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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

ABSTRACT

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 firm’s 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, Regent’sPark, London NW1 4SA. E-mail: [email protected]. 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 companies’bond 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.

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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 withKMV’s 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.

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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 treasury rate. We base this adjustment on two observations. Thefirst is that AAA debt has a very low chance of default. The second is that the componentsof the spread unrelated to default, such as taxes and bond-market risk factors, appear to berelatively constant across ratings categories. For the tax spread component discussed by Elton etal., we derive an explicit formula that demonstrates this independence. The adjustment for theAAA spread results in expected default spread estimates which are similar in magnitude to thosepredicted from historical default data. It appears to overcome, for the purposes of calibration,the commonly observed inability of structural debt models to explain spreads on investmentgrade debt (Jones, Mason and Rosenfeld (1984), Delianedis and Geske (2001), Huang and Huang(2002), Eom et al. (2002)).6

Third, to make the model more flexible, we ”endogenize” the time to maturity of the debt. TheMerton model assumes a single class of zero-coupon debt. Because of omitted factors, includingcoupons, default before maturity, strategic actions, and complex capital structures, the Mertonmodel is too simple to reflect reality. Therefore, the choice of maturity when implementing themodel is difficult. For instance, Huang and Huang use actual maturity, Delianedis and Geske useduration, and KMV use a procedure that mainly depends on liabilities due within one year. Toavoid this issue and give the model enough flexibility to fit actual yield spreads, we simply solvefor the value of maturity which makes the model consistent with observed spreads adjusted fornon-default factors.

There are many possible applications of the equilibrium expected default spreads that werecover. We illustrate these with the use in estimating a firm’s expected cost of debt for usein its cost of capital. The cost of capital is used in valuation, capital budgeting, goal-setting,performance measurement and regulation, and is perhaps the most important number in cor-porate finance. Its key inputs are the cost of equity and the cost of debt. Yet while the costof equity has been the subject of extensive debate, little attention has thus far been focused onestimating the cost of debt. Two common approaches are to use either the yield on the debt orthe riskless interest rate as proxies. Neither is correct when part of the yield spread is due toexpected default. The errors are most significant when the debt is risky. As Brealey and Myers(2000) say: ‘This is the bad news: There is no easy or tractable way of estimating the rate ofreturn on most junk debt issues’ (p. 548). Our method helps to overcome this problem.

Another application of the approach is related to equity premia estimates. In the form thatwe use it for most of the paper, the procedure derives the expected default rate on debt con-

6As discussed in the section on adjusting for factors other than default, we do not claim that this procedurehelps to explain investment grade spreads. It simply enables a calibration that is consistent with the data.

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ditional on the expected equity return. Alternatively, if expected default rates are known froman independent source, such as historical default data, the procedure can be reversed to giveestimates of the expected return on equity consistent with observed debt yields. We use thisapproach to provide a new set of equity risk premium estimates, based on data hitherto unusedfor this purpose. This method, based on bond yields, should contain information about consen-sus expectations of risk premia. An important advantage of our approach is that the resultingestimates are largely based on a forward-looking capital market variable.7 It may provide a use-ful benchmark comparison to estimates of equity premia derived in more standard ways. Otherestimates are typically based on the CAPM, APT or variants of the dividend growth model.8

All these methods generate large standard errors. Our method, based on different information,provides new insights into the equity premium.

Our main results are as follows. The technique we propose for estimating the equilibriumexpected default component of the spread appears quite robust. It recovers estimates whichare not very different from those obtained from historical average default and recovery datafor ratings categories. Our estimates also appear to be better behaved than estimates based onhistorical default frequencies in the following sense: The historical method gives expected returnson debt for our sample that are not monotonic as debt rating changes, whereas our proceduregives monotonic estimates. It also generates sensible estimates of the asset volatilities of firms,which are consistent with measures obtained in other ways.

In line with historical default rates, we find that only a small fraction of the spread for high-grade debt is due to expected default loss. For lower-grade debt, this component is larger, andour approach provides a method for adjusting yields to give expected debt returns. We find thatthe expected default component of the spread varies significantly within ratings categories, sousing average figures for ratings categories for individual companies may be misleading. Theestimates of equity risk premia that we obtain using the technique are well-behaved. They areconsistent with asset risk premia of about three percent and equity risk premia of between threeand nine percent.

The balance of the paper is organized as follows. The next section presents the calibrationapproach for the Merton (1974) model and the relationship between the cost of debt and equity.Section III discusses adjusting the yield spread for factors other than the risk of default. SectionIV provides a description of our data set. Section V discusses the calibration method and examinesits robustness. Section VI provides estimates of expected returns on debt. Section VII presentsestimates of equity risk premia. The following sections discuss various applications and extensionsof the method. Section X summarizes. Technical details are given in Appendices.

7The ’DCF’ method of relating equity prices to earnings forecasts relies on consensus expectations reflected inthe share price, but requires earnings forecasts which are obtained from surveys.

8See Welch (2000) for a survey of existing practices.

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II The Merton Model

The Merton model is the simplest equilibrium model of corporate debt. It assumes that thevalue of the firm’s assets follows a geometric Brownian motion:

dV

V= µdt + σdWt (1)

where V is the value of the firm’s assets, µ and σ are constants, and Wt is a standard Wienerprocess.9 The model further assumes that the firm has a single class of zero coupon risky debtof maturity T . Other assumptions include a constant flat risk-free yield curve and a very simplebankruptcy procedure.

Merton applies the Black-Scholes option pricing analysis to value equity as a call option onfirm’s assets. Merton’s formula can be written in a form that gives a relationship between thefirm’s leverage w, the maturity of the debt T, the volatility of the assets of the firm σ, and thepromised yield spread s (see Appendix A):

N(−d1)/w + esT N(d2) = 1 (2)

where N(·) is the cumulative normal distribution function and

d1 = [− lnw − (s− σ2/2)T ]/σ√

T (3)

d2 = d1 − σ√

T (4)

Of the variables in Equation (2), leverage and the spread are observable, and σ and T aregenerally unobservable. Another implication of the model is that the observable equity volatility10

σE satisfies:σE = σN(d1)/(1− w) (5)

We now have three inputs: w, s and σE , and two unknowns: σ and T . We solve equations (2)and (5) simultaneously to find values of σ and T that are consistent with the observed values ofw, s and σE .11 Thus, σ is computed as the implied volatility of the firm’s assets when the equityis viewed as a call option on the assets.

Once the values of σ and T are known, the relationship between the expected return on assets,equity and debt are related as follows. Since equity in this model is a call option on the assetsand therefore has the same underlying source of risk, the risk premia on assets π and equity πE

9The drift must be adjusted for cash distributions.10In contrast to the asset volatility, the short-term equity volatility is easily observable either from option-implied

volatilities or from historical returns data.11The system of equations is well-behaved, and we generally had no difficulties solving it applying standard

numerical methods. To assure a starting point for which standard algorithms quickly yield a solution, one cansolve equations (2) and (5) separately for σ for a few fixed values of T (or vice versa). This procedure alwaysconverged for any reasonable starting points. The intersection of the solution curves σ(T ) from equations (2) and(5) can then be used as a starting point for the system of these equations.

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are related as:π

πE≡ µ− r

µE − r=

σ

σE(6)

or:π = πEσ/σE (7)

Now the expected return premium on debt over the maturity period ΠD can be calculated as(see Appendix A):12

ΠD = s +1T

ln[e(π−s)T N(−d1 − π

√T/σ) /w + N(d2 + π

√T/σ)

](8)

and the spread which is due to expected default δ ≡ s−ΠD which should be excluded from theexpected return on debt is thus:

δ = − 1T

ln[e(π−s)T ) N(−d1 − π

√T/σ) /w + N(d2 + π

√T/σ)

](9)

The right-hand side of this expression is positive, and the expected return on debt is lower thanthe promised yield. Note also that the probability of default predicted by the model is:

P = N(−d2 − πE

√T/σE) (10)

If, on the other hand, the expected default loss on debt δ is known, then the expected equitypremium can be estimated. Equation (9) can be solved for π and combined with (7) to find πe

consistent with the expected default.13

III Adjusting for factors other than default risk

Before we apply the Merton model, we adjust for factors other than default risk by subtractingthe AAA spread. There is growing evidence that corporate yield spreads cannot be entirely dueto the risk of default. Huang and Huang (2002) and Delianedis and Geske (2001) measure thepart of the spread that is due to default risk and find that, for AAA bonds, very little of it can beexplained by default. Table I presents their estimates of the proportion of the AAA spread thatis due to default risk. All estimates suggest that very little of the AAA spread can be explainedby default risk.14 Thus, high grade spreads must be almost entirely due to other factors.15 Eltonet al. (2001) argue that a part of the spread for U.S. corporate bonds is due to the state tax oncorporate bond coupons which is not paid on government coupons. Collin-Dufresne et al. (2001)

12Note that, unlike the return on assets and equity, the calculated return on debt is an annualized compoundedreturn rather than an instantaneous return.

13If the probability of default is known, then it can also be used to estimate the equity premium.14Huang and Huang report that models such as Leland and Toft (1996) can explain up to half of the ten-year

spread. However, these models are for infinite maturity debt, so the comparison with ten year yields is, as Huangand Huang state, not very informative.

15We also tested the influence of default risk on AAA spread by regressing them on fundamental determinants ofdefault risk, including the equity volatility and leverage of the issuer, and found that these factors were statisticallyinsignificant determinants of AAA spreads.

