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Federal Reserve Bank of Chicago

Modeling Credit Contagion via the
Updating of Fragile Beliefs
Luca Benzoni, Pierre Collin-Dufresne,
Robert S. Goldstein, and Jean Helwege

WP 2012-04

Modeling Credit Contagion via the Updating
of Fragile Beliefs∗
Luca Benzoni†

Pierre Collin-Dufresne‡

Robert S. Goldstein§

Jean Helwege¶

This Version: December 26, 2011

Abstract
We propose a tractable equilibrium model for pricing defaultable bonds that are subject to
contagion risk. Contagion arises because agents with ‘fragile beliefs’ are uncertain about both
the underlying state of the economy and the posterior probabilities associated with these states.
As such, agents adopt a robust decision rule for updating that leads them to over-weight the
posterior probabilities of ‘bad’ states. We estimate the model using panel data on sovereign
Euro-zone CDS spreads during the recent crisis, and ﬁnd that it captures levels and dynamics
of spreads better than traditional aﬃne models with the same number of observable and latent
state variables.

∗

We thank Scott Brave, Richard Cantor, Sanjiv Das, Greg Duﬀee, Darrell Duﬃe, Lorenzo Garlappi, Alejandro
Justiniano, David Lando, Eric Wan, Tan Wang, Fan Yu, and seminar participants at Moody’s Advisory Research
Committee, the Bank of International Settlements, Duke University, Carnegie Mellon University, the FDIC, HEC
Montréal, London Business School, Groupe HEC, the Wharton School, the University of California at Berkeley, the
University of Illinois at Urbana-Champaign, the University of Texas at Dallas, the Western Finance Association,
the European Summer Symposia in Financial Markets and Economic Theory at Gerzensee, and the Day Ahead
Conference on Financial Markets and Institutions for helpful comments. Andrea Ajello, Olena Chyruk, Andy Fedak,
Paymon Khorrami, Kuan Lee, Harvey Stephenson, Guang Yang, and Ludovico Zaraga provided excellent research
assistance. Part of this work was completed while Benzoni was a visiting scholar at the Federal Reserve Board. The
most recent version of this paper is available at http://ssrn.com/abstract=2016579.
†
Federal Reserve Bank of Chicago, lbenzoni@frbchi.org
‡
Carson Family Professor of Finance, Columbia University, and NBER, pc2415@columbia.edu
§
C. Arthur Williams Professor of Insurance, University of Minnesota, and NBER, golds144@umn.edu
¶
J. Henry Fellers Professor of Business Administration, University of South Carolina, Jean.helwege@moore.sc.edu

Modeling Credit Contagion via the Updating of Fragile Beliefs

Abstract
We propose a tractable equilibrium model for pricing defaultable bonds that are subject to
contagion risk. Contagion arises because agents with ‘fragile beliefs’ are uncertain about both
the underlying state of the economy and the posterior probabilities associated with these states.
As such, agents adopt a robust decision rule for updating that leads them to over-weight the
posterior probabilities of ‘bad’ states. We estimate the model using panel data on sovereign
Euro-zone CDS spreads during the recent crisis, and ﬁnd that it captures levels and dynamics
of spreads better than traditional aﬃne models with the same number of observable and latent
state variables.

1

Introduction

During the recent (and ongoing) Euro-zone crisis, the risk of contagion has often been cited as one
of the major drivers of sovereign credit spreads. Indeed, typing the words “contagion and Euro”
into a Google search returns over 750,000 results, many of which refer to articles from the ﬁnancial
press that relate changes in sovereign spreads of European nations to the risk or ‘fear’ of contagion.
This dialogue raises many important questions, including:
• What is contagion risk, and what are its economic sources?
• Is there a risk-premium associated with contagion risk, and if so, what is its impact on
sovereign spreads?
• To what extent is the co-movement in sovereign spreads driven by contagion risk and its
risk-premium?
In this paper we propose a tractable equilibrium model in which contagion risk signiﬁcantly impacts the level and the dynamics of sovereign credit spreads. Contagion arises because agents are
uncertain about both the underlying state of the economy and their posterior probabilities associated with these states (they have ‘fragile beliefs’). Following Hansen and Sargent (2007, 2010), we
investigate agents that adopt a robust decision rule for updating that leads them to over-weight
the posterior probabilities of ‘bad’ states and under-weight those of ‘good’ states. Together, these
two ingredients (hidden states and fragile beliefs) can explain large sovereign spreads even if expected losses due to default are relatively small. Furthermore, the model can generate signiﬁcant
correlation in spreads even if common movements in macroeconomic fundamentals are relatively
modest.
To be more speciﬁc, we assume there is a hidden state of nature which, if known, would impact
the expected aggregate consumption growth as well as each country’s default probability. Agents
update the likelihood of each state based on all available information, but they are uncertain
about the true data-generating process, and therefore are uncertain about the updated posterior
probabilities assigned to each state. Following Hansen and Sargent (2007, 2010), we assume agents
adopt robust decision rules to mitigate this ‘model risk.’ In particular, we assume agents use
diﬀerent risk-sensitivity operators to account for i) uncertainty in the model speciﬁcation conditional
on the state, and ii) uncertainty regarding the correct posterior distribution of the state itself.
The ﬁrst component, preference for robustness regarding model parameters, has been more extensively studied in the literature. For example, it is well understood that, in a representative agent
framework, the decision rule of a log-utility investor who uses an entropy penalty is observationally
equivalent to that of an agent with recursive utility of the Epstein-Zin-Kreps-Porteus type (see, e.g.,
Barillas, Hansen and Sargent (2010)). However, the second component, which can be interpreted

1

as a preference for robustness towards mis-speciﬁcation in their beliefs, is less understood.1 As we
show, one beneﬁt of the ‘fragile beliefs’ speciﬁcation we adopt is that it is very tractable, even for
fairly complex models. Indeed, when an agent has fragile beliefs, she values long-lived securities by
ﬁrst estimating their value as if she knew the ‘true’ state, and then she takes a weighted average
of these values. This is very similar to the approach followed in the traditional time-separable
Bayesian setup (e.g., Detemple (1995), Genotte (1985), Veronesi (1998)), except that the weights
used by the fragile beliefs agent are not equal to the posterior probabilities associated with each
state. Instead, she distorts the probabilities to reﬂect her uncertainty about the estimated posterior
probabilities in an endogenous way, placing more weight on the models/states with lower utility.
This minor departure from the classical time-separable Bayesian setup2 result when has signiﬁcant
impact on equilibrium prices. Indeed, this updating of beliefs will generate correlations in credit
spreads that are signiﬁcantly higher than if spreads were functions of the macroeconomic conditions
only. Furthermore, since agents put higher weights on the states with lower utility, the model also
generates signiﬁcantly higher spreads (and credit risk-premia) than a traditional model based on
time-separable preferences, for example.
One of our theoretical contributions is to derive suﬃcient conditions for which the prices of
long-lived securities are equal to a weighted average of their conditional prices. We ﬁnd that if
these conditions do not hold, then the ‘model averaging pricing rule’ is not in general arbitragefree, implying that these prices are not consistent with a no-trade equilibrium. These conditions are
clearly related to the time-consistency of the preferences of agents with fragile beliefs (see Section
6.5 of Hansen and Sargent (2007)).
Another contribution of this paper is that we obtain closed-form solutions for bond prices
even though, to capture contagion, the default intensity process falls outside the popular “doubly
stochastic” (or Cox process) framework. Indeed, in a doubly stochastic setting, individual default
events are inherently precluded from impacting the intensities of the surviving entities. In contrast,
in our framework, agents update their beliefs by observing sovereign credit events (as well as other
news signals). If there is a default, agents increase the probability they assign to the ‘bad’ hidden
state. This updating raises the perceived default intensity (and in turn, credit spreads) of the other
countries.
We then estimate our model using panel data on sovereign CDS spreads from February 2004
to September 2010 for 11 Euro-zone countries. We use a two-stage procedure. First, we follow the
literature on sovereign risk3 to identify a list of variables that have been shown to predict a country’s
ability or propensity to repay its debt. For each Euro-zone country, we use a dynamic principal
1

One exception is the contemporaneous paper by Boyarchenko (2011) which investigates the impact of ambiguity
aversion on the US ﬁnancial crisis of 2007-08.
2
Indeed, we recover the time-separable Bayesian result when the tolerance parameter to model mis-speciﬁcation
becomes inﬁnitely large.
3
See, for example, Duﬃe et al. (2003), Edwards (1984), Hilscher and Nosbusch (2010), Longstaﬀ et al. (2010),
Min (1998), Pan and Singleton (2008).

2

component framework (e.g., Stock and Watson (1989, 1991)) to summarize the information in
these variables into a single country-speciﬁc Macroeconomic Conditions Index (MCI). Since the
data are observed at mixed frequencies, we rely on the ﬁltering method of Aruoba, Diebold, and
Scotti (2009). Assuming a multivariate AR(1) process, this approach provides us with both an
estimate of the time series of these underlying variables and a parameter vector that captures their
dynamics. We augment the set of explanatory variables to include the Chicago Board Option
Exchange (CBOE) VIX index as a measure of global economic uncertainty (e.g., Longstaﬀ, Pan,
Pedersen, and Singleton (2010), Pan and Singleton (2008)). Conditional default intensities are then
speciﬁed to be linear functions of the state vector that includes country-speciﬁc MCIs and the VIX
index.
In the second estimation step, we use sovereign CDS spreads panel data and time series of
default events to identify the rest of the model parameters and the time series of the ﬁltered
posterior probability of the hidden state. We cast the model in a state-space framework and
estimate it by quasi maximum likelihood in combination with the Kalman ﬁlter. Since both state
and measurement equations in the system contain non-linearities, we rely on a square-root unscented
ﬁlter (e.g., Wan and van der Merwe (2001), Christoﬀersen, Jacobs, Karoui, and Mimouni (2009)).
In the early part of the sample period, 2004-2007, we estimate the probability of the good
state to be nearly one. This changes at the end of 2007, when the posterior probability of the bad
economic state increases signiﬁcantly and its ﬂuctuations become more pronounced as the sovereign
crisis unfolds. Consistent with Hansen and Sargent (2007, 2010), we ﬁnd that the agent displays
a preference for robust beliefs, in that she slants the risk-adjusted probability of the hidden state
towards the model associated with the lowest continuation utility. That is, she attaches a higher
probability of being in the bad state under the risk-neutral measure than the physical measure.
Consequently, the level of risk-adjusted default intensities is higher than those computed under the
physical probability measure, i.e., our model generates positive jump-to-default (JTD) risk-premia.
Overall, the model ﬁts CDS spreads data well across Euro zone countries, both before and
during the crisis. To better gauge its performance, we compare the pricing errors of our model to
those of a (linear) aﬃne speciﬁcation with a state vector that includes country-speciﬁc MCIs and
the VIX (as in our model), and a single latent factor, which we estimate with principal component
analysis. In any arbitrage-free aﬃne framework, sovereign credit spreads are a linear function of
the state vector, with coeﬃcients determined by no-arbitrage restrictions. Therefore, in sample,
unrestricted ordinary least squares (OLS) regressions give an upper bound on the goodness of ﬁt
such an aﬃne model could achieve. We ﬁnd that our model signiﬁcantly outperforms the aﬃne
benchmark estimated via OLS regressions, with a 23-85% reduction in mean absolute pricing errors,
and a typical drop in maximum errors by a factor of two. This strongly suggests that there are
important nonlinearities in the behavior of credit spreads that elude aﬃne speciﬁcations and are
better captured by our model, which has nonlinearities in both the mapping of spreads onto the
state variables and state variable dynamics.

3

Related Literature.

Our paper builds on and combines two important strands of literature:

event risk and Bayesian updating of beliefs. Conditions for which jump-to-default is not priced have
been investigated by Jarrow, Lando and Yu (2005). However, recent empirical ﬁndings question
this doubly-stochastic assumption. For example, Das et al. (2006, 2007) report that the observed
clustering of defaults in actual data is inconsistent with this assumption. Duﬃe et al. (2009) use a
fragility-based model similar to ours to identify a hidden state variable consistent with a contagionlike response. Note that the focus of these papers is on estimating the empirical default probability,
whereas our focus is on pricing. Jorion and Zhang (2007) ﬁnd contagious eﬀects at the industry
level (see also related work by Jorion and Zhang (2009), and Lando and Nielsen (2010)).
Other papers investigating event risk include Jarrow and Yu (2001), who also provide a model
where the default of one ﬁrm aﬀects the intensity of another. However, their model remains
tractable only for a “small” number N of ﬁrms exposed to contagion-risk (e.g., Jarrow and Yu
(2001) investigate only N = 2). In contrast, our model remains tractable regardless of the number
of entities that share in the contagious response.4 Models of credit risk embedded within a macroeconomic setting include David (2008), Chen, Collin-Dufresne and Goldstein (2009), Chen (2010),
and Bhamra, Kuehn and Strebulaev (2011).
Our approach shares many common features with those in the learning and contagion literature
(e.g., David (1997), Detemple (1986), Feldman (1989), Veronesi (1999, 2000)).5 As in these papers,
the representative agent in our economy learns about a hidden state from observing aggregate
consumption and other “diﬀusive” signals. However, in our model the agent also learns from
observing the default history of an entities (ﬁrms or countries), i.e., information is revealed through
both diﬀusion processes and jump processes. Further, we identify a time-consistent model of a
representative agent that has fragile beliefs (Hansen and Sargent (2010), Hansen (2007)).6 Time
consistency allows us to price securities with long-dated cash ﬂows in a tractable manner. This
framework naturally generates a ‘ﬂight-to-quality’ like eﬀect (i.e., a drop in risk free rates) caused
by an unexpected default, consistent with observation.
Our information-based mechanism for contagion is similar to that proposed by King and Wadhwani (1990) and Kodres and Pritsker (2002), who investigate contagion across international ﬁnancial markets. There is also a large empirical literature that studies contagion in equity markets
(e.g., Lang and Stulz (1992)) and in international ﬁnance (e.g., Bae, Karolyi and Stulz (2003)).
Theocharides (2007) investigates contagion in the corporate bond market and ﬁnds empirical support for information-based transmission of crises.
The rest of the paper is as follows. In Section 2, we propose an intensity-based model of sovereign
risk with a hidden state and show how beliefs are updated from observing default events and other
4

Collin-Dufresne, Goldstein and Hugonnier (2004) simplify the bond pricing formula of Duﬃe, Schroeder and
Skiadas (1996). Note, however, that the formula itself does not identify a tractable framework for pricing contagion
risk. Other models of contagion include Davis and Lo (2001), Schönbucher and Schubert (2001), and Giesecke (2004).
5
See also related work by Aı̈t-Sahalia, Cacho-Diaz, Laeven (2010), Benzoni, Collin-Dufresne and Goldstein (2011).
6
See also related work on belief-dependent utilities by, e.g., Veronesi (2004).

