Economic Uncertainty, Disagreement, and Credit Markets Andrea Buraschi

Fabio Trojani

Andrea Vedolin

Imperial College London

University of Lugano

University of Lugano

27 August 2009

ȱ

c (2009) Buraschi, Trojani, Vedolin – 1 ⃝

⊳ Introduction Time Series Cross Section The Model Empirical Analysis Conclusion Appendix

Introduction

ȱ

c (2009) Buraschi, Trojani, Vedolin – 2 ⃝

Uncertainty-DiB: Time Series Economic Uncertainty usually surges in times of economic or financial crises.

Introduction Time Series Cross Section

⊳

The Model Empirical Analysis

Common DiB versus Default Spread 0.4

1.4 Speculation on European Currencies

Appendix

Common DiB / VIX

0.2

Asian Crisis & Russian Default

1.2

0

1

−0.2

0.8

−0.4 S&L Crisis −0.6 1990

ȱ

Credit Crunch

Mexican Crisis

1992

1994

0.6

DotCom Bubble

1996

1998

2000

Default Spread

Conclusion

NBER Recession Financial Crisis Common DiB VIX

2002

2004

2006

0.4 2008

c (2009) Buraschi, Trojani, Vedolin – 3 ⃝

Event Study: DiB & Default Spread Around Crises Average DiB and Default Spread (Baa - Aaa) before, at, and after a financial or economic crisis.

Introduction Time Series Cross Section

⊳

The Model Empirical Analysis Conclusion

Default Spread

Common DiB

0.2

1

Appendix 0.95 0.15 0.9

0.85

0.1

0.8 0.05

0.75

0.7 0 0.65

−0.05

ȱ

− 12

−6

0

+6

+ 12

0.6

− 12

−6

0

+6

+ 12

c (2009) Buraschi, Trojani, Vedolin – 4 ⃝

Uncertainty-DiB: Cross Section Asset Swap Spread and Uncertainty-DiB for six different sectors. ⊳

Technology

0.2

0 1995

2000

Financials

800 Asset Swap Spread

2005

0 2010

0.3

0 1995

2000

2005

0.125

2000

0 2010

400

1

200

0.5

0 1995

2000

ȱ

2005

0 2010

0 2010

400

1.5 βˆ = 0.48 t-stat = [6.52] R2 = 0.23

1 DiB

Asset Swap Spread

600

DiB

Asset Swap Spread

400

2000

2005

Energy 0.5

βˆ = 0.73 t-stat = [12.66] R2 = 0.53

0 1995

0 2010 1.5

βˆ = 0.58 t-stat = [8.35] R2 = 0.33

Capital Goods 800

2005

Banks

600

βˆ = 0.44 t-stat = [5.74] R2 = 0.19

400

300

0 1995

0.6

0.25

βˆ = 0.52 t-stat = [7.25] R2 = 0.27

DiB

Appendix

400

Asset Swap Spread

Conclusion

βˆ = 0.68 t-stat = [11.23] R2 = 0.47

DiB

Empirical Analysis

Services

600

DiB

Asset Swap Spread

The Model

0.4 Asset Swap Spread

800

DiB

Introduction Time Series Cross Section

200

0.5

0 1995

2000

2005

0 2010

c (2009) Buraschi, Trojani, Vedolin – 5 ⃝

Credit Spreads, DiB, and Exposure to Common DiB Common DiB versus Idiosyncratic DiB

Introduction Time Series Cross Section

1.4

⊳

The Model

1.2

Empirical Analysis Conclusion

Idiosyncratic DiB

Appendix

Technology Services Financials Banks Consumer Cyclical Capital Goods Energy

1

0.8

0.6

0.4

0.2

0 0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Common DiB

ȱ

c (2009) Buraschi, Trojani, Vedolin – 6 ⃝

Credit Spreads, DiB, and Exposure to Common DiB Credit Spreads versus Disagreement (Yearly Averages)

Introduction Time Series Cross Section

0.8

⊳

Technology Services

The Model

0.7

Empirical Analysis

Energy: 2008

Financials Banks Consumer Cyclical

Conclusion

Capital Goods

0.6

Disagreement

Appendix

Energy

Banks: 2008

0.5 Consumer Cyclical: 2008

0.4 0.3 Technology: 2001

0.2 0.1

50

100

150

200

250

300

350

Credit Spread

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c (2009) Buraschi, Trojani, Vedolin – 7 ⃝

Introduction

⊳ The Model DiB Intuition Definition Equilibrium Pricing Firm Value Credit Spreads Price of Equity

The Model

Empirical Analysis Conclusion Appendix

ȱ

c (2009) Buraschi, Trojani, Vedolin – 8 ⃝

The Economy’s State Variables Observed state space: Firm’s exogenous cash flows 𝐴(𝑡), with dynamics:

