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  • Using Yield Spreads to Estimate

    Expected Returns on Debt and Equity

    Ian A. Cooper Sergei A. Davydenko

    London Business School

    This version: 27 February 2003


    This paper proposes a method of extracting expected returns on debt and equityfrom corporate bond spreads. It is based on an easily implementable calibration ofthe Merton (1974) model to market debt spreads and other observable variables. Forrating classes, the approach generates robust expected default loss estimates verysimilar to historical default data. It also provides forward-looking estimates for in-dividual firms unavailable from historical data. The method can be used to adjustthe cost of the firms debt for the probability of default, which is essential for low-rated firms. The approach can also be applied to provide independent estimates ofexpected equity premia consistent with historical default experiences. These equityrisk premium estimates vary from three percent for typical investment-grade firms toover eight percent for the average junk bond issuer.

    Keywords: cost of debt, equity premium, credit spreads, expected default

    JEL Classification Numbers: G12, G32, G33

    Corresponding author. Please address correspondence to: London Business School, Sussex Place, RegentsPark, London NW1 4SA. E-mail: Tel: +44 020 7262 5050 Fax: +44 020 7724 3317. This isa revised version of our earlier paper The Cost of Debt. We thank Ilya Strebulaev for helpful comments.

  • I Introduction

    Corporate bond yields reflect a variety of factors, including liquidity, taxes, risk premia, andexpected losses from default.1 In many uses, such as cost of capital estimation, lending decisions,portfolio allocation, performance measurement, and bank regulation, estimates of expected re-turns on risky debt are required. These are equal to the promised debt yield minus the partof the yield that reflects expected default. To obtain expected returns, therefore, estimates ofexpected losses due to default are required.

    Most methods of estimating the expected default loss are based either on historical defaultdata, or accounting and equity market information (see Lao (2000), Elton et al. (2001), Crosbieand Bohn (2002)).2 However, such estimates ignore the most relevant variable incorporating themarket consensus expectation of default: the debt yield itself. In this paper we propose a methodof estimating the expected default loss and expected debt returns using individual companiesbond yields. It is based on calibrating the simplest structural model of corporate debt pricing,Merton (1974), to observed debt yield spreads. It allows to estimate how much of the observedmarket spread for individual bonds is due to expected default. From these, expected returns onthe bonds can be calculated. The procedure is simple and uses only easily observed variables.

    The proposed method has important advantages over alternative methods based on historicaldefault experiences as proxies for future default incidence (for example, Altman (1989), andBlume, Keim and Patel (1991)). Elton et al. (2001) use historical data on rating migrationsand recovery rates to estimate the expected default spread. Their approach uses data on ratingsclasses. Thus, it does not provide estimates for individual bonds unless they are typical of aratings class. Our method, on the other hand, recovers the part of the spread due to expecteddefault for individual bonds. Another disadvantage of the historical approach is that it doesnot provide forward-looking estimates. Asquith, Mullins and Wolff (1989) argue that historicaldefault frequencies may differ from future probabilities, because available historical data do notcover all likely future economic and market conditions (see also Waldman et al. (1998)). Incontrast, our method uses the observed market yield, which should reflect expectations of theeconomic and market conditions for the period to which it refers. The estimates that we deriveare, on average, consistent with historical default data for ratings classes. Where they differ, ourestimates appear to be better behaved than those based on historical averages.3

    The model that we use to split the yield spread between expected default and other com-ponents is the Merton (1974) model. The variables that we calibrate to are the yield spread,leverage and the equity volatility. The Merton model makes a number of simplifying assump-tions about capital structure and bankruptcy procedures. Many papers, including Black and

    1They may also reflect option features, such as call provisions, but these are assumed away in our analysis.2See also Driessen (2002) for a different approach to decomposing the credit spread.3An entirely different approach to estimating the expected return on debt is to apply an asset pricing model

    such as the CAPM to risky debt (Blume and Keim (1987)). However, this approach requires debt transactionprice series to estimate debt betas, which are often unavailable. Moreover, applying this method is complicated bythe fact that debt betas change significantly with changes in capital structure and over time. Also, it depends onthe CAPM being the correct equilibrium model, and using a correct estimate of the market risk premium. Takinginto account these difficulties, this approach is hard to implement.


  • Cox(1976), Leland (1994), Longstaff and Schwartz (1995), and Collin-Dufresne and Goldstein(2001) have extended the basic framework to incorporate more realistic assumptions about cor-porate bond markets. These models improve the fit to the general level of yields, but none givesa generally good cross-sectional fit to bond prices (see Eom et al. 2002). Despite the large varietyof structural models available, Huang and Huang (2002) show that very different models predictsimilar debt spreads when they are calibrated to fit observed default and recovery rates. In thissense, the choice of observed variables for calibration appears to be more important than theparticular model structure. Moreover, since our goal is not to predict the total spread, but ratherdetermine the fraction of the observed market spread which is due to the expected default loss,the importance of the choice of the model is likely to be reduced further. For these reasons, wechoose the simplest model for our calibration. If the Merton framework picks up first-order effectsrelevant to the relative valuation of risky debt and equity, then our estimates of the part of thespread due to expected default should not be overly sensitive to the model assumptions. We testrobustness of our estimates by varying the structure of the model and the parametrization. Wefind that the procedure, though simple, is robust in estimating the default loss component of thespread, confirming the result of Huang and Huang that predictions of these types of models donot vary much when calibrated to the same variables.

    Various other calibrations of structural models have been proposed.4 For individual firms,KMV (described in Crosbie and Bohn (2002)) calibrate a version of the Merton model to theface value and maturity of debt and a time series of equity values. They recover asset value andvolatility and use this to calculate a distance to default. This is then used in conjunction withKMVs proprietary default database to estimate the probability of default. Huang and Huang(2002) calibrate several structural models to historical default probabilities for rating classes,using average leverage, equity premia and debt maturity. They solve for the implies volatilityof assets and use it to calculate the sum of the expected default loss and risk premium dueto default. They conclude that different models have similar performance when calibrated tohistorical default data, and also that default cannot account for much of the spread for highgrade bonds. Delianedis and Geske (2001) calibrate a version of the Merton model to debt facevalue, maturity, equity value and equity volatility. Like Huang and Huang, they recover the partof the spread caused by default risk. They also show that this cannot explain the entire spreadfor high grade firms. An important difference from our approach is that these papers focus onthe total spread due to default risk, including the associated risk premium. We, on the otherhand, use the total market spread adjusted for non-default factors as an input, and split out thesingle component due to the expected default loss.

    We make three innovations in the calibration procedure. First, we calibrate the model to debtyield spreads. None of the above papers uses yield spreads for calibration.5 Of all capital marketvariables, bond yields should contain the most relevant information about consensus predictionsof default. Thus, estimates of default rates that do not use yields as inputs may be inconsistentwith market expectations, and the resulting inferences about default may be misleading. In

    4An alternative to structural models of risky debt is reduced-form models. These models are concerned onlywith pricing under the risk-adjusted probability measure and so cannot assist in the estimation of the actualdefault probability. See Madan (2000) for a review of these models.

    5Delianedis and Geske (2001) mention this as a possibility but do not implement it.


  • contrast, our approach assumes that bond are fairly priced, and backs out expected defaultadjustments that are consistent with observed spreads, leverage and volatility. This procedureallows to estimate the parameters of the firm asset value distribution implied by observed marketprices. This is then used to determine the part of the yield due to expected default, given theexpected return on equity.

    Second, we control for factors other than default risk by measuring spreads relative to theAAA rate rather than the treasu