Distressed Debt Prices and Recovery Rate Estimation

Transcription

1 Distressed Debt Prices and Recovery Rate Estimation Robert Jarrow Joint Work with Xin Guo and Haizhi Lin May 2008

2 Introduction Recent market events highlight the importance of understanding credit risk. Credit risk pricing and hedging involves understanding: 1. interest rates (stochastic discounting), 2. default process (when payments stop), and 3. recovery rate process (what happens after default). Points 1 and 2 well-studied. Point 3, less so...

3 Introduction Three sources of knowledge on recovery rates. 1. Industry papers: estimate recovery rates (not transparent, not validated by academic community), and study their properties (correlation with business cycle, dependence on rm characteristics,...) 2. Academic papers - use industry generated recovery rates to study their properties. 3. Academic papers - use pre-default debt and CDS prices to implicitly estimate recovery rates.

4 Introduction Potential problems with existing knowledge. Are we sure recovery rates are estimated correctly? then... If not, academic papers study mis-speci ed estimates, academic papers have no base to compare implicit estimates.

5 Introduction Purposes Primary- provide direct estimates of recovery rates using distressed debt prices. Secondary - t a model for defaulted debt prices. (it turns out, to solve one, must also solve the other)

6 Introduction Results 1. Recovery rate estimates are sensitive to the date selected for estimation (signi cant di erences between using the recorded default date and 30 days after). 2. Prices support the belief that the market often recognizes default before default is recorded. 3. An extended recovery rate model provides a poor t to distressed debt prices after the recorded default date (extension implicit in using 30 day after to estimate recovery rate). 4. We estimate a new recovery rate process and use it to price distressed debt. The model ts market prices well.

7 Prologue Structural models Use management s information set. Default can be viewed as the rst hitting time of the rm s asset value to a liability determined barrier. If the rm s asset value follows a continuous process, the value of a rm s debt does not exhibit a jump at default. No implications for risky debt prices subsequent to default. Reduced Form Models Use market s information set. Default modeled as the rst jump time of a point process. Debt prices exhibit a negative jump at default. No implications for risky debt prices subsequent to default.

8 Prologue 55 series Sep Feb Jul Dec 2005 Figure: Delta Airlines. Bankruptcy on September 14, Consistent with the standard structural model. 30-day di erent from default date.

9 Prologue 100 series Oct Nov Jan Mar 2005 Figure: Trico Marine Service Inc. Bankruptcy on December 18, Inconsistent with the structural model. (market recognized default earlier?) 30-day approximately same as default date.

10 Prologue 100 series Sep Feb Jul Dec 2005 Figure: Winn Dixie Stores. Bankruptcy on February 21, Consistent with the standard reduced form model. 30-day di erent from default date.

11 Prologue 90 series Apr Jun Sep Dec 2005 Figure: Northwest Airlines. Bankruptcy on September 14, Partially consistent with both the reduced form and structural. 30-day di erent from default date.

12 Fix a particular rm. Set Up Let B t denote the price of its risky debt (a particular issue with a given maturity, coupons ( oating or xed), and embedded options). De ne the economic default date τ as the time when the market knows default has happened. The recorded default date τ where τ τ is given in our data set. Let Bt d denote the risky debt price given economic default has already happened, i.e. for t τ, Bt d = B t. Let r t be riskless spot rate of interest. Let p t (T ) be price of a riskless coupon bond with the same maturity T and coupons as the risky bond under consideration.

13 Cross-Sectional Models 1. Recovery of Face Value (RFV): Bτ d = δ τ F where F is the face value of the debt (normalized to $100) 2. Recovery of Treasury (RT): Bτ d = δ τ p τ (T ) 3. Recovery of Market Value (RMV): B d τ = δ τ B τ

14 Cross-Sectional Models Purpose of these models is to provide the necessary inputs to price risky debt and credit derivatives prior to default. Recovery rate estimation procedure is: x a defaulted company x a date τ to observe debt prices, then estimate the recovery rate. Single point estimate of the recovery rate per company. Look cross-sectionally across companies to obtain estimate. For example, Moody s uses "30-day" post-default date for τ.