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demonstrate the presence of a systematic factor in credit spreads which appears to be unrelatedto equity markets. Lower liquidity of corporate bonds relative to government bonds is also likelyto be responsible for a part of the spread.

INSERT TABLE I HERE

The Merton model does not include factors such as tax, liquidity, and bond-market riskfactors unrelated to the equity market. Thus, the calibration of the model to credit spreads mustexclude the part of the spread unrelated to default risk. The magnitude of the tax, liquidityand bond-specific risk components of the spread are hard to estimate. For instance, Elton etal. estimate the component of the spread caused by differential state taxes on corporate andgovernment bonds. Their estimates range from 29 to 50 basis points for the tax component ofthe AA spread. Even this large range might be questioned, as any positive tax spread couldbe subject to arbitrage by institutions that are exempt from state taxes, such as pension funds.Uncertainties at least as great affect estimates of the bond-specific risk component. Elton et al.claim that equity-related risk factors can explain almost all of the spread unexplained by defaultand taxes, whereas Collin-Dufresne et al. identify a substantial bond-market risk factor.

For these reasons, direct estimation of the non-default components of the spread does notappear practical. We therefore need a variable that contains these components to adjust spreadsso that they reflect only default-related factors. The evidence that the AAA spread does notcontain a significant default risk component suggests that it reflects only non-default factors. Sowe could use the AAA spread to proxy these other factors as long as they are cross-sectionallyconstant. For the tax and bond-market risk components, there is evidence that this is the case.

The tax-induced spread is modelled by Elton et al. (2001). In the US, coupon payments oncorporate bonds are subject to state income taxes, while government bonds are not. Elton et al.(2001) do not give an explicit formula for the part of the spread induced by this tax effect. InAppendix B we derive such a simple formula. The yield spread due to tax is given by:

∆ytax =1

tMln

[1− τ

1− τe−rtMtM

](11)

where: ∆ytax is the spread due to tax, tM is the time to maturity, τ is the applicable tax rateand rtM

is the riskless interest rate. This formula holds for any bond, as long as capital gainsand losses are treated symmetrically and the capital gain tax rate is the same as the income taxon coupons.16 Table II shows estimates of the tax-induced spread based on the above formula

16In a more general case, when the income tax rate τI at which coupons are taxed does not coincide with the

capital gains tax τC , formula (11) becomes ∆ytax = 1tM

ln

[1−τI

1−e−rtM

tM (τC+(τI−τC)E∗[F̃ /B])

]. This is similar

to (11) when rtM tM is high or the risk-neutral expectation of the principal repayment is not very different fromthe purchasing price: E∗ [F ] ≈ B. Another nuance is that in reality taxation rules for bonds originally soldsignificantly below par (called original-discount bonds), such as zero-coupon bonds, are different from our model.Capital gains on such bonds are appreciated for tax purposes gradually throughout the life of the bond, so thatonly a small part of the tax is paid at maturity. For such bonds formula (11 will also be an approximation.

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for the average tax rates estimated in Elton et al. (2001) and maturity tM=10 years.

INSERT TABLE II HERE

The key feature for our application is that the tax-induced spread does not depend on the riskinessof the bond or the coupon rate. So this factor will influence the AAA yield by exactly the sameamount as lower grade yields. Even if we do not know the appropriate tax rate to use for theadjustment, subtracting the AAA yield will eliminate this factor.

A similar result regarding the equality across ratings categories of non-default factors is givenin Collin-Dufresne et al. (2001). They report that there is a systematic factor in bond returns thatappears to be unrelated to equity markets. The loadings of bonds from different rating categorieson this factor are very similar. They do not give a price of risk for this factor. However, ifthe factor loadings are constant across ratings categories, then subtracting the AAA yield willeliminate the effect of this factor, regardless of the price of risk.

The final component of yields that is not related to default is liquidity. The difference inliquidity between the corporate bond and government bonds may be responsible for part of thebond spread. Subtracting the corporate AAA spread will adjust for the difference in liquiditybetween corporate AAA and government bonds. It will not adjust for differences between theliquidity of the bonds we are analyzing and the AAA bonds we use as benchmarks. In ourestimations, this difference is limited by the data we use, which is for transactions involvinginsurance companies. The fact that the prices we use represent actual transactions ensures someliquidity in the bonds we analyze, and limits the difference between their liquidity and the liquidityof the AAA benchmark bonds we use. Therefore, we assume that relative liquidity effects are ofsecond-order importance across our sample.

IV Data

We use bond trade data supplied by the National Association of Insurance Companies (NAIC),which records all transactions in fixed-income securities by insurance companies in the US. Theoriginal dataset includes trade prices for more than six hundred thousand transactions in theperiod 1994-1999. We exclude all bonds other than senior unsecured fixed-coupon straight USindustrial corporate bonds without call/put/sinking fund provisions and other optionalities. Wealso exclude bonds for which we were able to unambiguously identify the promised cash flowstream and Moody’s or Standard and Poor’s rating at the date of trade, as well as find issuingcompany’s accounting data in Compustat for the fiscal year immediately preceding the date oftrade, and a 2-year history of its stock prices in CRSP. We use only senior unsecured debt, asthis is the type of debt on which Moody’s company ratings are based. To improve the matchingof the inputs, we retain in the sample only bonds with remaining maturity between 7.5 and 10years, where the trade is executed within three months after the fiscal year end of the issuingcompany. We use this maturity subsample because yield spreads vary with maturity, so we do

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not want to mix very different maturities with different term premia. We use a relatively longmaturity because we are interested in expected returns for relatively long horizons. We use thesubsample of trades shortly after the year end to better match the accounting information to thetrade data. Thus constructed, the final sample includes 2632 trades on 553 bonds of 292 issuers.

We estimate spreads on these bonds using data on U.S. Treasury STRIPS (risk-free zero-coupon securities) using the procedure suggested in Davydenko and Strebulaev (2002). We firstcompute the yield on each bond trade from the transaction price recorded by NAIC. We thencalculate the yield on a risk-free bond with the same promised cash flows using Treasury STRIPSprices as of the settlement date of trade.17 We subtract the estimated cash-flow matched risk-freerate from the yield on the bond to obtain the yield spread for the trade.

We measure leverage as a ratio of the Compustat-recorded book value of debt to the sum ofthe book value of debt and total market value of equity obtained from CRSP for the last businessday before the trade. We measure total debt by the book value of short term and long termdebt. Some other authors do not use observed leverage in structural debt models because thebook value of debt may not proxy for its market value. An alternative is to use the book value ofdebt to proxy for the face value of debt (Crosbie and Bohn (2002), Delianedis and Geske (2001)).This approach is also subject to criticism unless the structural model used is one that explicitlydeals with the coupon flows on the bonds. We follow Huang and Huang (2002) in using leveragebased on the book value of debt, but later find that our results are insensitive to the precisemeasurement of leverage. We estimate equity volatility as the volatility of daily equity returnsas recorded in CRSP over the two years prior to the bond trade. We also tried six months and1 and 3 years, with no significant change in the results. We use equity risk premium estimatesfor different ratings classes from Huang and Huang (2002). These are based on the empiricalrelationship between leverage and equity returns in Bhandari (1988).

Table III shows summary statistics for spreads and other fundamental variables for our sam-ple. All the variables behave as one would expect, with spread, equity volatility, and leverageincreasing on average as the rating deteriorates. There is, however, significant variation of thesevariables within ratings classes. The table also shows maturity and duration, which are similaracross ratings classes for the subsample. Finally, it gives the asset volatility calculated using theKMV method described in Crosbie and Bohn (2002) and also employed in Vassalou and Xing(2002), which we later use to benchmark our asset volatility. One interesting feature of thisvariable is that its average is relatively constant across ratings categories, apart from the single-Bcategory. The asset volatility estimate recovered from the KMV procedure for this class is doublethat for the other classes.

INSERT TABLE III HERE17For the majority of trades there are 4 annual STRIPS returns available. We use linear approximation of the

STRIPS yield curve to discount the corporate bond coupon payment which occurs between maturity dates on twoSTRIPS.

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To benchmark our procedure and to estimate equity premia, we need default loss estimatesbased on historical data. Elton et al. provide such estimates for different bond classes overdifferent maturities conditional on starting from a particular rating class. We use these as onebenchmark. However, the usefulness of these estimates for our purposes is limited. They assumethat recovery is a function of ratings class, whereas Altman and Kishore (1996) document thatrecovery rates are highly dependent on industry and bond seniority, but not on initial rating aftercontrolling for these variables. Moreover, Elton et al. also assume that annual ratings transititionsare Markovian and use these to estimate cumulative default probabilities for different horizons.Instead, we produce our own estimates of the yield equivalent of expected default using Moody’shistorical default frequencies reported in Keenan et al., also used by Huang and Huang. Theseare direct estimates of cumulative default probabilities. We complement these with a recoveryrate for senior unsecured bonds of 48.2% from Altman and Kishore (1996). This recovery rate isclose to the 51.3% recovery from Moody’s used by Huang and Huang (2002).