4

signals. In Section 3 we investigate the pricing implications of this model by incorporating it in a
general equilibrium framework where the representative agent has fragile beliefs. We then estimate
the model in Section 4 using six years of sovereign CDS prices. We conclude in Section 5. In the
Appendix, we identify necessary conditions for fragile beliefs preferences to generate time consistent
price processes.

2

Updating Beliefs by Observing Default Processes

Consider an economy in which the true state of nature S̃ is unknown and can be in any one of
s ∈ (1, M ) states. At date-t, investors do not know what state the economy is in, but form a prior
πs (t) ≡ Prob(S̃ = s|Ft ), where Ft is the investors’ information set at date-t. In this economy
there are n defaultable entities (ﬁrms, countries) indexed by i ∈ (1, n) with random default times
τi driven by point processes characterized by default intensities. In particular, conditional upon
being in state-s, the probability of default over the next interval dt is expressed via
[
Pr d1{τ

i <t}

= 1 S̃ = s, Ft

]

[
≡ E d1{τ

S̃ = s, Ft

i <t}

−

= λis (t ) 1{τ

i >t}

dt.

]
(1)

That is, we can interpret λis (t− ) as the date-t default intensity for country-i conditional upon being
in state-s. Below, we will assume that, conditioning both on the state-s and the paths λis (t− )|t=0
T

for some distant future date-T , the default events across countries (or ﬁrms) are independent. In
technical terms, we are assuming a doubly-stochastic, or Cox-process conditional upon being in a
particular state-s (e.g., Lando (1998)). We emphasize, however, because agents do not know the
correct state-s, our model falls outside of the Cox-process framework, as will be made clear below.
Since investors do not know the actual state of nature, their estimate of the actual default
P

intensity λi (t− ) is deﬁned implicitly through
P

λi (t− ) 1{τ

i

[
dt
≡
E
d1{τ
>t}
=

=

M
∑
s=1
M
∑
s=1

i <t}

Ft

]

[
πs (t) E d1{τ

i <t}

πs (t)λis (t− ) 1{τ

S̃ = s, Ft

i >t}

dt.

]

(2)

Thus, conditional on investors’ information, the default intensity of country-i is equal to a weighted
average of the conditional default intensities:
P

−

λi (t ) =

M
∑

πs (t)λis (t− ).

(3)

s=1

We assume that investors continuously update their estimates of the {πs (t)} process conditional
upon whether or not they observe a default event during the interval dt. A direct application of
5

Theorem 19.6 page 332 in Liptser and Shiryaev (2001) (see also their example 1, p. 333) gives the
updating equation for πs (t):
∑
dπs (t)
=
πs (t− )
N

i=1

(

λis (t− )
P

λi (t− )

)
−1

dMi (t),

(4)

where we have deﬁned the martingale process dMi (t) via:
(
dMi (t) ≡ d1{τ

P

i ≤t}

− λi (t)1{τ

i

)
dt
.
>t}

(5)

This process has many intuitive properties. First, if the prior πs (t) = 1 for some state-s (and thus
πs′ (t) = 0 for all other s′ ), then there is no updating. That is, in an economy where the agents
know for sure the intensity of the countries, then there is no learning to be done. Second, when no
default is observed over an interval dt, then investors revise downward the ‘high-default’ states of
P

nature (i.e., those s with λis (t− ) > λi (t− )), and in turn revise upward the ‘low-default’ states of
P

nature (i.e., those s with λis (t− ) < λi (t− )). Conversely, when a default is observed over an interval
dt, investors revise upward those high-default states of nature, and in turn revise downward those
[
]
low-default states of nature. Third, note that πs (t) ≡ E S̃ = s Ft is a P-martingale in that
E [dπs (t)|Ft ] = 0, as can be seen from equations (2), (4) and (5).

2.1

Updating also from Continuous Information

In addition to observing country default processes, investors also observe continuous signals that
provide information about the state. Speciﬁcally, we assume that investors observe K + 1 signals
with dynamics:
dΩk (t) = µk,s dt + σk dZk (t)

k ∈ (0, K),

(6)

where where the µk,s are constants,7 and {dZk (t)} are independent Brownian Motions adapted to
the full information ﬁltration Gt , which contains in particular the information on the ‘true’ state. In
particular, we have E [dZk (t)|Gt ] = 0. Using the ‘innovation’ approach to ﬁltering, we can rewrite
the signal dynamics as:
dΩk (t) = µ̄k (t) dt + σk dZk (t),
(

with
dZk (t) = dZk (t) +

µk,s − µ̄k (t)
σk

(7)

)
dt

(8)

and where we deﬁne the drift of the signal process based on the investors’ information ﬁltration to
be:
µ̄k (t) ≡
7

∑
1
E [dΩk (t)|Ft ] =
πs (t) µk,s .
dt
s

These could be stochastic processes as long as they are Ft predictable.

6

(9)

Note, in particular, that Zk (t) is a Brownian motion in the information ﬁltration of the investor
since it has quadratic variation t from (8) and satisﬁes E [dZk (t)|Ft ] = 0.
It is well-known (see, e.g., Liptser and Shiryaev (2001), David (1997), and Veronesi (1999)) that
the updating equation for the posterior probability of the state from this continuous information is
given by
dπs (t)
πs (t)

)
K (
∑
µk,s − µ̄k (t)
=
dZk (t).
σk

(10)

k=0

Given that the agent observes both continuous signals dΩk (t) and defaults d1τi >t , and that by
deﬁnition the two are orthogonal signals (since one is a pure diﬀusion and the other a pure jump
process, e.g., Protter (2001)), it follows from equations (4) and (10) that the updating equation is:
(
)
)
N
K (
∑
∑
µk,s − µ̄k (t)
dπs (t)
λis (t− )
=
− 1 dMi (t) +
dZk (t).
(11)
P
πs (t− )
σk
λ (t− )
i=1

2.2

k=0

i

Model for Intensity Conditional on the State

We specify the default intensity of entity i ∈ (1, N ) conditional on being in state s as:
′
λis (t) = αis + βis
X(t),

(12)

where X(t) is a state vector which contains both country-speciﬁc variables as well as a common
variables. We specify X(t) to follow a multi-dimensional Gaussian aﬃne process:
dX(t) = [ψ − κX(t)] dt + Σ dW (t),

(13)

where without loss of generality we specify the volatility matrix Σ to be lower diagonal and we
assume that W (t) is a vector Brownian motion process (independent of Z(t)) both in the general
Gt as well as the investor speciﬁc ﬁltration Ft .
Up to this point, the state variable dynamics have been speciﬁed under the historical measure.
In the following section we address the issue of pricing defaultable securities in the presence of
contagion risk when the representative agent has fragile beliefs.

3
3.1

General Equilibrium with Fragile Beliefs
Information Structure

It is useful to deﬁne more formally the information structure of the model we have setup. All
the uncertainty in our model is summarized by a ﬁltered reference probability space (Ω, G, {Gt }, P )
where Gt is the natural ﬁltration generated by (S, N, Z, W ) the ‘fundamental’ shocks in the economy.
Speciﬁcally, S is a multinomial G0 measurable random variable whose realization ‘selects’ the ‘true’
state s, N is the vector of default counting processes for each country Ni (t) , i = 1, . . . , n, which

7

each have Gt measurable default intensity λis (t), Z and W are both (respectively K + 1 and ddimensional) vectors of independent Brownian motions. Investors in our economy however do not
observe these ‘fundamental’ shocks. Instead, their ﬁltration Ft is generated by observing the history
of defaults (N ), the vector of continuous signals (Ωk for k = 0, . . . , J), and the state vector X. They
formulate a prior over the probabilities associated with the realization of the multinomial random
variable. As we have shown above, in the information ﬁltration Ft , the default counting vector N
P

has intensity (λi (t− ) for i = 1, . . . , n ). Also, W is a Ft -adapted Brownian motion. However, Z(t)
is not. Instead, the process Z(t) deﬁned in (8) above is a standard Ft -adapted Brownian motion.
Further, it is clear that Xt is both a Gt and Ft adapted Markov process and that (πs (t), N (t), Ω(t)))
have jointly Ft adapted Markov dynamics. Lastly, it is clear that once investors know the realization
∪
of the state S then all the uncertainty unravels, so that Gt has same information as Ft {S}.

3.2

Endowment Process

We assume that the aggregate endowment (which in equilibrium will be consumed by the representative agent) has the following dynamics:8
d log y = µ0,s dt + σ0 dZ0

(14)

Of course, the ‘innovations’ representation of log-endowment in the information ﬁltration of the
representative agent is:
d log y = µP0 (t) dt + σ0 dZ0 ,
∑
where the Ft -Brownian motion Z0 and µP0 (t) = s πs (t) µ0,s are deﬁned above.

3.3

(15)

Preferences

We assume that the representative agent displays fragile beliefs as described in Hansen and Sargent
(2007). More speciﬁcally, we assume that the agent will value consumption streams using a twostep approach. First, he values each stream conditional on knowing the true model/state. Second,
he takes an average of the model speciﬁc continuation values. Hansen and Sargent (2007) assume
that at both stages the agent worries about the possible mis-speciﬁcation of his model and therefore
uses a robust decision rule. Their novel insight is that agents may use a diﬀerent ‘risk-sensitive
operator’ to deal with mis-speciﬁcation of posterior beliefs associated with the model/state than to
deal with model speciﬁcation conditional on the state. Since we are interested in uncertainty about
the updating process and what consequences this may have for asset prices, we assume that in this
two-step procedure agents use standard time-separable log-utility function in the ﬁrst step, and use
a robust decision rule to ‘model-average’ the continuation values. More speciﬁcally, we assume:
8

We assume that the logarithmic aggregate endowment equals Ω0 for notational convenience, so it is already in
the agent’s information set.

8

• Conditional upon being in state s ∈ (1, M ), the agent has logarithmic-preferences. That is,
the agent ranks consumption lotteries in state-s according to the (state contingent) index
V ({C(·)}|F0 , s), which satisﬁes:
[∫

∞

V ({C(·)}|F0 , s) = E

β dt e

−βt

]
log C(t) F0 , s

(16)

0

• To rank consumption streams unconditionally, the agent displays fragile beliefs. In particular,
the agent weights the conditional utility indices V ({C(·)}|F0 , s) by using an entropy penalty
characterized by a preference for robustness parameter ζ via:
{M
(
)}
∑
V ({C(·)}|F0 ) = min
πs (0) ξs (0)V ({C(·)}|F0 , s) + ζ ξs (0) log ξs (0)
,
{ξs (0)}>0

(17)

s=1

subject to the constraint
∑

1 =

πs (0)ξs (0).

(18)

s

Solving the constrained minimization, Hansen-Sargent (2010) show:
ξs (0) =

∑

e
s′

−( ζ1 )V ({C(·)}|F0 ,s)

πs′ (0) e

−( ζ1 )V ({C(·)}|F0 ,s)

.

Plugging this back into equation (17), preferences simplify to
[M
]
∑
−( ζ1 )V ({C(·)}|F0 ,s)
πs (0) e
V ({C(·)}|F0 ) = −ζ log
.

(19)

(20)

s=1

It is worth noting that if the agent chooses to consume the endowment stream (which will
ultimately be the equilibrium in this exchange economy), we ﬁnd
[∫ ∞
]
−βt
V ({y(·)}|F0 , s) = E
β dt e
log y(t) F0 , s
0
∫ ∞
[
]
=
β dt e−βt log y(0) + µ0,s t
0

= log y(0) +

µ0,s
.
β

(21)

Under this scenario, the fragility parameters
ξs (t) =

e
∑

s′

−(

µ0,s
βζ

πs′ (t) e

)

−(

µ
0,s′
βζ

(22)
)

are independent of the state variables X(t), and hence change over time only through their dependence on the probabilities {πs′ (t)}. This feature is important for the model to be time-consistent
(as we show below).

