Time-varying asset volatility and the credit spread puzzle › facbios › file ›...

Time-varying asset volatility and the credit spread puzzle ∗

Du Du, Redouane Elkamhi, Jan Ericsson, Min Jiang

January 16, 2013

First version July 2011

Abstract

Extant structural credit risk models underestimate credit spreads while matching default

rates, recoveries, leverage, and equity risk premia - a shortcoming known as the credit spread

puzzle. We develop a new closed-form model with priced stochastic asset volatility that generates

realistic spreads. We evaluate the model using panel data on default swap spreads and realized

equity volatility. A GMM-based specification test shows that the model strongly outperforms

constant volatility models. We provide estimates of firm-specific asset volatility risk premia and

document that these are economically important and are key in allowing the model to explain

levels of credit spreads.

∗Du is with the HKUST Business School, Elkamhi with the Rotman School at the University of Toronto, Jiangwith State Street, Ericsson with the Desautels Faculty of Management at McGill University. Direct correspondenceto Jan Ericsson: [email protected], 1001 Sherbrooke Street West, Montreal, H3A 1G5, Quebec, Canada. Theusual disclaimer applies. The paper has benefited from comments by seminar participants at Case Western ReserveUniversity, HEC Montreal, Honk Kong University of Science and Technology, University of Houston, University ofIowa, Konstanz University, National University of Singapore, University of Toronto, and conference participants atthe 9th International Paris Finance Meeting 2011, ITAM 2012, Tremblant Risk Management 2012, Risk ManagementConference NUS 2011, SKK, IFSID 2012, Montreal. Special thanks to David Bates, Peter Christoffersen, SudiptoDasgupta, Jin Chuan Duan, Adlai Fisher, Santiago Garcıa Verdu, Kris Jacobs, Raunaq Pungaliya, Tom Rietz, AshishTiwari, Anand Vijh, Hao Wang, and Hao Zhou.

1 Introduction

Most structural credit risk models have met with significant difficulties in academic research. First,

attempts to empirically implement models on individual corporate bond prices have failed.1 Sec-

ond, subsequent efforts to calibrate models to observable moments including historical default rates

uncovered what has become known as the credit spread puzzle - the models are unable to match

average credit spreads levels.2 Finally, econometric specification tests further document the dif-

ficulties that existing models encounter in jointly explaining the dynamics of credit spreads and

equity volatilities3.

In this paper, we develop a structural model with time-varying asset volatility to address the levels

and dynamics of both credit spreads and equity volatilities. Our first contribution is to show by

means of a calibration exercise that the presence of a reasonable unlevered asset variance risk

premium allows the model to match spread levels without difficulty.4 Second, we implement a

GMM-based specification test and find that the presence of stochastic asset volatility dramatically

improves the model’s ability not only to account for time variation in equity volatility but also to

explain time series of default swap spread term structures at the firm level. Finally, we carry out

firm level estimation of both the physical and risk-adjusted dynamics of asset value and variance

jointly. We document economically significant asset variance risk premia.

The credit spread puzzle has come to refer to the inability of structural models, when calibrated to

default probabilities, loss rates and Sharpe ratios, to predict spread levels across rating categories

consistent with historical market spreads. Huang and Huang (2003, HH) perform this calibration

analysis for a broad and representative selection of models and find that, as an example, the latter

never predict spreads in excess of a third of the observed levels for 4- and 10-year debt issued by

A-rated firms. The performance is typically worse for more highly rated firms and somewhat better

for low grade firms. 5

1See Jones, Mason and Rosenfeld (1985) and Eom, Helwege and Huang (2003).

2See Huang and Huang (2003)

3See Huang and Zhou (2008).

4In a recent important paper, Cremers Driessen Maenhout (2008) study an alternative route to explaining creditspreads levels: priced jump risk. They find that jump risk premia do significantly help to bring spread levels closer toobserved levels in a similarly structured calibration. We abstract from jumps in this paper and simply note that theyare likely a complementary feature of the data to the volatility that we study. A joint study of jumps and volatilityis left to future work.

5Chen Collin-Dufresne and Goldstein (2009) argue that if one accounts for time variation in Sharpe ratios overthe business cycle, then spreads are more closely aligned with historical averages. Other papers have followed andreinforced the point that macroeconomic conditions can help explain spread levels (see Bhamra Kuehn and Strebulaev(2010), Chen (2012). Another stream of literature has relied on rare disasters to explain credit spreads, see e.g. Gabaix(2012).

1

Huang and Zhou (2008) test a broad set of structural models by designing a GMM-based specifica-

tion test that confronts the models with panels of CDS term structures and equity volatilities. In

addition to ranking the models by rejection frequency, their paper provides insights into the specific

shortcomings of the models. One important finding that emerges from their study is the models’

inability to fit the dynamics of CDS prices and equity volatilities. In particular, the models find

it difficult to generate time variation in the equity volatility of the same magnitude as is actually

observed, suggesting that an extension to allow for stochastic asset volatility is desirable.

In addition to these two important findings, recent empirical work on default swap spreads provides

evidence suggestive of an important role for stochastic and priced asset volatility in credit risk

modelling. Zhang, Zhou and Zhu (2005) perform an empirical study of the influence of volatility

and jumps on default swap prices. Although we abstract from jumps in this paper, their results

point to the importance of modelling time-varying volatility.6 Further evidence is provided in

Wang, Zhou and Zhou (2010) who show that in addition to volatility being important for the

price of default protection, the variance risk premium is a key determinant of firm-level credit

spreads. Both these regression-based studies provide evidence indicating that a structural credit

model with time-varying and priced asset risk may be better poised to explain spreads than its

constant volatility predecessors.

Along with this recent work on credit markets, there is a significant body of literature documenting

time variation in equity volatilities. Given this evidence, financial leverage would have to be the

sole source of variation in stock return volatility in order for asset volatility to be constant, as it is

assumed to be in the majority of structural credit risk models. In fact, recent empirical work by

Choi and Richardsson (2009) clearly documents time variability in asset risk as well.

Overall, stochastic asset volatility would appear to be a compelling extension to a class of models

that has been around for almost 40 years. However, likely for technical reasons, it is one that has

not garnered much attention. We present, in closed form (up to a Fourier Inversion), solutions

to debt and equity prices in a stochastic asset volatility framework where default is triggered by

a default boundary, as in Black & Cox (1976), Longstaff & Schwartz (1995) and Collin-Dufresne

Goldstein (2001). In doing so, we are, to the best of our knowledge, the first to solve the first

passage time problem of stochastic volatility asset value dynamics to a fixed boundary without

recourse to simulation or other numerical methods.

6The authors include both intra-day realized volatility and historical volatility as measures of short term andlong-term volatility, consistent with the notion that equity volatility varies both because of changes in leverage andbecause of changes in asset volatility. Their results suggest that disentangling the two sources of variation is importantfor explaining default swap prices.

2

In a model with stochastic volatility, three important additional determinants of spreads are the

volatility of volatility itself, the asymmetry of volatility, and the presence of a volatility risk pre-

mium. That the volatility itself is made an idiosyncratic risk source, uncorrelated with the level of

a firm’s asset value, does positively influence credit spreads. While it does so in a modest way for

longer-term credit spreads, it stands to make a significant impact on spreads for up to ten years

to maturity. The same is true for the correlation between shocks to asset values and variances. A

modest “leverage effect” at the unlevered firm value level leads to higher spreads with a pattern

similar to that of the volatility of asset risk: a non-trivial increase in long-term spreads, but an

important increase in relative spreads for maturities up to ten years. Finally, the dominant effect

stems from the market price of asset volatility risk, which can increase spreads by an order of

magnitude.

Comparative statics such as these are limited in that they do not reflect the constraints faced

when taking a model to the data. To illustrate this, we first replicate the Huang and Huang

(2003) calibration on rating based aggregates as a benchmark. This involves requiring the model

to simultaneously fit four moment conditions: the historical probability of default, recovery rates,

equity risk premia and leverage. We first confirm that, in the absence of stochastic asset risk,

our model replicates the credit spread puzzle - it can not generate spreads in line with historical

averages. However, we find that for reasonable parameter values governing volatility dynamics and

risk premia, our model has no difficulty in matching spread levels.

One potential concern with such a calibration exercise is that several parameters remain free.7 To

some degree this is mitigated by our finding that only the asset volatility risk premium can influence

spreads in a meaningful way in this exercise. The other two channels - volatility of asset risk and

an asset level “leverage effect” - are counteracted by the matching of empirical moments. Hence,

the volatility risk premium is the key ”free” parameter that determines how reasonable our spread

estimates are. We address this later by means of firm-level estimation.

We proceed to confront the model with firm level data on credit default swaps and equity volatil-

ities. We do this in two stages: first we evaluate the fit of our model using a formal GMM-based

specification test and, second, we estimate asset volatility risk premia implied by the data by means

of a simulated maximum likelihood strategy.

To evaluate the goodness of fit, we perform a specification test to understand the impact of stochas-

tic volatility on the model’s ability to fit time series of equity volatilities and default swap data

7Note that this issue is present also in the Huang and Huang (2003) study. Our model has 3 free parameters aftermatching the moment conditions. In Huang and Huang (2003) this number reaches 6 for some model specifications.

3

across maturities. To do so, we follow Huang and Zhou (2009) in relying on a GMM based test. We

first study the constant volatility version of our model and establish results very similar to those

reported in their paper. The model is rejected for 48 firms out of 49 and suffers from an inability

to generate sufficient time variation in equity volatility, while providing a poor time series fit of

default swap spreads.

In contrast, the model with time-varying volatility fits the spot volatility time series very well,

and, importantly, in doing so is able to explain much more of the default swap spread time series.

Absolute percentage pricing errors are smaller for 34 out of the 35 rating / model combinations that

Huang and Zhou study and are approximately halved in comparison. The model with stochastic

volatility could be rejected only for 3 firms out of 49 whereas for a constrained constant volatility

version, 48 out of 49 firms lead to a rejection of the model.8

As already noted, modelling stochastic asset volatility increases the number of parameters and thus

renders a rejection harder compared to the nested constant volatility model. It is then noteworthy

that the Collin-Dufresne and Goldstein (2001) model (CDG) which is part of the evaluation in

Huang and Zhou (2008), has the same number of additional parameters as our stochastic asset risk

model. Our non-rejection rates range between 94% and 98% while for the CDG model in Huang

and Zhou the corresponding range is 67% and 75%.

Next, we seek to measure firm-level volatility risk premia implicit in our default swap and equity

volatility panel. This is challenging as it requires the estimation of values and dynamics for both

asset values and volatilities, both of which are unobservable. To estimate both physical and risk-

adjusted firm value and volatility dynamics, we develop and implement a simulated maximum

likelihood methodology. Intuitively, we rely on the time series dimension of the data to identify

physical dynamics while the cross-section (term structures of default swaps) allow us to identify

the risk-adjustment.

To the best of our knowledge, we provide the first estimates of asset volatility risk premia in

the literature. Our estimates are positive across all firms, consistent with priced asset risk. The

estimates are quite stable across firms and are quantitatively consistent with metrics published

in the literature on equity volatility. To benchmark the level of our estimates we also translate

them into a metric comparable to ratios of option-implied to realized volatilities reported in the

literature. Our estimates range from 1.06 to 1.09 which is slightly lower than the average ratio of

implied to realized volatilizes for the SP100 index of 1.19 as reported in Driessen Maenhout and

8In addition, we document risk-adjusted mean reversion, a correlation between asset value and volatility shocksof -0.58, and asset risk volatility of 37% on average.

4

Vilkov (2009). Han and Zhou (2011) report one-month implied and realized volatilities for a panel

of 5000 individual stocks. The ratio of their reported averages is 1.17. Hence, it appears our model

is able to fit credit spread and equity volatility panels while relying on realistic levels of volatility

risk premia.

Given our estimates of both physical and risk-adjusted dynamics for firm value, we can decompose

credit spreads into a component due to average losses, a risk premium excluding volatility risk and

a component due to the asset variance risk premium. The volatility risk premium component is by

far the largest and can account for up to two thirds of total default swap spreads.

This paper is organized as follows: Section 2 describes the model and explains how we derive closed-

form solutions for our stochastic volatility credit risk model. Section 3 covers the comparative

statics, while Section 4 discusses the various calibration exercises. Section 5 reports on our time-

series specification tests, and finally Section 6 concludes.

2 The model

We model the firm’s unlevered asset value X as the primitive variable. Asset value and volatility

dynamics are described by the following two SDEs

dXt

Xt= (µ− δ)dt+

√VtdW1,

dVt = κ(θ − Vt)dt+ σ√VtdW2, (1)

where δ is the firm’s payout ratio and E(dW1dW2) = ρdt. Under the risk-adjusted measure, Xt

and Vt followdXt

Xt= (r − δ)dt+

√VtdW

Q1 ,

dVt = κ∗(θ∗ − Vt)dt+ σ√VtdW

Q2 , (2)

with κ∗ = κ+ λV and θ∗ = θκ/κ∗, where λV is the volatility risk premium.9

In what follows, we provide the solutions for the firm’s equity value and equity volatility. To solve

9Note we later assume that the price of volatility risk, λV V , is proportional to the volatility of asset variance,λV = kσ, where k is a constant.

5

for the firm’s equity value, we assume that the firm issues consol bonds as in Black & Cox (1976)

and Leland (1994).

Then the equity value can be written as the difference between the levered firm value (F) and the

debt value (D), i.e., E(X) = F (X)−D(X). The firm’s levered asset value is given by

F (X) = X +ηc

r(1− pD)− αXDpD, (3)

where X, η, c, α, XD and pD denote the initial unlevered asset value, the tax rate, the coupon

rate, the liquidation cost, the default boundary and the present value of $1 at default respectively.

