Introduction Credit risk

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 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 2of 34

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 3of 34

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) 4of 34

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 5of 34

Outline 1. Introduction 2. Credit risk models a) Review of the literature b) Risk management models, structural and reduced-form models 3. Hybrid model 4. Practical applications 5. Conclusion 6of 34

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 7of 34

Risk management models Moody s-kmv Based on the distance to default metric Distance to default (DD): DD E [ A ] = 1 DPT σ 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 8of 34

Structural models Suppose the debt matures in 20 years 900 800 Assets Liabilities Millions of dollars 700 600 500 400 300 200 100 0 0 2 4 6 8 10 12 14 16 18 20 Time 9of 34

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 10 of 34

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 11 of 34

Reduced-form models Moment of default r.v. is { } t t > 0 : H du E τ = inf where E 1 is an exponential r.v. of mean 1. Default probability (under the risk-neutral measure) Q t Example: H u follows a Cox-Ingersoll-Ross process 0 u > ( ) = Q τ T 1 E exp H du t T t 1 u 12 of 34

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 13 of 34

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 14 of 34

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+ 15 of 34

Hybrid model Framework Suppose the assets and liabilities of the firm are given by the stochastic processes {A t,t>0} and {L t,t>0} Lt Let us denote by X t its debt ratio X t = A Idea of the model is to represent the stochastic default intensity {H u,u>0} by H = h( X ) u u where h is a strictly increasing function t 16 of 34

Hybrid model Framework Examples: h(x) = c, h(x) = cx 2 and h(x) = cx 10 Default intensity 0,5 Constant 0,45 Increasing 0,4 Fast increasing 0,35 0,3 0,25 0,2 0,15 0,1 0,05 0 0% 50% 100% 150% 200% Debt ratio 17 of 34

Hybrid model Mathematics Assume that under the real-world measure, the assets of the firm follow a geometric Brownian motion (GBM) da t = µ A dt + σ Propose different debt structures Under constant risk-free rate Debt grows with constant rate L t = L 0 exp(bt) Debt is a GBM correlated with assets (hedging) Under stochastic interest rates A t Debt is a risk-free zero-coupon bond Assets are correlated with interest rates A A t db P t 18 of 34

Hybrid model Mathematics Assume the transformation h is strictly increasing with the specific form ( ) α 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 = µ t H dt + σ t H db t H ( ) ( ) P The drifts and diffusions change with the probability measures. t H t t 19 of 34

Hybrid model Mathematics It is possible to show that the survival probability can be written as a partial differential equation (PDE) H t S S + t + µ H S H 2 2 () t H + σ () t H = 0 t When µ H (t) and σ H (t) are constants, can use Dothan (1978) quasi-closed form equation. Otherwise, we have to rely on finite difference methods or tree approaches 1 2 H t 2 S H 2 20 of 34

Practical applications 21 of 34 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 shortterm Reason: firms with poor hedging that eventually survive have a long-term advantage because their debt ratio will have improved significantly

22 of 34 Practical applications Impact of hedging on credit risk Spreads in bps 50% 80% 95% 500 550 600 650 700 0 5 10 15 20 25 30 Time to maturity (in years)

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 A ( ) τ R τ = 1 κ min 1; Lτ Altman & Kishore (1996): Recovery rates between 40% to 70% Recovery rates decrease with default probability Recovery rates decrease during recessions 23 of 34

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 24 of 34

Practical applications Credit spreads term structure The price of defaultable zero-coupon bonds with endogenous recovery rate is ( ) ( ) ( ) [ ] r T t Q e Qt τ > T + Et Rτ 1τ 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) 25 of 34

Practical applications Credit spreads term structure 90 Spreads in bps 80 70 60 50 40 30 20 10 Increasing # 1 Increasing # 2 Hump shape Decreasing 26 of 34 0 0 5 10 15 20 25 30 Time to maturity

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 27 of 34

Practical applications Real-world default probabilities Cumulative default probability Cumulative default probability 0.03 0.025 0.02 0.015 0.01 0.005 Wal-Mart CDS implied real-world default probability Lower bound Upper bound Empirical default probabilities (S&P) 0 0 5 10 15 Time (in years) 0.06 0.05 0.04 0.03 0.02 0.01 Exxon Mobil CDS implied real-world default probability Lower bound Upper bound Empirical default probabilities (S&P) Cumulative default probability Cumulative default probability 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 General Motors CDS implied real-world default probability Lower bound Upper bound Empirical default probabilities (S&P) 0 0 5 10 15 Time (in years) 0.14 0.12 0.1 0.08 0.06 0.04 0.02 McDonalds CDS implied real-world default probability Lower bound Upper bound Empirical default probabilities (S&P) 28 of 34 0 0 5 10 15 Time (in years) 0 0 5 10 15 Time (in years)

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 29 of 34

Practical applications Credit VaR Radon-Nikodym derivative can be obtained for each debt structure For example, under constant interest rates and deterministically growing debt, dq dp ( T ) 2 µ A r P 1 µ A r = exp B T T σ A 2 σ A Consequently, the T-year horizon Value-at-Risk for a defaultable zero-coupon bond is VaR P 95% e r dq dp ( ) ( T t ) ( T ) 1 { τ > T } + R1 { τ T } where we recover a constant fraction R of the face value payable at maturity 30 of 34

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. 31 of 34

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 32 of 34

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. 33 of 34

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. 34 of 34