Transcript Document
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 Aa A Baa Ba
13.1% 21.2% 32.0% 43.3% 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 (down and-out call).
• In practice, as of today structural models are taken to be barrier models in the spirit of Black and Cox.
0.0100
0.0090
0.0080
0.0070
0.0060
0.0050
0.0040
0.0030
0.0020
0.0010
0.0000
0 No-Covenant Covenant 5 10 15 Maturità 20 25 30 35
Flaws of structural models
Structural models produce: 1) Undervaluation of the
default put
of the
credit spreads
; options and 2) Undervaluation of spreads which is particularly severe for short term maturities 3) Undervaluation of
credit spread
high standing obligors. particularly for
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
) 2
d
1
d
2 ln ln (
t
) (
t
)
N
V
1 2
V
V
T T
2
V
/ / 2
t t N
T
2
T
t t
Debt
• Defaultable bonds are prices as default free debt and a short
default put option d t T
v
1
t
N v
d
1
d
1
N
1
v
d
1
2
N
N
d
2
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:
t t T
0 , , 1
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 H
p H V V HH
(
t
0 ( ) 1
pV H p H
)
V HL
( 1
p
)
V L
and
V L
p L V HL
( 1
p L
)
V LL Final date:
the value of
V(T)
is publicly observed: firm market value is
V(T)
V
(
t
0 )
Figure 1 - Common knowledge info – Martingale measure
p H V H p
1
p H p L
1
p V L
1
p L t
0
t
1
T V LL V HH V HL
…a binomial example
•
Interim date: s
V
(
t
1 (see Figure 2) of high state as follows (Bayes’ rule): Pr(
H h
)
p
( 1 Pr(
s
π d h
) )
p
( 1
π p
( 1
d
)
π
( 1
d
)
p
)
π u
computes the value of the firm as a compound lottery (see Figure 3):
V
ˆ (
h
) Pr(
H h
)
V H
1 Pr(
H h
)
V L
Figure 2 - Balance sheet statement as a noisy signal
h
1
π d V H π d π u V L
1
π u h l l
Figure 3 - Market info after observing s=h
p H V HH
Pr
V H s
h
1
p H V HL p L
1 Pr
V L
1
p L t
1
T V LL
Figure 4 - True and market values
V H V
(
t
0 )
t
0
V t
1
L V HL T V LL V HH
Figure 4 - True and market values (s=h): full confidence
V
ˆ (
h
)
V H V HH p V
(
t
0 )
V HL t
0
V L t
1
T V LL
Figure 4 - True and market values (s=h): no confidence
V H V HH V
(
t
0 )
π u V
ˆ (
h
)
t
0
V L t
1
T V HL V LL
Figure 4 - True and market values (s=h): partial confidence
V H V HH
Pr(
s
h
)
V
ˆ (
h
)
V
(
t
0 )
V HL t
0
V L t
1
T V LL
Figure 4 - True and market values (s=l): partial confidence
V H V HH V
(
t
0 ) Pr(
s
l
)
t
0
V
ˆ (
l
)
V L t
1
V HL T V LL
Figure 4 - True and market values
V H V
(
t
0 )
t
0
V
ˆ (
h
) Pr(
s
h
) Pr(
s
l
)
V
ˆ (
l
)
V L t
1
V HL T V LL V HH
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.
Parmalat stock and CDS
2 5 0 0 2 0 0 0 1 5 0 0 1 0 0 0 5 0 0 0 2 1 /1 2 /0 1 0 4 /0 2 /0 2 0 4 /0 3 /0 2 0 2 /0 4 /0 2 2 9 /0 4 /0 2 2 7 /0 5 /0 2 2 4 /0 6 /0 2 2 2 /0 7 /0 2 2 6 /0 8 /0 2 2 3 /0 9 /0 2 2 1 /1 0 /0 2 2 5 /1 1 /0 2 0 3 /0 2 /0 3 1 7 /0 3 /0 3 1 4 /0 4 /0 3 1 2 /0 5 /0 3 0 9 /0 6 /0 3 0 7 /0 7 /0 3 0 4 /0 8 /0 3 0 1 /0 9 /0 3 2 9 /0 9 /0 3 2 7 /1 0 /0 3 2 4 /1 1 /0 3 2 2 /1 2 /0 3 4 ,5 4 1 ,5 1 0 ,5 0 3 ,5 3 2 ,5 2 C D S S p re a d M id 5 ye a rs S to c k
Maximum Likelihood Estimation Structural Models
• Assuming the market value of the firm is observed from a discrete sample at times {t 1 , t 2 ,…, t
N
} we may write the likelihood as ln
L
V
i
;
i
1 , 2 ,...,
N
, , i N 2 ln
V
i
N
2 1 ln 1 2
N
2 1 ln i N 2 ln
V V t
i i
1
i
2 where we define
i
= t
i
– t
i –
1
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 {t 1 , t 2 ,…, t
N
} ln
L
V
;
i i
1 , 2 ,...,
N
, ,
N
2 1 ln i N 2 ln
V
ˆ
t i
; i N 2 ln
Equity
V
ˆ
t N i
; 2 1
i
ln
Equity
i
V
ˆ
t
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
f
be in a situation of financial distress (V(t) < QuasiDebt) Denote by the probability of this event. The likelihood is ln
L
V
;
i i
1 , 2 ,...,
N
, , N i 2 ln
V
ˆ
t i
;
N
2 1 ln N i 2 ln 1
f
N
1 ln 2
Equity
V
ˆ
t i
;
i
Equity
i
1 1 2
f
N i 2 ln
t i
;
V
ˆ
V
ˆ
t t i i
1 ; ;
d
1
i
2
QuasiDebtN
2
3.500.000.000
3.000.000.000
2.500.000.000
2.000.000.000
1.500.000.000
1.000.000.000
500.000.000
0
Market value of debt…
Mrkt Price Mrkt Price
3.500.000.000
3.000.000.000
2.500.000.000
2.000.000.000
1.500.000.000
1.000.000.000
500.000.000
0
…that predicted by Merton model…
Merton Model Mrkt Price
3.500.000.000
…and the impact of garbling
3.000.000.000
2.500.000.000
2.000.000.000
1.500.000.000
1.000.000.000
500.000.000
0 Merton Model Mrkt Price Garbling model