Endogenous Liquidity and Defaultable Bonds Zhiguo He (Chicago Booth and NBER) Konstantin Milbradt (MIT Sloan/NU Kellogg and NBER)

SAET Conference Paris July 2013

Motivation: Default vs Liquidity �

Default and liquidity are interconnected as evident from recent financial crisis �

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Liquidity: funding liquidity, price impact, transaction costs, etc

Prevalent in theoretical literature: one-way linkages 1. Liquidity → default

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or

2. Default → liquidity

Today’s paper: liquidity�default, two-way feedback Endogenous liquidity solved jointly with default decision

Motivation: Default vs Liquidity �

Default and liquidity are interconnected as evident from recent financial crisis �

�

Liquidity: funding liquidity, price impact, transaction costs, etc

Prevalent in theoretical literature: one-way linkages 1. Liquidity → default

or

2. Default → liquidity

�

Today’s paper: liquidity�default, two-way feedback Endogenous liquidity solved jointly with default decision

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Important in understanding the liquidity and default premia for corporate bond credit spreads �

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Barclays Capital report (2009) shows high correlation between default and liquidity spreads, both in time-series and cross-section Dick-Nielsen, Feldh¨ utter, and Lando (2012), and Friewald, Jankowitsch, and Subrahmanyam (2012) Yet state-of-art empirical literature additively decomposes spreads into independent liquidity and default premium

Motivation: Corporate Bonds

CreditSpread 600 500 B

400 300

BB

200

BBB

100

AA 0 AAA 0.1 0.2

A 0.3

Source: Huang, Huang ’03

0.4

0.5

0.6

0.7

QL

Motivation: Corporate Bonds

BidAsk 80 70

BB

B

60 50 40

A AAA

0.1

BBB

AA 0.2

0.3

0.4

0.5

0.6

0.7

QL

Source: Edward, Harris, Piwowar ’06 (EHP; BA at median trade size)

Mechanism and Results Building blocks for interaction between fundamental and liquidity: �

How does bond illiquidity arise, and how is it affected by the state of the firm? �

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Over-the-counter market with search friction ` a la Duffie et al (2005)

How do corporate decisions interact with secondary market liquidity? �

Endogenous default and rollover channel ` a la Leland Toft (1996)

Mechanism and Results Building blocks for interaction between fundamental and liquidity: �

How does bond illiquidity arise, and how is it affected by the state of the firm? �

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Over-the-counter market with search friction ` a la Duffie et al (2005)

How do corporate decisions interact with secondary market liquidity? �

Endogenous default and rollover channel ` a la Leland Toft (1996)

Main results: �

Closed-form solution for bond values and bid-ask spreads, equity values, and default boundary

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Endogenous liquidity allows us to match the cross-sectional pattern of bid-ask spreads and credit spreads

Related Literature Search in asset markets: �

Duffie, Garleanu, Pedersen ’05 (DGP), ’07 OTC search market with simplified ’derivative’ security

Capital structure models: �

Leland, Toft ’96 (LT) Rollover increases exposure of equity holders to fundamental risk

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He, Xiong ’12 Exogenously given secondary market liquidity affects default decision

Empirical literature: �

Bao, Pan, Wang ’11; Edwards, Harris, Piwowar ’07; Hong, Warga ’00; Hong, Warga, Schultz ’01; Harris, Piwowar ’06; Feldh¨ utter ’11

Feedback models: �

Many many more papers...

