estimating the cost of risky debt by ian cooper

7/29/2019 Estimating the Cost of Risky Debt by Ian cooper

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90 Journal of Applied Corporate Finance Volume 19 Number 3 A Morgan Stanley Publication Summer 2007

Estimating the Cost of Risky Debt

by Ian A. Cooper, London Business School, and Sergei A. Davydenko,niversity of Toronto*

his article proposes an easily implementedmethod for estimating the expected return onrisky debt, which is an integral part of calculat-ing the weighted average cost of capital (WACC).

The WACC is the required return on the operating assets ofa firm. It is used in valuation, capital budgeting, goal setting,

performance measurement, and regulation. Its value is oneof the most important issues in corporate finance. Yet littleresearch attention thus far has focused on estimating oneof its key inputsthe cost of debt. Existing methods oftenoverlook a crucial factor for the cost of debtthe possibilityof defaultand thus the use of such methods may result insignificant errors in WACC estimates.

he WACC cannot be observed directly and thereforemust be estimated. The standard estimation method takes aweighted average of the estimated expected returns on debtand equity:

WACC = pD rD (1 T)+ (1 pD) rE, (1)

where pD

is the market-value proportion of the debt of thefirm (market leverage), r

Dis is the required return or cost of

debt, T is the tax rate, and rE

is the required return or costof equity. he expected return on equity is:

rE

= r +E, (2)

where r is the risk-free rate and E

is the equity risk premium.The cost of equity is typically estimated using the capital assetpricing model (CAPM), AP , or variants of the dividend

growth model. Its estimation has been the subject of contro-versy.1 Similarly, the appropriate tax rate to use has also beenhe subject of debate.2 In the remainder of this article, we sethe tax rate to zero for simplicity.

By contrast, although risky debt has been studied widely,

here is no model in a form that can be applied easily toestimate the cost of debt for an individual firm. he mostcommon approach is to use the promised yield on the newlyissued debt of the firm as an estimate of the cost of debt inhe WACC.3 In theory, however, the expected return on debt

should reflect the promised yield net of any expected default

loss, which in turn is a function of the expected probability ofdefault. As Kaplan and Stein pointed out, Because of defaultriskexpected returns[on highly leveraged corporate debt] areundoubtedly lower than the promised returns.4 Thus, at leastfor a company with a material probability of default, the useof the promised yield could significantly overstate both thecost of debt and the WACC. In extreme cases, the use of thepromised yield as the cost of debt could even result in theestimated cost of debt exceeding the cost of equity.

If the promised yield is not used as the cost of debt,an alternative is to assume that the beta of the debt is zero,implying that the debt has a zero risk premium.5 owever,

his method also defies economic logic, because the debt riskpremium must be greater than zero unless the default risk isentirely diversifiable by investors, which is unlikely.

he expected return on risky debt thus is neither thepromised yield nor the riskless interest rate but instead must liesomewhere in between. The problem with obtaining an estimateof this expected return arises because the spread between thepromised yields on risky debt and riskless debt (with the samematurity, liquidity, and tax characteristics) consists of two parts.The first part reflects the expected loss from default. The secondpart is due to the expected return premium, which reflects theundiversifiable risk of the debt, so that:

Promised yield spread = Expected default lossExpected return premium. (3)

The expected default loss should not be included in the

T

*We thank Don Chew, James Gentr y, Ilya Strebulaev, Mika Vaihekoski, and Fan Yu for

their helpful comments. We are grateful to participants at the European Finance Asso-iation, European Financial Management Association, Financial Management Associa-

tion Europe, INQUIRE Europe, and Lancaster University Finance Workshop.

1. See Ivo Welch, Views of Financial Economists on the Equity Premium and onProfessional Controversies,Journal of Business, Vol. 73 (2000), pp. 501537; John

raham and Campbell Harvey, The Theory and Practice of Corporate Finance: Evidence

from the Field,Journal of Financial Economics, Vol. 60 (2001), pp. 187243.2. See Ian Cooper and Kjell Nyborg, Valuing the Debt Tax Shield,Journal of Applied

orporate Finance, (Spring 2007), pp. 5059.

3. See, for example, Tim Koller, Marc Goedhart, and David Wessels, Valuation (New

York: Wiley, 2005).4. Steven Kaplan and Jeremy Stein, How Risky Is the Debt in Highly Leveraged

Transactions? Journal of Financial Economics, Vol. 27 (1990), pp. 215245 (p.

21).5. Typical examples of this approach appear in Benjamin Esty, Improved Techniques

for Valuing Large-Scale Projects,Journal of Project Finance Spring 1999), pp. 925,

nd Carliss Baldwin, Technical Note on LBO Valuation (B), Harvard Business Schoolase note 9-902-005 (2001).


