Distressed Debt Prices and Recovery Rate Estimation Robert Jarrow - - PowerPoint PPT Presentation

Distressed Debt Prices and Recovery Rate Estimation

Robert Jarrow Joint Work with Xin Guo and Haizhi Lin May 2008

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...

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.

Introduction

Potential problems with existing knowledge.

Are we sure recovery rates are estimated correctly? If not,

then...

academic papers study mis-speci…ed estimates, academic papers have no base to compare implicit estimates.

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)

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.

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.

Prologue

3 0 -Se p -2 0 0 4 2 7 -F e b -2 0 0 5 2 7 -J u l -2 0 0 5 2 4 -De c -2 0 0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 5 5 0 5 5 s e ri e s 1

Figure: Delta Airlines. Bankruptcy on September 14, 2005.

Consistent with the standard structural model. 30-day di¤erent from default date.

Prologue

0 3 -Oc t-2 0 0 4 3 0 -No v -2 0 0 4 2 7 -J a n -2 0 0 5 2 6 -M a r-2 0 0 5 4 0 5 0 6 0 7 0 8 0 9 0 1 0 0 s e ri e s 1

Figure: Trico Marine Service Inc. Bankruptcy on December 18, 2004.

Inconsistent with the structural model. (market recognized default earlier?) 30-day approximately same as default date.

Prologue

29-Se p-2004 26-Fe b-2005 26 -J u l-2 00 5 23-De c -2005 40 50 60 70 80 90 1 00 s e rie s 1

Figure: Winn Dixie Stores. Bankruptcy on February 21, 2005.

Consistent with the standard reduced form model. 30-day di¤erent from default date.

Prologue

0 3 -Ap r-2 0 0 5 3 0 -J u n -2 0 0 5 2 6 -Se p -2 0 0 5 2 3 -De c -2 0 0 5 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 s e ries 1

Figure: Northwest Airlines. Bankruptcy on September 14, 2005.

Partially consistent with both the reduced form and structural. 30-day di¤erent from default date.

Set Up

Fix a particular …rm.

Let Bt 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 Bd

t denote the risky debt price given economic default has

already happened, i.e. for t τ, Bd

t = Bt.

Let rt be riskless spot rate of interest. Let pt(T) be price of a riskless coupon bond with the same

maturity T and coupons as the risky bond under consideration.

Cross-Sectional Models

• 1. Recovery of Face Value (RFV):

Bd

τ = δτF

where F is the face value of the debt (normalized to $100)

• 2. Recovery of Treasury (RT):

Bd

τ = δτpτ(T)

• 3. Recovery of Market Value (RMV):

Bd

τ = δτBτ

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.

Data

December 2000 to October 2007. 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.

Data - Bankruptcy Time Analysis

To get a sense for duration of distressed debt market, considering

• nly issues that …le for bankruptcy.

N = 1902 Mean

• Std. Dev.

Median N λ Chapter 7 433.22 353.20 433 9 0.80 Chapter 11 454.34 427.08 354 631 0.84 Time in Bankruptcy in Days λ is average time spent in bankruptcy in years.

Cross-sectional Recovery Rates (RFV)

Di¤erence Count

• Avg. Price
• Std. Dev.
• Avg. Ratio
• 30

23 62.19 33.04 1.3199

• 20

58 48.99 28.06 1.3510

• 10

27 66.74 28.60 1.2382

• 5

51 40.42 25.81 1.2114

• 2

41 44.60 29.99 1.0541

• 1

61 45.05 29.55 0.9796 70 48.17 29.39 1 1 71 45.48 28.67 1.0292 2 63 41.27 28.85 1.0284 5 44 48.32 31.48 1.0341 10 45 54.62 28.83 1.0933 20 46 53.86 32.44 1.1473 30 64 42.31 29.30 1.0779

RFV at recorded default 48.17 statistically di¤erent from RFV at 30 day 42.31.