9

Given these preferences, and assuming complete markets, the representative agent chooses her
consumption stream to maximize her utility given by equations (20) and (16) subject to the budget
constraint

∫

∫

∞

dt

0 =

dωt A(ωt |ω0 ) [y(ωt ) − C(ωt )] ,

(23)

0

where the A(ωt |ω0 ) are the Arrow-Debreu prices (which the representative agent takes as exogenously speciﬁed). The agent’s ﬁrst order conditions with respect to consumption across all states
of nature C(ωt ) imply:
θA(ωt |ω0 ) =

∑

πs (0) ξs (0)βe−βt π(ωt |ω0 , s)

s

1
,
C(ωt )

(24)

where the Lagrange multiplier θ can be determined by taking the limit ωt ⇒ ω0 :
θ =

β
.
C(ω0 )

Combining these last two equations, we ﬁnd that the optimal consumption bundle satisﬁes
∑
C(ω0 )
A(ωt |ω0 ) =
πs (0) ξs (0)e−βt π(ωt |ω0 , s)
.
C(ωt )
s

(25)

(26)

Of course, in a representative agent endowment economy, when markets clear, the right-hand
side consumption is exogenously given and therefore this ﬁrst order condition deﬁnes equilibrium
state prices. As we show next, these state prices lead to a natural ‘algorithm’ to value long dated
claim. First, value the claims as if the true state s were known and the representative agent
had standard log-utility. Second, average these diﬀerent model-values by weighting them with
endogenously distorted posterior probabilities of the state. Finally, we show that this pricing rule
is arbitrage-free in that there exists a set of strictly positive state prices, for which the two stage
pricing rule described above holds at all times and states. Therefore these state prices support a
no-trade equilibrium in which a representative agent with fragile beliefs consumes the aggregate
endowment given in (14). This result is related to the time consistency of fragile beliefs preferences,
which is not guaranteed (see Hansen-Sargent (2007)). In fact, in Appendix A we give an example of
more general fragile beliefs preferences, where the state prices derived from the two-stage approach
would not be arbitrage-free. Instead these give rise to dynamic arbitrage opportunities. In that
exemple, the representative agent is not time-consistent. At time zero all claims, short and longdated, are valued such that he does not want to trade given his current and anticipated future
consumption. However, at future dates, if markets reopen for trading at the prices consistent with
time-zero state prices, the representative agent would like to trade, implying that the initial state
prices do not support a no-trade equilibrium.

3.4

Arrow-Debreu Equilibrium

For markets to clear in this endowment economy, Arrow-Debreu prices adjust until the optimal
consumption is equal to the exogenous endowment in each state. Thus, we ﬁnd equilibrium state
10

prices:
A(ωt |ω0 ) =

∑

πs (0) ξs (0) e−βt π(ωt |ω0 , s) e−[log y(ωt )−log y(ω0 )] .

(27)

s

Combining equations (14) and (27), we can express the Arrow-Debreu prices as
]
[( s )
∑
Λ (t)
Q
A(ωt |ω0 ) =
πs (t) E
1{ωf =ω } |F0 , s .
t
t
Λs (0)
s

(28)

Here we have deﬁned Λs (t) to be the ‘pricing kernel’ conditional on the state being s:
dΛs (t)
Λs (t)

= −rs dt − σ0 dZ0 (t),

(29)

where the state-contingent spot rates {rs } are constants,
rs

= β + µ0,s −

σ02
,
2

(30)

and the ‘distorted’ model-risk-adjusted probabilities are given by:
πsQ (t) ≡ πs (t) ξs (t).

(31)

D(ωT )

More generally, this suggests that the date-t price Vt

of a security with contingent cash

ﬂows D(ωT ) at date-T if state-ωT occurs is:
D(ω )
Vt T

=

∑

[(
Q

πs (t) E

s

=

∑

Λs (T )
Λs (t)

)

]
D(T )|Ft , s

πsQ (t) e−rs (T −t) EQs [D(T )|Gt , s] ,

(32)

s

where we have used the fact that when we condition on the realization of S then Ft and Gt
contain the same information, and we have deﬁned the measure Qs equivalent to P by the RadonNykodim derivative

dQs
dP

=

ers T Λs (T )
Λs (0) .

By Girsanov’s theorem then we know that Z0Qs (t) deﬁned by

dZ0Qs (t) = dZ0 (t) + σ0 dt is a Qs − Gt Brownian motion, and all other Brownian motions orthogonal
to dZ0 are unaﬀected by the change of measure.
We now show how this pricing rule is indeed consistent with absence of arbitrage, in that there
exists a well-deﬁned pricing kernel that supports this pricing function for all states and times.

3.5

The Pricing Kernel and Market Prices of Risk

In Appendix A, we show that the Ft adapted process Λt with the following dynamics:
dΛ(t)
Λ(t)

= −r(t) dt −

K
∑

ϕk (t) dZk (t) −

∑
i

k=0

11

Γi (t) dMi (t),

(33)

where
r(t) =

∑

πsQ (t)rs

s

ϕk = σk 1{k=0}
Q

Γi (t) =
Q

λi

µQ
− µPk
− k
σk

P

λi − λi
P

λi
∑
=
πsQ λis (t)
s

µQ
k

=

∑

πsQ µk,s

s

µPk

=

∑

πs µk,s ,

(34)

s

is a valid state price deﬂator for our economy, in the sense that for any arbitrary FT -measurable
payoﬀ D(T ) the following holds:
[

]
Λ(T )
D(T )
E
D(T )|Ft = Vt
,
Λ(t)

(35)

where VtD is deﬁned in equation (32) above.
In other words, this (strictly positive) state price density supports our conjectured pricing rule.
A direct implication of this is that in an economy where all prices are determined by the pricing
kernel Λ(t), an individual with the fragile beliefs described above who consumes the aggregate
consumption will not want to trade in any securities. We have thus identiﬁed a pricing system
consistent with a no-trade equilibrium for this fragile beliefs agent. We show in Appendix A that
this is directly related to the time consistency of the representative agent.
The fact that Γi (t) diﬀers from zero implies that sovereign jumps-to-default are priced in this
economy even though default of any country does not aﬀect aggregate consumption. This highlights
a diﬀerence with time-separable frameworks such as David (1997) and Veronesi (2000). In those
settings, jumps in πs would imply jumps in the expected growth rate of consumption. Such jumps,
however, would not carry a risk-premium, since with time-separable preferences changes in the
pricing kernel U ′ (c) ∼ C(t)γ are generated only by contemporaneous changes in consumption, and
not changes in the expected growth rate. However, in the fragile beliefs framework, these jumps
are priced.9
We now turn to the evaluation of long-dated risk-free and risky zero coupon bonds and the
solution for sovereign CDS spreads.
9

An alternative to having expected consumption growth carry a risk premium is to rely on long run risk; see, e.g.,
Benzoni, Collin-Dufresne and Goldstein (2011), Drechsler and Yaron (2011), and Eraker and Shaliastovich (2008).
However, this setting no longer admits a closed-form solution.

12

3.6

The Risk-Free Zero-Coupon Bond

Using equation (32) and (35) the price of the zero-coupon bond that pays D = 1 unit of consumption
in all states of nature at date-T is:

[

]
Λ(T )
P (π(t), T − t) = E
Λ(t)
∑
=
πsQ (t) e−rs (T −t) EQs [1|Gt , s]
s

∑

=

πsQ (t) e−rs (T −t) .

(36)

s

3.7

Defaultable Zero-Coupon Bond

The price of the zero-coupon bond that pays D = 1(τi >T ) one unit of consumption at date-T if
country-i does not default by that date, and zero otherwise, is:
[
]
∑
B i (π(t), X(t), T − t) =
πsQ (t) e−rs (T −t) EQs 1{τ >T } |Gt , s
i

s

= 1{τ

M
∑
i >t}

πsQ (t) e−rs (T −t) Bsi (X(t), T − t),

(37)

s=1

where we have deﬁned
]
[ ∫T
Bsi (X(t), T − t) ≡ EQs e− t du λi,s (Xu ) |Gt , s .

(38)

Note that risky bond price simpliﬁes to a weighted sum of ‘reduced-form’ risky bond prices
because, conditional on being in state-s, we are in a doubly stochastic framework.
Equation (38) implies that e−

∫t
0

du λi,s (X(u))

Bsi (X(t), t, T ) is a Qs -martingale, and, thus, that

the solution for Bsi (X(t), t, T ) satisﬁes the PDE (in this equation, we drop the (i,s) subscripts on
Bsi (X(t), t, T ) and λi,s to improve readability)
[
]
∑
∑
∑
1∑
0 = −λ(X(t)) B + Bt +
Bj ψj −
Bj,j ′
κjm Xm +
Σjm Σj ′ m .
2 ′
m
m
j

Here, we use the notation Bt ≡

∂
∂t B,

(39)

j,j

Bj ≡

∂
∂Xj

B, etc.

Given that λi,s (X(u)) is linear in the state vector X(t) via equation (12) and that the riskneutral dynamics are aﬃne, it is well known that the solution to this expectation takes the form:

(
with “initial conditions”

Bsi (X(t), T − t) = e

′ (T −t) X(t)
Mi,s (T −t)−Ni,s

,

(40)

)

′ (τ = 0) = 0 . Collecting terms linear and indepenMi,s (τ = 0) = 0, Ni,s

dent of X(t), we ﬁnd that the deterministic coeﬃcients satisfy
[
]
Ni,s (τ ) = (κ′ )−1 In − exp(−κ′ τ ) βi,s
]
[
∫ τ
1 ′
′
′
Mi,s (τ ) =
du −αis − Ni,s (u)ψ + Ni,s (u)ΣΣ Ni,s (u) .
2
0
13

(41)

3.8

The Risk-Neutral Survival Probability

The risk-neutral survival probabilities are deﬁned as:
S i (π(t), X(t), T − t) =
=

∑

[
]
πsQ (t) EQs 1{τ >T } |ωt , s

s
[M
∑

i

]
πs (t) Bs (X(t), T − t)
Q

i

s=1

1{τ

i >t}

.

(42)

Note that these can be obtained from equation (37) by setting the risk-free rate components to
zero.

3.9

Sovereign CDS Spreads

Here we obtain an expression for the sovereign CDS from the risky and riskless bond equations
(36)-(37). The present value of the payments in the fee leg of the CDS contract is:
[
]
n
n
∑
∑
tj + tj−1
∆
i
i
i
i
i
PvFee(c ) =
) S (πt , X(t), tj−1 ) − S (πt , X(t), tj ) ci ,
B (πt , X(t), tj ) c ∆ +
P (πt ,
2
2
j=1

j=1

(43)
where the second component is the present value of the accrued interest upon default (assumed
to occur half-way between tj−1 and tj for simplicity). Payments are made at pre-speciﬁed dates
t = t0 , t1 , t2 , . . . tn . ∆ = tj − tj−1 is the time between promised coupon payment (typically one
quarter).
The present value of the contingent default payment leg is:
PvDef =

n
∑
j=1

P (πt ,

]
tj + tj−1 [ i
) S (πt , X(t), tj−1 ) − S i (πt , X(t), tj ) L,
2

(44)

where L is the expected loss given default experienced upon a sovereign default. For example, Pan
and Singleton (2008) discuss the fact that market convention is to set L = 0.75 for sovereign risk
(as opposed to L = 0.6 for corporate bonds). They ﬁnd that their maximum likelihood estimates
are not too distinct for most countries from that market convention.
The fair credit default swap spread is the number ci that sets PvFee(ci ) = PvDef.

4

Model Estimation and Empirical Results

Here we estimate the model developed in the previous section using data on European sovereign
CDS from 2004 to 2010. The approach we follow is to ﬁrst estimate observable macro- and ﬁnancial
fundamental indicators that determine the willingness to pay of each individual country in the absence of a hidden state. This corresponds to the state vector Xt in our model. Then conditional on
these observable indicators we estimate from the time-series and cross-section of CDS spreads the

14

posterior probability of the hidden state (πs ) as well as the other parameters of the model (state dependent default intensity parameters and preference parameters) using quasi maximum likelihood,
and treating the posterior probability as a latent variable. We then compare the performance of our
model to a standard (linear) aﬃne model with the same number of observable and latent variables
as our benchmark. For simplicity we focus on the case in which there are only two states (which
corresponds to one latent variable). We ﬁrst describe the data. Then we explain the methodology
adopted to determine a comprehensive set of observable macro and ﬁnancial indicators. Next, we
develop an aﬃne benchmark case with observable and latent state variables. We go on to present
the likelihood function for our model and the econometric methodology used to estimate it. Finally,
we discuss results.

