dividend yields, and default probabilities: linking the credit risk and equity literatures

7/30/2019 Dividend Yields, and Default Probabilities: Linking the Credit Risk and Equity Literatures

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Time Varying Expected Returns, StochasticDividend Yields, and Default Probabilities:

Linking the Credit Risk and Equity Literatures1

George Chacko Peter Hecht Jens Hilscher

Kite Partners Allstate Investments Brandeis University

First draft: September 2002

This version: September 2007

Abstract

Motivated by the equity literature, which has argued that returns are timevarying and that the dividend yield predicts future returns, this paper intro-duces time varying expected returns into a structural model of rm value.Our model produces a stochastic dividend yield in equilibrium. It impliessignicant dierences in the reaction of the default probability to dierentsources of news; a reduction in asset value due to discount rate news has

a smaller impact on the default probability than a reduction due to cashow news. In general, model implied default probabilities are less volatilethan those generated from a standard Merton (1974) model. The model isconsistent with the standard credit rating agencies view that focuses on fun-damental analysis, in addition to market information, when assessing creditrisk. It is particularly relevant in periods of overvaluation and low expectedreturns such as the recent technology bubble.

JEL classications: G13, G33, G35Keywords: Stochastic dividend yield, Default probabilities, Merton model,Bond pricing


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1 Introduction

What is the best way to measure changes in a particular rms default probability?Traditionally, credit rating agencies, such as Moodys and S&P, perform fundamentalanalysis to determine creditworthiness. This usually entails analyzing a rms account-ing nancials, calculating nancial ratios, and understanding its business model. Sincefundamental analysis relies heavily on accounting information, it is susceptible to pro-ducing a stale and potentially backward looking measure of credit risk.1

A second approach is to assess credit risk by interpreting market and accountinginformation in a manner consistent with the structural form Merton (1974) bond pricingmodel. In the model, expected returns and the growth rate of rm value are constant.At maturity, the rm defaults if rm value lies below the face value of debt. Thisapproach yields a default probability that moves with market based information athigh frequency,2 while fundamental analysis provides no incremental information aboutdefault.

In this paper we introduce time varying expected returns and dividend yields into astructural rm value model and consider the implications for default probabilities. Re-laxing the assumption of constant expected returns, which is common to most corporatebond pricing models, is motivated by the ndings in the equity literature which arguesthat expected returns and dividend yields are time varying and persistent and thatthe dividend yield predicts future returns.3 In this setting rm value and the defaultprobability move in response to two types of news discount rate and cash ow news.Distinguishing between these price changes is important because they may have very

dierent impacts on the default probability: If expected returns are time varying andmean-reverting, a reduction in asset value due to discount rate news will have a smallerimpact on the default probability than a reduction due to cash ow news. Defaultprobabilities in our model are less volatile, overcoming a major weakness of standardMerton-type models emphasized by the credit rating agencies. The model is consistentwith the standard credit rating agencies view that focuses on fundamental analysis, inaddition to market information, when assessing credit risk.

We begin our analysis by pointing out the restrictions a constant dividend yield placeson the possible causes of price movements. Either expected returns and dividend growthrates must be constant, or movements in expected returns must be exactly oset bydividend growth rate movements, keeping the dividend yield constant. Expected returnnews can never independently cause price movements. Therefore, only fundamentalsinduced price changes mo e the probabilit of default in models ith a constant di idend


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Since price movements are no longer a sucient statistic for changes in default proba-

bilities when the model allows for time varying expected returns via a stochastic dividendyield, our results suggest a role for both fundamental- and market-based measures ofcredit risk. The role of fundamental-based measures of credit risk will be most impor-tant for rms with discount rates (dividend yields) that display low persistence. It willalso be particularly relevant in periods of overvaluation and low expected returns suchas the recent technology bubble.

Our model also has implications for risk neutral default probabilities, i.e. bondprices. Unlike the constant dividend yield, continuous time pricing models of Black andScholes (1973) and Merton (1974), we calculate risk neutral default probabilities andbond prices in a discrete time model.5 In standard structural bond pricing models, therm defaults if the market value of assets lies below the face value of debt at maturity.In the Merton model the rm is costlessly liquidated in the case of default. Risky debtis then equivalent to the combination of a safe bond and a short put option on the rmsassets with a strike price equal to the face value of debt. We calculate bond prices usingour models rm value process while maintaining the assumption of default at maturity

and costless liquidation.

Similar to the true default probability results, we nd that movements in the un-derlyings asset value can contain a varying degree of information about changes in riskneutral default probabilities, depending on how much of the underlyings price move-ment was due to news about cash ows (vs. the discount rate). However, in contrastto the true default probability results, our results suggest underlying asset price changesdue to expected return news play a more important role when calculating risk neutral

default probabilities since expected return news moves the dividend yield and, thus, theasset value risk neutral drift. News about cash ows only aects the price level withoutchanging the dividend yield.6

Our model brings insights from the equity literature and applies them to credit riskand risky debt pricing. If expected dividend growth rates are constant, time varyingexpected returns must imply a time varying dividend yield. The related equity literatureis large and rich. It argues that expected returns and dividend yields are time varying

and persistent, that the dividend yield predicts future returns, that price movements arethe result of both cash ow (dividend) and expected return news, and that stock pricevolatility is dierent across dierent horizons.7 Another important insight is that theempirical ndings in the equity literature are internally consistent with standard presentvalue models, which produce the mechanical relationship between the current dividend


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yield and future expected returns and dividend growth rates.8

There is also a large related credit risk literature. Our work is related to manystudies that have considered the determinants of default probabilities.9 The rm valueprocess we specify is related to the large and rich literature on structural bond pricingmodels.10 One thing that all of the these models have in common is the assumption ofconstant expected returns and/or constant dividend yield.11

Our approach to modeling the rm value process is also related to Bekaert andGrenadier (2001) who price stocks by specifying processes for state variables and the

stochastic discount factor in an a discrete time, ane economy. Our approach ismore reduced form since we specify a price process. However, similar to Bekaert andGrenadier, we make use of the no arbitrage condition and nd that the dividend yieldwill be stochastic in equilibrium if expected returns are time varying.

