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 University of Chicago: 11/25/2013

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

4 This Paper 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 Asset return series from observed equity series I Assets have time-varying volatility at high frequencies I Two applications: the leverage effect puzzle and SRISK

5 Theoretical Foundation

6 Structural Models of Credit I Under relatively weak assumptions on the vol process, structural models say E t = f (A t,d t,s A,t,t,r t ) I I I At = market value of assets Dt = book value of debt s A,t = stochastic asset volatility I Generic dynamics for assets and asset variance: da t = µ A (t)dt + s A,t db A (t) A t dsa,t 2 = µ v (t,s A,t )dt + s v (t,s A,t )db v (t) I B A (t) and B v (t) potentially correlated

7 Equity Return Dynamics Apply Itō s Lemma I Ignore O(dt) terms, since daily equity returns 0 I Equity returns: de t A t = t s A,t db A (t)+ n t s v (t,s A,t ) db v (t) E t E t E t 2s A,t I t = f / A t is just our familiar in option pricing I n t = f (A t,d t,s A,t,t,r)/ s A,t is vega of the option

8 The Leverage Multiplier Invert Call Option Formula I Assume f ( ) is homogenous degree 1 in first two arguments, and invertible: A t D t = g (E t /D t,1,s A,t,t,r t ) I Define the leverage multiplier as: LM (E t /D t,1,s A,t,t,r t ) I Just the %-Delta of the option f 1 (E t /D t,1,s A,t,t,r t ) t g E t /D t,1,sa,t f,t,r t Dt E t

9 Equity Returns The Leverage Multiplier I Rewrite equity returns as: de t E t = LM t s A,t db A (t)+ n t s v (t,s A,t ) db v (t) E t 2s A,t so, LM t amplifies asset shocks and volatility I Two questions: 1. What does LM t look like? 2. How much does the higher order term contribute?

10 Theoretical Results Preview 1. What does LM t look like? I Similar across many different option pricing models 2. How much do the higher order terms contribute? I I I Not much. Simple intuition... Volatility mean reversion speed typical debt maturities, so total volatility over option is basically constant We verify in paper for variety of option pricing models

11 Equity Volatility as a Function of Leverage I Vol of asset vol contributes little to equity dynamics I Thus, straightforward expressions for equity returns and instantaneous volatility: de t LM t s A,t db A (t) E t det vol t LM t s A,t E t I LM t describes how leverage interacts with equity volatility

12 What Does the Leverage Multiplier Look Like? Simple Case: Black-Scholes-Merton World (s = 0.15,r = 0.03,t = 5) Leverage Multiplier Debt to Equity

13 What Does the Leverage Multiplier Look Like? Simple Case: Black-Scholes-Merton World r = 0.03; Varying 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

14 What Does the Leverage Multiplier Look Like? Other Option Pricing Models Leverage Multiplier BSM Debt to Equity Leverage Multiplier MJD Debt to Equity Lever a ge M u l t i p l i er Heston Lever age M u l t i p l i er SVJ Debt to Equity Debt to Equity

15 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 use simple transformations of 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 inverse BSM call function. BSM t I f 6= 1isthedeparturefromtheMertonmodel is BSM delta I Dividing equity returns by LM t gives us asset returns

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

17 GARCH Option Pricing and Non-Normal Shocks Monte Carlo Exercise I What s the right leverage multiplier under GARCH 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

18 Leverage Multiplier with GARCH/Non-Normality GARCH Parameters s.t. Unconditional Asset Volatility = t = 2,r = 0 Leverage Multiplier B-S GARCH-N GARCH-t GJR-N GJR-t φ =1. 27 φ = Debt to Equity

19 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 appropriate SV/GARCH/non-normal environments

20 Structural GARCH

21 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 BSM t 1 g BSM 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)

22 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 The g parameter governs asymmetry in asset volatility

23 Estimation Details I QMLE I Iterate over t 2 [1,30] I 88 financial firms I Equity returns and balance sheet information from Bloomberg I D t is exponentially smoothed book value of debt I smoothing parameter = 0.01, so half-life of weights 70 days

24 Estimation Results

25 Dynamic Forecast vs Constant Forecast I Estimate two types of models: 1. Using a dynamic forecast for asset volatility over life of the option 2. Using unconditional volatility of GJR process I Then take the model that delivers the highest likelihood I 46/88 firms had higher LLF using constant forecast model I A few outliers where f hits lower bound (exclude from subsequent analysis): I SCHW,LM,NTRS,CME,CINF,NYB,UNH

26 Parameter Values Example: Citibank Parameter GJR Value SGARCH Value SGARCH t-stat w 2e-6 3.5e a g b f t I Optimal model: constant forecast

27 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

28 Bank of America Results f = 1.4 (t = 11.4) Annualized Volatility Assets Equity Leverage Multiplier Date

29 Lehman Results f = 1.96 (t = 20.9) Annualized Volatility Assets Equity Leverage Multiplier Date

30 American Express Results f = 0.4 (t = 3.9) Annualized Volatility Assets Equity Leverage Multiplier Date

31 Application: The Leverage Effect

32 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)

33 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

34 Higher Leverage and Higher Asymmetry? I Definitions: I I I ga,i from Structural GARCH ge,i is same parameter from GJR on equity returns i indexes firm I Firms with more leverage should have larger (g E,i g A,i ) I So, run the following regression: g E,i g A,i = a + b D/E i + error i

35 Equity Asymmetry versus Asset Asymmetry Regressions results for: g E,i g A,i = a + b D/E i + error i Variable Coefficient Value t-stat R 2 b % I Higher leverage reduces g E,i by larger amount X

36 Asset Asymmetry Another Qualitative Check I Remaining asset asymmetry due to the risk-premium story I Risk-premium story (mkt. index): I positive shock to current vol ) future vol rises ) risk-premium rises ) current price falls I Higher market betas should mean higher asset asymmetry I So, run the following 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

37 Asset Asymmetry Risk-Premium Story? I Regression results for: 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 Coefficient Value t-stat R 2 f % I Higher b A mkt,i means higher asset vol asymmetry X

38 Application: SRISK

39 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

40 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

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

42 Asset Based Systemic Risk: Preliminary Numbers

43 Bank of America LRMES: Full Sample

44 Bank of America LRMES:

45 Bank of America Capital Shortfall:

46 Citigroup LRMES:

47 Citigroup Capital Shortfall:

48 Structural GARCH vs Regular GARCH Simulation for BAC on 8/29/2008: Bankruptcy Paths Cumulative Return S GJR Cumulative Return GJR

49 Structural GARCH vs Regular GARCH Simulation for BAC on 8/29/2008: Bankruptcy Summary Statistics Structural GARCH Regular GARCH # of Bankruptcies 45 9 Avg. Time to Bankruptcy Min Time to Bankruptcy Max Time to Bankruptcy

50 What s Next

51 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