Structural Models of Credit Risk are Useful:

Evidence from Hedge Ratios on Corporate Bonds∗

Stephen M. Schaefer Ilya A. Strebulaev

London Business School

This version: 10 November 2003

ABSTRACT

It is well known that structural models of credit risk provide poor predictions of bondprices. We show that they may perform much better as a predictor of debt returnsensitivities to equity. This is important since it gives us an opportunity to identifymuch better the reasons for model failure. The main result of this paper is thateven the simplest of the structural models (Merton (1974)) produces hedge ratiosthat are in line with those observed empirically. As well as providing insight intothe determinants of corporate bond prices our results are also useful to practitionerswho wish to hedge their positions in corporate debt. The paper also shows thatcorporate bond prices are sensitive to some variables – e.g., VIX – in a way thatappears unrelated to credit risk.

Keywords: Credit risk, structural models, hedge ratios, credit spreads

JEL Classification Numbers: G12, G13

∗We would like to thank Crispin Southgate and Joseph Nehoraj from Merrill Lynch and European Credit Managementfor help with the data. We are responsible for all remaining errors. Address for correspondence: Institute of Finance andAccounting, London Business School, Regent’s Park, London NW1 4SA, UK. E-mail: istrebulaev@london.edu.

Structural Models of Credit Risk are Useful:Evidence from Hedge Ratios on Corporate Bonds

Abstract

It is well known that structural models of credit risk provide poor pre-dictions of bond prices. We show that they may perform much better asa predictor of debt return sensitivities to equity. This is important sinceit gives us an opportunity to identify much better the reasons for modelfailure. The main result of this paper is that even the simplest of thestructural models (Merton (1974)) produces hedge ratios that are in linewith those observed empirically. As well as providing insight into thedeterminants of corporate bond prices our results are also useful to prac-titioners who wish to hedge their positions in corporate debt. The paperalso shows that corporate bond prices are sensitive to some variables –e.g., VIX – in a way that appears unrelated to credit risk.

Keywords: Credit risk, structural models, hedge ratios, credit spreads.

I Introduction

It is commonly agreed that structural models of credit risk over-value corporate bonds.1

Structural models employ the contingent claims approach to value the put option inherent

in the contract between lenders and equityholders. Contingent claims models are, of

course, widely used in practice and are seen as one of the major successes of financial

theory. The failure of such models to explain satisfactorily actual corporate debt prices

and spreads is therefore surprising. The poor performance of these models in this area

has been recognized for many years but their failure continues to surprise.

This paper makes two simple but important points. First, while structural models

provide a poor prediction of prices and returns, they may perform much better as a

predictor of the sensitivity – or hedge ratio – of debt to equity. This is important because

hedge ratios determine the composition of the replicating portfolio which, according to

the theory, determines the price. Thus, if we find that a model provides a good prediction

of hedge ratios but a poor prediction of the price, we are better able to identify the reasons

for model failure. In fact, we find that even the simplest structural model (Merton (1974))

predicts hedge ratios that are in line with those observed empirically. This leads us to

reconsider the possible explanations for the failure of structural models to explain better

the level of prices and yields.

At present, there are two main explanations for this failure. First, that structural

models fail to predict accurately the probability of default and/or the recovery rate.

This explanation has weight because all current models violate at least some known

facts about capital structure and/or the circumstances of corporate default. Conse-

quently, the development of much of the theory has been in pursuit of improved ways to

model credit events. As a result, we possess an arsenal of models that include stochastic

default boundaries, dynamic capital structure and opportunistic behavior on the part1For early empirical investigation of the Merton model see Jones, Mason, and Rosenfeld (1984). For

a more recent analysis see Eom, Helwege and Huang (2003) who study the empirical performance of a

number of structural models.

2

of claimholders (e.g., Anderson and Sundaresan (1996), Collin-Dufresne and Goldstein

(2001)). Empirical tests have showed, however, that these modifications do not substan-

tially improve the ability of the models to explain the level of corporate bond prices

(Eom, Helwege and Huang (2002), Huang and Huang (20002)).

A second explanation is that the pricing of corporate bonds is influenced by factors

that are not related to credit risk, and are therefore not included in structural models.

It could even be the case that structural models account very well for the credit risk

component of bond prices and returns while, at the same time, credit risk is actually

responsible for only a part, perhaps not even a very large part, of spreads and returns.

Some recent results tend to support this view. Elton, Gruber, Agrawal, and Mann (2001)

find that differences in taxation account for about a third of credit spreads. Huang and

Huang (2002) estimate that credit risk accounts only for a small fraction of the observed

credit spread. Collin-Dufresne, Goldstein, and Martin (2001) find that the variables

present in structural models can not explain changes in spreads.

On the positive side, a recent paper by Leland (2002) shows that the default proba-

bility prediction of structural models are indeed roughly consistent with observed default

frequency.2,3

Our paper contributes to this on-going debate but, unlike many previous authors

(e.g., Huang and Huang (2002)), we do not focus on the level of prices or the size of

the spread. Instead we investigate the ability of structural models to explain rates of

return and we ask is whether these models can be used to hedge corporate bond returns.

Using data on monthly returns for a large sample of U.S. corporate bonds over a five-year

period, we find that the variables present in structural models explain a large fraction

of the returns on investment grade bonds and a smaller but significant fraction for high

yield bonds. This result is in itself not surprising, since it has been known for some2This finding means that those models which try to find the ways to make predicted spreads closer

to the actual ones, can substantially overpredict default frequency.3This result is also in line with the eponymous KMV method that used a Merton-based approach to

predict “distance to default”.

3

time that investment grade and government bonds have similar returns (Campbell and

Ammer, 1993). We also find that debt returns are significantly related to returns on the

underlying asset and that the pattern of sensitivities is broadly consistent with the level

of credit exposure.

In other words, this paper focusses attention on the second-moment predictions of

the model. In structural models, any change in the value of a credit risky bond credit is

a result of a change in the value of the assets that collateralize the debt or a change in

riskless rates. In our empirical analysis we employ the change in the value of the equity

together with descriptors of the change in riskless rates and ask whether the sensitivities

of corporate bond returns to equity and riskless bond returns are consistent with the

model. Our main result is that even the Merton (1974) model produces equity sensitiv-

ities that are roughly in line with those observed empirically. Our test is supportive of

the view that structural models account well for the credit risk component of corporate

bond returns. This positive result is also consistent with Leland’s favorable and recent

findings on the default probability predictions of structural models.

