Beyond Structural Models
Advanced Methods of Risk Management
Umberto Cherubini
Learning Objectives
• In this lecture you will learn
1. Credit risk as a short position in a put
option
2. The seniority structure of a firm as an
option spread
3. How to hedge corporate bonds with
common stock
Merton model and data:
the 10 year maturity (US)
Rating
Leverage
Aaa
13.1%
Aa
21.2%
A
32.0%
Baa
43.3%
Ba
53.5%
B
65.7%
Source: Wang ang Wang (2000).
Volatility
27.8%
23.4%
19.7%
18.8%
25.2%
35.2%
Predicted
credit spread
8.0
10.0
14.3
32.0
137.9
363.3
Observed
credit spread
63
91
123
194
299
408
% explained
12.6%
11.0%
11.6%
16.5%
46.1%
89.0%
Covenants (Black e Cox, 1976)
• The Merton model was extended to the case of
default before maturity by Black and Cox. Default
before maturity is obtained by introducing covenants.
Covenants are limits set to special variables. When
such limits are reached, debt is called back.
• In the structural model all information is contained in
the value of the firm, which is assumed to be
observed on the markets. It is evident that in this
case the value of equity is a call option with (downand-out call).
• In practice, as of today structural models are taken to
be barrier models in the spirit of Black and Cox.
0.0100
No-Covenant
Covenant
0.0090
0.0080
0.0070
Credit Spread
0.0060
0.0050
0.0040
0.0030
0.0020
0.0010
0.0000
0
5
10
15
Maturità
20
25
30
35
Flaws of structural models
Structural models produce:
1) Undervaluation of the default put options and
of the credit spreads;
2) Undervaluation of spreads which is particularly
severe for short term maturities
3) Undervaluation of credit spread particularly for
high standing obligors.
Low credit spreads
• The problem with low credit spreads is
that calibration would require a volatility
of assets too high to be consistent with
the historical default probabilities
• Solutions
– Asset substitution: asset volatility may change
– Absolute priority violations: strategic debt service
(Anderson and Sundaresan, 1996)
– Conservative assessment of the value of assets,
and the probability of default (Cherubini and Della
Lunga, 2001)
– Other risk factors: market liquidity.
Short term credit risk
• Merton model is based on the assumption that
– The value of the firm is observed in continuous
time: technically, it is a process adapted to the
information set
– The value of the firm follows diffusion process.
Technically, default is predictable and there exists
an “announcing sequence” of the default event
• Three solutions
– Including a jump in the value of the firm
– Introducing noise in the default barrier
– Introducing “noise” in the value of the firm
Notation and definitions
• v(t) = the value of assets measured in terms
of nominal discounted value of debt.
• e(t) = the value of equity measured in terms
of nominal discounted value of debt.
• d(t) = the value of debt measured in terms of
nominal discounted value of debt.
Equity
• The value of equity is a call option:
ev, t   v(t ) N d1   N d 2 

2
V

2
V

ln v(t )    / 2 T  t 
d1 
V T  t

ln v(t )    / 2 T  t 
d2 
V T  t
Debt
• Defaultable bonds are prices as defaultfree debt and a short default put option
d t , T   vt N  d1   N d 2 
 1  vt N  d1   N  d 2 
 1   vt N  d1   N  d 2 
Accounting noise
Duffie and Lando (2001)
• Balance sheet data are observed at discrete
times and are assumed to be noisy, but
unbiased.
• The entrepeneur can decide to default on the
project at any time
• If the value of the firm is close to the default
barrier investors may fear that the firm be
already technically in default or that a default
event could take place before the next arrival
of accounting data, and require a higher short
term credit spread for this.
Accounting Fraud
Our model
• Balance sheet data are observed at discrete
times and are assumed to be biased: there is
a positive probability that any firm could be
already in a default state, despite good
accounting figures. We call this “fraud risk”.
• Notice: this model does not rely on, but does
not exclude, any strategic default behavior
from the entrepeneur. This is made to
concentrate the focus on “fraud risk”. In
principle, a realistic model should include
strategic debt service, strategic default and
accounting noise.
A binomial example
• Three dates: t0 , t1 , T
• Initial date: “Manager” and “market” share the
following info about the “fundamental” or
“true” evolution of firm asset value V (see Figure
1):
firm market value is
where
V (t0 )  pVH  (1  p)VL
VH  pHVHH  (1  pH )VHL
VL  pLVHL  (1  pL )VLL
• Final date: the value of V(T) is publicly
observed: firm market value is V(T)
and
Figure 1 - Common knowledge
info – Martingale measure
VHH
pH
VH
p
1  pH
VHL
V (t0 )
pL
1 p
VL
1  pL
t0
t1
VLL
T
…a binomial example
• Interim date:
– the manager observes V (t1 ) and issues an
“accounting signal” s  h, l  (see Figure 2)
– suppose s  h : the market updates the probability
of high state as follows (Bayes’ rule):
p(1  π d )
p(1  π d )
Pr( H h) 