In equation (3), the first term is the unlevered asset value, the second term is the tax benefit and

the third term is the bankruptcy cost. The debt value is the present value of the coupon payments

before default and recovered firm value at default, which is given by

D(X) =c

r+[(1− α)XD −

c

r

]pD.

Thus, the equity value is given by

E(X) = X − (1− η)c

r+[(1− η)

c

r−XD

]pD. (4)

Applying Ito’s lemma, we obtain the stochastic process for the equity value as follows:

dEtEt

= µE,t +Xt

Et

∂Et∂Xt

√XtdW1t +

1

Et

∂EtVt

σ√VtdW2t,

where µE,t is the instantaneous equity return.

σE,t =

√√√√[(Xt

Et

∂Et∂Xt

)2

+

(σ

Et

∂Et∂Vt

)+ ρσ

Xt

E2t

∂EtXt

∂Et∂Vt

]Vt. (5)

From equation (4), we know the firm’s equity value given pD. Under the risk-adjusted measure Q,

pD = EQ[e−rτ ], (6)

with τ = inf{s > 0, Xs 6 XD}, denotes the default time. To solve for pD, we need to compute the

probability density function of the stopping time τ under the risk-adjusted probability measure Q.

To do so we will rely on Fortet’s lemma, as we discuss next.

6

2.1 Closed-form pricing

To solve our model we rely on Fortet’s lemma which was first introduced into the finance literature

by Longstaff and Schwartz (1995). The later work of Collin-Dufresne and Goldstein (2001) extends

their work to the case where the state variables (leverage ratio and interest rate) follow a general two-

dimensional Gaussian Markov process. In both papers, the state variables are (jointly) Gaussian.

However, this is not the case given our volatility dynamics. In order to apply Fortet’s lemma in our

framework, we first need to solve for the joint probability density of the asset value and variance,

which we do by means of an inversion of the characteristic function.

Define zt = ln (Xt/XD), a measure of the distance to default, which serves as another state in

our model besides the asset return volatility Vt. Let pQ(zT , VT ; zt, Vt, h) denote the Q-measure

transition density function of the two states (zT , VT ) conditional on their starting values at t,

where h = T − t. Define Ψ(.) as the joint characteristic function of (zT , VT ) conditional on (zt, Vt),

i.e., Ψ(ϕ1, ϕ2; zt, Vt, h) ≡ EQt[ei(ϕ1zT+ϕ2VT )|zt, Vt

], where EQt denotes the expectation under the

Q-measure. It can be shown that for ∀σ > 0, the closed-form solution for Ψ(t) is given by

Ψ(ϕ1, ϕ2; zt, Vt, h) = ef1(h;ϕ1,ϕ2)+f2(h;ϕ1,ϕ2)Vt+iϕ1zt , (7)

where i2 = −1;

f1 (h;ϕ1, ϕ2) = h

(r − δ − κ∗θ∗ρ

σ

)iϕ1 +

κ∗θ∗

σ2

[hκ∗ + h

√G+ 2ln

H + 1

Heh√G + 1

], (8a)

f2 (h;ϕ1, ϕ2) =1

σ2

[κ∗ − iρσϕ1 −

√GHeh

√G − 1

Heh√G + 1

], (8b)

with

G(ϕ1) = σ2(1− ρ2)ϕ21 + (σ2 − 2ρσκ∗)iϕ1 + κ∗

2,

H(ϕ1, ϕ2) = − iρσϕ1 − κ∗ −√G+ iσ2ϕ2

iρσϕ1 − κ∗ +√G+ iσ2ϕ2

.

By Fourier inversion,

pQ(zT , VT ; zt, Vt, h) =1

4π2

∫ ∞−∞

e−iVTϕ2

[∫ ∞−∞

e−izTϕ1Ψ(ϕ1, ϕ2; zt, Vt, h)dϕ1

]dϕ2. (10)

We apply the Gauss-Laguerre Quadrature with the integration range between zero and infinity,

which provides more accurate results than that using Gauss-Hermite Quadrature for which the

7

range is from -infinity to infinity. To match the range for Gauss-Laguerre Quadrature, we define

g(zT , ϕ2 ; zt, Vt, h) ≡ 1

2π

∫ ∞−∞

e−izTϕ1Ψ(ϕ1, ϕ2; zt, Vt, h)dϕ1

=1

2π

[∫ ∞0

e−izTϕ1Ψ(ϕ1, ϕ2; zt, Vt, h)dϕ1 +

∫ ∞0

eizTϕ1Ψ(−ϕ1, ϕ2; zt, Vt, h)dϕ1

]. (11)

Expressed in terms of g (.) , pQ (.) defined by (10) can be rewritten as:

pQ(zT , VT ; zt, Vt, h) =1

2π

∫ ∞−∞

e−iVTϕ2g(zT , ϕ2 ; zt, Vt, h)dϕ2

=1

π

∫ ∞0

Re[e−iVTϕ2g(zT, ϕ2 ; zt,Vt,h)

]dϕ2,

where the second equality holds because pQ (.) is real.10

Denote by H(zτ = 0, Vτ , τ |zt, Vt, t) the probability density for the first passage time of the asset

value to the default boundary. Since (z, V ) follow a two-dimensional Markov process in our setup,

applying Fortet’s lemma yields:

pQ(zT , VT ; zt, Vt, h) =

∫ t

0dτ

∫ ∞0

dVτH(zτ = 0, Vτ , τ |zt, Vt, t)pQ(zT , VT ; zτ , Vτ , T − τ). (12)

for zt > 0 > zT .11 To numerically solve (15) for H(.), it is convenient to introduce

Ψ (VT , T ) ≡∫ 0

−∞dzT p

Q(zT , VT ; z0, V0, T ) (13)

ψ (VT , T, Vt, t) ≡∫ 0

−∞dzT p

Q(zT , VT ; zt, Vt, h). (14)

Setting t = 0 in (12) without the loss of generality and integrating both its sides by∫ 0−∞ dzT , we

obtain:

Ψ (VT , T ) =

∫ t

0dτ

∫ ∞0

dVτH(zτ = 0, Vτ , τ |zt, Vt, t)ψ (VT , T, Vτ , τ) . (15)

10In practice, we find that Gauss-Laguerre Quadrature converges more quickly if we first compute pQ (α1, α2) ≡eα1x1t+α2x2tpQ(zT , VT ; zt, Vt, h) for α1, α2 > 0, where we depress the dependence of pQ (.) on other variables. Thecorresponding characteristic function is Ψ(α1, α2) ≡

∫∞−∞ dVT

∫∞0dzT e

i(ϕ1zT+ϕ2VT )pQ (α1, α2) = Ψ(ϕ1 − iα1, ϕ2 −iα2; zt, Vt, h). To compute it, we define g(α1, α2) ≡ e−α1x1t−α2x2t

2π

∫∞−∞ e

−izTϕ1Ψ(ϕ1 − iα1, ϕ2 − iα2; zt, Vt, h)dϕ1 =e−α1x1t−α2x2t

2π

[∫∞0e−izTϕ1Ψ(ϕ1 − iα1, ϕ2 − iα2; zt, Vt, h)dϕ1 +

∫∞0eizTϕ1Ψ(−ϕ1 − iα1, ϕ2 − iα2; zt, Vt, h)dϕ1

]. By

Fourier inversion, pQ(zT , VT ; zt, Vt, h) = e−α1x1t−α2x2t pQ (α1, α2) = e−α1x1t−α2x2t

2π

∫∞0

Re[e−iVTϕ2 g(α1, α2)

]dϕ2.

Numerically, we find that setting α1 = 0.5 and α2 = 1.5 generally delivers fairly accurate results compared tosimulation.

11One main intuition behind Fortet’s lemma is that given a continuous process, if it starts at zt which is higherthan a fixed boundary (zD = 0), it has to cross the boundary to reach a point below the boundary (zT ).

8

Again, we use Gauss-quadrature method to compute integrations in (13)–(14). To fit in the range

of Gauss-Laguerre Quadrature which is from zero and inf, numerically we compute Ψ (.) and ψ (.)

by

Ψ (VT , T ) ≡∫ ∞

0dzT p

Q(−zT , VT ; z0, V0, T ),

ψ (VT , T, Vt, t) ≡∫ ∞

0dzT p

Q(−zT , VT ; zt, Vt, h).

Given Ψ (.) and ψ (.) , the next step is to solve H(.) specified by (15). Denote by V and V the

maximum and minimum for the asset variance, respectively. First, we discretize the time interval

T and the variance V into nT and nV equal steps, respectively. Second, we define q(Vi, tj) =

4t · 4V · H(ztj=0, Vtj = Vi, tj |z0, V0, 0), where tj = j TnT

= j4t for j ∈ {1, 2, · · ·, nT }; Vi =

V + i4V = V + i V−VnVfor i ∈ {1, 2, · · ·, nV }. The discretized version of equation (15) is thus:

Ψ (Vi, tj) =

j∑m=1

nV∑u=1

q(Vu, tm)ψ (Vi, tj , Vu, tm) , (16)

for ∀i ∈ {1, 2, · · ·, nV } and ∀j ∈ {1, 2, · · ·, nT }, where

1

4t · 4Vq(Vu, tm) = H(ztm=0, Vtm = Vu, tm|z0, V0, 0)

which denotes the probability density that the default time is tm at which the realized asset variance

is Vu. From (16), we obtain q(Vi, tj) recursively as follows:

q(Vi, t1) = 4VΨ (Vi, t1) , (17)

q(Vi, tj) = 4V

[Ψ (Vi, tj)−

j−1∑m=1

nV∑u=1

q(Vu, tm)ψ (Vi, tj , Vu, tm)

]for ∀j ∈ {2, 3, · · ·, nT }. (18)

In our numerical procedure, we select V close to zero and restrict V to not exceed the observed

equity variance. Furthermore, we find that choosing V to include four standard deviations about

the long-run average is sufficient for convergence of (17)–(18). In the actual implementation of

(17)–(18), we apply the two-dimensional collocation method (e.g., Miranda and Fackler, 2002)

along both the variance and the time dimension. More specifically, we use Chebyshev interpolation

and evaluate q(.) at{(V node, tnode

)}, where

{(V node, tnode

)}are referred to as the interpolation

nodes. For two-dimensional Chebyshev polynomials defined in ℵ ≡([V , V

], [0, T ]) with the degree

of m by n, we have m·n interpolation nodes that set to zero all the involved Chebyshev polynomials.

q(.) at any other points within ℵ are evaluated by interpolation. Numerically, we find that setting

m = n = 10 is sufficient to yield accurate results. Since our estimation scheme (described in

9

the following) involves backing out the states from observables using a large scale grid search,

interpolation is vital to dramatically reducing the computation time.

Given q (.) , we approximate the present value of $1 at default pD using:

pD =N∑j=1

nV∑i=1

e−rtjq(Vi, tj). (19)

We find that tN = 60 years is sufficient for convergence. Once pD is found, we solve for firm’s equity

value Et and the implied equity volatility σE,t according to (4)–(5). In addition, the risk-neutral

probability that the default (first passage) time is less than T is computed by

G(0, T ; z0, V0) =

nT∑j=1

nV∑i=1

q(Vi, tj). (20)

The model-implied CDS spread is given by

cdst(T ) =(1−R)

∑4Ti=1B(t, t+ Ti) [G(t, t+ Ti)−G(t, t+ Ti−1)]∑4Ti=1B(t, t+ Ti) [1−G(t, t+ Ti)] /4

, (21)

where T is the time to maturity; R denotes the (constant) recovery; B(t, t+ Ti) is the default-free

discount function which is simply e−rTi in our model; we depress the dependence of G(.) on the

two states for simplicity. We verify the accuracy of the above ”semi closed-form” computation of

σE,t and cds (.) by simulation at various states and maturity. By comparison, our method is much

faster than simulation which is essential to the tractability of ML estimation.

3 Model Implications

3.1 Comparative Statics

Introducing stochastic volatility into a credit risk model adds three new potential channels for

asset risk to influence credit spreads. First, the very fact that volatility is random may impact

spreads directly. As we shall see, this is particularly true for short-term credit spreads. Second,

volatility may be correlated with shocks to asset value. For example, there may be a “leverage

effect” or asymmetry at the unlevered volatility level - that is, the asset risk may increase as the

value decreases. This would work over and above the traditional financial leverage effect that is

already present in Merton (1974) and subsequent models. Third, volatility risk may be systematic

10

and carry a risk premium that is eventually reflected in credit spreads. Figure 1 summarizes these

comparative statics.

Uncertainty about the volatility level generates fatter tails in the asset value distribution, which, all

else equal, increases the likelihood of distressed scenarios and thus spreads. The top panel in Figure

1 demonstrates this effect by retracing the yield spread curves for different levels of the volatility

of the asset variance, nesting the constant volatility case, corresponding to the original Black and

Cox (1976) model. Note that the effect can be quite significant but is more so for maturities less

than 10 years. This is even more noticeable in the lower panel of the figure that reproduces the

same data in terms of ratios of spreads to the Black and Cox (1976) case. For maturities less than

7 years, it is quite straightforward for the model to more than double spreads. The relative effect

dissipates further out on the term structure and seems to reach stable levels after the 10-year tenor

- a spread increases in the range of 10% to 20% of the constant volatility spread.