The rest of paper is divided into ve sections. Since the stochastic dividend yield iscritical to our results, Section 2 explains why time-varying expected returns forces us toabandon the standard constant dividend yield assumption from a theoretical perspec-

tive. Section 3 motivates our models main assumption from an empirical perspectiveby documenting the dividend yield time series behavior for the rms within the S&P500. Section 4 describes the simple model used to generate the underlying asset pricemovements and, thus, estimate true and risk neutral default probabilities. Section 5discusses the analytical and numerical results, and Section 6 concludes.

2 Implications of a constant dividend yield for ex-pected returns

Is a Constant Dividend Yield Consistent With Time-Varying Expected Returns andConstant Expected Dividend Growth?

Before we move to the actual model that will generate prices and default proba-bilities, it is important to understand why we move away from the typical constant

dividend yield assumption. On the one hand, we would like our economy to exhibittime-varying expected returns (discount rates) and constant expected dividend growth,properties that are both reasonable on economic grounds and consistent with manyempirical ndings. On the other hand, we would like our model to be as similar as

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possible to the standard Merton model. From a pure statistical standpoint, there is no

problem with simultaneously having time-varying expected returns, constant expecteddividend growth, and a constant dividend yield. However, from an economic stand-point, is a model with time-varying expected returns and constant expected dividendgrowth economically consistent with a constant dividend yield? What types of economicconstraints does a constant dividend yield assumption place on a model? We brieyinvestigate these questions in the following section.

Using only the denition of return, we rst derive necessary conditions for a constantdividend yield model. Rewriting the denition of return,

Rt+1 =Pt+1+Dt+1

Pt

Rt+1 =Dt+1

Dt

Pt+1Dt+1

DtPt

+ DtPt

If the dividend yield is constant, this reduces to

Rt+1 =Dt+1

Dt

1 +DP

log(Rt+1) = log

Dt+1Dt

+ log

1 + D

P

(1)

Thus, when the dividend yield is constant, realized returns and realized dividendgrowth are perfectly positively correlated. The same is true for log returns and logdividend growth. For logs, the volatilities are identical, and revisions in future expecteddividend growth rates and revisions in future expected returns are identical too. So, any

news about future expected dividend growth rates are exactly oset by future expectedreturns, by construction. Thus, 100% of return variation is explained by the contempo-raneous dividend growth. It is impossible to even entertain the possibility that a shockto expected returns moved prices since this shock is always exactly oset by a shockto future expected dividend growth. News about the future is irrelevant. Only thecurrent shock to dividends can move prices. All models with a constant dividend yieldassumption place these extreme restrictions on the underlying economic processes.

If we add in the assumption that expected log dividend growth is constant, then itis impossible to have an economy with time-varying expected returns and a constantdividend yield. Even without the constant expected dividend growth assumption, asshown above, it is impossible to identify price movements that are due to movementsin expected returns (discount rates). In eect, all movements in prices must be "fun-damental" when the constant dividend yield assumption is imposed on an economic


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expected returns will cause the dividend yield to change. Thus, the stochastic dividendyield assumption used in this model should not be viewed as a standalone exogenousassumption. It is a result - a result that must follow from a model that wants tobe economically consistent with time-varying expected returns and constant expecteddividend growth. Since the dividend yield will pick up movements in discount rates,it will be important to watch both price movements and movements in the dividendyield. Movements in prices that occur without a change in the dividend yield are, in asense, "fundamental". In contrast, movements in prices that occur in combination witha movement in dividend yields will partially reect the movement in expected returns,

a "non-fundamental" source of price variation. In the extreme case where prices movewithout any move in the level of the dividend, the price move will be entirely "non-fundamental". These identications provide the foundation for our numerical results.

3 Dividend yields for S&P 500 rms

In the last section, we showed, from a theoretical perspective, that a model with timevarying expected returns must have a stochastic dividend yield - the key feature of ournew model. In this section, we look at the time series properties of the asset dividendyield for the aggregate S&P 500 and the asset dividend yield13 for individual S&P 500rms actually observed in the data. In addition to empirically justifying the mainfeature of our model, we will use the rm level statistics as inputs to the calibrationexercises performed in Section 5. For both the aggregate S&P 500 and the typical S&P

500 rm, we nd that the asset dividend yield displays large variation through time andis highly autocorrelated.

In order to construct the asset dividend yield series for the S&P 500 rms, we con-struct payout and asset value series. Payout is measured as the sum of dividends oncommon equity, dividends on preferred equity, and interest payments. Asset value isequal to the sum of common equity value, preferred equity value, and debt value. Weuse monthly data from CRSP and annual data from COMPUSTAT to construct these

series. We use book value of preferred equity as well as debt to proxy for their marketvalues. Book value of debt is equal to long term debt plus debt in current liabilities.We measure variables at an annual frequency and cover the period from 1980-2002. Theasset dividend yield is dened as the payout over the period divided by the end of periodasset value. Because this data is likely to be more familiar, we also calculate the equity


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dividend yield series. It is equal to dividend payments paid during the period dividedby the end of period value of common equity. To deal with outliers in the rm levelseries, we winsorize the payout yield at the 0.5% level of the entire distribution, i.e. wereplace values below the 0.5th percentile with the 0.5th percentile, and values above the99.5th percentile with the 99.5th percentile.