Introduction The Model DiB Intuition Definition Equilibrium Pricing Firm Value Credit Spreads Price of Equity

𝑑 log 𝐴(𝑡) = 𝜇𝐴 (𝑡)𝑑𝑡 + 𝜎𝐴 𝑑𝑊𝐴 (𝑡), 𝑑𝜇𝐴 (𝑡) = (𝑎0𝐴 + 𝑎1𝐴 𝜇𝐴 (𝑡))𝑑𝑡 + 𝑒𝐴 𝑑𝑊𝜇𝐴 (𝑡), and a signal 𝑧(𝑡), with dynamics: 𝑑𝑧(𝑡) 𝑑𝜇𝑧 (𝑡)

Empirical Analysis Conclusion

= (𝛼𝜇𝐴 (𝑡) + 𝛽𝜇𝑧 (𝑡))𝑑𝑡 + 𝜎𝑧 𝑑𝑊𝑧 (𝑡), = (𝑎0𝑧 + 𝑎1𝑧 𝜇𝑧 (𝑡))𝑑𝑡 + 𝑒𝑧 𝑑𝑊𝜇𝑧 (𝑡).

Appendix

The growth rate of the firm cash flows and the signal are unobserved by agents in the economy. ⇛ The subjective expected growth rate of cash flows and signals is: ( ) 𝑖 𝑖 𝑖 ′ 𝑖 ′ 𝑌 𝑚 (𝑡) := (𝑚𝐴 (𝑡), 𝑚𝑧 (𝑡)) := 𝐸 (𝜇𝐴 (𝑡), 𝜇𝑧 (𝑡)) ∣ℱ𝑡

where ℱ𝑡𝑌 := ℱ𝑡𝐴,𝑧 . ⇛ Agents might interpret the same information about 𝐴(𝑡) and 𝑧(𝑡) differently. ȱ

c (2009) Buraschi, Trojani, Vedolin – 9 ⃝

Investors’ Disagreement Introduction The Model DiB Intuition Definition Equilibrium Pricing Firm Value Credit Spreads Price of Equity

We define the Disagreement Process as follows:

Empirical Analysis Conclusion

Ψ(𝑡) :=

Appendix

(

Ψ𝐴 (𝑡) Ψ𝑧 (𝑡)

)

=

(

(𝑚1𝐴 (𝑡) − 𝑚2𝐴 (𝑡))/𝜎𝐴 (𝑚1𝑧 (𝑡) − 𝑚2𝑧 (𝑡))/𝜎𝑧

) .

(1)

Ψ𝐴 (𝑡) (Ψ𝑧 (𝑡)) measures the disagreement about the expected growth rate of firm cash flows (signals). Both components are normalized by their risk.

ȱ

c (2009) Buraschi, Trojani, Vedolin – 10 ⃝

Economic Uncertainty and Difference in Beliefs Rational learning, but with different agents. Introduction

Kalman-Bucy solution: Let 𝑏1 = 𝑑𝑖𝑎𝑔(𝜎𝜇1 𝐴 , 𝜎𝜇1 𝑧 ), defined as Economic Uncertainty: In our case subjective and agent specific. ( ) −1 1 1 𝑑𝑚 (𝑡) = 𝑎0 + 𝑎1 𝑚 (𝑡) 𝑑𝑡 + 𝛾 1 (𝑡)𝐴′ (𝐵𝐵 ′ ) 𝑑𝑊𝑌1 (𝑡),

The Model DiB Intuition Definition Equilibrium Pricing Firm Value Credit Spreads Price of Equity

⊳

1

1

𝑑𝛾 (𝑡)/𝑑𝑡 = 𝑎1 𝛾 (𝑡) + 𝛾

1

(𝑡)𝑎′1

Empirical Analysis

−1

Define the DiB as Ψ(𝑡) = 𝐵 ⎛

Conclusion Appendix

𝑑Ψ(𝑡)

1 1′

+ 𝑏 𝑏 − 𝛾 1 (𝑡)𝐴′ (𝐵𝐵 ′ )−1 𝐴𝛾 1 (𝑡).

( 1 ) 2 𝑚 (𝑡) − 𝑚 (𝑡) .

⎞

⎟ ⎜ = 𝐵 −1 ⎝𝑎1 𝐵 + 𝛾 2 (𝑡) 𝐴′ 𝐵 −1 ⎠ Ψ(𝑡)𝑑𝑡 | {z } Uncertainty

+𝐵

−1

( 1 ) ′ −1 2 𝛾 (𝑡) − 𝛾 (𝑡) 𝐴 𝐵 𝑑𝑊𝑌1 (𝑡). | {z } Diff. in Uncertainty

⇒ Uncertainty affects the average DiB Ψ.