4.1

Data

Figure 1 shows a panel of daily ﬁve-year sovereign CDS spreads for eleven Euro zone countries:
Austria, Belgium, Finland, France, Germany, Greece, Ireland, Italy, the Netherlands, Portugal,
and Spain. The data are from Markit Financial Information Services and span the period from
February 12, 2004, to September 30, 2010. Markit’s coverage of the ﬁve-year sovereign CDS market
is fairly comprehensive over the entire sample period, with daily observations typically available
for each country. Notable exceptions are Finland, Ireland, and the Netherlands, for which data are
unavailable over the periods 09/19/2005-05/01/2006 (Finland); 02/12/2004-05/18/2004 (Ireland)
and 02/12/2004-06/23/2006 (Netherlands). In contrast, the coverage of sovereign CDS contracts
with maturities other than ﬁve years is spotty, especially in the early part of the sample period.
Thus, we do not include those maturities in our analysis.
Table 1 contains summary statistics for the CDS series. A breakdown of the sample into a
pre-crisis period (from 02/2004 to 12/2007) and a crisis period (from 01/2008 and on) conﬁrms
the patterns already evident in Figure 1. That is, prior to 2008 sovereign CDS spreads are low
across Euro zone countries, with little time series variation (mean spreads range from 2.06 to 10.60
bps, with a 0.72-3.34 standard deviation). This all changes starting from 2008, when the spreads
of Portugal, Ireland, Italy, Greece, and Spain (the so-called PIIGS countries) soar. The rest of the
Euro zone trails behind them, also showing much higher and more volatile CDS spreads compared
to the pre-crisis period.
To explain these ﬂuctuations in CDS prices within our model, we ﬁrst specify a state vector X
that determines the default intensities for Euro zone countries, as in equation (12). The literature
has identiﬁed a large set of country-speciﬁc political and macro economic indicators that relate to
a country’s ability or propensity to repay its sovereign debt (e.g., Duﬃe et al. (2003), Edwards
(1984), Hilscher and Nosbusch (2010), Longstaﬀ et al. (2010), Min (1998), Pan and Singleton
(2008)). To obtain a parsimonious speciﬁcation for the state vector X, for each Euro-zone country
we summarize the information in these variables into a single country-speciﬁc MCI, as detailed in
Section 4.2 below. We then augment our state vector X to include the Chicago Board Option
15

Exchange VIX index as a proxy for global economic uncertainty (e.g., Longstaﬀ, Pan, Pedersen,
and Singleton (2010), Pan and Singleton (2008)).

4.2

Macroeconomic Conditions Indices for Euro Zone Countries

Following Stock and Watson (1989, 1991), we consider a dynamic factor framework to estimate an
indicator of the latent macroeconomic and political state of each Euro zone country, as well as the
Euro zone area. We explicitly incorporate macroeconomic variables and political risk indicators
observed at various frequencies (monthly, quarterly, and yearly). To deal with these mixed frequencies, we follow the approach of Aruoba, Diebold, and Scotti (2009) (see also Harvey (1990), Section
6.3). In particular, for each country i we estimate the following model at the daily frequency using
maximum likelihood in combination with the Kalman ﬁlter:
yi,t = βi M CIi,t + Γi Wi,t + εi,t , εi ∼ N (0, Qi )
M CIi,t+1 = ρi M CIi,t + ηi,t+1 ,

ηi ∼ N (0, Ri ) .

(45)

The latent macroeconomic conditions indicator M CIi is an AR(1) process with Gaussian i.i.d.
errors. For identiﬁcation, we normalize Ri = 1. The vector yi contains country-speciﬁc variables.
Similar to Aruoba, Diebold, and Scotti (2009), the vector Wi includes lagged values of yi . The
sample period goes from January 3, 2001, to September 30, 2010.
We ﬁrst estimate the model on aggregate data for the whole Euro zone and we follow the
existing literature on sovereign credit risk to guide the choice of variables to include in yEU . We
experiment with various speciﬁcations and exclude some of the variables that have an insigniﬁcant
factor loading βEU and do little to help identify the latent Euro zone indicator, MCIEU . Similar to
Aruoba, Diebold, and Scotti (2009), we also consider an extension that allows the error term η in
the MCI transition equation (45) to be auto-correlated. We do not ﬁnd statistical support for this
speciﬁcation and rule it out. We then go on to estimate the same speciﬁcation using data for each
Euro zone country and thus obtain the associated MCIi indicators.
Table 2 summarizes the results. The ﬁrst column lists the variables yij , j = 1, . . . , 10, that
constitute the vector yi . To construct these measures, we use the series described in Online Appendix along with the corresponding data sources (we accessed the data through Haver Analytics).
We treat all elements of yi as stock variables with the exception of GDP per capita, which is a
ﬂow variable. To handle temporal aggregation of GDP per capita, we augment the state vector to
include a ‘cumulator’ variable as in Harvey (1990), Section 6.3.3. Moreover, prior to estimation,
we detrend GDP per capita.
The other columns of Table 2 show the sign and statistical signiﬁcance of the loading βij on the
MCIi index for the economic/political risk variable yij of country i. The EU column has the results
for the Euro zone area. Real economic activity is an important determinant of the ﬂuctuations in
the MCIEU indicator. Both real GDP growth and GDP per capita load on MCIEU with a negative
and signiﬁcant coeﬃcient. This implies that higher MCIEU values are associated with worse eco16

nomic conditions in Europe. This is also evident from the positive and signiﬁcant loading of the
unemployment (UE) rate on the index. Consistent with this interpretation, the exports to GDP
ratio has a negative and signiﬁcant β. Inﬂation also loads on the MCIEU indicator with a negative
and signiﬁcant coeﬃcient. The Euro zone economy performed well prior to 2008, then poorly during the ﬁnancial crisis, and then started to rebound in 2009. Inﬂation followed a similar pattern
in our sample – higher consumer price growth coincided with better economic conditions. The
ratios of Government surplus and debt to GDP have plausible loadings on MCIEU , suggesting that
stronger public ﬁnances go together with a better economic situation. The coeﬃcients, however, are
insigniﬁcant. Since we observe these variables at a yearly frequency, this is not entirely surprising.
We retain them in the preferred speciﬁcation due to their economic importance. Political stability
is also associated with lower MCIEU values. This is intuitive, although the coeﬃcient is insigniﬁcant. Finally, previous studies have stressed the importance of liquidity measures (e.g, M3/GDP
and Reserves/GDP) to explain sovereign credit ratings (e.g., Jaramillo (2010)). We include these
variables in the model, but ﬁnd them to be insigniﬁcant. Also, the results show that these measures
of ﬁnancial intermediation are positively related to the index. This could be speciﬁc to our sample
period, e.g., central banks were pouring liquidity in the economy at the peak of the crisis.
The remaining columns of Table 2 contain results for individual countries. For the most part,
the evidence is similar to our ﬁndings for the Euro zone. A notable exception concerns variables
that reﬂect the state of a country’s public ﬁnances. For instance, the ratios of Government surplus
and debt to GDP have a signiﬁcant loading on the MCIs of Greece and Ireland.
Figure 2 plots the MCIs for the Euro zone and its individual countries. As mentioned previously,
higher values are associated with a deterioration in economic conditions. It is evident from the plots
that the indices are persistent. We ﬁnd the AR(1) coeﬃcient ρ to range from a low of 0.9931, with
a 0.0035 standard error, for Portugal, to a high of 0.9994 with a 0.0005 standard error, for Ireland
(with daily scaling). Consequently, the unconditional standard deviation of the processes is also
high, as seen in the wide ﬂuctuations of the MCI series.
The indices share a similar pattern through the ﬁrst part of the ﬁnancial crisis, when they
increase rapidly. In 2009 they start to improve across the Euro zone, largely because of higher
growth. Ireland’s recovery is much slower. Greece stands out as the only country that shows no
sign of recovery as of September 2010.

4.3

Linear Regression Benchmarks

Our model explains sovereign spreads with a combination of macroeconomic and ﬁnancial variables,
summarized in the state vector X, and a latent variable π that measures the posterior probability
of the hidden state of the economy. In this setting, defaultable bond prices are the sum of nonlinear
functions of X, weighted by the probability π (equation (37)). Thus, sovereign spreads are nonlinear transformations of the state variables.
A natural competitor for our model is a linear (aﬃne) speciﬁcation that includes the same state
17

vector X augmented with a single latent variable that captures a common component in credit
spreads, possibly related to the notion of contagion. In any linear model of defaultable bonds, the
spread between yields on risky and riskfree bonds is a linear function of the state vector. Here we
explore the performance of such a model via linear OLS regressions that relate the CDS spreads
for each Euro-zone country to the vector of state variables X. In the simplest case, the only
explanatory variable is the country-speciﬁc MCI. We then augment these regressions to include
aggregate political and macroeconomic information for the Euro zone region as well as measures
of global economic uncertainty. Finally, we include a proxy for a latent variable that helps explain
the covariation in CDS spreads across Euro zone countries that is not spanned by macroeconomic
and ﬁnancial variables.
The most general version of these regressions is given below:
CDSi = b0,i + b1,i MCIi + b2,i MCIEU⊥ + b3,i log VIX + b4,i BB-BBB spread + b5,i PC1CDS−i + εi (46)
i

∆CDSi = b1,i ∆MCIi + b2,i ∆MCIEU⊥ + b3,i ∆log VIX + b4,i ∆BB-BBB spread + b5,i ∆PC1CDS−i + εi .
i

(47)
Equation (46) relates the levels of CDS spreads to the levels of the explanatory variables, consistent with our bond pricing model. A potential concern with this speciﬁcation has to do with the
persistence of the variables included in the regressions. For instance, Granger and Newbold (1974)
warn that autocorrelation in the regression residuals could produce spuriously high R2 statistics.
Further, it is well known that ignoring the autocorrelation in the error term makes signiﬁcance
tests on the coeﬃcients invalid. As a partial remedy to these concerns, we still rely on consistent
OLS estimates of the model, but compute coeﬃcients’ t-ratios using standard errors that are robust to autocorrelation and heteroskedasticity. Moreover, in equation (47) we also estimate the
same speciﬁcation in diﬀerences. This alternative approach is common in the empirical credit risk
literature, but is also subject to criticism. For instance, Maeshiro and Vali (1988) document a loss
of estimation eﬃciency caused by the adoption of a diﬀerenced model when the disturbances of the
original (levels) linear regression are autocorrelated. Doshi, Ericsson, Jacobs, and Turnbull (2011)
discuss these issues in more detail and conclude that a model expressed in levels might be preferred
when ﬁtting daily CDS data, as we do in our application.
When estimating regressions (46)-(47), our measure for the dependent CDSi variable is the daily
5-year credit default swap basis-point spread for country i. MCIi is the macroeconomic conditions
index for country i, while MCIEU⊥ is the residual of an OLS regression of the daily Euro zone MCIEU
i

against the daily country-i MCIi and a constant. The log VIX variable is the logarithmic daily S&P
500 percentage implied volatility index published by the CBOE. The BB-BBB spread variable is the
diﬀerence between Bank of America Merrill Lynch corporate bond eﬀective percentage yield indices,
sampled at the daily frequency. To compute the PC1CDS−i variable, for each Euro zone country j ̸= i
we ﬁrst regress CDSj against the MCIj and the logarithmic VIX. PC1CDS−i is the ﬁrst principal
component extracted from the panel of the regression residuals (we exclude Finland, Ireland, and
18

the Netherlands due to data availability). This approach allows us to extract common information
in CDS spreads across Euro zone countries that is not captured by global ﬁnancial uncertainty
and local macroeconomic conditions. When estimating the regressions in diﬀerences, we measure
∆CDSi with the series of daily overlapping changes in the 5-year CDS spreads for country i relative
to the prior month, ∆CDSi (t) = CDSi (t) − CDSi (t − 21). The sample period goes from 02/12/2004
to 09/30/2010, with the exception of Finland, Ireland, and the Netherlands, for which data are
unavailable over the periods 09/19/2005-05/01/2006 (Finland); 02/12/2004-05/18/2004 (Ireland)
and 02/12/2004-06/23/2006 (Netherlands).
We report OLS estimates for France, Germany, Greece, and Italy in Table 3 (regressions in
levels include a constant, which we omit from the table). Here, we focus on these four countries
as they provide a diverse representation of the Euro zone; a complete set of results is in the
Online Appendix. In each panel, column (1.1) has the results for the simplest model, with the
country-speciﬁc MCI being the only independent variable. The associated coeﬃcient is positive
and statistically signiﬁcant, consistent with the interpretation that a deterioration in the country’s
economic conditions leads to higher CDS spreads (coeﬃcient t-ratios in brackets are based on
Newey-West heteroskedasticity and autocorrelation robust standard errors). The explanatory power
of the regression is typically high, with R2 coeﬃcients ranging from 11% for Italy to 72% for
Greece.10
We then add MCIEU⊥ in column (1.2). The associated regression coeﬃcient is typically posi

itive and signiﬁcant, but the improvement in ﬁt is generally small. This suggests that economic
conditions in the rest of the Euro zone have some impact, albeit moderate, on local CDS spreads.
An exception is France, where the coeﬃcient is negative and insigniﬁcant.
Consistent with Longstaﬀ et al. (2011), global economic uncertainty, measured by the logarithmic VIX, helps explain the variation in the spreads (column 1.3). This is evident in the error
diagnostics in the two bottom rows of each panel, where we report mean and max of the absolute
residuals. Compared to the results in column (1.1), the regression with the log-VIX alone produces
similar mean absolute errors. For strong Euro zone economies like Germany and France, the mean
absolute error in column (1.3) is lower than the one for model (1.1), suggesting that global economic uncertainty plays a signiﬁcant role in explaining credit risk for these countries. The reverse
applies to troubled countries like Greece, for which country-speciﬁc macroeconomic conditions do
a better job at explaining CDS data. When we combine the log-VIX with the country-speciﬁc MCI
in column (1.4), both variables typically have positive and signiﬁcant coeﬃcients. Nonetheless,
collinearity in these two measures can cloud the interpretation of this conclusion. Augmenting the
regression to include another measure of global economic uncertainty, the High-yield (BB-BBB)
spread, leads to a further moderate increase in ﬁt (see, e.g., the lower mean absolute error in column
1.5). The results for Greece in columns (1.4)-(1.5) stand out; local economic conditions drive most
10

A notable exception is the coeﬃcient for Portugal’s MCI (Panel J of Table 2 in the Online Appendix); while
positive, it is insigniﬁcant, and the regression’s R2 is nearly zero.

