The estimation of fair market value for thinly traded or non-traded instruments containing credit risk is an essential component in the management of financial institutions. International Accounting Standard 39, for example, requires that there be a high correlation between the value of the hedging instruments and the value of the instruments being hedged, even if these instruments are not traded in such a way that market prices are continuously observable. In a similar vein, Financial Accounting Standard 157 sets out the hierarchy of rules on how “fair value” should be determined in markets of varying liquidity and transparency.
Default probabilities and losses in the event of default are critical determinants of the fair market value for credit sensitive securities, but they are not the only such determinants. Risk aversion, differential information, market liquidity and market frictions (e.g. transaction costs and institutional constraints) are at least as important. Financial theory argues that it is the combination of all of these components that determines credit spreads, not just the default probabilities and recovery rates alone.
Contrary to this view, it is still common to hear traders argue that observable spreads in the credit default swap market are equal to the product of the default probability for the underlying credit times the loss given default. But, this simple trader’s argument obviously ignores supply and demand considerations in the market for credit insurance. Indeed, if CDS spreads only reflect expected loss with no risk premium, then why would institutional investors be willing to be providers of credit insurance in the CDS market?
Consistent with financial theory, and contrary to the simple trader’s argument, researchers have consistently found that bond spreads and credit default swap quotations are higher than historical credit loss experience for those particular credits (see Van Deventer, Imai and Mesler , Chapter 18, for a recent review). Empirically, a credit risk premium appears to exist.
Nonetheless, the simple trader’s argument is still common in the industry. Consequently, the purpose of this post is two-fold. First, to help dispel the trader’s argument, we explain in simple terms and using standard economic theory why credit spreads should exceed expected losses (the default probability times loss given default). Second, given the existence of a credit risk premium, we then provide a statistical model for estimating credit spreads in the credit default swap market that uses other explanatory variables (in addition to default probabilities and losses given default) to capture credit risk premium. The statistical model performs quite well. The estimation is performed using a large database of credit default swap prices provided by the broker GFI.
The Credit Risk Premium - the Supply and Demand for Credit
The degree of competition among lenders for a particular financial instrument varies dramatically with the attributes of the instrument and the attributes of the lenders. A hedge fund trader recently commented:
“Bond prices in the market place anticipate ratings changes with one exception. When a firm is downgraded to ‘junk,’ a very large number of institutions are compelled to sell immediately. This causes a step down in bond prices of large magnitude, unlike any other change in ratings.”
This statement reflects market segmentation, generated by heterogeneity in lenders’ risk aversion, trading restrictions, and private information. Implicitly this quotation says that the number of potential lenders to a “junk” credit is much less than the number of potential lenders to an investment grade credit, even if the default risk for these companies are the same. There is less demand for the financial liabilities of the junk credit, and spreads therefore must widen. In the limit, a small and risky business may have only one lender willing to lend it monies. In this case, the fact that the lender is a monopoly supplier (with their own risk aversion, private information, and institutional constraints) has at least as much impact on the level of the spread as does the default probability or loss given default of the borrower.
The graph below illustrates this point for a very large hypothetical company. On the x-axis is plotted the one-year default probabilities that proxy for the credit risk of the company. On the y-axis is plotted both the dollar supply and demand for borrowed funds.
On the demand side, as the company’s true one-year default probability increases, its need for funds increases. It needs more funds because as the default probability rises, its revenues are declining and its expenses are increasing. Conversely, when the company has strong cash flows, it has very low default risk and it has little need for additional funds.
On the supply side, when the company is a strong credit, the potential supply of funds to the company is very large. More lenders are willing to participate in loans to the company. This is due to the heterogeneity of lenders’ risk aversion, private information, institutional constraints, and transaction costs. Indeed, those lenders that are very risk averse, or who are not privy to private information concerning this company’s cash flows, are willing to lend because the risk is low and private information is less relevant. Also, any institutional investment restrictions regarding junk lending will not be binding. As the company becomes more risky, the number of lenders drops, due to the desire to avoid risk by an increasing number of lenders and related concerns regarding private information about the borrower’s prospects. Institutional constraints (as mentioned above) will also kick in at an appropriate level. The maximum potential supply of funds drops steadily until, in the limit, it reaches zero.