ȱ

c (2009) Buraschi, Trojani, Vedolin – 11 ⃝

Difference in Beliefs & Uncertainty First Moment of Steady-State Distribution of Ψ

Second Moment of Steady-State Distribution of Ψ

0.376

0.35

0.374

0.3 0.25

0.372

0.2 0.37 0.15 0.368

0.1

0.366

0.05

0.364 0.1

0 0.1 0.08

0.01 0.06

0.01 0.06

0.04

σ ¯µz

ȱ

0.08

0.005 0.02 0

0

∆σµz

0.04

σ ¯µz

0.005 0.02 0

0

∆σµz

c (2009) Buraschi, Trojani, Vedolin – 12 ⃝

Main Intuition In a standard economy with common beliefs 𝑑𝑄 we have: ∫ ∫ sup 𝑈 (𝑐1 (𝑡))𝑑𝑄 + 𝑈 (𝑐2 (𝑡))𝑑𝑄.

Introduction The Model DiB Intuition Definition Equilibrium Pricing Firm Value Credit Spreads Price of Equity

⊳

𝑐1 +𝑐2 =𝐴

Optimal allocation condition implies that: 𝑈 ′ (𝑐1 (𝑡)) = 𝑈 ′ (𝑐2 (𝑡)).

Empirical Analysis Conclusion

However, if agents disagree , then

Appendix

] ∫ [ 2 𝑑𝑄 sup 𝑈 (𝑐1 (𝑡)) + 𝑈 (𝑐2 (𝑡)) 1 𝑑𝑄1 𝑑𝑄 𝑐1 +𝑐2 =𝐴 which implies that 𝑈 ′ (𝑐1 (𝑡)) = 𝜆(𝑡)𝑈 ′ (𝑐2 (𝑡)), where 𝜆(𝑡) is a function of difference in beliefs .

ȱ

c (2009) Buraschi, Trojani, Vedolin – 13 ⃝

Asset Pricing Implications Aggregation yields:

Introduction The Model DiB Intuition Definition Equilibrium Pricing Firm Value Credit Spreads Price of Equity

𝑊 = 𝑈 (𝑐1 (𝑡)) + 𝜆(𝑡) 𝑈 (𝑐2 (𝑡)). |{z}

⊳

stochastic

Changes in difference in beliefs have real effects :

Empirical Analysis Conclusion

𝜉(𝑡) is affected by 𝜆(𝑡) =

□

Implications for Hansen-Jagannathan bounds: If 𝜆(𝑡) is volatile, asset prices can be violated.

□

Agents have different beliefs, thus different efficient frontiers: In general, the CAPM will be violated.

□

We are interested about the implications for structural models.

Appendix

ȱ

𝑑𝑄2 𝑑𝑄1 :

□

Uncertainty is priced.

c (2009) Buraschi, Trojani, Vedolin – 14 ⃝

Financial Markets and Equilibrium Preferences: Two groups of investors with life-time utility: (∫ ∞ ) 1−𝛾 𝑐𝑖 (𝑡) 𝑉 𝑖 = sup 𝐸 𝑖 𝑒−𝜌𝑡 𝑑𝑡 ℱ0𝑌 , 1−𝛾 𝑐𝑖 0

Introduction The Model DiB Intuition Definition Equilibrium Pricing Firm Value Credit Spreads Price of Equity

⊳

(2)

where 𝑐𝑖 (𝑡) is the consumption of agent 𝑖 = 1, 2 and 𝜌 ≥ 0 is the time preference rate. Financial market: An incomplete market, completed by the firm’s capital structure:

Empirical Analysis Conclusion Appendix

□

A risk-free bond and a European stock option (in zero net supply)

□

A senior, a junior corporate bond and a stock (in positive supply).

Definition 1 An equilibrium consists of a unique SDF s.t. 1. given equilibrium prices, all agents in the economy solve the optimization problem (2), subject to their budget constraint. 2. Good and financial markets clear. ȱ

c (2009) Buraschi, Trojani, Vedolin – 15 ⃝

Equilibrium: State Price Densities and Optimal Consumption Proposition 1 In equilibrium, the individual state price densities of agent one and two are:

Introduction The Model DiB Intuition Definition Equilibrium Pricing Firm Value Credit Spreads Price of Equity

⊳

Empirical Analysis

𝜉 1 (𝑡)

=

𝜉 2 (𝑡)

