A Binomial Lattice Method for Pricing Corporate Debt and

Modeling Chapter 11 proceedingsAuthor:Mark Braodie and ÖzgÜr Kaya*

報告者 :柯婷瑱

• Basic setup• Bankruptcy with Immediate Liquidation• Bankruptcy with Grace Period and Bargaining• Bankruptcy with Grace Period, Automatic Stay

and Arrears Account

2

• It develop a binomial lattice method that can be used to handle complex structural models such as ones that include Chapter 11 proceeding of the U.S bankruptcy code.

• In a structural model, the value of firm’s asset or the firm’s earnings process is modeled as a primitive variable, and all other variables are derived from this basic variable.

3

• It treat bankruptcy and liquidation as the different event.

• While in bankruptcy, the coupon payments are stopped and are recorded in an arrears account. The earning of the firm are collected in a separated account.

• The collected earnings are then used to pay the arrears when the firm comes out of bankruptcy.

• It shows that the issue of the limited liability of equality holders can be handled.

4

Basic Setup

t

The risk-neutralized version of the firm asset's diffusion process is

( )

is a standard Brownian motion is the risk-free rate is the volatility of firm asset.= V is the insta

tt

t

t

t

dVr q dt dW

VwhereWr

q

ntaneous cash generated by the firm C total coupon payment at time t

5

• To Build a lattice based on the binomial method of Cox, Ross, and Rubinstein(1979)

( )

t

t

r q t

u e

d ea dp where a eu d

6

7

The instantaneous cash flow produced by the firm at t is given by

On the binomial lattice,the total firm cash flow is given by

( -1)

t t

q tt t

V q

V e

Finite Maturity Case

Terminal nodes:If (1 )

: (1 )::

If (1 ): 0: (1 )( ): (1 )( )

T T t

T T t

t

T T t

T T t

T T

T T

V C PE V C PD C PF V CV C P

ED VF V

8

Backward Induction

9

Internal nodes:

( (1 ) )

( (1 ) )

If (1 )

: (1 )

:

:

If (1 ): 0: (1 )( ): (1 )( )

t

t

ru d

ru d

t t

t t

t

t t

t t

t t

t t

E e pE p E

D e pD p D

E C

E E C

D D C

F D E C

E CED VF V

Eu

Du

Ed

Dd

ED

Infinite Maturity Case

Terminal nodes:

If

:

:

:

If

: 0: (1 )( ): (1 )( )

T

T

T

T

T

T

CVrCE Vr

CDr

F VCVr

ED VF V

10

timeT

2 1(1 ) (1 ) (1 ) (1 )n nC C r C r C r C r

This model can be used to price a consol with infinite maturity.

Bankruptcy and Grace Period and Bargaining

• It consider the approach of FM(2004) and assume that, at a certain level of the firm asset value VB, equity holders decide to declare bankruptcy.

• There is a distress cost ω that reduces the net firm cash flow when then firm is in bankruptcy. Thus, the firm cash flow rate is reduced from q to q-ω.

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• It follow Fan and Sundaresan (2000) to determine the debt service using a Nash bargaining game.

• It assume that the bargaining power of the equity holders is η and the bargaining power of the debt holders is 1-η.

• If the firm is not liquidated, its value will be FB and this amount will be shared between equity holders and debt holders.

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• It denotes the sharing rule at the bankruptcy point as θ, then the incremental value gain by equity holders is and the incremental value gain by debt holders is

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BF(1 ) (1 )B BF V

B

B

B B

liquidate at bankruptcy pointE:0D:(1- )Vnot liquidate at bankruptcy pointE: F (incremental value) D:(1- )F -(1- )V (incremental value)

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* 1-

*

*

*

The optimal sharing rule satisfies

arg max{[ ] [(1 ) (1 ) ] }(1 )(1 )

As a result,

equity holders: ( (1 ) )

debt holders:(1- ) (1- )( (1 ) ) (1 )