We would like to consider data for both the index as well as individual rms. How-ever, we cannot directly observe the aggregate payout of S&P 500 rms. In addition,the composition of the S&P 500 changed over the 1980-2002 period. We therefore col-lect data on rms that were in the S&P in 1990. We then construct aggregate series bytaking cross-sectional averages. We use data from CRSP and/or COMPUSTAT overthe 1980-2002 period. We include rms with scal years ending in December. Thisleaves us with a little over 7,000 rm year observations. We construct the aggregateS&P 500 data series by calculating cross sectional means of this data.14

Figures 1 and 2 plot the aggregate equity and asset dividend yield, respectively. Wecan immediately see that both the equity and asset dividend yield are highly seriallycorrelated and persistent. Data points at the beginning and end of the series suggestthat both series are mean reverting. Table 1, Panel A provides summary statistics.The mean asset and equity aggregate dividend yield are large at 4.52% and 3.38%,respectively. The unconditional asset and equity aggregate dividend yield volatilityare 1.50% and 1.25%, respectively. Table 2, Panel A reports summary statistics fromrunning AR(1) regressions of the log asset dividend yield. The log asset dividend yieldprocess is highly persistent with an AR(1) coecient of 0.97. Shocks to the dividendyield process are volatile at 8.48%.

Figures 3, 4, and 5 graph the rm level asset dividend yield for three S&P 500 rms:FirstEnergy, Merrill Lynch, and PACCAR. Similar to the aggregate series, the rmlevel dividend yield is highly serially correlated and persistent. The entire pooled rmlevel summary statistics are reported in Panel B of Tables 1 and 2. When looking at therm level, the mean asset and equity dividend yield across all rm years are similar inmagnitude to the aggregate results at 4.72% and 3.27%, respectively. The unconditionalasset and equity aggregate dividend yield volatility across all rm years are 3.05% and

2.57%, respectively, displaying both within rm and across rm variation. Although thelog asset dividend yield process is less persistent at the individual rm level, the medianrm is still highly persistent with an AR(1) coecient of 0.66. Relative to the aggregateresults, the shocks to the median rm level dividend yield process are much more volatileat 23.61%. This result should not be surprising considering that the aggregate index isa portfolio and thus exhibits diversication-type eects


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substantially more volatile than its aggregate counterpart. While modeling the assetdividend yield as an AR(1) is a simplication, it provides a reasonable, parsimoniousapproximation.

Clearly, the constant dividend yield assumption is inconsistent with the data in arst-order manner. Allowing the dividend yield to be stochastic and persistent providesa much better t. These empirical features are the foundation of our theoretical modeland calibration exercises.

4 A simple model of rm value

This section provides the processes, distributional assumptions, and parameter restric-tions that generate the data in our economy. It is important to keep in mind that thismodel is not meant to be all encompassing. For example, the model does not incorpo-rate stochastic interest rates and stochastic volatility. This exclusion does not intend tosay that these properties are unimportant. The model is simply a vehicle to highlightthe stochastic dividend yield (induced by time-varying expected returns) channel andalso help provide a transparent map between dierent types of price movements anddefault probabilities.

4.1 Firm value process

In order to generate true default probabilities, risk neutral default probabilities, andbond prices, we assume the log rm price growth (pt+1 = log

Pt+1

Pt

), log dividend yield

(t+1 = log

Dt+1Pt+1

), and log stochastic discount factor (log SDF, mt+1 = log (Mt+1)) are

conditionally multivariate normal with the following distributional parameters:

pt+1 = p;t + "p;t+1 ; V art[pt+1] = 2p

t+1 = + (t

) + ";t+1 ; V art[t+1] = 2

mt+1 = rf 122m + "m;t+1 ; V art[mt+1] = 2mCovt[mt+1; pt+1] = mp;t ; Covt[mt+1; t+1] = m

Covt[t+1; pt+1] = p

(2)

The processes for the log price growth and log dividend yield imply a process for the log

dividend growth (dt+1 = log

Dt+1

) via an accounting identity


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0 < < 1. Out of all of the specied covariances, only the covariance between the logSDF and log price growth is time-varying. The risk free rate and expected log dividendgrowth are assumed to be constant.

Before moving further, it is important to note that this reduced form model forthe price and dividend yield process can be justied from a more general equilibriumperspective. Given the dividend, SDF, and market price of dividend risk processes,the price process is exactly determined by the expected innite sum of stochasticallydiscounted future dividends. If the log dividend growth, log SDF, and market priceof dividend risk processes are conditionally multivariate normal with the market priceof dividend risk following an AR(1) process, it can be shown that the equilibrium logdividend yield process follows a conditionally normal AR(1) process to a rst order ap-proximation.15 The log dividend yield process inherits the dynamics of the market priceof dividend risk process, consistent with the discussion in Section 2 of this paper. Sincethe log dividend growth and log dividend yield processes are conditionally multivariatenormal to a rst order approximation, by denition, the log price process also followsa conditionally normal process to a rst order approximation. This more primitive

underlying model justies the reduced form model used throughout this paper.Many of the parameters from our reduced form model are constrained by distribu-

tional assumptions, the no-arbitrage condition associated with the existence of an SDF,and accounting identities. First, by no-arbitrage, the expected value of the rms returnmultiplied by the SDF must be 1.

1 = Et [Mt+1Rt+1]

1 = Eth

Mt+1 Pt+1

Pt

1 + Dt+1

Pt+1

i(4)

Rearranging and using our distributional assumptions,

1 = Et [exp(mt+1 + pt+1)] + Et [exp(mt+1 + pt+1 + t+1)]1 = exp rf + p;t + mp;t +

12

2p 1 + exp + (t ) +12

2 + m + p0 = rf + p;t + mp;t + 122p + log 1 + exp + (t ) + 122 + m + p

p;t = rf mp;t 122p log

1 + exp

+ (t ) + 122 + m + p

(5)

This is one restriction placed on the parameters of the model. Another restric-tion results from the constant expected log dividend growth assumption. Using the


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pt+1 = dt+1

t+1 + tp;t = d ( + (t )) + t

p;t = d + (1 ) (t )(6)

Equating the two expressions for the expected log price growth, we nd that ourmodel implies that movements in the log dividend yield are directly related to the marketprice of price risk, mp;t. Given the discussion in Section 2 of the paper relatingdividend yields and expected returns in addition to the underlying general equilibrium

discussion above, this nding is not too surprising.