Panel B visualizes the relationship between asset volatility asymmetry and spreads. For this exper-

iment, we make conservative assumptions regarding the volatility of asset risk, setting σ = 0.3 and

κ = 4. The case where ρ = 0 corresponds to a Black & Cox model extended to allow for asset-risk

dynamics independent of asset value dynamics. The resulting spreads are barely higher than in the

constant volatility case.

Setting the asymmetry parameter to ρ = −0.3, which implies positive shocks to asset risk on

average when asset values suffer negative shocks, has a limited effect on spreads, in the range

of 2-10 basis points. For short term spreads (less than five years), this amounts to a non-trivial

relative increase - spreads are approximately doubled. The absolute size of the increase is relatively

stable so that for longer tenors such as 15-20 years, the percentage stabilizes around 5% of the total

spread. Increasing the correlation between asset volatility and value shocks to ρ = −0.6 provides

a more significant boost in spreads. With this level of correlation, 5-year spreads essentially triple

as compared to the zero correlation case. At 15 years, spreads increase by about a fifth of the

no-asymmetry spreads.

The pattern for the relative spread increases is quite similar to the one reported for the volatility

of volatility parameter σ. It seems that the impact of asymmetry might be quantitatively slightly

more important than volatility risk itself, although not dramatically.

11

0 1 5 10 15 200

20

40

60

Time to maturity

Yie

ld s

pre

ad

(b

ps)

A. Impact of Volatility of Asset Volatility (σ)

σ=0.3

σ=0.6

σ=0.9

Black−Cox

0 5 10 15 200

20

40

60

Time to maturity

Yie

ld s

pre

ad

(b

ps)

B. Impact of Leverage Effect (ρ)

ρ=0

ρ=−0.3

ρ=−0.6

ρ=−0.9

Black−Cox

0 5 10 200

50

100

150

200

Time to maturity

Yie

ld s

pre

ad

(b

ps)

C. Impact of Market Price of Volatility Risk (k)

k=2

k=4

k=8

Black−Cox

Figure 1. Comparative statics. The top panel shows the impact of the volatility of asset

volatility on the yield spread when the market price of volatility risk is zero. The solid curve

corresponds to the Black-Cox (1976) setting, where the asset volatility is a constant. In Panel

B, the values in the Y-axis are normalized by (or relative to) the corresponding values from the

Black-Cox case. The initial asset value X0 = 100, the default boundary XB = 35, the initial

asset volatility is 21%, the yearly interest rate is 8% and the asset payout ratio is 6%. The other

parameter values for the stochastic volatility model are: κ = 4, ρ = −0.1, θ = 0.212.

Finally, panel C illustrates the impact of volatility risk premia on credit spreads without any

asymmetry effect (ρ = 0). As can be seen, the effect of the risk premium parameter k is first-

order. Spreads can be be increased dramatically by allowing for systematic asset volatility risk.

For example, when k = 3, spreads more than triple for some maturity segments. In a relative sense,

it is clear from the lower panel of Figure 1 that, like in the case of volatility risk (σ) and asymmetry

12

(ρ), the effect dominates the shorter part of the term structure, up to about 10 years. However,

the sheer magnitude of the impact makes the effect significant for all maturities. While short-term

spreads can be inflated tenfold, long-term spreads can easily double, if not triple.

Obviously, the difficulty at this stage will be to determine reasonable values for the unlevered

volatility risk premium. We address this below, and we will see that this third effect of stochastic

volatility on credit spreads is in fact the dominant one, and will survive the calibrations to empirical

moments.

3.2 Stochastic Asset Volatility and the credit spread puzzle

We have now documented that in a comparative static setting, three channels exist that may have

an important effect on credit spreads: the volatility of asset risk itself, a “leverage” effect at the

unlevered firm level and risk premia associated with shocks to asset risk. As we shall see, only one of

these effects, the volatility premium, can help structural models achieve the levels of credit spreads

necessary to address what has recently become known as the credit spreads puzzle. Huang and

Huang (2003) calibrate a selection of different structural models to historical default rates, recovery

rates, equity risk premia and leverage ratios. They find that all models are consistently incapable

of simultaneously matching credit spreads while calibrated to these four moments. Their results

are striking since they compare models with quite different features: stochastic interest rates, time-

varying leverage, jump risk, counter-cyclical risk premia, endogenous default and strategic debt

service. None of these extensions of the basic Merton (1974) framework is able to more than

marginally bring market and model spreads closer to each other. This finding forms the basis for

the credit spread puzzle.

We now ask whether the three channels through which stochastic asset volatility may influence

spreads in our model, can help reconcile model with market spreads on average. In order to do so,

we perform a calibration experiment closely following the methodology used in Huang and Huang

(2003), while relying on their data for comparability. In other words, we require our model to

match, for different rating categories, the following observables

1. The historical default probability

2. The equity risk premium

3. The leverage ratio

4. The recovery rate

13

Table 1 reports on this exercise. To highlight the marginal contribution of stochastic volatility in

our model, the top panel in Table 1 performs the calibration exercise with all parameters related to

time-varying volatility set to zero (σ = ρ = κ = k = 0). The model thus recovered, corresponding

to Black and Cox (1976), behaves very similarly to those studied in Huang and Huang (2003).

For high grade bonds, the model cannot explain any significant part of the spread and reaches a

maximum of 68% for the lowest grade bonds.

In panel B, we assume values for the additional parameters to be ρ = −0.1, σ = 0.3, κ = 4, and

k = 7. The asymmetry at the asset level is chosen to be modest to allow financial leverage effect to

have a significant impact on the asymmetry obtained for stocks. By means of comparison, in equity

markets, Heston (1993) uses ρ = −0.5, while Broadie Chernov and Johannes (2009) use ρ = −0.52

and Eraker Johannes and Polson (2003) find values for ρ between -0.4 and -0.5. Pan (2002) uses a

value for λV = kσ equal to 7.6, while Bates (2006) uses a λV equal to 4.7. Our choice of k implies

a volatility risk premium λV = 2.1. Bates (2006) documents estimates of κ in the range of 2.8 and

5.9 for a selection of models, whereas Pan (2002) estimates values ranging between 5.3 and 7.

For the Aaa category we are able to explain about 96% of average historical spread levels. For Aa to

Baa we explain between 73% and 82% while for the two lowest we actually overestimate spreads by

13% and 24% respectively. This compares to 16% for Aaa, 29% for Baa and a maximum of 83% for B

rated firms in Huang and Huang (2003). It is clear from this table that for this set of parameters, we

can address the spread underestimation for high and low rating categories and reduce it significantly

for the intermediate ones. Our framework, for reasonable and conservative inputs, does not suffer

from the same systematic underestimation problem that all the models studied in Huang and Huang

(2003) are subject to. In fact, it is straightforward to find combinations of values for the parameters

such that spreads are exactly matched. We repeat the same analysis (not reported here) where we

selectively shut down asymmetry and volatility risk premia to identify which of the three channels

drives our results. Only the volatility risk premium has the ability to allow the model to match

spreads.12

Unfortunately, we do not have rating-specific estimates of the new parameters related to our stochas-

tic volatility model. Hence, we cannot, at this stage, claim that our model is able to match historical

spreads given historical moment restrictions, only that it has no difficulty in generating the required

12The reason for this results is discussed in Chen Collin-Dufresne Goldstein (2009). One can write theprice P of a defaultable discount bond as P = E [Λ(At, t)(1 − 1τ≤T (At, Bt, t) · Lτ (At, Bt, t)], where A and Bare common and idiosyncratic state variables respectively, τ is the default time, Λ is the pricing kernel, andL is the recovery at default. This can be rewritten as E [Λ(At, t)] · E [(1 − 1τ≤T (At, Bt, t) · Lτ (At, Bt, t)] −cov (Λ(At, t), (1 − 1τ≤T (At, Bt, t) · Lτ (At, Bt, t)). In forcing the model to match historical default rates and recovery,the first product cannot be influenced, Only parameters that impact the covariance term can do so and in our modelthis will be the volatility risk premium.

14

spread levels.

We next consider a GMM-based specification test following Huang and Zhou (2010) who document

the ability of a selection of earlier models to explain time series of CDS term structures and equity

volatilities. They conclude that the evaluated models exhibit significant difficulties in doing so.

Adapting this exercise to as stochastic volatility setting will permit us to isolate the contribution

of variance dynamics to the pricing performance of our framework.

4 Stochastic asset volatility and pricing performance

4.1 Data

The CDS spread is the premium paid to insure the loss of value on the underlying realized at

pre-defined credit events. This contrasts with the yield spread of a corporate bond, which reflects

not only default risk but also the choice of risk-free benchmark yield, the differential tax treatment

and liquidity of corporate bonds vs. Treasury bonds. Further, while bonds age over time, CDS

spreads are quoted daily for a fixed maturity. In addition, CDS contracts trade on standardized

terms and while CDS and bond spreads are quite in line with each other in the long run, in the

short run CDS spreads tend to respond more quickly to changes in credit conditions. For all these

reasons, it is plausible that the CDS spread is a cleaner and more timely measure of the default

risk of a firm than bond spreads. As a result, they may be better suited for specification tests of a

structural credit risk model.

We collect single-name CDS spreads from Markit. Daily CDS spreads reflect the average quotes

contributed by major market participants. This database has already been cleaned to remove

outliers and stale quotes. We require that two or more banks should have contributed spread

quotes in order to include an observation (Cao, Yu, and Zhong, 2010). The data sample that is

available to us includes only the firms that constituted the CDX index from January 2002 to March

2008.

Our sample includes US dollar-denominated five-year CDS contracts written on senior unsecured

debt of US firms. While CDS contracts range between six months and thirty years to maturity, we

use the 1, 3, 5,7 and 10 years only because that are relatively more liquid than other maturities (6

months, 2 years, 20 years).

The range of restructurings that qualify as credit events vary across CDS contracts from no restruc-

15

turing (XR) to unrestricted restructuring (CR). Modified restructuring (MR) contracts that limit

the range of maturities of deliverable instruments in the case of a credit event are the most popular

contracts in the United States. We therefore include only US dollar-denominated contracts on

senior unsecured obligations with modified restructuring (MR), which also happen to be the most

liquid CDS contracts in the US market (Duarte, Young, and Yu, 2007).

Together with the pricing information, the dataset also reports average recovery rates used by

data contributors in pricing each CDS contract. In addition, an average rating of Moody’s and

S&P ratings as well as recovery rates are also included. Following Huang and Zhou (2008) we

perform our test on monthly data. Their sample is restricted to 36 monthly intervals as their

sample ends in 2004. Instead, we require that the CDS time series has at least 62 consecutive

monthly observations to be included in the final sample. Another filter is that CDS data have

to match equity price (CRSP), equity volatility computed from (TAQ) and accounting variables

(COMPUSTAT). We also exclude financial and utility sectors, following previous empirical studies

on structural models. After applying these filters, we are left with 49 entities in our study.

In testing structural models, the asset return volatility is unobserved and is usually backed out

from the observed equity return volatility. Often, a rolling window of daily returns volatility is

used to proxy for equity volatility. In order to benchmark our results to the specification tests of

alternative models covered in Huang and Zhou (2008) we use a more accurate measure of equity

volatility from high-frequency data. Following Huang and Zhou (2008) we use bi-power variation

to compute volatility. As shown by Barndorff-Nielsen and Shephard (2003), such an estimator

of realized equity volatility is robust to the presence of rare and large jumps. The data on high

frequency prices are provided by the NYSE TAQ data base, which contains intra-day (tick-by-tick)

transaction data for all securities listed on NYSE, AMEX, and NASDAQ. The monthly realized

variance is the sum of daily realized variances, constructed from the squares of log intra-day 5-

minute returns. Then, monthly realized volatility is the square-root of the annualized monthly

realized variance.

4.2 GMM Estimation of the Model

Let cds(t, t+T ) and cdsobs(t, t+T ) denote the model-implied and empirically observed CDS spreads

of a CDS contract at time t for which the maturity date is t + T respectively. Let σE,t and σobsE,t

denote the model-implied and empirically observed equity volatilities at time t respectively. The

16

model-implied CDS spread is given by

cds(t, t+ T ) =(1−R)

∑4Ti=1B(t, t+ Ti) [Q(t, t+ Ti)−Q(t, t+ Ti−1)]∑4Ti=1B(t, t+ Ti) [1−Q(t, t+ Ti)] /4

, (22)

where R is the recovery, B(t, t + Ti) is the default-free discount function and Q(t, t + Ti) is the

risk-neutral default probability. The model-implied equity volatility is given by

σE,t =

√√√√[(Xt

Et

∂Et∂Xt

)2

+

(σ

Et

∂Et∂Vt

)+ ρσ

Xt

E2t

∂EtXt

∂Et∂Vt

]Vt. (23)

We define ft(Θ) as the overidentifying restrictions, which is given by

ft(Θ) =

cds(t, t+ T1)− cdsobs(t, t+ T1)

· · · · ··cds(t, t+ Tj)− cdsobs(t, t+ Tj)

σE,t − σobsE,t

, (24)

where Θ = (ρ,XD, κ, θ, σ) is the parameter vector to be estimated. 13 The term structure of CDS

spread includes five maturities: 1, 2, 3, 5, 7 and 10 years. Thus, we apply the seven moment

conditions (from six CDS tenors and equity volatility) to estimate five parameter values in the

GMM test. Given that the model is correctly specified, we obtain that E [ft(Θ)] = 0. We define

the sample mean of the moment conditions as gT (Θ) = 1/T∑T

t=1 ft(Θ), where T is the number of

time series observations. Following Hansen (1982), the GMM estimator is given by

Θ = arg min gT (Θ)′W (T )gT (Θ), (25)

where W (T ) is the asymptotic covariance matrix of gT (Θ).14 With some regularity conditions, the

GMM estimator Θ is√T consistent and asymptotically normally distributed given that the model

is correctly specified (null hypothesis). The J-statistic is given by

J = T gT (Θ)′W (T )gT (Θ). (26)

The J-statistic is asymptotically distributed as a Chi-square with the degree of freedom being equal

13There are two latent variables in the estimation: asset value and asset variance at each observation date. FollowingHuang and Zhou (2008), we back out the asset value from the observed leverage ratio, which is defined as the ratioof face value over the asset value. We estimate the initial asset variance in the two steps to be discussed later in thissection.

