Using Yield Spreads to Estimate Expected Returns on Debt ...

[Pages:34]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 equity from corporate bond spreads. It is based on an easily implementable calibration of the Merton (1974) model to market debt spreads and other observable variables. For rating classes, the approach generates robust expected default loss estimates very similar to historical default data. It also provides forward-looking estimates for individual firms unavailable from historical data. The method can be used to adjust the cost of the firm's debt for the probability of default, which is essential for lowrated firms. The approach can also be applied to provide independent estimates of expected equity premia consistent with historical default experiences. These equity risk premium estimates vary from three percent for typical investment-grade firms to over 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's Park, London NW1 4SA. E-mail: icooper@london.edu. Tel: +44 020 7262 5050 Fax: +44 020 7724 3317. This is a 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, and expected 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 returns on risky debt are required. These are equal to the promised debt yield minus the part of the yield that reflects expected default. To obtain expected returns, therefore, estimates of expected losses due to default are required.

Most methods of estimating the expected default loss are based either on historical default data, or accounting and equity market information (see Lao (2000), Elton et al. (2001), Crosbie and Bohn (2002)).2 However, such estimates ignore the most relevant variable incorporating the market consensus expectation of default: the debt yield itself. In this paper we propose a method of 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 observed market spread for individual bonds is due to expected default. From these, expected returns on the 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 historical default experiences as proxies for future default incidence (for example, Altman (1989), and Blume, Keim and Patel (1991)). Elton et al. (2001) use historical data on rating migrations and recovery rates to estimate the expected default spread. Their approach uses data on ratings classes. Thus, it does not provide estimates for individual bonds unless they are typical of a ratings class. Our method, on the other hand, recovers the part of the spread due to expected default for individual bonds. Another disadvantage of the historical approach is that it does not provide forward-looking estimates. Asquith, Mullins and Wolff (1989) argue that historical default frequencies may differ from future probabilities, because available historical data do not cover all likely future economic and market conditions (see also Waldman et al. (1998)). In contrast, our method uses the observed market yield, which should reflect expectations of the economic and market conditions for the period to which it refers. The estimates that we derive are, on average, consistent with historical default data for ratings classes. Where they differ, our estimates 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 components 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 assumptions 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 transaction price series to estimate debt betas, which are often unavailable. Moreover, applying this method is complicated by the fact that debt betas change significantly with changes in capital structure and over time. Also, it depends on the CAPM being the correct equilibrium model, and using a correct estimate of the market risk premium. Taking into 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 corporate bond markets. These models improve the fit to the general level of yields, but none gives a generally good cross-sectional fit to bond prices (see Eom et al. 2002). Despite the large variety of structural models available, Huang and Huang (2002) show that very different models predict similar debt spreads when they are calibrated to fit observed default and recovery rates. In this sense, the choice of observed variables for calibration appears to be more important than the particular model structure. Moreover, since our goal is not to predict the total spread, but rather determine 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, we choose the simplest model for our calibration. If the Merton framework picks up first-order effects relevant to the relative valuation of risky debt and equity, then our estimates of the part of the spread due to expected default should not be overly sensitive to the model assumptions. We test robustness of our estimates by varying the structure of the model and the parametrization. We find that the procedure, though simple, is robust in estimating the default loss component of the spread, confirming the result of Huang and Huang that predictions of these types of models do not 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 the face value and maturity of debt and a time series of equity values. They recover asset value and volatility and use this to calculate a 'distance to default'. This is then used in conjunction with KMV'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 volatility of assets and use it to calculate the sum of the expected default loss and risk premium due to default. They conclude that different models have similar performance when calibrated to historical default data, and also that default cannot account for much of the spread for high grade bonds. Delianedis and Geske (2001) calibrate a version of the Merton model to debt face value, maturity, equity value and equity volatility. Like Huang and Huang, they recover the part of the spread caused by default risk. They also show that this cannot explain the entire spread for high grade firms. An important difference from our approach is that these papers focus on the total spread due to default risk, including the associated risk premium. We, on the other hand, use the total market spread adjusted for non-default factors as an input, and split out the single component due to the expected default loss.