mp;t = d (1 ) (t ) + rf 122p log

1 + exp

+ (t ) + 122 + m + p

(7)

This relationship between mp;t and t is easier to see if we linearize the log dividendyield around its long run mean.

mp;t = d (1 ) (t ) + rf 122p (a1 + a2 (t ))mp;t = d (1 ) t + (1 ) + rf 122p a1 a2t + a2

mp;t = (1 + a2) t d + (1 + a2) + rf 122p a1(8)

where a1 and a2 are linearization constants

a1 = log

1 + exp

+12

2 + m + p

a2 = exp(+ 122+m+p)1+exp(+ 122+m+p)

(9)

Since 1 + a2 > 0, a higher mp;t (lower risk) corresponds to a lower log dividendyield. The log dividend yield is perfectly correlated with risk (mp;t) to a rst orderapproximation. Thus, to a rst order approximation, since the log dividend yield followsan AR(1) with persistence , the process for risk (mp;t) will also follow an AR(1) withpersistence . By assuming a constant expected log dividend growth, assuming aprocess for the log dividend yield is equivalent to assuming a process for risk (mp;t).Reenforcing the discussion from Section 2 and the general equilibrium discussion above,time-varying risk has a one-to-one map with the dividend yield and, thus, is economically


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4.2 Adjusting the rm value process to the risk neutral mea-

sure

All of the analysis from the previous section relates back to the natural measure. Inorder to calculate risk neutral probabilities and bond prices, we will need to specifythe risk neutral processes for the log price growth and log dividend yield. Based onthe conditional multivariate normality assumption from before, it can be shown (e.g.Brennan (1979)) that the log price growth and log dividend yield are still conditionallymultivariate normal with the same conditional covariance matrix under the risk neutral

measure. However, the new conditional means are equal to the original conditionalmeans plus a risk correction term, i.e. the covariance between the particular shockand the log SDF.

EQt [pt+1] = p;t + mp;t

EQt [t+1] = ( + (t )) + m = +

m1 + t +

m1

(10)

Under the risk neutral measure, the log dividend yield still follows a stationaryAR(1), but it now has a new long run mean, +

m1

. The no-arbitrage conditionderived before for the rm allows us to conveniently substitute out the expression forthe log price growths risk neutral conditional mean.

p;t = rf mp;t 1

2

2

p log 1 + exp + (t ) +1

2

2

+ m + p

p;t + mp;t = rf 122p log

1 + exp

+ (t ) + 122 + m + p

(11)

Now, all of the relevant processes have been specied under the natural and riskneutral measure.

4.3 Calculating default probabilities

We now consider a zero coupon bond where default is dened by the normal Merton(1974) boundary conditions. In the standard Merton setup, the bondholders receivetheir promised payment (I) at maturity t + j if the value of the rm is greater than orequal to I. Otherwise, the rm liquidates, and bondholders receive the value of the


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ution of rm value at the bonds maturity date conditional on todays information set.Working in logs and using the accounting identity for log prices,

log(Pt+j ) = log (Pt) +Pj

k=1 pt+k= log (Pt) +jd + (1 )

Pj1k=0 (t+k ) +

Pjk=1 "p;t+k

= log (Pt) +jd + (1 )Pj1

k=0 k (t ) +

Pj1k=1

1jk

1

";t+k

+Pj

k=1 "p;t+k

= log (Pt) +jd + (1 )(t )1j

1 + Pj1k=1

1jk

1 ";t+k + Pj

k=1 "p;t+k:

(13)We can reduce this further:

log(Pt+j ) = log (Pt) +jd + (t )

1 j

+Pj1

k=1

1 jk

";t+k +

Pjk=1 "p;t+k:

(14)Since the log price at maturity is just the sum of i.i.d. multivariate normal shocks, the

log price at maturity is normal too. Lets calculate the mean and variance.

Et [log (Pt+j )] = log (Pt) +jd + (t )

1 j

V art [log (Pt+j )] = 2

Pj1k=1

1 jk

2+j2

p+ 2p

Pj1k=1

1 jk

(15)Given the mean and variance of the normally distributed log price at maturity, we caneasily calculate default probabilities in closed form.

T DP =

log(I)Et[log(Pt+j)]p

V art[log(Pt+j)]

(16)

where () represents the standard normal cdf and T DP represents the true defaultprobability.

We can go through a similar exercise to calculate the risk neutral probability. How-

ever, since the risk neutral conditional mean for the log price (refer to Section 3.2) is anonlinear function of a normally distributed random variable (t), the log price at matu-rity will not be normally distributed. Thus, we will have to simulate the processes underthe risk neutral measure and estimate the risk neutral default probability (RNDP) fromthe simulations.


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As with the risk neutral default probability calculation, we can not solve for the bondprice in closed form since the log price at maturity does not have a normal distribution.The bond price can be obtained via simulation methods too.

5 Economic magnitude of changes in default proba-

bilities

This section analytically measures the reaction of the true and risk neutral probabilityof default in response to cash ow versus expected return news. In order to gaugethe economic signicance of the results, we also calibrate our model using the S&P 500rm level parameters estimated in Section 3. Cash ow news is identied as a pricemovement with no change in the dividend yield while expected return news is identiedas a price movement with no change in the level of the dividend, i.e. an opposite move inthe dividend yield. In general, we nd that true default probabilities are more sensitiveto cash ow news relative to expected return news, especially when the dividend yieldprocess is less persistent. In contrast, we nd that risk neutral default probabilities aremore sensitive to expected return news relative to cash ow news since expected returnnews moves the dividend yield and, thus, the risk neutral drift.

5.1 Sensitivity of the default probability to dierent sources of

news

In the last section, we showed that under the natural measure

Et [log (Pt+j )] = log (Pt) +jd + (t )

1 j

V art [log (Pt+j )] = 2

Pj1k=1

1 jk

2+j2

p+ 2p

Pj1k=1

1 jk

(18)Given the mean and variance of the normally distributed log price at maturity, in thelast section, we also calculated default probabilities in closed form.