=

( )𝛾 𝑒−𝜌𝑡 𝐴(𝑡)−𝛾 1 + 𝜆(𝑡)1/𝛾 , 𝑦1 ( )𝛾 𝑒−𝜌𝑡 𝐴(𝑡)−𝛾 1 + 𝜆(𝑡)1/𝛾 𝜆(𝑡)−1 , 𝑦2

where the weighting process 𝜆(𝑡) = 𝑦1 𝜉 1 (𝑡)/(𝑦2 𝜉 2 (𝑡)) follows the dynamics: ( ) 𝑑𝜆(𝑡) 𝜎𝐴 = −Ψ𝐴 (𝑡)𝑑𝑊𝐴1 (𝑡) − 𝛼Ψ𝐴 (𝑡) + 𝛽Ψ𝑧 (𝑡) 𝑑𝑊𝑧1 (𝑡) . 𝜆(𝑡) 𝜎𝑧

Conclusion Appendix

(3)

The individual optimal consumption policies are: ( )−1 1/𝛾 𝑐1 (𝑡) = 𝐴(𝑡) 1 + 𝜆(𝑡) ,

𝑐2 (𝑡) = 𝐴(𝑡)𝜆(𝑡)

1/𝛾

( )−1 1/𝛾 1 + 𝜆(𝑡) .

↶ ȱ

c (2009) Buraschi, Trojani, Vedolin – 16 ⃝

Pricing of Financial Assets Introduction

Equilibrium firm value: ↷

The Model DiB Intuition Definition Equilibrium Pricing Firm Value Credit Spreads Price of Equity

𝑉 (𝑡) = 𝐴(𝑡)𝐸𝑡1

⊳

(∫

∞

𝑒−𝜌(𝑢−𝑡) 𝑡

1

)

𝜉 (𝑢) 𝐴(𝑢) 𝑑𝑢 , 1 𝜉 (𝑡) 𝐴(𝑡)

Price of senior bond: (

𝐵 𝑠 (𝑡, 𝑇 ) = 𝐾1 𝐵(𝑡, 𝑇 ) − 𝐸𝑡1 𝑒−𝜌(𝑇 −𝑡)

Empirical Analysis Conclusion

1

)

𝜉 (𝑇 ) + (𝐾 − 𝑉 (𝑇 )) , 1 1 𝜉 (𝑡)

Appendix

Price of junior bond (mezzanine): 𝐵 𝑗 (𝑡, 𝑇 ) = 𝐶𝑎𝑙𝑙(𝑉 ; 𝐾1 ) − 𝐶𝑎𝑙𝑙(𝑉 ; 𝐾1 + 𝐾2 ), Price of equity: 𝑆(𝑡) = 𝑉 (𝑡) − 𝐵 𝑠 (𝑡, 𝑇 ) − 𝐵 𝑗 (𝑡, 𝑇 ) ,

ȱ

c (2009) Buraschi, Trojani, Vedolin – 17 ⃝

Firm Value and Firm Value Volatility Firm Value

Firm Value Volatility Firm Value Volatility

Firm Value

Firm Value Volatility

Firm Value

160

159

158 0

Risk-neutral Skewness

Risk-Neutral Skewness (Firm Value Returns)

161

0.13

0.10

0.07 0

0.2

0.2 0.05

0.05 0.1

Ψz

0.05 0.2 0

𝜉𝑖 (𝑡) =

0.1

0.1

Ψz ΨA

1 −𝜌𝑡 𝐴(𝑡)−𝛾 𝑠𝑖 (𝑡)−𝛾 , 𝑦𝑖 𝑒

0 −0.1 −0.2 −0.3 −0.4 −0.5 0

0.2 0.05

0.15 0.1

0.15

0.15

0.15

Firm Value Skewness

0.1 0.15

Ψz

0.1 0.15

0.05 0.2 0

ΨA

0.05 0.2 0

ΨA

this is the stochastic discount factor for the optimist.

In good (bad) cash flow states the marginal utility of the optimist (pessimist) is lower, the present value is lower, which implies a lower equilibrium firm value.

ȱ

c (2009) Buraschi, Trojani, Vedolin – 18 ⃝

Corporate Credit Spreads and Equity Volatility Credit Spreads

Equity Volatility Equity Volatility (high Leverage)

1.5 1.4

0.30

Equity Volatility

Senior Bond Credit Spread (in %)

Senior Bond Credit Spread (high Leverage)

1.3 1.2 1.6 1.5 1.4 1.3 1.2 0

0.2 0.05

0.15

0 0

0.2 0.05

0.15

0.15 0.1

Ψz

0.1

0.1

0.1 0.15

Ψz

0.05 0.2

0

ΨA

0.15

0.05 0.2

0

ΨA

Very important: Credit spreads and implied option volatility (endogenously) are positively correlated (Campbell and Taksler (2003), Cremers, Driessen, and Maenhout (2007))

ȱ

c (2009) Buraschi, Trojani, Vedolin – 19 ⃝

Equity = V - Debt 5 ZCB Firm Value Delta Vega Equity Skewness

Introduction 4

The Model DiB Intuition Definition Equilibrium Pricing Firm Value Credit Spreads Price of Equity

