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 (NYSE:BAC) 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 the movements in the credit spreads, default probabilities, and trading volume in the senior fixed-rate non-call bonds of these energy and commodity companies:

Vale Overseas Ltd. (NYSE:VALE)

Devon Energy Corporation (NYSE:DVN)

ConocoPhillips (NYSE:COP)

Hess Corporation (NYSE:HES)

Anadarko Petroleum Corporation (NYSE:APC)

**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 the 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.

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), X 1, X 2,â€¦.X n]

Quarter 2: PD=f[PD(-2),NI/A(-2), R(-2), X 1, X 2,â€¦.X n]

Quarter 3: PD=f[PD(-3),NI/A(-3), R(-3), X 1, X 2,â€¦.X n]

Quarter 4: PD=f[PD(-4),NI/A(-4), R(-4), X 1, X 2,â€¦.X n]

Quarter 5: PD=f[PD(-5),NI/A(-5), R(-5), X 1, X 2,â€¦.X n]

Quarter 6: PD=f[PD(-6),NI/A(-6), R(-6), X 1, X 2,â€¦.X n]

Quarter 7: PD=f[PD(-7),NI/A(-7), R(-7), X 1, X 2,â€¦.X n]

Quarter 8: PD=f[PD(-8),NI/A(-8), R(-8), X 1, X 2,â€¦.X n]

Quarter 9: PD=f[PD(-9),NI/A(-9), R(-9), X 1, X 2,â€¦.X n]

Quarter 10: PD=f[PD(-10),NI/A(-10), R(-10), X 1, X 2,â€¦.X n]

Quarter 11: PD=f[PD(-11),NI/A(-11), R(-11), X 1, X 2,â€¦.X n]

Quarter 12: PD=f[PD(-12),NI/A(-12), R(-12), X 1, X 2,â€¦.X n]

Quarter 13: PD=f[PD(-13),NI/A(-13), R(-13), X 1, X 2,â€¦.X n]

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 X i for i = 1,n.

Econometrically, we can use any one of a number of methods to fit the parameters of this formula after we specify the functions X i 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 on this issue will be distributed shortly. The KRIS default probability service includes more than 500,000 individual econometric relationships linking default probabilities to macroeconomic factors. An example is given here for energy firm Teck Resources (TCK), 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 (NYSE:RDS.A) (NYSE:RDS.B) is given here, graphing the fit of the actual and estimated default probabilities instead of the actual and predicted logit of the default probabilities:

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.

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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.

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J. Hilscher and M. Wilson, "Credit Ratings and Credit Risk: Is One Measure Enough?" Working Paper, Brandeis University and Oxford University, 2013.

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R. Jarrow, "Default Parameter Estimation using Market Prices," *Financial Analysts Journal,* 2001*.*

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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, 2009**.**

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van Deventer, Donald R. " Building a Default Model: Lessons Learned About How Much Data Is Necessary," Kamakura blog, 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, kamakuraco.com, April 22, 2009.

van Deventer, Donald R. "A Ratings Neutral Investment Policy," Kamakura blog, kamakuraco.com, May 12, 2009. Redistributed on riskcenter.com, May 15, 2009.

van Deventer, Donald. R. "Credit Portfolio Models: The Reduced Form Approach," Kamakura blog, kamakuraco.com, June 5, 2009. Redistributed on riskcenter.com on June 9, 2009.

van Deventer, Donald R. "Mapping Credit Models to Actual Defaults: Key Issues and Implications," Kamakura blog, kamakuraco.com, June 11, 2009. Redistributed on 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, kamakuraco.com, June 16, 2009. Redistributed on riskcenter.com on June 18, 2009.

van Deventer, Donald R. " Out of Sample Performance of Reduced Form and Merton Default Models," Kamakura blog, kamakuraco.com, July 9, 2009. Redistributed on 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, kamakuraco.com, July 28, 2009. Redistributed on riskcenter.com on July 29, 2009.

van Deventer, Donald R. "The Merton Model of Risky Debt: Confessions of a Former True Believer," Kamakura blog, kamakuraco.com, July 30, 2009. Redistributed on riskcenter.com on July 31, 2009.

van Deventer, Donald R. "Comparing Sovereign and Corporate Default Models: Facts and Figures." Kamakura blog, kamakuraco.com, August 6, 2009. Redistributed on riskcenter.com on August 7, 2009.

van Deventer, Donald R. "Comparing the Credit Risk Term Structure of Corporations and Sovereigns," Kamakura blog, kamakuraco.com, September 16, 2009. Redistributed on riskcenter.com on September 17, 2009.

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

**Disclosure:** I/we have no positions in any stocks mentioned, and no plans to initiate any positions within the next 72 hours.

I wrote this article myself, and it expresses my own opinions. I am not receiving compensation for it. I have no business relationship with any company whose stock is mentioned in this article.