# distressed debt valuation

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Distressed Debt Prices and Recovery RateEstimationRobert JarrowJoint Work with Xin Guo and Haizhi LinMay 2008Introduction Recent market events highlight the importance ofunderstanding credit risk. Credit risk pricing and hedging involves understanding:1. interest rates (stochastic discounting),2. default process (when payments stop), and3. recovery rate process (what happens after default). Points 1 and 2 well-studied. Point 3, less so...IntroductionThree sources of knowledge on recovery rates.1. Industry papers:estimate recovery rates (not transparent, not validated byacademic community), andstudy their properties (correlation with business cycle,dependence on rm characteristics, ...)2. Academic papers - use industry generated recovery rates tostudy their properties.3. Academic papers - use pre-default debt and CDS prices toimplicitly estimate recovery rates.IntroductionPotential problems with existing knowledge. Are we sure recovery rates are estimated correctly? If not,then... academic papers study mis-specied estimates, academic papers have no base to compare implicit estimates.IntroductionPurposesPrimary- provide direct estimates of recovery rates usingdistressed debt prices.Secondary - t a model for defaulted debt prices.(it turns out, to solve one, must also solve the other)IntroductionResults1. Recovery rate estimates are sensitive to the date selected forestimation (signicant dierences between using the recordeddefault date and 30 days after).2. Prices support the belief that the market often recognizesdefault before default is recorded.3. An extended recovery rate model provides a poor t todistressed debt prices after the recorded default date(extension implicit in using 30 day after to estimate recoveryrate).4. We estimate a new recovery rate process and use it to pricedistressed debt. The model ts market prices well.PrologueStructural models Use managements information set. Default can be viewed as the rst hitting time of the rms assetvalue to a liability determined barrier. If the rms asset value follows a continuous process, the value of arms debt does not exhibit a jump at default. No implications for risky debt prices subsequent to default.Reduced Form Models Use markets 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.Prologue3 0 -Se p -2 0 0 4 2 7 -Fe b -2 0 0 5 2 7 -J u l -2 0 0 5 2 4 -De c -2 0 0 51 01 52 02 53 03 54 04 55 05 5s e ri e s 1Figure: Delta Airlines. Bankruptcy on September 14, 2005.Consistent with the standard structural model.30-day dierent from default date.Prologue0 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 54 05 06 07 08 09 01 0 0s e ri e s 1Figure: 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.Prologue29-Sep-2004 26-Feb-2005 26-J ul -2005 23-Dec -2005405060708090100s eri es 1Figure: Winn Dixie Stores. Bankruptcy on February 21, 2005.Consistent with the standard reduced form model.30-day dierent from default date.Prologue03-Apr-2005 30-J un-2005 26-Sep-2005 23-Dec -20052030405060708090s eri es 1Figure: Northwest Airlines. Bankruptcy on September 14, 2005.Partially consistent with both the reduced form and structural.30-day dierent from default date.Set UpFix a particular rm. Let Bt denote the price of its risky debt (a particular issuewith a given maturity, coupons (oating or xed), andembedded options). Dene the economic default date as the time when themarket knows default has happened. The recorded default date + where + _ is given in ourdata set. Let Bdt denote the risky debt price given economic default hasalready happened, i.e. for t _ , Bdt = Bt. Let rt be riskless spot rate of interest. Let pt(T) be price of a riskless coupon bond with the samematurity T and coupons as the risky bond underconsideration.Cross-Sectional Models1. Recovery of Face Value (RFV):Bd = Fwhere F is the face value of the debt (normalized to $100)2. Recovery of Treasury (RT):Bd = p(T)3. Recovery of Market Value (RMV):Bd = BCross-Sectional Models Purpose of these models is to provide the necessary inputs toprice risky debt and credit derivatives prior to default. Recovery rate estimation procedure is: x a defaulted company x a date to observe debt prices, then estimate the recovery rate. Single point estimate of the recovery rate per company. Lookcross-sectionally across companies to obtain estimate. For example, Moodys uses "30-day" post-default date for .Data December 2000 to October 2007. Debt Price Data - Advantage Data Corporation - Trade dataand broker quotes to get end of day prices 4:45 p.m. Filter data: have 50 prices over a 60 day window surrounding recordeddefault 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 lefor bankruptcy (potential selection bias). Bond Characteristics - Mergent Fixed Income Database.Default is when a debt issue violates a bond covenant, missesa coupon or principal payment, or les for bankruptcy. Agrace period of 30 days must usually pass before default isrecognized for a missed coupon.Data - Bankruptcy Time AnalysisTo get a sense for duration of distressed debt market, consideringonly issues that le for bankruptcy.N = 1902 Mean Std. Dev. Median N Chapter 7 433.22 353.20 433 9 0.80Chapter 11 454.34 427.08 354 631 0.84Time in Bankruptcy in Days is average time spent in bankruptcy in years.Cross-sectional Recovery Rates (RFV)Dierence 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 0.97960 70 48.17 29.39 11 71 45.48 28.67 1.02922 63 41.27 28.85 1.02845 44 48.32 31.48 1.034110 45 54.62 28.83 1.093320 46 53.86 32.44 1.147330 64 42.31 29.30 1.0779RFV at recorded default 48.17 statistically dierent from RFV at 30 day42.31.Cross-sectional Recovery Rates (RT)N = 96 RT EstimatesMean 0.4062Median 0.3452Standard Deviation 0.2528First Quartile 0.1692Third Quartile 0.6374Lower than the RFV estimates because otherwise identical defaultfree bonds trade at a premium (> $100).Cross-sectional Recovery Rates (RMV)N = 96 Pre-Default Default Date RMV EstimatesMean 48.4 48.6 1.0230Median 39 38.5 1.0013Standard Deviation 30.6 30.7 0.1824First Quartile 21.5 22 0.9681Third 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 defaultdate.Time-Series ModelsBdt = m e

