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					    Liquidation Triggers and the
      Valuation of Equity and
                Debt

                    May 2007

        Dan Galai, Alon Raviv and Zvi Wiener



1
        Net-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.
2
             Liquidation trigger
•   Liquidation 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).
3
4
      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.


5
               Our 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.


6
                      Outline

    1.   Literature review
    2.   Model and Assumptions
    3.   Calibration of the model to market data
    4.   Empirical implications




7
          Pricing corporate liabilities

    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)


8
    Structural Approach - Merton (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.

9
                  Payoffs at Maturity
•        With absolute priority, we have the following payoffs at
         maturity T:
                              Bonds  Equity
                  VT  F       F    VT  F
                   VT  F      VT      0

• The bond payoff is:          B(T )  min( VT , F )  F  max( F  VT ,0)

• The equity payoff is:        E (T )  max( VT  F ,0)

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

    10
              First-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 K:  inf  t  0 K  V 
                                             t                                                                                             t   t


              120.0


              100.0


               80.0


               60.0

                                                                              First passage time: Liquidation
               40.0


               20.0


                0.0
                 0
                      4
                           8
                                3
                                     7
                                          1
                                               5
                                                    9
                                                         3
                                                              8
                                                                   2
                                                                        6
                                                                             0
                                                                                  4
                                                                                       8
                                                                                            2
                                                                                                 7
                                                                                                      1
                                                                                                           5
                                                                                                                9
                                                                                                                     3
                                                                                                                          7
                                                                                                                               2
                                                                                                                                       6
                                                                                                                                      .0
                0.
                      0.
                           0.
                                1.
                                     1.
                                          2.
                                               2.
                                                    2.
                                                         3.
                                                              3.
                                                                   4.
                                                                        4.
                                                                             5.
                                                                                  5.
                                                                                       5.
                                                                                            6.
                                                                                                 6.
                                                                                                      7.
                                                                                                           7.
                                                                                                                7.
                                                                                                                     8.
                                                                                                                          8.
                                                                                                                               9.
                                                                                                                                    9.
                                                                                                                                    10
                                                                         Years
    11                                          Value of the firm assets                Absorbing threshold
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.


    12
    The consecutive excursion method
      (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.


    13
Liquidation procedures as basketball
          penalty method




14
    The deficiencies of the consecutive
            excursion method
•        Each time firm value falls below the threshold level an additional grace
         period is granted without reference to previous instances of insolvency
            120.0
            110.0
            100.0
             90.0
             80.0
             70.0
             60.0
             50.0
             40.0
                                                                                          No liquidation till
             30.0                                                                         debt maturity!!!
             20.0
             10.0
              0.0
               0
                    4
                         8
                              3
                                   7
                                        1
                                             5
                                                  9
                                                       3
                                                            8
                                                                 2
                                                                      6
                                                                           0
                                                                                4
                                                                                     8
                                                                                           2
                                                                                               7
                                                                                                     1
                                                                                                          5
                                                                                                              9
                                                                                                                   3
                                                                                                                         7
                                                                                                                             2
                                                                                                                                  6
                                                                                                                                         0
              0.
                    0.
                         0.
                              1.
                                   1.
                                        2.
                                             2.
                                                  2.
                                                       3.
                                                            3.
                                                                 4.
                                                                      4.
                                                                           5.
                                                                                5.
                                                                                     5.
                                                                                          6.
                                                                                               6.
                                                                                                    7.
                                                                                                         7.
                                                                                                              7.
                                                                                                                   8.
                                                                                                                        8.
                                                                                                                             9.
                                                                                                                                  9.

                                                                                                                                          .
                                                                                                                                       10
                                                                       Years
                                         Value of the firm's assets                                 Distress threshold
    15                                   Liquidation state variable                                 Grace period
   The cumulative excursion method
           (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.
  16
     Liquidation procedures as
       soccer penalty method




17
            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).
                140.0


                120.0


                100.0


                 80.0


                 60.0


                 40.0                                                                              Liquidation is
                 20.0
                                                                                                    triggered!!!

                  0.0
                   0
                        4
                             8
                                   3
                                        7
                                            1
                                                 5
                                                      9
                                                           3
                                                                8
                                                                     2
                                                                          6
                                                                               0
                                                                                    4
                                                                                          8
                                                                                              2
                                                                                                    7
                                                                                                        1
                                                                                                             5
                                                                                                                  9
                                                                                                                       3
                                                                                                                            7
                                                                                                                                 2
                                                                                                                                      6
                                                                                                                                             0
                  0.
                        0.
                             0.
                                  1.
                                       1.
                                            2.
                                                 2.
                                                      2.
                                                           3.
                                                                3.
                                                                     4.
                                                                          4.
                                                                               5.
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                                                                                                             7.
                                                                                                                  7.
                                                                                                                       8.
                                                                                                                            8.
                                                                                                                                 9.
                                                                                                                                      9.

                                                                                                                                              .
                                                                                                                                           10
                                                                           Years
    18                            Value of the firm's assets                             Distress threshold
                                  Granted period                                         Liquidation state variable
                      To summarize…
•        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.




    19
         The adjustable excursion method

•        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.



    20
             Model‟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: dVt  (r   )Vt dt  Vt dWt
•        Where: Vt - The value of the firm's assets
                    - The instantaneous standard deviation of the rate of return of the firm.
                    - The firm's payout ratio
                   W - A standard Brownian motion

    4. The firm has outstanding only equity and a single bond
       issue with a promised final payment of P, that mature at
       time T.
    21
        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:


                  K t  Fe  r (T t ) where : 0    1




 22
            The 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:


                               gtK  sup s  t Vs  K s 
where:
            g tk  is the last time before t that firm value crossed threshold K s .