19

of the variation in its CDS spreads; proxies for global ﬁnancial uncertainty are insigniﬁcant.
The regressions in column (1.6) contain only three explanatory variables: the country-speciﬁc
MCI, the logarithmic VIX, and our proxy for a latent factor, PC1CDS−i . This model gives the
best overall ﬁt. In particular, the mean and maximum absolute errors in the two bottom rows of
each panel are much lower in model (1.6) compared to other speciﬁcations. This evidence suggests
that regional information latent in Euro-zone spreads helps explain country-speciﬁc ﬂuctuations in
sovereign credit risk.
Columns (2.1)-(2.6) report estimates for the same regressions, estimated in changes (equation
(47)). Consistent with Longstaﬀ et al. (2011), in this case the explanatory power of changes in
local and regional macroeconomic conditions is very limited. Changes in global ﬁnancial indicators
improve the ﬁt slightly. Also in this case, regressions that include our proxy for a latent factor,
∆PC1CDS−i , provide the best ﬁt.
In sum, this analysis serves two purposes. First, it illustrates the ability of a parsimonious set
of variables, country-speciﬁc MCIs in combination with a global ﬁnancial indicator such as the VIX
index, to explain sovereign credit risk in the Euro zone. The explanatory power of macroeconomic
and ﬁnancial indicators is more evident in regression for spread levels, consistent with recent work by
Doshi, Ericsson, Jacobs, and Turnbull (2011) that focuses on CDS contracts on corporate bonds.
Second, the results provide a benchmark for the performance of our CDS pricing model. In an
arbitrage-free (linear) aﬃne framework, sovereign credit spreads would be a linear function of the
state vector, where the loadings of credit spreads on the state variables would be determined by
no-arbitrage restrictions. Therefore, in sample, unrestricted OLS regressions give an upper bound
on the goodness of ﬁt such an aﬃne model could achieve (given that it uses the same state variable).
We construct this benchmark to also include information that is common across Euro zone countries
and independent of macroeconomic conditions and ﬁnancial uncertainty; by construction our proxy
PC1CDS−i should capture the most important latent factor. Therefore, if our model performs better
than this aﬃne benchmark, then this strongly suggests that there are important non-linearities in
the behavior of credit spreads better captured by our model. In Section 4.5 below, we compare
the errors produced by our non-linear CDS pricing models to the OLS residuals of the preferred
regression in column (1.6).

4.4

Model Specification and Estimation

Based on the results in the previous section, we specify our state vector X to include the N =11
MCIs as well as the logarithmic VIX index, X = [ M CI1 , . . . , M CI11 , log V IX ]′ . We assume that
M = 2 and the hidden economic state s can be either ‘good’ or ‘bad.’ That is, the agent views
the probability of a default event to be higher in the bad economic state than in the good one,
λiG < λiB . We assume that the default intensity λiB for country i, i = 1, . . . , N , only depends on
the country-speciﬁc macroeconomic index as well as global uncertainty. That is, we restrict the
elements of the βiB vector in equation (12) to be zero, except for the two coeﬃcients that load on
20

MCIi and the logarithmic VIX index. Further, we ﬁx βiG = 0.
The latent variable π, which measures the probability that the economy is in the good state,
completes the set of model state variables. Thus, we deﬁne an augmented state vector X = [X ′ , π]′ ,
with π dynamics given in equation (11). In particular, we assume that the agent updates her
posterior using information on default events d1{τ

i ≤t}

of the Euro zone countries, i = 1, . . . , 11, as

well as a single continuous signal dΩ, with dynamics given in equation (7).
Model estimation proceeds in two stages. In the first step, we specify and estimate the dynamics
of the X vector. We map the discrete-time AR(1) process for the MCIs, given in equation (45),
into the continuous-time dynamics in equation (13). We assume that the MCIs are independent
across countries, with mean-reversion coeﬃcients κi,i = − log(ρi ), zero long-run means, i.e., ψi = 0,
and diﬀusion terms normalized to unity, Σi,i = 1. The independence assumption across the MCIs
greatly simpliﬁes estimation, as we can use the ρi coeﬃcients computed separately for each MCI
process in Section 4.2.
Prior to estimation, we demean the logarithmic VIX series and therefore restrict ψV IX = 0.
Moreover, we allow the VIX process to depend on lagged realization of the MCIs. We explore this
linkage in linear regressions that have the daily log-VIX on the left-hand side and include lagged
log-VIX and MCI realizations among the explanatory variables. We ﬁnd a strong and persistent
auto-regressive component in the VIX. Interestingly, including the MCIs of troubled Euro zone
economies (the PIIGS countries) increases the explanatory power of the regressions. This suggests
that the turmoil in Europe has had an impact on global economic uncertainty. Interpreting the
point estimates and statistical signiﬁcance of these coeﬃcients is however diﬃcult, as there is
collinearity among the MCIs. In contrast, we do not ﬁnd the MCIs of Euro zone economies outside
of the PIIGS circle to improve the explanatory power of the regressions, and therefore we exclude
them from our speciﬁcation. In this framework, the lagged log-VIX remains signiﬁcant but the
persistence of the autoregressive component declines.
We also explore the possibility that shocks to the VIX correlate with shocks to the MCIs. For
each MCIi , we correlate the residuals of the AR(1) process in equation (45) with the residuals of
our linear regression model for the log-VIX. We ﬁnd such correlations to be insigniﬁcant and ﬁx
them at zero in the rest of the analysis. With this additional restriction, the log-VIX equation in
the X dynamics (13) takes the form
d log V IX(t) = −( κV IX log V IX(t) +

∑

κV IX,i M CIi (t) ) dt + ΣV IX dWV IX (t) .

(48)

i∈P IIGS

We ﬁx κV IX , κV IX,i , i ∈ P IIGS, and ΣV IX at the OLS estimates of equation (48). Finally, we ﬁx
µG = 0.018, µB = 0.005, and σ0 = 0.03 in the endowment dynamics (14). These parameters also
determine the value of the state-contingent spot rate (equation (30)), which has to be constant
to ensure that preferences are time-consistent, as discussed in the Appendix. The robustness
parameter ζ and the discount rate β enter equation (22) as a product; thus, for identiﬁcation we
set β = 0.01.
21

The second estimation step relies on the panel of sovereign CDS spreads and default events to
identify the remaining model parameters and the time series of the ﬁltered posterior probability of
the hidden economic state. We cast the model in a state-space framework. The set of measurement
equations includes the sovereign CDS spreads for each Euro zone country, which we assume to be
determined by the model up to a Gaussian i.i.d. pricing error,
] i,t = CDSi (X t ) + υi,t , υi,t ∼ N (0, Vi ), i = 1, . . . , 11 .
CDS

(49)

Moreover, we assume that the MCIs and the log-VIX are measured without error. Equations (11)
and (13) deﬁne the state dynamics for X = [X ′ , π]′ and complete the state-space representation.
There are two sources of non-linearity in the system. First, the CDS pricing formula in the
measurement equation (49); see equation (37) and the discussion in Section (3.9). Second, the
probability of the hidden state π has non-linear dynamics (11). To accommodate these features, we
discretize the model and estimate it by quasi maximum likelihood in combination with the squareroot unscented Kalman ﬁlter (UKF); see, e.g., Wan and van der Merwe (2001) and Christoﬀersen
et al. (2009).
We denote by Y the vector of CDS spreads, MCI indices and log-VIX. In addition to the elements
of Y , we also observe sovereign default events. Thus, the transition density of the observable
variables is given by
P ( Yt , Nt | Yt−1 , Nt−1 ; Θ) = P ( Nt | Yt−1 , Nt−1 ; Θ) × P ( Yt | Nt , Yt−1 , Nt−1 ; Θ) ,

(50)

where Θ is the vector of model coeﬃcients. The ﬁrst term on the right-hand side of equation
(50) is the default probability for the Euro zone countries. In our sample, we do not observe
defaults, i.e., for all dates t and countries i = 1, . . . , 11, d1{τ <t} = 0, and P (Nt |Yt−1 ; Θ) =
i
∏11 ∑M
−
−
i=1
s=1 πs (t) exp(−λis (t ) dt), where λis (t ) is given in equation (12). Conditional on time
t default information and time t − 1 values of Y , we approximate the distribution of Y (t) on the
right-hand side of equation (50) with a multivariate Gaussian density, and compute its conditional
mean and variance using the UKF. We then maximize the likelihood function,
P ( Yt , Nt , t = 0, . . . , T | Θ ) = P ( Y0 , N0 | Θ ) ×

T
∏

P ( Yt , Nt | Yt−1 , Nt−1 ; Θ ) .

(51)

t=1

4.5

Empirical Results

Time series of estimated probability of the hidden state.

Figure 3 shows default intensities

estimated over the period from February 12, 2004 to September 30, 2010. During the pre-crisis
period, sovereign defaults of member countries are very unlikely, as evident from the low default
intensity estimates. Consistent with this interpretation, our ﬁltered posterior probability of the
hidden state, πGood , is almost one during that period (Figure 4). That is, agents are nearly sure
that the Euro zone economy is in the good state.

22

This situation starts to change at the end of 2007, when we estimate a signiﬁcant decrease
in the posterior probability of the good economic state. Since then, πGood estimates start to
ﬂuctuate in connection with credit spreads. This suggests that much of the co-variation in sovereign
spreads is due to the ‘contagion’ variable (in our model the posterior probability of the unobservable
hidden state), rather than to observable common and country-speciﬁc macroeconomic and ﬁnancial
covariates (the state variable X in equation (12)). Movements in the posterior probability of the
states (or credit spreads, for that matter) do not appear to be random and unexplainable, however.
Indeed, on days of major turning points in πGood we ﬁnd ex-post a lot of contemporaneous news
reports (discussed in the Online Appendix) which are qualitatively consistent with the directional
change in our estimated posterior probability of the good state. This is reassuring since our πGood
variable is a latent variable inverted from prices, and thus not directly estimated from such news.11
Fragile belief preference parameters.

Consistent with Hansen and Sargent (2010), we ﬁnd

that the agent displays model uncertainty aversion and slants the risk-adjusted probability of the
hidden state towards the model associated with the lowest continuation utility. We estimate the
robustness parameter ζ at 1.79, with a standard error of 0.08. This determines a risk adjustment
ξ in the posterior probability that the agent assigns to being in the bad economy (equations (22)
and (31)). It is evident from Figure 4 that under the risk-neutral measure the agent attaches
a higher probability of being in the bad economy. Consequently, the levels of the risk-adjusted
Q

default intensities, λi , are higher than those computed under the physical probability measure, λi
(see Figure 3 and magniﬁed versions of the same plots in the Online Appendix).
Estimates of default risk premia.
the two intensities,

Q
λi /λi .

The right-hand side axes in Figures 3 show the ratios of

Across countries and throughout the sample period, the ratio ranges

from one to two. Prior to the crisis, the ratios have a mild downward sloping pattern and are lower
than in the post-2007 period, especially for strong European economies like Germany and France.
Starting from the end of 2007 they increase signiﬁcantly and ﬂuctuate between 1.5 and 2. The
Q

increase in λi /λi coincides with the drop in our estimate of the probability that the economy is
in the good state (Figure 4). Since the end of 2007, the posterior π declines from nearly one to a
low of approximately 0.4 in June 2010. This greater uncertainty results in a higher risk premium
on sovereign bonds.
These results are in line with empirical papers (Berndt et al. (2005), Driessen (2005)) that
estimate the jump to default risk-premium, as measured by the ratio of risk-neutral to historical
default intensities, and ﬁnd it to be of the order two to ﬁve (i.e., short term credit spreads should be
two to ﬁve times higher than historical default rates with constant recovery rates). In contrast to
these models, however, we provide a theoretical explanation for such a JTD risk-premium. First, the
updating mechanism described above, makes default event not conditionally diversiﬁable. Second,
11

Of course, one would expect CDS spreads to be related to such news, and our posterior probability is clearly
linked to spreads as is evident from the ﬁgure.

23

because the representative agent displays fragile beliefs, any small shift in the probability of the
bad states is magniﬁed when it comes to pricing. The combination of Bayesian updating of hidden
states and fragile beliefs generates a sizable and time-varying JTD risk-premium.
Decomposition of credit spreads.

To better gauge the importance of model uncertainty pre-

mia in the presence of fragile beliefs, we compute the component of CDS spreads associated with
uncertainty aversion. We measure it by the diﬀerence between (1) CDS prices predicted by the
model in the presence of fragile beliefs and (2) CDS prices computed under the same coeﬃcients,
except for turning oﬀ uncertainty aversion (ζ ⇒ ∞). Figure 5 shows that this diﬀerence ﬂuctuates with a pattern similar to that of our πGood estimate in Figure 4 and exhibit a great deal of
co-movement across countries. Prior to the crisis, the cross-country average of these measures is
approximately 19% of the average level in CDS spreads across the Euro zone. This estimate increases to 36% during the period from January 2008, consistent with the agent demanding higher
compensation for model uncertainty during the crisis.
Model ﬁt of CDS spreads.