A number of authors have found that the returns on corporate bonds are also related

to a number of factors that are not present in structural models. These include the Fama-

French SMB and HML factors (Elton et. al. (2001)), returns on a broad index of equity

prices and implied volatility from options on equity indices, “VIX” (Collin-Dufresne et.

al. (2001)). In the second half of the paper we include these variables in our analysis of

hedged corporate bond returns.

The results represent the second main finding of this paper. We find that returns

on corporate bonds are significantly related to changes in the VIX implied volatility

index but that the sensitivities are not related to any of the standard measures of credit

exposure such as rating, leverage or asset volatility. Our results on the Fama-French

SMB factor are similar. Thus VIX and SMB have significant effects on the prices of

corporate bonds but are not related to their credit risk..

Our results also have potential interest for practitioners who wish to hedge corporate

4

debt positions. Our findings suggest that structural models are in fact more useful for

this purpose than might appear from their performance in explaining the size of credit

spreads.

Other authors have previously studied these issues. Blume, Keim, and Patel (1991)

also study the behavior of corporate bond returns but not within the framework of struc-

tural models. Huang and Huang (2002) use a variety of models to determine whether

structural models explain the average level of yield spreads but do not study hedge ra-

tios. Collin-Dufresne, Goldstein, and Martin (2001) analyze changes in yield spreads in

a regression framework where the choice of regressors is motivated by structural mod-

els. They do not, however, examine whether the size (as distinct from the sign) of the

estimated coefficients is consistent with the theory.

The closest paper to ours in spirit is Leland (2002) in sense that we investigate the

predictions of structural models that do not directly touch on the level of bond prices

(or yield spreads). While Leland (2002) studies whether structural models are able to

predict the presence of corporate default, we study their implications for hedge ratios

and so attempt to define more narrowly the reasons why these models fail to explain the

level of spreads.

Our paper proceeds as follows. Section II provides a description of the data set and

the sample selection procedure and also gives descriptive statistics. Section III describes

some preliminary regression analysis of ability of structural models to explain returns

on corporate bonds. In section IV the procedure is refined and the regressions take into

account the effect of changes in asset values, volatility and leverage on the hedge ratios

predicted by the structural model. Section V examines the sensitivity of bond returns

to other variables such as VIX and the Fama-French SMB and HML factors. Section VI

concludes.

5

II Data, Sample Selection and Descriptive Statistics

II.1 Data

We use monthly prices on corporate bonds that are included either in the Merrill Lynch

Corporate Master index or the Merrill Lynch Corporate High Yield index. These indices

include most rated U.S. publicly issued corporate bonds. The data covers the period from

December 1996 to September 2002. Table I provides descriptive statistics on the bonds

in the data set. The data set contains more than 323000 bond-month observations, with

about 2900 issuers and 9000 issues. Matching with CRSP and COMPUSTAT allows us

to use about 50% of the total number of observations and all rating categories (from

AAA to CCC) are represented. As we would expect, as we move down the ratings the

average time-to-maturity decreases and the average coupon rate increases. The median

size at issuance is $200 million dollars. Detailed information on each bond is obtained

from the Fixed Income Securities Database (FISD) as provided by LJS Global Services)

and equity and treasury bond returns are from CRSP.

II.2 Sample selection

The specific bonds included in the analysis satisfy the following criteria: (1) the bond

is not convertible, exchangeable, callable or putable; (2) the bond is issued by a U.S.

company and denominated in $U.S.;4 (3) it is possible to match unambiguously the

bond issuer with a company in CRSP using the CUSIP; (4) the bond is issued by a non-

financial corporation; (5) the bond has an initial maturity of at least four years and (6)

the bond has at least 25 consecutive monthly price observations. Table I gives summary

statistics for the entire sample and Table II for the remaining sample of 1362 bonds.

II.3 Descriptive statistics on returns

For each bond j we calculate the return between months t and t− 1 as follows:4More specific, the company is of the U.S. origin according to the FISD definition. In particular, its

headquarters should be located in the U.S. and it is subject to the U.S. legal practice.

6

rj,t =Pj,t + AIj,t + Ij,tCj/Nj

Pj,t−1,

where Pj,t is the price of bond j at the end of month t and AIj,t is the change in

the accrued interest between t − 1 and t. Since the calculation of the accrued interest

restarts with each coupon payment, if the coupon date falls between t−1 and t, Cj/Nj is

added to the price, where Cj is the annual coupon rate and Nj is the coupon frequency

per annum of bond j. Ij,t is an indicator function taking the value of 1 if the coupon is

due between t− 1 and t. The excess return is then calculated as

rj,t = rj,t − rf1m,t,

where rf1m,t is the return on one-month Treasury bills up to month t.

Table III provides summary statistics on individual raw returns and excess returns

for the whole data set and by rating (where rating, as provided by the S&P, is defined as

the rating on the first date that a bond is present in the dataset). The realized average

monthly return on corporate bonds is 0.50% and the average excess return is 0.11%.5

Over this period the low-grade bonds had a lower average return than investment-grade

bond and a substantially higher standard deviation of rate of return. for the period

1977-1989.

III Structural models and returns

In this section we study the determinants of corporate bond returns implied by structural

credit risk models. We take initially a conservative view and include in the the regressions

only those factors which are explicitly stochastic in the Merton model. Later other

variables are included – specifically VIX and the Fama-French factors – and the sensitivity

of debt returns to equity re-examined.5We include these statistics on mean returns only for comparison. The period covered by our data is

too short to be able to identify means with any precision. The focus of our study, however, is on second

moments (e.g. hedge ratios) and for this purpose the data is adequate.

7

The value of the firm, V , is the driving state variable in most structural models;

indeed the presence of V is the distinguishing feature of the structural approach. In this

section we estimate the sensitivity of debt values to changes in the value of the firm by

regressing excess rates of return on corporate bonds against the excess return on the

equity of the issuing firm.6 In a one-factor model the elasticity of the value of debt to

equity is related to the delta of the debt against V by:

∂D

∂E

E

D=

(∂D∂V∂E∂V

)E

D=

(1

∂E∂V

− 1

)E

D, (1)

Clearly, this elasticity is a function of both V and interest rates and therefore varies

over time. These regressions are, therefore, simply a first step. Later in the paper we use

the Merton model to account for time variation in the elasticity. In our regressions we

also control for changes in the riskless term structure by including returns on a ten-year

constant maturity Treasury bond. While in the first generation of structural models

the risk-free rate was held constant, more recent versions include a stochastic interest

rate (e.g., Kim, Ramaswamy, and Sundaresan (1993), Longstaff and Schwartz (1995)).

An increase in the risk-free rate has two opposing effects on the price of debt. First,

it decreases the value of debt by decreasing the present value of all future cash flows.