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91Journal of Applied Corporate Finance Volume 19 Number 3 A Morgan Stanley Publication Summer 2007

cost of debt because it is not part of the expected return.he cost of debt therefore is the promised yield, adjusted for

expected default losses as follows:

Cost of debt = Promised yield Yield equivalent of expected default loss. ( )

As noted above, the yield spread in (3) should be measuredrelative to a riskless bond with the same maturity, liquidity,and tax characteristics. A simple way of measuring this spreadis to calculate debt spreads relative to the AAA rate ratherhan the Treasury rate. Alternatively, the spread due to default

risk can be estimated from the credit default swap market ifCDS rates are available.

The Importance of Accurate Estimation of the

Cost of Risky Debthe bias in the WACC resulting from use of the promised

yield as the cost of debt depends on the proportion of theyield spread that is an expected return premium. If the entirespread is an expected return premium, it is correct to use thepromised yield. But because there must be some chance ofdefault for the debt to be risky, and because some part of thespread must reflect the expected default, the true cost of debties somewhere between the two extremes of the promised

yield and the riskless rate.To demonstrate that the difference between the promised

yield and the expected return on debt can have a material

impact on valuation, consider a highly leveraged transactionfunded with 70% debt, p

D= 70%. Suppose further that the

riskless real rate is 3%, the risk premium on the equity of thefirm is 6%, and the promised real return on the debt is 7%.The conventional WACC calculation would give:

Real WACC (1) = 0.7 % + 0.3 (3% + 6%) = 7.6%.

But if half of the debt premium of 4% (7% 3%) is reallycompensation for the probability of default, the true expectedreturn on the debt is 5%, and the WACC becomes:

Real WACC (2) = 0.7 5% + 0.3 (3% + 6%)=6.2%.

With a real WACC of 7.6%, the multiplier for a realperpetuity growing at 3% is 22, whereas with a real WACCof 6.2%, it is 31. Thus, the use of the wrong cost of debtestimate can have a material effect on valuation.

These errors in the WACC that can arise from usingconventional cost of debt estimates are most significant when

he debt is risky, which is also when the problem of estimatingexpected return is greatest. As Brealey and Myers put it: Thisis the bad news: There is no easy or tractable way of estimatinghe expected rate of return on most junk debt issues.

Alternative Approaches to Estimating theExpected Return on Risky DebtWhen promised yields do not provide a sufficiently accurateapproximation for the cost of debt, several other approachescan be used.

One possibility is to apply standard asset pricing modelsike the CAPM to risky debt. o illustrate, one study has

reported betas for high-yield debt of approximately 0.30.7 Ifhe market risk premium is 6%, these estimates imply debt

risk premia of approximately 180 basis points. However, thisapproach requires traded debt price series to estimate betas,

and such prices are often unavailable. Moreover, applying thismethod to an individual companys debt requires great carebecause debt betas naturally decline as debt matures. Finally,it is not clear that the standard CAPM provides the relevantrisk premium model. Taking these issues and the questionableprecision of market risk premium estimates into account, it isclear that implementing this approach is difficult.

A second approach estimates the frequency of defaultsand average recovery rates empirically, and then uses thoseestimates to adjust the promised yield to obtain the expectedreturn. This approach is typically used to study returns ondifferent bond rating classes.8 For example, one study used

historical data on recovery rates and rating migrations to splithe debt spread into three parts: expected default, taxes, and a

risk premium. The researchers argued that the risk premiumargely could be explained by common risk factors similaro those of equity risk premia. For a typical 10-year BBB

industrial bond, they found that 1 (or 3 %) basis points of aotal spread of 118 bp could be explained by expected default,eaving 77 bp as the yield spread that should be included inhe cost of capital.

Although it provides interesting guidelines on the cost ofdebt, this approach based on historical estimates can be usedonly when there is a long enough history of debt returns for a

sufficient number of similar bonds. Because they use no firm-specific information, the reliability of such estimates dependscritically on the assumption that similar bonds have thesame expected default rate and rate of return. Another disad-vantage of this method is that it is based solely on historicaldata and thus is not forward-looking. As many observers ofhigh-yield debt markets have suggested, historical defaultfrequencies can be very different from future probabilities,

6. Richard Brealey and Stewart Myers, orporate Finance, 6th ed. (New York: Mc-

raw-Hill, 2006) p. 515.7. Marshall Blume and Donald Keim, Lower-Grade Bonds: Their Risks and Returns,

Financial Analysts Journal, Vol. 43, (July/August 1987), pp. 2633.

