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Liquidation Triggers Liquidation Triggers and the Valuation of and the Valuation of

Equity and DebtEquity and Debt

May 2007

Dan Galai, Alon Raviv and Zvi Wiener

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Net-worth covenantsNet-worth covenants

• Net-worth covenants - provide the firm’s bondholders

with the right to force reorganization or liquidation if

the value of the firm falls below a certain threshold.

• Typical for collateralized loans, non-recourse loans and

other cases with observable assets.

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Liquidation triggerLiquidation trigger• Liquidation triggerLiquidation trigger - depends on the nature of the

bankruptcy codes and on its enforcement.

• Empirical studies have shown that the criteria for liquidation of a firm after the onset of financial distress vary substantially across:

• Countries (Thorburn (2000): US versus Sweden bankruptcy procedures).

• Bankruptcy procedures (Bris, Welch and Zhu (2006) : Chapter 11 versus Chapter 7).

• Time (Covitz, Han and Wilson (2006): A significant decline in the length of time spent in default between the 80ies and the 90ies).

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Motivation: Valuation of Corporate Securities

As a result of the different relationship between default and liquidation the value of corporate securities, especially equity and debt, should reflect the nature of the existing liquidation procedures.

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Our ContributionOur Contribution

• We suggest a general model for pricing corporate securities applicable to a wide array of legal regimes and contractual arrangements.

• The liquidation decision may depend on:

1. The length of past distress events

2. Consecutiveness of past and current distress events.

3. The severity of the distress event.

4. The distance of past distress events from current time.

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OutlineOutline

1. Literature review2. Model and Assumptions3. Calibration of the model to market

data4. Empirical implications

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Pricing corporate Pricing corporate liabilitiesliabilities

1. Structural approach.• Merton 1974 classic option• Black and Cox 1976 first passage.

2. Reduced form/intensity based approach

• Jarrow and Turnbull (1995)• Duffie and Singleton (1998)

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Structural Approach - Merton Structural Approach - Merton (1974)(1974)

• A firm is financed by equity and a single issue of zero-coupon debt with face value F maturing at T.

• The firm defaults if at debt maturity, T, the value of the firm’s assets VT are not sufficient

to fully payoff the bond holders (absolute priority).

• In this case the equity investors surrender the firm to the bond investors which then make use of the remaining assets.

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Payoffs at MaturityPayoffs at Maturity

• With absolute priority, we have the following payoffs at maturity T:

Bonds EquityFVT F FVT FVT TV 0

• The bond payoff is: )0, max() , min()( TT VFFFVTB

• The equity payoff is: )0, max()( FVTE T

• Equity is valued as a call option on the value of the firm’s assets.

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First-Passage Default ModelFirst-Passage Default Model• Black-Cox (1976) had recognized that the firm

may default well before T.

• Default and liquidation takes place at the first time the assets fall below some threshold Kt: 0 inf tt VKt

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Years

Value of the firm assets Absorbing threshold

First passage time: Liquidation

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Discrepancy between default and control transfer

• Empirical studies have found that financial distress does not mean an immediate transfer of rights/assets to debt holders:

• The average time period between the indication of financial distress and its resolution ranges between 2 to 3 years at the 80ies and between 1 to 2 years at the 90ies

• Firms that improve their operating performance when still in financial distress usually survive, while those who keep presenting poor operating performance eventually are liquidated.

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The consecutive excursion The consecutive excursion method method

(François and Morellec 2002)(François and Morellec 2002) • Liquidation is triggered when the value of the firm’s

assets dips below the distress threshold and

remains below that level for an interval exceeding a

pre-determined ‘grace’ period.

• If the firm’s asset value rebounds and rises above

the distress threshold before the pre-determined

grace period has elapsed the liquidation state

variable is reset to zero.