Model: The Firm Preferences: Everyone risk-neutral with common discount rate r Cash flows: �

Cash-flow rate δ = e y , dy = µdt + σdZtQ

Model: The Firm Preferences: Everyone risk-neutral with common discount rate r Cash flows: �

Cash-flow rate δ = e y , dy = µdt + σdZtQ

Debt structure: �

Debt in place with aggregate face value p and coupon c

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Stationary principal & staggered maturity (as in LT): � �

Uniform maturity structure ⇒ Mass 1/T matures every instant Maturing bonds reissued with identical contract terms (c, p, T )

Model: The Firm Preferences: Everyone risk-neutral with common discount rate r Cash flows: �

Cash-flow rate δ = e y , dy = µdt + σdZtQ

Debt structure: �

Debt in place with aggregate face value p and coupon c

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Stationary principal & staggered maturity (as in LT): � �

Uniform maturity structure ⇒ Mass 1/T matures every instant Maturing bonds reissued with identical contract terms (c, p, T )

Rollover: �

Primary market with transaction costs κ, debt reissued at DH Mass maturing y

NetCashFlowt = ���� e − (1 − π ) c + � �� � � CF

Coupon

���� 1/T

Reissue

� �� � [(1 − κ ) DH (yt , T ) − p ] �� �

Rollover gain/loss

Schematic Representation: Leland-Toft Primary market No Arbitrage: H indifferent

Reissue

D: c dt

Resale A�DH

Firm: dy�Μdt�ΣdZ �Log cashflow�

Maturity 1�T Maturity

Ξ

Competitive Interdealer Market

Liq. Shock

DL : �c�Χ�dt

Λ Intermediation

B�DL �Β�DH �DL �

Secondary market

NetCashFlowt =

ey

− (1 − π ) c + 1/T [(1 − κ ) D (y ; T ) − p ]

Schematic Representation: Leland-Toft Primary market No Arbitrage: H indifferent

Reissue

D: c dt

Resale A�DH

Firm: dy�Μdt�ΣdZ �Log cashflow�

Default at y b : D�Αvb

Maturity 1�T Maturity

Ξ

Competitive Interdealer Market

Liq. Shock

DL : �c�Χ�dt

Λ Intermediation

B�DL �Β�DH �DL �

Secondary market

Equity optimally defaults at yb when absorbing further losses unprofitable

Model: Investors, Liquidity Shocks & Search Idiosyncratic liquidity shock to bond investors: � �

Asset holding restriction {0, 1} as in DGP ’05 Uninsurable i.i.d. liquidity shock results in two types of agents: �

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H type: subject to liq shock with intensity ξ before default, ξ b > ξ after default L type: currently in liquidity shock state, holding cost χ pre-default y (χb r −(µe−bσ2 /2) post-default) until asset sold

Schematic Representation: Rollover & Liquidity Shocks Primary market No Arbitrage: H indifferent

Reissue

DH : c dt

Resale A�DH

Firm: dy�Μdt�ΣdZ �Log cashflow�

Maturity 1�T Maturity

Ξ

DL : �c�Χ�dt

Default at y b

Competitive Interdealer Market

Liq. Shock

Λ Intermediation

B�DL �Β�DH �DL �

Secondary market

NetCashFlowt =

ey

− (1 − π ) c + 1/T [(1 − κ ) DH (y ; T ) − p ]

Model: Investors, Liquidity Shocks & Search Idiosyncratic liquidity shock to bond investors: � �

Asset holding restriction {0, 1} as in DGP ’05 Uninsurable i.i.d. liquidity shock results in two types of agents: �

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H type: subject to liq shock with intensity ξ before default, ξ b > ξ after default L type: currently in liquidity shock state, holding cost χ pre-default y (χb r −(µe−bσ2 /2) post-default) until asset sold

Trade & search friction: � �

L sellers, H buyers, all meet OTC dealers with intensity λ Competitive interdealer market, no inventory, transaction price M

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Agents have bargaining power β vis-a-vis a dealer

Schematic Representation: Intermediation Primary market No Arbitrage: H indifferent

Reissue

DH : c dt

Resale A�DH

Firm: dy�Μdt�ΣdZ �Log cashflow�

Maturity 1�T Maturity

Ξ

DL : �c�Χ�dt

Default at y b

Competitive Interdealer Market

Liq. Shock

Λ Intermediation

B�DL �Β�DH �DL �

Secondary market

NetCashFlowt =

ey

− (1 − π ) c + 1/T [(1 − κ ) DH (y ; T ) − p ]

