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An Introduction to Stress Testing Oil Prices, Default Probabilities and Credit Spreads Donald R. van Deventer March 29, 2016 Over the last 18 months, the dramatic fall in oil prices has triggered a dramatic widening of credit spreads and default probabilities for oil-related firms. To a slightly lesser degree, the same kind of macro factor sensitivity in the credit spreads and default probabilities in firms closely associated with other basic commodities. This note explains the “reduced reduced form” modeling approach used in Kamakura’s KRIS default probability service to link forward looking macro factors to simulated default probabilities. We refer readers interested in more detail to the recent note from Kamakura “Bank of America and CCAR 2016 Stress Testing: A Simple Model Validation Example” and the references at the end of this note. Background Information on Movements in Credit Spreads and Default Probabilities In this section, we graphically summarize movements in the credit spreads, default probabilities, and trading volume in the senior fixed-rate non-call bonds of these energy and commodity companies:

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Vale Overseas Ltd.

Devon Energy Corporation

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ConocoPhillips

Hess Corporation

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Anadarko Petroleum Corporation

Credit Spreads and Default Probabilities Many market participants naively believe that there is a one to one mapping of default probabilities in the bond market and credit default swap markets. In a recent paper “Credit Spreads and Default Probabilities: A Simple Model Validation Example” we proved that this model is strongly rejected from a model validation perspective. In fact, in a series of publications and in another Appendix to the KRIS Technical Guide, Kamakura has proven econometrically that almost 60 factors in addition to default probabilities are statistically significant in predicting credit spreads. The Kamakura approach to modeling credit spreads is evolving but is strongly related to default probability estimation techniques. In what follows, we concentrate on an oil-related stress test of default probabilities. Clarifying the Distinction between the Best Available Current Default Probability Estimates and Simulated Default Probabilities Conditional on Macro Factors In the recent note on Bank of America, we explain that Kamakura’s KRIS default probabilities, the best available default probability estimates, are based on observable macro factors, company specific inputs including financial ratios and stock price history, and other inputs.

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When simulating default probabilities forward on the false assumption that the macro values are known today, there are basically two methods to do the analysis:

• Use macro factors only, with no company specific inputs. • Use both macro factors and only those company specific inputs known at time

zero, since their future values are unknowable. In the Bank of America note, the second approach can be summarized this way if we are simulating default probabilities 13 quarters forward under the Federal Reserve’s CCAR Stress Tests. Consider the case where the potential explanatory variables include macro factors, lagged default probabilities, lagged net income to assets ratio, and the lagged one year return on the firm’s common stock, R. The macro factors are not lagged unless the econometric process reveals that lags are helpful. Negative numbers in parentheses denote lagged values, which effectively cause the time 0 values to be used as explanatory variables: Quarter 1: PD=f[PD(-1),NI/A(-1), R(-1), X1, X2,….Xn] Quarter 2: PD=f[PD(-2),NI/A(-2), R(-2), X1, X2,….Xn] Quarter 3: PD=f[PD(-3),NI/A(-3), R(-3), X1, X2,….Xn] Quarter 4: PD=f[PD(-4),NI/A(-4), R(-4), X1, X2,….Xn] Quarter 5: PD=f[PD(-5),NI/A(-5), R(-5), X1, X2,….Xn] Quarter 6: PD=f[PD(-6),NI/A(-6), R(-6), X1, X2,….Xn] Quarter 7: PD=f[PD(-7),NI/A(-7), R(-7), X1, X2,….Xn] Quarter 8: PD=f[PD(-8),NI/A(-8), R(-8), X1, X2,….Xn] Quarter 9: PD=f[PD(-9),NI/A(-9), R(-9), X1, X2,….Xn] Quarter 10: PD=f[PD(-10),NI/A(-10), R(-10), X1, X2,….Xn] Quarter 11: PD=f[PD(-11),NI/A(-11), R(-11), X1, X2,….Xn] Quarter 12: PD=f[PD(-12),NI/A(-12), R(-12), X1, X2,….Xn] Quarter 13: PD=f[PD(-13),NI/A(-13), R(-13), X1, X2,….Xn] There are a number of econometric techniques one can use to fit these relationships. For purposes of today’s example, we use the logistic function to predict the default probability at time t based on n functions of macro factors Xi for i = 1,n:

∑+

==

−−n

iii tX

e

tP1

][

1

1][βα

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Econometrically, we can use any one of a number of methods to fit the parameters of this formula after we specify the functions Xi of the m macro factors that we will use as candidate variables. Some common choices are

General linear methods Maximum likelihood estimation

Non-linear least squares estimation Fractional regression Ordinary least squares on the transformed logit Z of the default probabilities:

For a fortunate analyst, the methods will often yield similar stress test results, but the ordinary least squares approach can occasionally produce strongly biased results. A paper in the works will be distributed shortly. The KRIS default probability service includes more than 500,000 individual econometric relationships linking default probabilities to macro-economic factors. An example is given here for energy firm Teck Resources, in which we compare the actual logit transformation Z of the historical default probabilities and the in-sample fitted values of the logit of the historical default probabilities:

A similar analysis for Royal Dutch Shell is given here, graphing the fit of the actual and estimated default probabilities instead of the actual and predicted logit of the default probabilities:

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For more information, clients of the KRIS default probability service can access the KRIS Technical Guide on Reduced Reduced Form Models available from your Kamakura Corporation representative. We encourage clients to review the discussion of “forbidden models” in Angrist and Pischke and cited in the references below: References Joshua D. Angrist and Jorn-Steffen Pischke, Mostly Harmless Econometrics: An Empiricist’s Companion, Princeton University Press, Princeton, 2009. Sreedhar T. Bharath and Tyler Shumway, “Forecasting Default with the Merton Distance to Default Model,” Review of Financial Studies, May 2008. Original paper dated 2004. Derek H. Chen, Harry H. Huang, Rui Kan, Ashok Varikooty, and Henry N. Wang, “Modelling and Managing Credit Risk,” Asset & Liability Management: A Synthesis of New Methodologies, RISK Publications, 1998. J. Y. Campbell, Jens Hilscher, and Jan Szilagyi, “In Search of Distress Risk,” Journal of Finance, December 2008. Original working paper was dated October 2004. Gregory Duffee, “Estimating the Price of Default Risk,” The Review of Financial Studies, 12 (1), 197-226, Spring 1999. D. Duffie, M. Schroder and C. Skiadas, “Recursive Valuation of Defaultable Securities and the Timing of Resolution of Uncertainty,” working paper, Stanford University, 1995. D. Duffie and K. Singleton, “Modeling Term Structures of Defaultable Bonds,” Review of Financial Studies, 12(4), 1999, 197-226.

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J. Hilscher, Robert A. Jarrow, and Donald R. van Deventer, “Measuring the Risk of Default, A Modern Approach“, RMA Journal, July-August, 2008, pp. 60-65.

J. Hilscher and M. Wilson, “Credit Ratings and Credit Risk: Is One Measure Enough?” Working Paper, Brandeis University and Oxford University, 2013. J. Ingersoll, Theory of Modern Financial Decision Making (Rowman & Littlefield Publishers, Inc., New York), 1987. R. Jarrow, “Default Parameter Estimation using Market Prices,” Financial Analysts Journal, 2001. R. Jarrow, D. Lando and S. Turnbull, “A Markov Model for the Term Structure of Credit Risk Spreads,” The Review of Financial Studies, 10(2), 1997, 481-523. R. Jarrow, David Lando, and Fan Yu, “Default Risk and Diversification: Theory and Applications,” Mathematical Finance, January 2005, 1-26 R. Jarrow and S. Turnbull, “Pricing Derivatives on Financial Securities Subject to Credit Risk,” Journal of Finance, 50(1), 1995, 53 – 85. R. Jarrow and S. Turnbull, Derivative Securities, 2nd edition, South-Western College Publishing: Cincinnati, Ohio, 2000. R. Jarrow, M. Mesler and D. van Deventer, Kamakura Public Firm Default Probabilities Technical Guide, Kamakura Risk Information Services, Version 2.0, May, 2003. R. Jarrow, M. Mesler and D. van Deventer, Kamakura Public Firm Default Probabilities Technical Guide, Kamakura Risk Information Services, Version 3.0, December, 2003. R. Jarrow, M. Mesler and D. van Deventer, Kamakura Public Firm Default Probabilities Technical Guide, Kamakura Risk Information Services, Version 4.1, Edition 10.0, January 26, 2006. R. Jarrow, S.P. Klein, M. Mesler and D. van Deventer, Kamakura Public Firm Default Probabilities Technical Guide, Kamakura Risk Information Services, Version 5.0, Edition 12.0, March 3, 2011. R. Jarrow, J. Hilscher, T. Le, M. Mesler and D. van Deventer, Kamakura Public Firm Default Probabilities Technical Guide, Kamakura Risk Information Services, Version 6.0, Edition 7.0, June 30, 2015. R. Jarrow and Donald R. van Deventer, “Integrating Interest Rate Risk and Credit Risk in ALM,” Asset & Liability Management: A Synthesis of New Methodologies, RISK Publications, 1998. R. Jarrow and Donald R. van Deventer, "Practical Use of Credit Risk Models in Loan Portfolio and Counterparty Exposure Management: An Update," Credit Risk: Models and Management, Second Edition, David Shimko, Editor, RISK Publications, 2004.