3

∆S/∆Ψ

2

⊳

1

0

Empirical Analysis −1

Conclusion Appendix

−2 0

0.0625

0.125

0.1875

0.25

0.3125

0.375

0.4375

0.5

Leverage

Delta: +

Vega: +

Skewness: +

z }| { z }| { z }| { [ 𝑑𝑉 𝑑𝑍𝐶𝐵 𝑑𝑆𝑘 𝑉 ] 𝑑𝑆 𝑑𝑃 𝑑𝑉 𝑑𝑃 𝑑𝜎𝑉 𝑑𝑃 = −𝐾1 ⋅ + ⋅ + ⋅ + ⋅ . 𝑑Ψ 𝑑Ψ 𝑑Ψ 𝑑𝑉 𝑑Ψ 𝑑𝜎𝑉 𝑑Ψ 𝑑𝑆𝑘𝑉 𝑑Ψ

+/−

−

−

−

−

+

+

−

−

𝑉 is monotonically decreasing in 𝜓(𝑡). Put option effect dominates for low leverage because of skewness effect! ȱ

c (2009) Buraschi, Trojani, Vedolin – 20 ⃝

Equity Price and Risk-neutral Skewness Equity Price (high Leverage)

Equity Price (low Leverage)

Introduction The Model DiB Intuition Definition Equilibrium Pricing Firm Value Credit Spreads Price of Equity

120

82.75

Equity Price

Equity Price

83

82.5 82.25 82

118

117

0.2

0

⊳

119

Ψz

0.2

Ψz

ΨA

0

0.05

0.15 0.2

0

ΨA

Risk-Neutral Skewness (Equity Returns), low Leverage

0

Risk-neutral Skewness

Risk-neutral Skewness

Risk-neutral Skewness (Equity Returns), high Leverage

−0.25 −0.5 −0.75 −1 0 0.05

0.2

1 0.75 0.5 0.25 0 0

0.2 0.05

0.15

0.15

0.1

0.1

0.1

Ψz

0.15

0.05 0.2

ȱ

0.1

0.1 0.05

0.15

Appendix

0.2 0.15

0.05

0.1

0.1

Empirical Analysis Conclusion

116 0

0.15

0.05

0

ΨA

Ψz

0.1 0.15

0.05 0.2

0

ΨA

c (2009) Buraschi, Trojani, Vedolin – 21 ⃝

Introduction The Model

⊳ Empirical Analysis Results Mini-Case No-Arbitrage Violations Logit Stock Returns Conclusion

Empirical Analysis

Appendix

ȱ

c (2009) Buraschi, Trojani, Vedolin – 22 ⃝

Empirical Results: Credit Spreads

Constant Individual DiB Common DiB Implied Volatility Implied Volatility Skew

(1)

(2)

(3)

(4)

(5)

(6)

0.756★★★ [8.45] 0.976★★★ [8.56] 0.718★★★ [3.85] 0.713 [1.43] −0.256★ [-1.65]

0.613★★★ [6.38] 0.837★★ [8.93] 0.476★★★ [4.09]

0.594★★★ [7.39] 0.837★★★ [8.31] 0.521★★★ [5.10]

0.420★★★ [5.29] 0.880★★★ [7.73] 0.469★★★ [5.28]

0.639★★★ [6.38] 0.820★★★ [7.27] 0.560★★★ [5.83] 0.672★ [1.77] −0.231 [-1.38] −1.746★ [-1.67] −0.429★★★ [-3.43] −0.561★★★ [-3.56] −1.939 [-1.63] 0.564★★★ [3.34] 0.035★★ [2.45] −0.102★★ [-1.99] -0.541 [-1.01] 0.321★★ [1.99] 0.491★ [1.67] 0.391★★ [2.30] 0.80

0.508★★★ [5.99] 0.735★★★ [7.11] 0.489★★★ [5.00] 0.523 [1.43]

Slope of Term Structure Risk-free Rate Non-Farm Payroll NBER Dummy

−0.412★ [-1.71] −0.532★★★ [-4.35] −0.632★★★ [-4.33] −2.545★ [-1.73]

Leverage Firm Size

0.375★★★ [4.13] 0.045★★★ [3.40]

Liquidity 𝑅𝑚 − 𝑅𝑓 SMB HML Earnings Volatility Adjusted 𝑅2

ȱ

0.54

0.68

0.70

−0.124★★ [-2.51] −0.632 [-1.02] 0.427★★★ [3.22] 0.321★ [1.77] 0.400★★ [2.47] 0.68

−1.652★ [-1.87] −0.493★★★ [-3.84] −0.461★★★ [-3.98] 0.488★★★ [3.43] 0.027★★ [2.31] −0.120★★ [-2.01] 0.308★ [1.76] 0.410★ [1.64] 0.418★★ [2.34] 0.80

c (2009) Buraschi, Trojani, Vedolin – 23 ⃝

Money. So they say. Is the Root of All Evil Today. Pink Floyd Financials & Banks: Asset Swap Spreads, Stock Index and DiBs 600 Bank DiB