14Following Newey and West (1987), we use a heteroskedasticity robust estimator for W (T ).

17

to the difference between the number of moment conditions and the length of Θ, which is equal to

two in our setup.

We estimate our model in two steps. First, we set the initial asset volatility (√Vt) at each obser-

vation date t to be (1 − leveraget)σobsE,t and then obtain one GMM estimator (Θ) for Θ. In the

second step, we obtain the updated asset volatility (

√V updatet ) at each observation date t such that

the model implied equity volatility to be equal to the observed one, i.e., σmodelE (V updatet , Θ) = σobsE,t.

Then use V updatet to obtain the updated GMM estimator (Θupdate) for Θ.

4.3 Pricing performance

Table 2 reports summary statistics for the 49 firms (4116 default swap quotes in total) in our sample

which spans the period 2002-2008 and intentionally contains many of the firms that are also part

of the sample in the Huang and Zhou (2008) study we rely on as a benchmark.

Rating-based averages for equity volatilities range from 22% to 41% and leverage ratios from 26%

to 77%. Asset payout rates are also quite similar to those in Huang and Zhou (2008) typically just

above 2%. CDS spreads are similar as well. Panel C in Table 2 reports on the standard deviations

of CDS spreads.

Table 3 contains the parameter estimates resulting from the GMM implementation. We note first

that the degree of volatility asymmetry - that is, the correlation between shocks to asset values

and asset variances (ρ) is similar across firms and ratings and averages -0.58. This is similar to

values reported in the literature on equity volatilities (see Eraker Johannes and Polson (2003) and

the discussion in section 4 above). This is higher than the value we assumed in the comparative

statics above (-0.1) and may provide evidence of a “leverage” effect at the asset value level. In

other words, the asymmetry observed in equity markets stems both from mechanical changes in

financial leverage as stock prices fluctuate and a negative correlation between the levels of asset

values and volatility.

We find that for most rating categories, the instantaneous correlation between asset value and

variance shocks is lower than for equity and equity variance shocks. However, the magnitude of

this difference is small. This suggests that, quantitatively, financial leverage only plays a minor

role in the asymmetry observed at the equity return level.

The estimated speed of mean reversion under the risk-adjusted probability measure (κ∗) ranges

from 0.5 and 1.2 averaged within rating categories, lower than the levels we used in the compar-

18

ative statics. Given that higher mean reversion speeds will tend to reduce variance volatility, our

assumptions in the comparative statics, like those make for the correlation parameter, also appear

conservative. The asset variance volatility parameter (σ), is estimated to values in the range of 24%

to 45%, averaging 37%, somewhat higher than our choice of parameter value in the comparative

statics (30%).

We find that the default boundary is estimated to between 62% and 75% of the book value of

debt. This entails that a BBB firm, whose default boundary is 67% of debt, would default at an

asset value level of about 32% of its current non-distressed value. This is broadly consistent with

estimates in Davydenko (2011) and Warner (1977). Firms often operate at significantly negative

net worth levels before defaulting, reflecting the valuable optionality of equity when faced with

financial distress. Note that the ratio of the default point to liabilities is smaller (greater) for the

lower (higher) grade firms - a B (AA) firm defaults at 75% (62%) of book debt where leverage is

around 77% (26%). Thus they would default at an asset value level 42% (84%) lower than current

value.

Long run risk-adjusted variance levels are estimated to lie between 3% and 9%, corresponding to

volatility levels of 17% and 30% respectively.

The reported mean J-stats in Table 3 for the stochastic volatility model are well below the critical

values at conventional significance levels. This stands in stark contrast with the findings in Huang

and Zhou (2008) who find that almost all the models they study: the Merton (1974), Black and Cox

(1976), and Longstaff and Schwartz (1995) are consistently rejected whereas the Collin-Dufresne

and Goldstein (2001) model is rejected in half of the cases. The last two columns of Table 3 report

on the number of firms for which our model can be rejected. At the 1% (5%) level only 1 (3) firm

out of 49 leads to a rejection of the model.

To provide a more specific benchmark by which to judge these results, Table 4 reports on the same

exercise with the stochastic asset volatility channel shut down. These results are similar to the

findings of Huang and Zhou (2008) for the same model. Here, the Black and Cox (1976) model

can be rejected for 45 out of the 49 firms (compared to at best 87 out of 93, in HZ). Clearly, the

addition of stochastic volatility renders the rejection of the model significantly harder. Note that

the number of free parameters is greater when we introduce stochastic asset risk, and that this will

make a rejection harder. A fairer comparison in this regard is the Collin-Dufresne Goldstein (2001)

model (CDG) evaluated in Huang and Zhou (2008), which has the same number of additional

parameters with respect to the Black and Cox (1976) model. In that case between and 67% and

75% of the firms lead to non rejections, compared to between 94% and 98% in Table 3. The CDG

19

model yields somewhat lower pricing errors on the defaults swaps but, not surprisingly, faces more

resistance in fitting the time series of equity volatilities.

The pricing errors are reported in Panels B and C of Table 3. The first finding is that spreads

are underestimated by between 3 (A rated firms) and 61 (B rated firms) basis points with an

average of 18. The direction of the bias is reminiscent of the findings across 29 of 35 rating/model

combinations in Huang and Zhou (2009). However, the level is significantly lower than for the

Merton, Black and Cox, Longstaff and Schwartz models as reported by HZ.

For the BB (B) rating categories we underestimate spreads by 59 (61) basis points while for the

models in Huang and Zhou report underestimation ranges of 14 to 143 basis (22 to -608) basis points

respectively. Moreover the dispersion of the errors is smaller. Huang and Zhou report absolute

pricing errors in the range 13 to 1381 basis points across ratings (averaging 101 basis points) for

the Black and Cox model. In contrast, we find a range between 7 and 96 basis points.

A better understanding of the findings can be had by comparing to the Black & Cox (1976)

model estimated on our sample (by shutting down stochastic asset risk). Our overall average

underestimation with stochastic volatility is by 18 basis points (or 10 percent of the average total

spread) as compared to 48 basis points (or 65% of the average total spread) with constant volatility.

The average absolute pricing error is 26 basis points with stochastic volatility and 50 basis points

without. Not surprisingly, the model with stochastic asset volatility does a much better job at fitting

the time series of equity volatilities, generating average pricing errors (absolute pricing errors) of

15 (35) basis points as compared to -396 (1085) basis points.

Figure 2 summarizes average model implied and market spreads over time for the sample by rating

groupings. In comparison to Huang and Zhou (2009), these figures provide a much more encouraging

summary of the model’s performance. Note also that one of the conclusions in Huang and Zhou

is that their model finds it hard to fit both CDS and equity volatility time series. Our model, of

course, matches equity volatilities by construction (see Figure 3), but it appears that in doing so,

it is also better able to fit the price of default insurance across maturities.

In this estimation exercise, we only consider risk-adjusted asset dynamics, whereas the credit spread

puzzle discussed above concerns also the physical dynamics. In order to estimate jointly dynamics

under both probability measures, we consider an alternative estimation strategy below.

20

Jan2002 Jan2003 Jan2004 Jan2005 Jan2006 Jan2007 Jan2008 Dec20080

0.51

1.52

A. Rating A to AAA


0.51

1.52

B. Rating BBB


5

10

15C. Rating B to BB

ObservedStochastic Vol ModelBarrier

Figure 2. Observed and model-implied 5-year CDS spreads. This figure shows the time

series of observed 5-year CDS spreads and those estimated from the stochastic volatility model and

the Black-Cox (1976) model. One unit in the Y axis corresponds to 100 basis points.

21


40

80

A. Rating A to AAA


40

80

120

B. Rating BBB


40

80

120

C. Rating B to BB

ObservedStochastic Vol ModelBarrier

Figure 3. Observed and model-implied equity volatility. This figure shows the time series

of the realized volatility, which is estimated from 5-minute intraday stock returns, and the model-

implied equity volatility from the stochastic volatility model and the Black-Cox (1976) model.

5 Estimating volatility risk premia

A limitation of the analysis above is that it does not permit joint identification of physical and risk-

adjusted dynamics. As a result it is difficult to measure the importance of volatility risk premia for

spread levels and dynamics. In this section, we develop an estimation methodology which addresses

this issue. In essence, the estimation relies on the cross-sectional aspects of the data to identify

parameters associated with the change of probability measure required to price default swaps, while

pinning down the physical dynamics of state variables by means of the time series of default swap

22

spreads.

For a given firm, we have six observables: the CDS spreads at 2, 3, 5,7, and 10 years, and the equity

volatility.15 Since we have only two states: zt and Vt, we follow the literature (e.g., Duffee, 2002) by

assuming that two variables, jointly denoted by Y(a)t , are accurately observed, while the other four

variables, jointly denoted by Y(m)t , are observed with measurement error. Specifically, we choose

Y(a)t = {cdst(5), σE,t} and Y

(m)t = {cdst(2), cdst(3), cdst(7), cdst(10)} . Our choice reflects that i)

compared to the other spreads, spreads at 5 years have the highest liquidity and hence are most

accurately measured; ii) σE,t is also accurately measured using the high-frequent data (e.g., Zhang,

Zhou,and Zhu (2008)), and it is instrumental for identifying Vt. For notational convenience, denote

by Yt ={Y

(a)t , Y

(m)t

}.

To generate quantitative results from the model, we first set (r, η, α, L) = (4%, 15%, 15%, 49%)

which is consistent with numerous empirical studies, where r, η, α, and L denote the short term

rate, the tax rate, the liquidation cost, and the default loss, respectively. For a given firm, we obtain

time series for the payout ratio and coupon rate, and use their averages in the calibration for δ and

c. Denote the remaining parameters to be estimated by Θ = {λV , σ, κ, µ, θ, ρ,Xd} , where µ denotes

the expected growth rate (excluding payout) of firm value; κ, θ, σ characterize the variance process;

λV determines (κ∗, θ∗) , the Q-measure counterpart of (κ, θ) ,through κ∗ = κ− λV and θ∗ = κθ/κ∗;

ρ is the ”leverage” parameter denoting the correlation between firm asset value and its volatility;

Xd is the (constant) default boundary.

In order to identify Θ that contains parameters under both the P and the Q-measure, it is essential

to apply maximum likelihood (ML) estimation with respect to the P-distribution of the observables

Yt. The reasons are twofolds. First, ML estimation requires the computation of CDS spreads which

helps identify the Q-measure parameters. Second, the derivation of P-distribution of Yt depends on

the P-distribution of the two states (zt, Vt) which helps identify the P-measure parameters. A similar

methodology has been used previously for estimating models for default-free bonds (e.g., Singleton

(2001)). To the best of our knowledge, we are the first to apply this estimation methodology in a

credit risk application.

Define St ≡ {zt, Vt} which jointly denotes the two states in our model. Given Θ, Y(a)t can be

inverted using the procedure described in last subsection to form an implied state vector St. To

facilitate the subsequent exposition, we write

15We omit one year CDS spreads due to concerns about that contract’s liquidity.

23

St = F(Y

(a)t

), (27)

where F (.) denotes the functional form that maps the two observables without measurement error

into the two states. To see the identification of St from Y(a)t , first note that cdst(5) = Y

(a)t (1) is

monotone in zt = St (1) : intuitively, a higher zt (≡ ln (Xt/XD)) implies a greater distance to default

which is associated with lower credit spreads. Second, σE,t = Y(a)t (2) is monotone in Vt = St (2)

since a higher asset value variance is associated with a higher firm equity volatility. Numerically,

we back out St from Y(a)t using a large scale grid search which greatly accelerates the computation

speed while maintaining the accuracy.

Given St, implied spreads for the other four CDS contracts can be calculated. Stack these into Y(m)t

and denote by εt ≡ Y (m)t −Y (m)

t ∈ R4 the implied measurement error. To address the concern that

these error terms may not be scale-independent and tend to depend on the levels of credit spreads,

we also introduce the relative measurement error defined by εRt ≡(Y

(m)t − Y (m)

t

)/Y

(m)t = εt/Y

(m)t .

We assume that the variance-covariance matrix of εRt is time-invariant which implies the Cholesky

decomposition: E(εRt(εRt)′)

= CC ′. As a result, the variance-covariance matrix of εt will vary

across time which and credit ratings. We randomly choose a firm and try the estimation under

various specifications for C, but the results are largely unchanged. Given this finding, we choose

C = σeI4×4 where I4×4 denotes the 4 by 4 identity matrix, and perform the estimations for all

firms in our sample under this specification.

What remains is to derive the P-distribution of Y(a)t , which depends on the P-distribution of St.

In our model, the P-measure density of ST conditional on St is not in available in closed form.

Consequently, we use simulation to approximate the transition density p(Stn+1 , tn+1|Stn , tn) for two

adjacent discrete time tn and tn+1. Note that p (.) differs from pQ(.), the latter of which is under

the Q-measure and is derived below.