We make three innovations in the calibration procedure. First, we calibrate the model to debt yield spreads. None of the above papers uses yield spreads for calibration.5 Of all capital market variables, bond yields should contain the most relevant information about consensus predictions of default. Thus, estimates of default rates that do not use yields as inputs may be inconsistent with 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 only with pricing under the risk-adjusted probability measure and so cannot assist in the estimation of the actual default 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 default adjustments that are consistent with observed spreads, leverage and volatility. This procedure allows to estimate the parameters of the firm asset value distribution implied by observed market prices. This is then used to determine the part of the yield due to expected default, given the expected return on equity.

Second, we control for factors other than default risk by measuring spreads relative to the AAA rate rather than the treasury rate. We base this adjustment on two observations. The first is that AAA debt has a very low chance of default. The second is that the components of the spread unrelated to default, such as taxes and bond-market risk factors, appear to be relatively constant across ratings categories. For the tax spread component discussed by Elton et al., we derive an explicit formula that demonstrates this independence. The adjustment for the AAA spread results in expected default spread estimates which are similar in magnitude to those predicted 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 investment grade 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. The Merton model assumes a single class of zero-coupon debt. Because of omitted factors, including coupons, default before maturity, strategic actions, and complex capital structures, the Merton model is too simple to reflect reality. Therefore, the choice of maturity when implementing the model is difficult. For instance, Huang and Huang use actual maturity, Delianedis and Geske use duration, and KMV use a procedure that mainly depends on liabilities due within one year. To avoid this issue and give the model enough flexibility to fit actual yield spreads, we simply solve for the value of maturity which makes the model consistent with observed spreads adjusted for non-default factors.

There are many possible applications of the equilibrium expected default spreads that we recover. We illustrate these with the use in estimating a firm's expected cost of debt for use in 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 corporate finance. Its key inputs are the cost of equity and the cost of debt. Yet while the cost of equity has been the subject of extensive debate, little attention has thus far been focused on estimating the cost of debt. Two common approaches are to use either the yield on the debt or the riskless interest rate as proxies. Neither is correct when part of the yield spread is due to expected 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 of return 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 that we 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 procedure helps 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 from an independent source, such as historical default data, the procedure can be reversed to give estimates of the expected return on equity consistent with observed debt yields. We use this approach to provide a new set of equity risk premium estimates, based on data hitherto unused for this purpose. This method, based on bond yields, should contain information about consensus expectations of risk premia. An important advantage of our approach is that the resulting estimates are largely based on a forward-looking capital market variable.7 It may provide a useful benchmark comparison to estimates of equity premia derived in more standard ways. Other estimates 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 equilibrium expected default component of the spread appears quite robust. It recovers estimates which are not very different from those obtained from historical average default and recovery data for ratings categories. Our estimates also appear to be better behaved than estimates based on historical default frequencies in the following sense: The historical method gives expected returns on debt for our sample that are not monotonic as debt rating changes, whereas our procedure gives 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 highgrade debt is due to expected default loss. For lower-grade debt, this component is larger, and our approach provides a method for adjusting yields to give expected debt returns. We find that the expected default component of the spread varies significantly within ratings categories, so using average figures for ratings categories for individual companies may be misleading. The estimates of equity risk premia that we obtain using the technique are well-behaved. They are consistent with asset risk premia of about three percent and equity risk premia of between three and nine percent.

The balance of the paper is organized as follows. The next section presents the calibration approach 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. Section IV provides a description of our data set. Section V discusses the calibration method and examines its robustness. Section VI provides estimates of expected returns on debt. Section VII presents estimates of equity risk premia. The following sections discuss various applications and extensions of 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 in the 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 the value of the firm's assets follows a geometric Brownian motion:

dV

V = ?dt + dWt

(1)

where V is the value of the firm's assets, ? and are constants, and Wt is a standard Wiener process.9 The model further assumes that the firm has a single class of zero coupon risky debt of maturity T . Other assumptions include a constant flat risk-free yield curve and a very simple bankruptcy procedure.