1

400

Asset Swap Spread Financials Asset Swap Spread Banks

DiBs

Asset Swap Spread / Stock Index

Stock Index

Financial DiB

0.5 200

0

05/03/06

11/19/06

July 07: Bear Stearns unwinds two hedge funds Ben Bernanke warns that U.S. sub-primecrisis could cost up to USD 100 bn Aug 07: BNP Paribas tells it cannot value two of its structured product funds ECB pumps 95bn euro into banking sector Fed cuts rate and warns credit crunch could be a risk to economic growth

06/07/07

12/24/07

0 12/31/08

Mar 08: Collapse of Bear & Stearns DJIA hits lowest level since 2006 July 08: IndyMac fails (second largest financial bankruptcy in US history). Fannie Mae and Freddie Mac share prices drop 50%. Emergency intervention to help Fannie Mae and Freddie Mac.

Sept 07: Northern Rock asks emergency financial support from the BoE. A day later depositors withdraw GBP 1 bn: Biggest run on a British bank for more than a century

ȱ

07/11/08

Sept 08: Collapse of Lehman Brothers Federal Takeover of Fannie Mae and Freddie Mac Rescue of AIG Nov 08: Remaining investment banks convert to bank holding companies U.S. government rescues Citigroup Emergency Economic Stabilitzation Act signed into law

c (2009) Buraschi, Trojani, Vedolin – 24 ⃝

Predictability Across Sectors Horizon 0 Information Technology: Coeff 0.687★★★ t-stat [8.15] 2 𝑅 0.47

1

3

6

9

12

0.742★★★ [6.75] 0.54

0.788★★★ [5.56] 0.53

0.799★★★ [4.37] 0.47

0.741★★★ [4.77] 0.37

0.735★★★ [3.85] 0.36

Services: Coeff 0.521★★★ t-stat [4.13] 2 𝑅 0.27

0.544★★★ [3.12] 0.28

0.578★★ [2.49] 0.26

0.504★ [1.85] 0.16

0.491★ [1.71] 0.13

0.685★ [1.83] 0.24

Financials: Coeff 0.435★★★ t-stat [3.20] 𝑅2 0.19

0.405★★★ [2.78] 0.16

0.349★★★ [2.84] 0.12

0.371★★★ [6.77] 0.13

0.777★★★ [5.37] 0.58

0.934★★ [2.23] 0.39

Banks: Coeff t-stat 𝑅2

0.575★★★ [4.82] 0.33

0.658★★★ [3.88] 0.43

0.853★★★ [4.67] 0.66

1.106★★★ [9.09] 0.68

1.560★★★ [2.64] 0.48

1.372★ [1.77] 0.26

Capital Goods: Coeff 0.730★★★ t-stat [8.48] 2 𝑅 0.53

0.845★★★ [5.64] 0.50

1.061★★★ [3.35] 0.41

1.176★★ [2.19] 0.31

0.386 [1.32] 0.02

0.309 [1.06] 0.01

0.465★★ [2.36] 0.20

0.528★ [1.88] 0.23

0.495 [1.44] 0.16

0.466 [1.24] 0.12

0.390 [1.09] 0.07

Energy: Coeff t-stat 𝑅2

ȱ

0.482★★★ [3.09] 0.23

c (2009) Buraschi, Trojani, Vedolin – 25 ⃝

Pre-Crisis versus Crisis Introduction The Model Empirical Analysis Results Mini-Case No-Arbitrage Violations Logit Stock Returns

⊳

□

Effect of Uncertainty-DiB in crisis period seems to be larger.

□

Idiosyncratic Uncertainty-DiB cluster: Systematic component becomes more important in crisis periods.

Pre-Crisis and Crisis Regressions:

Pre Credit Crisis

Credit Crisis

0.671★★★ [4.18] 0.148★★★ [4.93] 0.428★★ [2.23] 0.525★★★ [4.21] 0.564★★★ [4.05] 0.482★★★ [6.52] 0.27

0.709★★★ [4.18] 0.691★★★ [4.93] 0.437★★ [2.23] 0.580★★★ [3.55] 0.758★★★ [9.16] 0.857★★★ [5.21] 0.30

Conclusion Appendix

Information Technology Services Financials Banks Capital Goods Energy Average adjusted 𝑅2

ȱ

c (2009) Buraschi, Trojani, Vedolin – 26 ⃝

No-Arbitrage Violations Introduction The Model Empirical Analysis Results Mini-Case No-Arbitrage Violations Logit Stock Returns

According to Merton’s (1974) model, a rise in credit spreads should go pari passu with a decrease in the stock price.