More specifically, we normalize the length of the interval [tn, tn+1] to one and divide this interval

into M subintervals of length h = 1/M. The discretization of St = (zt, Vt) in [tn, tn+1], denoted by(ztn+mh, Vtn+mh

)for m = 0, 1, ...,M − 1, is a Gaussian process:

[ztn+(m+1)h

Vtn+(m+1)h

]=

[ztn+mh

Vtn+mh

]+ ν (.)h+ Σ (.)

√h

[εztn+(m+1)h

εVtn+(m+1)h

],

24

where

ν (.) =

[ztn+mh

Vtn+mh

]; Σ (.) =

√Vtn+mh 0

σρ

√Vtn+mh σ

√1− ρ2

√Vtn+mh

;

[εztn+(m+1)h, ε

Vtn+(m+1)h

]′is the two-dimensional standard normal random variates; we’ve used that

the Brownian driving zt is correlated with the Brownian driving Vt by ρ. Thus, the probability of

Stn+(m+1)h =[ztn+(m+1)h, Vtn+(m+1)h

]′= y conditional on Stn+mh =

[ztn+mh, Vtn+mh

]′= x, is

pM (y, tn + (m+ 1)h|x, tn +mh) = φ (y;x+ ν (.)h, V (.)h) ,

where φ (y; mean,variance) denotes a bivariate normal density; V (.) = Σ (.) Σ (.)′ . pM is the ap-

proximation of p(y, tn+(m+1)h|x, tn+mh), and the approximation is exact as h→ 0. By recursive

integration,

pM (Stn+1 , tn+1|ztn , Vtn , tn) =

∫ ∫φ(Stn+1 ;u+ ν (.)h, V (.)h

)×

pM (u, tn + (M − 1)h|Stn , tn)du, (28)

where u ∈ R2 denote the realizations of (z, V ) one step before tn+1. Following Brandt and Santa-

Clara (2002), we view (28) as the expectation of a function φ of the random variable u. Consequently,

we approximate pM as the average of the function φ evaluated at a large number of the simulated ul

for l = 1, 2, ..., L. As L → ∞ and M → ∞, the approximated density, denoted by pM,L, converges

to the transition density of the continuous time process. Numerically, we find that at the monthly

frequency, an M of 30–40 and an L of 50,000–100,000 are sufficient to capture the shape of the

original transition densities.

Given the simulated p(Stn+1 , tn+1|Stn , tn), the distribution of Y(1)tn+1

conditional on Y(1)tn is

pY (Y(a)tn+1

, tn+1|Y (a)tn , tn) =

1

|det (J)|p(Stn+1 , tn+1|Stn , tn),

where

J ≡

∂Y(a)t (1)

∂St(1)∂Y

(a)t (1)

∂St(2)

∂Y(a)t (2)

∂St(1)∂Y

(a)t (2)

∂St(2)

=

[∂cdst(5)∂zt

∂cdst(5)∂Vt

∂σE,t∂zt

∂σE,t∂Vt

]. (29)

In (29), the relation betwen Y(a)t and St is summarized by (27). Numerically it is more convenient

to deal with Y(a)t as the function of St. Also, assume that the relative measurement error εt is

jointly normally distributed with distribution denoted by fε

(εt;Y

(m)t

). From the above discus-

sion, the variance-covariance matrix of εt depends on the contemporaneous credit spreads that are

25

inaccurately observed. The log likelihood of observation tn is then:

ltn (Θ) = log pY (Y(a)tn , tn|Y (a)

tn−1, tn−1) + log fε

(εtn ;Y

(m)tn

), (30)

where pY (Y(a)t1, t1|Y (a)

t0, t0) is set equal to the unconditional distribution of Y

(a)t by simulation. Due

to a relatively long time series of our sample, we find that assuming a degenerated pY (Y(a)t1, t1|Y (a)

t0, t0)

incurs very limited information loss compared to that using the unconditional distribution of Y(a)t .

Our estimation is thus based on

maxΘL (Θ) =

N∑n=2

ltn (Θ) .

Specifically, we use monthly data, and N is 84 for most firms.

To address the concern that our model may have a large number of local maxima, we resort to

our GMM estimates from the specification tests discussed above for the choice of initial values

for (σ, ρ,Xd). Second, λV determines the relation between the Q-measure parameters (κ∗, θ∗) and

their P-measure counterpart (κ, θ), which is not identifiable under the GMM. We thus try various

initial values of λV within its feasible range estimated in the previous literature until the implied

L (Θ) is no longer improved. Third, we compute initial values of (κ, θ) using the GMM estimates

of (κ∗, θ∗) and our choice of the initial λV . Finally, we set the inital µ at the average growth rate

of equity multipled by the average leverage for the given firm. Conditional on finding out the

”appropriate” initial λV , there is little improvement in the L (Θ) values for most firms after a few

hundred iterations of numerical search.

This exercise will allow us to better understand spread and volatility dynamics in the following

way. We first compare market spreads to spreads that would obtain with idiosyncratic unpriced

stochastic asset volatility and second to spreads that would obtain in the absence of any risk premia

on either volatility or asset value shocks. The first hypothetical spread permits us to identify the

contribution of the volatility risk premium. Together these two hypothetical spread measures allow

us to decompose spreads into three components: the spread that compensates for average losses

but does not provide any compensation for bearing the risk associated with them, the component

that provides a risk premium related to asset value shocks and finally an additional risk premium

component associated with stochastic asset volatility.

26

5.1 Estimation Results

The last observed credit rating for firms in our sample ranges from AAA to B and their 5 year

CDS spread spans the range 22 basis points (WalMart) to 214 basis points (Eastman Kodak). The

lowest (highest) leverage ratios and equity volatilities are 16% (94%) and 18% (40%) respectively.

Payout rates range from about 0.3% to just less than 4%.

Table 5 provides summary statistics per firm while Table 6 reports on the estimates we obtain on

a firm by firm basis. We obtain estimates of the variance risk premium parameter λv ranging from

0.708 to 3.166, averaging 2.241. The positivity highlights that unlevered volatility risk is priced and

carries a non-negligible risk premium. The level is in fact quite comparable to the 2.1 we assumed in

the calibration exercise reported on in Table 1. That number, in turn, was motivated by a desire to

be conservative given that there are no prior empirical estimates available in the literature, except

for equity markets. The corresponding risk premia for equity volatility tends to be at least twice as

large.16 This level is nevertheless, as shown above, sufficient to have a significant impact on credit

spreads.

The volatility of the variance is estimated to average 0.408. This is a little higher than for example

in Pan (2002) where estimates for the corresponding parameter in a model of equity volatility

range from 0.28 to 0.38. Mean reversion speeds range from 1.869 to 4.223, averaging 3.381 which

is in line with estimates from S&P 500 returns in e.g. Bates (2006). The average of the long run

volatility (√θ) is 0.118, which is again very similar to the estimates in Bates (2006). Estimates for

the correlation parameter range from -0.669 to 0.20, with a mean of -0.497. This is slightly lower

than estimates in Bates (2006) and similar to the -0.53 reported in Pan (2002).

The remaining estimated parameter, the default boundary, averages 72.3% of the face value of

debt, with lowest and highest estimates of 49% and over 150% respectively. This may be compared

to estimates compiled from actual defaults in Davydenko (2011) that average 66% and 5th and

exhibits 95th percentiles of 30% and 120% respectively.

Table 7 reports the standard errors for our estimates firm-by-firm. Overall, these are quite small

with the exception of the asset drift µ. There is very limited information in CDS spreads, stock

prices and volatilities to pin this parameter down.

Table 8 summarizes the model fit resulting from the MLE estimation. The overall average absolute

percentage pricing error is approximately 19% which compares to about 100% for the Merton model

16See Pan (2002) and Bates (2006) who report values of 7.6 and 4.7 respectively.

27

and 49% for the Collin-Dufresne Goldstein model as measured in Huang and Zhou (2009). Our

pricing errors are largest for the two year tenor (47%) and lowest for the 7 and 10 year tenors (9%

and 14% respectively).17

Figure 4 provides a visual summary of the 5 year default swap spread component estimation across

rating categories. Note that in our estimation we take 5 year spreads to be estimated without error

and that model and market spreads will virtually always coincide. The expected loss spread only

accounts for a small fraction of the total spread. This reflects the findings of Elton et al. (2001) and

Huang and Huang (2003). What is striking is that it is the volatility risk premium which permits the

model to match market spreads. As found in Huang and Huang (2003) the risk premium generated

by constant volatility structural credit risk models is insufficient to explain spreads and leaves us

with the credit spread puzzle. This is no longer the case when we allow for priced stochastic asset

volatility - market spread levels are attainable without resorting to unrealistic inputs.

In Figure 5, we consider the time series of equity volatility. The plots summarize market mea-

sured equity volatility together with model implied volatilities for the cases when asset volatility

is stochastic and when it is constrained to be a firm-specific constant. Note that even when asset

volatility is constant, our model generates an equity volatility which varies through time as the

leverage ratio is altered by shocks to equity values. The main insights gained from these plots is

that equity volatility is tracked well by our full model and that forcing a constant volatility makes

it very difficult for the model to track equity volatility while also trying to explain default swap

spreads.

In order to gauge whether our estimated risk premium parameters are reasonable, we compute

ratios of expected risk-adjusted to physical equity volatilities. Intuitively, these ratios should be

quantitatively similar to ratios of implied to realized volatilities in stock and stock option markets.

Figure 6 reports averages across rating categories for these ratios.18 There is limited evidence on

these types of metrics but a recent paper by Han and Zhou (2011) reports on model-free implied

one-month equity volatilities and their high frequency realized counterparts in a panel of more than

5000 firms. The ratio of these two is 1.17. Over time the ratio varies between 0.967 (in 1996) to

1.266 in 2003. We find that at the one-month horizon, the ratios range from 1.026 to 1.119 with

an average of 1.083. This suggests that our estimated volatility risk premia are quite similar if not

somewhat lower than what is found directly in stock and option markets.

An interesting pattern emerges across rating categories. Firms with higher credit ratings tend to

17Of course the error is even smaller for the 5 year tenor which we assume is estimated without error in the MLEestimation.

18Firm level estimates are not reported for space reasons.

28

have higher ratios of risk-adjusted to physical volatilities. This is reminiscent of recent findings in

structured credit markets by Coval Jurek and Stafford (2009). They find that compensation for risk,

as measured by the ratio of yield spreads to loss rates is higher for more senior (and more highly

rated) CDO tranches. The intuition is that although these tranches have lower default probability,

they are likely to default in more extreme economic conditions than a more junior tranche.

29

2003 2004 2005 2006 2007 2008 20090

0.5

1

1.5

Rating AAA to A

2003 2004 2005 2006 2007 2008 2009

0.5

1

1.5

Rating BBB

2003 2004 2005 2006 2007 2008 2009

2

4

6

8

10

12

Rating BB to B

data

full model

no risk premium on V

no risk premium

Figure 4. Decompostion of 5 year CDS spreads In all four cases, we use the same time

series for the two states: firm value X and firm value variance V; which are backed out from the

full model. In computing spreads under no risk premium on V , we set λV to zero. In computing

spreads with no risk premia, we set λV to zero and rely on the P -dynamics for X.

30

2003 2004 2005 2006 2007 2008 2009

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Rating AAA to A

2003 2004 2005 2006 2007 2008 2009

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Rating BBB

2003 2004 2005 2006 2007 2008 2009

0.4

0.6

0.8

1

1.2

Rating BB to B

data

full model

Black−Cox

Figure 5. Decomposition of σE In the first two cases (data and full model), we use the same

time series of the two states: firm value X and firm value variance V; which are backed out from

the full model. There is no variation in V in the third case of Black-Cox (hence λV , see footnote

9). As a result, we set V as the constant equalling the average of the its time series from the full

model when computing equity values under Black-Cox. Note that we still use the same series of X

for BC.

31

1.040

1.055

1.070

1.085

1.100

AAA AA A BBB BB B

Figure 6. Ratios of one month expected equity volatilities under Q and P. Averages

of the square root of the ratio of one-month expected equity return variances,

√EQ[V eT ]

EP [V eT ], are taken

across rating categories for the 49 firms.

6 Concluding remarks

We have developed, evaluated and estimated a first-passage time structural credit risk model with

priced stochastic volatility as a means of addressing the credit spread puzzle documented in Huang

and Huang (2003) and further studied in Collin Dufresne Goldstein (2009). We show that although

such a model has various ways of generating higher credit spreads than constant volatility models,

in a calibration where the model is required to match default and recovery rates, leverage ratios

32

and equity risk premia, the key driver of spreads ends up being the volatility risk premium.

Having first evaluated the cross-sectional properties of spreads implied by our model, we proceed

to also study the ability of our model to jointly explain the dynamics of credit spreads and equity

volatilities, a task which has been shown to be out of the reach of constant volatility structural

credit risk models. The additional factor for asset volatilities permits our model to fit equity

volatilities well while improving the fit for CDS prices relative to the findings for constant volatility

models studied in Huang and Zhou (2009). In addition, this exercise provides interesting empirical

evidence on the dynamics of firms’ unlevered assets. We find evidence of a significant non-financial

leverage effect - asset value and variances shocks are significantly negatively correlated.

After assessing the empirical fit of the model to market data, our MLE estimation provides estimates

of the wedge between risk-adjusted and physical dynamics. Our estimates provide evidence of a

significant volatility risk premium which is quite stable in the cross section. We benchmark ratios of

risk-adjusted to physical volatilities against observed levels in a large panel study by Han and Zhou

(2011). Our ratios are similar although slightly lower, suggesting that our model does not require

unreasonable levels of asset volatility risk premium to explain credit spread levels. Nevertheless,

the contribution of this risk premium to credit spread levels is dramatic.