B B B

B

B

B B B

B B B B

F F VV

F

F F V

F F V V

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B

B

B

B

B

B

BBBB

BBB

BBBBBBBB

BBBBBBBB

BBB

FV

FV

FV

FVFFFVF

FVFFVFFF

FVFFVFFFddf

VFFf

11

11

11

0

1101110

111110

11111

11

11

11

1

case 1

• nodes with Vt>VB

• For these nodes, we update the equity and debt values in the following way:

( (1 ) )

( (1 ) )

If (1 )

: (1 )

:

:

t

t

ru d

ru d

t t

t t

t

t t

E e pE p E

D e pD p D

E C

E E C

D D C

F D E C

If (1 ): 0: (1 )( ): (1 )( )

t t

t t

t t

E CED VF V

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Eu

Du

Ed

Dd

ED

case 2

• nodes with Vt<VB

• The debt is served strategically based on the outcome of the bargaining game.

• There is no tax benefits for payments to debt holders.• We only need to keep track of the firm value F when

the firm is in bankruptcy.• The total time spent in bankruptcy needs to be

recorded so that it can be checked against the allowed graced period G.

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18

t

Let denotes the maximum of time steps that firm can spend in bankruptcy.

We have , where G is the grace period and is the time step.

Then g will take values in [0,1, , -1, ][ ] will denote the

t

gGg

g gF i

( )

firm value at the current node when g=i

( [ 1] (1 ) [ 1]) for i=1, ,g 1[ ]

(1 )( ) for i=g

[ ] ( [ ] (1 ) )[ ] (1

t

t

rt u d

t t

qt t t

t

e pF i p F iF i

V

V e VE i F i VD i

- )( [ ] (1 ) ) (1 )t tF i V V

case 3

• nodes with Vt=VB

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[0] ( (1 ) [1])

[ ] ( (1 ) [1]) for i=1, ,g

( [0] (1 ) )(1- )( [0] (1 ) ) (1 )

t

t

rt u d

rt u d

B

B B

F e pF p F

F i e pF p F

E F VD F V V

Using Bisection method to find optimal bankruptcy boundary

• Choose an arbitrary bound. one natural choice is the Leland liquidation boundary on equity has value zero.

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Pricing Finite Maturity Debt

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Terminal nodes:

: (1 )If (1 ) : :

:

: 0If (1 ) : : (1 )( )

: (1 )( )

[ ] (1 )( ) for i=1, ,g

T B

T T t

T T t t

T T t

T T t T T

T T

T B

T T

V V

E V C PV C P D C P

F V C

EV C P D V

F VV V

F i V

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(1 )

If (1 ) :[0]

[ ] for i=1, ,g

0(1 )( )

If (1 ) :[0] (1 )( )

[ ] (1 )( )

T B

T T t

tT T t

T T t

T T t

T TT T t

T T

T T

V V

E V C PD C P

V C PF V C

F i V C

ED V

V C PF V

F i V

Optimal bankruptcy Boundary

• The optimal VB in the finite maturity setting will not be constant, but rather be time dependent since the remaining value of the bond is changing over time.

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is the bankruptcy boundary at an intermediate time t is the risk free bond at time t

The equity holders can choose a pramater for to maximize the equity value.

tB t

tB

t

V P

VP

• Bankruptcy boundary– it will be an increasing function of time t for a

discount bond, decreasing function of time t for a premium bond.

– it will be constant of time t for a par bond

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lower bankruptcy boundary for a cheap bondhigher bankruptcy boundary for a expensive bond

tB tV P

• face value =100

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Bankruptcy with Grace Period, Automatic Stay and Arrears Account• It consider the approach of BCS(2005) and

uses as its primitive variable the earnings before interest and taxes(EBIT).

• It consider a consol bond with infinite maturity. While the firm is in bankruptcy, all coupon payments are stopped, and unpaid coupons are recorded in an arrears account At.

• The entire firm cash flow or EBIT is accumulated in a separate account St.

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• If the firm reaches a healthy state at a future time T, the amount ST is used to pay the arrears θAT.