⊳

⇒Common factor is driving both: Difference in beliefs.

Conclusion

Capital structure arbitrage: One could buy / sell credit default swaps (CDS) and use equities (or a derivative on the equity) to dynamically delta hedge the position.

Appendix

ȱ

c (2009) Buraschi, Trojani, Vedolin – 27 ⃝

A Crude Example: May 2005 General Motors (GM) and Ford get downgraded to junk status by S&P.

Introduction The Model

Before the downgrade:

Empirical Analysis Results Mini-Case No-Arbitrage Violations Logit Stock Returns

⊳

Conclusion

□

Many hedge funds shorted CDS on GM and hedged their exposure by shorting the equity.

□

Wider credit spreads were expected to be accompanied by a drop in 𝑆(𝑡) and / or an increase in implied option volatility.

Appendix

After the downgrade: □

Spreads on a 10 year CDS increased by 200 bp in one month.

□

But the stock price rose almost 25% up to USD 32.75.

At the same time, DiB on GMs future earnings: more than doubled from 0.21 to 0.49.

ȱ

c (2009) Buraschi, Trojani, Vedolin – 28 ⃝

No-Arbitrage Violations Definition 2 A violation is observed when Δ𝐶𝑆Δ𝑆 > 0, where Δ𝐶𝑆 is the change in the credit spread and Δ𝑆 the change in the corresponding individual stock price.

Introduction The Model Empirical Analysis Results Mini-Case No-Arbitrage Violations Logit Stock Returns

⊳

Panel A: Simulated Violations

Conclusion Appendix

Low

Average

High

15.3

14.2

12.2

Low

Average

High

18.9

15.4

14.3

Panel B: Empirical Violations

ȱ

c (2009) Buraschi, Trojani, Vedolin – 29 ⃝

Logit Regressions The probability that a violation event occurs is specified as: Introduction

𝑃 (𝑦(𝑖𝑡) = 1) = 𝐹 (𝛽0 + 𝛽1 log Ψ𝑖 (𝑡) + 𝛽2 log Ψ𝑧 (𝑡) +

The Model

2 ∑

𝛿𝑗 𝐹𝑖𝑗 (𝑡) +

𝑗=1 Leverage

Low

Average

High

Constant

Conclusion

Individual DiB

Appendix

Common DiB

−0.21★★ [-2.39] 0.17★★★ [3.32] 0.12★★ [2.40] 0.19★ [1.68] 0.05★ [1.67] 0.13★★ [2.02] −0.16 [-1.22] −0.17★ [-1.88] 0.05 [1.32] 0.24★ [1.90] 0.18 [1.47] 0.40★ [1.69] 0.19

−0.32★★ [-2.19] 0.18★★★ [3.83] 0.18★★ [2.36] 0.26★ [1.75] 0.04★ [1.66] 0.18★★ [2.30] −0.12 [-1.58] −0.20★★ [-2.01] 0.10 [1.57] 0.17 [1.43] 0.23 [1.62] 0.49★ [1.91] 0.19

−0.37★★ [-2.43] 0.23★★★ [4.27] 0.13★★ [2.43] 0.18★ [1.90] 0.02 [1.38] 0.21★ [1.71] −0.11★ [-1.73] −0.21★★ [-2.29] 0.12 [1.63] 1.28★ [1.91] 0.48★★ [2.52] 0.42★ [1.75] 0.22

Implied Volatility VIX Leverage PS Liquidity FG Liquidity 𝑅𝑚 − 𝑅𝑓 SMB HML Earnings Volatility Pseudo 𝑅2

ȱ

𝛾𝑘 𝑇 (𝑡)).

𝑘=1

Empirical Analysis Results Mini-Case No-Arbitrage Violations Logit Stock Returns

⊳

7 ∑

c (2009) Buraschi, Trojani, Vedolin – 30 ⃝

Stock Returns We expect a different sign for low leverage firms.

Introduction The Model Empirical Analysis Results Mini-Case No-Arbitrage Violations Logit Stock Returns

⊳

Conclusion Appendix

ȱ

Dependant

Credit Spreads

Stock Returns

Constant Individual DiB (LL) Individual DiB (AL) Individual DiB (HL) Common DiB Implied Volatility (LL) Implied Volatility (AL) Implied Volatility (HL) Implied Volatility Skew (LL) Implied Volatility Skew (AL) Implied Volatility Skew (HL) Slope of Term Structure Risk-free Rate Non-Farm Payroll NBER Dummy Leverage (LL) Leverage (AL) Leverage (HL) Liquidity 𝑅𝑚 − 𝑅𝑓 SMB HML Earnings Volatility Adjusted 𝑅2