8. See the vast related literature, beginning with Edward Altman, Measuring Corpo-

rate Bond Mortality and Performance, Journal of Finance, Vol. 44 (1989), pp. 909

22.9. Edward Elton, Martin Gruber, Deepak Agrawal, and Christopher Mann, Explaining

the Rate Spread on Corporate Bonds, Journal of Finance, Vol. 56 (2001), pp. 247

77.


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particularly when future economic and market conditionsikely will differ from those in the past.10

Proposed Method: Inferring the Cost of Debt from

Other Inputs to the WACChe approach to estimating the cost of debt that we recom-

mend is different. Specifically, it uses a risky debt pricingmodel to impute the expected rate of return on debt fromhe standard inputs into the WACC. Whereas other studies

proposed estimation methods that are similar in spirit, ourapproach has several distinct advantages when used in cost ofcapital estimation.11 Some methods rely on the assumptionhat the debt pricing model can correctly predict the level of

yields, although at present no model does. Others provideonly the risk-adjusted probability of default, not the trueprobability. Still others require complicated inputs that are

ypically not available when estimating the cost of capital.In contrast, our approach uses a procedure that, by

construction, is consistent with the observed debt yield ofhe company for which the cost of capital is being estimated.

Our proposed method sidesteps the well-known problem ofhe inaccuracy of existing debt pricing models (a problemhat doesnt trouble us because we are not concerned with

debt valuation per se but rather with inferring the proportionof the actual market spread attributable to expected default)

while at the same time enabling us to use inputs that are easilyobservable. Except for equity volatility, all the inputs requiredfor our procedure are standard inputs for the WACC. he

method is thus explicitly designed to yield the expected costof debt for use in the WACC.

The risky debt pricing model that we use was proposed byRobert Merton.12 When combined with estimates of expectedequity returns, the Merton model enables us to back out theexpected distribution of the value of the firms assets impliedby the observed market prices of debt and equity. This distri-bution can then be used to decompose the debt yield spreadinto compensation for the expected default and the expectedreturn prem um.

The Cost of Debt Implied by the Merton Model

The Merton model is the simplest equilibrium model of therelationship between corporate interest rates and inputs tohe WACC. It assumes that the value of the firms productive

assets, V, follows geometric Brownian motion:

dV/V = dt + dWt, (5)

where is the expected return on the firms assets, is thevolatility of assets, and W

tis a standard Wiener process.

In the Merton model, the firm has a single class of zerocoupon risky debt of maturity . In addition, the model

assumes a constant interest rate and a simple bankruptcyprocedure; namely, if at maturity the value of the assets isower than the liability, the assets are handed over to the

bondholders without costs or violation of priority rules.The simplicity of the model has led to difficulties in using

it to explain the relationship between the absolute level ofdebt spreads, capital structure, and asset volatility. In thecontext of this paper however, we are not interested in theabsolute level of the spread. Instead, we use the model simplyo divide the observed market spread between the portionhat represents expected default and the portion that is the

expected return premium. If the Merton model reflects at

east the first-order effects relevant for splitting the spreadon risky debt, it can be used to estimate the expected return,given the promised yield. For this purpose, it has the merit ofbeing a relatively simple equilibrium model.13, 14

Neither the expected return on assets nor the assetvolatility can be directly observed. Moreover, in thepresence of multiple issues of debt and debt with coupons andcall provisions, the value of debt maturity in the Mertonmodel is not well defined. Te idea behind our procedure iso find the values of , , and that reconcile the Merton

model with observed debt spreads, given the firms capitalstructure and other characteristics. After finding the implied

parameters of the asset distribution, we can compute theexpected return on debt, consistent with this distributionand the return on equity.

Merton applied the Black-Scholes option pricing formulao value equity as a call option on a firms assets. Mertons

formula can be written in a form that expresses the relation-ship among the firms leverage p

D, the maturity of the debt

, the volatility of the assets of the firm , and the promisedyield spread s:15

(1 pD

) = N(d1) p

De sT N(d

2, (6)

where N() is the cumulative normal distribut ionfunction, and:

d1

= ln pD

(s 2/2) / (7)

d2= d

1 . (8)

10. See Paul Asquith, David W. Mullins, and Eric D. Wolff, Original Issue High Yield

Bonds: Aging Analyses of Defaults, Exchanges, and Calls,Journal of Finance, Vol. 441989), pp. 923952.

11. A comprehensive list of related references appears in Long Chen, Pierre Collin-

Dufresne, and Robert Goldstien, On the Relation Between the Credit Spread Puzzle andthe Equity Premium Puzzle, Working paper, electronic copy available at http://ssrn.com/

bstract=687473 (2006).