Figures 6 compares model-implied CDS spreads with actual data

(magniﬁed versions of these displays are in the Online Appendix). The plots illustrate the ability
of the model to ﬁt the time series of CDS spreads across regimes. In particular, it captures the low
level and variability of pre-crisis data, and it does well at matching the wild ﬂuctuations in spreads
during the ﬁnancial crisis. Moreover, a comparison across countries shows that the model can ﬁt
the cross section of Euro zone spreads.
Comparison with a (linear) aﬃne benchmark. Table 4 lends additional support to these
conclusions. The ﬁrst two rows compare the mean absolute CDS pricing errors produced by our
contagion model to those produced by a linear model that includes the country-speciﬁc MCIs, the
log-VIX, and a single latent factor, as explained in Section 4.3. The last two rows show a similar
comparison based on the maximum absolute CDS pricing errors. The improvement over the linear
speciﬁcation is evident across countries. For Greece, the mean absolute error is almost one third
of the error associated with the linear regressions. For other countries the improvement is also
signiﬁcant, ranging from 23-85% and with an average improvement of 67% across the Euro zone.
The maximum pricing deviation improves considerably as well, with an average 47% reduction
across countries.

5

Conclusions

We investigate a general equilibrium framework that captures contagion risk in defaultable bonds.
The model has two important ingredients: a hidden state of nature which impacts expected consumption growth and default probabilities, and a representative agent with fragile beliefs (Hansen
and Sargent (2010)). Even though our reduced-form model is not ‘doubly stochastic,’ bond prices
24

remain tractable, in turn facilitating empirical investigation. The model can generate large and
highly correlated credit spreads even when default probabilities and correlations in macroeconomic
fundamentals are low. Finally, we identify conditions for which the marginal utility of the agent
with fragile beliefs generate time-consistent state prices.
We apply the model to a panel of sovereign CDS spreads for Euro zone countries going from
February 12, 2004 to September 30, 2010. Default intensities depend on (1) indices that summarize country-speciﬁc macroeconomic conditions; (2) the VIX, as a measure of global ﬁnancial
uncertainty; and (3) a latent variable that captures contagion risk. Estimation via the unscented
Kalman ﬁlter shows that agents update their posterior probability of contagion as the Euro-zone
sovereign crisis unfolds. Prior to the crisis, our estimate of the probability that the economy is in
the good state is nearly one. Starting from December 2007 this estimate decreases signiﬁcantly,
with ﬂuctuations that match the ﬂow of news closely. Default intensities increase dramatically
over the same period with a great deal of comovement across countries, driven by the agent’s assessment that the economy is more likely to be in the bad state. We ﬁnd the agent to be averse
to model uncertainty, as under the risk-neutral measure she slants the probability of the hidden
state towards the model with higher default intensities. This model uncertainty premium pushes
risk-neutral default intensities up during the crisis and accounts for a large portion of CDS spreads.
The model ﬁts CDS spreads data well across Euro zone countries before and through the crisis,
and it signiﬁcantly outperforms aﬃne speciﬁcations that include the same number of observable
and latent factors.
Our sample period ends on September 30, 2010. Since then, the European sovereign crisis has
continued to unfold. A default by Greece on its sovereign debt has become progressively more likely,
and concerns about the solvency of other Euro zone countries have escalated as well. Nonetheless,
there seems to have been some decoupling of the fate of Greece from the rest of the Euro zone,
as measured by a decline in correlations between CDS spreads for Greece and other countries. On
October 27, 2011, Euro zone leaders agreed upon a new package of measures aimed at containing
the crisis. The deal included a ‘voluntary’ restructuring of Greek debt held by the private sector
with a 50% haircut on the bonds’ value. Yet, ISDA ruled that such ‘voluntary’ renegotiation,
no matter how much arm-twisting by European Governments was involved, did not constitute a
default event and, therefore, would not trigger CDS protection.
These recent events highlight a limitation of our empirical analysis, which assumes that at
most two hidden states describe the underlying economy. First, the presence of additional hidden
economic states could help capture scenarios in which the default of one or more member countries
is mainly idiosyncratic in nature and therefore has limited contagious eﬀect on other sovereign
entities. Second, it has become apparent during the recent crisis that the CDS market could
reﬂect technical legal risks that are quite separate from the risks driving cash-bond default risk.
In principle, this could be captured within our framework by allowing for additional hidden states.
As the crisis unfolds and more data becomes available, it may be useful to extend the empirical

25

analysis in this direction. We leave this for future research.

26

A

Appendix

A.1

Proof that Robust and Fragile Preferences are not Time-Consistent

Here we demonstrate that equilibrium state prices in an economy in which the representative agent
has preferences for both robustness and fragility are not time-consistent. Deﬁne A(ωT |ωt ) as the
date-t price of an Arrow-Debreu (A/D) security that pays $1 at date-T iﬀ state ωT occurs. We
examine the necessary conditions for the following formula to hold:
?

A(↑↑ |0) = A(↑↑ | ↑) A(↑ |0).

(A.1)

Intuitively, the left-hand side (LHS) is the date-0 cost of a buy and hold strategy that pays $1
at date-2 if (ω2 =↑↑). In contrast, the right-hand side (RHS) can be thought of as the cost of a
dynamic trading strategy which purchases, at date-0, A(↑↑ | ↑) shares of the A/D security that pays
$1 at date-1 if (ω1 =↑). Then, if ↑ occurs, the strategy pays oﬀ A(↑↑ | ↑), which is just enough at
date-1 in state-↑ to purchase one share of the A/D security that pays $1 if (ω2 =↑↑) occurs. Thus,
both sides of equation (A.1) imply two diﬀerent methods of obtaining $1 at date-2 iﬀ the (↑↑) state
occurs, and hence, by absence of arbitrage, the present value of these two portfolios should be the
same. We show that, in an economy in which state prices are derived from the marginal utility of
an agent with fragile beliefs, in general this relation does not hold. Consequently, in general the
preferences are not time-consistent. Then we prove that, for our speciﬁcation, the fragile beliefs
economy is actually arbitrage-free.
A.1.1

The Economy

Consider an inﬁnite-period, discrete-time economy in which the exogenously speciﬁed dividend,
which in equilibrium equals consumption, follows the binomial process:
{
√
log C(t) + σ dt
if ↑ occurs
√
log C(t + dt) =
log C(t) − σ dt
if ↓ occurs .

(A.2)

We emphasize that, even though we use the notation dt, we are investigating a discrete-time
economy, and only in some instances specialize to the continuous time limit.
The agent is uncertain which of s ∈ (1, S) states the economy is in, but she has priors π(S̃ =
s|ωt ) ≡ πs (t). Conditional upon being in state-s, the probability of an up state is
π(↑ |s, ωt ) =

1 µ0,s √
+
dt.
2
2σ

(A.3)

Note that the probability of an up (or down) move is time invariant in that π(↑ |s, ωt ) = π(↑ |s) ∀ωt .
Standard Bayesian updating implies that, conditional upon observing an ↑ event at date-(t + 1),

27

the probability that the economy is in state-s updates to:
πs (↑) ≡ π(s|ωt ∪ ↑)
π(↑ |s) π(s|ωt )
=
π(↑ |ωt )
π(↑ |s) π(s|ωt )
= ∑
′
′
s′ π(↑ |s ) π(s |ωt )
π(↑ |s) πs (t)
≡ ∑
.
′
s′ π(↑ |s ) πs′ (t)
A.1.2

(A.4)

Preference for Robustness

We ﬁrst assume that the economy is known to be in state-s. Generalizing the conditional logpreferences of equation (16), here we specify the agent’s conditional utility as having preference for
robustness:

[

V (t|s) = (1 − e

−βdt

−β dt

) log C(t) + e

min
ξ>0,E[ξ]=1

]

Et ξV (t + dt|s) + ζ1 ξ log ξ |s .

(A.5)

The Lagrangian for this constrained minimization is:
L = π(↑ |s) [ξ(↑ |s)V (↑ |s) + ζ1 ξ(↑ |s) log ξ(↑ |s)]

[

]

+π(↓ |s) [ξ(↓ |s)V (↓ |s) + ζ1 ξ(↓ |s) log ξ(↓ |s)] + λ 1 − π(↑ |s)ξ(↑ |s) − π(↓ |s)ξ(↓ |s) .
The ﬁrst order condition gives:
∂L
: 0 = π(↑ |s) [V (↑ |s) + ζ1 log ξ(↑ |s) + ζ1 − λ] ,
∂ξ(↑ |s)
implying that

[(
ξ(↑ |s) = exp

λ − ζ1
ζ1

)

]
V (↑ |s)
−
.
ζ1

(A.6)

(A.7)

To identify λ, we plug back into constraint that E[ξ] = 1 to ﬁnd
ξ(↑ |s) =
=

e−V (↑|s)/ζ1
E[e−V (t+dt|s)/ζ1 ]
e−V (↑|s)/ζ1
,
π(↑ |s) e−V (↑|s)/ζ1 + π(↓ |s) e−V (↓|s)/ζ1

(A.8)

with an analogous equation for ξ(↓ |s).
Plugging this back into the original equation yields:
[
]
V (t|s) = (1 − e−βdt ) log C(t) − ζ1 e−β dt log π(↑ |s) e−V (↑|s)/ζ1 + π(↓ |s) e−V (↓|s)/ζ1 . (A.9)
Recursively, we also ﬁnd12
[
]
V (↑ |s) = (1 − e−βdt ) log C(↑) − ζ1 e−β dt log π(↑ |s) e−V (↑↑|s)/ζ1 + π(↓ |s) e−V (↑↓|s)/ζ1 .
(A.10)
(
)
12
This is a slight abuse of notation: V (↑ |s) should really be written as V ωt+1 = (ωt ∪ ↑)|s .

28

In what follows, two points are important. First:
∂V (t|s)
∂V (↑ |s)

= e−β dt π(↑ |s) ξ(↑ |s).

(A.11)

In particular, note that the RHS is independent of date-t. The interpretation of this result is that,
conditional upon being in a state-s, things work as they would in a Black-Scholes binomial tree,
with, e.g., π(↑↑ |s, 0) = [π(↑ |s, 0)]2 and A(↑↑ |s, 0) = [A(↑ |s, 0)]2 .
Second, the solution to equation (A.9) is
V (t|s) = log C(t) + Bs .

(A.12)

Indeed, plugging this proposed solution into equation (A.9), and then using equation (A.2), we ﬁnd
the functional form of equation (A.12) to be self-consistent (that is, the log Ct term cancels), and
further identify the functional form of the constants Bs :
( −β dt )
(
√
√ )
e
−σ dt
σ dt
log
π(↑
|s)
e
Bs = −ζ1
+
π(↓
|s)
e
.
1 − e−β dt

(A.13)

Using equation (A.3) and taking the continuous-time limit, we get
Bs(dt→0) =

µ0,s
σ2
−
.
β
2βζ1

(A.14)

This can be seen as generalizing the results of the log-utility framework inherent in equation (21)
to the case in which the robustness parameter ζ1 is ﬁnite.
A.1.3

Fragile Beliefs

The agent must still decide how to weight the diﬀerent states. We assume she does this by maximizing the following objective:
V (t) =

m
∑

[

]

πs (t) ξs (t)V (t|s) + ζ2 ξs (t) log ξs (t) ,

(A.15)

s=1

subject to the constraint

(
0 = λ 1−

m
∑

)
πs (t)ξs (t) .

(A.16)

s=1

Setting up the Lagrangian, the ﬁrst order condition gives
∂
: 0 = πs (t) [V (t|s) + ζ2 log ξs (t) + ζ2 − λ] .
∂ξs

(A.17)

Applying the expectation constraint to identify λ, we obtain
ξs (t) = ∑

e−V (t|s)/ζ2
.
−V (t|s′ )/ζ2
s′ πs′ (t) e

(A.18)

Using equation (A.12), we can rewrite this as
ξs (t) = ∑

s′

e−Bs /ζ2
πs′ (t) e−Bs′ /ζ2
29

.

(A.19)

It is worth noting that ξs (t) depends on date-t only through {πs′ (t)}. Therefore, with slight abuse
of notation, we have
e−Bs /ζ2

ξs (↑) = ∑

s′

πs′ (↑) e−Bs′ /ζ2

.

(A.20)

Also, we emphasize that equation (A.4) yields
(

πs (↑) ξs (↑) =

)

e−Bs /ζ2
∑ ( π(↑|s′ ) πs′ (t) ) −B ′ /ζ
e s 2
s′
π(↑|ω )
π(↑|s) πs (t)
π(↑|ωt )
t

=
=

∑

π(↑ |s) πs (t) e−Bs /ζ2

π(↑ |s′ ) πs′ (t) e−Bs′ /ζ2
π(↑ |s) πs (t) ξs (t)
∑
.
′
s′ π(↑ |s ) πs′ (t) ξs′ (t)
s′

(A.21)

The fact that both the numerator and denominator are linear in πsQ (t) ≡ πs (t)ξs (t) is important in
what follows and implies that fragility is well-speciﬁed.
Getting back to the issue at hand and plugging equation (A.18) into equation (A.15), we ﬁnd
[
]
∑
πs (t) e−V (t|s)/ζ2 ,
V (t) = −ζ2 log
(A.22)
s

where, as explained above,
(
)
[
]
V (t|S) =
1 − e−βdt log C(t) − ζ1 e−β dt log π(↑ |s)e−V (↑|s)/ζ1 + π(↓ |s)e−V (↓|s)/ζ1 ,
(A.23)
and
∂V (t)
∂V (t|s)

A.2

= πs (t) ξs (t).

(A.24)

Arrow-Debreu Prices

To identify the Arrow-Debreu prices, we consider the agent starting at the optimal controls, and
then modifying those controls by purchasing ϵ shares of the Arrow-Debreu security that pays
$1(ω1 =↑). As such, her current consumption drops by ϵA(↑ |0), and her date-1 consumption in
the ↑ state increases by ϵ, with all other consumption in time and event space held constant:
[C(t), C(↑)] ⇒ [C(t) − ϵA(↑ |0), C(↑) + ϵ] .