Second, it leads to increase in the risk-neutral drift of the value or the firm, V , and so

increases the value of debt by decreasing the likelihood of default. The second effect will

be relatively more important for bonds with a higher likelihood of default.

Thus we regress the excess return on each bond, rj,t, on the excess return on the

issuing firm’s equity, rE,t, and return on riskless bonds, rf10y,t:

rj,t = αj,0 + αj,ErE,t + αj,rfrf10y,t, (2)

6An increase in E causes an increase in D in case this increase is caused by an increase in V , i.e. an

increase in the total pie to be divided between equityholders and debtholders. It also can be the case that

an increase in E is a result of a wealth transfer from debtholders to equityholders (with V kept constant).

The latter issue has been investigated in the structural framework by the strategic debt service models

such as Anderson and Sundaresan (1996) and Mella-Barral and Perraudin (1997).

8

Table IV reports the average value of the coefficients and their t-statistics calculated

from their cross-sectional standard deviation.7

A number of points are worth noting. Both factors are significant for the whole

sample and for most of the rating categories. For the whole sample, a one percent

return on the riskless bond leads to 0.45% increase in the corporate debt price. The

standard deviation of treasury bond returns is about 1.5% per month and so the average

impact on one-month returns on corporate bonds of a one standard deviation return on

government bonds is about 0.75%. The impact of the riskless rate becomes smaller for

lower credit rating categories: the loading is significantly positive for investment-grade

bonds and negative but insignificant for B and CCC grade bonds. This suggests, quite

interestingly, that for very low quality bonds the effect of an increase in the riskless rate

– and thus the risk-neutral drift of V – in reducing credit exposure may outweigh the

resulting decline in the present value of firm’s liabilities.

Ignoring the effects of credit risk and assuming parallel shifts in the riskless term

structure, the coefficient on the return on Treasuries would equal the ratio of the duration

of the corporate bond to the duration of the Treasury. As the average maturity of our

sample of corporate bonds is just under 13.5 years (see Table II) and the Treasuries have

a constant maturity of 10 years, the actual coefficient of 0.75 is clearly lower than the

ratio of the durations. This is consistent with a negative correlation between changes in

the yield spreads on corporate debt and changes in Treasury yields.

For the full sample, the return impact of a 1% return on equity is 0.042%. If we

assume that the standard deviation of monthly equity returns is 12%, then a one stan-

dard deviation return on equity increases a bond’s return by 0.4% on average. This is

smaller than, but of the same order of magnitude as the effect of the risk-free rate. The

sensitivities of bond returns to equity are convex in the credit rating: a one percent

increase in the stock price increases returns by 1-2 basis points (bp) for AA-A bonds, 47Similar procedure was employed by Collin-Dufresne, Goldstein, and Martin (2001) in the context of

studying the changes in credit spreads.

9

bp for BBB bonds, and 7-11 bp for BB-B bonds. The average coefficient on equity is

significant for each of the ratings (apart from AAA).8

In univariate regressions (not reported here) returns on the riskless bond are the

major determinant of returns on high-grade bonds, while for low-grade bonds equity

is the more important. The two factors combined explain about half of a variation

in returns on AAA-A-grade bonds and 20% for low-grade bonds. We show later that,

even when other regressors are included and the results are computed for subperiods,, the

sensitivity of corporate bond returns to both equity and riskless bonds remain significant

and the coefficients exhibit the same relationship with credit ratings.

IV Debt sensitivity to equity

IV.1 Sensitivity in structural models

Having established that the sensitivity of corporate debt returns to the underlying equity

and riskless debt are significant, we now ask whether the magnitude of these sensitiv-

ities is consistent with the predictions of structural models. In this paper we employ

the Merton (1974) model. This may be a surprising choice because, not only are its

assumptions regarding capital structure clearly oversimplified, but it is also well known

to underestimate credit spreads by a wide margin (Jones, Mason, and Rosenfeld, 1984).

However, it remains an open question whether this simple model is able to explain the

sensitivity of debt returns to equity and this is the question we now address.

In the Merton model the value of equity is simply the value of a European call option

on the assets of the firm with exercise price equal to the face value of the debt. Using

equation (1) we may therefore write the sensitivity of the return on a credit risky bond

to the return on equity as:8This is a different result from Collin-Dufresne, Goldstein, and Martin (2001) who find that changes

in equity and quasi-market leverage do not have a significant impact on changes in credit spreads.

10

hE =(

1∆E

− 1)(

1L− 1

), (3)

where L is market leverage (defined as market value of debt as a percentage of the

market value of the firm) and ∆E is the delta of a European call option on the value

of the firm (equity). In the Merton model the parameters of interest are book leverage,

B, the volatility of the firm’s assets, σA, time to maturity, T − t, and the risk-free rate,

r. The table below shows the comparative statics of hE (assuming that the option to

default is out-of-the-money).

Parameter hE

B +

σA +

T − t +/−r −

For short maturity bonds the effect of an increase in time to maturity on the dispersion

of asset values at maturity is greater than the effect on the risk-neutral expected value

of the firm’s assets and the hedge ratio increases. For long maturity bonds the second

effect dominates and an increase in time to maturity decreases the hedge ratio. Our

calculations suggest that, for reasonable parameter values, the actual times-to-maturity

in our sample are too small for the second effect to dominate.

Table V shows the values of hE for the Merton model for asset volatilities between

10% and 50% and values of L between 10% and 70%. The risk-free rate is 5 %per annum

and the time-to-maturity is 10 years. These values have been chosen to reflect those

encountered in our sample. The table shows that the sensitivity varies significantly from

zero (less than 0.01), when both leverage and volatility are low, to about 0.3 when both

leverage and volatility are high.

While in the remainder of the paper we focus on the Merton model, it would be

straightforward to carry out the same analysis using other debt pricing models. To

11

illustrate, we consider two other models of corporate debt pricing. In the Leland (1994)

model, where debt is assumed to be perpetual and to pay a constant coupon, the bond

price is given by:

Pt =cF

r

[1−

(Vt

V ∗

)−x]

+ max(wV ∗ −K, 0)(

Vt

V ∗

)−x

,

where c is the coupon rate, F is the principal, K and w are fixed and proportional

liquidation (bankruptcy) costs, V ∗ is the time-invariant default boundary,

V ∗ =(

cF

r

)1

1 + x−1,

and x is constant and a function of other parameters, x = x(σ, δ, r), where δ is the

firm’s payout ratio.

Taking the derivative of Pt with respect to Vt, we find that

∂Pt

∂Vt=

(1

V ∗

)−x

V −x−1t x

[cF

r−max(wV ∗ −K, 0)

].