T DP =

log(I)Et[log(Pt+j)]p

V art[log(Pt+j)]

(19)

where () represents the standard normal cdf and T DP represents the true default


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Et [log (Pt+j )] = log (Dt)

t +jd + (t

) 1

jEt [log (Pt+j )] = log (Dt) + (t) +jd ((t) + ) 1 j (20)

We are interested in measuring the default sensitivity to a cash ow-induced pricechange, i.e. movement in log(Dt), and an expected return-induced price change, i.e.movement in t. Thus, we need to calculate

@TDP@log(Dt)

and @TDP@[t]

. Since we are consideringprice increases, we take the partial derivative with respect to the negative log dividendyield. Holding the level of the dividend constant, a decrease in the dividend yieldcorresponds to an increase in price. For the cash ow-induced price move,

@TDP@log(Dt)

= n

log(I)Et[log(Pt+j)]pV art[log(Pt+j)]

@Et[log(Pt+j)]

@log(Dt)

= n


(21)

where n () represents the standard normal pdf. For the expected return-induced price

move,

@TDP@[t]

= n


@Et[log(Pt+j)]

@[t]

= n


1

1 j

= n


j

(22)

The default probability is more sensitive to a cash ow-induced price movement.Changes in expected returns are temporary, having a smaller impact on default prob-abilities. For rms with more persistent expected return processes, i.e. higher , thiseect is mitigated since the mean reversion is slower. Also, for rms with shorter termdebt, i.e. lower j, this eect is mitigated since less mean reversion can occur over ashorter time period.

Unfortunately, we cannot perform a similar closed form analysis for the risk neutralprobability of default. Since the risk neutral conditional mean for the log price isa nonlinear function of a normally distributed random variable (t), the log price atmaturity will not be normally distributed. However, if we linearize the conditional mean,the log price at maturity will be normally distributed to a rst order approximation. To

id i i ht d t l t th t d f lt b bilit lt f


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EQt [log (Pt+j)] = log (Pt) +j

rf 122p a1 a2 t + m1 1j1

= log (Dt) t +j

rf 122p a1 a2

t

+

m1

1j

1

= log (Dt) + (t) +j

rf 122p a1

+ a2

(t) +

+

m1

1j

1

V arQt [log (Pt+j )] = (a

2)2

2Pj1

k=1

1jk

1

2+j2p 2a2p

Pj1k=1

1jk

1

(23)

where

= +m1

a1 = log (1 + A exp(

))

a2 =A exp()

1+A exp()A = exp

m + p +

12

2

(24)

Botha1 and

a2 are the linearization constants.

Similar to the true default probability analysis, we can easily calculate risk neutraldefault probabilities in closed form.

RNDP =

log(I)EQt [log(Pt+j)]p

V arQt [log(Pt+j)]

(25)

where RNDP represents the risk neutral default probability. The sensitivity of the

risk neutral default probability to a cash ow-induced price movement ( @RNDP@log(Dt)) andexpected return-induced price movement ( @RNDP

@[t]) is the following:

@RNDP@log(Dt)

= n

log(I)EQt [log(Pt+j)]p

V arQt [log(Pt+j)]

@E

Qt [log(Pt+j)]

@log(Dt)

= n

log(I)EQt [log(Pt+j)]pV ar

Qt [log(Pt+j)]

(26)

and

@RNDP@[t]

= n

log(I)EQt [log(Pt+j)]pV ar

Qt [log(Pt+j)]

@E

Qt [log(Pt+j)]

@log(Dt)

n

log(I)EQt [log(Pt+j)]

1 + a

1j

(27)


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yield is now lower. This eect does not occur for a cash ow-induced price movement.For rms with more persistent expected return processes, i.e. higher , this eect isexacerbated since future risk neutral price drifts are more likely to be aected by theoriginal expected return shock that moved the dividend yield. Also, for rms withlonger term debt, i.e. higher j, this eect is greater since the original expected returnshock that moved the dividend yield will have a larger cumulative eect over a longertime period.

5.2 Predicted changes in default probabilities for S&P 500rms

In the last section, we analytically derived the true and risk neutral default probabilitysensitivities to a cash ow and expected return-induced price movement. Cash ow-induced price movements are more important for the true default probability whileexpected return-induced price movements are more important for the risk neutral defaultprobability, i.e. bond prices. In this section we calibrate our model to the actualdata, providing a means to judge and measure the economic signicance of our results.Also, since the analytical results for the risk neutral default probabilities were onlyapproximate, this section provides additional support for our sensitivity results.

Most of the parameters were estimated using the actual rm level S&P 500 datadescribed in Section 3 and reported in Tables 1 and 2, Panel B. For all of the numericalresults, we used the following median annualized parameters: time to maturity (j) =5 years, exp(rf)

1 = 2.0%, m = 0, p = 30%, = 24%,

p

p

= -0.63, =

-3.16, d = 0.054, face value of debt (I) = 2,722, = 0.66, Pt = 10,533. For theholding D/P constant results, i.e. cash ow-induced price movement, we use a t= -3.42, the December 2002 value for the median S&P 500 rm. For the holding Dconstant results, i.e. expected return-induced price movement, we use log(Dt) = 5.84,the implied number from the December 2002 value of t (= -3.42) and Pt (= 10,533)for the median S&P 500 rm. We set the market price of dividend yield risk (m) tozero in order to emphasize that this new risk premium was not important to our main

results. The 30% annual rm level asset price volatility can be implied from the medianS&P 500 rm leverage, a stock price volatility of 40%, a risky bond price volatility of10%, and a 0.1 correlation between stock and bond prices.