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Liquidation procedures as Liquidation procedures as basketball penalty methodbasketball penalty method

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The deficiencies of the The deficiencies of the consecutive excursion methodconsecutive excursion method

• Each time firm value falls below the threshold level an additional grace period is granted without reference to previous instances of insolvency

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Years

Value of the firm's assets Distress threshold

Liquidation state variable Grace period

No liquidation till debt maturity!!!

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The cumulative excursion The cumulative excursion method method

(Moraux, 2002)(Moraux, 2002)

• liquidation is triggered when the total time

that the firm’s asset value spends under the

distress threshold (“excursion time”)

exceeds a pre-determined grace period.

• In this method the liquidation model

becomes highly path-dependent, since it

accumulates the entire history of a firm’s

financial distress.

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Liquidation procedures as Liquidation procedures as soccer penalty methodsoccer penalty method

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The deficiencies of the cumulative excursion method

• This liquidation process might have “too strong memory”. A firm may be liquidated even if the value of the firm’s assets has recently spent only a very short period of time under the distress threshold (since years ago it had spent an extensive period below the threshold).

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Years

Value of the firm's assets Distress thresholdGranted period Liquidation state variable

Liquidation is triggered!!!

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• Bankruptcy codes are usually not identical to soccer or basketball penalty methods.

• Bankruptcy codes (and soccer penalty method) diverge from the real liquidation procedures and depend on the sovereign (referees) enforcement.

To summarizeTo summarize……

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The adjustable excursion The adjustable excursion methodmethod

• Liquidation is executed when a liquidation state variable exceeds a pre-determined level.

• The state variable accumulates the weighted distress periods, which are defined as any period that the value of the firm’s assets has spent under the distress threshold (“excursion time”).

• By applying the process one can increase the weight of recent and/or severe distress periods over remote and/or mild distress periods.

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Model’s Assumptions: ConventionalModel’s Assumptions: Conventional

1. Assets are continuously traded in an arbitrage-free and complete market

2. The interest rate level, r, is assumed to be constant.

3. The value of the firm’s assets is independent of the capital structure of the firm, and is well described under the risk neutral process by:

tttt dWVdtVrdV )(

• Where:

motion Brownian standardA -

ratiopayout sfirm' The -

firm. theofreturn of rate theofdeviation standard ousinstantane The -

assets sfirm' theof valueThe -

W

Vt

4. The firm has outstanding only equity and a single bond issue with a promised final payment of P, that mature at time T.

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Model’s Assumptions (cont’d)Model’s Assumptions (cont’d)

5. The bondholders are theoretically allowed to force liquidation in one of two ways:

A. If the value of the firm’s assets falls below a time dependent threshold level, denoted by Kt, at any time prior to debt maturity.

B. If the value of the assets is less than some constant F at debt maturity.

6. The time dependent threshold level Kt is defined by:

10 : where)( tTrt FeK

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The cumulative distress timeThe cumulative distress time

7. Since default and liquidation are distinct events, liquidation is declared when the liquidation state variable exceeds a pre-determined grace period, denoted by d .

• In order to determine the value of the liquidation state variable we define the following random variable: ss

Kt K Vt sg sup

where:

. thresholdcrossed valuefirm that before last time theis skt Kt g

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The cumulative distress time The cumulative distress time (Cont’d)(Cont’d)

• The liquidation state variable is calculated at each day t as:

dsVfedsVfeI s

t

g

sts

gstK

t

t

t

)()(K

K

)(

0

)(

where:

variable.staten liquidatio on theevent distress theofseverity theofimpact theDefines - )(

period. distresslast for thefactor Decay -

periods. distresspast for factor Decay -

tVf

Time decay factor

Severity of the distress

event

Effect of current

distress period

Effect of past distress periods

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The severity of the distress The severity of the distress eventevent

0

• The function f(Vt) defines the impact of the severity of the distress event on the liquidation state variable . We model f(Vt) as follows:

To make certain that the liquidation state variable would increase with the severity of the distress event we set 0

tt

tt

K

VK

t

KV

KVeVf t

tt

0

)(

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The liquidation eventThe liquidation event • Liquidation occurs in the first time that the

liquidation state variable exceeds d. The liquidation time is denoted by , and it is defined mathematically by:

K

ttKt

K KVdIt , 0 inf

• Where the severity of the distress period has no impact on the liquidation procedure and the liquidation state variable is defined by:

dsedseIs

Kt

s

Kt t

g

stg

stKt ss K V

)(K V

0

)( 11

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Previous contributions as special Previous contributions as special cases of the adjustable excursion cases of the adjustable excursion

methodmethod • Example 1. When and , liquidation

procedure occurs at the first point in time that the firm value process has spent consecutively more than the pre-specified grace period below the threshold Kt, and we receive the François and Morellec (2002) liquidation procedure.

• When d=0, we receive, as a special case, the standard modeling of default and liquidation [ Leland (1994)].

• When d > (T-t), default never leads to liquidation and we receive, as a special case, the standard model for default and reorganization [Anderson and Sundaresan (1996) or Fan and Sundaresan (2000)].

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Previous contributions as Previous contributions as special cases of the adjustable special cases of the adjustable

excursion method (Cont’d)excursion method (Cont’d) • Example 2. When and , , liquidation

occurs the first time the firm value spends a total time greater than the pre specified grace period below Kt, and we receive the Moraux (2002) liquidation procedure.

• When d=0, default leads to immediate liquidation of the firm’s assets and we receive, as a special case, the Black and Cox (1976) liquidation model.

• When d > (T-t), liquidation can occur only at debt maturity, and the model collapses to the basic structural approach introduced by Merton (1974).

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The value of the firm’s equityThe value of the firm’s equity

• The value of the equityholders claim at any time prior to debt maturity is expressed by:

]1)[(),,,( TT

Qt

rTKtt KPVEeTItVS

• The governing partial differential equation that should be solved to value the firm’s stocks as a function of the two state variables V and I is:

0)(2 2

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I

SrS

V

SVr

V

SV

t

S

• The boundary conditions are as follows:

dI forPVTITVS KTT

KTT 0 )0 ,( max),,,(

0),,, ( TdVS kt

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The value of the firm’s debtThe value of the firm’s debt

• The value of the zero-coupon bond is decomposed to two sources of value:

1. The value at maturity, assuming the firm is not prematurely liquidated.

2. The value if the firm is liquidated before debt maturity, since the pre-determined grace period was violated by the weighted excursion distress time.

]1[]1),[min(),,,( T

rQtT

rTT

Qt

Ktt K

K

KK eVePVTItVB

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An example of calibrating the model to An example of calibrating the model to

market datamarket data • The unique model parameters ( and ) are calculated by minimizing the

mean-absolute- error (MAE) between the observed historical credit spread of four bonds that are typical for their rating category and the calculated model’s spreads:

4

1,

1 min

N

iii SppS

N

• Where:

Spi - the observed credit spread of the bond which belong to the ith rating

category

- the spread which is calculated by the model

N - the number of rating categories

ipS

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General parameters for the base caseGeneral parameters for the base case

Parameter Name Value Source/ Calculation method

Interest rate 6% Average yield on ten years Treasury bonds between 1998-2006

Dividend yield 0 Typical to B-rated bond

Grace period 1 year Covitz, Han and Wilson (2006).

Time to Maturity 10 years A typical IPO of corporate bond

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Specific parameters for each rating Specific parameters for each rating categorycategory

Rating Category

Leverage ratio

Assets Volatility

Average Market spread

Average Credit Spread

A 0.3198 22.40% 1.23% 0.22%

BBB 0.4328 23.00% 1.94% 0.67%

BB 0.5353 27.00% 3.20% 1.11%

B 0.657 29.00% 4.70% 1.63%

• Leverage ratio: Standard & Poor’s (1999) used by Huang and Huang (2003)

• Assets Volatility: Strebulaev and Schaefer (2005)

• The average bond spread is based on average yield spreads as calculated by Caouette, Altman and Narayanan (1998). Since our model explains only the credit spread component, we multiply the total yield spread with the percent of yield spread due to default as calculated by Elton, Gruber, Agrawal and Mann (2001).

