Leverage and Asymmetric

Volatility: The Firm Level

Evidence

Presented by Stefano Mazzotta

Kennesaw State University

joint work with

Jan Ericsson, McGill University

and

Xiao Huang, Kennesaw State University

Outline

Introduction

Asymmetric volatility and leverage

The panel data model

Time varying risk premium and leverage

Empirical results

Conclusion

Two explanations of asymmetric

volatility:

Leverage effect (e.g. Black (1976) Christie (1982)):

Stock price↓⇒Leverage↑⇒Risk↑⇒Volatility↑Time-varying risk premium, (e.g. French et al. (1987) Campbell and Hentschel (1992), Bekaert and Wu (2000)

Persistent volatility↑⇒Required rate of return↑⇒Stock price↓

Leverage and volatility

The traditional leverage effect argument states that as stock prices drop, increased leverage pushes up the equity volatility for a given asset volatility (business risk).

This is the spirit of the test in Christie (1982). But from a firm’s perspective, the choice of leverage and volatility is a joint one, at least in the medium to long term.

Leverage will depend on volatility and vice versa. Thus a two way relationship needs to be modeled

The leverage effect revisited

We ask what can be learnt

By reconsidering the leverage effect at

the individual firm level in the presence

of an alternative explanation (time

varying risk premia)

By allowing leverage and volatility to

influence each other in a two-way system

Captures also long term effects.

Our contributions

A panel vector autoregression (PVAR)

approach to study the dynamics of leverage,

volatility, and risk premium jointly

Control firm level heterogeneity

We find leverage effect is more important

than volatility feedback effect and it

accumulates over the time

Volatility feedback exists but does not

reinforce over the time.

The data

CRSP quarterly database

COMPUSTAT daily database merged using the identifiers CUSIP, and CNUM.

Sample period: 1971 - 2005.

116,861 firm-quarter observations

The debt is computed as the sum of total liabilities (data54) and preferred stock (data55).

Equity is computed as the product of common shares outstanding (data61) and price at the end of the quarter (data14).

The leverage variable QRit is the ratio of debt over equity at time t for firm i.

Quartiles are obtained based on the average QRit. We consider firms with QR>100 outliers and eliminate them.

Empirics

Leverage: QRit = Debt/Equity for firm i at time t

Realized volatility: σit =qPQ

h=1 r2ith

where

Q = number of days in each quarter

rit = return of stock i in excess of T-Bill

rmt = market return in excess of T-Bill

Risk premium: rit =PQ

h=1 rm,t−1,hPQh=1 r

2m,t−1,h

PQh=1[ri,t−1,hrm,t−1,h]

Stats:

QR

Panel 1: All Sample

Q 1 Q 2 Q 3 Q 4 All

Mean 0.184 0.540 1.315 6.984 2.315Median 0.130 0.431 1.095 2.448 0.708

Maximum 4.264 8.882 39.678 26377.000 26377.000Minimum 0.000 0.000 0.001 0.003 0.000Std. Dev. 0.192 0.477 1.091 235.211 118.637Skewness 3.912 3.546 4.715 99.435 197.145Kurtosis 39.669 29.989 79.637 10342.310 40659.870

Jarque-Bera 1427190 1025705 7964801 1.34E+11 8.13E+12Probability 0 0 0 0 0

Observations 24365 31613 32061 30018 118057

Panel 2: No Outliers

Q 1 Q 2 Q 3 Q 4 All

Mean 0.182 0.533 1.288 3.739 1.483Median 0.129 0.425 1.066 2.395 0.698

Maximum 4.264 8.882 39.678 98.387 98.387Minimum 0.000 0.000 0.001 0.003 0.000Std. Dev. 0.190 0.473 1.089 5.284 3.068Skewness 3.812 3.599 4.829 6.779 10.633Kurtosis 38.352 30.871 81.692 74.022 195.074

Jarque-Bera 1311161 1082949 8279855 6.50E+06 1.82E+08Probability 0 0 0 0 0

Observations 24060 31367 31614 29820 116861

-.2

-.1

.0

.1

.2

.3

.4

.5

.6

.7

1975 1980 1985 1990 1995 2000 2005

Q1

-0.4

0.0

0.4

0.8

1.2

1.6

2.0

1975 1980 1985 1990 1995 2000 2005

Q2

-2

-1

0

1

2

3

4

5

1975 1980 1985 1990 1995 2000 2005

Q3

-8

-4

0

4

8

12

16

20

1975 1980 1985 1990 1995 2000 2005

Q4

-6

-4

-2

0

2

4

6

8

10

1975 1980 1985 1990 1995 2000 2005

All

Figure 1. Cross Sectional Average QR , with +/- One Standard Deviation Bands.