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

N (-d1)/w + esT N (d2) = 1

(2)

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

d1 = [- ln w - (s - 2/2)T ]/ T

(3)

d2 = d1 - T

(4)

Of the variables in Equation (2), leverage and the spread are observable, and and T are

generally unobservable. Another implication of the model is that the observable equity volatility10

E satisfies:

E = N (d1)/(1 - w)

(5)

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

Once the values of 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 assets

and therefore has the same underlying source of risk, the risk premia on assets and equity E

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 can solve equations (2) and (5) separately for for a few fixed values of T (or vice versa). This procedure always converged for any reasonable starting points. The intersection of the solution curves (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:

?-r

=

(6)

E ?E - r E

or:

= E/E

(7)

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

1 D = s + T ln

e(-s)T N (-d1 - T /) /w + N (d2 + T /)

(8)

and the spread which is due to expected default s - D which should be excluded from the expected return on debt is thus:

1 = - ln

T

e(-s)T ) N (-d1 - T /) /w + N (d2 + T /)

(9)

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

P = N (-d2 - E T /E)

(10)

If, on the other hand, the expected default loss on debt is known, then the expected equity

premium can be estimated. Equation (9) can be solved for and combined with (7) to find e 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 subtracting the AAA spread. There is growing evidence that corporate yield spreads cannot be entirely due to the risk of default. Huang and Huang (2002) and Delianedis and Geske (2001) measure the part of the spread that is due to default risk and find that, for AAA bonds, very little of it can be explained by default. Table I presents their estimates of the proportion of the AAA spread that is due to default risk. All estimates suggest that very little of the AAA spread can be explained by default risk.14 Thus, high grade spreads must be almost entirely due to other factors.15 Elton et al. (2001) argue that a part of the spread for U.S. corporate bonds is due to the state tax on corporate 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 compounded return 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 Huang and Huang state, not very informative. 15We also tested the influence of default risk on AAA spread by regressing them on fundamental determinants of default risk, including the equity volatility and leverage of the issuer, and found that these factors were statistically insignificant determinants of AAA spreads.

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demonstrate the presence of a systematic factor in credit spreads which appears to be unrelated to equity markets. Lower liquidity of corporate bonds relative to government bonds is also likely to 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 risk factors unrelated to the equity market. Thus, the calibration of the model to credit spreads must exclude the part of the spread unrelated to default risk. The magnitude of the tax, liquidity and bond-specific risk components of the spread are hard to estimate. For instance, Elton et al. estimate the component of the spread caused by differential state taxes on corporate and government bonds. Their estimates range from 29 to 50 basis points for the tax component of the AA spread. Even this large range might be questioned, as any positive tax spread could be 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 default and 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 not appear practical. We therefore need a variable that contains these components to adjust spreads so that they reflect only default-related factors. The evidence that the AAA spread does not contain a significant default risk component suggests that it reflects only non-default factors. So we could use the AAA spread to proxy these other factors as long as they are cross-sectionally constant. 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 on corporate 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. In Appendix B we derive such a simple formula. The yield spread due to tax is given by:

ytax = 1 ln tM

1- 1 - e-rtM tM

(11)

where: ytax is the spread due to tax, tM is the time to maturity, is the applicable tax rate and rtM is the riskless interest rate. This formula holds for any bond, as long as capital gains and losses are treated symmetrically and the capital gain tax rate is the same as the income tax on 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 I at which coupons are taxed does not coincide with the

capital

gains

tax

C,

formula

(11)

becomes

ytax

=

1 tM

ln

1- I

1-e-rtM tM ( C +( I - C )E[F /B])

. This is similar

to (11) when rtM tM is high or the risk-neutral expectation of the principal repayment is not very different from the purchasing price: E [F ] B. Another nuance is that in reality taxation rules for bonds originally sold

significantly 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 that

only a small part of the tax is paid at maturity. For such bonds formula (11 will also be an approximation.

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