As a benchmark, we compute the expected default loss spread using historical data on defaultsand recoveries. This spread, called in Elton et al. the risk-neutral spread, is the coupon withwhich the bond would trade at par in a risk-neutral world. This coupon, C, is defined implicitlyby the relationship:

∑ (P ct − P c

t−1)R + (1− P ct )C

(1 + r)t+

1− P cT

(1 + r)T= 1

where R is the recovery rate and P ct is the cumulative probability of defaulting over t years. The

spread due to expected default risk is C − r. The resulting estimates are discussed in the nextsection.

V Calibrating the model

In the form that we use it, the Merton model gives two equations relating five variables:leverage, yield spread, equity volatility, debt maturity and asset volatility. In addition, there isanother equation relating the expected default loss on debt and the expected equity premium un-der the real probability measure. We calibrate the model using leverage, yield spread, and equityvolatility as observable inputs, and solve the first two equations simultaneously to give implieddebt maturity and asset volatility. Asset volatility cannot be observed directly, so imputing itsvalue from a model using equity volatility as an input is standard.18 Once these parameters arefound, we use the formula relating the expected return on debt to the expected return on equity.

18See, for instance, Vasicek (1984).

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V.1 Calibration for rating classes

Before we use the model to estimate the expected default for individual firms, we test thecalibration primarily based on aggregate data for ratings used in Huang and Huang (2002). TableIV shows in columns (1) to (3) the values of spreads, leverage and equity expected return givenby Huang and Huang for six ratings groups for bonds of ten years maturity. These are drawn bythem from a variety of sources, not all for the same period of time. We also need an estimateof equity volatility, which Huang and Huang do not use. As the Huang and Huang data are formixed periods, and we use them only to test the general properties of the calibration, we simplyobtain the volatility from our dataset. This is given in column (4).

Table IV shows three calibrations using different spread adjustments. The first, called ’none’,uses the spread against treasuries as the input variable. The second, called ’tax’, deducts 20basis points for the estimate of the tax-induced spread. This is calculated using formula (11)with the estimate of the relevant tax rate of 4.875% from Elton et al., an interest rate of 5%, anda maturity of 10 years. The third adjustment, called ’AAA’, deducts the average AAA spread of63 basis points to control for the non-default component not accounted for in the Merton (1974)model.

The calibrated model parameters T, σ, and π are given in columns (5) to (7). The asset riskpremium, π, is generally about 4.5%, and constant across ratings groups. This suggests that ourprocedure is not generating any systematic bias in the relationship between equity and asset risk.The calibrated values of asset volatility, σ, also appear reasonable. Although they should notnecessarily be equal across ratings classes, they are sufficiently similar to suggest that most of thevariation in equity volatility is coming from differences in leverage between ratings classes ratherthan differences in the nature of the assets. We later test whether our procedure recovers assetvolatilities that are similar to those produced by the KMV procedure, which is rather differentand uses time series of equity prices.

The maturity parameter, T , that we recover is given in column (5). This reflects not only theactual ten year maturity of the debt, but also any other factors such as complex capital structures,distress costs, strategic behavior by equity holders, and other complications not included in theMerton model but reflected in the adjusted spread s. The values of T found are typically higherthan the true debt maturity. We later test whether this is a problem for the procedure by alsocalibrating a variant of the strategic debt service model of Anderson and Sundaresan (1996),which has liquidation costs in bankruptcy as another free parameter that we can use to matchthe actual maturity of the debt.

INSERT TABLE IV HERE

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V.2 Estimated default loss for rating classes

The estimated values of the default loss spread are given in column (8) of Table IV. Column(9) shows them as a proportion of the spread relative to treasuries, and column (10) gives thepredicted probability of default from the fitted model. Interestingly, the estimated spread due todefault is not highly sensitive to the deduction of the AAA spread. However, we believe that thereasons for making this adjustment are so compelling that we use it throughout the rest of thepaper.

The estimated spreads due to default for AAA, AA, A, BBB, BB and B ratings categoriesare 4, 4, 9, 21, 78, and 237 basis points out of total spreads of 63, 91, 123, 194, 320, and 470b.p., respectively. The corresponding estimates based on historical data, reported in column (11),are 4, 5, 8, 24, 132, and 353 b.p. Apart from the non-investment grades, our model generatesestimates of default spreads which are very close to those estimated with historical default andrecovery rates. For the junk grades, uncertainties about the estimates based on historical dataare quite large, so the correspondence between the fitted and historical default risk componentsappears reasonable. Indeed, it is unlikely that a B spread of 470 basis points can be consistentwith the expected default spread of 353 basis points computed on the basis of historical data.The historical estimate would leave only 117 basis points for liquidity, tax, bond-specific riskpremia and the default risk premium. This is lower than the corresponding quantity for the BBBspread, which is difficult to justify. So our estimate based on calibration appears more reasonablein the cases where it differs from the estimate based on historical data.

Our estimates of the proportion of the spread due to expected default share one importantproperty with the historical estimates. This is that the proportion is increasing as the debtquality deteriorates. This means that using a constant value for the default risk spread withina ratings category, or even a constant proportion of the spread, is unlikely to be correct, anda procedure such as ours is necessary to estimate the default spread accurately, even within acategory.

Elton et al. also provide historical estimates of the spread due to default risk for AA, A andBBB categories. These are given in column (14) and are 5, 14, and 41 b.p. respectively. Thesediffer somewhat from our calibrated estimates of 4, 9 and 21 b.p., but they also differ from ourhistorical estimates. For reasons given above, we believe that our historical estimates are moreappropriate in this context.

As well as the spread due to default risk, which reflects the probability of default and therecovery rate, we also report the predicted and historical probability of default. The fitted valueof this variable is given in column (10). This differs markedly from the historical estimate, givenin column (13). Even though our procedure matches the expected spread due to default well, itgives a generally higher probability of default than the historical data. This is because continuousmodels of the Merton type tend to have a high probability of small losses in default. It is difficultto have a high loss rate in conjunction with a low probability of default in such models. So ourcalibration of the model fits the spread due to expected default but not the default probability.

13

V.3 Sensitivity to inputs and model specifications

Table V presents sensitivity analysis and robustness checks for the calibration procedure. Weuse AA and BB ratings categories and vary the input parameters. These are given in columns(1) to (5). The parameters are each varied individually up or down by 10 percent of their basevalue. The results from calibrating the basic version of the model are given in columns (6) to(10).

For both ratings categories, the estimated values of asset volatility and asset risk premia arevery robust with respect to variation in the inputs and the structure of the model. Even whenequity volatility is varied, the asset volatility estimates remain stable. The expected defaultspread, δ, is also quite robust. Table IV showed that it is not sensitive to the subtraction ofthe AAA spread. Equation (9) shows that it does not depend at all on the risk-free interestrate. The sensitivity analysis in Table V also shows that it is insensitive to leverage. Thus, therelatively crude measure of leverage that we use is unlikely to be an issue. The value of δ issomewhat sensitive to the estimate of the equity risk premium. We use this sensitivity belowwhen we invert the procedure to estimate equity premia. The one variable to which the expecteddefault spread shows high elasticity is equity volatility, σE . Second moments such as σE canbe estimated quite accurately for equity returns. Thus, the procedure has the merit of giving aresult that is sensitive only to a parameter that can be observed relatively accurately.19

For AA bonds, the estimates of the expected default spread, δ, are all between two andseven basis points out of a total spread of 91 basis points. Thus, the expected default is a verysmall proportion of the spread regardless of the parameters or the form of the model. For BBdebt, across all the parameter values and different models, the expected default component ofthe spread is between 50 and 120 basis points out of a total spread of 320 basis points. Sothe expected default component is between 16% and 38% of the spread. Although this is quitesubstantial variation, it reflects a very wide range of parameter values and model structures. Sothe split of the spread between expected default and other components is quite robust.

To test robustness to model specifications, we also use a simple variation of the Merton modelwhich allows for liquidation costs and strategic debt service of the Anderson and Sundaresan(1996) type. This is described in Appendix C. It requires another parameter θ, which maybe thought of as the proportional deadweight loss in liquidation. When liquidation costs andstrategic debt service are introduced into the model, the implied value of maturity T inverselydepends on the assumed liquidation costs. In particular, one can solve for the value of Altering themodel in this way results in a reduction in the implied value of T . The amount of the reductiondepends on the assumed value of bankruptcy costs. In particular, we can solve for the level ofbankruptcy costs θ which make the implied value of T to equal the actual ten year maturity ofthe debt.

19Note that this suggests that measures of default probability derived models of the Merton type are verysensitive to volatility. These measures have recently been shown by Vassalou and Xing (2002) to explain someimportant factor risk premia in the equity market. Because of the close relationship between volatility and thedefault measure, however, it may be difficult in practice to distinguish between the effect of volatility per se andthe effect of volatility through the default measure.

14

Columns (11) to (15) of Table V give the results of these estimates, first using bankruptcy costof 5% of the debt face value suggested by Anderson and Sundaresan, and then solving for θ usingthe actual 10 year maturity for T . In most cases, these adjustments result in 15-30 basis pointchanges in the estimated default loss for the BB bond out of total spread of 320 basis points, andmuch smaller changes for the AA bond. These are small proportions of the total spread, and sothe results are not highly sensitive to the choice of model.

INSERT TABLE V HERE

V.4 Sensitivity to dividend payouts

The version of the Merton model that we use does not include distributions in the form ofdividends or coupons on debt. We deal with the debt structure by allowing the maturity of debtto be endogenous. To test for sensitivity to dividends, we amend the standard Merton model byassuming that the firm pays continuous dividends that are a constant proportion γ of the valueof the firm V . This version of the model is given in Appendix B. We set the level of γ from theinstantaneous dividend yield g by the relationship γ = (1− w)g.