Stats: Realized Volatility

No OutliersQ 1 Q 2 Q 3 Q 4 All

Mean 0.594 0.574 0.521 0.488 0.542Median 0.515 0.481 0.425 0.352 0.447

Maximum 10.360 7.399 6.728 7.893 10.360Minimum 0.000 0.000 0.000 0.000 0.000Std. Dev. 0.375 0.380 0.409 0.450 0.408Skewness 3.332 2.794 3.112 3.382 3.105Kurtosis 39.915 23.590 23.750 25.274 26.442

Jarque-Bera 1410678 594877.4 618184.5 6.73E+05 2.86E+06Probability 0 0 0 0 0Observations 24060 31367 31614 29820 116861

0.0

0.4

0.8

1.2

1.6

2.0

2.4

2.8

1975 1980 1985 1990 1995 2000 2005

Q1

0.0

0.4

0.8

1.2

1.6

2.0

2.4

2.8

1975 1980 1985 1990 1995 2000 2005

Q2

0.0

0.4

0.8

1.2

1.6

2.0

2.4

2.8

1975 1980 1985 1990 1995 2000 2005

Q3

0.0

0.4

0.8

1.2

1.6

2.0

2.4

2.8

1975 1980 1985 1990 1995 2000 2005

Q4

0.0

0.4

0.8

1.2

1.6

2.0

2.4

2.8

1975 1980 1985 1990 1995 2000 2005

All

Figure 2. Cross Sectional Average Realized Volatility, with +/- One Standard Deviation Bands.

Stats: CAPM required returns

No OutliersQ 1 Q 2 Q 3 Q 4 All

Mean 0.036 0.032 0.024 0.020 0.028Median 0.016 0.013 0.011 0.009 0.012

Maximum 5.420 7.859 7.623 4.332 7.859Minimum -3.544 -3.057 -4.630 -4.697 -4.697Std. Dev. 0.381 0.346 0.312 0.280 0.329Skewness -0.471 -0.109 -0.136 -0.365 -0.260Kurtosis 11.513 19.347 25.290 22.351 19.081

Jarque-Bera 73534.3 349308.4 654536.4 465905.9 1260513Probability 0 0 0 0 0Observations 24060 31367 31614 29820 116861

The PVAR model

wit = αi + Φwi,t−1 + εit

wit =

⎛⎝ QRitσitrit

⎞⎠ ,αi =⎛⎝ αi1

αi2αi3

⎞⎠ ,Φ =⎛⎝ φ11 φ12 φ13

φ21 φ22 φ23φ31 φ32 φ33

⎞⎠

εi,ti.i.d.v (0,Ωε) and Ωε =

⎛⎝ σε11σε21 σε22σε31 σε32 σε11

⎞⎠i = 1, · · · , N and t = 1, · · · , T.

Pesaran’s QMLE Estimation

The log likelihood function is

l(θ) = −mN(T − 1)2

log (2π)− N2log |Σ∆η|− 1

2

NXi=1

∆η0iΣ−1∆η∆ηi

where θ = (φ11,φ12,φ13,φ21,φ22,φ23,φ32,φ31,φ33,ω11,ω21,ω31,ω22,ω32,ω33)0

Ωε =

⎛⎝ ω211 ω11ω21 ω11ω31ω11ω21 ω211 + ω222 ω21ω31 + ω22ω32ω11ω31 ω21ω31 + ω22ω32 ω211 + ω222 + ω233

⎞⎠

∆wit = Φ∆wi,t−1 +∆εit

Take first differences

Trivariate PVAR∆QRi,t = φ11∆QRi,t−1 + φ12∆σi,t−1 + φ13∆ri,t−1 +∆εit1∆σit = φ21∆QRi,t−1 + φ22∆σi,t−1 + φ23∆ri,t−1 +∆εit2∆ri,t = φ31∆QRi,t−1 + φ32∆σi,t−1 + φ33∆ri,t−1 +∆εit3

Trivariate Panel VAR. No outliers.