Cross-sectional Recovery Rates (RT)

N = 96 RT Estimates Mean 0.4062 Median 0.3452 Standard Deviation 0.2528 First Quartile 0.1692 Third Quartile 0.6374 Lower than the RFV estimates because otherwise identical default free bonds trade at a premium (> $100).

Cross-sectional Recovery Rates (RMV)

N = 96 Pre-Default Default Date RMV Estimates Mean 48.4 48.6 1.0230 Median 39 38.5 1.0013 Standard Deviation 30.6 30.7 0.1824 First Quartile 21.5 22 0.9681 Third Quartile 67.55 69.375 1.0597

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.

Time-Series Models

Bd

t = m δτe R t

τ rsds

where m = 8 < : F if RFV pτ(T) if RT Bτ if RMV . Assumes that risky debt position is sold at τ, and the model prices debt as the value of this position. Equivalently, Bd

t = Bd τe R t

τ rsds for t τ.

This form is independent of model type. Use this to:

• 1. Determine economic default date.
• 2. Test accuracy of valuation model.

Time-Series Models - Economic Default Date

Given our de…nition of the economic default date, using debt

prices, our estimator is: b τ = inf

τ180tτft : Bt Bd τe R τ

t

rsdsg.

Bound below by 180 days before the recorded default date. Our estimator depends on the information up to time τ.

Time-Series Models - Economic Default Date

20 40 60 80 100 120 140 160 180 5 10 15 20 25

Days

C a s e s Time Between Economic Default Date and Announced Default Date

Figure: N = 73.

Time-Series Models - Revised Recovery Rates

b δτ = 8 > < > :

Bτ F

if RFV

Bτ p(τ,T )

if RT

Bτ Bτ

if RMV .

Time-Series Models - RFV

N = 73 Economic Default Recorded Default Mean 0.4879* 0.5283 Median 0.45 0.5782 Standard Deviation 0.3044 0.3151 First Quartile 0.2 0.2225 Third Quartile 0.76 0.8425 *P value essentially zero.

Time-Series Models - RT

N = 73 Economic Default Recorded Default Mean 0.3970* 0.4335 Median 0.3291 0.4776 Standard Deviation 0.2461 0.2600 First Quartile 0.1578 0.1803 Third Quartile 0.6031 0.6610 *P value essentially zero.

Time-Series Models - RMV

N = 73 Economic Default Recorded Default Mean 0.8314* 1.0653 Median 0.9094 1.0217 Standard Deviation 0.1775 0.1729 First Quartile 0.7267 0.9976 Third Quartile 0.9649 1.0854 *P value essentially zero. This is consistent with a jump on the economic default date.

Time-Series Models - RMV

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 1 2 3 4 5 6 7

Recov ery Rate Estimates Density Recov ery of Market Value Estimates Based on Economic Def aults Recov ery of Market Value Estimates Based on Announced Def ault

Time-Series Models - Pricing Tests

Bd

t = m b

δτe

R t

τ rsds + ǫt for t τ.

"Good" if the residuals have zero mean, and are i.i.d. N = 20,942 Pricing Errors Mean 17.81 Median 9.31 Standard Deviation 28.06 First Quartile 0.11 Third Quartile 30.00 Very large pricing errors.

Time-Series Models - Pricing Tests

Run for each bond issue the time-series regression

ǫt = α + βt and test if α = 0 and β = 0.

103 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? Ignores

information on default resolution after τ.

Provides additional rejection of using the 30-day recovery.

The Recovery Rate Model

Database limitations - model the resolution of the bankruptcy

…ling.

Restrict to t τ. Let τ0 represent the time to resolution of bankruptcy. Exponential distribution with parameter λ. Dollar payo¤ equal to m δτ0 0 where

m = 8 < : F if RFV pτ(T) if RT Bτ if RMV .

The Recovery Rate Model

Assume that distressed debt trades in the standard continuous time arbitrage free setting. Bd

t

= mE

• δτ0e R τ0

t

rsds jFt

• =

mE Z ∞

t

δse R s

t ruduλeλ(st)ds jFt

• where E() is expectation under equivalent martingale probability

measure.