Table VI shows the sensitivity of our estimates to the level of dividend yields. The firstcolumn shows the dividend yield varying from zero to three percent. The final column showsthe impact of this on the proportion of the spread due to expected default. In all cases otherthan BB debt, the change from zero to three percent dividend yield changes the expected defaultestimate by less than four percent of the spread. In the case of BB bonds, the impact is less thanseven percent of the spread. So substantial variation in the dividend yield has little impact onthe proportion of the spread that is due to expected default.

INSERT TABLE VI HERE

The dividend yield does, however, have a major impact on the implied maturity of the debt,T. It brings this down from the high levels shown in Tables IV and V to levels that match those ofthe bonds used for the observed spreads. This means that the procedure can match the impliedmaturity of the debt to the actual maturity. However, since the inclusion of dividends doesnot significantly affect the expected default that we recover, we use the standard version of theMerton model in the rest of the paper.

VI Estimates of the expected return on debt

To estimate the expected default loss for individual transactions, for each trade in our databasewe estimate the bond spread and match it with leverage information from Compustat and histor-ical equity volatility from CRSP. We use the equity risk premia for ratings categories described

15

in the previous section. We first adjust for the AAA spread in each year to account for varyingsystematic factors and factors other than expected default. Spreads vary with maturity andover time. Therefore, we make the adjustment using a AAA spread that is matched on thesecharacteristics. We then solve the model to determine the level of the expected default spread.20

The calibrations produce well-behaved estimates of asset volatility and risk premia. In par-ticular, the asset volatility estimates are similar across ratings classes, and do not exhibit thevery high estimate for the single-B class that is given by the KMV estimate shown in Table III.This suggests that our procedure is incorporating different information from that of the KMVprocedure. To check this, we measured the correlation of the two measures, and it was 0.65 acrossthe whole sample, confirming that the two measures are somewhat different.

The estimates of δ show large variation within ratings classes. The coefficients of variation aregreater than one for all but the B class. For the B class, the standard deviation of δ within theclass is 124 basis points, which is large in absolute magnitude but lower than the mean spread.These results indicate that average expected default for a ratings class is misleading if appliedto the individual bonds in the class. Because of the large variation within each ratings class,methods which largely rely on ratings to estimate the expected default of bonds are unreliable.A slightly better assumption is that the ratio of δ to the spread is constant within a ratingsclass. This combines the information on rating with the spread in a simple way. It has a lowercoefficient of variation than the absolute value of δ for all ratings classes. However, it also exhibitssignificant variation within ratings classes.

Interestingly, the averages of the ratio of δ to the spread for the ratings classes in Table VII arevery similar to those obtained in Table IV. Table VII is based on individual bonds for the NAICdatabase, whereas Table IV is based on aggregate data from Huang and Huang for a differentperiod. The similarity of the results is another indication of the robustness of our technique.

INSERT TABLE VII HERE

VII Equity risk premia estimates

In this section we use our method to estimate equity risk premia from debt yields. The equityrisk premium is one of the most important parameters in corporate finance and valuation, and itsvalue is the subject of great controversy (Welch (2000)). Even small differences in estimates canhave a major effect on valuations, and estimates ranging from 0% to more than 9% have beenadvocated. The reason for the disagreement is that all estimates are potentially subject to largeerrors.21 For instance, historical returns data are often used as the basis of the estimate. Thisleads to large standard errors because equity returns data are very noisy and the risk premium

20We also computed the results with an adjustment for the average AAA spread over the period rather than ineach year, and the results were very similar.

21Estimates of equity risk premia used in valuation typically do not have standard errors attached. A notableexception is Fama and French (1997).

16

may be time-varying. This has led to extensive criticism of the use of historical data as the basisof equity risk premia estimates, and many other estimation methods have been proposed, eachof which has its own drawbacks.

As one alternative, some advocate the use of expectational or “ex-ante” methods of estima-tion. These rely on the observation of expectational data from surveys or models of expectationsformation (see Harris and Marston (1992) for a good example). In these methods, errors arisebecause the relevant expectations are only indirectly observable, the relevant weighting for dif-ferent agents’ expectations are unknown, and the appropriate method to convert expectations toa risk premium estimate is debatable.

The value of our approach is that it uses the corporate debt spread, a market variable thataggregates agents’ expectations and is observable, therefore avoiding the measurement problemsof other expectational variables. And since the debt spread incorporates a risk premium for thefuture rather than the past, it avoids the problems associated with the use of historical returns toestimate the equity risk premia. The equity risk premium is such an important and controversialparameter that any extra information that assists in its estimation has value. Our methodprovides new information that is based on data (bond yields) which have not been employed forthis purpose before. Furthermore, these data are current capital market data that should reflectmarket consensus expectations. No other methods commonly used to estimate equity risk premiaare based on such data.

Our method of estimating the equity premium relies on obtaining an estimate of expecteddefault from historical data. This is set equal to the Merton model estimate to impute the equityrisk premium. Historical data can be used only for ratings classes, so we use the method toestimate equity risk premia for ratings classes. To estimate equity premia we use the data forindividual trades, as described in the previous section. We calibrate the model in the same way,but then solve the equation relating the equity premium and the expected default spread foundfrom the historical default and recovery data to determine the level of the equity premium. TableVIII shows the results.

The mean equity premium across our sample of firms is 4.8% and the asset premium 3.3%.These estimates are lower than estimates based upon an unadjusted use of historical averages.For instance, Brealey and Myers report an average premium for equities relative to treasury billsof 9.1% based on the period 1926-2000. Our estimates are more consistent with those of Dimsonet al. (2001), who use a different historical period and make various adjustments to the rawhistorical averages. For ratings classes, the equity premia range from 3.1% for AA companies to8.5% for B companies, reflecting the different leverage in the different ratings classes. The assetrisk premia are similar across classes, indicating that differences in risk premia for different ratingclasses are primarily driven by differences in leverage .

INSERT TABLE VIII HERE

These estimates depend on the validity of the historical data for ratings classes as predictions

17

of the future, and also the robustness of the procedure used to impute from them equity riskpremia. Our tests suggest that the procedure is robust. So the main issue is the use of historicaldata to estimate the average expected default for ratings classes. This is justifiable if past datacan be expected to give a good indication of expected default over our horizon, which is eightyears.

VIII Applications

The expected return on debt plays a central role in many applications. Here we discuss three:applications in corporate finance, banking, and portfolio management.

The corporate finance application is adjusting the cost of debt for use in the cost of capital.Standard methods of cost of debt estimation assume that the cost of debt is equal to either thepromised yield on newly-issued debt of the firm (Erhardt, 1994), or the risk-free rate (Bodie,Kane and Marcus, 1993). Both approaches fail to make a proper adjustment for the possibilityof default. Using them may result in serious errors in the cost of capital and value estimates.

As a simple example, consider a company with 60% leverage and a promised yield spreadof 4%. Suppose that half of the spread is due to default. Suppose that real interest rate is 2%,and the equity risk premium for this firm is 6%.22 Using the riskless rate or the yield into thestandard weighted-average cost of capital formula, excluding taxes, gives:

WACCriskless rate = 0.6× 2% + 0.4× (2% + 6%) = 4.4%

WACCpromised yield = 0.6× 6% + 0.4× (2% + 6%) = 6.8%

The true WACC, using the promised real yield of 6% minus the expected default of 2% asthe cost of debt, is:

WACCtrue = 0.6× (6%− 2%) + 0.4× (2% + 6%) = 5.2%

These differences can have a material impact on valuations. The multipliers for a real per-petuity growing at 2.5% are 53 times, 23 times and 37 times, for the WACC estimated with theriskless rate, the debt yield and the true expected return on debt, respectively.

An important banking application of the spread due to expected default loss is to controlthe risk and measure the performance of lending. In some uses, such as making the initiallending decision, it may be desirable to speculate on a bank’s private view regarding futuredefault. For risk measurement and performance measurement, however, it is often useful to havea non-speculative market-based benchmark against which to measure outcomes. For instance,

22These illustrative numbers correspond roughly to the figures for B debt in Tables IV and VII.

18

measuring whether a portfolio of low grade debt has a higher or lower incidence of default thancould have been expected on the basis of our measure of expected default would be an indicatorof whether there is predictive ability with respect to default.

A similar issue of using market consensus beliefs as a neutral expectation arises in portfoliomanagement. In deciding how much risky debt to hold in a portfolio, the expected return playsa key role. Our method enables one to extract the market consensus expected return, whichshould assist in determining the neutral holding of each bond. Holdings based on speculativeviews will be deviations from this neutral holding and should be based on the difference betweenthe speculative view and the market consensus view.

These three applications illustrate the importance of a technique to estimate the consensusexpected default and expected return on debt. In addition, we have provided a new method ofestimating the expected return on equity, which is the central number in many corporate financeand valuation applications.

IX Extensions

The method we use in this paper could be extended and applied in many ways. For instance,we calibrate to the yield spread using a novel method which assumes that leverage is observableand generates an implied value of maturity. We could, alternatively, calibrate to the yield spreadusing the procedure of Delianedis and Geske (2001), which assumes that the market value of debtis unobservable and generates an implied firm value. As another example of possible extensions,our procedure for estimating the equity premium uses an historical estimate of the spread dueto default as an input. It could also be implemented using the historical default frequency as aninput, in a way similar to Huang and Huang (2002).