Model: Bid-Ask Spreads Nash-bargaining: �

Let Π be generic surplus. Then Nash-bargaining splits it βΠ → Investor (1 − β) Π → Dealer

Model: Bid-Ask Spreads Nash-bargaining: �

Let Π be generic surplus. Then Nash-bargaining splits it βΠ → Investor (1 − β) Π → Dealer

Seller’s market Assumption: � Mass sellers µL1 smaller than mass buyers µH0 , i.e., µL1 < µH0

Model: Bid-Ask Spreads Nash-bargaining: �

Let Π be generic surplus. Then Nash-bargaining splits it βΠ → Investor (1 − β) Π → Dealer

Seller’s market Assumption: � Mass sellers µL1 smaller than mass buyers µH0 , i.e., µL1 < µH0 Pre-default market: �

L-dealer (seller) surplus ΠL , H-dealer (buyer) surplus ΠH

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Bertrand competition in interdealer market erodes H-dealer surplus �

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Why? Any positive surplus would be outbid as there is more potential buyers than sellers

Ask price A (H is buying at), bid price B (L is selling at) � �

Buy side: A = DH − βΠH = M = DH and ΠH = 0 Sell side: B = DL + βΠL and Π ≡ ΠL = DH − DL > 0

Schematic Representation: Bid-Ask Spreads Primary market No Arbitrage: H indifferent

Reissue

DH : c dt

Resale A�DH

Firm: dy�Μdt�ΣdZ �Log cashflow�

Default at y b

Maturity 1�T Maturity

Ξ

Competitive Interdealer Market

Liq. Shock

DL : �c�Χ�dt

Λ Intermediation

B�DL �Β�DH �DL �

Secondary market

Bid-Ask A − B = (1 − β) (DH − DL ) proportional to valuation wedge

Schematic Representation: Default Primary market No Arbitrage: H indifferent

Reissue

DH : c dt

Resale A�DH

Firm: dy�Μdt�ΣdZ �Log cashflow�

Maturity 1�T Maturity

Ξ

DL : �c�Χ�dt

Default at y b : DH �ΑH vb DL �ΑL vb

Competitive Interdealer Market

Liq. Shock

Λ Intermediation

B�DL �Β�DH �DL �

Secondary market e yb

Di (yb , τ ) = αi r −(µ−σ2 /2) from frictional post-bankruptcy market

Schematic Representation: The Primary Market Primary market No Arbitrage: H indifferent

Reissue

DH : c dt

Resale A�DH

Firm: dy�Μdt�ΣdZ �Log cashflow�

Maturity 1�T Maturity

Ξ

Competitive Interdealer Market

Liq. Shock

DL : �c�Χ�dt

Λ Intermediation

Secondary market

B�DL �Β�DH �DL �

Schematic Representation: The Secondary Market Primary market No Arbitrage: H indifferent

Reissue

DH : c dt

Resale A�DH

Firm: dy�Μdt�ΣdZ �Log cashflow�

Maturity 1�T Maturity

Ξ

Competitive Interdealer Market

Liq. Shock

DL : �c�Χ�dt

Λ Intermediation

Secondary market

B�DL �Β�DH �DL �

Analytic Solutions and Comparative Statics Closed form solutions: �

Closed form solutions for debt DH/L (mix of two LT solutions), equity E and optimal default boundary yb

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Consequently, closed form solutions for absolute and proportional bid-ask spread, A − B = (1 − β) ΠL and ∆ = 1 A−B1 , respectively 2 A+ 2 B

Analytic Solutions and Comparative Statics Closed form solutions: �

Closed form solutions for debt DH/L (mix of two LT solutions), equity E and optimal default boundary yb

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Consequently, closed form solutions for absolute and proportional bid-ask spread, A − B = (1 − β) ΠL and ∆ = 1 A−B1 , respectively 2 A+ 2 B

Analytic comparative statics: 1. If wedge at default, Π = (αH − αL ) at (y , τ ) → (∞, ∞), Π =

χ r +ξ +λβ ,

e yb , µ−(µ−σ2 /2)

greater than wedge

then ∂y (A − B ) < 0.