Contrary to the diagram, however, the equilibrium is NOT where the supply of funds as a function of the one-year default probabilities exactly equals the funding needs (where the lines intersect on the graph). This is because there are other considerations to an economic equilibrium in the credit risk market, not built into these curves. For example, loss given default is a relevant and missing consideration in the determination of a borrower’s risk, as are third party bankruptcy costs (e.g. legal fees) and transactions costs incurred in the market for borrowed funds. From the borrower’s side, a financial strategy that retains the option to seek additional funds at a later date holds a real option of considerable value and this option provides a reason not to borrow the maximum at any moment in time. These missing considerations make the analysis more complex then this simple diagram indicates, but the downward sloping nature of the supply curve for lenders’ funds will still hold. That is, as the credit risk of a borrower increases, the volume of available loans declines.
An individual lender, realizing the downward sloping nature of the aggregate supply curve, will decide on the volume of their lending based on their marginal costs and revenues. The graph below depicts a lender’s percentage marginal costs and revenues per dollar loan (y-axis) versus loan volume (x-axis).
Marginal cost is depicted as a horizontal line. It represents the expected loss component of the spread plus the marginal costs of servicing the credit. The expected loss component, equal to PD*LGD, depends on the lender’s private information regarding a borrower’s default probabilities and loss given default.
Marginal revenue represents the lender’s risk adjusted percentage revenue per additional dollar of loan. Due to risk aversion1, the risk adjusted marginal revenue curve will be downward sloping. To induce a risk averse lender to hold more of a particular loan in a portfolio, increasing idiosyncratic risk, the lender must be compensated via additional spread revenue. Alternatively, and independent of risk aversion, if the lender’s loan volume is sufficient to affect the market spread (i.e. the lender is not a price taker), then the marginal revenue curve will be downward sloping based on this consideration alone. For either reason, as the loan volume increases, the percentage revenue per dollar of additional lending declines.
Standard economic reasoning then implies that the lender will extend loans until their risk adjusted marginal revenue equals their marginal cost. Ignoring for the moment market frictions (the difference between marginal costs and the expected loss), we see that for the marginal dollar lent, the marginal credit spread equals the expected loss (PD*LGD). Next, adjusting for market frictions, on the margin, the marginal credit spread equals (marginal costs + PD*LGD). But, the observed credit spread reflects the average spread per dollar loan, and not the marginal spread. Since the marginal revenue spread curve is downward sloping, the average revenue spread per dollar loan exceeds the marginal revenue spread curve. Thus, the observed credit spread per dollar loan is given by:
Credit spread = (average-marginal revenue spread + marginal costs + PD*LGD).
The first component is due to risk aversion and or market liquidities, i.e. the fact that the supply curve for loans is downward sloping. The second component is due to market frictions - the marginal costs of servicing loans. And, the third component is the expected loss itself.
This simple economic reasoning shows that a trader who argues that the CDS quote equals the default probability times the loss given default is really assuming an extreme market situation. The extreme market situation requires that the first and second components of the credit spread (as depicted above) are zero. The first component is zero only if the supply curve for loans is horizontal. That is, there is an infinite supply of funds at the marginal cost of credit – markets are perfectly liquid and there is no credit risk premium. And, the second component is zero only if there are no costs in servicing loans. These are unrealistic conditions, and their absence explains why the simple trader’s argument is incorrect.
A Statistical Model for Credit Risk Spreads
Generally speaking, there are two methods available to estimate credit risk spreads. The first is to build an economic model, akin to that described above, and fit the model to observed credit risk spreads. The second is to fit a statistical model. A statistical model is often useful when building an (equilibrium) economic model is too complex. And, furthermore, a statistical model can be thought of as an approximation to the economic model. For example, linear regression analysis can be viewed as providing a linear approximation to a more complex, and non-linear, economic model.
As suggested by the discussion in the previous section, building a realistic equilibrium model for credit spreads with lender heterogeneity in risk aversion, private information, and institutional restrictions, is a complex task. Furthermore, such a construction is subject to subjective assumptions regarding lender preferences, endowments, institutional structures as well as the notion of an economic equilibrium (e.g. a non-cooperative game versus a Walrasian equilibrium). To avoid making these subjective assumptions, which limit the applicability of the model selected, and, in order to obtain a usable model for practice, we elect to build a statistical model for credit risk spreads instead.
