America Capital Shortfall:

70 Citigroup LRMES:

71 Citigroup Capital Shortfall:

72 What s Next

73 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

74 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

75 Appendix

76 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

77 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 = e 4 I Plot all terms that contribute to equity variance

78 Decomposition of Equity Variance: JPM On average, LM term 12 times the size of vega terms 12 x LM Term Cov Term Vega Term Contribution to Equity Variance Date Back