12. Robert Merton, On the Pricing of Corporate Debt: The Risk Structure of Interest

Rates,Journal of Finance, Vol. 29 (1974), pp. 449470.13. A well-known industry application of the Merton (1974) model is used by MKMV

to infer the distance to default from observed equity prices.

14. A related approach was applied to obtain equity expected returns in Murilloampello, Long Chen, and Lu Zhang, Expected Returns, Yield Spreads, and Asset Pricing

Tests, Working paper, available electronically at http://papers.ssrn.com/sol3/papers.

fm?abstract_id=491403 (2006).


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Equation (6) includes two unknowns, and T. However,he Merton model also implies that the equity volatility

E

satisfies16

E = N(d1)/(1 pD . (9)

Therefore, we have three observable inputspD, s, and

E and two unknowns and T. We solve equations (6)

and (9) simultaneously to find values of and T that areconsistent with the observed values of s, p

D, and

E17 ssen-

ially, we compute as the implied volatility of the firmsassets when equity is viewed as a call option on the assets.The parameter T reflects not only the actual maturities ofdifferent debt issues in complex capital structures but alsohe presence of distress costs and other complications not

included in the Merton model but reflected in the observed

spreads.18After determining the values of and T, we can combine

hem with the estimate of the expected return on equity givenby equation (2) to calculate the expected return on assetsand debt. Because equity in this model is a call option onhe assets and therefore has the same underlying source of

risk as the assets, the risk premia on assets and equityE

are related:

/E=(-r)/(

E-r)= /

E. (10)

ubstituting equation (9) for E

yields:

= E 1 pD /N(d ). (11)

Now the part of the spread that is due to expected defaultcan be calculated as:19

= (1/ )ln e(-s)TN(d1

E /

E)/ p

D+

N (d2

+ E /

E) . (12)

This formula gives the annual yield due to expecteddefault losses on the debt. he cost of debt can be obtainedby subtracting it from the promised yield:

Cost of debt = Promised yield . (13)

In summary, the resulting cost of debt estimate is consis-ent with the promised yield on the firms debt and inputs to

its cost of capital.

Numerical Examplesable 1 shows the breakdown of the promised yield spread

between the expected return premium and the default risk forrepresentative values of p

D, s,

E, and

E. The values of p

Dand

s are similar to those used in a previous study of debt in 12arge leveraged recapitalizations.20 he equity risk premium

values are those commonly used, ranging between 5% and%. he volatility of equity is set at a typical level, ranging

from 30% to 50% per annum.Panel A of able 1 shows the results for an investment-

grade issuer, which roughly corresponds to A/BBB ratedbonds. he first row shows the base case with a proportionof debt in the capital structure of 30% and equity volatil-ity of 30% per annum. he equity risk premium is 6%per annum, and the debt spread relative to AAA bonds is100 basis points per annum. he expected default loss is

16 basis points, which is only a small part of the promisedyield spread (16%). As a result, the error from using thepromised yield as the cost of debt would be low in this case,consistent with evidence from studies of actual defaults oninvestment-grade bonds, such as those on the Standard &Poors and Moodys Web sites.

Panel B shows the results for a high-leverage firm with aproportion of debt at 70%. he volatility of equity is higher,at 50% per annum, and the debt spread is higher, at 400 basispoints. These values roughly correspond to bonds rated B. Forhe base case, less than half of the debt spread is an expected

return premium. Although the promised yield spread is four

imes greater than that in Panel A, the expected default lossis almost 16 times as high. The proportion of the yield spreadhat represents expected default is 62%. Therefore, the error

resulting from using the promised yield on debt as the costof debt would be substantial for a highly leveraged firm withhese parameter values.

Sensitivity AnalysisThe proportion of the spread that reflects expected returnis not very sensitive to inputs other than

E. In particular,

equation (12) shows that it does not depend at all on therisk-free interest rate. According to the results in able 1, it

also is relatively insensitive to realistic levels of variation inpD, s, and

E. he insensitivity to p

Dand

Es mportant,

because these variables are difficult to est imate precisely.Although it is sensitive to the volatility of equity, secondmoments of returns such as

Ecan be estimated relatively

15. Detailed derivations can be found in Ian Cooper and Sergei Davydenko, The Cost

f Debt, Working paper (2001), available electronically at: http://papers.ssrn.com/sol3/

papers.cfm?abstract_id=254974.16. In contrast to asset volatility, the short-term equity volatility is easily observable

from either option implied volatilities or analysis of historical returns data.