    23
      The cumulative distress time (Cont’d)
 •     The liquidation state variable is calculated at each day t as:

                                                                                   Severity of
          Time decay
                                                                                   the distress
            factor
                                                                                      event
                             g tK                           t
                     I tK   e   (t  s ) f (Vs )ds   e  (t  s ) f (Vs )ds
                             
                            0                            g tK



                      Effect of past                                          Effect of
                     distress periods                                          current
where:                                                                     distress period
   - Decay factor for past distress periods.
   - Decay factor for thelast distress period.
f (Vt ) - Defines the impact of the severity of the distress event on the liquidatio n state variable.

 24
          The severity of the distress event
•        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:
                                   Kt Vt  
                                             
                                e  Kt  
                                    
                                             
                                                           Vt  K t
                      f (Vt )  
                                0                         Vt  K t
                                
                                                     0

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

    25
                  The liquidation event
•        Liquidation occurs in the first time that the liquidation state
                                                             K
         variable exceeds d. The liquidation time is denoted by         ,
         and it is defined mathematically by:

                                           
                           K  inf t  0 I tK  d , Vt  K t                     
•        Where 0, the severity of the distress period has no
         impact on the liquidation procedure and the liquidation
         state variable is defined by:
                              g tK                                t
                     I tK     e   (t  s ) 1 Vs  Ks  ds   e  ( t  s ) 1 Vs  Ks  ds
                               0                                 gtK




    26
    Previous contributions as special cases of
        the adjustable excursion method
•        Example 1. When   and 0 , 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)].
    27
         Previous contributions as special
         cases of the adjustable excursion
                 method (Cont’d)
•        Example 2. When  0 and 0 , , 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).
    28
                    The value of the firm‟s equity
        •     The value of the equityholders claim at any time
              prior to debt maturity is expressed by:

                           S (Vt , t , I tK ,T )  e  rT EtQ [(VT  P)  1  K T ]

•        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:
                            S  2V 2  2 S             S        S
                                            (r   )V     rS     0
                            t   2 V     2
                                                        V        I
    •        The boundary conditions are as follows:

                    S (VT , T , I TK , T )  max (VT  P, 0)                 for 0  I TK  d

                    S ( Vt , k , d , T )  0
        29
                    The 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.

                                                    rT                              r K
           B (Vt , t , I , T )   [min(VT , P)e
                     t
                      K        Q
                               t                          1 K T ]   [V K e
                                                                        Q
                                                                        t                    1  K T ]



    30
An example of calibrating the model to market 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:


                                      1   N 4   
                             min
                               
                                ,     N
                                          
                                          i 1
                                                 Spi  Spi

      •   Where:
          Spi - the observed credit spread of the bond which belong to the ith rating
          category
           
          Spi - the spread which is calculated by the model
           N - the number of rating categories
 31
 General 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




  32
Specific parameters for each rating category
              Rating    Leverage    Assets        Average       Average
             Category     ratio    Volatility   Market spread 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
    33
       calculated by Elton, Gruber, Agrawal and Mann (2001).
                             The Calibration results

                                        Model’s credit spreads           Model’s statistics

     Calibration Parameters       A       BBB          BB          B     MAE        RMSE
     Calibration with a and b
                                0.23%    0.51%       1.12%       1.63%    0.45        0.8
         (a=1 , b=0 .8 )
        Calibration with b
                                0.22%     0.49%      1.12%       1.63%    0.52        0.92
         (a=0 , b=0 .4 1 )
    The Cumulative excursion
             method             0.21%     0.47%      0.99%       1.43%    1.33        1.56
           (a=0 , b=0 )
    The consecutive excursion
             method             0.24%     0.55%      1.23%       1.83%    1.19        1.36
         (a=0 ., b® ¥)

         Market Spreads         0.22%    0.67%       1.11%       1.63%



•      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)
34
          Empirical implications of the model
1.   An increase of the grace                2.    Modeling the bankruptcy procedure
     period leads to an increase                   becomes more important for high
     in credit spreads.                            leverage ratios.
                                                   Equity   Debt    Credit
                              Scenario       b
                                                    value   Value   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%
                 LR=0.657                    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%
                  LR=0.9                     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%
35
                            Merton (1974)          44.71    55.29   3.85%
      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.


                           2.50%
                           2.25%
                           2.00%
           Credit spread




                           1.75%
                           1.50%
                           1.25%
                           1.00%
                           0.75%
                           0.50%
                                   0   0.5        1            1.5       2   2.5
                                                          b

                                         s=0.24       s=0.29    s=0.34


36
      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))

                              60.00

                              55.00

                              50.00
                Stock price




                              45.00

                              40.00

                              35.00

                              30.00
                                 10.0%      15.0%         20.0%       25.0%       30.0%           35.0%
                                                           Assets volatility
                                         Merton                        beta=0 (Moraux 2002)
                                         beta--> infinity (FM 2002)    beta=0.41(our base case)
37
      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.



                               2.60%
                               2.40%
                               2.20%
               Credit spread




                               2.00%
                               1.80%
                               1.60%
                               1.40%
                               1.20%
                               1.00%
                                       0   0.5        1               1.5            2
                                                       b
                                           LR=0.779        LR=0.657         LR=0.9

38
                     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.




39

				
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