Q 1 t-stats Q 2 t-stats Q 3 t-stats Q 4 t-stats All t-stats

φ1,1 0.8459 51.47 0.8329 80.65 0.8601 56.13 0.8830 36.35 0.8909 40.28φ1,2 0.0083 2.35 0.0377 2.10 0.1363 4.77 0.6157 5.34 0.2079 6.07φ1,3 -0.0085 -0.88 -0.1229 -0.15 -0.2469 -1.79 -1.1362 -0.63 -0.2805 -3.63φ2,1 0.1291 4.06 0.0695 8.70 0.0442 11.73 0.0127 8.59 0.0171 11.67φ2,2 0.3543 9.01 0.3916 20.07 0.4126 20.12 0.4931 16.53 0.4252 30.61φ2,3 0.0411 1.64 -0.0571 -0.37 -0.0169 -7.54 -0.0594 -0.16 -0.0228 -1.36φ3,1 -0.0002 -0.06 0.0006 0.27 0.0013 10.71 0.0004 5.61 0.0005 2.30φ3,2 0.0038 1.46 0.0016 0.24 0.0004 1.20 0.0052 8.29 0.0028 2.55φ3,3 -0.0370 -1.14 -0.0346 -1.55 -0.0302 -1.83 -0.0276 -5.29 -0.0331 -11.75ω1,1 0.1036 23.18 0.2732 29.66 0.6349 18.13 2.7925 17.46 -1.4602 -18.26ω2,1 0.0164 3.11 0.0248 3.48 0.0246 6.82 0.0395 7.14 -0.0241 -8.86ω3,1 -0.0068 -15.08 -0.0076 -12.68 -0.0077 -28.54 -0.0061 -27.78 0.0044 9.43ω2,2 0.2932 27.35 0.2871 34.90 0.2916 36.50 0.2974 29.73 0.2936 63.79ω3,2 -0.0088 -18.40 -0.0056 -13.91 -0.0033 -16.37 -0.0039 -16.36 -0.0054 -8.35ω3,3 -0.0955 -27.09 0.0865 33.29 -0.0781 -31.94 -0.0697 -30.14 -0.0825 -95.45

Main results

• φ21 gives a much larger leverage effect compared to estimates in previousresearch.

• φ21 and φ12 show leverage effect accumulates.

• φ32 confirms volatility feedback

• φ32 and φ23 show volatility feedback does not accumulate.

Wald Test

Ho: φ13,φ23,φ31,φ32,φ33 are jointly = 0

Quartile Q 1 Q 2 Q 3 Q4 All

Wald statistics 342.155 295.482 41.270 326.315 908.035p-val 0.000 0.000 0.000 0.000 0.000

Contemporaneous Correlations

Q1 t-stats Q2 t-stats

ρ(QR,σ) 0.056 554.74 0.086 487.98ρ(r,σ) -0.071 -788.15 -0.088 -812.20

ρ(QR, r) -0.096 -733.27 -0.071 -251.86

Q3 t-stats Q4 t-stats All t-stats

ρ(QR,σ) 0.084 605.24 0.132 409.50 0.082 981.78ρ(r,σ) -0.098 -695.05 -0.087 -508.72 -0.053 -1641.44

ρ(QR, r) -0.050 -195.49 -0.066 -188.75 -0.070 -1094.12

Contemporanous Correlation

• ρ (QR,σ) > 0 ⇒contemporaneous leverage effect• ρ (r,σ) < 0 ⇒ no contemporaneous volatility feedback

Impulse response function

• A shock from QR affects QR, σ and r persistently.

• Volatility shock affects leverage when leverage ratio is high.• Shock from required return affects all variables negatively.

2 4 6 8 10 120

0.02

0.04

0.06

0.08

0.1

2 4 6 8 10 120

0.05

0.1

0.15

0.2

2 4 6 8 10 12-4

-2

0

2

4x 10-3

2 4 6 8 10 120

0.1

0.2

0.3

0.4

2 4 6 8 10 120

0.05

0.1

0.15

0.2

2 4 6 8 10 12-15

-10

-5

0

5x 10-3

2 4 6 8 10 120

0.2

0.4

0.6

0.8

2 4 6 8 10 120

0.05

0.1

0.15

0.2

2 4 6 8 10 12-0.02

-0.01

0

0.01

2 4 6 8 10 120

1

2

3

2 4 6 8 10 120

0.1

0.2

0.3

0.4

2 4 6 8 10 12-0.1

-0.05

0

0.05

0.1

Figure 1 - Impulse response: Response to one std shock. Legend: * blue = QR; + green = σ ; o red = E[r]

Q 1

Q 2

Q 3

Q 4

Shock to QR Shock to σ Shock to E[r]

0 2 4 6 8 10 120

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Q10 2 4 6 8 10 12

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Q2

0 2 4 6 8 10 120

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Q3

Cumulative Effects of one sdt shock to QR on σ

0 2 4 6 8 10 120.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

Q4

0 2 4 6 8 10 121.4

1.5

1.6

1.7

1.8

1.9

2x 10-3

Q10 2 4 6 8 10 12

6

7

8

9

10x 10-4

Q2

0 2 4 6 8 10 122

3

4

5

6

7

8

9

10x 10-4

Q3

Figure 4 Cumulative Effects of an Orthogonal Shock ( 1 s.d. of σ ) to σ on rbar

0 2 4 6 8 10 121.5

2

2.5

3

3.5

4

4.5

5x 10-3

Q4

Concluding remarks

We reconfirm the relationship between equity volatility and the debt ratio presented in Christie (1982) across the four leverage quartiles.

Our main finding is that a dynamic set up is important to capture the cumulative leverage effect.

Financial leverage is an economically more significant determinant of equity volatilities than previous work has documented, and its effect accumulates over time.

The accumulation of the leverage effect over time renders it at least up to five times larger than previously thought.

Our study suggests that past results may be due to not fully allowing for the endogenous nature of the relationship between capital structure and business risk.