The analysis indicates that equity volatility is an important input to the estimation. In fact,equity volatility plays two roles: One is that it enters the equation relating the instantaneousvolatilities of equity and assets. The other is that it enters the equation determining the totalvolatility of the asset value over the life of the bond. It may be an improvement in the imple-mentation of this type of model to use separate estimates for these two volatilities. They couldbe related to each other by one of the standard stochastic volatility models. Although this wouldnot be strictly consistent with the use of a simple contingent claim model such as Merton, itmight improve the performance of the model, and could be justified as a first approximation toa more complex model.

Our analysis is based on using firm-specific variables as the basis of the expected defaultspread estimate. An alternative would be to use these to determine the relative characteristicsof different firms and rely on average historical results for ratings classes for the typical firm ina class. This would give a way of using the evidence on expected default for ratings classes inconjunction with a relative adjustment based on our method.

19

The method proposed in this paper produces expected default probabilities that are consistentwith yield spreads. Delianedis and Geske (1998) present evidence that structural models can helpto predict ratings transitions, even in their risk-neutral form, without the use of the information inyield spreads. An obvious extension of their work would be to see if the incremental informationprovided by using information in yields in the way proposed here can add to this predictiveability.

X Summary

This paper uses a new calibration approach to calculate the expected default loss on corpo-rate bonds using the observed market debt spread. It first adjust the spread for factors otherthan default risk by using an explicit formula for the tax spread, or subtracting the yield ona matched AAA-rated corporate bond under the assumption that it is largely default-free. By”endogenizing” the value of the bond maturity, the method allows to apply the Merton modelto complex capital structures and implicitly to account for many real-life characteristics of thebond market ignored in the Merton model.

The adjustment for the AAA spread enables us to recover expected spreads due to defaultthat are similar in magnitude to observed frequencies. It appears to overcome, for the purposesof calibration, the commonly observed inability of structural debt models to explain spreadson investment grade debt. We test robustness by varying the structure of the model and theparametrization, and find that the procedure, though simple, is robust in estimating the defaultloss component of the spread. We believe that this robustness comes from the fact that all modelsof risky debt must preserve the basic structure of debt and equity: that debt is senior to equity.This makes the choice of the particular structural model secondary when the goal is splitting theobserved market spread into default and non-default components.

Our estimates suggest that only a small fraction of the spread for a high-grade firm is dueto expected default. For lower-grade firms, this component is larger, and our approach providesa method for adjusting yields to give expected debt returns. The expected default componentof the spread varies significantly within ratings categories, so using average figures for ratingscategories for individual firms could be misleading.

There are many possible applications of the equilibrium expected default spreads that werecover. We illustrate these with the use in determining a firm’s expected cost of debt for usein its cost of capital. We also use the procedure to provide an entirely new set of equity riskpremium estimates, based on data hitherto unused for this purpose. They are consistent withasset risk premia of about three percent and equity risk premia of between three and nine percent.

20

XI REFERENCES

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[2] Altman, E., and V. Kishore, 1996, ”Almost Everything You Wanted To Know About Re-coveries on Defaulted Bonds”, Financial Analysts Journal, November/December, 57-64.

[3] Anderson Ronald W., and Suresh M. Sundaresan, 1996, “The Design and Valuation of DebtContracts”, Review of Financial Studies, 9, 37-68.

[4] Asquith, P., D. Mullins, and E. Wolff, 1989, ”Original Issue High Yield Bonds: AgeingAnalysis of Default, Exchanges and Calls”, Journal of Finance 44, 923-52.

[5] Bhandari, C., 1988, ”Debt/Equity Ratio and Expected Common Stock Returns: EmpiricalEvidence”, Journal of Finance 43, 507-528.

[6] Black Fischer, and John C. Cox, 1976, “Valuing Corporate Securities: Some Effects of BondIndenture Provisions”, Journal of Finance, 31, 351-367.

[7] Blume M E and D B Keim, 1987, “Lower-grade bonds: Their risks and returns”, FinancialAnalysts Journal 43, July/August, 26-33.

[8] Blume M E, D B Keim and S A Patel, 1991 “Returns and volatility of low-grade bonds1977-1989 ”, Journal of Finance 46, No.1, March, 49-74.

[9] Bodie Z, A Kane and Markus, 1993 Investments, Irwin, Homewood.

[10] Brealey R A and S C Myers, 2000, “Principles of Corporate Finance”, McGraw Hill, 6thEdition.

[11] Collin-Dufresne Pierre, and Robert S. Goldstein, 2001, “Do Credit Spreads Reflect Station-ary Leverage Ratios?”, Journal of Finance, 56, 1929-1957.

[12] Collin-Dufresne Pierre, Robert S. Goldstein, and J.Spencer Martin, 2001, “The Determinantsof Credit Spread Changes”, Journal of Finance, 56, 2177-2207.

[13] Crosbie Peter J., and Jeffrey R. Bohn, 2002, “Modeling Default Risk”, KMV LLC.

[14] Davydenko S A, and I A Strebulaev, 2002, “Strategic Behavior and Credit Spreads: AnEmpirical Investigation”, Mimeo, London Business School.

[15] Delianedis G and R Geske, 1999, ”Credit risk and risk neutral default probabilities: In-formation about rating migrations and defaults” , working paper, The Anderson School,UCLA.

[16] Delianedis, G. and R. Geske, 2001, ”The components of corporate credit spreads: Default,recovery, tax, jumps, liquidity, and market factors”, Working Paper 22-01, Anderson School,UCLA.

[17] Dimson, Elroy, Paul Marsh, and Michael Staunton, 2001, ”Millennium Book II: 101 Yearsof Investment Returns”, ABN Amro/London Business School.

21

[18] Driessen Joost, 2002, “Is Default Event Risk Priced in Corporate Bonds?”, working paper,University of Amsterdam.

[19] Ehrhardt M C, 1994, “Measuring the Company’s Cost of Capital”, Harvard Business SchoolPress.

[20] Elton Edwin J., Martin J. Gruber, Deepak Agrawal, and Christopher Mann, 2001, “Explain-ing the Rate Spread on Corporate Bonds”, Journal of Finance, 56, 247-277.

[21] Eom Young Ho, Jean Helwege, and Jing-zhi Huang, 2002, “Structural Models of CorporateBond Pricing: An Empirical Analysis”, Working paper, Penn State University.

[22] Fama E and K French, 1993, “Common risk factors in the returns on stocks and bonds”,Journal of Financial Economics, 33, 3-57.

[23] Fama, E., and K. French, 1997, ”Industry costs of equity”, Journal of Financial Economics43, 153-93.

[24] Harris, R. and F. Marston, 1992, ”Estimating Shareholder Risk Premium Using Analysts’Growth Forecasts”, Financial Management, 63-70.

[25] Huang, Jing-zhi, and Ming Huang, 2002, “How Much of the Corporate-Treasury Yield Spreadis Due to Credit Risk?”, working paper, Pennsylvania State University.

[26] Jones P.E., S.P. Mason, and E. Rosenfeld, 1984, “Contingent Claims Analysis of CorporateCapital Structures: An Empirical Analysis”, Journal of Finance, 39, 611-25.

[27] Kao, D., 2000, ”Estimating and Pricing Credit Risk: An Overview”, Financial AnalystsJournal, 50-66.

[28] Kaplan S N and J C Stein, 1990, “How Risky is the Debt in Highly Leveraged Transactions”,Journal of Financial Economics, 27, 215-245.

[29] Keenan Sean C., Igor Shtogrin, and Jorge Soberhart, 1990, “Historical Default Rates ofCorporate Bond Issuers, 1920-1998”, Moody’s Investor Service.

[30] Leland Hayne E., 1994, “Corporate Debt Value, Bond Covenants, and Optimal CapitalStructure”, Journal of Finance, 49, 1213-1252.

[31] Leland Hayne E., and Klaus Bjerre Toft, 1996, “Optimal Capital Structure, EndogenousBankruptcy, and the Term Structure of Credit Spreads ”, Journal of Finance, 51, 987-1019.

[32] Longstaff Francis A., and Eduardo S. Schwartz, 1995, “A Simple Approach to Valuing RiskyFixed and Floating Rate Debt”, Journal of Finance, 50, 789-819.

[33] Madan Dilip B., 2000, “Pricing the Risks of Default”, Working paper, University of Marylandat College Park.

[34] Merton Robert C., 1974, “On the Pricing of Corporate Debt: The Risk Structure of InterestRates”, Journal of Finance, 29, 449-470.

22

[35] Vassalou Maria, and Yuhang Xing, 2002, “Default Risk in Equity Returns”, Journal ofFinance, forthcoming.

[36] Vasicek O A, 1984, “Credit Valuation”, Working paper, KMV Corporation.

[37] Waldman R A , E I Altman and A R Grinsberg, 1998, “Defaults and Returns on High YieldBonds, Analysis through 1997”, Salomon Smith Barney, New York.

[38] Welch I, 2000, ”Views of Financial Economists in the Equity Premium and Other ProfessionalControversies”, Yale University Working Paper.