• In order to do the calculations, we need to distinguish among three types of nodes:– nodes with Vt>VB

– nodes with Vt<VB

– nodes with Vt=VB

(VB is the time independent default boundary)

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case1

• nodes with Vt>VB

• For these nodes, we update the equity and debt values in the following way:

( (1 ) )

( (1 ) )

If

:

:

:

t

t

ru d

ru d

t t

t t

t

t

E e pE p E

D e pD p D

E C

E E C

D D C

F D E

If : 0: (1 )( ): (1 )( )

t t

t t

t t

E CED VF V

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Eu

Du

Ed

Dd

ED

case2

• nodes with Vt<VB

• Firm is in bankruptcy.• The coupon payments are accumulated in an arrears

account A.• The firm cash flows are accumulated in a automatic stay

payoff account S.• There is a grace period G, which is the maximum amount

of time that the firm can spend in bankruptcy.• We will add two state variables each node to represent

the values of g and S for each node.

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• To keep track of the automatic stay payoffs S, we will use a discrete grid for its values and then use linear interpolation. An upper bound on the value of S is given by

• Assume we want to use M values in the discrete grid. Then S is represented by the values on the grid , where j takes values in the set[0,1,…,M-1,M]

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( 1)q tBS V e G

/jS jS M

• E[i,j] denotes the equity value in the current node when g=i and S=Sj.

• Assume we know the values Eu[i,j] in the up state and Ed[i,j] in the down state in the next time step for all i and j.

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If in the current node,then it will be

for up and down moves

j

r tu j u

r td j d

S S

S S e

S S e

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0 S

1 S

.

.

.

.

.

.

G-1 S

G S

0 S

1 S

.

.

.

.

.

.

G-1 S

G S

0 S

1 S

.

.

.

.

.

.

G-1 S

G S

r tu j uS S e

r td j dS S e

Sj

Su

0 1S*/M

2S*/M

M-1(S*)/M

M

E[i+1,j]Sj

E[i+1,j+1]Sj+1

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1

We find the equity values that connect to the current node in the next time step by linearly interpolating on the value of S

-[ 1, ] [ 1, ] ( [ 1, 1]- [ 1, ])

-

[ 1, ] [ 1, ]

u ju u u u

j j

d d

S SE i j E i j E i j E i j

S S

SE i j E i j

1

-

-( [ 1, 1]- [ 1, ])

-

Next, we compute the present value of equity at the current node

[ , ] ( [ 1, ] (1 ) [ 1, ])

d jd d

j j

r tu d

SE i j E i j

S S

E i j e pE i j p E i j

• Before we finalize the values, we need to check for liquidation. Liquidation may occur because of the distress cost ω.

If [ , ]

[ , ] [ , ]

[ , ]If [ , ]

[ , ] 0[ , ] (1- )( )

t t

t t

t t

t j

E i j V

E i j E i j V

D i j DE i j V

E i jD i j V S

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case3

• nodes with Vt=VB

• The firm may have reached the nodes with Vt<VB either from above, being in a healthy state, or from below, being in bankruptcy.

( (1 ) [1,0])

If

[0,0]

[0,0] ( (1 ) [1,0])

If [0,0] 0[0,0] (1- )( )

t

t

rdu

t t

t t

rdt u

t t

t j

E e pE p E

E C

E E C

D C e pD p D

E CED V

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Eu

Ed[1,0]

E[0,0]

• For a firm to continue in a healthy state, it needs to clear the arrears. Therefore the sum of automatic stay payoffs Sj and the present value of equity should be larger than Ai.

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( (1 ) [1,0])

If

[ , ]

[ , ] ( (1 ) [1,0])

If

[ , ] 0[ , ] (1- )( )

t

t

rdu

j i

j i

rdi u

j i

t j

E e pE p E

E S A

E i j E S A

D i j A e pD p D

E S A

E i jD i j V S

Eu

Ed[1,0]

E[i,j]