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The Calibration resultsThe Calibration results

Calibration Parameters A BBB BB B MAE RMSE

Calibration with and

(1.8)

Calibration with (.41)

The Cumulative excursion method

()

The consecutive excursion method

(. ® )

Market Spreads 0.22% 0.67% 1.11% 1.63%

1.12% 1.63%

Model’s credit spreads Model’s statistics

0.23% 0.51% 1.12% 1.63% 0.45 0.8

0.52 0.92

0.21% 0.47% 0.99% 1.43% 1.33 1.56

0.22% 0.49%

1.19 1.360.24% 0.55% 1.23% 1.83%

• We search for the parameters and that minimize the MAE (mean absolute error) between the credit spread of typical A, BBB, BB and B rated bonds and the model spreads (the MAE and the RMSE are multiplied by 1,000)

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Empirical implications of the modelEmpirical implications of the model

1. An increase of the grace period leads to an increase in credit spreads.

2. Modeling the bankruptcy procedure becomes more important for high leverage ratios.

Scenario Equity value

Debt Value

Credit spread

0 42.99 57.01 1.43%

Base case 0.41 44.18 55.82 1.63%

(d= 1 ) 1.5 45.01 54.99 1.78%

45.34 54.66 1.84%

0 38.8 61.2 0.71%

d= 0 . 25 0.41 39.21 60.79 0.78%

1.5 39.71 60.29 0.86%

40.37 59.63 0.97%

Merton (1974) 49.19 50.81 2.57%

0 25.28 74.72 1.86%

d= 1 0.41 27.6 72.4 2.18%

(One year) 1.5 29.34 70.76 2.42%

29.63 70.37 2.46%

0 17.75 82.25 0.90%

d= 0 . 25 0.41 18.49 81.51 0.99%

(3 months) 1.5 19.46 80.54 1.11%

20.65 79.35 1.26%

Merton (1974) 44.71 55.29 3.85%

LR=0.657

LR=0.9

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Empirical implications of the model (Cont’)Empirical implications of the model (Cont’)

3. Volatility has greater impact the higher is the liquidation procedure’s memory, i.e., better bondholder protection (small ), has smaller impact on the absolute change of credit risk.

0.50%

0.75%

1.00%

1.25%

1.50%

1.75%

2.00%

2.25%

2.50%

0 0.5 1 1.5 2 2.5

Cre

dit

sp

read

.24 .29 .34

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Empirical implications of the model (Cont’)Empirical implications of the model (Cont’)

4. The sensitivity of the stock values to asset’s volatility increases with and as a result the incentive for assets substitution increases as well (see Jensen and Meckling (1976))

30.00

35.00

40.00

45.00

50.00

55.00

60.00

10.0% 15.0% 20.0% 25.0% 30.0% 35.0%

Assets volatility

Sto

ck p

ric

e

Merton beta=0 (Moraux 2002)

beta--> infinity (FM 2002) beta=0.41(our base case )

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Empirical implications of the model (Cont’)

5. Financial leverage has greater impact the higher is the liquidation procedure’s memory, i.e., better bondholder protection (small ), has smaller impact on the absolute change of credit risk.

1.00%

1.20%

1.40%

1.60%

1.80%

2.00%

2.20%

2.40%

2.60%

0 0.5 1 1.5 2

Cre

dit

sp

rea

d

LR=0.779 LR=0.657 LR=0.9

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Summary

• A flexible model of default based on triggers that accumulate during distress periods but are forgiven over time, allows to value corporate securities for different legal and contractual regimes.