Our statistical model uses various explanatory variables (to be discussed), selected to capture the relevant components of the credit spread, as discussed in the previous section. In particular, we need explanatory variables that will capture the credit risk premium, market liquidity, institutional constraints, and expected losses. Before providing these explanatory variables, it is important to first discuss the functional form of the explanatory variables fit to the data.
Previous authors (see, for example, Campbell et al , Collin-Dufresne et al , Huang et al , and Elton et al ) fitting a statistical model to credit spreads typically use a linear function to link the credit spread to explanatory variables via ordinary least squares regression. Implicit in the linear regression structure, however, is the possibility that when the model is used in a predictive fashion, the statistical model may predict negative credit spreads. Negative credit spreads, of course, are inconsistent with any reasonable economic equilibrium.
To show that predicting negative credit spreads is not just a conceptual problem, but an actual one, we fitted a linear regression linking 4,410 observations of CDS spreads (bid side) for seven major auto makers during the period January 1, 2004 to August 3, 2005. The credit spreads were provided by GFI. During this period, the auto sector was the most actively traded sector in the CDS market. We fitted a linear function linking credit default swaps for five-year maturities to three default probability inputs and five other company specific variables. While this relationship explained almost 91% of the variation in actual CDS spreads, the graph below shows that a relatively large number of predicted CDS spreads are negative. This is unacceptable for reasonable mark-to-market calculations.
Of course, the fix is easy. One only needs to fit a functional form that precludes negative credit spreads. Fitting a linear function to ln[credit spread(t)] is one such transformation. Another transformation is to use the logistic function,
where alpha and beta for i = 1,…n are constants and X1 for i = 1,…n are the relevant explanatory variables.
Unlike the natural logarithm, the logistic formula has the virtue that predicted CDS spreads always lie between 0 and 100%. And, similar to the use of the natural logarithm, one can estimate the alphas and betas in the logistic formula by using the transformation, (-ln[(1-credit spread[t])/credit spread[t]]) to fit a linear function in the explanatory variables via ordinary least squares regression. Alternatively, one can use a general linear model for the derivation. In this post, we use the logistic regression approach to model credit risk spreads.
The broker GFI supplied the CDS quotes used in our estimation. Our database includes daily data from January 2, 2004 to November 3, 2005, which includes more than 500,000 credit default swap bid, offered, and traded price observations. Bid prices, offered prices and traded prices were all estimated separately. There were 223,006 observations of bid prices, 203,695 observations of offered prices, and 19,822 observations of traded prices. Interestingly, traded CDS prices were only 1/10th as numerous as the bid and offered quotations. CDS quotes where an upfront fee was charged were excluded from the estimation because conversion of the upfront fee to a spread equivalent requires a joint hypothesis about the term structure and relationship of default probabilities and credit spreads, and that is what we are trying to derive in this post.
The explanatory variables used to fit CDS prices include the credit rating, estimated default probabilities, and company specific attributes. Two types of estimated default probabilities were used: one from the Jarrow - Chava hazard rate model, and the second from a Merton type structural model. All of these estimates were obtained from version 3.0 of the Kamakura Risk Information Services default probability service. The CDS maturities, agency ratings, and all of the individual macro-economic factors and company specific balance sheet ratios that are inputs to the Jarrow-Chava hazard rate model were also included as inputs. These macro variables and company specific factors are listed below. These macro- and micro- variables are intended to capture the credit risk premium, market liquidities, institutional constraints, and market frictions as previously discussed.
47 variables were found to be statistically significant in predicting CDS spreads. For bid prices, the explanatory variables included:
- 7 maturities of KRIS version 3.0 Jarrow - Chava default probabilities
- KRIS version 3.0 Merton default probabilities
- Dummy variables for each rating category
- A dummy variable indicating whether or not the company is a Japanese company
- 10 company specific financial and equity ratios (see Jarrow and Chava  for a list of the relevant variables)
- Dummy variables for each CDS maturity
- Dummy variables for senior debt
- Dummy variables for the restructuring language of the CDS contract
- Selected macro-economic factors (see Jarrow, Li, Matz, Mesler, and van Deventer  for details)
The best fitting relationship for each of the three series (bid, offered and traded) explained more than 80% of the variation in the transformed CDS quotations. The best fitting relationship also explains more than 90% of the variation in the raw CDS quotations (after reversing the log transformation explained above). The t-scores of the explanatory variables ranged from 2 to 236. The five-year maturity Jarrow - Chava default probability had the highest t-score among all the different maturity default probabilities included. The Japanese dummy variable was statistically significant with a t-score equivalent of 133 standard deviations, indicating that CDS spreads on Japanese names are much narrower than the otherwise equivalent non-Japanese name. Company size had a t-score equivalent of 100 standard deviations from zero, implying that CDS spreads are narrower for large companies, everything else constant. As indicated, our statistical model fits CDS market prices quite well, validating the existence of credit risk premium in CDS prices.