0.653★★★ 0.756★★★ 0.786★★ 0.859★★ 0.721★★ 0.625★ 0.520 0.492 −0.261 −0.324 −0.529 −0.377 −0.506★★★ −0.597★★ −1.657 0.427★★★ 0.287★★★ 0.265★★★ −0.129★★ −0.532 0.417★★ 0.313★ 0.373★ 0.75

0.001★★★ −0.012★★ 0.001★ 0.002★★ 0.007★★ −0.001 −0.001 −0.002 −0.026 −0.017 −0.015

0.013 -0.002 0.017★ 0.015 0.013 0.029★ 0.001★★★ 0.002★★★ 0.001★★★ −0.002★ 0.05

c (2009) Buraschi, Trojani, Vedolin – 31 ⃝

Introduction The Model Empirical Analysis

⊳ Conclusion Model Predictions Appendix

Conclusion

ȱ

c (2009) Buraschi, Trojani, Vedolin – 32 ⃝

Theoretical Findings Introduction The Model Empirical Analysis Conclusion Model Predictions

⊳

Appendix

What is the economic importance of divergence of opinions for credit spreads? We extend the most recent literature on the “credit spreads puzzle”. In a general equilibrium with divergence of opinions, we can support more realistic credit spreads, even for low levels of RRA and a reasonable level of the default probability. Why do corporate credit spreads and the volatility of stock returns co-move? The model offers a simple structural explanation for the positive empirical link between the volatility of stock returns, the implied volatility of individual stock options, and corporate credit spreads. Are no-arbitrage violations of one-factor models puzzling? Beliefs disagreement might explain no-arbitrage violations by single-factor models for credit spreads.

ȱ

c (2009) Buraschi, Trojani, Vedolin – 33 ⃝

Empirical Findings We find a strong counter-cyclical systematic component in Uncertainty-DiB which tends to be larger in times of financial or economic crises.

Introduction The Model Empirical Analysis Conclusion Model Predictions

⊳

Appendix

⇛ The systematic Uncertainty-DiB tends to increase with the default spread. Across different sectors, we find some lead-lag relationship between the different sector-wide Uncertainty-DiBs. We find a strong positive relation between divergence of opinions and credit spreads. ⇛ Uncertainty-DiB dominates other commonly used variables, in terms of explanatory power, such as option-implied volatilities and proxies for pure cash flows uncertainty. We find that the relation between divergence of opinions and equity prices indeed depends on the leverage of the firm. ⇛ Beliefs disagreement dominates in terms of explanatory power other proxies of pure cash flows uncertainty. The main model predictions are supported by the data.

ȱ

c (2009) Buraschi, Trojani, Vedolin – 34 ⃝

Introduction The Model Empirical Analysis Conclusion

⊳ Appendix Risk Factors Probability of Default

Appendix

ȱ

c (2009) Buraschi, Trojani, Vedolin – 35 ⃝

Idiosyncratic vs. Systematic Risk Introduction

□

Dispersion does not measure idiosyncratic risk (c.f. Merton (1987)). See Johnson (2004) and Anderson et al. (2005).

□

An extension to multiple assets is work in progress.

□

Crucial: The main asset price predictions remain the same! ⇒ Co-movement of Difference in beliefs.

□

Here: Existence of a representative agent whose beliefs are a weighted beliefs of analysts, and hence the impact of heterogeneity is channelled through its effect on average beliefs (Detemple and Murthy (1994)).

□

Aggregation of beliefs: Calvet et al. (2004) and Chiarella et al. (2007). Consensus beliefs as in Jouini and Napp (2006): Weighted geometric average of the individual beliefs. Weights are proportional to wealth as this determines risk-bearing.

The Model Empirical Analysis Conclusion Appendix Risk Factors Probability of Default

⊳

ȱ

c (2009) Buraschi, Trojani, Vedolin – 36 ⃝

Probability of Default We generate a higher firm value and a lower default risk premium, this induces higher corporate credit spreads at reasonably low default probabilities.

Introduction The Model Empirical Analysis Conclusion Appendix Risk Factors Probability of Default

⊳

Chen, Collin-Dufresne, and Goldstein (2006): Taking expected default rates as given, one way to increase credit spreads is to increase the correlation between a corporate bond and stock returns. ⇒ Stock-Bond correlations in an economy with disagreement are endogenously driven! When leverage is low (high), stock-bond correlation is negative (positive). ⇒ There is no need for a counter-cyclical default boundary in our economy. We generate high credit spreads with reasonable low probability of default which is 0.04 (0.07) for low (high) leverage firms. This is comparable to an average 0.043 default probability of senior secured bonds (Moody’s default report). The dynamics of the distance-to-default are non-monotonic in our economy. Pre-mature vs. default at maturity.

ȱ

c (2009) Buraschi, Trojani, Vedolin – 37 ⃝