A structural credit risk model with
a reduced-form default trigger
Applications to finance and insurance
Mathieu Boudreault, M.Sc., F.S.A.
Ph.D. Candidate, HEC Montréal
2007 General Meeting
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Montréal, Québec
Introduction – Credit risk
• General definition of credit risk
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• Potential losses due to:
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• Default;
• Downgrade;
• Many examples of important defaults
• Enron, WorldCom, many airlines, etc.
• Need tools/models to estimate the
distribution of losses due to credit risk
Introduction – Credit risk
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• Credit risk models can be used for:
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– Pricing credit-sensitive assets (corporate
bonds, CDS, CDO, etc.)
– Evaluate potential losses on a portfolio of
assets due to credit risk (asset side)
– Measure the solvency of a line of business
(premiums flow, assets backing the liability)
(liability side)
• Risk theory models (ruin probability) are an
example of credit risk models
Introduction – Classes of models
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• Models oriented toward risk management
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– Based on the observation of defaults, ratings
transitions, etc.
– Goal: compute a credit VaR (or other tail
risk measure) to protect against potential
losses
• Models oriented toward asset pricing
– Based on financial and economic theory
– 2 classes of models
• Structural models
• Reduced-form (intensity-based)
Introduction - Contributions
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• As part of my Ph.D. thesis, I introduce:
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– An hybrid (structural and reduced-form)
credit risk model
– Can be used for all three purposes
• Characteristics of the model
– Default is tied to the sensitivity of the credit
risk of the firm to its debt
– Endogenous and realistic recovery rates
– Model is consistent in both physical and
risk-neutral probability measures
– Quasi closed-form solutions
Outline
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1. Introduction
2. Credit risk models
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a) Review of the literature
b) Risk management models, structural and
reduced-form models
3. Hybrid model
4. Practical applications
5. Conclusion
Risk management models
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• CreditMetrics by J.P. Morgan
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– Based on credit ratings transitions
– Revalue assets at each possible transition
– Compute credit VaR
• CreditRisk + by CreditSuisse
– Actuarial model of frequency and severity
– Frequency (number of defaults): Poisson
process
– Severity (losses due to default): some
distribution
Risk management models
• Moody’s-KMV
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– Based on the distance to default metric
– Distance to default (DD):
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DD 
E A1   DPT

– Using their database, they relate the distance to
default to an empirical default probability
– Can be used to determine a credit rating transition
matrix
– Can be the basis of revaluation of the portfolio for
credit VaR computations
Structural models
• Suppose the debt matures in 20 years
900
Assets
Liabilities
700
Millions of dollars
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800
600
500
400
300
200
100
0
0
2
4
6
8
10
Time
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12
14
16
18
20
Structural models
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• Idea: default of the firm is tied to the
value of its assets and liabilities
• Main contributions:
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– Merton (1974): equity is viewed as a call
option on the assets of the firm, debt is a
risk-free discount bond minus a default put
– Black & Cox (1976): default occurs as soon
as the assets cross the liabilities
– Longstaff & Schwartz (1995), CollinDufresne, Goldstein (2001): stochastic
interest rates
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Reduced-form models
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• Default is tied to external factors and take
investors by surprise
• Parameters of the model are obtained using
time series and/or cross sections of prices of
credit-sensitive instruments
– Corporate bonds, CDS, CDO
• Main contributions: Jarrow & Turnbull (1995),
Jarrow, Lando & Turnbull (1997), Lando
(1998).
• Idea: directly model the behavior of the default
intensity
Reduced-form models
• Moment of default r.v. is

t
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  inf t  0 :  H u du  E1
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0

where E1 is an exponential r.v. of mean 1.
• Default probability (under the risk-neutral
measure)
T


