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Don founded Kamakura Corporation in April 1990 and currently serves as its chairman and chief executive officer where he focuses on enterprise wide risk management and modern credit risk technology. His primary financial consulting and research interests involve the practical application of leading edge financial theory to solve critical financial risk management problems. Don was elected to the 50 member RISK Magazine Hall of Fame in 2002 for his work at Kamakura. Read More

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kamakura blog
Nov 18

Written by: Donald van Deventer
11/18/2018 11:50 PM 



This paper derives a tractable, arbitrage-free valuation model for corporate coupon bonds that includes a more realistic recovery rate process than that used in the existing literature. The existing literature uses a recovery rate process that is misspecified because it includes the recovery rates for coupons to be paid after the default date. Pricing errors resulting from assuming recovery on coupons can be substantial in size. They are larger if recovery rates, coupons, maturity and default probabilities are larger. We present evidence that market prices of traded coupon bonds reflect the different recovery rates that our model predicts and that our model provides a good fit to market prices.

1     Introduction
An important and still debated issue in the fixed income literature is the exact decomposition of a coupon bond’s credit spread into its various components: the expected loss, a default risk premium, a liquidity risk premium, and an adjustment for the deductibility of government bond income for state taxes. See Duffee [10], Elton, Gruber, Agrawal, and Mann [13], Collin-Dufresne, Goldstein, and Martin [8], Duffie, Pedersen, and Singleton [11], Driessen [9], and Longstaff, Mithal, and Neis [16]. All of these papers make the assumption, either implicitly by using credit spreads or explicitly in the construction of a reduced form model, that a coupon bond is equivalent to a portfolio of risky zero-coupon bonds from a single term structure where the number of each zero-coupon bond held in the portfolio corresponds to the promised dollar cash payments (coupons and principal), and the maturity of each zero-coupon bond corresponds to the cash payment dates (see expression (10) in the text).

As shown by Jarrow [18], this assumption of a single term structure of risky zero-coupon bonds for valuing coupon bonds is valid if and only if all of the risky zero-coupon bonds are of equal seniority and all have the same recovery rate in the event of default. Unfortunately, this assumption is inconsistent with practice. After default, only the bond’s principal becomes due, and no additional coupon payments will be made on or after the default date. Consequently, in the “equivalent coupon bond portfolio,” the risky-zero-coupon bonds corresponding to coupon payments and the risky zero-coupon bond corresponding to the principal repayment have different recovery rates, violating the necessary and sufficient condition. This model misspecification implies that, in the existing literature, the parameter estimates and the decomposition of a coupon bond’s credit spread into its various components are biased.

The purpose of this paper is to derive a valuation model for risky coupon bonds that avoids this model misspecification. We refer to our model as the “reduced form” model. We use a more realistic recovery rate process consistent with market practice where the recovery rate on coupon payments differs both before and after default. We show that the reduced form model can be computed using two different issuer and maturity specific discount rate functions – spread curves – one for coupons and one for principal, rather than using the traditional one curve for both. The spreads appropriate for discounting coupon and principal payments can be quite different.

We calculate pricing errors relative to a “full coupon recovery” model which assumes that the recovery rate on all coupon payments is the same as on the principal. This model generates prices that are too large since it erroneously assumes a positive recovery value associated with coupon payments after default, when in reality it is zero. The pricing errors that result from this assumption are larger if recovery rates, default probabilities, maturity or coupon payments are larger. We calculate exact pricing errors and also provide an approximate formula that can be used to estimate the pricing error magnitudes. In this estimate, pricing errors are proportional to the recovery rate and the coupon size; they are approximately proportional to the default probability and the square of the number of coupon payments.

We illustrate the reduced form model by applying it to daily market prices for a collection of liquidly traded bonds over the period from September 1, 2017 through August 31, 2018. We first illustrate the estimation process with examples from August 30, 2018. The results are encouraging and show that the reduced form model prices match market prices reasonably well. We next perform a full sample analysis, which confirms the close link between predicted pricing errors calculated using our approximation and those that we observe in the data. For a large share of the liquidly traded bonds analyzed, the pricing errors due to assuming positive recovery on coupons after default are substantial. The misspecified full coupon recovery model has the most difficulty fitting the data when its pricing errors have the largest dispersion. In that case there are simply not enough degrees of freedom for the model to fit the data well. This is in constrast to our reduced for model, which fits the data quite well.

As a further test of the model, we perform a time-series comparative analysis of this reduced form model against the ratings-based valuation model that underlies the Basel Committee on Banking Supervision’s regulations (Basel Committee on Banking Supervision [2, 3]). The ratings-based model values a coupon bond as if it is a portfolio of zero-coupon bonds (as discussed above). In addition, we estimate the parameters of a “full coupon recovery” model which incorrectly assumes that the recovery includes the same portion of unpaid coupons as that applied to principal. We compare the performance of the three valuation models in matching market prices, daily, from September 1, 2017 through August 31, 2018. The reduced form valuation model fits market prices better than both the rating-based model and the full coupon recovery model, providing support for the necessity of using this alternative methodology. The reduced form model fits market prices uniformly better than the ratings based model, but only on average for the full coupon recovery model. The reduced form model performs better than the full coupon recovery model for the riskiest of the traded bonds.

The outline of the paper is as follows. Section 2 presents the model for valuing risky coupon bonds, while Section 3 discusses the model’s empirical parameterization. Section 4 discusses the estimation procedures, while Section 5 presents some illustrative pricing results for six representative bond issuers on August 30, 2018. Section 6 presents a comparative analysis of two alternative pricing models, Section 7 provides some specification tests, and Section 8 concludes.

Please click here for the full Paper.


Helpful comments from John Y. Campbell and seminar participants at the Bank of England, the Bank of Finland, the Board of  Governors of the Federal Reserve, the European Bank for Reconstruction and Development, the International Monetary Fund, and the World Bank are gratefully acknowledged.

University of California, Davis, California 95616 and Kamakura Corporation, Honolulu, Hawaii 96815. email:

Samuel Curtis Johnson Graduate School of Management, Cornell University, Ithaca, N.Y. 14853 and Kamakura Corporation, Honolulu, Hawaii 96815. email:

§ Kamakura Corporation, Honolulu, Hawaii 96815. email: