Liquidation Triggers and the Valuation of Equity and Debt

1. Liquidation Triggers and the Valuation of Equity and Debt May 2007 Dan Galai, Alon Raviv and Zvi Wiener

2. 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.

3. 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).

5. 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.

6. 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: • The length of past distress events • Consecutiveness of past and current distress events. • The severity of the distress event. • The distance of past distress events from current time.

7. Outline • Literature review • Model and Assumptions • Calibration of the model to market data • Empirical implications

8. Pricing corporate liabilities • Structural approach. • Merton 1974 classic option • Black and Cox 1976 first passage. • Reduced form/intensity based approach • Jarrow and Turnbull (1995) • Duffie and Singleton (1998)

9. 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 VTare 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.

10. Payoffs at Maturity • With absolute priority, we have the following payoffs at maturity T: • The bond payoff is: • The equity payoff is: • Equity is valued as a call option on the value of the firm’s assets.

11. 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 Kt:

12. 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 80iesand 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.

13. 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.

14. Liquidation procedures as basketball penalty method

15. 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 No liquidation till debt maturity!!!

16. 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.

17. Liquidation procedures as soccer penalty method

18. 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). Liquidation is triggered!!!

19. 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.

20. 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.

21. Model’s Assumptions: Conventional • Assets are continuously traded in an arbitrage-free and complete market • The interest rate level, r, is assumed to be constant. • 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: • Where: • The firm has outstanding only equity and a single bond issue with a promised final payment of P, that mature at time T.

22. Model’s Assumptions (cont’d) • The bondholders are theoretically allowed to force liquidation in one of two ways: • 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. • If the value of the assets is less than some constant F at debt maturity. • The time dependent threshold level Kt is defined by:

23. The cumulative distress time • 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: where:

24. The cumulative distress time (Cont’d) • The liquidation state variable is calculated at each day t as: Severity of the distress event Time decay factor Effect of past distress periods Effect of current distress period where:

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

26. The 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: • Where a=0, the severity of the distress period has no impact on the liquidation procedure and the liquidation state variable is defined by:

27. Previous contributions as special cases of the adjustable excursion method • Example 1. When b -> and g=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 theFrançois 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)].

28. Previous contributions as special cases of the adjustable excursion method (Cont’d) • Example 2. When b =0 and g=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 theMoraux (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).

29. The value of the firm’s equity • The value of the equityholders claim at any time prior to debt maturity is expressed by: • 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: • The boundary conditions are as follows:

30. The value of the firm’s debt • The value of the zero-coupon bond is decomposed to two sources of value: • The value at maturity, assuming the firm is not prematurely liquidated. • The value if the firm is liquidated before debt maturity, since the pre-determined grace period was violated by the weighted excursion distress time.

31. 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 An example of calibrating the model to market data • The unique model parameters (a and b) 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:

32. General parameters for the base case

33. Specific parameters for each rating category • 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).

34. The Calibration results • We search for the parameters a and bthat 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)

35. Empirical implications of the model • An increase of the grace period leads to an increase in credit spreads. • Modeling the bankruptcy procedure becomes more important for high leverage ratios.

36. Empirical implications of the model (Cont’) • Volatility has greater impact the higher is the liquidation procedure’s memory, i.e., better bondholder protection (small b), has smaller impact on the absolute change of credit risk.

37. Empirical implications of the model (Cont’) • The sensitivity of the stock values to asset’s volatility increases with b and as a result the incentive for assets substitution increases as well (see Jensen and Meckling (1976))

38. Empirical implications of the model (Cont’) • Financial leverage has greater impact the higher is the liquidation procedure’s memory, i.e., better bondholder protection (small b), has smaller impact on the absolute change of credit risk.

39. 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.