Substituting the above formulas into equation (3) we obtain the corresponding value of

hE .

In the strategic debt service models of Anderson and Sundaresan (1996) and Mella-

Barral and Perraudin(1997), and if we also assume that the bond is a constant coupon

perpetuity, the result is similar to Leland’s, the only difference being the value of the

boundary:

V ∗ =cF/r + K

w (1 + x−1)

IV.2 Preliminary comparison of sensitivity in the Merton model and

actual data

As a first step in investigating the relationship between the Merton model sensitivities

and those estimated empirically using regressions we perform a simulation. The object of

this simulation is to calculate the mean value of the sensitivity of bond returns to equity

in our data when (a) the Merton model holds and (b) the sensitivities are estimated

12

using linear regression on monthly data that has the same characteristics as our sample.

For a given rating class, the difference between the mean sensitivity we obtain in this

way and the sensitivity calculated from Merton model using the mean values by rating

class, leverage, volatility, and time to maturity can be attributed to (a) non-linearity and

(b) the discrete time interval between observations.

We aggregate the seven principal rating classes into three: “AAA-A” (including AAA,

AA, and A bonds), “BBB”, and “Junk” class (BB, B, and CCC bonds). For each of

the three classes we find the 5% and 95% quantiles of leverage,9 volatility, and time to

maturity and we take these to be the minimum and maximum points of unform distri-

butions from which we draw values in the simulation.10 The table below gives the upper

and lower limits for the three variables for each rating class.

Parameter AAA-As BBB JunkLeverage

min 0.15 0.2 0.3max 0.35 0.45 0.7

Asset volatilitymin 0.1 0.15 0.2max 0.3 0.35 0.5

Time to maturitymin 5 5 5max 20 20 20

Frequency 0.48 0.35 0.17

We now generate 2000 time series of “bond returns” as follows. First, we assign a

rating class to each time series according to the proportions found in the actual data.

Second, again for each time series, we randomly draw values for leverage, volatility, and

time to maturity from the distributions for the relevant credit class and generate a time

series of asset values. Using the Merton model we then calculate monthly equity and9The ratio of the face value of the firm’s debt to the market value of assets.

10We also tried to account better for the distribution patterns of leverage and volatility within the

rating class; it did not produces any significant changes.

13

bond prices and, from these, monthly returns. Finally, we estimate the hedge ratio, hE ,

by running a regression of the simulated bond returns on simulated equity returns (i.e.,

a regression similar to (2) but excluding the return on the riskless bond.11)

Table VI reports the mean values of hE and R2 for these regressions. Comparing

these results and those in Table IV we see that the sensitivities are surprisingly similar.

For high quality (AAA-A) bonds, the average sensitivity is found to be about 0.01 for

simulations and 0.0003–0.02 for the actual the data; for BBB bonds both for the model

and the actual data produce a sensitivity of 0.04 and for junk bonds we find 0.15 for the

simulations and 0.07–0.11 bp for the actual data. The mean values of the simulated and

empirical hedge ratios are not significantly different at a 5% confidence for the entire

sample and the “AAA-A” and “BBB” subsamples and at the 10% level for the “Junk”

subsample. These results are surprising since, for the same simulated data, the model

underestimates the observed level of credit spreads by more than 80%, or by more than

50bp in absolute terms.

IV.3 Preliminary Analysis of Hedge Ratios

The results in the previous section raise the possibility that, although the Merton model

leads to poor predictions of credit spreads, it may perform better as a predictor of hedge

ratios. This result, if substantiated, would be important because, in contingent claims

pricing theory, the hedge ratios define the composition of the replicating portfolio which,

in turn, defines price of the contingent claim. Thus, if we find that the model provides

good predictions of hedge ratios but poor predictions of the bond price, we are better

able to identify the reasons for model failure.

To address this issue, we next test the second-moment prediction of the model in a

more rigorous manner. Note that if the model is correct then in equation (2) αj,E is

11We also run regressions on the actual data using only equity returns to be consistent. The results

are broadly similar with the equity sensitivities slightly larger.

14

equal to hE and we can therefore rewrite the regression as

rj,t = αj,0 + βj,EhE,j,trE,t + αj,rfrf10y,t, (4)

where hE,j,t is the model hedge ratio for firm j at time t and under the null hypothesis

that the Merton model holds, βj,E is equal to one. To implement (2) the following

parameters need to be estimated for each firm: the ratio of the book value of debt to the

market value of assets, B/V , the volatility of assets, σA, time to maturity, T − t, and

the riskless rate.

To estimate BV , we take the ratio of the book value of debt (sum of COMPUSTAT

items 9 (long-term debt) and 34 (debt due within a year)) to the quasi-market value

of assets (sum of COMPUSTAT items 9 and 34 plus the number of shares outstanding

times stock price (both CRSP)). The COMPUSTAT data are taken at the date of the

last annual accounting report and the CRSP data are taken on the date of observation.

The estimation of asset volatility is a challenging task and here we consider a number

of alternatives. First, we compute upper and lower bounds on asset volatility as follows.

The maximum equates asset and equity volatility, σE , i.e., assuming zero leverage. The

minimum is calculated as σE(1−L), where L is the market leverage, i.e., assuming that

the debt bears no asset risk. In this case, the theoretical hedge ratio is zero.

A more realistic estimate of asset volatility recognizes that debt bears some asset risk

and that equity and debt covary. For firm j at time t we have:

σ2Ajt = (1− Ljt)2σ2

Ejt + Ljt2σ2

Djt + 2Ljt(1− Ljt)σED,jt, (5)

where σDjt is the time t volatility of firm j’s debt, and σED,jt is the time t covariance

between returns on firm j’s debt and equity.12 We could estimate firm j’s debt volatility

using the returns on each of firm j’s bonds but this approach has two drawbacks. First,

it assumes that all of a firm’s outstanding debt has the same volatility as its publicly

traded debt. Second, much corporate debt is relatively illiquid and some of the observed

volatility may be spurious.12This calculation assumes again that leverage is measured instantaneously.

15

We therefore estimate equation (5) as follows. Firm j’s equity volatility at time t is

estimated as the time series volatility of returns on firm j’s equity using three years of

data up to month t. For the volatility of returns on firm j’s debt we first calculate the

average volatility of debt returns by credit rating. Thus, for rating category BBB and

firm j, for example, we take the returns on firm j’s debt for each month that the debt

was rated BBB at the start of the month. If the bond in question is rated BBB in at

least 15 months we then compute the time series volatility. Averaging these volatilities

over all firms we obtain the average volatility for BBB debt. The volatility of firm j’s

debt in month t is then set equal to the average volatility for the rating category of

firm j at month t.13. The covariance of equity and debt returns, σED,jt, is estimated as

ρED,jtσD,jtσE,jt where ρED,jt is estimated in a similar way to σD,jt.