23

Appendix A The Merton Model and the ExpectedReturn on Debt

The standard form of the Merton model is:

B = V N(−d1)− Fe−rT N(d2) (A1)

where

d1 = [ln[V/F ] + (r + σ2/2)T ]/σ√

T (A2)

d2 = d1 − σ√

T (A3)

V is the value of the assets of the firm, B is the value of the debt, and F is the promised debt payment(the face value of the debt for a zero-coupon bond). The other variables are as in the main text.

If the continuously compounded promised yield on debt is y then:

F = BeyT (A4)

Promised debt spread and financial leverage are defined by:

s = y − r (A5)

w = B/V (A6)

Substitution of (A4)-(A6) into (A1)-(A3) yields equations (2)-(4) in the main text.

It follows from (1) that the asset returns are log-normally distributed:

ln(VT /V ) ∼ N((µ− σ2/2)T, σ2T

)where VT is the value of the firm’s assets at date T . Therefore,

VT = V e(µ−σ22 )T+σ

√TZ (A7)

where Z is a standard normal variable.

The compound return on debt over T , denoted by ΠD, is determined by:

BeΠDT =

VT =F∫−∞

VT dN(Z) +

∞∫VT =F

F dN(Z)

Evaluating the integrals and taking into account (A6), we obtain:

wV eΠDT = V eµT N(−K1) + F N(K2) (A8)

where F is given by (A4) and:

K1 =[ln [V/F ] + (µ + σ2/2)T

]/σ√

T

K2 = K1 − σ√

T

Rearranging terms in (A8) and noting that (see (3), (4) and (6)):

K1 = d1 + (µ− r)√

T/σ = d1 + π√

T/σ

K2 = d2 + π√

T/σ

we obtain equation (8) of the main text.

24

Appendix B Tax and the yield spread

Elton et al. (2001) claim that a significant proportion of yield spreads is due to tax. Their focus isthe fact that corporate bonds are subject to state taxes in the US, whereas government bonds are not.They employ a data-fitting procedure to estimate the contribution to spread due to tax. We derive asimple expression for the tax-induced spread using assumptions very similar to theirs.

Consider a security with maturity tM , at which a random ‘principal’ F̃ is repaid. The security mayalso deliver random ‘coupon’ payments C̃ti at intermediate moments 0 < ti < tM . Let rt denote therisk-free discount rate from time 0 to t. Then before-tax value of such security is the expectation of itsbefore-tax cash flows taken under the equivalent martingale measure:

B̄ = E∗[∑

i

e−rtiti C̃ti

]+ e−rtM

tM E∗[F̃

](B1)

Assume that income tax is paid on ‘coupons’, and that the capital gain tax must be paid on the ‘principal’at maturity, with a symmetric treatment of capital gains and losses. Further, assume that relevant incomeand capital gain tax rates are the same and equal to τ .23 Then the after-tax value of the bond is:

B = E∗[∑

i

e−rtiti (1− τ) C̃ti

]+ e−rtM

tM E∗[B + (1− τ) (F̃ −B)

]Substituting for E∗

[∑e−rti

ti C̃t

]from (B1):

B = (1− τ)(B̄ − e−rtM

tM E∗[F̃

])+ e−rtM

tM τB + (1− τ) e−rtMtM E∗

[F̃

]=

= (1− τ) B̄ + e−rtMtM τB

from which it follows that

B =1− τ

1− τe−rtMtM

B̄ (B2)

Thus, the yield spread due to tax can be computed from the simple formula:

∆ytax =1

tMln

[1− τ

1− τe−rtMtM

](B3)

This depends on the level of interest rates, so we cannot use the Elton et al. results with our data, whichrefer to a different period. Also, we have slightly different assumptions, in that they assume principalacceleration upon early bond default, and deductibility of the principal amount, whereas we have noacceleration and deductibility of the purchase price. Their estimated tax spreads are almost constantacross ratings, the variations being within 1-2 basis points.

Appendix C Bankruptcy costs and strategic default inthe Merton model

A simple way to introduce bankruptcy costs strategic bankruptcy into the Merton model is to usea one-period specification of the Anderson-Sundaresan (1996) model. In this model bankruptcy resultsin a fixed cost, H. However, both borrowers and lenders act strategically, and bankruptcy does notautomatically happen when the borrower fails to repay the debt face value, F, to bondholders at maturity.Instead, the borrower offers the bondholders a payoff which makes them indifferent between acceptingthe payment and filing for bankruptcy when F is not repaid in full. In this model costly bankruptcy

23The relevant tax rate for the tax spread between corporate and government bonds is τ = ts(1− tg) where tsand tg are the state and federal tax rate, respectively.

25

never occurs in equilibrium. The payoff to debt at maturity is:

BT = min{F, max{VT −H, 0}}

The value of such a security is:

B = Call(H)− Call(F + H) =

= V[N(dH

1 )−N(dF+H1 )

]+ Fe−rT

[(1 + Θ)N(dF+H

2 )−ΘN(dH2 )

]where Θ = H/F is the bankruptcy costs expressed as a fraction of the debt par value, and

dH1 =

[− ln w − (s− σ2/2)T

]/σ√

T − lnΘ/σ√

T

dF+H1 =

[− ln w − (s− σ2/2)T

]/σ√

T − ln(1 + Θ)/σ√

T

di2 = di

1 − σ√

T , i = H, F + H

Dividing by B yields an equation for T and σ in terms of w and s:

1 =[N(dH

1 )−N(dF+H1 )

]/w + esT

[(1 + Θ)N(dF+H

2 )−ΘN(dH2 )

](C1)

It follows that:

σE/σ =∂E

∂VV/E =

[1−N(dH

1 ) + N(dF+H1 )

]/(1− w) (C2)

For a given value of Θ we solve system (C1)-(C2) to determine T and σ which are consistent with theobserved firm’s characteristics w, σE , and s. The asset return premium on debt is:

π = πEσ/σE (C3)

The derivation of the expected default loss on the debt is similar to that described in II. The expectedbasis points loss over the maturity period is:

δ = − 1

Tln

[e(π−s)T N(KH

1 )−N(KF+H1 )

w+ (1 + Θ)N(KF+H

2 )−ΘN(KH2 )

](C4)

Kij = di

j − π√

T/σ, i = H, F + H, j = 1, 2

which is the analog of formula (9) for this case.

Appendix D Dividends in the Merton model

To take into account dividend payments, we amend the Merton model by assuming that the firmpays continuous dividends that are a constant proportion γ of the value of the firm V .

Repeating the analysis of similar to described in Appendix Aunder this assumption, equations (2)–(9)become:

e−γT N(−d1)/w + esT N(d2) = 1 (D1)

where

d1 = [− ln w − (s + γ − σ2/2)T ]/σ√

T (D2)

d2 = d1 − σ√

T (D3)

(1− w)σE = σ[1− e−γT N(−d1)] (D4)

26

Asset and equity return premia (including distributions) π ≡ µ − r and πE ≡ µE − r are related asbefore:

π

πE=

σ

σE(D5)

The spread which is due to expected default and should be excluded from the expected return on debtis thus:

δ ≡ s−ΠD = − 1

Tln

[e(π−γ−s)T N

(−d1 − (π − γ)

√T/σ

)/w + N

(d2 + (π − γ)

√T/σ

)](D6)

27

Table I. The Default Risk Component of AAA Spreads

The table shows the average AAA spread and the part of the spread due to default.Estimates are from Huang and Huang (2002) and Delianedis and Geske (2001).

Source Model Maturity Spread Default DefaultYears b.p. b.p. %

Huang and Huang Longstaff/Schwartz I 10 63 10 16Huang and Huang Longstaff/Schwartz II 10 63 6 10Huang and Huang Collin-Dilfresne/Goldstein 10 63 11 18

Delianedis and Geske Merton/Geske 1-10 36 2 5

Table II. Corporate bond tax spread

The table gives estimates of spreads which is due to differentialtaxation of corporate and government bond coupons on the statelevel. T is the maturity of the bond in years. rTm

is the risk-less interest rate over the life of the bond, and τ is the averageapplicable state tax rate. Assumed bond maturity is 10 years.

r% τ = 4% τ = 4.875% τ = 6.7%

3.0 10.7 13.2 18.44.0 13.6 16.8 23.45.0 16.3 20.0 27.96.0 18.6 22.9 31.97.0 20.8 25.5 35.5

28

Table III. Summary Statistics on Credit Risk Variables

This table reports summary statistics for the final studied sample, which includestrades on bonds between 7.5 and 10 years to maturity, executed within 3 calendarmonths after the last fiscal year end. Leverage is the ratio of book value of debt tothe book value of debt plus the market value of equity on the last business day beforethe trade date. Equity volatility is the volatility of daily share price returns over 252business days before the trade date. Asset volatility is the volatility of asset returnsimplied by the Merton (1974) model estimated using the KMV procedure describedin Crosbie and Bohn (2002). Volatilities and spreads are annualized.