Given the median S&P 500 rm level parameters described above, we calculate thechange in default probability to various instantaneous moves in the underlying asset


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i.e. volatility increases with the square root of time. This exercise produces a 1.89%,4.16%, and 8.66% for the daily, weekly, and monthly price volatility, respectively.16 In

addition to the median S&P 500 rm level persistence parameter = 0.66, we alsoshow results for lower and higher values: = 0.00, 0.25, 0.75, 0.95. When calculatingthe risk neutral default probability, we simulate 1,000,000 draws from the actual riskneutral distribution. No rst order approximation is used. Consistent with the previoussections, cash ow news is identied as a price movement with no change in the dividendyield while expected return news is identied as a price movement with no change inthe level of the dividend, i.e. an opposite move in the dividend yield.

Table 3 reports the calibration results for the true default probability (TDP). Foreach of the dividend yield persistence parameters, the true default probability is moresensitive to a cash ow-induced price move. The percentage change in the defaultprobability is always higher for the cash ow-induced price move. This general resultis most dramatic when the persistence parameter is low. In fact, when = 0.00, anexpected return-induced price movement has no impact on the default probability. Evenwith = 0.25, an expected return-induced price movement still has almost no impact

on the default probability.The results for = 0.66, the median rm level S&P 500 value, are most relevant for

gauging economic signicance. For a price drop equivalent to a one standard deviationdaily move, the TDP increases 9.7682% in response to a cash ow shock while onlyincreasing 1.1797% for an expected return shock. In other words, the TDP sensitivityto a cash ow-induced price movement is 8.28 times higher. These results becomeeven more striking when considering a price drop equivalent to a one standard deviation

annual move. The TDP increases 382.6429% in response to a cash ow shock whileonly dropping 24.1806% for an expected return shock, making the TDP sensitivity to acash ow-induced price movement 15.82 times higher. From the perspective of a bondrating agency or any institution interested in predicting default, there is a signicantdierence between a 3.3519% and 0.8624% default probability, which correspond to the"cash ow" TDP and "expected return" TDP level for the "annual shock" to the price,respectively. Figure 6 provides a graphical representation of our main result: truedefault probabilities are more sensitive to cash ow-induced price movements.

Table 4 reports the calibration results for the risk neutral default probability (RNDP).In contrast to the results for the TDP, for each of the dividend yield persistence para-meters (excluding = 0.00), the risk neutral default probability is more sensitive to aexpected return-induced price move. There is no dierence between the default sen-sitivities for the = 0 00 case since any move in the current dividend yield due to an


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= 0.25 case, it is interesting to note that the dierences between the "cash ow" and"expected return" RNDP sensitivities are quite small. In contrast to the TDP results,

the RNDP general result is most dramatic when the persistence parameter is high. Forexample, for the = 0.95 case, a one standard deviation annual price decline increasesthe RNDP 181.6663% for the expected return-induced movement and 143.9657% forthe cash ow-induced movement. All of the numerical results for the RNDP reinforceand complement the approximate rst order analytical results reported in the previoussection.

Similar to the TDP results, the results for = 0.66, the median rm level S&P 500

value, are most relevant for gauging economic signicance. For a price drop equivalentto a one standard deviation daily move, the RNDP increases 5.7206% in response to aexpected return shock while only increasing 5.2479% for a cash ow shock. In otherwords, the RNDP sensitivity to a expected return-induced price movement is only 1.09times higher. In absolute terms, these dierences are more pronounced when consideringa price drop equivalent to a one standard deviation annual move. The RNDP increases161.6567% in response to a expected return shock while only increasing 146.3626% for a

cash ow shock. In relative terms, however, the RNDP sensitivity to a expected return-induced price movement is only 1.10 times higher. From a bond valuation perspective,there is still signicant dierence between a 16.9373% and 15.9473% risk neutral defaultprobability, which correspond to the "expected return" RNDP and "cash ow" RNDPlevel for the "annual shock" to the price, respectively. Under the assumption of zerorecovery for simplicity, these RNDP dierences (16.9373% vs. 15.9473%) correspond tooverpaying for the bond by 1.19% if the price move is incorrectly assumed to be cashow-induced. Relative to the TDP results, the RNDP results are less dramatic, butstill economically signicant.

6 Conclusion

Most credit risk measures produced by traditional credit agencies were too slow to reactto the multiple, high prole, corporate bankruptcies of the recent past Many see this

as a failure of fundamental-based measures of credit risk.

During the tech boom of the 1990s, asset valuations sky-rocketed, predicting un-believably low default rates via a standard Merton model. Many see this as a failure ofmarket-based measures of credit risk.

Thi d l f d f l h l i d id l i


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the dividend yield identify changes in the discount rate. Asset prices are no longer asucient statistic for fundamentals.

The model predicts that asset price movements that are driven by fundamentals,i.e. price moves that occur with no change in the dividend yield, have a greater impacton true default probabilities than asset price movements due to changes in discountrates, i.e. price moves that occur with no change in the level of the dividend and, thus, achange in the dividend yield. Intuitively, movements in the discount rate are expectedto mean-revert, causing expected price growth to move in the opposite direction of thecurrent price change. This osetting move in expected price growth dampens the eect

that the current price change has on the default probability. The osetting expectedprice growth is largest when the change in discount rate is less persistent. In the extremecase where discount rate changes display no persistence, movements in asset prices dueto a discount rate shock contain no information about changes in default probability.

Our stochastic dividend yield model also has implications for risk neutral defaultprobabilities, i.e. bond prices. Similar to the true default probability results, we ndthat movements in the underlyings asset value can contain a varying degree of informa-

tion about changes in risk neutral default probabilities, depending on how much of theunderlyings price movement was due to news about cash ows (vs. the discount rate).However, in contrast to the true default probability results, our results suggest under-lying asset price changes due to expected return news (or "non-fundamentals") play amore important role when calculating risk neutral default probabilities since expectedreturn news moves the dividend yield and, thus, the asset value risk neutral drift. Newsabout cash ows only aects the price level without changing the dividend yield.

The results for both the true and risk neutral default probabilities are more thanjust analytically signicant. Using parameters estimated from actual S&P 500 rmlevel data, the model produces economically signicant results too. However, the truedefault probability results are much more dramatic.

