A Market-Based Framework for
Bankruptcy Prediction
Alexander S. Reisz
Claudia Perlich
Thursday, April 12, 2007
Third International Conference on Credit and Operational Risks
HEC Montreal
Alexander Reisz
Motivation
• According to Black & Scholes (1973)/Merton (1974), the
equity of a corporation can be seen as a standard call
option written on the assets of the firm with strike equal to
face value of debt
• Put-Call parity: SH can also be seen as holding the firm
and owing PV(F), but also having the (put) option to walk
away, in effect selling the firm to BH for F, the face value
of debt:
C=V-PV(F)+P
• BH are long the assets of the firm, but are short a call
option; alternatively, they are long riskless debt, but short
a put option:
V-C=PV(F)-P
Alexander Reisz
The KMV Approach
• From the pricing equation,
Et  Vt e (T t )(d1 )  e r (T t ) F(d1   A T  t )
Vt
and Ito’s lemma,  E   A , derive MVA (Vt) and σA
Et
– NB: F=CL+0.5*LTD
• Compute a distance-to-default (DD), equal to
ln Vt / F   ( r     A2 / 2)(T  t )
A T t
• At this point, KMV departs from normality assumption and
looks these DDs up in historical default tables
• For more on the topic, www.moodyskmv.com
Alexander Reisz
Problems with the BSM Paradigm
• 1)
Et
 Vt e  d (T t )  (d1 )
 A
Shareholder-aligned managers would therefore maximize
the volatility of the firm’s assets (new investments)
Et

 r ( T t )
 d ( T t )
 d ( T t )

Fre

(
d
)

V
de

(
d
)

V
e

(
d
)
• 2)
2
t
1
t
1

2 
which is positive for small d. Shareholder-aligned
managers would therefore maximize the maturity of the
firm’s debt
• 3) Underestimates the probability of bankruptcy 
overoptimistic credit rating
Our framework
• A (European) down-and-out barrier option is a contract
that gives its holder the right, but not the obligation, to
purchase an underlying asset at a prespecified price (strike)
at a prespecified date (maturity), provided the underlying
asset has not crossed a lower bound at any time before
maturity
V-DOC=PV(F)-P+DIC (the right to pull the plug)
• Choosing very volatile projects raises the probability to
end up in-the-money (and the extent to which you are inthe-money), but also the probability to cross the
(exogenous) bankruptcy barrier; C/0 leads to an
interior solution (you may even end up with a John and
Brito (2002) problem)
Alexander Reisz
Alexander Reisz
Alexander Reisz
Alexander Reisz
Results
• Equity prices do reflect embedded barriers (on
average, 30% of MVA)
• Better performance of predicted default
probabilities than in the BSM or KMV
frameworks, both in terms of discriminatory
power (ranking) and in terms of calibration
• However, even in the barrier model, probabilities
have to be recalibrated (no big deal); explains
KMV’s modified strike price and departure from
normality
• But our power (ranking accuracy) is achieved
without departing from the model’s assumptions!
Alexander Reisz
Results (contd.)
• The Old Man ain’t dead yet: in one-year-ahead
predictions, Altman (1968, 1993) scores
outperform structural models!
• Combine accounting-based scores and PD’s from
structural models in a logistic regression to
achieve highest power.
Alexander Reisz
Assumptions galore
• Exogenous bankruptcy bound B
• Markets are dynamically complete
(existence of an MMA may not even be
necessary)
• No bankruptcy costs, APR holds (no rebate)
• Constant interest rate
Alexander Reisz
Stock value (when F>B)
Et  Vt e (T t ) (d1 )  Fe r (T t ) (d1   T  t )
2( r  )
2( r  )


1
1
2
2



 (T  t )  B 
 r (T  t )  B 
 Vt e
  d 2   Fe
 d2   T  t
 
 
Vt 
Vt 




with
and
ln(Vt / F )  ( r     2 / 2)(T  t )
d1 
 T t
d2 
ln  B 2 /(Vt F )   ( r     2 / 2)(T  t )
 T t
• See it as a standard BS option minus the ex-ante
costs of the covenant
• Not increasing in  for all other parameters
Alexander Reisz