AAA AA A BBB BB B All

Spread Mean 0.28 0.45 0.69 1.07 2.08 3.96 1.01Median 0.28 0.47 0.66 0.95 1.82 3.69 0.81Std. Dev. 0.10 0.21 0.39 0.50 1.11 1.19 0.80

Leverage Mean 0.06 0.13 0.25 0.35 0.49 0.63 0.31Median 0.06 0.12 0.24 0.35 0.44 0.61 0.29Std. Dev. 0.03 0.07 0.14 0.14 0.20 0.15 0.17

Equity volatility Mean 0.33 0.30 0.33 0.33 0.39 0.62 0.34Median 0.33 0.28 0.29 0.30 0.38 0.62 0.30Std. Dev. 0.17 0.10 0.12 0.11 0.12 0.18 0.12

Maturity Mean 8.53 8.81 9.00 8.86 8.91 8.87 8.92Median 8.53 8.81 9.12 8.88 9.15 9.01 8.99Std. Dev. 0.21 0.75 0.75 0.74 0.69 0.87 0.74

Duration Mean 6.56 6.69 6.76 6.57 6.47 6.14 6.64Median 6.56 6.71 6.79 6.58 6.48 5.99 6.65Std. Dev. 0.16 0.48 0.52 0.47 0.56 0.54 0.52

KMV asset Mean 0.24 0.25 0.25 0.25 0.26 0.49 0.25volatility Median 0.24 0.25 0.22 0.23 0.25 0.39 0.23

Std. Dev. 0.07 0.06 0.09 0.10 0.10 0.30 0.10

N 2 231 1088 1003 266 42 2632

29

Table

IV.Est

imat

edex

pec

ted

def

ault

loss

es

The

tabl

egi

ves

mod

el-p

redi

cted

expe

cted

defa

ult

loss

esan

dpr

obab

iliti

esfo

rge

neri

cbo

nds

in4

rati

nggr

oups

.T

hree

met

hods

,in

dica

ted

inth

ese

cond

colu

mn,

are

used

toad

just

the

obse

rved

spre

adfo

rno

n-de

faul

tfa

ctor

s:no

adju

stm

ent,

subt

ract

ing

ata

xsp

read

,and

subt

ract

ing

the

aver

age

spre

adon

AA

Abo

nds.

The

inpu

tsar

em

ean

valu

esin

each

rati

ngcl

ass

ofth

epr

omis

edsp

read

onde

bts,

leve

rage

w,

and

assu

med

equi

tyri

skpr

emiu

m,

πE

,as

used

inH

uang

and

Hua

ng(2

002)

;an

dth

em

edia

nvo

lati

lity

ofeq

uity

σE

from

our

stud

ied

sam

ple.

T,σ

and

πar

em

odel

-im

plie

dm

atur

ity,

vola

tilit

yof

asse

tsan

dre

turn

onas

sets

.T

heou

tput

sar

eth

eex

pect

edpe

rcen

tage

defa

ult

loss

onde

btδ

and

the

expe

cted

prob

abili

tyof

defa

ult

P.

Ave

rage

hist

oric

alde

faul

tlo

sses

and

defa

ult

freq

uenc

ies

δ han

dP

har

ere

port

edfo

rco

mpa

riso

n.

Mod

elM

odel

Def

ault

loss

Spre

adin

puts

para

met

ers

Mod

elH

isto

rica

lE

GR

atin

gad

just

-s

ET

σπ

δδ/

sP

δ hδ h

/sP

hδ E

G

men

tb.

p.%

yrs.

%b.

p.%

b.p.

%b.

p.(1

)(2

)(3

)(4

)(5

)(6

)(7

)(8

)(9

)(1

0)(1

1)(1

2)(1

3)(1

4)

AA

AN

one

63.0

00.

135.

380.

2751

.08

0.24

4.83

5.36

0.09

6.11

3.95

0.06

0.77

Tax

43.0

437

.22

0.24

4.78

4.01

0.06

3.90

AA

Non

e91

.00

0.21

5.60

0.28

51.9

70.

234.

749.

450.

1010

.37

5.41

0.06

0.99

4.80

Tax

71.0

439

.81

0.23

4.67

7.98

0.09

7.67

AA

A28

.00

19.6

00.

224.

513.

870.

042.

71

AN

one

123.

000.

325.

990.

2949

.86

0.23

4.68

15.5

80.

1315

.75

8.33

0.07

1.55

14.0

0Tax

103.

0439

.63

0.22

4.58

13.9

20.

1112

.65

AA

A60

.00

22.9

70.

214.

379.

460.

086.

82

BB

BN

one

194.

000.

436.

550.

3165

.38

0.23

4.91

26.8

50.

1428

.40

24.4

00.

134.

3940

.90

Tax

174.

0453

.75

0.22

4.78

25.3

20.

1324

.51

AA

A13

1.00

35.2

20.

214.

5121

.28

0.11

17.2

4

BB

Non

e32

0.00

0.54

7.30

0.38

43.0

50.

275.

0793

.02

0.29

51.9

413

2.24

0.41

20.6

3Tax

300.

0438

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0.26

4.96

88.4

40.

2847

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AA

A25

7.00

29.2

50.

254.

7378

.15

0.24

39.3

7

BN

one

470.

000.

668.

760.

579.

870.

304.

5427

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0.58

51.0

835

2.87

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43.9

1Tax

450.

049.

340.

294.

4826

0.40

0.55

48.9

6A

AA

407.

008.

270.

284.

3323

6.70

0.50

44.3

7

30

Table

V.Sen

sitivity

and

Rob

ust

nes

sof

Est

imat

edD

efau

ltLos

s

The

tabl

ere

port

sth

ese

nsit

ivity

ofth

em

odel

-pre

dict

edex

pect

edde

faul

tlo

sses

and

prob

abili

ties

toin

puts

and

mod

elas

sum

ptio

ns,fo

rty

pica

lA

and

BB

bond

s.O

bser

ved

spre

ads

are

adju

sted

bysu

btra

ctin

gth

eA

AA

spre

adof

63ba

sis

poin

ts.

Rep

orte

dar

ere

sult

sfo

rth

eba

sic

mod

el,

asw

ellas

the

And

erso

n-Su

ndar

esan

proc

edur

efo

rtw

ole

vels

ofba

nkru

ptcy

cost

s.T

hein

puts

are

sam

ple

med

ian

valu

esin

each

rati

ngcl

ass

ofth

epr

omis

edsp

read

onde

bts,

leve

rage

w,as

sum

edeq

uity

risk

prem

ium

E,an

dth

evo

lati

lity

ofeq

uity

σE

.T

and

πar

em

odel

-im

plie

dm

atur

ity,

vola

tilit

yof

asse

tsan

dre

turn

onas

sets

.T

heou

tput

sar

eth

eex

pect

edpe

rcen

tage

defa

ult

loss

onde

btδ

and

the

expe

cted

prob

abili

tyof

defa

ult

P.

Mod

elB

asic

mod

elB

ankr

uptc

yco

sts

and

stra

tegi

cde

faul

tin

puts

θ=

=0.

05,E

ndog

enou

sT

T=

10,E

ndog

enou

ss

adj.

ET

σπ

δδ/

sT

σπ

δδ/

σπ

δδ/

sb.

p.b.

p.%

%b.

p.%

%b.

p.%

%b.

p.%

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

(15)

(16)

(17)

(18)

(19)

(20)

Pan

elA

:T

ypic

alA

A-r

ated

bond

s

91.0

280.

215.

600.

2819

.60

0.22

4.51

3.87

4.25

17.6

40.

224.

514.

164.

570.

360.

224.

495.

996.

5881

.918

.915

.80

0.22

4.48

2.77

3.38

14.3

40.

224.

482.

973.

620.

250.

224.

473.

774.

6010

0.1

37.1

23.4

40.

224.

554.

874.

8620

.92

0.22

4.54

5.26

5.26

0.43

0.22

4.52

8.37

8.36

0.19

20.8

80.

234.

623.

633.

9918

.90

0.23

4.62

3.90

4.28

0.42

0.23

4.61

5.91

6.49

0.23

18.5

10.

224.

404.

094.

4916

.57

0.22

4.40

4.41

4.85

0.30

0.22

4.38

6.07

6.67

5.04

19.6

00.

224.

064.

855.

3317

.64

0.22

4.06

5.16

5.68

0.36

0.22

4.05

7.09

7.80

6.16

19.6

00.

224.

963.

063.

3717

.64

0.22

4.96

3.33

3.66

0.36

0.22

4.94

5.04

5.54

0.25

27.6

10.

204.

542.

092.

3024

.70

0.20

4.53

2.32

2.55

0.54

0.20

4.50

4.87

5.36

0.30

14.5

30.

244.

495.

836.

4013

.13

0.24

4.49

6.15

6.76

0.19

0.24

4.49

7.09

7.79

Pan

elB

:T

ypic

alB

B-r

ated

bond

s

320

257

0.54

7.30

0.38

29.2

50.

254.

7378

.15

24.4

216

.94

0.24

4.60

92.7

128

.97

0.12

0.23

4.45

108.

3133

.85

288

225

24.0

30.

244.

5670

.10

24.3

415

.04

0.23

4.45

81.5

928

.33

0.11

0.23

4.35

92.4

532

.10

352

289

35.5

60.

264.

9085

.86

24.3

918

.74

0.25

4.74

104.

1429

.59

0.13

0.24

4.55

124.

5535

.38

0.48

30.2

30.

265.

0175

.96

23.7

418

.53

0.26

4.91

88.3

727

.62

0.15

0.25

4.76

106.

1233

.16

0.59

28.7

10.

234.

4480

.10

25.0

315

.51

0.22

4.27

97.2

830

.40

0.10

0.22

4.14

110.

6334

.57

6.57

29.2

50.

254.

2690

.00

28.1

316

.94

0.24

4.14

104.

1332

.54

0.12

0.23

4.01

119.

1837

.24

8.03

29.2

50.

255.