Finally, we take time to maturity as equal to the median time to maturity for each

rating class and the riskless rate equal 5%.

Table VII reports summary statistics for these estimates. As expected, leverage is

higher for lower rating categories and, similarly, so is equity volatility. The average values

of equity volatility and quasi-market leverage are broadly consistent with similar results

reported in other studies. Deleveraging equity volatility using L but taking no account

of the asset risk borne by debtholders (the third panel of Table VII) results in estimates

of asset volatility that are relatively constant across the rating categories.

The fourth panel of Table VII gives estimates of σAt using equation (5) and the

method described above. Here the mean values of asset volatility are quite similar for

investment grade bonds (22% to 24%) but noticeably higher for junk bonds: 27% for

BB, 29% for B and 30% for C. The range of values, as measured by the 5% and 95%

quantiles, is also much wider for the lower rated bonds, e.g.,12-33% for AA vs. 14-56%

for B.

Table VIII shows estimates of the hedge ratio hE(σE). The first two panels set σA

13One can think of this as a form of “switching regime” where the volatility for firm j switches between

the volatilities of the different rating categories

16

equal to σE and (1 − L)σE respectively. The final panel shows estimates of the hedge

ratio using the estimates of σA derived using equation (5). As expected, these rise

monotonically as the rating category declines.

IV.4 Testing Merton Model Predictions of Hedge Ratios

We now use our estimates of asset volatility to test more formally whether hedge ratios

from the Merton model are consistent with the empirical relation between equity and

corporate bond returns.

For firm j we take the estimate of time t asset volatility described above and use this

as an input to the calculation of the time t hedge ratio, hE,jt(σA).14 We then estimate

the following regression for each firm, j:

rj,t = αj,0 + βj,EhE,jt(σA)rE,t + αj,rfrf10y,t. (6)

Under the null hypothesis that the Merton model correctly estimates the sensitivity

of returns on firm j’s debt to firm j’s equity, the coefficient βj,E should be unity.

The results are given in Table IX. For the entire sample the mean estimate of βj,E

is 1.206. The t-statistic against unity, the value of βj,E under the null is 1.156. For the

six rating categories the mean value of βj,E is different from unity in only two cases: BB

where the mean value if 2.498 and CCC where the mean is 0.415. For the other four

categories the mean value ranges from 0.55 (AA) to 1.54 (B) and none is significantly

different from one.

These results are supportive of the structural approach, and the Merton model in

particular, in a way that previous analyses of the level of prices or credit spreads have

not been. They are also complementary to the results recently obtained by Leland (2002)

who shows that the default frequency predictions of structural models are also broadly

consistent with the data.

Apart from the size of the yield spread, there is a further prediction of structural14The other inputs – book leverage, time to maturity and the riskless rate – are as described earlier.

17

models that is inconsistent with the results in Table IX. This is the R2 in the regression

which, if the Black-Scholes conditions supporting the structural approach apply, should

be much higher. In our simulations reported in Table VI the R2 varied from 0.65% for

AAA-A to 0.936% for BB-CCC.15 In Table IX the R2 are much lower (11% for CCC and

37% for AA).

The low variation in the fraction of rate of return volatility accounted for by equity

returns and interest rates has a number of possible explanations. One is simply that it

reflects noise in the bond return data (or, possibly in the equity or riskless bond data).

This almost certainly accounts for some of the unexplained variability of corporate debt

returns. Another is that the model is mis-specified and that either the functional form of

the hedge ratio is incorrect or that other variables are necessary to account for the credit

exposure of corporate debt. For example, the Merton could hold except that volatility

is stochastic and other variables are necessary to predict volatility. Finally, returns on

corporate debt could be related to other variables in a way that is not directly related

to credit risk. For example, returns on corporate debt might be related to variables that

proxy for changes in liquidity.

V Other Determinants of Returns on Corporate Bonds

In this section we consider the impact of variables that other authors have found to be

significant explanators of returns on corporate bond. These include (i) changes in the

10-year minus 2-year yield spread on US Treasuries, (ii) the return on the S&P 500 index,

(iii) changes in the VIX index of implied volatility of options on the S&P 100 index and

(iv & v) the Fama-French SMB and HML factors. All five factors are included in the

recent study by Collin-Dufresne et. al. (2001) and the Fama-French factors are included

by Elton et. al. (2001).

The results are shown in Table X. The mean coefficients on 10-year Treasury returns

are very similar to those in Table IV and those on equity are also similar but little lower.15Note that in this procedure interest rate is held constant.

18

The mean coefficients on the S&P are not significant except for the A credit category.

The mean coefficients on HML are significant for the whole sample and for three out of

the six rating subsamples while those on SMB are significant for the whole sample and

all the subsamples except for CCC.

Perhaps the most interesting results, however, are those for changes in the VIX

volatility index. Collin-Dufresne et. al. (2001) had previously found this variable to be

significant in regressions of changes in yield spreads on a very similar list of regressors.

The mean coefficients for ∆(V IX) are significant in every case except for CCC; in fact,

the t-statistics are higher than those on changes in the firm’s own equity for all the

investment grade subsamples. A natural interpretation of the role of ∆(V IX) in the

regression is as a proxy for changes in the volatility of equity. In this interpretation VIX

is related to a bond’s credit exposure via its effect on the default put. However, when we

examine the magnitude of the coefficients for the different rating subcategories we see

that they are essentially the same. For example, the mean coefficient for AAA and AA

are -0.043 and -0.060 respectively and for BBB and BB are -0.056 and -0.054 respectively.

If ∆(V IX) were acting as a proxy for changes in equity volatility the sensitivity for the

lower credit categories, as in the case of the coefficients on equity returns, would be much

larger.16

We find, therefore, that the change in the VIX index is a variable that, while strongly

related to returns returns on credit risky bonds, is apparently unrelated to credit risk.

The R2 in Table X are significantly higher than in Table IV but particularly so for the

non-investment grade bonds: the average R2 for BB and B increases to 28% from 20%.

The precise role of ∆(V IX) and the other variables is, at this point, unclear but it seems

highly unlikely that is connected with credit exposure.16We have also carried out cross-sectional regressions of the individual coefficients on ∆(V IX) on

simple descriptors of credit risk, such as equity volatility and leverage, and found no relation.