Pr( s  h) p(1  π d )  (1  p)πu
computes the value of the firm as a compound
lottery (see Figure 3):
Vˆ (h)  Pr( H h)VH  1  Pr( H h)VL
Figure 2 - Balance sheet statement as a noisy signal
h
h
1  πd
πu
VL
VH
1  πu
πd
l
l
Figure 3 - Market info after observing s=h
pH
VHH
VH
PrH h
1  pH
sh
VHL
pL
1 PrH h
VL
1  pL
VLL
t1
T
Figure 4 - True and market values
VHH
VH
VHL
V (t0 )
VL
t0
t1
VLL
T
Figure 4 - True and market values (s=h): full confidence
Vˆ (h)  VH
VHH
p
VHL
V (t0 )
VL
t0
t1
VLL
T
Figure 4 - True and market values (s=h): no confidence
VHH
VH
πu
V (t0 )
VHL
Vˆ ( h )
VL
t0
t1
VLL
T
Figure 4 - True and market values (s=h): partial
confidence
VH
VHH
Vˆ (h)
Pr( s  h)
VHL
V (t0 )
VL
t0
t1
VLL
T
Figure 4 - True and market values (s=l): partial
confidence
VH
V (t0 )
VHH
VHL
Pr( s  l )
Vˆ (l )
VL
t0
t1
VLL
T
Figure 4 - True and market values
VHH
VH
Vˆ ( h )
Pr( s  h)
VHL
V (t0 )
Pr( s  l )
Vˆ (l )
VL
t0
t1
VLL
T
Parmalat
• The typical case of accounting fraud in
Europe is represented by Parmalat.
• For several years, analysts had been raising
doubts on the fact that Parmalat were
endowed with liquidity (around 4 billions).
• In 2003, when Parmalat decided to issue a
300 mio bond, the marked asked: what are
they doing with all that money if they sit on a
lot of cash. Maybe what they are sitting on is
not cash…
Parmalat case: 2003
• February: Parmalat announces 300 mio issue for institutionals.
The stock falls 9% and the bond is withdrawn
• March: 80 mio increase in capital announced to repay a bond.
Assogestioni calls for transparency.
• April: Parmalat announces a debt/capital ratio of 83%. The new
stakeholder Philips Pensionfunds ask a better governance
• June: Philips and Nextra (Intesa) reduce their exposure below
2%, Nextra underwrites the 300 mio bond.
• September: issued new 350 mio bonds underwritten by
Deutsche Bank, outlook downgrade (positive to stable) by S & P
(covenant for Nextra)
• November and December are the final act…
Parmalat: 2003 (Nov.)
• November: recap envisaged for 400-500 mio.
CONSOB calls for clarity concerning repayment of
bonds due in December. Parmalat answers it will use
liquidity. On 11-11 Deloitte expresses doubts on
investment in a hedge fund called Epicurum, and S &
P revises outlook to negative, concerning doubts on
Parmalat accounting and real liquidity held by
Parmalat. On 12 Parmalat announces future
unwinding of Epicurum and the stock soars. The CFO
resigns. Deutsche Bank increases stockholding to
5.15%. The Assembly gives ok to Epicurum
unwinding.
Parmalat 2003 (Dec)
• December: on 8 a 150 mio bond gets to maturity and
it is not paid. CONSOB asks Parmalat to reassure
the market. Parmalat answers Epicurum did not pay.
Trading of the stock is suspended. On 9 the Board
reassures that the bond will be repaid on the 15.
Exchange of the stock reopens and the stock falls by
40%. On 12 it is announced that the bond was repaid
(with help from banks, 25 mio). On 15, Tanzi resigns.
Mediobanca and Lazard are advisors. On 18, deal
with Epicurum in stall. On 19, BoA reveals that 3.9
billions that were assumed to be deposited with it did
not exist. On 27, Parmalat files for Amministrazione
Controllata (Chapter 11).
Parmalat as “Peso problem”
• A peso problem is a case in which the market assigns
some small probability to a major event which is not
included in the sample. That may introduce a bias in
the estimates of a stochastic process.
• As a result a “peso problem” may induce “mirages” of
arbitrage opportunities that are simply not there.
• Recent models on imprecision and outright fraud in
accounting data foresight the possible relevance of a
“peso problem” in corporate liabilities data.
Credit risk information
• The major sources of infomation on credit risk are equity
markets and CDS markets.
• Equity and bond markets for “public” firms are mainly retail
markets, and they collect most of the information of the general
public
• CDS markets are more professional markets involving financial
intermediaries and institutional investors.
• An interesting questions is whether the two markets carry the
same information, or whether the CDS market has some
advantage, so that information arrives in this market earlier than
the equity and bond markets. If this were the case, there would
be market inefficiencies to be exploited by taking arbitrage
positions on the different markets.
22
24
27
29
01
04
07
09
12
14
17
03
25
21
23
26
22
24
27
29
02
04
04
21
/1
/1
/1
/0
/0
/0
/0
/0
/0
/0
/0
/0
/1
/1
/0
/0
/0
/0
/0
/0
/0
/0
/0
/1
2/
1/
0/
9/
9/
8/
7/
6/
5/
4/
3/
2/
1/
0/
9/
8/
7/
6/
5/
4/
4/
3/
2/
2/
03
03
03
03
03
03
03
03
03
03
03
03
02
02
02
02
02
02
02
02
02
02
02
01
Parmalat stock and CDS
2500
4 ,5
4
2000
3 ,5
3
1500
2 ,5
C D S S p re a d M id 5 ye a rs
S to c k
2
1000
1 ,5
500
1
0 ,5
0
0
Maximum Likelihood Estimation
Structural Models
• Assuming the market value of the firm is
observed from a discrete sample at times {t1,
t2,…, tN} we may write the likelihood as
N 1
N 1
ln LV t i ; i  1,2,..., N ,  ,    
ln 2  
ln  2 
2
2
 