2067

.47

21.0

816

.94

0.24

5.06

82.2

425

.70

0.12

0.23

4.90

98.2

230

.69

0.34

52.2

60.

245.

0651

.14

15.9

824

.94

0.23

4.86

69.4

521

.70

0.17

0.22

4.57

97.0

630

.33

0.42

18.3

50.

264.

4910

1.86

31.8

311

.80

0.25

4.39

113.

7835

.56

0.07

0.25

4.35

118.

4637

.02

31

Table VI. Sensitivity of Estimated Default Loss to Dividend Yield

The table reports the sensitivity of the model-predicted expected default losses forgeneric bonds in 4 rating groups to the assumed instantaneous equity dividend yieldg. Mean values of the promised spread on debt s in each rating class, adjusted bysubtracting the median AAA spread of 63 b.p., are used as inputs. Other inputs aremean leverage w and assumed equity risk premium πE used in Huang and Huang(2002), and the median volatility of equity σE from our sample. γ, T , σ and π are themodel-implied asset payout ratio, maturity, volatility of assets and return on assets.The outputs are the expected basis point loss in debt yield due to default δ, and theproportion of the promised spread which is due to default, δ/s.

g γ T σ π δ δ/s% % % b.p. %

Panel A: Typical AA-rated bondss=91 bp, s adj.=28 bp, w = 0.21, πE = 5.60%, σE = 0.28

0.00 0.00 19.60 0.22 4.51 3.87 4.251.00 0.79 16.26 0.22 4.51 4.08 4.482.00 1.58 14.02 0.22 4.50 3.02 3.323.00 2.36 12.40 0.22 4.50 0.42 0.46

Panel A: Typical A-rated bondss=123 bp, s adj.=60 bp, w = 0.32, πE = 5.99%, σE = 0.29

0.00 0.00 22.39 0.21 4.36 9.78 7.951.00 0.68 17.37 0.21 4.34 11.03 8.972.00 1.36 14.41 0.21 4.32 10.22 8.313.00 2.04 12.41 0.21 4.31 7.38 6.00

Panel C: Typical BBB-rated bondss=194 bp, s adj.=131 bp, w = 0.43, πE = 6.55%, σE = 0.31

0.00 0.00 33.28 0.21 4.51 22.68 11.691.00 0.57 22.14 0.21 4.42 28.22 14.542.00 1.14 17.07 0.21 4.37 29.68 15.303.00 1.71 14.07 0.21 4.33 28.31 14.59

Panel D: Typical BB-rated bondss=320 bp, s adj.=257 bp, w = 0.54, πE = 7.30%, σE = 0.38

0.00 0.00 29.25 0.25 4.73 78.15 24.421.00 0.46 19.36 0.24 4.60 90.40 28.252.00 0.93 14.91 0.24 4.51 96.71 30.223.00 1.40 12.28 0.23 4.45 99.78 31.18

Panel E: Typical B-rated bondss=470 bp, s adj.=407 bp, w = 0.66, πE = 8.76%, σE = 0.57

0.00 0.00 8.22 0.28 4.30 237.36 50.501.00 0.34 7.03 0.28 4.24 243.80 51.872.00 0.68 6.20 0.27 4.19 248.48 52.873.00 1.02 5.57 0.27 4.15 251.90 53.60

32

Table

VII

.Expec

ted

loss

esfo

rin

div

idual

bon

ds

inth

esa

mple

The

tabl

egi

ves

mod

el-p

redi

cted

expe

cted

defa

ult

loss

esan

dpr

obab

iliti

esfo

rbo

nds

inth

esa

mpl

eby

rati

ngs.

The

sam

ple

cons

ists

ofse

nior

unse

cure

dbo

nds

wit

h7.

5-10

year

sto

mat

urity.

The

obse

rved

spre

adw

asad

just

edfo

rno

n-de

faul

tfa

ctor

bysu

btra

ctin

gth

esa

mpl

em

edia

nsp

read

onA

AA

bond

sfo

rth

eye

arof

trad

e.

Mod

elin

puts

Mod

elpa

ram

eter

sM

odel

esti

mat

ess

sad

j.w

πE

σE

πδ

δ/s

b.p.

b.p.

%b.

p.%

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

AA

Mea

n57

.06

11.0

80.

175.

600.

2819

.67

0.24

4.71

1.78

2.73

Med

ian

53.2

47.

900.

175.

600.

2714

.24

0.22

4.68

0.54

1.19

Std.

Dev

.12

.08

9.99

0.08

0.00

0.09

16.8

70.

080.

412.

983.

97N

90.0

0

AM

ean

78.0

627

.63

0.27

5.99

0.31

19.8

70.

234.

496.

196.

40M

edia

n70

.40

21.7

90.

265.

990.

2814

.10

0.21

4.52

2.31

3.41

Std.

Dev

.29

.36

25.0

60.

140.

000.

1118

.81

0.10

0.79

10.3

97.

53N

825.

00

BB

BM

ean

110.

1358

.56

0.36

6.55

0.33

28.2

20.

234.

5513

.83

10.7

0M

edia

n96

.14

47.1

40.

356.

550.

3015

.03

0.21

4.58

7.03

7.56

Std.

Dev

.45

.63

43.4

80.

140.

000.

1068

.19

0.09

0.76

20.7

910

.23

N94

9

BB

Mea

n20

8.13

155.

070.

497.

300.

3928

.57

0.23

4.49

55.5

822

.04

Med

ian

182.

1613

0.42

0.45

7.30

0.37

19.0

10.

214.

8531

.75

20.0

4St

d.D

ev.

110.

7510

9.85

0.19

0.00

0.12

33.2

60.

081.

2069

.17

16.1

8N

255

BM

ean

395.

6533

5.46

0.63

8.76

0.62

16.3

30.

294.

3519

5.74

47.2

8M

edia

n36

8.59

315.

650.

618.

760.

624.

440.

294.

1015

7.15

48.9

0St

d.D

ev.

119.

3512

1.24

0.15

0.00

0.18

23.6

40.

091.

3712

4.44

18.3

6N

42

All

Mea

n11

2.79

61.5

50.

346.

430.

3324

.49

0.23

4.52

18.8

810

.78

Med

ian

88.2

538

.00

0.32

6.55

0.30

14.6

70.

214.

585.

676.

44St

d.D

ev.

78.6

376

.24

0.17

0.55

0.12

48.4

40.

090.

8444

.04

12.5

0N

2161

33

Table

VII

I.Est

imat

edco

stof

equity

for

indiv

idual

bon

ds

inth

esa

mple

The

tabl

egi

ves

cost

ofeq

uity

valu

esco

nsis

tent

wit

hM

oody

’shi

stor

ical

defa

ult

data

and

anav

erag

ehi

stor

ical

reco

very

for

seni

orun

secu

red

bond

sof

48%

.T

hesa

mpl

eco

nsis

tsof

seni

orun

secu

red

bond

sw

ith

7.5-

10ye

ars

tom

atur

ity.

The

obse

rved

spre

adw

asad

just

edfo

rno

n-de

faul

tfa

ctor

bysu

btra

ctin

gth

esa

mpl

em

edia

nsp

read

onA

AA

bond

sfo

rth

eye

arof

trad

e.

Mod

elin

puts

Mod

elpa

ram

eter

sM

odel

ss

adj.

πes

tim

ate

b.p.

b.p.

b.p.

E,%

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

AA

Mea

n61

.13

15.8

40.

180.

285.

4522

.80

0.24

2.51

3.06

Med

ian

57.9

813

.02

0.20

0.27

5.46

15.5

40.

222.

092.

68St

d.D

ev.

11.9

69.

460.

080.

090.

0718

.51

0.08

2.28

2.95

N58

AM

ean

82.5

832

.01

0.28

0.31

7.74

21.6

80.

233.

785.

30M

edia

n74

.14

25.3

00.

280.

287.

7815

.32

0.20

2.61

3.42

Std.

Dev

.29

.51

24.9

20.

140.

100.

4319

.64

0.09

3.88

5.51

N69

6

BB

BM

ean

116.

6865

.53

0.37

0.33

23.5

231

.04

0.23

3.02

4.49

Med

ian

103.

3252

.13

0.37

0.30

23.5

816

.59

0.20

2.03

2.98

Std.

Dev

.45

.83

43.0

30.

140.

100.

8073

.08

0.08

3.22

4.87

N81

5

BB

Mea

n28

0.53

226.

460.

550.

4213

4.93

35.6

00.

242.

314.

30M

edia

n24

4.47

185.

090.

500.

4113

4.48

19.2

20.

231.

382.

44St

d.D

ev.

125.

3112

5.20

0.21

0.11

1.40

43.7

10.

092.

724.

31N

119.

00

BM

ean

534.

8647

5.14

0.61

0.73

367.

0514

.91

0.35

3.83

8.53

Med

ian

496.

5043

8.34

0.60

0.82

368.

472.

350.

372.

916.

36St

d.D

ev.

96.3

090

.08

0.14

0.22

4.50

21.7

20.

083.

478.

34N

14

All

Mea

n11

5.74

64.7

50.

340.

3327

.07

27.1

20.

233.

274.

79M

edia

n91

.45

41.4

00.

320.

3022

.38

16.3

80.

212.

203.

18St

d.D

ev.

80.1

577

.64

0.17

0.11

44.0

553

.73

0.09

3.48

5.12

N17

02

34