17. The system of equations is well behaved and generally can be solved by applyingstandard numerical methods. To ensure a starting point for which standard algorithms

uickly yield a solution, we can solve equations (6) and (9) separately for for a few

fixed values of T(or vice versa). This procedure always converges for any reasonable

starting points. The intersection of the solution curves T) from equations (6) and (9)

then can be used as the starting point for the system of equations.

18. The values of T derived from the system (6) (9) typically are higher than the trueebt maturity, a result of the Merton models failure to take into account bankruptcy

osts, strategic debt service, and other important aspects relevant for the expected return

n debt.19. Unlike the return on assets and equity, the calculated return on debt is an annual-

ized compounded return rather than an instantaneous return.

20. Kaplan and Stein (1990), cited previously.


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accurately. Thus, the proposed procedure has the merit ofbeing sensitive only to a parameter that can be observedrelatively accurately.21

ConclusionThe cost of equity and the cost of debt are two key inputs intohe weighted average cost of capital (WACC). The former

has been the subject of extensive debate, but little attentionhas focused on the latter. Two common ways to estimatehe cost of debt use the promised yield or the riskless rate,

but both yield biased results, and the errors could be mate-rial. Other estimation methods, such as using the CAPM oradjusting the promised yield for the expected frequency ofdefault, are hard to implement or lead to errors because theyfail to capture firm-specific information or current marketc rcumstances.

This article proposes a practical way to estimate the truecost of debt that sidesteps some of these problems. Using theMerton model of risky debt, the proposed approach splits

he promised yield spread into one portion due to expecteddefault and another portion that represents an expectedreturn premium. The inputs required are standard inputso the WACC and the volatility of equity, which are easily

observable. The proposed method uses these inputs to imputehe parameters of the Merton risky debt model and computehe expected return on debt. Although the Merton model

is a stylized version of real debt structures, it should reflectfirst-order effects that are relevant to the cost of debt, andherefore can be used to estimate the expected return relative

o the promised yield.We illustrate the approach by estimating the cost of debtfor parameter values typical to a firm with investment-gradedebt and another firm with high leverage, or below-invest-ment-grade debt. In the former scenario, most of the promisedyield spread is an expected return premium. In contrast, fora highly leveraged firm with low-grade debt, most of thepromised yield spread is expected default. he standardapproach of using the promised yield as the cost of debt

21. Some further evidence regarding the robustness of the Merton model appears intephen Schaefer and Ilya Strebulaev, Structural Models of Credit Risk Are Useful: Evi-

ence from Hedge Ratios on Corporate Bonds, Working paper, London Business School

2003).

Table 1 Promised yield spread and expected return on debtThe table shows the relationship between the promised debt yield and the expected default loss in the Merton model.

The inputs are the leverage ratio p , the promised spread on debt in basis points s, the equity risk premium , and the volatility

of equity . The outputs are the expected default loss in basis points and the proportion of the promised debt spread due to the

expected default /s. NA means that these input values are incompatible with the model. Sensitivity tests are underlined.

pD

s (b.p.) E

(% p.a.) E

(% p.a.) (b.p.) /s (%)

Panel A: Investment-grade debt firm

0.3 00 6.0 0.3 6 6.4

0.4 00 6.0 0.3 9 8.7

0.2 00 6.0 0.3 3 3.4

0.3 0 6.0 0.3 9 9.0

0.3 50 6.0 0.3 2 4.5

0.3 00 .0 0.3 3 3.4

0.3 00 7.0 0.3 1 1.1

0.3 00 6.0 0.2 NA NA0.3 00 6.0 0.4 2 41.5

Panel B: High-leverage firm

0.7 00 6.0 0.5 48 61.9

0.8 00 6.0 0.5 52 62.9

0.6 00 6.0 0.5 44 60.9

0.7 300 6.0 0.5 88 62.7

0.7 00 6.0 0.5 307 61.3

0.7 00 .0 0.5 70 67.5

0.7 00 7.0 0.5 27 6.7

0.7 00 6.0 0.4 73 43.3

0.7 00 6.0 0.6 92 73.0


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herefore may be adequate for companies with high-gradedebt, but for firms with lower-rated debt, it is likely to causea significant overstatement of the cost of debt and, hence, ofhe WACC. In these cases, the approach proposed in this

paper can be used to adjust the WACC for the probability ofdefault on the companys debt.

ian cooperis a Professor of Finance at London Business School.

sergei davydenko is an Assistant Professor of Finance at the Univer-

sity of Torontos Rotman School of Management.


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estimating the cost of risky debt by ian cooper

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