1   V t i  


  ln V t i    ln 
  i 

2 i  2   V t i 1  
i2

N
N
where we define i = ti – ti – 1
2
Maximum Likelihood Estimation
on transformed data (Duan)
• Assume now that the market value of the firm is not
observed, but the price of a liability, say equity is
observed instead at discrete times {t1, t2,…, tN}
N 1
N 1
ln LV t i ; i  1,2,..., N ,  ,    
ln 2  
ln  2  
2
2
Equityt i 
ˆ
  ln V t i ;     ln

ˆ
V t i ;  
i2
i2
N


N

1   Vˆ t i ;   
   i 
  ln 
2 i  2   Vˆ t i 1 ;   

N
2
Equityt i   Vˆ t i ;  N d1   QuasiDebtNd 2 
Maximum Likelihood Estimation
on a model with garbling
• Assume now that the market price of equity allows for the
possibility that accounting figures be actually biased, and the firm
be in a situation of financial distress (V(t) < QuasiDebt) Denote by
f the probability of this event. The likelihood is
ln LV t i ; i  1,2,..., N ,  ,    
N 1
N 1
ln 2  
ln  2  
2
2
Equityt i 
  ln Vˆ t i ;     ln 1  f 

ˆ
V t i ;  
i2
i2
N


N

1   Vˆ t i ;   


  ln 
  i 

ˆ
2 i  2   V t i 1 ;   

Equityt i   1  f  Vˆ t i ;  N d1   QuasiDebtNd 2 
N

2

21/12/2003
21/11/2003
21/10/2003
21/09/2003
21/08/2003
21/07/2003
21/06/2003
21/05/2003
21/04/2003
21/03/2003
21/02/2003
21/01/2003
21/12/2002
21/11/2002
21/10/2002
21/09/2002
21/08/2002
21/07/2002
21/06/2002
21/05/2002
21/04/2002
21/03/2002
21/02/2002
21/01/2002
21/12/2001
Market value of debt…
Mrkt Price
3.500.000.000
3.000.000.000
2.500.000.000
2.000.000.000
Mrkt Price
1.500.000.000
1.000.000.000
500.000.000
0
21/12/2003
21/11/2003
21/10/2003
21/09/2003
21/08/2003
21/07/2003
21/06/2003
21/05/2003
21/04/2003
21/03/2003
21/02/2003
21/01/2003
21/12/2002
21/11/2002
21/10/2002
21/09/2002
21/08/2002
21/07/2002
21/06/2002
21/05/2002
21/04/2002
21/03/2002
21/02/2002
21/01/2002
21/12/2001
3.500.000.000
…that predicted by Merton
model…
3.000.000.000
2.500.000.000
2.000.000.000
Merton Model
Mrkt Price
1.500.000.000
1.000.000.000
500.000.000
0
21/12/2003
21/11/2003
21/10/2003
21/09/2003
21/08/2003
21/07/2003
21/06/2003
21/05/2003
21/04/2003
21/03/2003
21/02/2003
21/01/2003
21/12/2002
21/11/2002
21/10/2002
21/09/2002
21/08/2002
21/07/2002
21/06/2002
21/05/2002
21/04/2002
21/03/2002
21/02/2002
21/01/2002
21/12/2001
…and the impact of garbling
3.500.000.000
3.000.000.000
2.500.000.000
2.000.000.000
Merton Model
Mrkt Price
Garbling model
1.500.000.000
1.000.000.000
500.000.000
0