The Recovery Rate Model

De…ne Rs δse R s

τ rudu

where dRt = a(b Rt)dt + σdWt with a, b, σ are constants and Wt is a standard B.M. under the martingale measure. Distressed debt price: Bd

t

= me

R t

τ rudu

• Rt

λ (a + λ) + ba (a + λ)

• =

m

• δt

λ (a + λ) + ba (a + λ)e

R t

τ rudu

Recovery Rate Model - Estimation Methodology

Estimate λ = 0.80 using bankruptcy data. Given τ = τ, estimate (a, b, σ, ρ) and (Rt)tτ using Kalman Filter: Bd

t

m e R t

τ rudu = At + HtRt + ǫt

where ǫt N(0, ρ) is i.i.d. observation error. Rt = Ct + FtRt1 + ηt where ηt N(0, σ2

2a(1 2ea) and

At ba (a + λ), Ht λ (a + λ), Ct b(1 ea), Ft ea Given (a, b, σ, ρ, λ) and (Rt)tτ, estimate τ = inf

τ180tτft : Bt Bd τe R τ

t

ruduea(τt) + mb(1 ea(τt))g

Recovery Rate Model - Economic Default Date

20 40 60 80 100 120 140 160 180 5 10 15 20 25 30 35

Day s C a s e s Time Between Economic Def ault Date and Announced Def ault Dat

Figure: N = 82.

Recovery Rate Model - Parameters

N = 21,083 a b σ ρ Mean 5.03 0.5558 1.33 0.009232 Median 2.04 0.6183 0.23 0.008507 Std Dev 14.10 0.2778 5.97 0.001082 First Quartile 0.65 0.3052 0.047 0.000834 Third Quartile 4.28 0.7966 0.84 0.01319

Recovery Rate Model - Recovery Rates

50 100 150 200 250 300 350 400 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75

Number of Days After Economic Default Recovery Rate Estimates Average Recovery Rate Estimates

Recovery Rate Model - Recovery Rates

RFV RT RMV N = 69 EM KFM EM KFM EM KFM Mean 0.4814 0.5068* 0.3919 0.4134* 0.8272 0.8868** Median 0.4006 0.55 0.3204 0.4698 0.9 0.9299 StdDev 0.3029 0.2985 0.2442 0.2423 0.1780 0.1255 25 % 0.2 0.23 0.1578 0.1830 0.7267 0.8074 75 % 0.71 0.785 0.6020 0.6150 0.9577 0.9743 At economic default date.

Recovery Rate Model - Pricing Tests

N = 21,083 Pricing Errors in Dollars Mean 0.0018 Median 0.0000 Standard Deviation 0.4611 First Quartile .0124 Third Quartile 0.0153 Face Value $100

Recovery Rate Model - Pricing Tests

Run for each bond issue the time series regression

ǫt = 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 0.01.

We perform a Durbin-Watson autocorrelation test. For 62 out of the 103 issues, we fail to reject the null

hypothesis that corr(ǫt, ǫt1) = 0 with signi…cance level 0.01.

For most issues, recovery rate model …ts data well.

Epilogue

29-Sep-2004 09-Aug-2005 19-Jun-2006 29-Apr-2007 10 20 30 40 50 60 70 80 series1

Figure: Delta Airlines

Epilogue

03-Oct-2004 30-Nov-2004 27-Jan-2005 26-Mar-2005 40 50 60 70 80 90 100 series1

Figure: Trico Marine Service Inc.

Epilogue

30-Sep-2004 30-Jun-2005 30-Mar-2006 28-Dec-2006 40 50 60 70 80 90 100 110 series1

Figure: Winn Dixie Stores

Epilogue

03-Apr-2005 22-Dec-2005 11-Sep-2006 01-Jun-2007 20 30 40 50 60 70 80 90 100 110 series1

Figure: Northwest Airlines

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.