As noted above, jump risk premia have been shown to help explain credit spread levels (Cremers

Driessen and Maenhout (2009)), although we know little of their ability to explain the time series

behaviour of spreads and equity volatilities. We leave to future research the question as to whether

jump risk and volatility risk can be thought of as complements or substitutes in their ability to

explain spreads. It would also be interesting to estimate all firms’ volatility dynamics jointly,

allowing for a single estimate of the market price of volatility risk and may shed light on the role

of idiosyncratic volatility.

The technical contribution of our paper - closed form analytics for a first passage time stochastic

volatility model - has many obvious applications in the credit risk literature. More generally,

we believe there are numerous applications in the real options literature, where investment and

volatility are closely related.

33

Appendix A: details on the calibration procedure

We calibrate the model in three different ways. In the first one, we calibrate the initial asset value,

long-run mean of asset volatility and asset risk premium to match the target leverage ratio, equity

premium and cumulative default probability. In the second one, we calibrate the initial asset value,

long-run mean of asset volatility, market price of volatility risk and asset risk premium to match

the target leverage ratio, equity premium, cumulative default probability and historical average

yield spread. In the third one, we calibrate the initial asset value, long-run mean of asset volatility,

market price of volatility risk, asset risk premium and mean-reversion parameter to match the target

leverage ratio, equity premium, cumulative default probability, historical average yield spread and

equity volatility. For all the three calibration, we assume that the firm recovers 51.31% of the face

value given default.

To calibrate our model, we need to specify the asset premium and the leverage ratio. The asset

premium is given by

πX = (1− L)πE + LπD, (31)

where πE is the equity premium, πD is the bond risk premium and L is the firm’s leverage ratio.

We use the yield spread of the corporate bond over a comparable default-free bond as a proxy for

the bond risk premium. The leverage ratio is given by

L = P/X, (32)

where P is the face value and X is the unlevered asset value.

Also to calibrate the model to the term structure of yield spreads, we need to price a corporate bond

with finite maturity. For a corporate bond with maturity T and semi-annual coupon payment, the

bond price is given by

D0,T =c

2

2T−1∑i=1

[1− ωQ(0, Ti)] /(1 + r)Ti +(P +

c

2

)[1− ωQ(0, T )] /(1 + r)T , (33)

where P is the face value of the bond, c is the annual coupon payment, Ti is the ith coupon date,

ω is loss rate given default, and Q(0, Ti) is the risk-neutral default probability before time Ti. We

assume that the corporate bond is priced at par. Thus we can back out the annual coupon payment

from equation (33). Also the bond yield is the same as the coupon rate.

Finally, when we perform the second and third calibrations as mentioned earlier, we use equation

34

(5) to calibrate the model to the historical average equity volatility for different credit ratings.

35

Appendix B: a Solution for the Default Probability: One Special

Case

We denote S(z, V, h) as the risk-neutral probability that the log asset value z has never crossed the

default boundary zD = 0 before T = t + h, given that zt = z and Vt = V . Obviously, the default

probability Q(z, V, h) = 1 − S(z, V, h). Below we show the procedure to obtain the closed-form

solution for S when S satisfies the smooth pasting condition at z = 0.

Define the default time as τt = inf{s ≥ t, zs ≤ zD}. Then P (τt < T |zt = z, Vt = V ) = 1−S(z, V, h).

Since we can rewrite S(z, V, h) as a conditional expectation, it follows a martingale and satisfies

the following backward Kolmogorov equation.

Sh =1

2V Szz + ρσV SzV +

1

2σ2V SV V + (r − 1

2V )Sz + κ∗ (θ∗ − V )SV , (34)

with the initial condition S(z, V, 0) = 1 and the boundary condition S(0, V, h) = 0. Define

S(ω, V, h) =∫∞

0 e−ωzS(z, V, h)dz with the real part of ω being positive. Given the solution for

S, we obtain that S(z, V, h) = 12πi

∫ C+i∞C−i∞ ezωS(ω, V, h)dω with C being a positive constant. To

solve for S(ω, V, h), we use the following equations:

S(ω, V, 0) =

∫ ∞0

e−ωzdz =1

ω, (35a)

∫ ∞0

e−ωz∂S

∂zdz = ωS, (35b)

∫ ∞0

e−ωz∂2S

∂z∂Vdz = ω

∂S

∂V, (35c)

∫ ∞0

e−ωz∂2S

∂z2dz = ω2S. (35d)

When getting equation (35d), we assume that the “smooth pasting” (we need to find the correct

wording or the economic intuition here since it is the same as in Leland (1994)) condition is satisfied

by S at z = 0, i.e., ∂S∂z |z=0 = 0. Applying the transform∫∞

0 e−ωz · dz on both sides of equation (34)

and plugging equations (35a)-(35d), we obtain that

∂S

∂h=

1

2σ2V

∂2S

∂V 2+ [κ∗θ∗ + (ρσω − θ∗)V ]

∂S

∂V+

[ωr +

(1

2ω2 − 1

2ω

)V

]S. (36)

36

Guessing that the solution for S is S(ω, V, h) = 1ωe−A(ω,h)−B(ω,h)V , we obtain

−A′ −B′V =1

2σ2V B2 − [κ∗θ∗ + (ρσω − κ∗)V ]B + ωr +

(1

2ω2 − 1

2ω

)V. (37)

Thus A and B satisfy

−A′ = −κ∗θ∗B + ωr, (38a)

−B′ = 1

2σ2B2 − (ρσω − κ∗)B +

1

2ω2 − 1

2ω, (38b)

with A(ω, 0) = 0 and B(ω, 0) = 0. Note that we use ′ to denote the first-order derivative. For

example, A′ = dAdh and B′ = dB

dh . We first solve for B from equation (38b) and then solve for A

from equation (38a). Essentially, equation (38b) is a Riccati equation. Let B(h) = q′(h)q(h) ·

2σ2 and

plug it into equation (38b), we obtain

q′′ − (ρσω − κ∗)q′ − 1

2σ2

(−1

2ω2 +

1

2ω

)q = 0.

Thus the general solution for q is q(h) = C1eλ1h + C2e

λ2h, where C1 and C2 are constants, and λ1

and λ2 solve

λ2 − (ρσω − κ∗)λ− 1

4σ2(−ω2 + ω) = 0.

Thus λ1,2 = ρσω−κ∗±d2 with d =

√(ρσω − κ∗)2 + σ2(−ω2 + ω). The solution for B(ω, h) is

B(ω, h) =2

σ2· C1λ1e

λ1h + C2λ2eλ2h

C1eλ1h + C2eλ2h. (39)

Since B(ω, 0) = 0, we obtain that C2 = −C1λ1/λ2. Plugging equation the expression for C2 into

equation (39), we obtain

B(ω, h) =ρσω − κ∗ + d

σ2· 1− e−dh

1− ge−dh, (40)

with d =√

(ρσω − κ∗)2 + σ2(−ω2 + ω) and g = ρσω−κ∗+dρσω−κ∗−d . Now we solve for A(ω, h). First,

plugging equation (40) into equation (38a) yields

A′ = κ∗θ∗ · ρσω − κ∗ + d

σ2· 1− e−dt

1− ge−dt− ωr. (41)

Let u(h) = e−dh, then u′ = −du. Plugging A′ = A′u · u′ into equation (41) yields

A′u · u′ = −A′udu = κ∗θ∗ · ρσω − κ∗ + d

σ2· 1− e−dt

1− ge−dt− ωr.

Thus

A′u = κ∗θ∗ · ρσω − κ∗ + d

σ2d·(−1

u+

1− g1− gu

)+ωr

du. (42)

37

The solution for equation (42) is

A = κ∗θ∗ · ρσω−κ∗+d

σ2d·[−ln(u) + (1−g)ln(1−gu)

−g

]+ ωr

d ln(u) + C3

= κ∗θ∗ · ρσω−κ∗+d

σ2 · h+ κ∗θ∗ · ρσω−κ∗+d

σ2d· g−1

g ln(1− ge−dh

)− ωrh+ C3,

(43)

where C3 is a constant. Since A(0) = 0, we obtain C3 = −κ∗θ∗ · ρσω−κ∗+d

σ2d· g−1

g ln (1− g). Thus

A(ω, h) = −ωrh+ κ∗θ∗ · ρσω − κ∗ + d

σ2· h+ κ∗θ∗ · ρσω − κ

∗ + d

σ2d· g − 1

gln

[1− ge−dh

1− g

]. (44)

Since (ρσω − κ∗ + d)g−1g = (ρσω − κ∗ + d) 2d

ρσω−κ∗−d ·ρσω−κ∗−dρσω−κ∗+d = 2d, we rewrite the solution for A

as

A(ω, h) = −ωrh+ κ∗θ∗ · ρσω − κ∗ + d

σ2· h+

2κ∗θ∗

σ2ln

[1− ge−dh

1− g

]. (45)

Thus the solution for S(z, V, h) is given by

S(z, V, h) =1

2πi

∫ C+i∞

C−i∞ezωS(ω, V, h)dω, (46)

with S(ω, V, h) = 1ωe−A(ω,h)−B(ω,h)V , and the solutions for A(ω, h) and B(ω, h) are given by equa-

tions (45) and (40).

Appendix C: Volatility asymmetry for equity and asset value

In this appendix, we translate the correlation between asset value and variance shocks to a corre-

lation between equity and equity variance shocks. The stochastic process that the unlevered asset

value follows is given bydXt

Xt= (µ− δ)dt+

√VtdW1, (47)

dVt = κ(θ − Vt)dt+ σ√VtdW2, (48)

where δ is the firm’s payout ratio and E(dW1dW2) = ρdt. Applying Ito’s lemma, we obtain that

dEtEt

= µE,t +Xt

Et

∂Et∂Xt

√XtdW1t +

1

Et

∂EtVt

σ√VtdW2t, (49)

where µE,t is the instantaneous equity return. The equity variance is given by

VE,t = C · Vt, (50)

38

where C =(XtEt

∂Et∂Xt

)2+(σEt

∂Et∂Vt

)+ ρσXt

E2t

∂EtXt

∂Et∂Vt

. Applying Ito’s lemma to VE,t given by equation

(50), we obtain that

dVE,t = µVE ,tdt+ σVE,tdW2t. (51)

Given the specification of the processes for equity and equity variance as in equations (49) and

(51), we obtain that the correlation between equity and equity variance is

ρE,t =

(Xt

Et

∂Et∂Xt

ρ+σ

Et

∂Et∂Vt

)/C, (52)

where C =(XtEt

∂Et∂Xt

)2+(σEt

∂Et∂Vt

)+ ρσXt

E2t

∂EtXt

∂Et∂Vt

.

39

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42

Table 1: Calibrated Yield Spreads with and without variance risk premium This tablereports the calibration results for the model in which the firm value follows either a constantvolatility or a stochastic volatility process with priced variance risk. The yield spread is calculatedas the difference between the bond yield and risk-free rate. Following Huang and Huang (2003), wecalibrate the initial asset value, long-run mean of asset volatility and asset risk premium to matchthe target leverage ratio, equity premium and cumulative default probability (columns 2-4). Givendefault, the firm recovers 51.31% of the face value. The average yield spreads in the last columnare the historically observed yield spreads for the bonds as reported in Huang and Huang (2003).In panel B, we select the initial asset volatility to the same as the long-run mean.

Panel A: ρ = 0, σ = 0, κ = 0 and k = 0

TargetCredit Leverage Equity Cumul. Implied Calculated Average % ofRating Ratio Premium Def. Prob. Asset Vol. Spread Spread Spread due to

(%) (%) (%) (%) (bps) (bps) Default

Aaa 13.1 5.38 0.77 31.9 0 63 0Aa 21.2 5.60 0.99 28.1 0 91 0A 32.0 5.99 1.55 25.0 5.9 123 4.8Baa 43.3 6.55 4.39 24.7 32.9 194 16.9Ba 53.5 7.30 20.63 30.1 148.3 320 46.3B 65.7 8.76 43.91 35.2 320.8 470 68.3

Panel B: ρ = −0.1, σ = 0.30, κ = 4 and k = 7

TargetCredit Leverage Equity Cumul. Implied Calculated Average % ofRating Ratio Premium Def. Prob. Asset Vol. Spread Spread Spread due to

(%) (%) (%) (%) (bps) (bps) Default

Aaa 13.1 5.38 0.77 31.3 60.3 63 95.7Aa 21.2 5.60 0.99 27.4 67.5 91 74.2A 32.0 5.99 1.55 24.8 89.4 123 72.7Baa 43.3 6.55 4.39 25.3 159.8 194 82.4Ba 53.5 7.30 20.63 32.3 361.2 320 112.8B 65.7 8.76 43.91 39.8 584.7 470 124.4

43

Table 2: Summary Statistics by Ratings. This table reports the summary statistics on theCDS spreads and the underlying firms from January 2002 to December 2008. Equity volatility isestimated using 5-minute intraday returns. Leverage ratio is calculated as the ratio of the totalliabilities over the total asset, which is the sum of the total liability and equity market value. Assetpayout ratio is the weighted average of dividend payout and interest expense over the total asset.