t rsdswherem =

F if RFVp(T) if RTB if RMV.Assumes that risky debt position is sold at , and the model pricesdebt as the value of this position. Equivalently,Bdt = Bd+ e

t+ rsdsfor t _ .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 denition of the economic default date, using debtprices, our estimator is: = inf+180_t_+t : Bt _ Bd+ e

+t rsds. Bound below by 180 days before the recorded default date. Our estimator depends on the information up to time +.Time-Series Models - Economic Default Date0 20 40 60 80 100 120 140 160 1800510152025DaysCasesTi me Between Economic Default Date and Announced Default DateFigure: N = 73.Time-Series Models - Revised Recovery Rates =

BF if RFVBp(,T) if RTBB if RMV.Time-Series Models - RFVN = 73 Economic Default Recorded DefaultMean 0.4879* 0.5283Median 0.45 0.5782Standard Deviation 0.3044 0.3151First Quartile 0.2 0.2225Third Quartile 0.76 0.8425*P value essentially zero.Time-Series Models - RTN = 73 Economic Default Recorded DefaultMean 0.3970* 0.4335Median 0.3291 0.4776Standard Deviation 0.2461 0.2600First Quartile 0.1578 0.1803Third Quartile 0.6031 0.6610*P value essentially zero.Time-Series Models - RMVN = 73 Economic Default Recorded DefaultMean 0.8314* 1.0653Median 0.9094 1.0217Standard Deviation 0.1775 0.1729First Quartile 0.7267 0.9976Third Quartile 0.9649 1.0854*P value essentially zero.This is consistent with a jump on the economic default date.Time-Series Models - RMV0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 201234567Recov ery Rate EstimatesDensityRecov ery of Market Value Estimates Based on Economic Def aultsRecov ery of Market Value Estimates Based on Announced Def aultsTime-Series Models - Pricing TestsBdt = m e

t rsds+ t for t _ ."Good" if the residuals have zero mean, and are i.i.d.N = 20,942 Pricing ErrorsMean 17.81Median 9.31Standard Deviation 28.06First Quartile 0.11Third Quartile 30.00Very large pricing errors.Time-Series Models - Pricing Tests Run for each bond issue the time-series regression

t = + t and test if = 0 and = 0. 103 bond issues in our sample. For 87 we reject the null hypothesis that = 0 and = 0with a signicance level of 0.01 (for 79 we have negligibleP-values). 77 out of 103 issues produce positive slopes. Rejects distressed debt pricing model. Why? Ignoresinformation on default resolution after . Provides additional rejection of using the 30-day recovery.The Recovery Rate Model Database limitations - model the resolution of the bankruptcyling. Restrict to t _ . Let 0 represent the time to resolution of bankruptcy. Exponential distribution with parameter . Dollar payo equal

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