2007 General Meeting

Assemblée générale 2007

Montréal, Québec

2007 General Meeting

Assemblée générale 2007

Montréal, Québec

A structural credit risk model with A structural credit risk model with a reduced-form default triggera reduced-form default trigger

Applications to finance and insuranceApplications to finance and insurance

A structural credit risk model with A structural credit risk model with a reduced-form default triggera reduced-form default trigger

Applications to finance and insuranceApplications to finance and insurance

Mathieu Boudreault, M.Sc., F.S.A.Mathieu Boudreault, M.Sc., F.S.A.Ph.D. Candidate, HEC MontréalPh.D. Candidate, HEC Montréal

Mathieu Boudreault, M.Sc., F.S.A.Mathieu Boudreault, M.Sc., F.S.A.Ph.D. Candidate, HEC MontréalPh.D. Candidate, HEC Montréal

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Introduction – Credit riskIntroduction – Credit risk

• General definition of credit risk• Potential losses due to:

• 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

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Introduction – Credit riskIntroduction – Credit risk

• Credit risk models can be used for:– 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

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Introduction – Classes of modelsIntroduction – Classes of models

• Models oriented toward risk management– 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)

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Introduction - ContributionsIntroduction - Contributions

• As part of my Ph.D. thesis, I introduce:– 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

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OutlineOutline

1. Introduction

2. Credit risk modelsa) Review of the literature

b) Risk management models, structural and reduced-form models

3. Hybrid model

4. Practical applications

5. Conclusion

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Risk management modelsRisk management models

• CreditMetrics by J.P. Morgan– 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

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Risk management modelsRisk management models

• Moody’s-KMV– Based on the distance to default metric

– Distance to default (DD):

– 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

DPTAEDD

1

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Structural modelsStructural models

• Suppose the debt matures in 20 years

0

100

200

300

400

500

600

700

800

900

0 2 4 6 8 10 12 14 16 18 20

Time

Mil

lio

ns

of

do

llar

sAssets

Liabilities

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Structural modelsStructural models

• Idea: default of the firm is tied to the value of its assets and liabilities

• Main contributions:– 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), Collin-Dufresne, Goldstein (2001): stochastic interest rates

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Reduced-form modelsReduced-form models

• 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

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Reduced-form modelsReduced-form models

• Moment of default r.v. is

where E1 is an exponential r.v. of mean 1.

• Default probability (under the risk-neutral measure)

• Example: Hu follows a Cox-Ingersoll-Ross process

10:0inf EduHt

t

u

T

t uQtt duHETQ exp1

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2007ComparisonComparison

• Structural models– 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

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Hybrid model – IdeasHybrid model – Ideas

• Hybrid model (presented in my Ph.D. thesis):– 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

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Hybrid model – IdeasHybrid model – Ideas

• Idea # 2: firms do not necessarily default immediately when assets cross liabilities– 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 – FrameworkHybrid model – Framework

• Suppose the assets and liabilities of the firm are given by the stochastic processes {At,t>0} and {Lt,t>0}

• Let us denote by Xt its debt ratio

• Idea of the model is to represent the stochastic default intensity {Hu,u>0} by

where h is a strictly increasing function

t

tt A

LX

uu XhH

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Hybrid model – FrameworkHybrid model – Framework

• Examples: h(x) = c, h(x) = cx2 and h(x) = cx10

0

0,05

0,1

0,15

0,2

0,25

0,3

0,35

0,4

0,45

0,5

0% 50% 100% 150% 200%

Debt ratio

Def

ault

inte

nsi

ty

Constant

Increasing

Fast increasing

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Hybrid model – MathematicsHybrid model – Mathematics

• Assume that under the real-world measure, the assets of the firm follow a geometric Brownian motion (GBM)

• 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

PttAtAt dBAdtAdA

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Hybrid model – MathematicsHybrid model – Mathematics

• Assume the transformation h is strictly increasing with the specific form

• 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.

– The drifts and diffusions change with the probability measures.

0,1 xcxxh

PttHtHt dBHtdtHtdH

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Hybrid model – MathematicsHybrid model – Mathematics

• It is possible to show that the survival probability can be written as a partial differential equation (PDE)

• 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

02

12

222

H

SHt

H

SHt

t

SSH tHtHt

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Practical applicationsPractical applications

• Impact of hedging on credit risk– 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 short-

term– Reason: firms with poor hedging that

eventually survive have a long-term advantage because their debt ratio will have improved significantly

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Practical applicationsPractical applications

• Impact of hedging on credit risk

0 5 10 15 20 25 30

50

05

50

60

06

50

70

0

Time to maturity (in years)

Sp

rea

ds

in b

ps

50%80%95%

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Practical applicationsPractical applications

• Endogenous recovery rate distribution– 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

– Altman & Kishore (1996):• Recovery rates between 40% to 70%

• Recovery rates decrease with default probability

• Recovery rates decrease during recessions

L

AR ;1min1

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Practical applicationsPractical applications

• Endogenous recovery rate distribution

• Obtained using 100 000 simulations• Asset volatilities of 10% and 15%• Initial debt ratios of 60% and 90%• No liquidation costs

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Practical applicationsPractical applications

• Credit spreads term structure– The price of defaultable zero-coupon bonds with

endogenous recovery rate is

• 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)

TQtt

tTr RETQe 1

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Practical applicationsPractical applications

• Credit spreads term structure

0 5 10 15 20 25 300

10

20

30

40

50

60

70

80

90

Time to maturity

Spr

eads

in b

ps

Increasing # 1

Increasing # 2Hump shape

Decreasing

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Practical applicationsPractical applications

• 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

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Practical applicationsPractical applications

• Real-world default probabilities

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Practical applicationsPractical applications

• Credit VaR– 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

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Practical applicationsPractical applications

• Credit VaR– Radon-Nikodym derivative can be obtained for

each debt structure– For example, under constant interest rates and

deterministically growing debt,

– Consequently, the T-year horizon Value-at-Risk for a defaultable zero-coupon bond is

where we recover a constant fraction R of the face value payable at maturity

T

rB

rT

dP

dQ

A

APT

A

A

2

2

1exp

TT

tTrP RTdP

dQeVaR 11%95

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Practical applicationsPractical applications

• Credit VaR– Caution: there is dependence between the Radon-

Nikodym 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.

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ConclusionConclusion

• Intuitive model that provides results consistent with the literature– 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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2007BibliographyBibliography

• 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, 351-367.

– 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.

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2007BibliographyBibliography

• Other referenced papers (continued)– 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, 789-819.

– Merton, R. (1974), "On the Pricing of Corporate Debt: The Risk Structure of Interest Rates", Journal of Finance 29, 449-470.