Panel A: Firm Characteristics

Credit Firms Equity Leverage AssetRating Volatility (%) Ratio (%) Payout (%)

AAA a 1 24.29 63.67 2.83AA 1 21.99 25.81 1.36A 16 27.19 37.95 2.33BBB 22 27.72 48.51 2.12BB 7 35.18 51.01 2.50B 2 41.24 77.10 2.66

Panel B: CDS Spreads (%)

Credit Rating 1-year 2-year 3-year 5-year 7-year 10-year

AAA 0.45 0.49 0.52 0.58 0.60 0.63AA 0.12 0.14 0.17 0.22 0.25 0.30A 0.28 0.33 0.37 0.45 0.51 0.57BBB 0.40 0.47 0.54 0.67 0.75 0.85BB 0.75 0.86 1.00 1.35 1.44 1.57B 3.73 4.22 4.49 4.80 4.78 4.77

Panel C: CDS Spreads Std. Dev. (%)

Credit Rating 1-year 2-year 3-year 5-year 7-year 10-year

AAA 1.16 1.12 1.08 0.98 0.90 0.85AA 0.16 0.16 0.17 0.18 0.17 0.17A 0.39 0.40 0.40 0.39 0.37 0.36BBB 0.49 0.49 0.50 0.49 0.47 0.45BB 0.86 0.89 0.88 0.88 0.83 0.85B 7.78 7.02 6.40 5.70 5.17 4.70

aNote that the only AAA-rated company in our sample is GE.

44

Table 3: Specification Test of the Stochastic Volatility Model. This table reports the GMMtest results of the stochastic volatility model. Panel A reports the parameter estimates, p-values(in parentheses) and J-statistic. Panel B reports the average and absolute pricing errors of CDSspreads and equity volatility. Panel C reports the average and absolute percentage pricing errorsof CDS spreads and equity volatility. The pricing errors are calculated as the time-series average,absolute, average percentage and absolute percentage differences between the model-implied andobserved values.

Panel A: Parameter Estimates

Credit Rating ρ κ∗ σ XB/100 θ∗(%) J-stat Proportion of Not RejectedSig. level=0.01 0.05

Overall -0.58 0.98 0.37 0.67 5.34 1.637 48/49 46/49(0.007) (0.008) (0.007) (0.013) (0.011)

AAA -0.47 1.2 0.24 0.70 3.06 0.924 1/1 1/1(0.006) (0.008) (0.005) (0.014) (0.008)

AA -0.54 0.8 0.36 0.62 6.40 0.672 1/1 1/1(0.008) (0.012) (0.005) (0.018) (0.011)

A -0.56 1.2 0.42 0.68 4.58 1.008 17/17 16/17(0.006) (0.009) (0.007) (0.014) (0.007)

BBB -0.61 0.9 0.31 0.67 5.20 2.352 21/22 20/22(0.008) (0.007) (0.008) (0.010) (0.011)

BB -0.59 0.8 0.45 0.70 6.92 1.260 6/6 6/6(0.007) (0.009) (0.004) (0.015) (0.012)

B -0.65 0.5 0.32 0.75 9.24 1.092 2/2 2/2(0.012) (0.009) (0.007) (0.018) (0.017)

45

Panel B: Average and Absolute Pricing Errors

Credit Average Pricing Error (%) Absolute Pricing Error (%)Rating CDS Spreads Equity Volatility CDS Spreads Equity Volatility

Overall -0.18 0.15 0.26 0.35AAA -0.15 0.03 0.18 0.14AA -0.04 0.08 0.07 0.12A -0.03 0.12 0.17 0.06BBB -0.13 0.10 0.19 0.55BB -0.59 0.36 0.55 0.45B -0.61 0.23 0.96 0.34

Panel C: Average and Absolute Percentage Pricing Errors

Credit Average Percentage Pricing Error (%) Absolute Percentage Pricing Error (%)Rating CDS spreads Equity Volatility CDS spreads Equity Volatility

Overall -10.16 0.40 29.61 0.96AAA -27.42 0.12 32.90 0.58AA -20.02 0.36 35.04 0.55A -7.16 0.41 36.65 0.20BBB -11.16 0.28 25.25 1.55BB -19.28 0.78 24.43 0.97B 22.34 0.56 35.06 0.82

46

Table 4: Specification Test of the Black and Cox (1976) Model. This table reports the GMMtest results of the Black and Cox (1976) model. Panel A reports the parameter estimates, p-values(in parentheses) and J-statistic. Panel B reports the average and absolute pricing errors of CDSspreads and equity volatility. Panel C reports the average and absolute percentage pricing errorsof CDS spreads and equity volatility. The pricing errors are calculated as the time-series average,absolute, average percentage and absolute percentage differences between the model-implied andobserved values.

Panel A: Parameter Estimates

Credit Rating Asset Vol(%) XB/100 J-stat Proportion of Not RejectedSig. level=0.01 0.05

Overall 12.41 1.31 15.112 4/49 1/49(0.008) (0.036)

AAA 10.01 1.067 11.413 0/1 0/1(0.004) (0.016)

AA 11.75 1.795 8.487 1/1 0/1(0.010) (0.051)

A 10.15 1.586 16.396 0/16 0/16(0.008) (0.042)

BBB 12.45 1.201 15.263 2/22 0/22(0.008) (0.034)

BB 14.35 1.125 11.545 1/7 1/7(0.007) (0.031)

B 20.17 0.804 20.822 0/2 0/2(0.006) (0.028)

47

Panel B: Average and Absolute Pricing Errors

Credit Average Pricing Error (%) Absolute Pricing Error (%)Rating CDS Spreads Equity Volatility CDS Spreads Equity Volatility

Overall -0.48 -3.96 0.50 10.85AAA -0.27 3.04 0.29 10.35AA 0.08 -1.56 0.27 8.41A -0.31 -5.01 0.31 11.22BBB -0.43 -1.81 0.43 8.14BB -0.74 -8.09 0.76 12.99B -1.77 -9.46 2.20 31.94

Panel C: Average and Absolute Percentage Pricing Errors

Credit Average Percentage Pricing Error (%) Absolute Percentage Pricing Error (%)Rating CDS spreads Equity Volatility CDS spreads Equity Volatility

Overall -65.44 -6.56 67.31 29.51AAA -25.11 17.63 40.69 51.64AA -61.11 6.55 65.67 35.91A -73.02 -4.64 73.73 32.32BBB -68.09 -9.53 68.08 24.15BB -62.17 -3.74 64.16 32.61B -9.36 -17.69 32.64 41.08

48

Table 5: Summary Statistics of Individual Names. This table reports the ratings, 5-yearCDS spread, equity volatility, leverage ratio and asset payout ratio for each of the 49 firms.

Company Last 5-year Equity Leverage AssetName Rating CDS(%) Volatility (%) Ratio (%) Payout(%)

Amgen Inc A 0.392 26.94 15.99 0.42Arrow Electronics Inc BBB 1.357 35.48 55.89 1.76

Boeing Co A 0.441 29.89 50.51 1.58Baxter Intl Inc A 0.343 26.66 26.42 1.59

Bristol Myers Squibb Co A 0.297 27.28 25.09 3.95Boston Scientific Corp BB 0.816 35.41 24.53 0.75

Conagra Inc BBB 0.440 21.10 41.21 3.56Caterpillar Inc A 0.359 26.43 54.47 2.45

Cigna Corp BBB 0.680 30.36 77.62 0.27Comcast Corp New BBB 1.138 31.93 61.15 1.91Campbell Soup Co A 0.277 21.98 32.10 2.49

Computer Sciences Corp BBB 0.613 28.78 46.44 1.17Du Pont Co A 0.275 24.43 36.84 2.81Deere & Co A 0.394 29.40 58.44 2.61

Disney Walt Co A 0.492 27.27 34.68 1.49Dow Chemical Co A 0.658 27.37 45.31 3.21

Darden Restaurants Inc BBB 0.689 32.41 31.42 1.69Devon Energy Corp BBB 0.553 33.31 44.82 1.87Eastman Kodak Co B 2.136 32.64 60.35 2.12

Eastman Chemical Co BBB 0.658 29.04 53.05 3.13Ford Motor Co B 7.471 41.29 93.85 3.21

Fortune Brands Inc BBB 0.570 21.48 36.96 2.31General Electric Co AAA 0.584 24.29 63.68 2.83General Mills Inc BBB 0.432 17.76 40.72 3.03Goodrich Corp BBB 0.780 28.43 52.39 2.55

Honeywell Intl Inc A 0.359 28.39 40.17 2.10Intl Paper Co BBB 0.932 28.74 56.38 3.08

Ingersoll Rand Co BBB 0.414 28.12 39.39 1.91J C Penney Co Inc BB 2.09 39.93 50.98 1.94

Kraft Foods Inc BBB 0.464 21.12 52.20 2.76Kroger Company BBB 0.641 29.25 52.34 1.93

49

Company Last 5-year Equity Leverage AssetName Rating CDS(%) Volatility (%) Ratio (%) Payout(%)

Marriott Intl Inc New BBB 0.894 27.06 32.10 1.29Mcdonalds Corp A 0.285 25.45 25.70 2.31

Mckesson Inc BBB 0.564 27.67 53.14 0.76Northrop Grumman Corp BBB 0.459 20.99 46.26 2.08Newell Rubbermaid Inc BBB 0.572 26.50 41.99 3.03

Olin Corp BB 1.247 37.83 48.51 3.11Radioshack Corp BB 1.136 37.90 29.09 1.64

Raytheon Co BBB 0.599 28.09 43.06 2.33Sherwin Williams Co A 0.486 29.66 31.15 1.99

Supervalu Inc BB 1.526 27.78 60.52 3.21Safeway Inc BBB 0.646 30.68 48.76 2.01

A T&T Corp A 1.249 28.29 43.81 3.50Target Corp A 0.392 30.47 35.16 1.48

Temple Inland Inc BB 1.411 31.01 81.40 3.75Tyson Foods Inc BB 1.254 36.37 62.01 3.11

Verizon Communications Inc A 0.621 25.19 51.37 3.40Whirlpool Corp BBB 0.683 31.38 59.95 2.16

Wal Mart Stores Inc AA 0.215 21.97 25.81 1.36

50

Table 6: Parameter estimates by simulated MLE.

Based on estimating equations (19) and (21) using the methodology explained in section 5; specif-ically the maximization of the log likelihood in equation (30). For most firms the sample lengthis 84 monthly term structures of CDS spreads with corresponding equity data. Last rating refersto the last reported (most recent) rating for a given firm’s history in our sample, VRP refers tothe parameter λV = kσ which, as discussed in section 2, defines the wedges between risk-adjustedand physical mean reversion speeds and long run variance levels. The third column reports on thevolatility of the variance dynamics in equations (1) and (2). Next σe denotes the measurement errorin the MLE estimation. The following four columns report on the remaining parameters governingthe dynamics of asset variance under P , in equation (1). The last column reports on the estimateddefault boundary, reported as a fraction of total liabilities.

Company Last VRP Vol. of Meas. Mean rev. Asset Long run Correl. Bound.

Name Rating λV var. σ error σe speed κ drift µ var. θ ρ to debt XD100

Amgen A 2.396 0.440 0.101 3.465 0.007 0.023 -0.630 0.740Arrow Electr. BBB 3.166 0.541 0.084 4.223 0.000 0.011 -0.197 1.012

AT&T A 2.367 0.397 0.102 3.641 0.094 0.014 -0.581 0.623Baxter Intl. A 2.400 0.410 0.108 3.675 0.020 0.020 -0.580 0.630

Boeing A 2.520 0.420 0.102 3.800 0.029 0.013 -0.550 0.670Boston Sci. BB 0.708 0.669 0.117 3.508 0.029 0.031 -0.508 0.800

Bristol M.Sq. A 2.400 0.450 0.113 3.675 0.072 0.022 -0.610 0.630Campbell Soup A 2.477 0.437 0.098 3.413 0.024 0.012 -0.546 0.729

Caterpillar A 2.455 0.378 0.094 3.780 0.012 0.008 -0.470 0.680Cigna BBB 2.239 0.326 0.101 3.572 0.046 0.004 -0.472 0.657

Comcast BBB 2.276 0.302 0.102 3.441 0.008 0.013 -0.508 0.670Computer Sci. BBB 2.413 0.291 0.101 3.196 0.037 0.017 -0.617 0.621

Conagra BBB 1.848 0.396 0.116 3.890 -0.016 0.014 -0.497 0.738Darden Rest. BBB 1.868 0.295 0.108 3.303 0.034 0.026 -0.597 0.720

Deere A 2.410 0.396 0.100 3.204 0.041 0.006 -0.492 0.690Devon Energy BBB 2.203 0.446 0.075 4.041 0.041 0.024 -0.560 0.848

Disney A 2.403 0.470 0.100 3.502 0.025 0.017 -0.573 0.636Dow Chemical A 2.420 0.463 0.099 3.438 0.033 0.012 -0.520 0.612

Du Pont A 2.357 0.440 0.101 3.382 0.012 0.011 -0.476 0.723Eastman Chem. BBB 1.530 0.482 0.078 3.204 0.056 0.016 -0.445 0.620Eastman Kodak B 1.268 0.482 0.136 2.342 0.030 0.019 0.037 1.009

Ford Motors B 1.403 0.595 0.206 1.869 0.029 0.006 0.300 1.543Fortune Brands BBB 2.253 0.328 0.101 3.263 -0.018 0.013 -0.593 0.623General Electric AAA 2.515 0.271 0.109 3.990 0.058 0.010 -0.470 0.715

General Mills BBB 2.386 0.248 0.099 3.614 0.007 0.010 -0.528 0.643Goodrich BBB 2.289 0.355 0.101 3.223 0.004 0.013 -0.640 0.752Honeywell A 2.324 0.414 0.101 3.624 0.005 0.016 -0.600 0.666

Ingersoll Rand BBB 2.129 0.304 0.102 3.149 0.000 0.017 -0.655 0.627Intl. Paper BBB 2.442 0.421 0.102 3.320 0.019 0.008 -0.217 1.045Kraft Foods BBB 2.346 0.245 0.102 3.477 0.007 0.009 -0.568 0.651

Kroger BBB 2.343 0.313 0.102 3.048 -0.016 0.014 -0.627 0.652

51

Parameter estimates by simulated MLE. Continued.