Brockman and Turtle (2003): a critique
• When you assume that MVA=BVD+MVE, you force B>F.
Indeed, the barriers backed out by BT can be fairly well
replicated by just solving DOC(V,F,B)=V-F for reasonable
parameters (Wong and Choi, 2005)
• Although it is possible, it is suspicious when it holds for
almost all firms (uniformly riskless corporate debt)
• Good luck in front of a court of law (maybe it works in
Germany…)
• Contradicts KMV’s use of a strike price of CL+0.5*LTD<F;
visitation and excursion times structural models; and Leland
and Toft’s (1996) gamble for resurrection
Alexander Reisz
Backing out Vt, A, and B:
a generalized market-based approach
• Start with 60,110 firm-years (1988-2002)
• Theory: A and B are constant for the life of the firm
• Empirically: Allow A and B to vary from year to year for
a given firm (amount of liabilities varies anyway from year
to year; trivially allows for variation in leverage ratios)
• However, force in a first step A and B to be constant over
two consecutive years; the price and Ito equations for t-1
and t are solved for Vt-1, Vt, A and B
• Keep Vt, A and B. Vt-1, A and B for time t-1 will be
estimated from equations for t-1 and t-2 (so as to avoid a
hindsight bias)
• Left with 33,037 firm-years (5,784 unique firms)
Alexander Reisz
Summary statistics
Variable
Market value of equity (in
millions)
Annualized equity volatility
Total liabilities (in millions)
Book value of assets
Implied market value of
assets
Implied
(annual)
asset
volatility
Payout ratio
Leverage
Implied barrier (% of MVA)
Implied barrier (% of TL)
One-year leverage
Three-year leverage
Five-year leverage
Number of
observations
33,037
Average
Median
Minimum
Maximum
108.08
Standard
deviation
7,119.3
1,142.1
0.1749
274,429
33,037
33,037
33,037
0.5926
390.76
701.43
0.5525
49.736
119.43
0.2974
1,303.8
2,284.5
0.0626
0.0050
0.1260
1.5998
33,528
73,037
33,037
1,267.1
135.31
6,775.5
0.2495
227,981
33,037
0.4358
0.3813
0.2619
0.0346
1.5791
33,037
33,037
33,037
33,037
25,582
20,256
16,514
0.0338
0.5162
0.3053
0.6616
0.3088
0.3938
0.4654
0.0224
0.3900
0.2758
0.6236
0.2233
0.2863
0.3492
0.0460
0.4722
0.2447
0.3977
0.3061
0.3712
0.4146
0.0000
0.0001
0.0000
0.0000
0.0001
0.0035
0.0056
2.3112
14.017
0.9682
1.1402
11.090
12.682
13.141
Alexander Reisz
(Physical) probabilities of bankruptcy
• Early bankruptcy:
 ln( B / Vt )  (      2 / 2)(T  t ) 
P (t  T )   


T

t
A


*
( B / Vt )
2(   ) /  A2 1
 ln( B / Vt )  (      2 / 2)(T  t ) 


A T t


• Bankruptcy at maturity:
 ln Vt / B   (      A2 / 2)(T  t ) 
 ln Vt / F   (      A2 / 2)(T  t ) 
P ( B  VT  F , min V ( s )  B )   
   

t  s T

T

t

T

t
A
A




( B / Vt )
2(   ) /  A2 1


  ln  B / V   (      2 / 2)(T  t ) 
 ln B 2 /( FVt )  (      A2 / 2)(T  t )  
t
A

   
 



T

t

T

t
A
A

 


Alexander Reisz
Total probability of bankruptcy
 ln(Vt / F )  (      2 / 2)(T  t ) 
P(bankruptcy )  1   


T

t


( B / Vt )2(   ) /
2
2
2

ln(
B
/(
V
F
))