19

VI Conlusion

This paper studies the ability of structural models to explain excess returns on corporate

bonds and the main question we ask is whether these models provide accurate predictions

of hedge ratios. Using data on monthly returns for a large sample of U.S. corporate bonds

over a five-year period, we find those variables included in structural models – returns

on the issuing firm’s equity and on riskless bonds – explain a large fraction of returns

on investment grade bonds and a smaller but significant fraction for high yield bonds.

Further, and this is the main result of the paper, we find that, for most rating categories,

the equity ratios predicted by the Merton model are not rejected by time series data.

The next step is to account for other factors. We include in our regression variables

that in previous studies have been shown to influence corporate bond prices. The vari-

ables we use are: (i) changes in the 10-year minus 2-year yield spread on US Treasuries,

(ii) the return on the S&P 500 index, (iii) changes in the VIX index of implied volatility

of options on the S&P 100 index and (iv & v) the Fama-French SMB and HML factors.

We find that none of the included variables undermines the significance of either the

risk-free rate or equity. Our main result here, and the second main result of the paper,

is that changes in the VIX index have an impact on corporate bond returns that is both

significant and apparently unrelated to a bonds exposure to credit risk. It seems clear,

therefore, that returns on credit risky bonds are systematically related to at least one

factor that lies outside standard measures of “credit risk”. Whether there are other

factors, and the precise role of ∆(V IX) in the determination of risky bond prices, is a

question for further research.

20

References

[1] Anderson Ronald W., and Suresh M. Sundaresan, 1996, “The Design and Valuation

of Debt Contracts”, Review of Financial Studies, 9, 37-68.

[2] Blume, M. E., D. B. Keim, and S. Patel, 1991, “Returns and Volatility of Low-Grade

Bonds: 1977–1989”, Journal of Finance, 46.

[3] Campbell, John Y., and John Ammer, 1993,“What Moves the Stock and Bond

Markets? A Variance Decomposition for Long-Term Asset Returns,” Journal of

Finance, 48, 3–37.

[4] Collin-Dufresne Pierre, and Robert S. Goldstein, 2001, “Do Credit Spreads Reflect

Stationary Leverage Ratios?”, Journal of Finance, 56, 1929-1957.

[5] Collin-Dufresne Pierre, Robert S. Goldstein, and J.Spencer Martin, 2001, “The

Determinants of Credit Spread Changes”, Journal of Finance, 56, 2177-2207.

[6] Elton Edwin J., Martin J. Gruber, Deepak Agrawal, and Christopher Mann, 2001,

“Explaining the Rate Spread on Corporate Bonds”, Journal of Finance, 56, 247-277.

[7] Eom Young Ho, Jean Helwege, and Jing-zhi Huang, 2003, “Structural Models of

Corporate Bond Pricing: An Empirical Analysis”, Review of Financial Studies,

forthcoming.

[8] Huang, Jing-zhi, and Ming Huang, 2002, “How Much of the Corporate-Treasury

Yield Spread is Due to Credit Risk?”, working paper, Pennsylvania State University.

[9] Jones P.E., S.P. Mason, and E. Rosenfeld, 1984, “Contigent Claims Analysis of

Corporate Capital Structures: An Empirical Analysis”, Journal of Finance, 39,

611-25.

[10] Kim, J., Ramaswamy, and S. Sundaresan, 1993, “Does Default Risk in Coupons

Affect the Valuation of Corporate Bonds?: A Contingent Claims Model”, Financial

Management, 117–131.

21

[11] Leland, Hayne E., 2002, “Predictions of Expected Default Frequencies in Structural

Models of Debt”, Venice Conference on Credit Risk, Sept 2002

[12] Longstaff Francis A., and Eduardo S. Schwartz, 1995, “A Simple Approach to Valu-

ing Risky Fixed and Floating Rate Debt”, Journal of Finance, 50, 789–819.

[13] Mella-Barral Pierre, and William R.M. Perraudin, 1997, “Strategic Debt Service”,

Journal of Finance, 52, 531–556.

[14] Merton Robert C., 1974, “On the Pricing of Corporate Debt: The Risk Structure

of Interest Rates”, Journal of Finance, 29, 449-470.

22

Table ISummary statistics for the entire data sample

This table reports summary statistics for the entire sample of corporate debt returns overperiod 12.1996–07.2002. The number of observations is given in thousands. T − t is the timeto maturity remaining on the date of each observation. The coupon rate is in %. Volume isin million $US dollars. % in CRSP reports the fraction of observations that are matched withCRSP and COMPUSTAT.

All AAA AA A BBB BB B CCCObservations 323.53 7.23 32.20 117.17 93.38 26.93 35.18 11.41Issuers 2894 77 240 824 1035 667 1084 568Issues 9049 301 1154 3811 3332 1515 1738 879Mean T − t 10.60 11.90 11.13 10.94 12.10 9.36 6.94 6.74Median T − t 7.04 7.34 6.50 6.80 7.88 6.67 6.97 6.30Mean Coupon 7.88 7.37 7.07 7.31 7.58 8.42 10.09 10.69Median Coupon 7.50 7.20 6.88 7.13 7.38 8.36 10 10.63Mean Volume 281.52 338.62 337.90 293.11 269.72 247.53 246.87 250.92Median Volume 200 250 200 200 200 200 179.80 200% in CRSP 49.96 13.80 39.52 50.41 55.92 52.26 49.97 43.50

23

Table IISummary statistics for the final sample

This table reports summary statistics for the final sample (the selection procedure is described inthe paper) of corporate debt returns over period 12.1996–07.2002. The number of observationsis given in thousands. T − t is the time to maturity remaining on the date of each observation.The coupon rate is in %. Volume is in million $US dollars. % in CRSP reports the fraction ofobservations that are matched with CRSP and COMPUSTAT.

All AAA AA A BBB BB B CCCObservations 58.49 0.27 4.03 18.73 23.07 5.03 6.20 1.16Issuers 493 4 30 149 226 121 148 57Issues 1362 6 110 560 687 265 229 83Mean T − t 13.46 20.53 13.87 16.10 14.15 8.71 7.13 8.38Median T − t 8.49 24.16 8.63 12.80 8.96 6.79 7.13 6.88Mean Coupon 7.76 7.26 6.97 7.41 7.49 8.27 9.56 9.86Median Coupon 7.40 7.15 6.75 7.13 7.25 8 9.50 9.75Mean Volume 259.10 321.88 321.03 266.06 258.68 247.08 213.59 220.90Median Volume 200 300 250 200 200 200 175 200% in CRSP 100 100 100 100 100 100 100 100

Table IIISummary statistics on returns

This table reports summary statistics for excess and raw returns for the final sample (theselection procedure is described in the paper) of corporate bond returns over period 12.1996–07.2002. We calculate for each corporate bond j the return between months t and t − 1 asfollows:

rj,t =Pj,t + AIj,t + Ij,tCj/Nj

Pj,t−1,

where Pj,t is the price of bond j at the end of month t and AIj,t is the accrued interest betweent− 1 and t. Since the calculation of the accrued interest restarts with each coupon payment, ifthe coupon falls between t−1 and t, Cj/N is added to the price, where Cj is the annual couponrate and Nj is the coupon frequency per annum of bond j. Ij,t is an indicator function takingthe value of 1 if the coupon is due between t− 1 and t. The excess return is then calculated as

rj,t = rj,t − rf1m,t,

where rf1m,t is the return on one-month Treasury bills. Returns are first calculated for eachindividual bond and then averaged across bonds. Excess returns are given in parentheses. Nis the number of bonds.