(A.25)

Such an inﬁnitesimal change has no eﬀect on optimal utility:
0 = δV (t)
∑
=
πs (t) ξs (t) δV (t|s)
s

=

∑
s

[(
πs (t) ξs (t)

1−e

(A.26)
−βdt

)(1)
(
)( 1 ) ]
−βdt
−βdt
[−ϵA(↑ |0)] + e
π(↑ |s)ξ(↑ |s) 1 − e
(ϵ) ,
Ct
C↑
30

where equation (A.26) comes from equation (A.24). The solution to this is
( )
∑
Ct
e−β dt
πs (t) ξs (t) π(↑ |s) ξ(↑ |s)
A(↑ |0) =
C↑
s
√
∑
−σ dt −β dt
= e
e
πs (t) ξs (t) π(↑ |s) ξ(↑ |s).

(A.27)

s

It is important to note that the only t-dependence on the right-hand side is through the πs (t) ξs (t).
In particular, if at date-1 an ↑-state occurs, the price the A/D security that pays $1(ω2 =↑↑) (with
some abuse of notation) is:
A(↑↑ | ↑) = e−σ
≡ e−σ

√

dt −β dt

e

√

∑

dt −β dt

e

∑
s

= e

√
∑
−σ dt −β dt

e

s

= e

πs (ωt ∪ ↑) ξs (ωt ∪ ↑) π(↑ |s)ξ(↑ |s)

s

√
−σ dt −β dt

e

∑

πs (↑) ξs (↑) π(↑ |s) ξ(↑ |s)
(

π(↑ |s) πs (t) ξs (t)
∑
′
s′ π(↑ |s ) πs′ (t) ξs′ (t)

)
π(↑ |s) ξ(↑ |s)

|s) ξ(↑ |s)
,
′
s′ πs′ (t) ξs′ (t) π(↑ |s )

πs (t) ξs (t)π
s∑

2 (↑

(A.28)

where we have used equation (A.21) in the second-to-last line.
Finally, consider the two-period inﬁnitesimal variation:
[C(t), C(↑↑)] ⇒ [C(t) − ϵA(↑↑ |0), C(↑↑) + ϵ] .

(A.29)

We ﬁnd
A(↑↑ |0) = e−2σ

√
∑
dt −2β dt

e

πs (t) ξs (t) π 2 (↑ |s)ξ 2 (↑ |s).

(A.30)

s

A.3

Time Consistency

The LHS of equation (A.1) is equation (A.30). Combining equations (A.27) and (A.28), we ﬁnd
the RHS of equation (A.1) is
]
∑
[
][
√
√
∑
πs (t) ξs (t)π 2 (↑ |s) ξ(↑ |s)
−σ dt −β dt
−σ dt −β dt
s∑
A(↑↑ | ↑)A(↑ |0) = e
e
e
πs (t) ξs (t) π(↑ |s) ξ(↑ |s) .
e
′
s′ πs′ (t) ξs′ (t) π(↑ |s )
s
Dividing both sides by e−2σ
obtain

[
“LHS” =
[
“RHS” =

√

∑

dt e−2β dt

and multiplying both sides by
][

πs (t) ξs (t) π 2 (↑ |s) ξ 2 (↑ |s)

s

∑

][
πs (t) ξs (t) π 2 (↑ |s) ξ(↑ |s)

s

∑

∑

s′

πs′ (t) ξs′ (t) π(↑ |s′ ), we
]

πs (t) ξs (t) π(↑ |s)

s

∑

]
πs (t) ξs (t) π(↑ |s) ξ(↑ |s) .

(A.31)

s

Now, since the priors {πs (t)} are completely arbitrary (except that they sum to unity), it follows
that fragility combined with robustness typically generates an economy with time-inconsistent state
prices. However, there are at least two cases where the LHS equals the RHS:
31

• There is no hidden state. That is, πs (t) = 1 for one value of s and zero for all others.
• ξ(↑ |s) = ξ(↓ |s) = 1, which implies that preference for robustness has been ‘turned oﬀ’ (i.e.,
ζ1 = ∞).
The ﬁrst case rediscovers the well-known result that recursive preferences are time-consistent. The
second case implies that a combination of fragility and conditional time-separable preferences may
possibly be time-consistent. Note that all we have identiﬁed here are necessary conditions. In the
next section, we show that the model we investigate in the text is in fact time-consistent.

A.4

Derivation of the Pricing Kernel in our Economy (Necessary Conditions)

Here we identify the stochastic discount factor Λ(ωT ) that determines the price of a generic asset
V D(ωT ) (ωt ) with state-contingent cash ﬂows D(ωT ). The pricing kernel is deﬁned via:
∫
D(ωT )
Λ(ωt ) V
(ωt ) =
dωT π(ωT |ωt ) Λ(ωT ) D(ωT ).

(A.32)

It is convenient to notionally specify pricing kernel dynamics as:
dΛ(t)
Λ(t)

= −r(t) dt −

K
∑

ϕk (t) dZk (t) +

n
∑

[
Γi (t) d1(τ

i <t)

]
P
− λi (t) dt .

(A.33)

i=1

k=0

We want to identify the risk-free rate r(t), the market prices of Brownian motion risk {ϕk (t)}, and
jump risk Γi (t). To this end, note that

Λ(t+dt)
Λ(t)

= 1+

dΛ
Λ ;

this implies that we can express the price

of the asset with cash ﬂows D(ωt+dt ) paid out at time-(t + dt) as:
∫
D(ωt+dt )
V
(ωt ) =
dωt+dt π(ωt+dt |ωt )
[
n
[
∑
× 1 − r(t) dt − ϕ0 (t) dZ0 (t) +
Γi (t) d1(τ

(A.34)
P

i <t)

− λi (t) dt

]
]
D(ωt+dt ).

i=1

From equations (29) and (32), the security price has the expression
∫
∑
D(ωt+dt )
Q
V
(ωt ) =
πs (t)
dωt+dt π(ωt+dt |ωt , s) [1 − rs dt − σ0 dZ0 (t)] D(ωt+dt ).

(A.35)

s

We now use these two diﬀerent expressions to identify r(t), {ϕk (t)}, and {Γi (t)}. In particular, we
consider four securities:
1. Consider a risk-free security that pays D(ωt+dt ) = 1 in all states of nature. Comparing
equations (A.34) and (A.35), we obtain:
∑

1 − r(t) dt =

πsQ (t) [1 − rs dt] ,

(A.36)

s

which implies that the risk-free rate r(t) satisﬁes
r(t) =

∑
s

32

πsQ (t) rs .

(A.37)

(
2. Consider a security that pays D(ωt+dt ) ≡ dZ0 = dZ0 +

µ0,s −µP
(t)
0
σ0

)
dt, where we have used

equation (8). Comparing equations (A.34) and (A.35), we see that the price of risk on dZ0 is
(
)
P (t)
∑
µ
−
µ
0,s
0
− ϕ0 dt = −σ0 dt +
πsQ (t)
dt
σ
0
s
)
(
µQ (t) − µP0 (t)
≡ −σ0 dt +
dt,
(A.38)
σ0
where we have deﬁned µQ (t) =

∑

Q
s πs (t)µ0,s .

ϕ0

This expression yields:
(
)
µQ (t) − µP0 (t)
= σ0 −
.
σ0

(A.39)
(

3. Consider a security that pays, for some k ∈ (1, K), D(ωt+dt ) ≡ dZk = dZk +

µk,s −µP (t)
k
σk

)
dt,

where we have used equation (8). Comparing equations (A.34) and (A.35), we observe that
the price of risk on dZk is
(

∑

µk,s − µPk (t)
πsQ (t)
− ϕk dt =
σk
s
)
(
µQ
(t) − µP0 (t)
k
dt,
≡
σk
where we have deﬁned µQ
(t) =
k

∑

Q
s πs (t)µk,s .

(
ϕk

= −

)
dt
(A.40)

Simplifying, we ﬁnd

µQ
(t) − µPk (t)
k
σk

4. Finally, consider a security that pays D(ωt+dt ) = d1{τ

)
.

i <t}

(A.41)

. Comparing equations (A.34)

and (A.35), we obtain:
P

λi (t) (1 + Γi (Xt )) dt =

∑

πsQ (t) λi,s (Xt ) dt.

(A.42)

s

Deﬁning the risk-neutral intensity via
∑

Q

λi (Xt ) =

πsQ (t) λi,s (Xt ),

(A.43)

s

we get
Q

Γi (Xt ) =

P

λi (Xt ) − λi (Xt )
P

λi (Xt )

33

.

(A.44)

A.5

Proof that our Fragile Beliefs Economy is Arbitrage-Free

Our candidate pricing kernel has following dynamics:
dΛ(t)
Λ(t)

= −r(t) dt −

K
∑

ϕk (t) dZk (t) −

∑ λQ − λP
i

i

k=0

i

P
λi

dMi (t),

(A.45)

(
)
where dMi (t) ≡ dNi (t) − λi (t) . Note that the pricing kernel is deﬁned with respect to the
ﬁltration of the agent. That is, the {Zk (t)} are Ft -Brownian motions and Ni (t) has Ft -intensity
λi (t)). Further, the candidate market price of Brownian risk is given by:
ϕk = σk 1{k=0} −
∑
µQ
=
πsQ µk,s
k

P
µQ
k − µk
σk

(A.46)
(A.47)

s

µPk

∑

=

πs µk,s .

(A.48)

s

Now, according to our calculations,
∑

r(t) =

πsQ (t)rs

(A.49)

πs (t)λis (t)

(A.50)

πsQ (t)λis (t)

(A.51)

s

P
λi

∑

=

s

Q

λi

∑

=

s

χ π (t)
∑s s
s χs πs (t)

πsQ (t) =
for constants χs = e

−

µ0,s
βζ

and a πs (t) process with dynamics

dπs (t) = πs (t)

K
∑
µk,s − µP (t)
k

σk

k=0

A.5.1

(A.52)

dZk (t) +

N
∑

αis (t− ) dMi (t).

(A.53)

i=1

Pricing the Risk-Free Bond

We ﬁrst consider the pricing of the risk-free bond. We want to show that the calculated value
of a zero-coupon bond using the marginal utility of an agent with fragile beliefs is equal to the
calculated value when the pricing kernel of equation (A.45) is used. In other words, we want to
show that:
[

Λ(T )
E
| Ft
Λ(t)

]
=

∑

[
πsQ (t) E

s

]
Λs (T )
s, Ft .
Λs (t)

(A.54)

Alternatively, since:
[

Λ(T )
E
| Ft
Λ(t)

]

[ ∫T
]
− t r(u) du
= EQ
t e
[ ∫T ∑ Q
]
− t ( s πs (u)rs ) du
= EQ
e
,
t
34

(A.55)

and since
[

Λs (T )
E
s, Ft
Λs (t)

]

[
]
= E Qs e−rs (T −t) Ft , s = e−rs (T −t) ,

(A.56)

we need to show that:
−
EQ
t [e

∫T ∑
t

s

πsQ (u)rs du

|Ft ] =

∑

πsQ (t) e−rs (T −t) .

(A.57)

s

To prove this, it is suﬃcient to show that M (t) deﬁned as:
M (t) = e−

∫t∑
s

0

πsQ (u)rs du

∑

πsQ (t) e−rs (T −t)

(A.58)

s

is a {Q, F}-martingale.
Applying Itô’s lemma to M (t) we ﬁnd:
dM (t) = e

−

∫t∑
0

Q
s πs (u) rs du

∑

{
πsQ (t) e−rs (T −t)

−

∑

′
πsQ′ rs

dt + rs dt +

s′

s

Therefore, a suﬃcient condition for EQ [dM (t) | Ft ] = 0 is that
[
]
(
)
Q
∑ Q ′
dπ
(t)
s
EQ
Ft
=
πs′ rs − rs dt.
πsQ (t)
s′

dπsQ (t)
πsQ (t)

}
. (A.59)

(A.60)

Since in our equilibrium we have
rs

= constant + µ0,s ,

we see that necessary and suﬃcient condition for EQ [dM (t) | Ft ] = 0 is
[
]
(
)
Q
∑ Q
Q dπs (t)
E
=
Ft
πs′ µ0,s′ − µ0,s dt.
πsQ (t)
′
s

(A.61)

(A.62)

This result also shows that we cannot freely parameterize rs independent of the updating equation.
That is, even with ‘robustness’ turned oﬀ and the conditional preferences modeled as time-separable,
in order for the fragile beliefs utility to be time-consistent, additional restrictions on the model are
required.
To determine the appropriate market price of risk for our economy, here we derive the dynamics
of π Q and see what the restriction (A.62) implies for ϕ.
A.5.2

Dynamics of π Q

Recall πsQ (t) =

∑χs πs (t) ,
s χs πs (t)

where the χs = e

Also recall that
dπs (t) = πs (t)

−

µ0,s
βζ

are constants, as can be seen from equation (22).