2. If additionally ∂y DH > 0 (condition provided), then also ∂y ∆ < 0. 3. If αH > αL , then ∂τ (A − B ) > 0. Interpretation: 1.+ 2. Controlling for time-to-maturity, both abs and prop bid-ask spreads decreasing in δ (pro-cyclical liquidity) 3. Controlling for dist-to-default, abs bid-ask spread is increasing in τ.

Liquidity and Default: Full Feedback Loop Equilibrium feedback loop: �

Compare to counterfactual constant transaction costs :'*2;<3=% >%+",(-."*%

!"#$%&'()"*% +",(-."%

9-0)-+-$1% +",4"'*"*%

!"#$%43((3&"4% 634"%"78".*-&"%

/0)-$1%23(+"4*% +"5')($%"'4(-"4% �

Fixed point (default threshold) δb = e yb outcome of this “spiral”

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For calibration, map y into quasi leverage QL (y ) ≡

p p +E (y )

Calibration: Liquidity BidAsk 80 70

BB

B

60 50 40

A AAA

0.1

BBB

AA 0.2

0.3

0.4

0.5

0.6

0.7

QL

Solid: Adjust c so issued at par (newly issued bonds); Dashed: Constant c (stale bonds)

Calibration: Liquidity BidAsk 80 70

BB

B

60 50 40

A AAA

0.1

BBB

AA 0.2

0.3

0.4

0.5

0.6

0.7

QL

Solid: Adjust c so issued at par (newly issued bonds); Dashed: Constant c (stale bonds)

Calibration: Credit Spread CreditSpread 600 500 B

400 300

BB

200 100

AA 0 AAA 0.1 0.2

BBB A 0.3

0.4

0.5

0.6

0.7

QL

Solid: Adjust c so issued at par (newly issued bonds); Dashed: Constant c (stale bonds)

Calibration: Credit Spread CreditSpread 600 500 B

400 300

BB

200 100

AA 0 AAA 0.1 0.2

BBB A 0.3

0.4

0.5

0.6

0.7

QL

Solid: Adjust c so issued at par (newly issued bonds); Dashed: Constant c (stale bonds)

Model-Based Decomposition: Methodology �

Longstaff et al ’05: CDS back out default component yˆDEF . How much of default component is caused by liquidity?

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Structural model allows finer decomposition of credit spread :

yˆ �

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=

Default Component yˆDEF � �� � yˆpureDEF + yˆLIQ →DEF

+

Liquidity Component yˆLIQ � �� � yˆpureLIQ + yˆDEF →LIQ

Pure default yˆpureDEF : fully liquid secondary bond market (LT 96), ∗ default at δLT Liquidity-driven Default yˆLIQ →DEF : additional default due to earlier ∗ (but full liquidity in trading) default at δb∗ > δLT Pure Liquidity yˆpureLIQ : riskless bond spread with illiquid secondary bond market (DGP 05) Default-driven Liquidity yˆDEF →LIQ : additional illiquidity part due to default

Goal: Separate causes from consequences

Conclusion Fully solved non-stationary dynamic search model: �

Closed form solution for debt, equity, default boundary

Liquidity-default spiral: �

Lower liquidity in secondary market lowers the distance to default, which further lowers liquidity in secondary market,...