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R. Jarrow, Donald R. van Deventer and Xiaoming Wang, “A Robust Test of Merton’s Structural Model for Credit Risk,” Journal of Risk, Fall 2003, pp. 39-58. J. Johnston, Econometric Methods, 2nd edition, McGraw-Hill Book Company, 1972. E.P. Jones, S. P. Mason, and E. Rosenfeld, “Contingent Claims Analysis of Corporate Capital Structure: An Empirical Investigation,” Journal of Finance 39, 611-626, 1984. F. A. Longstaff and E. S. Schwartz, “A Simple Approach to Valuing Risky Fixed and Floating Rate Debt”, Journal of Finance 50, 789-820, 1995. G. S. Maddala, Introduction to Econometrics, 2nd edition, Prentice Hall, 1992. R. Merton, Theory of Rational Option Pricing, Bell Journal of Economics and Management Science 4, 141-183, 1973. ----------------, “An Intertemporal Capital Asset Pricing Model,” Econometrica 41, 867-887, September 1973. ----------------, On the Pricing of Corporate Debt: The Risk Structure of Interest Rates, Journal of Finance 29, 1974, 449-470. ----------------, An Analytic Derivation of the Cost of Deposit Insurance and Loan Guarantees: An Application of Modern Option Pricing Theory, Journal of Banking and Finance 1, 1977, 3-11. ----------------, On the Cost of Deposit Insurance When There are Surveillance Costs, Journal of Business 51, 1978, 439-452. ----------------, Continuous Time Finance, Basil Blackwell Inc., Cambridge, Massachusetts, 1990. L. Nielsen, J. Saa-Requejo and P. Santa-Clara, “Default Risk and Interest Rate Risk: The Term Structure of Default Spreads,” working paper INSEAD, 1993. D. Shimko, N. Tejima and van Deventer, “The Pricing of Risky Debt when Interest Rates are Stochastic,” Journal of Fixed Income, 58-66, September 1993. van Deventer, Donald R. and K. Imai, Financial Risk Analytics: A Term Structure Model Approach for Banking, Insurance and Investment Management, McGraw-Hill, 1996. van Deventer, Donald R. and K. Imai, Credit Risk Models and the Basel Accords, John Wiley & Sons, 2003. van Deventer, Donald R., K. Imai and M. Mesler, Advanced Financial Risk Management: Tools and Techniques for Integrated Credit Risk and Interest Rate Risk Management., second edition, John Wiley & Sons, Singapore, 2013. van Deventer, Donald R.” ‘Point in Time’ versus ‘Through the Cycle’ Credit Ratings: A Distinction without a Difference,” Kamakura blog, www.kamakuraco.com, March 24,

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2009. van Deventer, Donald R. “Default Risk and Equity Portfolio Management: A Mission Critical Combination” Kamakura blog, www.kamakuraco.com, April 7, 2009. van Deventer, Donald R. “Building a Default Model: Lessons Learned About How Much Data Is Necessary,” Kamakura blog, www.kamakuraco.com, April 14, 2009. Redistributed on www.riskcenter.com, April 16, 2009. van Deventer, Donald R. “Why Would An Analyst Cap Default Rates in a Credit Model?” Kamakura blog, www.kamakuraco.com, April 22, 2009. van Deventer, Donald R. “A Ratings Neutral Investment Policy,” Kamakura blog, www.kamakuraco.com, May 12, 2009. Redistributed on www.riskcenter.com, May 15, 2009. van Deventer, Donald. R. “Credit Portfolio Models: The Reduced Form Approach,” Kamakura blog, www.kamakuraco.com, June 5, 2009. Redistributed on www.riskcenter.com on June 9, 2009. van Deventer, Donald R. “Mapping Credit Models to Actual Defaults: Key Issues and Implications,” Kamakura blog, www.kamakuraco.com, June 11, 2009. Redistributed on www.riskcenter.com on June 15, 2009. van Deventer, Donald R. “Does a Rating or a Credit Score Add Anything to a Best Practice Default Model?” Kamakura blog, www.kamakuraco.com, June 16, 2009. Redistributed on www.riskcenter.com on June 18, 2009. van Deventer, Donald R. “Out of Sample Performance of Reduced Form and Merton Default Models,” Kamakura blog, www.kamakuraco.com, July 9, 2009. Redistributed on www.riskcenter.com on July 15, 2009. van Deventer, Donald R. “The Search for Significance in Default Modeling: The Long and the Short of It,” Kamakura blog, www.kamakuraco.com, July 28, 2009. Redistributed on www.riskcenter.com on July 29, 2009. van Deventer, Donald R. “The Merton Model of Risky Debt: Confessions of a Former True Believer,” Kamakura blog, www.kamakuraco.com, July 30, 2009. Redistributed on www.riskcenter.com on July 31, 2009. van Deventer, Donald R. “Comparing Sovereign and Corporate Default Models: Facts and Figures.” Kamakura blog, www.kamakuraco.com, August 6, 2009. Redistributed on www.riskcenter.com on August 7, 2009. van Deventer, Donald R. “Comparing the Credit Risk Term Structure of Corporations and Sovereigns,” Kamakura blog, www.kamakuraco.com, September 16, 2009. Redistributed on www.riskcenter.com on September 17, 2009.

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van Deventer, Donald R. “Modeling Correlated Default in a Reduced Form Model: A Worked Example,” Kamakura blog, www.kamakuraco.com, September 24, 2009. Redistributed on www.riskcenter.com on September 28, 2009.