Models for Credit Risk in a Network Economy

Transcription

1 Models for Credit Risk in a Network Economy Henry Schellhorn School of Mathematical Sciences Claremont Graduate University

2 An Example of a Financial Network Autonation Visteon Ford United Lear Lithia GM TRW

3 Outline Prognostics and Financial Mathematics Single-Firm Credit Risk Models: Structural Reduced form Multiple-Firm Credit Risk Models: Statistical models Network models Austrian Fed model Jarrow and Yu (2001) Cossin-Schellhorn (2007) Application to: US Automotive Industry US Banking Sector Conclusion

4 Prognostics and Financial Mathematics Two Main Areas of Financial Mathematics Asset pricing and trading Risk management Forecasting in Asset Pricing and Trading If the model price of a security is higher than the observed price, then buy the security now Hedge the security over time At some point in the future cancel the position We are thus forecasting that in the future, in (most) states of the world we will make a profit

5 Prognostics and Financial Mathematics Forecasting in Risk Management Banks forecast their risk profile Value-at-Risk (VaR) is the 95% quantile of the loss probability distribution at a certain date Basel accords: VaR is the main metric to determine capital requirements 0 VaR

6 Some Finance Definitions Credit spread = difference between the rate on a risky bond and the risk-free rate Bond prices are inversely proportional to bond rates Arbitrage: an investment strategy that gives a positive return in all states of the world (a free lunch ) Contagion: the propensity of the default of one firm triggering the default of another firm CDO (Collateralized Debt Obligation): a securitized portfolio of defaultable assets (loans, bonds, mortgages); the security is sold in different tranches, with a varying degree of risk Call option: the right but not the obligation to buy a specific security a pre-specified strike price Exposure: how much a firm lent to another firm

7 Single Firm Credit Risk Models Option Pricing Models Structural Reduced-form Goals: Asset Pricing Risk Management Merton s (structural) Model (1974): Liabilities: one bond and equity Assets: follow geometric Brownian motion

8 Structural Credit Risk Models: Merton Since Assets=Liabilities, then at maturity of the debt Equity value=max (Asset value Debt principal, 0) Debt principal Asset value

9 Structural Credit Risk Models: Merton Equity is a European call on the value of the assets of the firm, with strike price =debt principal The value of equity at time zero is obtained by the Black- Scholes formula Problem: underlying is not observable What are we trying to achieve? What asset is rightly priced? Stock or debt? Problem: how to handle bankruptcy?

10 Structural Credit Risk Models: Leland Model Equity as an American option Bankruptcy can occur at any time Formula for the value of the debt Debt value in Leland s (1994) model is lower than debt value in Merton s model Problem: structural models are poor predictors of debt prices (Eom et al 2004)

11 Reduced-Form Credit Risk Models Goal: not to price Debt but to price Debt Derivatives Bond options CDOs, CLOs Debt Price is the Underlying Modeled as jump process References: Jarrow and Turnbull (1995) Duffie and Singleton (1997) etc

12 Financial Contagion: the Subprime Crisis US Properties in Foreclosure. Source: RealtyTrac

13 Statistical Models Goal: Pricing of CDOs CDOs have thousands of underlying variables: each homeowner equity value Difficulty is not of conceptual nature but in Statistical inference Monte Carlo simulation Methodologies Copulae (Li 2000) Markov Chain Monte Carlo (e.g., Frey and McNeil 2003)

14 Network Credit Risk Models Two Types of Models: Risk Management Models Pricing Models Risk Management Models Central Bank Models Example: Elsinger et al 2006 Given exposure matrix What is the distribution of bank s losses How does contagion affect the network

15 Network Model: Austrian Central Bank Source: Elsinger et al 2006

16 Network Pricing Models Reduced-Form Jarrow and Yu (2001) Structural Form Cossin and Schellhorn (2007)

17 Cossin-Schellhorn (CS 2 )Model Revenue Reinvestment Debt pmt to firm Cash account Dividends Debt pmt from firm

18 CS 2 Model Assumptions Fixed counterparty relationships Independent role of cash management Exogenous reinvestment policy Endogenous bankruptcy All firms survive bankruptcy Debt has infinite maturity

19 CS 2 Model: Formula for Debt Price The value of bankruptcy costs of firm iis: The value of debt of firm iis: ) 1 ( )) ( ( ) ( ˆ ) ( n O t V p w K t t BC i B i i i i i + = α ) 1 ( )) ( ( ) (1 ))) ( ( (1 ) ( ˆ ) ( n O t V p K w t V p r t t F i B i i i i B i i j ij i i + + = λ α

20 Benchmark Structural credit risk models explain only 30% of the credit spread (Huang and Huang, 2003) Compare single-firm model with multiple firm model Single-firm: directly infer future revenue from the stock price Multiple firm: our model

21 Automotive Industry: Exposure to and from Ford Autonation 1.1% 0.58% Visteon 3.6% Ford 0.51% United 1.45% 0.44% Lear Lithia TRW

22 Automotive Industry: Exposure to and from GM Autonation Visteon 1.1% 0.5% United 3.5% 0.4% Lear Lithia 1.4% GM 0.4% TRW

23 Automotive Industry: CS 2 Results Visteon credit spreads Q3Y2001-Q3Y time observed multi-firm single-firm

24 US Banking Industry Source: Bloomberg on September 17, Values in U.S. million dollars.

25 Conclusion Evolution from Single-Firm Models to Multiple-Firm Models Within Multiple-Firm Models, Evolution from Statistical Models To Network Models Network Information is becoming Public Particular Case of Exposure Data The Future: from a Social Network to a Financial Network?

26 References Cossin, D., and H. Schellhorn, "Credit Risk in a Network Economy". Management Science, Volume 53(10), , Duffie, J.D., and K.J. Singleton, "Modeling Term Structures of Defaultable Bonds." Review of Financial Studies, 12, Elsinger, H.; Lehar, A.; Summer, M. Risk Assessment for Banking Systems. Management Science pp Volume 52, Issue 9, Eom, Y.H., J. Helwege, and J.Z. Huang, 2004, "Structural Models of Corporate Bond Pricing: An Empirical Analysis. Review of Financial Studies, 17, 2, Frey, R. and McNeil, A., Dependent defaults in models of portfolio credit risk. Journal of Risk, 6(1): Jarrow, R.A., and S.M. Turnbull, "Pricing Derivatives on Financial Securities Subject to Credit Risk." Journal of Finance, 50, 1, Jarrow, R.A., and F. Yu, "Counterparty Risk and the Pricing of Defaultable Securities." Journal of Finance, 56, 5, Leland, H., "Corporate Debt Value, Bond Covenants, and Optimal Capital Structure." Journal of Finance, 49, 4, Merton, R.C, "On the Pricing of Corporate Debt: The Risk Structure of Interest Rates." Journal of Finance, 44,