Company Last VRP Vol. of Meas. Mean rev. Asset Long run Correl. Bound.

Name Rating λV var. σ error σe speed κ drift µ var. θ ρ to debt XD100

L A 2.400 0.449 0.101 3.284 0.016 0.019 -0.568 0.708Marriott BBB 2.365 0.421 0.134 3.433 0.015 0.021 -0.500 0.491

McDonalds A 2.321 0.411 0.101 3.501 0.014 0.016 -0.620 0.689McKesson BBB 2.352 0.398 0.096 3.136 -0.014 0.011 -0.590 0.773

Newell Rubb. BBB 1.816 0.360 0.109 3.314 0.008 0.013 -0.604 0.699Northrop Gr. BBB 2.311 0.361 0.101 3.297 -0.014 0.009 -0.542 0.765

Olin BB 1.504 0.518 0.108 3.139 0.001 0.021 -0.566 0.805Radioshack BB 1.492 0.583 0.136 2.862 0.027 0.031 -0.198 0.747Raytheon BBB 2.219 0.298 0.102 3.059 0.003 0.013 -0.654 0.660Safeway BBB 2.322 0.338 0.100 3.192 0.018 0.018 -0.669 0.682

Sherwin Will. A 2.031 0.481 0.105 3.791 0.020 0.022 -0.584 0.688Supervalu BB 1.828 0.554 0.101 2.772 0.040 0.008 -0.080 0.651

Target A 2.047 0.429 0.107 3.666 0.071 0.021 -0.589 0.706Temple Inland BB 2.000 0.328 0.100 2.900 0.033 0.002 -0.441 0.680Tyson Foods BB 1.959 0.564 0.104 3.084 0.036 0.010 -0.545 0.725

Verizon A 2.400 0.410 0.100 3.885 0.022 0.009 -0.510 0.730Walmart AA 2.511 0.344 0.102 3.465 -0.000 0.016 -0.540 0.635

Whirlpool BBB 2.188 0.339 0.104 3.600 0.015 0.013 -0.571 0.636

52

Table 7: Parameter estimate standard errors by simulated MLE.

These standard errors correspond to the parameter estimates reported in Table 6. Based on esti-mating equations (19) and (21) using the methodology explained in section 5; specifically the max-imization of the log likelihood in equation (30). For most firms the sample length is 84 monthlyterm structures of CDS spreads with corresponding equity data. VRP refers to the parameterλV = kσ which, as discussed in section 2, defines the wedges between risk-adjusted and physicalmean reversion speeds and long run variance levels. The third column reports on the volatility ofthe variance dynamics in equations (1) and (2). Next σe denotes the measurement error in theMLE estimation. The following four columns report on the remaining parameters governing thedynamics of asset variance under P , in equation (1). The last column reports on the estimateddefault boundary, reported as a fraction of total liabilities.

Company VRP Vol. of Meas. Mean rev. Asset Long run Correl. Bound.

Name λV var. σ error σe speed κ drift µ var. θ ρ to debt XD100

Amgen 0.0191 0.0047 0.0202 0.018 1.0949 0.0001 0.0049 0.01Arrow Electr. 0.3707 0.1162 0.1286 0.3863 1.9756 0.0042 0.0822 0.4239

Boeing 0.0309 0.0021 0.0227 0.0254 0.8978 0.0001 0.0022 0.8385Baxter Intl. 0.0099 0.0018 0.0174 0.0153 0.946 0.0001 0.0034 0.5133

Bristol M.Sq. 0.018 0.0037 0.0145 0.0168 0.7178 0.0004 0.0039 0.0477Boston Sci. 0.117 0.0452 0.0324 0.0379 1.2532 0.001 0.0587 0.0825

Conagra 0.0383 0.0106 0.0154 0.0338 1.5062 0.0002 0.0072 0.0056Caterpillar 0.0189 0.0028 0.0227 0.0404 1.1009 0.0002 0.0066 0.0131

Cigna 0.1967 0.0356 0.0698 0.1413 1.3159 0.0013 0.0564 0.1092Comcast 0.1116 0.0571 0.0541 0.0527 1.8672 0.0022 0.0712 0.1768

Campbell Soup 0.0148 0.0039 0.0154 0.0219 0.6941 0 0.0043 0.0555Computer Sci. 0.1125 0.0138 0.0176 0.1104 0.9082 0.0009 0.05 1.7135

Du Pont 0.0544 0.0122 0.0214 0.0543 0.588 0.0002 0.0336 0.658Deere 0.0112 0.0045 0.0256 0.044 0.7584 0.0002 0.0099 0.0166Disney 0.0113 0.0022 0.017 0.0079 0.662 0 0.0051 0.006

Dow Chemical 0.0434 0.0212 0.0236 0.0402 0.4999 0.0005 0.0175 0.3771Darden Rest. 0.1390 0.0391 0.0495 0.1218 1.3096 0.0015 0.0448 0.4055Devon Energy 0.3017 0.0802 0.0756 0.4008 3.003 0.0033 0.1357 1.7469

Eastman Kodak 0.4813 0.0639 0.0427 0.2623 0.8589 0.0039 0.0924 0.0095Eastman Chem. 0.1313 0.0789 0.0297 0.1028 1.6042 0.0012 0.0478 1.6841

Ford Motors 0.2734 0.1041 0.0548 0.2341 0.7229 0.0049 0.1706 0.4463Fortune Brands 0.0069 0.0014 0.0169 0.0143 0.2977 0.0001 0.0026 0.004General Electric 0.0058 0.0015 0.0138 0.0153 0.4556 0.0001 0.0022 0.0032

General Mills 0.0092 0.0079 0.0178 0.0141 0.5552 0.0003 0.0038 0.0285Goodrich 0.0921 0.0428 0.0517 0.1542 1.4539 0.0012 0.0427 1.3024Honeywell 0.0225 0.0068 0.0361 0.0313 0.7957 0.0001 0.0105 0.0184Intl. Paper 0.2722 0.1014 0.0709 0.1418 0.9237 0.0018 0.2338 0.0141

Ingersoll Rand 0.0143 0.0161 0.0342 0.0172 0.6674 0.0009 0.005 0.0062Kraft Foods 0.0224 0.0036 0.0272 0.0264 0.5053 0.0002 0.0096 0.0142

Kroger 0.2631 0.0633 0.0752 0.2521 3.1491 0.0019 0.1377 0.2458

53

Parameter estimate standard errors by simulated MLE. Continued.

Company VRP Vol. of Meas. Mean rev. Asset Long run Correl. Bound.

Name λV var. σ error σe speed κ drift µ var. θ ρ to debt XD100

L 0.151 0.0309 0.0696 0.1565 1.5845 0.0006 0.0476 3.9169Marriott 0.2473 0.0427 0.0671 0.2276 2.2774 0.0013 0.0818 0.3275

McDonalds 0.0167 0.003 0.0443 0.0318 0.8225 0.0001 0.0043 0.0127McKesson 0.0781 0.013 0.0264 0.0412 0.6982 0.0005 0.0274 0.0205

Northrop Gr. 0.0103 0.0043 0.0049 0.0131 0.9092 0.0002 0.0041 0.592Newell Rubb. 0.5072 0.058 0.1655 0.556 3.1975 0.0087 0.1153 1.5466

Olin 0.0151 0.0047 0.0798 0.0192 0.6345 0.0003 0.0091 0.0056Radioshack 0.2859 0.0675 0.0405 0.4051 3.093 0.0025 0.1331 0.3259Raytheon 0.0085 0.0017 0.0043 0.0038 0.4128 0.0004 0.0234 0.0111

Sherwin Will. 0.0427 0.0181 0.0405 0.0143 0.84 0.0007 0.0218 0.3397Supervalu 0.1171 0.0871 0.0302 0.1127 0.7292 0.0026 0.0782 0.1183Safeway 0.1988 0.1476 0.0934 0.3366 3.2942 0.004 0.086 0.3891AT&T 0.014 0.0015 0.0205 0.0186 0.5144 0.0003 0.0062 0.007Target 0.0128 0.0037 0.0096 0.0086 0.4699 0.0003 0.0094 0.0094

Temple Inland . . . . . . . .Tyson Foods 0.3145 0.0962 0.0482 0.3278 1.3364 0.0013 0.0714 0.0584

Verizon 0.0042 0.0014 0.0112 0.006 0.2847 0 0.0019 0.0041Whirlpool 0.1576 0.0482 0.0271 0.1306 1.0949 0.0006 0.057 0.1313Walmart 0.0029 0.0012 0.0097 0.0137 0.459 0.0001 0.0018 0.0022

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Table 8: Pricing errors for simulated MLE estimation.

These are computed as xmodel−xactualxactual

×100, i.e. expressed in percentage points; and x denotes eithera CDS spread for a particular maturity or the equity volatility. The average across all maturitiesand firms is 18%. For the equity volatility the average is 2.5%.

Company Absolute percentage pricing errorsName 2y 3y 5y 7y 10y average Equity vol.

Amgen 81.191 30.512 13.280 20.256 24.984 34.045 3.403Arrow Electr. 23.844 13.423 0.112 6.095 10.554 10.806 6.352Boeing 58.711 20.068 0.172 9.568 12.576 20.219 4.939Baxter Intl. 76.857 28.638 0.340 10.178 14.278 26.058 3.589Bristol M.Sq. 86.445 34.777 7.973 20.908 30.715 36.163 6.635Boston Sci. 31.199 15.033 0.385 8.456 12.914 13.597 2.226Conagra 22.686 13.287 0.273 8.607 12.077 11.386 0.130Caterpillar 45.724 14.082 1.026 10.359 14.665 17.171 5.329Cigna 30.196 28.729 0.183 14.403 30.252 20.753 35.710Comcast 36.884 14.221 0.379 6.360 12.370 14.043 2.003Campbell Soup 53.348 20.713 2.414 8.704 12.970 19.629 5.291Computer Sci. 69.834 20.009 0.655 8.925 13.330 22.551 6.233Du Pont 50.001 19.204 0.216 9.690 12.432 18.309 8.614Deere 54.385 25.279 11.694 15.829 18.837 25.205 8.071Disney 56.595 19.587 0.267 8.389 11.661 19.300 4.577Dow Chemical 45.180 17.416 0.227 6.988 10.425 16.047 4.252Darden Rest. 62.044 18.349 0.122 8.205 12.674 20.279 10.531Devon Energy 50.561 20.782 0.170 8.148 9.833 17.899 4.622Eastman Kodak 36.519 19.686 0.142 7.629 13.374 15.470 1.795Eastman Chem. 24.881 12.634 0.595 7.645 11.393 11.430 1.866Ford Motors 15.270 9.594 1.716 6.095 10.575 8.650 3.221Fortune Brands 54.613 19.213 0.687 8.783 15.319 19.723 1.887General Electric 66.623 31.365 0.637 16.700 24.891 28.043 2.166General Mills 48.693 19.305 0.308 7.641 10.723 17.334 0.232Goodrich 48.363 10.509 0.204 6.775 9.099 14.990 6.546Honeywell 42.368 12.435 0.187 9.689 13.232 15.582 7.013Intl. Paper 31.675 19.457 0.149 7.419 11.237 13.987 4.645Ingersoll Rand 58.584 20.246 1.076 7.293 9.348 19.309 4.036Kraft Foods 43.124 16.438 0.202 6.644 10.103 15.302 1.858Kroger 54.231 16.222 3.804 9.376 14.068 19.540 0.457

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Pricing errors continued.

Company Absolute pricing errorsName 2y 3y 5y 7y 10y average Equity vol.

L 39.753 13.690 0.086 8.150 13.676 15.071 10.284Marriott 45.274 19.534 0.884 7.971 11.900 17.113 0.093McDonalds 56.776 20.234 1.240 8.122 12.442 19.763 3.369McKesson 42.141 13.699 0.087 7.935 16.696 16.112 9.893Northrop Gr. 69.810 15.967 0.184 6.190 9.484 20.327 4.410Newell Rubb. 27.101 13.757 0.196 6.654 9.126 11.367 3.564Olin 23.999 16.854 0.098 7.364 12.483 12.160 4.061Radioshack 29.732 17.956 0.125 9.543 15.930 14.657 2.954Raytheon 92.969 26.180 0.191 6.944 7.739 26.805 3.508Sherwin Will. 36.565 11.166 0.151 7.671 12.387 13.588 2.577Supervalu 28.428 17.002 0.609 8.649 12.775 13.493 1.818Safeway 44.425 16.632 4.142 10.083 14.597 17.976 0.140AT&T 55.685 16.275 0.197 8.185 11.604 18.389 4.100Target 42.077 15.604 0.381 9.423 12.804 16.058 4.646Temple Inland 34.576 19.852 0.163 8.628 15.285 15.701 22.839Tyson Foods 25.728 19.162 0.502 7.947 13.977 13.463 2.301Verizon 62.258 29.196 0.234 12.473 17.972 24.427 6.878Whirlpool 37.066 20.937 0.177 10.479 15.254 16.783 3.929Walmart 83.105 64.705 1.383 13.102 20.612 36.581 3.238

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