The tech boom of the 1990s was a time where prices and dividend yields movedin opposite directions, suggesting a discount rate-induced price change and, thus, aprice change that contained little information about default probabilities. The recent,

high prole bankruptcies were preceded by price drops with little movements in cashow yields, suggesting a "fundamental"-induced price change and, thus, a price changethat contained substantial information about default probabilities. In sum, since pricechanges contain varying degrees of information about default, our model provides a rolefor both fundamental-based and market-based measures of credit risk, all within a ratio-

l M d l I l id i h h f d l l i ill


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References

Altman, Edward I., 1968, Financial ratios, discriminant analysis and the prediction ofcorporate bankruptcy, Journal of Finance 23, 589-609.

Bekaert, Geert, and Steven R. Grenadier, 2001, Stock and bond pricing in an aneeconomy, Working Paper: Columbia University.

Bharath, Sreedhar and Tyler Shumway, 2004, Forecasting default with the Merton

model, unpublished paper, University of Michigan.Black, Fischer, and Myron Scholes, 1973, The pricing of options and corporate liabili-

ties, Journal of Political Economy 81, 637654.

Black, Fischer, and John C. Cox, 1976, Valuing corporate securities: Some eects ofbond indenture provisions, Journal of Finance 31, 351367.

Brennan, Michael J., 1979. The pricing of contingent claims in discrete time models,

Journal of Finance 34, 5368.

Campbell, John Y., 1991 A variance decomposition for stock returns, Economic Journal101, 157179.

Campbell, John Y., Jens Hilscher, and Jan Szilagyi, 2007, In Search of Distress Risk,Journal of Finance, forthcoming.

Campbell, John Y., and Robert J. Shiller, 1988a, The dividend-price ratio and expec-tations of future dividends and discount factors, Review of Financial Studies 1,195228.

Campbell, John Y., and Robert J. Shiller, 1988b, Stock prices, earnings, and expecteddividends, Journal of Finance 43, 661676.

Chava, Sudheer and Robert A. Jarrow, 2004, Bankruptcy prediction with industry

eects, Review of Finance 8, 537569.

Cochrane, John H., 1991, Volatility tests and ecient markets: A review essay, Journalof Monetary Economics 27, 463485.

Cochrane, John H., 1992, Explaining the variance of price-dividend ratios, Review of


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Figure 1: Aggregate S&P 500 Equity Dividend Yield Over Time

This figure graphs the annual equity dividend yield for the aggregate S&P 500 over the 1980 2002

period. The annual aggregate equity dividend yield is computed by taking the cross-sectional sumof firm level common dividends divided by the cross-sectional sum of firm level common equityvalues. To be consistent with our firm level sample, we only include firms that were in the S&P500in 1990. The firm level equity data is computed from CRSP's RET and RETX series.

S&P 500 Aggregate Equity Dividend Yield

0.015

0.020

0.025

0.030

0.035

0.040

0.045

0.050

0.055

0.060

19

80

19

81

19

82

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83

19

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19

99

20

00

20

01

20

02

Year

Equ

ity

DP


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Figure 2: Aggregate S&P 500 Asset Dividend Yield Over Time

This figure graphs the annual asset dividend yield for the aggregate S&P 500 over the 1980 2002

period. The annual aggregate asset dividend (payout) yield is computed by taking the cross-sectional sum of firm level payouts divided by the cross-sectional sum of firm level assets. To beconsistent with our firm level sample, we only include firms that were in the S&P500 in 1990. Thefirm's asset dividend (payout) yield is defined as dividend +preferred dividend +interest (fromCOMPUSTAT) paid during the year divided by the end of year market value of equity +bookpreferred equity +book debt (from CRSP and COMPUSTAT), where book debt includes short andlong term maturities.

S& P 500 Agg regate Asset Dividend Yield

0.020

0.030

0.040

0.050

0.060

0.070

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

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1994

1995

1996

1997

1998

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2000

2001

2002

Year

AssetDP


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Figure 4: Merrill Lynch Firm Level Asset Dividend Yield Over Time

This figure graphs the annual asset dividend yield for Merrill Lynch, an S&P 500 firm, over the

1980 2002 period. The firm's asset dividend (payout) yield is defined as dividend +preferreddividend +interest (from COMPUSTAT) paid during the year divided by the end of year marketvalue of equity +book preferred equity +book debt (from CRSP and COMPUSTAT), where bookdebt includes short and long term maturities.

Merrill Lync h Asset Dividend Yield

0.030

0.040

0.050

0.060

0.070

0.080

0.090

0.100

0.110

0.120

0.130

19

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As

setDP


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Figure 5: PACCAR Firm Level Asset Dividend Yield Over Time

This figure graphs the annual asset dividend yield for PACCAR, an S&P 500 firm, over the 1980

2002 period. The firm's asset dividend (payout) yield is defined as dividend +preferred dividend +interest (from COMPUSTAT) paid during the year divided by the end of year market value ofequity +book preferred equity +book debt (from CRSP and COMPUSTAT), where book debtincludes short and long term maturities.

PACCAR Asset Dividend Yield

0.020

0.030

0.040

0.050

0.060

0.070

0.080

0.090

0.100

19

80

19

81

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AssetDP


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Figure 6: True Default Probability vs. Cash Flow and Expected Return-Induced PriceChanges

This figure graphs the true default probability (TDP) as a function of the underlying asset price.The underlying asset price moves either due to news about cash flows or expected returns. TheTDP is calculated based on the model presented in Section 4, where the parameters of the model arechosen to match the actual median values found in the firm level S&P 500 data reported in Tables 1and 2 (discussed in Section 3 of the paper). The base line underlying asset price, denoted by a staron the graph, is $10,533, the 2002 S&P 500 median value in millions. Four instantaneous pricedecrease sizes from the base line are highlighted in the graph: $199.07, $438.17, $912.16, and$3,159.90. These four perturbations (shocks) can be interpreted as a one standard deviation daily,

weekly, monthly, and annual price movement, respectively. The Cash Flow-Induced line graphsthe TDP under the assumption that the entire price movement is cash flow-induced. The ExpectedReturn-Induced line graphs the TDP under the assumption that the entire price movement isexpected return-induced. The bond is assumed to be zero coupon and matures in five years.