All AAA AA A BBB BB B CCCMean 0.50 0.59 0.59 0.56 0.49 0.48 0.34 -0.22

( 0.11) ( 0.20) ( 0.20) ( 0.16) ( 0.10) ( 0.09) (-0.05) (-0.62)5% -0.18 0.38 0.38 0.20 0.04 -1.40 -2.51 -4.07

(-0.57) ( 0.00) (-0.00) (-0.19) (-0.34) (-1.80) (-2.91) (-4.49)95% 0.90 0.73 0.75 0.76 0.83 0.97 1.29 3.14

( 0.51) ( 0.34) ( 0.36) ( 0.37) ( 0.44) ( 0.59) ( 0.91) ( 2.73)std 2.87 1.72 1.85 2.15 2.72 3.36 5.16 9.73

( 2.87) ( 1.70) ( 1.84) ( 2.14) ( 2.72) ( 3.36) ( 5.16) ( 9.74)N 1362 6 102 475 473 124 172 10

24

Table IVRegressions of Excess Returns

This table reports regressions of excess returns on corporate bonds over period 12.1996–07.2002.rfret

10y is the excess return on the 10-year constant maturity U.S. Treasury bond. Eret is theexcess return on the issuer’s equity and N is the number of observations.

All AAA AA A BBB BB B CCCIntercept -0.001 -0.001 -0.001 -0.001 -0.001 0.002 0.001 -0.006

(-4.158) (-3.630) (-8.244) (-13.142) (-4.951) (2.876) (1.961) (-1.073)rf ret

10y 0.451 0.748 0.690 0.611 0.507 0.201 -0.066 -0.380(36.655) (6.125) (28.087) (49.690) (27.826) (3.816) (-1.845) (-1.475)

Eret 0.042 0.003 0.016 0.021 0.039 0.074 0.100 0.109(18.796) (0.685) (4.635) (10.322) (11.038) (6.725) (10.895) (2.794)

R2 0.375 0.658 0.553 0.471 0.352 0.254 0.153 0.216N 42.943 51.500 49.039 43.137 42.655 41.968 40.244 38.500

(1362) (6) (102) (475) (473) (124) (172) (10)

Table VSensitivities of bond returns to equity in the Merton (1974)

model

This table reports the sensitivities of bond returns to equity in the Merton (1974) model fordifferent values of leverage and asset volatility. Leverage is the ratio of the face value of debtto the market value of assets. σA is asset volatility. Leverage and asset volatility are given in%. The riskless rate is assumed to be equal to 5%, time to maturity 10 years. Sensitivities aregiven in 0.01%. The procedure is described in the paper. t-statistics are in parentheses.

σA 10 15 20 25 30 40 50Leverage10 0 0 0 0.05 0.37 3.66 8.70

(18.40) ( 2.42) ( 7.90) ( 9.88) (16.33) (20.55) (27.54)20 0 0 0.07 0.66 2.44 7.91 14.18

( 4.39) ( 7.93) (13.12) (13.67) (18.82) (27.80) (38.65)30 0 0.02 0.42 2.35 5.19 12.02 20.10

( 9.66) (12.67) (18.21) (17.17) (25.23) (32.86) (50.08)40 0 0.15 1.35 4.31 7.72 15.57 22.64

(10.74) (10.28) (20.91) (26.88) (33) (40.93) (53.12)50 0.01 0.55 2.96 6.23 10.31 18.16 24.83

( 7.57) (15.88) (23.96) (32.97) (39.25) (48.93) (58.37)60 0.04 1.33 4.44 8.63 12.70 20.36 26.64

(15.08) (17.82) (27.88) (35.39) (42.87) (54.62) (61.85)70 0.20 2.41 6.33 10.31 14.75 21.53 28.72

(12.61) (23.32) (34.45) (41.62) (46.37) (56.57) (74.88)

25

Table VISensitivities of debt returns to equity implied by the Merton

(1974) model for the actual data set

This table reports sensitivities of bond returns to equity in the Merton (1974) model. Thesensitivities were obtained using a simulation, as described in the paper. Sensitivities are givenin 0.01%. t-statistics are in parentheses.

All AAA-A BBB BB-CCChE 0.046 0.011 0.040 0.153

(29.50) (17.95) (22.87) (32.54)R2 0.74 0.65 0.78 0.93

Table VIILeverage and Volatilities

This table reports results of the first step of a more extensive analysis of hedge ratios. σE isthe historical equity volatility. L is market leverage. σA is the estimated asset volatility, hE

is the hedge ratio calculated for three different estimates of asset volatility. Volatilities andleverage are given in % and hedge ratios are given in 0.01%. N is the number of observations.The details of the estimation procedure are given in the paper.

All AAA AA A BBB BB B CCCLMean 0.346 0.047 0.162 0.239 0.364 0.439 0.587 0.751Median 0.301 0.026 0.110 0.208 0.342 0.449 0.614 0.817Std. Dev. 0.232 0.143 0.169 0.165 0.204 0.222 0.219 0.2095% quantile 0.035 0.014 0.028 0.036 0.057 0.064 0.192 0.34395% quantile 0.804 0.042 0.458 0.564 0.754 0.802 0.886 0.975σE

Mean 0.361 0.251 0.268 0.291 0.348 0.438 0.543 0.803Median 0.326 0.241 0.258 0.277 0.328 0.421 0.517 0.753Std. Dev. 0.165 0.045 0.074 0.092 0.117 0.142 0.193 0.3765% quantile 0.186 0.188 0.156 0.177 0.193 0.240 0.285 0.50495% quantile 0.669 0.346 0.402 0.468 0.585 0.716 0.905 1.205(1− L)σE