(





/ 2)(T  t ) 
1
t


 T t


• First line: BS probability of assets ending up short
of F, the promised repayment
• Second line denotes the increase in the probability
of bankruptcy due to the possibility of early
passage (well…almost)
• Allows one to stay within the model’s framework
and fit B to a training sample
Alexander Reisz
Alexander Reisz
Alexander Reisz
Evaluating the performance of
probability estimates
• Accuracy does not reflect quality of probabilities:
– No discrimination for extreme priors (a dumb model
can achieve very high accuracy)
– Equal penalty for prediction of 0.01 or 0.499 in case of
default (although the latter is very large in credit risk):
accuracy is not designed to judge continuous
probabilities, but 0/1 predictions
– But in credit risk, we are not interested in 0/1
predictions, but in a continuous variable (at what rate
should we lend?) ; even more so the case because of
asymmetry of costs (lending to Enron vs. denying credit
to small businesses that would have deserved it)
Alexander Reisz
What we want
• An evaluation metric that reflects how well a model
ranks firms (discriminatory power)
– Did our model predict a larger PD for firms that actually
defaulted?
• In particular, we want to know how many of the true
defaults we catch for an arbitrary Type I error rate we are
willing to tolerate
• A metric that reflects whether the predicted PD’s
correspond to ex-post frequencies of default (calibration)
– Pb: you need many “similar” firms to judge ex-post whether
your average PD over a given group corresponds to the ex post
true frequency of default for that group
• In general: recalibration is easy, more power is hard to
achieve: favor the more powerful model
Alexander Reisz
Panel A: 1 Year
Model
BSM
KMV
DOC
Altman Zscore
Altman Z’’score
Number of
Observations
25,582
25,582
25,582
22,462
22,462
Prior Survival
Rate
0.9930
0.9930
0.9930
0.9944
0.9944
Area Under ROC
0.7170
0.7478
0.7636†‡
0.7794*
0.7769
Accuracy
(Concordance)
0.9738
0.9244
0.9903
NA
NA
Log-likelihood
-9,819.17
-5,483.97
-4,486.29
NA
NA
Number of
Observations
18,216
18,216
18,216
15,892
15,892
Prior Survival
Rate
0.9515
0.9515
0.9515
0.9555
0.9555
Area Under ROC
0.7369
0.7597
0.7625†*
0.6890
0.7279
Accuracy
(Concordance)
0.9063
0.8542
0.9092
NA
NA
Log-likelihood
-11,804.74
-10,090.4
-9,200.75
NA
NA
Panel B: 3 Year
Significant differences in AUC between DOC and BSM are identified by †, between DOC and KMV by
‡, and between DOC and the better of the two Altman scores by * (the symbol is entered next to the
AUC of the dominating model).
Alexander Reisz
Alexander Reisz
• Need to recalibrate!
Alexander Reisz
Panel A: 1 Year; estimation period: 1988-1999; evaluation period: 2000-2001
BSM
KMV
DOC
Altman Z-score
Altman Z’’-score
Number of Observations
25,582
25,582
25,582
22,462
22,462
Prior Survival Rate
0.9930
0.9930
0.9930
0.9944
0.9944
Area Under ROC
0.7170
0.7478
0.7636†‡
0.7794*
0.7769
Accuracy (Concordance)
0.9930
0.9930
0.9930
0.9942
0.9943
-2,033.54
-2,006.28
-1,992.04
-1,431.87
-1,461.10
-0.0795
-0.0784
-0.0779
-0.0637
-0.0650
Number of Observations
2,761
2,761
2,761
2,456
2,456
Prior Survival Rate
0.9931
0.9931
0.9931
0.9939
0.9939
Area Under ROC
0.7308
0.7653
0.7865†‡
0.8063*
0.7268
Accuracy (Concordance)
0.9931
0.9931
0.9931
0.9927
0.9935
Log-likelihood
-215.88
-211.03
-208.77
-174.51
-177.81
Average Log-likelihood
-0.0782
-0.0764
-0.0756
-0.0711
-0.0724
Model
In-sample recalibration
Log-likelihood
Average Log-likelihood
Out-of-sample
recalibration
Significant differences in AUC between DOC and BSM are identified by †, between DOC and KMV by ‡, and between
DOC and the better of the two Altman scores by * (the symbol is entered next to the AUC of the dominating model).
Alexander Reisz
The picture KMV does not want you to see
From Saunders and Allen (2002)
Alexander Reisz
Panel B: 3 Year; estimation period: 1988-1997; evaluation period: 1998-1999
In-sample recalibration
Number of Observations
18,216
18,216
18,216
15,892
15,892
Prior Survival Rate
0.9515
0.9515
0.9515
0.9555
0.9555
Area Under ROC
0.7369
0.7597
0.7625†*
0.6890
0.7279
Accuracy (Concordance)
0.9515
0.9515
0.9515
0.9554
0.9550
-6,513.08
-6,437.45
-6,421.25
-5,617.30
-5,493.12
-0.3575
-0.3534
-0.3525
-0.3534
-0.3457
Number of Observations
2,592
2,592
2,592
2,267
2,267
Prior Survival Rate
0.9117
0.9117
0.9117
0.9153
0.9153
Area Under ROC
0.7380
0.7664
0.7795†*
0.7055
0.6997
Accuracy (Concordance)
0.9117
0.9117
0.9117
0.9162
0.9127
Log-likelihood
-1491.61
-1,468.16
-1,444.00
-1,361.07
-1,385.83
Average Log-likelihood
-0.5755
-0.5664
-0.5571
-0.6004
-0.6113
Log-likelihood
Average Log-likelihood
Out-of-sample
recalibration
Significant differences in AUC between DOC and BSM are identified by †, between DOC and KMV by ‡, and between
DOC and the better of the two Altman scores by * (the symbol is entered next to the AUC of the dominating model).
Alexander Reisz
Future research I: Fitting better models
• Debt structures form a PF of American (outside;
compound) barrier options
• Debt covenants may specify that a certain time has
to be spent consecutively (cumulatively) below the
bankruptcy barrier (excursion (visitation) time):
Parisian (Parasian) options
• Allow for severity of excursion to play a role as
well: Galai, Wiener, and Raviv (2005)
• General problem: backing out a parameter when
only numerical pricing is available
Alexander Reisz
Future research II: let’s be realistic
• Force in a first step A and B to be constant over three
consecutive years; the price and Ito equations for t-2, t-1
and t are solved for Vt-2, Vt-1, Vt, A, B and R, the option
rebate
• Provides a new estimate of violations of the APR rule on
the shareholders’ side
• Estimate bond prices using both early bankruptcy and
bankruptcy at maturity; in both cases, LGD is given
endogenously by the model
• Alternatively, use the recalibrated total PD’s to estimate
equilibrium bond yields on new issues.
Alexander Reisz
Alexander Reisz
Alexander Reisz