Comparing the Explanatory Power of Different Explanatory Variables
Although the previous section provides the best fitting CDS spread model, it is interesting to investigate a related question. The question is this: how do ratings, Merton default probabilities, hazard rate default probabilities and the hybrid (combined) approach outlined in the previous section compare in their ability to explain CDS quotes?
To answer this question we estimated four different logistic-based ordinary least squares models. The results are graphed below.
The Merton default probabilities explain 12-28% of the variation in the transformed CDS quotes. Ratings explain 36-42% of the transformed CDS quotes. The Kamakura Risk Information Services version 3.0 Jarrow - Chava default probabilities and all of their explanatory variables explain 56-61% of the transformed variables. The super hybrid approach outlined in the previous section explains 81-83% of the variation in the transformed CDS quotes. From this analysis, it appears that if only one set of default probabilities are included, the Jarrow-Chava default probabilities provide the best predictions. However, the Merton structural default probabilities provide additional, and different, information that together with the Jarrow – Chava default probabilities, provides the best predictive model for credit spreads, even though Merton default probabilities provide negligible, at best, incremental explanatory power in predicting default itself (see Jarrow, Mesler and van Deventer ).
Credit spreads are driven by the supply and demand for borrowed funds, and not just the default probability and loss given default. This is true for all borrowers from retail to corporate to sovereign. When fitting a statistical model to corporate sector CDS spreads, hazard rate default probabilities and their inputs dominate ratings, and ratings dominate Merton default probabilities, in their ability to explain movements in CDS quotes. A super hybrid approach provides the best over-all explanatory power for CDS quotes.
1It might be argued that lenders – banks – are risk neutral and not risk averse. However, it can be shown that risk neutral lenders, subject to either regulatory capital constraints and/or subject to deadweight losses in the event of their own default, will behave as if they are risk averse (see Jarrow and Purnanandam ).
Robert A. Jarrow, Li Li, Mark Mesler, and Donald R. van Deventer
Honolulu, September 23, 3009
Campbell, John Y. and Glen B. Taksler, “Equity Volatility and Corporate Bond Yields”, Harvard University memorandum, May 2002.
Collin-Dufresne, Pierre, Robert S. Goldstein, J. Spencer Martin, “The Determinants of Credit Spread Changes,” June 26, 2000.
Elton, Edwin J., Martin J. Gruber, Deepak Agrawal, and Christopher Mann, “Explaining the Rate Spread on Corporate Bonds,” Journal of Finance, February 2001.
Huang, Jing-zhi and Weipeng Kong, “Explaining Credit Spread Changes: Some New Evidence from Option-Adjusted Spreads of Bond Indexes.” June 2003.
Jarrow, Robert and Sudheer Chava, “Bankruptcy Prediction with Industry Effects,” Review of Finance, 8 (4), 2004.
Jarrow, Robert and Amiyatosh Purnanandam, “The Valuation of a Firm’s Investment Opportunities: A Reduced Form Credit Risk Perspective.” Working paper, Cornell University, 2005.
Jarrow, Robert, Li Li, Leonard Matz, Mark Mesler, and Donald R. van Deventer, Kamakura Risk Information Services Technical Guide, Version 4.1, Appendix D: Mapping Default Probabilities and Ratings to Implied Spreads, Kamakura Corporation memorandum, May 25, 2006.
Jarrow, Robert, Mark Mesler, and Donald R. van Deventer, Kamakura Default Probabilities Technical Report, Kamakura Risk Information Services, Version 4.1, Kamakura Corporation memorandum, January 25, 2006.
Jarrow, Robert, Li Li, Mark Mesler, and Donald R. van Deventer, “The Determination of Corporate Credit Spreads,” RISK Magazine, September, 2007.
Van Deventer, Donald R., Kenji Imai, and Mark Mesler, Advanced Financial Risk Management: An Integrated Approach to Credit Risk and Interest Rate Risk Management, John Wiley & Sons, 2004.