What about adverse selection? �

Definitely reasonable but challenging. Probably generates similar empirical illiquidity pattern (Crotty, Back ’13)

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For understanding the role of liquidity in credit spreads, search framework (simple, easy to be integrated) delivers first-order effects

Empirical implementation: �

Targeting liquidity, we match bond credit spreads and are then able to decompose into liquidity and default components

Microfoundation of Bankruptcy Wedge Bankruptcy payout delay: δb r −µ

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Bankruptcy recovery α < 1 of unlevered firm value

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Recovery payout at exponential (θ) time due to legal delay

Microfoundation of Bankruptcy Wedge Bankruptcy payout delay: δb r −µ

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Bankruptcy recovery α < 1 of unlevered firm value

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Recovery payout at exponential (θ) time due to legal delay

Post-default market: � � �

Search market characterized by (θ, ξ b , λb , χb , β b , δb ) Ask price Ab = DHb , bid price B b = DLb + (1 − β) ΠbL

Seller’s market assumption: Competitive interdealer price M b erodes all surplus of buyers

Microfoundation of Bankruptcy Wedge Bankruptcy payout delay: δb r −µ

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Bankruptcy recovery α < 1 of unlevered firm value

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Recovery payout at exponential (θ) time due to legal delay

Post-default market: � � �

Search market characterized by (θ, ξ b , λb , χb , β b , δb ) Ask price Ab = DHb , bid price B b = DLb + (1 − β) ΠbL

Seller’s market assumption: Competitive interdealer price M b erodes all surplus of buyers

Effective bankruptcy recovery for H and L investors: � �

δb δb b Closed form DHb = αH r − µ > DL = α L r − µ ⇒ Pre-default liquidity, via δb , affects post-default liquidity

Interpretation of default as firm-wide liquidity event that is endogenously triggered

Optimal Maturity: Rollover Risk vs Liquidity Negative: Short-term debt leads to earlier default �

Higher rollover frequency increases equity’s exposure to δ Rollover gain/losst =

1/T ����

×

Rollover frequency �

[(1 − κ ) DH (δt , T ) − p ] � �� � Repricing

Higher exposure to δ leads to higher default boundary δB

Optimal Maturity: Rollover Risk vs Liquidity Negative: Short-term debt leads to earlier default �

Higher rollover frequency increases equity’s exposure to δ Rollover gain/losst =

1/T ����

×

Rollover frequency �

[(1 − κ ) DH (δt , T ) − p ] � �� � Repricing

Higher exposure to δ leads to higher default boundary δB

⇒ LT, He and Xiong ’12: Infinite maturity debt always optimal ex-ante

Optimal Maturity: Rollover Risk vs Liquidity Negative: Short-term debt leads to earlier default �

Higher rollover frequency increases equity’s exposure to δ Rollover gain/losst =

1/T ����

×

Rollover frequency �

[(1 − κ ) DH (δt , T ) − p ] � �� � Repricing

Higher exposure to δ leads to higher default boundary δB

⇒ LT, He and Xiong ’12: Infinite maturity debt always optimal ex-ante Positive: Short-term debt provides liquidity �

Short maturity improves bargaining outcome between seller & dealer

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Issuing to H types more frequently improves allocative efficiency as it ’recycles’ L types to H types quicker

Optimal Maturity: Rollover Risk vs Liquidity Negative: Short-term debt leads to earlier default �

Higher rollover frequency increases equity’s exposure to δ Rollover gain/losst =

1/T ����

×

Rollover frequency �

[(1 − κ ) DH (δt , T ) − p ] � �� � Repricing

Higher exposure to δ leads to higher default boundary δB

⇒ LT, He and Xiong ’12: Infinite maturity debt always optimal ex-ante Positive: Short-term debt provides liquidity �

Short maturity improves bargaining outcome between seller & dealer

�

Issuing to H types more frequently improves allocative efficiency as it ’recycles’ L types to H types quicker

⇒ Finite maturity T ∗ < ∞ optimal if moderate initial leverage; T ∗ lower the less liquid secondary market (i.e. the lower λ)