Qt   T   1  E exp    H u du 
  t

Q
t
• Example: Hu follows a Cox-Ingersoll-Ross
process
Comparison
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• Structural models
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– Default is predictable given the value of assets and
liabilities
– Short-term spreads are too low
– Recovery rates generated too high
• Reduced-form models
– Default is unpredictable but not tied to debt of firm
– Spreads can be calibrated to instruments
– Recovery assumptions are exogenous
• Risk management models
– Cannot price credit sensitive instruments
Hybrid model – Ideas
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• Hybrid model (presented in my Ph.D. thesis):
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– Model the assets and liabilities of the firm, as with
structural models
– Different debt structures are proposed
• Idea # 1: Default is related to the sensitivity of
the credit risk of the company to its debt
– McDonald’s (BBB+) vs Exxon Mobil (AA+)
– Similar debt ratio, other characteristics are good for
McDonald’s
– Spreads of both companies very different
– Industry in which the firm operates is important
Hybrid model – Ideas
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• Idea # 2: firms do not necessarily default
immediately when assets cross liabilities
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– Ford (CCC) and General Motors (BB-) have
very high debt ratios and still operate
• Idea # 3: firms can default even if their
financial outlooks are reasonably good
(surprises occur)
– Recovery rates very close to 100%
– Enron’s rating a few months before its
phenomenal default was BBB+
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Hybrid model – Framework
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• Suppose the assets and liabilities of the firm
are given by the stochastic processes {At,t>0}
and {Lt,t>0}
Lt
• Let us denote by Xt its debt ratio X t 
At
• Idea of the model is to represent the stochastic
default intensity {Hu,u>0} by
H u  h X u 
where h is a strictly increasing function
Hybrid model – Framework
• Examples: h(x) = c, h(x) = cx2 and h(x) = cx10
0,5
Constant
Increasing
Fast increasing
0,4
0,35
Default intensity
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0,45
0,3
0,25
0,2
0,15
0,1
0,05
0
0%
50%
100%
Debt ratio
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150%
200%
Hybrid model – Mathematics
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• Assume that under the real-world measure, the
assets of the firm follow a geometric Brownian
motion (GBM)
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dAt   A At dt   A At dBtP
• Propose different debt structures
– Under constant risk-free rate
• Debt grows with constant rate Lt = L0exp(bt)
• Debt is a GBM correlated with assets (hedging)
– Under stochastic interest rates
• Debt is a risk-free zero-coupon bond
• Assets are correlated with interest rates
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Hybrid model – Mathematics
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• Assume the transformation h is strictly
increasing with the specific form
hx   cx 1 , x  0
• Assume the assets and liabilities of the firm
are traded
– We proceed with risk neutralization
• Property: with h, most of the time, the
default intensity remains a GBM i.e.
dH t   H t H t dt   H t H t dBtP
– The drifts and diffusions change with the
probability measures.
Hybrid model – Mathematics
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• It is possible to show that the survival
probability can be written as a partial
differential equation (PDE)
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2
S
S 1 2

S
2
 Ht S 
  H t H t
  H t H t
0
2
t
H 2
H
• When µH(t) and σH(t) are constants, can use
Dothan (1978) quasi-closed form equation.
• Otherwise, we have to rely on finite difference
methods or tree approaches
Practical applications
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• Impact of hedging on credit risk
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– Use a stochastic debt structure
– Impact of correlation between assets and
liabilities on the level of spreads
• Result
– Depends on the initial condition of the firm
– Impact of hedging is positive over shortterm
– Reason: firms with poor hedging that
eventually survive have a long-term
advantage because their debt ratio will have
improved significantly
Practical applications
650
600
550
50%
80%
95%
500
Spreads in bps
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700
• Impact of hedging on credit risk
0
5
10
15
20
Time to maturity (in years)
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25
30
Practical applications
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• Endogenous recovery rate distribution
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– Firm can survive (default) when its debt ratio is
higher (lower) than 100%
– Assets over liabilities at default, minus liquidation
and legal fees can be a reasonable proxy for a
recovery rate
 A 
R  1    min 1; 
 L 
– Altman & Kishore (1996):
• Recovery rates between 40% to 70%
• Recovery rates decrease with default probability
• Recovery rates decrease during recessions
Practical applications
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• Endogenous recovery rate distribution
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•
•
•
•
Obtained using 100 000 simulations
Asset volatilities of 10% and 15%
Initial debt ratios of 60% and 90%
No liquidation costs
Practical applications
• Credit spreads term structure
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– The price of defaultable zero-coupon bonds with
endogenous recovery rate is
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