Mean 0.221 0.240 0.222 0.219 0.219 0.246 0.222 0.194Median 0.204 0.236 0.222 0.207 0.200 0.217 0.189 0.135Std. Dev. 0.110 0.056 0.071 0.081 0.110 0.138 0.148 0.1935% quantile 0.080 0.183 0.111 0.117 0.086 0.077 0.060 0.02095% quantile 0.420 0.332 0.328 0.365 0.448 0.524 0.524 0.549σA

Mean 0.239 0.240 0.224 0.223 0.230 0.270 0.287 0.300Median 0.219 0.237 0.223 0.211 0.211 0.244 0.258 0.262Std. Dev. 0.106 0.055 0.069 0.080 0.106 0.128 0.133 0.1845% quantile 0.118 0.184 0.116 0.125 0.108 0.117 0.136 0.09595% quantile 0.440 0.333 0.329 0.366 0.453 0.530 0.557 0.622N 58488 272 4027 18729 23068 5031 6203 1158

26

Table VIIIHedge Ratios

This table reports results of the first step of a more extensive analysis of hedge ratios. σE isthe historical equity volatility. L is market leverage. σA is the estimated asset volatility, hE

is the hedge ratio calculated for three different estimates of asset volatility. Volatilities andleverage are given in % and hedge ratios are given in 0.01%. N is the number of observations.The details of the estimation procedure are given in the paper.

All AAA AA A BBB BB B CCChE(σE)Mean 0.095 0.006 0.023 0.047 0.092 0.138 0.221 0.334Median 0.053 0 0.001 0.011 0.064 0.133 0.230 0.343Std. Dev. 0.105 0.022 0.055 0.071 0.089 0.098 0.097 0.0745% quantile 0 0 0 0 0 0.001 0.045 0.20895% quantile 0.306 0.029 0.129 0.213 0.265 0.312 0.367 0.438hE((1− L)σE)Mean 0.020 0.002 0.005 0.012 0.018 0.030 0.045 0.053Median 0.003 0 0 0.001 0.005 0.013 0.023 0.021Std. Dev. 0.037 0.006 0.014 0.026 0.032 0.042 0.056 0.0735% quantile 0 0 0 0 0 0 0 095% quantile 0.095 0.017 0.034 0.066 0.083 0.118 0.163 0.196hE(σA)Mean 0.028 0.002 0.006 0.013 0.021 0.040 0.083 0.123Median 0.007 0 0 0.001 0.008 0.027 0.071 0.116Std. Dev. 0.044 0.006 0.014 0.027 0.033 0.045 0.055 0.0735% quantile 0 0 0 0 0 0 0.011 0.01695% quantile 0.121 0.017 0.036 0.070 0.091 0.135 0.191 0.250N 58488 272 4027 18729 23068 5031 6203 1158

Table IXHedge ratio regressions

This table reports results of a regression analysis of hedge ratios. The regression is

rj,t = αj,0 + βj,EhE,j,trE,t + αj,rfrf10y,t,

where hE,j,t is the hedge ratio for firm j at time t as implied by the model. Under the nullhypothesis that the Merton model holds, βj,E is equal to one (100 bp). rfret

10y is the excessreturn on the 10-year constant maturity U.S. Treasury bond. The details of the estimationprocedure are given in the paper. N is the number of observations.

All AAA AA A BBB BB B CCCIntercept -0.001 (—) -0.002 -0.002 -0.001 0.001 -0.001 -0.007

(-4.554) (—) (-6.764) (-8.601) (-4.978) (1.312) (-1.207) (-0.854)βE 1.206 (—) 0.552 1.173 0.787 2.498 1.540 0.415

(1.156) (—) (-1.304) (0.608) (-0.629) (2.968) (1.762) (-2.159)rf ret

10y 0.369 (—) 0.815 0.688 0.479 0.140 -0.116 -0.408(18.180) (—) (25.734) (34.809) (22.968) (2.172) (-2.165) (-1.386)

R2 0.266 (—) 0.372 0.371 0.289 0.203 0.128 0.108N 42.076 35 48.536 42.902 41.569 42.976 40.471 41.714

(686) (1) (28) (143) (288) (83) (136) (7)

27

Table XRegression of Excess Returns

This table reports results of regression analysis of excess returns on corporate bonds over period12.1996-07.2002. rfret

10y is the excess return on the 10-year constant maturity U.S. Treasurybond. Eret is the excess return on the issuer’s equity. ∆(rf10y − rf2y) is the change in theslope of the term structure (the difference between the yield on ten-year and two-year constant-maturity U.S. Treasury bonds). S&P ret is the return on the S&P index. VIX is the impliedoption volatility on the S&P100 index options. SMB and HML are the Fama-French Smallminus Big and High minus Low factors. N is the number of observations. t-statistics are givenin parentheses.

All AAA AA A BBB BB B CCCInt -0.001 -0.001 -0.002 -0.002 -0.002 0 0.001 -0.004

(-8.628) (-4.049) (-10.237) (-17) (-7.561) (1.077) (0.905) (-0.931)rf ret

10y 0.509 0.796 0.743 0.659 0.573 0.291 -0.018 -0.416(41.557) (6.137) (27.578) (53.560) (31.885) (6.223) (-0.468) (-1.805)

∆(Spd) -0.015 -0.006 -0.008 -0.012 -0.009 -0.027 -0.027 -0.131(-11.239) (-0.958) (-3.419) (-7.504) (-4.369) (-6.261) (-4.526) (-3.941)

Eret 0.030 0.002 0.009 0.008 0.028 0.054 0.091 0.110(13.196) (0.426) (2.423) (4.249) (7.377) (5.304) (9.198) (2.502)

S&P ret 0.008 0.023 -0.002 0.019 -0.002 0.033 -0.014 0.146(1.691) (1.487) (-0.253) (4.157) (-0.178) (1.913) (-0.593) (0.892)

∆(V ix) -0.053 -0.043 -0.060 -0.045 -0.056 -0.054 -0.059 -0.095(-13.073) (-4.414) (-9.083) (-10.712) (-8.282) (-3.314) (-3.071) (-1.035)

SMB 0.104 0.052 0.077 0.092 0.104 0.182 0.100 0.124(19.429) (2.518) (8.165) (15.424) (13.536) (6.929) (3.986) (1.020)

HML 0.023 0.021 0.017 0.012 0.008 0.080 0.040 0.303(5.558) (1.532) (4.154) (3.241) (1.145) (5.400) (1.923) (4.098)

R2 0.463 0.700 0.631 0.553 0.455 0.347 0.221 0.247N 42.943 51.500 49.039 43.137 42.655 41.968 40.244 38.500

(1362) (6) (102) (475) (473) (124) (172) (10)

28