**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 Assemblée générale 2007 Montréal, Québec

**Introduction – 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

**Introduction – 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

**Introduction – 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)

**Introduction - 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

**Outline** • Introduction • Credit risk models a) Review of the literature b) Risk management models, structural and reduced-form models • Hybrid model • Practical applications • Conclusion

**Risk 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

**Risk 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

**Structural models** • Suppose the debt matures in 20 years

**Structural 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

**Reduced-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

**Reduced-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

**Comparison** • 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

**Hybrid 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

**Hybrid 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+

**Hybrid 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

**Hybrid model – Framework** • Examples: h(x) = c, h(x) = cx2 and h(x) = cx10

**Hybrid 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

**Hybrid 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.

**Hybrid 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

**Practical 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

**Practical applications** • Impact of hedging on credit risk

**Practical 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

**Practical 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

**Practical 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)

**Practical applications** • Credit spreads term structure

**Practical 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

**Practical applications** • Real-world default probabilities

**Practical 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

**Practical 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

**Practical 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.

**Conclusion** • 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

**Bibliography** • 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.

**Bibliography** • 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.