**Abstract **

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

**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. This literature can be partitioned into two streams. The ﬁrst stream estimates credit spreads directly (see Elton, Gruber, Agrawal, and Mann ^{[18]}, Collin-Dufresne, Goldstein, and Martin ^{[13]}), and the second stream prices bonds or related securities using a reduced form model (see Duﬀee ^{[15]}, Duﬃe, Pedersen, and Singleton ^{[16]}, Driessen ^{[14]}, and Longstaﬀ, Mithal, and Neis ^{[23]}). A careful reading of these papers shows that this literature makes the assumption (either implicitly or explicitly) that a coupon bond is equivalent to a portfolio of risky zero-coupon bonds valued using a single term structure. The number of zero-coupon bonds held in the portfolio corresponds to the promised coupons and principal with their maturities corresponding to the payment dates (see expression (10) in the text). For the credit spread estimation literature, this implicit assumption follows because all promised coupons and principal are included when computing a bond’s credit spread. In the reduced form model literature, the recovery rate process utilized is the “recovery of market value (RMV)” introduced by Lando [33] and Duﬃe and Singleton [17], which implies this result. This pricing approach assumes that when discounting, coupon and principal cash ﬂows are treated the same, and, therefore, that both promised payments entitle the holder to recovery in default. For subsequent discussion, we therefore call this approach the “full-coupon recovery” model.

As shown by Jarrow ^{[25]}, a single term structure of risky zero-coupon bonds that can be used 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. However, this assumption is inconsistent with industry practice. After default, as evidenced by ﬁnancial restructurings and default proceedings, only the bond’s principal becomes due, and no additional coupon payments are made on or after the default date. This implies that coupon and principal payments cannot be valued using the same (single) credit spread or spread term structure and that basing a bond valuation model on the erroneous assumption of equal seniority will produce predicted model prices that have misspeciﬁcation errors.

Industry practice has been conﬁrmed in the recovery rate estimation literature where it has been shown that alternative recovery rate processes,^{1} either the “recovery of face value (RFV)” or the “recovery of Treasuries (RTV)” formulations, provide a better approximation to realized recovery rates than does RMV (see Guha and Sbuelz ^{[21]} and Guo, Jarrow and Lin ^{[20]}). And, it is also well known that both the RFV and RTV recovery rate processes are consistent with a zero recovery on coupons promised after default. Therefore, these recovery rate processes do not imply the full-coupon recovery model. See Jarrow and Turnbull ^{[32]}, Bielecki and Rutkowski ^{[5]}, chapter 13, and Collin-Dufresne and Goldstein ^{[12]} for models with zero recovery on coupons promised after default.^{2}

The purpose of this paper is to explore, both theoretically and empirically, the eﬀect on bond prices assuming zero recovery on coupons promised after default. To do so we derive an empirically tractable reduced form bond pricing model, the form of which is new to the literature. For subsequent discussion, we refer to it as the “reduced form” model. We derive an intuitive and straightforward-to-implement pricing formula in which model prices depend only on the risk-free term structure, the term structure of default probabilities, and two parameters that we estimate – the recovery rate and a parameter capturing liquidity. We also show that the bond can be valued using the following “building block” securities: zero recovery zero coupon bonds, which pay oﬀ if there is no default, and digital recovery bonds, which pay oﬀ only in the event of default. The decomposition is useful both to build pricing intuition and for the empirical implementation.

We adapt the common practice of pricing bonds by calculating credit spreads and show that our 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 single curve for both.

We perform a calibration of the model to demonstrate that the spreads appropriate for discounting coupon and principal payments can be quite diﬀerent. We calculate misspeciﬁcation errors relative to a full-coupon recovery model, which generates prices that are too large since it erroneously assumes a positive recovery associated with coupon payments due after default, when in reality they are zero. The misspeciﬁcation errors that result from this assumption are larger if recovery rates, default probabilities, maturity or coupon payments are larger. For example, for a 10-year bond with a recovery rate of 40%, a coupon of 2.61%, and an annual default probability of 1%, the full-coupon recovery model will assign a price that is $0.50 too large. If it is a 30-year bond, the price is $3.61 too large, a substantial diﬀerence relative to the correct price, which is equal to par. We calculate exact misspeciﬁcation errors and also provide an approximate formula that can be used to estimate the misspeciﬁcation error magnitudes. In this estimate, misspeciﬁcation 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, which results in a close relationship to maturity.

We begin our empirical work by providing direct evidence of a diﬀerence in seniority between principal and coupons. We provide three examples of issuers that have ﬁled for bankruptcy: Lehman Brothers, Paciﬁc Gas and Electric (PG&E), and Weatherford International. We use both the misspeciﬁed full-coupon recovery model and our reduced form model to price the bonds. We ﬁnd that pricing errors from using the full coupon recovery model are between ﬁve and ten times as large as the reduced form model pricing errors. Observed prices are thus consistent with market participants assuming zero recovery on coupons and they are inconsistent with the assumption of equal recovery. This analysis provides independent evidence in support of the validity of the industry pricing practice implying zero recovery on coupons discussed above.

Next, we investigate if and when the diﬀerences in seniority is reﬂected in traded bond prices prior to default. To do so we perform a comparative analysis of the reduced form model against the full-coupon recovery model and a credit ratings-based valuation model that underlies the Basel Committee on Banking Supervision’s regulations (Basel Committee on Banking Supervision ^{[2, 3]}). Both of these models value a coupon bond as if it is a portfolio of zero-coupon bonds (as discussed above). Our sample consists of daily market prices for a collection of liquidly traded bonds over the period from September 1, 2017 through June 30, 2019.

We show that the reduced form model outperforms both the full coupon recovery and the ratings based models. First, we ﬁt the reduced form model to the data to recover unbiased estimates of the model parameters. Second, we calculate predicted full-coupon recovery model misspeciﬁcation errors based on these parameters. Misspeciﬁcation errors are large (5% are larger than $1.43) and they are highly correlated with our simple approximation formula. We next run a horse race between the models; we use both models for pricing and then compare pricing errors. The reduced form model again outperforms the coupon recovery model. This is true for the full sample. In particular, the out performance is larger for large default probability issuer-days, exactly those cases where we expect the erroneous assumption of equal seniority to have the largest impact on misspeciﬁcation errors. The outperformance is also larger on those days where the full-coupon recovery model’s misspeciﬁed assumptions imply that ﬁtting the data becomes more difﬁcult relative to the reduced form model. When including the more restrictive assumption that credit spreads are the same for bonds with the same rating (the credit ratings-based model), performance drops further. In sum, this evidence provides strong support for the necessity of using the alternative methodology of our reduced form model.

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 three companies that ﬁled for bankruptcy. Section 6 presents a comparative analysis of two alternative pricing models, Section 7 provides a time series comparison of these models and the ratings-based model, and Section 8 presents some speciﬁcation tests. Section 9 concludes.

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^{∗}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, the World Bank, Brandeis University, and the 2019 GEA meetings at the Frankfurt School of Finance and Management are gratefully acknowledged.

^{†}University of California, Davis, California 95616 and Kamakura Corporation, Honolulu, Hawaii 96815. email: jhilscher@ucdavis.edu.

^{‡}Samuel Curtis Johnson Graduate School of Management, Cornell University, Ithaca, N.Y. 14853 and Kamakura Corporation, Honolulu, Hawaii 96815. email: raj15@cornell.edu.

^{§} Kamakura Corporation, Honolulu, Hawaii 96815. email: dvandeventer@kamakuraco.com.

^{1}See Bielecki and Rutkowski ^{[5]}, Chapter 8 for a discussion of these diﬀerent recovery rate processes.

^{2}Unlike our paper, these studies do not explore empirically the pricing eﬀect of zero coupon recovery.