15 Data December 2000 to October Debt Price Data - Advantage Data Corporation - Trade data and broker quotes to get end of day prices 4:45 p.m. Filter data: have 50 prices over a 60 day window surrounding recorded default date. Remove bond issues with missing data on maturity, coupons. This leaves 96 issues remaining for recovery rate estimation. Filters imply that all our defaulted debt issues eventually le for bankruptcy (potential selection bias). Bond Characteristics - Mergent Fixed Income Database. Default is when a debt issue violates a bond covenant, misses a coupon or principal payment, or les for bankruptcy. A grace period of 30 days must usually pass before default is recognized for a missed coupon.

16 Data - Bankruptcy Time Analysis To get a sense for duration of distressed debt market, considering only issues that le for bankruptcy. N = 1902 Mean Std. Dev. Median N λ Chapter Chapter Time in Bankruptcy in Days λ is average time spent in bankruptcy in years.

17 Cross-sectional Recovery Rates (RFV) Di erence Count Avg. Price Std. Dev. Avg. Ratio RFV at recorded default statistically di erent from RFV at 30 day

18 Cross-sectional Recovery Rates (RT) N = 96 RT Estimates Mean Median Standard Deviation First Quartile Third Quartile Lower than the RFV estimates because otherwise identical default free bonds trade at a premium (> $100).

19 Cross-sectional Recovery Rates (RMV) N = 96 Pre-Default Default Date RMV Estimates Mean Median Standard Deviation First Quartile Third Quartile Debt prices do not jump on the default date. Implies that, on average, the debt is "riskless." Anomalous result: either the RMV is a poor model for recovery rates, or the recorded default date does not equal the economic default date.

20 Time-Series Models B d t R t = m δ τ e τ r s ds where 8 < F if RFV m = p τ (T ) if RT : B τ if RMV. Assumes that risky debt position is sold at τ, and the model prices debt as the value of this position. Equivalently, B d t = B d τ e R t τ r s ds for t τ. This form is independent of model type. Use this to: 1. Determine economic default date. 2. Test accuracy of valuation model.

21 Time-Series Models - Economic Default Date Given our de nition of the economic default date, using debt prices, our estimator is: bτ = inf τ 180tτ ft : B t Bτ d R e τ t r s ds g. Bound below by 180 days before the recorded default date. Our estimator depends on the information up to time τ.

22 Cases Time-Series Models - Economic Default Date 25 Time Between Economic Default Date and Announced Default Date Days Figure: N = 73.

23 Time-Series Models - Revised Recovery Rates 8 >< bδ τ = >: B τ F if RFV B τ p(τ,t ) if RT B τ B τ if RMV.

24 Time-Series Models - RFV N = 73 Economic Default Recorded Default Mean * Median Standard Deviation First Quartile Third Quartile *P value essentially zero.

25 Time-Series Models - RT N = 73 Economic Default Recorded Default Mean * Median Standard Deviation First Quartile Third Quartile *P value essentially zero.

26 Time-Series Models - RMV N = 73 Economic Default Recorded Default Mean * Median Standard Deviation First Quartile Third Quartile *P value essentially zero. This is consistent with a jump on the economic default date.

27 Density Time-Series Models - RMV Recovery of Market Value Estimates Based on Economic Defaults Recov ery of Market Value Estimates Based on Announced Def aults Recov ery Rate Estimates

28 Time-Series Models - Pricing Tests B d t = m bδ τ er t τ r s ds + ɛ t for t τ. "Good" if the residuals have zero mean, and are i.i.d. N = 20,942 Pricing Errors Mean Median 9.31 Standard Deviation First Quartile 0.11 Third Quartile Very large pricing errors.

29 Time-Series Models - Pricing Tests Run for each bond issue the time-series regression ɛ t = α + βt and test if α = 0 and β = bond issues in our sample. For 87 we reject the null hypothesis that α = 0 and β = 0 with a signi cance level of 0.01 (for 79 we have negligible P-values). 77 out of 103 issues produce positive slopes. Rejects distressed debt pricing model. Why? information on default resolution after τ. Ignores Provides additional rejection of using the 30-day recovery.

30 The Recovery Rate Model Database limitations - model the resolution of the bankruptcy ling. Restrict to t τ. Let τ 0 represent the time to resolution of bankruptcy. Exponential distribution with parameter λ. Dollar payo equal to m δ τ0 0 where 8 < F if RFV m = p τ (T ) if RT : B τ if RMV.

31 The Recovery Rate Model Assume that distressed debt trades in the standard continuous time arbitrage free setting. Bt d R τ0 = me δ τ0 e t r s ds jf t Z R s = me δ s e t rudu λe λ(s t) ds jf t t where E () is expectation under equivalent martingale probability measure.

32 De ne where The Recovery Rate Model R s δ s e dr t = a(b R s τ r udu R t )dt + σdw t with a, b, σ are constants and W t is a standard B.M. under the martingale measure. Distressed debt price: Bt d R t = me τ r udu = m δ t R t λ (a + λ) + ba (a + λ) λ (a + λ) + ba R (a + λ) e t τ r udu

33 Recovery Rate Model - Estimation Methodology Estimate λ = 0.80 using bankruptcy data. Given τ = τ, estimate (a, b, σ, ρ) and (R t ) tτ using Kalman Filter: Bt d R m e t τ rudu = A t + H t R t + ɛ t where ɛ t N(0, ρ) is i.i.d. observation error. R t = C t + F t R t 1 + η t where η t N(0, σ2 2a (1 A t ba (a + λ), H t 2e a ) and Given (a, b, σ, ρ, λ) and (R t ) tτ, estimate τ = λ (a + λ), C t b(1 e a ), F t e a inf τ 180tτ ft : B t Bτ d R e τ t r u du e a(τ t) + mb(1 e a(τ t) )g

34 Cases Recovery Rate Model - Economic Default Date 35 Time Between Economic Def ault Date and Announced Def ault Date Days Figure: N = 82.

35 Recovery Rate Model - Parameters N = 21,083 a b σ ρ Mean Median Std Dev First Quartile Third Quartile

36 Recovery Rate Estimates Recovery Rate Model - Recovery Rates 0.75 Average Recovery Rate Estimates Number of Days After Economic Default

37 Recovery Rate Model - Recovery Rates RFV RT RMV N = 69 EM KFM EM KFM EM KFM Mean * * ** Median StdDev % % At economic default date.

38 Recovery Rate Model - Pricing Tests N = 21,083 Pricing Errors in Dollars Mean Median Standard Deviation First Quartile.0124 Third Quartile Face Value $100

39 Recovery Rate Model - Pricing Tests Run for each bond issue the time series regression ɛ t = α + βt and test if α = 0 and β = 0. For 83 out of 103 issues, we fail to reject the null hypothesis that α = 0 and β = 0 with signi cance level We perform a Durbin-Watson autocorrelation test. For 62 out of the 103 issues, we fail to reject the null hypothesis that corr(ɛ t, ɛ t 1 ) = 0 with signi cance level For most issues, recovery rate model ts data well.

40 Epilogue 80 series Sep Aug Jun Apr 2007 Figure: Delta Airlines

41 Epilogue 100 series Oct Nov Jan Mar 2005 Figure: Trico Marine Service Inc.

42 Epilogue 110 series Sep Jun Mar Dec 2006 Figure: Winn Dixie Stores

43 Epilogue 110 series Apr Dec Sep Jun 2007 Figure: Northwest Airlines

44 Conclusions Economic default date di ers from reported default date. Recovery rate estimates based on 30 days after recorded default are misspeci ed. Recovery rate estimates based on recorded default date signi cantly di erent from economic default date. Implies that: Existing academic studies using 30 day recovery rates, results quantitatively incorrect, but qualitatively? Recovery rate estimates can be improved. Time to economic default estimates can be improved. Prices of credit derivatives and pre-default risky debt can be improved. Exploring these implications await subsequent research.