e  r T t  Qt   T   EtQ R 1 T 
• The following is obtained with a random debt
structure and endogenous recovery (10%
liquidation costs)
• Levels and shapes of credit spreads are
consistent with literature
– Three possible shapes
– See Elton, Gruber, Agrawal, Mann (2001)
Practical applications
• Credit spreads term structure
90
Increasing # 1
Increasing # 2
Hump shape
Decreasing
70
60
Spreads in bps
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80
50
40
30
20
10
0
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0
5
10
15
Time to maturity
20
25
30
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Practical applications
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• Model is defined under both physical and
risk-neutral probability measures
• Default probabilities can be computed in
both probability measures
• Can use accounting information to
estimate parameters of the capital
structure
• Can use prices from corporate bonds and
CDS to infer the sensitivity of the credit
risk to the debt
Practical applications
• Real-world default probabilities
Wal-Mart
General Motors
0.9
Cumulative default probability
Cumulative default probability
CDS implied real-world default probability
Lower bound
Upper bound
Empirical default probabilities (S&P)
0.025
0.02
0.015
0.01
0.005
0
0.8
0.7
0.6
0.5
0.4
CDS implied real-world default probability
Lower bound
Upper bound
Empirical default probabilities (S&P)
0.3
0.2
0
5
10
0
15
0
5
10
15
Time (in years)
Time (in years)
Exxon Mobil
McDonalds
0.06
0.14
CDS implied real-world default probability
Lower bound
Upper bound
Empirical default probabilities (S&P)
CDS implied real-world default probability
Lower bound
Upper bound
Empirical default probabilities (S&P)
0.12
Cumulative default probability
0.05
0.04
0.03
0.02
0.01
0
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1
0.1
Cumulative default probability
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0.03
0.1
0.08
0.06
0.04
0.02
0
5
10
Time (in years)
15
0
0
5
10
Time (in years)
15
Practical applications
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• Credit VaR
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– Need to use the distribution of losses under
the real-world measure
– Cash flows occur over a long-term time
period: need to discount
– Which discount rate is appropriate ?
– Answer: Radon-Nikodym derivative
– Interpreted as the adjustment to the risk-free
rate to account for risk aversion toward the
value of assets
Practical applications
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• Credit VaR
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– Radon-Nikodym derivative can be obtained for
each debt structure
– For example, under constant interest rates and
deterministically growing debt,
2
  r



dQ
1


r
P
A
A

 T 
T   exp  
BT  
dP
A
2 A  


– Consequently, the T-year horizon Value-at-Risk
for a defaultable zero-coupon bond is
dQ

T 1 T   R1 T  
VaR95P % e  r T t 
dP


where we recover a constant fraction R of the face
value payable at maturity
Practical applications
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• Credit VaR
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– Caution: there is dependence between the RadonNikodym derivative and the payoff of the bond.
– Preferable to use simulation for example
– Current framework works for a single company
only (multi-name extensions will be studied in my
following paper)
– CreditMetrics uses 1-year horizons for their VaR.
– It is also possible to do so with the model.
Conclusion
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• Intuitive model that provides results consistent
with the literature
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– Shape and level of credit spread curves, especially
over the short-term;
– Endogenous recovery rates;
– Interesting calibration to financial data;
• Possible to use the model for risk management
purposes
– Real-world default probabilities;
– Credit VaR and other tail risk measures;
• Future research
– Correlated multi-name extensions
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Bibliography
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• Main paper
– Boudreault, M. and G. Gauthier (2007), « A
structural credit risk model with a reduced-form
default trigger », Working paper, HEC Montréal,
Dept. of Management Sciences
• Other referenced papers
– Altman, E. and V. Kishore (1996), "Almost Everything You Always
Wanted to Know About Recoveries on Defaulted Bonds", Financial
Analysts Journal, (November/December), 57-63.
– Black, F. and J.C. Cox (1976), "Valuing Corporate Securities: Some
Effects of Bond Indenture Provisions", Journal of Finance 31, 351367.
– Collin-Dufresne, P. and R. Goldstein (2001), "Do credit spreads
reflect stationary leverage ratios?", Journal of Finance 56, 1929-1957.
– Dothan, U.L. (1978), "On the term structure of interest rates", Journal
of Financial Economics 6, 59-69.
Bibliography
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• Other referenced papers (continued)
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– Elton, E.J., M.J. Gruber, D. Agrawal and C. Mann (2001),
"Explaining the Rate Spread on Corporate Bonds", Journal of
Finance 56, 247-277.
– Jarrow, R. and S. Turnbull (1995), "Pricing Options on Financial
Securities Subject to Default Risk", Journal of Finance 50, 53-86.
– Jarrow, R., D. Lando and S. Turnbull (1997), "A Markov model for
the term structure of credit risk spreads", Review of Financial Studies
10, 481-523.
– Lando, D. (1998), "On Cox Processes and Credit Risky Securities",
Review of Derivatives Research 2, 99-120.
– Longstaff, F. and E. S. Schwartz (1995), "A simple approach to
valuing risky fixed and floating debt", Journal of Finance 50, 789819.
– Merton, R. (1974), "On the Pricing of Corporate Debt: The Risk
Structure of Interest Rates", Journal of Finance 29, 449-470.