0.000

0.005

0.010

0.015

0.020

0.025

0.030

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7,

373

7,577

7,781

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045

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085

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13,493

Asset Pr ice

TrueDefaultProbability

Cash Flow-Induced Expected Return-Induced

BaseLine

Daily

Shock

WeeklyShock

MonthlyShock

AnnualShock


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Table 1: S&P 500 Firm Level Asset Dividend Yield Summary Statistics

This table reports annual summary statistics for S&P500 firms. EquityDP is the equity dividend

yield, computed from CRSP's monthly RET and RETX series, DP is the firm's payout yield (assetdividend), where payout is equal to dividend +preferred dividend +interest (COMPUSTAT annualdata items 19, 15) paid during the year and assets are measured as end of year market value ofequity +book preferred equity +book debt (CRSP prc*shrout, COMPUSTAT data items 130,9+34), where book debt is measured as long term debt +debt in current liabilities. Variables aremeasured at the end of each calendar year. logDP is the natural log of DP. LEV is the firmsfinancial leverage, defined as book debt divided by assets. logD Grow is the log growth rate of thefirm's payout, and P is the value of assets in millions. Panel A reports aggregate level statistics.

The aggregate asset dividend yield is computed by taking the cross-sectional sum of firm levelpayouts divided by the cross-sectional sum of firm level assets. The aggregate equity dividendyield is computed similarly. Panel B reports the pooled firm level statistics. We includeobservations from 1980-2002 for firms that were in the S&P500 in 1990. For logDP and P, we alsoinclude statistics for only 2002. We report mean and standard deviation as well as percentiles of thedistribution.

Panel A: Aggregate Level S&P 500, Across Time, 1980 - 2002

Statistics Equity DP DP logDP LEV logD Grow

Mean 0.0338 0.0452 -3.1544 0.3954 0.0698

SD 0.0125 0.0150 0.3519 0.0557 0.0580

Min 0.0156 0.0247 -3.7029 0.2815 -0.0299

p25 0.0230 0.0304 -3.4931 0.3488 0.0272

p50 0.0317 0.0456 -3.0872 0.4152 0.0723

p75 0.0457 0.0561 -2.8804 0.4361 0.1101

Max 0.0581 0.0722 -2.6282 0.4617 0.1961

N 23 23 23 23 22

Panel B: Firm Level S&P 500, Across Firms and Time, 1980 - 2002

2002 ONLY 2002 ONLY

Statistics Equity DP DP logDP LEV logD Grow logDP P

Mean 0.0327 0.0472 -3.3036 0.3144 0.0676 -3.5593 34199

SD 0.0257 0.0305 0.8100 0.2295 0.3701 0.7140 80738

Min 0.0000 0.0009 -7.0069 0.0000 -6.3238 -7.0069 190p25 0.0153 0.0254 -3.6726 0.1318 -0.0280 -3.9175 3810

p50 0.0278 0.0423 -3.1641 0.2730 0.0541 -3.4193 10533

p75 0.0447 0.0632 -2.7612 0.4545 0.1507 -3.0735 30390

Max 0.1356 0.1608 -1.8276 0.9915 10.5132 -1.8276 917720

N 7160 7160 7160 7093 6677 211 211


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Table 2: S&P 500 Firm Level Asset Dividend Yield AR(1) Estimation Summary Statistics

This table reports summary statistics for annual log asset dividend yield AR(1) estimations

performed on S&P500 firms. The firm payout yield, DP, and firm assets, P, are defined as in Table1. logDP is the natural log of DP. Corr(logP,logDP) is the conditional correlation betweenchanges in log asset growth and the log payout yield (using our model, we estimate this as theunconditional correlation between the shock to logDP and log dividend growth minus shock tologDP). Theta is the AR(1) coefficient from the log payout yield regression; we refer to it as thepersistence parameter. SD(logDP Shock) is the standard deviation of the log payout yield AR(1)residual. Panel A reports statistics for aggregate level variables. The annual aggregate assetdividend yield is equal to the cross-sectional sum of firm level payouts divided by the cross-

sectional sum of firm level assets. Panel Breports the firm level statistics. The AR(1) is estimatedover the 1980-2002 period for those firms that were in the S&P500 in 1990. In order to beincluded, a firm is required to have logDP data for at least 10 years. We report statistics for thosefirms with persistence parameters between 0 and 1 in order to be consistent with our model.

Panel A: Aggregate Level AR(1) Parameters: S&P 500 Firms, 1980 - 2002

Statistics DP SD(DP) SD(logDP) SD(logDP Shock) Corr(logP,logDP) Theta

Mean 0.0452 0.0150 0.3519 0.0848 -0.7536 0.9668

Panel B: Firm Level AR(1) Parameters: S&P 500 Firms, 1980 - 2002

Statistics DP SD(DP) SD(logDP) SD(logDP Shock) Corr(logP,logDP) Theta

Mean 0.0475 0.0194 0.4205 0.2832 -0.5660 0.6309

SD 0.0212 0.0113 0.2332 0.1763 0.2809 0.2211

Min 0.0015 0.0006 0.1023 0.0686 -0.9226 0.0187

p25 0.0323 0.0113 0.2814 0.1791 -0.7645 0.5064

p50 0.0482 0.0174 0.3656 0.2361 -0.6310 0.6649

p75 0.0630 0.0265 0.4911 0.3331 -0.4355 0.7990

Max 0.1053 0.0603 1.5247 1.1861 0.5740 0.9929N 327 327 327 327 327 327


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dividend yields, and default probabilities: linking the credit risk and equity literatures

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