K
N
∑
∑
µk,s − µPk (t)
dZk (t) +
αis (t− ) dMi (t).
σk
i=1

k=0

35

(A.63)

For simplicity, here we set N = 1 and drop the i-subscripts in the following. We then give the
general result below, which follows trivially. We ﬁnd
dπsQ (t)
πsQ (t)

=



∑
∑
∑
λs
c (t)
c (t))2
c (t)
χ
π
s
s
P
χ
dπ
(t)
χ
dπ
(
χ
dπ
dπsc (t)
dπ
1
s
s
s
s
λ
∑s s s
− 1 dNt ,
− ∑s
+ ∑s
− s
+ Q ∑
2
λs
πs (t)
χ
π
(t)
(
π
(t)
χ
π
(t)
χ
π
(t))
χ
π
s
s
s
s
s
s
s
π
s
s
s
s
s s s P
λ

(A.64)
where the superscript ‘c’ denotes the continuous part. Note that the jump component simpliﬁes to
(
)
λs
dNt .
Q − 1
λ
∑
Now we determine the dynamics of s χs πs (t):
∑
χ dπ (t)
∑s s s
s χs πs (t)

K
∑
(µQ − µP )
k

k

=

σ

k=0

(
P

dZk (t) +

Q

λ

)
P

P

λ

− 1 (dNt − λ dt).

Combining equations (A.64) and (A.65) and simplifying terms yields:
(
)
Q
K
P
∑
µk,s − µQ
µQ
dπsQ (t)
λs − λ
Q
k
P
k − µk
(dNt − λ dt).
=
dZ
(t)
−
dt
+
k
Q
Q
σ
σ
k
k
πs (t)
λ

(A.65)

(A.66)

k=0

Note that, interestingly, the jump component of π Q is a Q-martingale. However, under Q the drift
of π Q is in general not equal to zero. In fact, since the market price of Brownian risk is ϕk we have
the following:
[
EQ

dπsQ (t)
πsQ (t)

]
Ft

K
∑
µk,s − µQ
k
= −
σk
k=0

(

µQ
− µPk
ϕk + k
σk

)
dt.

(A.67)

Recall that we want to ﬁnd the market price of risk ϕ such that equation (A.62) holds, i.e.,
[
]
Q
(
)
dπ
(t)
s
Q
EQ
=
µ
−
µ
dt.
(A.68)
F
0,s
t
0
πsQ (t)
Combining this expression with equation (A.67), we obtain the restriction:
σk 1{k=0}

= ϕk +

P
µQ
k − µk
, ∀k = 0, 1, . . . , K .
σk

(A.69)

In turn, the Q-dynamics of π Q are given by
dπsQ (t)
πsQ (t)
A.5.3

= (µQ
0 − µ0,s ) dt +

Q
K
∑
µk,s − µQ
λs − λ
Q
k
dZkQ (t) +
(dNt − λ dt).
Q
σk
λ
k=0

(A.70)

Pricing an Arbitrary Contingent Claim

The last section proved that the price of the risk-free bond is consistent with the market price
dynamics implied by our pricing kernel given in equation (A.45) above. More generally, here
we show that the price of any arbitrary contingent claim priced oﬀ of the fragile beliefs agent’s
marginal utility is consistent with the prices given by the economy with this pricing kernel. In
36

other words, the fragile beliefs economy is arbitrage-free, and there exists a no-trade equilibrium
with a representative agent, endowed with the fragile beliefs preferences we have speciﬁed, who
consumes the aggregate consumption.
Consider a claim to an arbitrary FT measurable payoﬀ XT . We want to show that the value
of the payoﬀ in the economy where prices are determined by the pricing kernel of equation (A.45)
agrees with the price of the claim evaluated oﬀ of the gradient of the fragile beliefs agent. In other
words, we want to show that:
[
] ∑
[ s
]
Λ(T )
Λ (T )
Q
E
XT Ft =
XT Ft , s .
πs (t)E
Λ(t)
Λs (t)
s
Alternatively, since

[

Λ(T )
E
XT Ft
Λ(t)

]

= EQ [e−
= EQ [e−

∫T
t

r(u)du

∫T ∑
s

t

(A.71)

XT | Ft ]

πsQ (u)rs du

XT | Ft ],

]
Λs (T )
XT Ft , s = E Qs [e−rs (T −t) XT |Ft , s],
E
Λs (t)

(A.72)

[

and

(A.73)

we need to show that
EQ [e−

∫T ∑
s

t

πsQ (u)rs du

∑

XT | F t ] =

πsQ (t)E Qs [e−rs (T −t) XT |Ft , s].

(A.74)

s

It suﬃces to verify that M (t) deﬁned as
M (t) = e−

∫t∑
0

s

πsQ (u)rs du

∑

πsQ (t) E Qs [e−rs (T −t) XT |Ft , s]

(A.75)

s

is a {Q, F}-martingale. Note that from the deﬁnition of the Qs measure, using the martingale
representation theorem, there exist {Ft } adapted processes ψ·,· (t) so that
Ht := EQs [XT |Ft , s]
=

E

Qs

[XT |F0 , s] +

∫ t {∑
K
0

}
ψk,s (u) dZkQs (u)

+ ψW,s (u) dW (u) + ψN,s (dN (u) − λs dt) ,

k=0

(A.76)
where for simplicity we assume the jump process (N ) is one-dimensional (the extension to multidimensional i = 1, . . . , n is straightforward). Applying Itô’s lemma to M (t) we ﬁnd:
∫t∑ Q
∑
dM (t) = e− 0 s πs (u)rs du
πsQ (t) e−rs (T −t) ×
{
−Ht

(

s

∑

′
πsQ′ rs

dt + rs dt +

s′

+

∑
k

µk,s −
ψk,s (t)
σk

dπsQ (t)

)

πsQ (t)

µQ
k

dt +

+

∑

ψk,s (t) dZkQs (t) + ψW,s (t) dW (t)

k

ψN,s (u)λs
Q

λ (t)

}
Q

(dN (t) − λ (t)) .
(A.77)

37

Now, since by deﬁnition of the risk-neutral measure we have
(
)
P
µQ
−
µ
Q
k
dZk (t) = dZk (t) + σk 1{k=0} − k
dt.
σk

(A.78)

Further, by deﬁnition of the Qs measure we have:
dZkQs (t) = dZk + σk 1{k=0} dt.

(A.79)

Finally, we have deﬁned
dZk (t) = dZk (t) +

µk,s − µPk
dt.
σk

(A.80)

Combining, we obtain:
dZkQs (t)
Thus

µk,s − µQ
k
+
dt = dZkQ (t).
σk

[
EQ
t

ψk,s (t)dZkQs (t)

Previously, we have proved that
[
EQ
t

dπsQ (t)
πsQ (t)

µk,s − µQ
k
dt | Ft
+ ψk,s (t)
σk
]

Ft

= (

∑

(A.81)

]
= 0.

′

πsQ′ rs − rs ) dt.

(A.82)

(A.83)

s′

Therefore, since W (t) is a Ft -Brownian motion and N (t) is has Ft jump intensity λ(t), we have
indeed proved that
EQ
t [dM (t)|Ft ] = 0.
This implies that we have determined a valid pricing kernel for our economy.

38

(A.84)

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42

Figures and Tables
1200
Austria
Belgium
Germany
Finland
France
Greece
Ireland
Italy
Netherlands
Portugal
Spain

1000

bps

800

600

400

200

0

2004

2006

2008

2010

Dates

Figure 1: Sovereign CDS Spreads. The plots show 5-year sovereign CDS spreads for Euro zone
countries. The sample period is 02/12/2004-09/30/2010. Source: Markit Financial Information Services.

43

MCI

Eurozone

Austria

50

50

50

0

0

0

−50

−50

−50

02

05

07

10

02

MCI

Finland

07

10

02

France
50

0

0

0

−50

−50

−50

05

07

10

02

05

07

10

02

Ireland
50

50

0

0

0

−50

−50

−50

05

07

10

02

Netherlands

05

07

10

02

Portugal
50

50

0

0

0

−50

−50

−50

05

07

10

02

10

05

07

10

05 07
Dates

05

07

10

07

10

Spain

50

02

07

Italy

50

02

05

Germany

50

Greece
MCI

05

50

02

MCI

Belgium

10

02

05

Figure 2: Macroeconomic Conditions Indices. The plots show the macroeconomic conditions
indices for Euro zone countries. The sample period is 01/03/2001-09/30/2010.

44

10

2 1200
800
1 400
0
0

2 1200
800
1 400
0
0

2

05

bps

Portugal
1200
800
400
0

05

07

07

10

10

3

3

2 1200
800
1 400
0
0

2

05

07

10

bps

1
10

0

300
200
100
0

2
1
05

07

10

bps

0

3
300
200
100
0

2
1
05

Germany
2
1

07
Date

0

Finland

10

0

07

10

3
300
200
100
0

2
1
05

07
Date

0

Netherlands

3

05

0

3

France
300
200
100
0

10

Q

Belgium
2

07

07

λi
λi
Q
λi /λi

1

3

05

05

Spain

Austria
300
200
100
0

1

Q

3
λi /λi

3

Q

07

3

λi /λi

05

Italy

10

0

3
300
200
100
0

2
1
05

07
Date

10

Q

1200
800
400
0

Ireland

λi /λi

bps

Greece

0

Figure 3: Default Intensities. The plots show the P - and Q-measure estimates for the default
Q
intensities, λi and λi , for Euro zone countries. The left-hand side axis shows default intensities
Q
measured in basis points. The right-hand side axis shows the λi / λi ratio. We estimate the model
via quasi maximum likelihood in combination with the unscented Kalman ﬁlter. The sample period
is 02/12/2004-09/30/2010.
45

1
πGood
πQ

0.9

Good

0.8

0.7

πGood

0.6

0.5

0.4

0.3

0.2

0.1

0

2004

2006

2008

2010

Date

Figure 4: Hidden State Probability. The red line shows the unscented Kalman ﬁlter estimate of
the probability that the economy is in the ‘good’ state, πGood . The blue line shows the risk-adjusted
Q
measure that the economy is in the ‘good’ state, πGood
. The sample period is 02/12/2004-09/30/2010.
350
Austria
Belgium
Finland
France
Germany
Greece
Ireland
Italy
Netherlands
Portugal
Spain

300

250

bps

200

150

100

50

0
2004

2006

2008

2010

Date

Figure 5: Fragility Risk Premium. The plots show the component of CDS spreads that the model
attributes to the agents’ aversion to model uncertainty, measured by the diﬀerence between (1) CDS
prices predicted by the model in the presence of fragile beliefs and (2) CDS prices computed under
the same coeﬃcients, except that the parameter of uncertainty aversion ζ ⇒ ∞. The sample period
is 02/12/2004-09/30/2010.

46

Greece

Ireland

Italy

1000

1000

500

500

500

0

0

0

bps

1000

05

07

10

05

Portugal

500

500

0

0

bps

1000

07

10

05

bps

bps

100
07

10

bps

bps

100
07
Date

10

Finland
200

100

100

0

0

05

07

10

05

Germany

200

05

07

10

200

France

0

07
Belgium

200

05

05

CDS Model
CDS data

Austria

0

10

Spain

1000

05

07

10

200

100

100

0

0

07
Date

10

Netherlands

200

05

07

10

05

07
Date

10

Figure 6: Sovereign CDS spreads: Model vs. Data. The plots contrast model-implied 5-year
sovereign CDS spreads with actual data for Euro zone countries. We estimate the model via quasi maximum likelihood in combination with the unscented Kalman ﬁlter. The sample period is 02/12/200409/30/2010.
47

Table 1: Sovereign CDS Spreads: Summary Statistics. The table shows summary statistics
for the daily 5-year sovereign CDS spreads across Euro-zone countries. The data are from Markit
Financial Information Services and span the period from 02/12/2004 to 09/30/2010, with the exception
of Finland, Ireland, and the Netherlands, for which data are unavailable over the periods 09/19/200505/01/2006 (Finland); 02/12/2004-05/18/2004 (Ireland) and 02/12/2004-06/23/2006 (Netherlands).
Mean

Max

Std. Dev.

Skewness

Kurtosis

Full sample period: 02/12/2004-09/30/2010
Austria
31.47
268.88
47.94
1.88
Belgium
27.24
155.53
37.71
1.62
Finland
14.32
92.23
17.50
1.74
France
18.09
99.25
24.38
1.53
Germany
13.92
91.38
17.56
1.63
Greece
114.68
1015.24
210.20
2.63
Ireland
68.64
486.69
100.10
1.47
Italy
48.47
249.73
58.60
1.38
Netherlands
26.10
127.83
28.85
1.42
Portugal
50.79
461.22
81.43
2.46
Spain
44.07
273.62
63.03
1.71

6.85
4.66
6.08
4.16
5.32
9.32
4.40
3.70
4.79
8.84
5.25

Pre-crisis sample period: 02/12/2004-12/31/2007
Austria
2.54
6.08
0.72
1.81
Belgium
3.19
13.20
1.49
4.33
Finland
2.57
5.75
1.00
0.17
France
2.81
7.54
1.02
1.25
Germany
2.91
6.50
1.08
0.58
Greece
10.60
21.70
3.34
0.50
Ireland
3.22
14.69
1.92
3.92
Italy
9.57
21.21
2.47
1.60
Netherlands
2.06
6.34
1.03
2.45
Portugal
6.54
19.35
2.14
2.43
Spain
3.85
19.70
2.50
4.17

8.91
25.17
2.35
6.49
2.98
2.97
20.07
7.13
9.17
12.51
22.29

Euro crisis sample period: 01/01/2008-09/30/2010
Austria
71.69
268.88
52.08
0.96
Belgium
60.69
155.53
38.41
0.74
Finland
28.01
92.23
17.71
1.21
France
39.34
99.25
25.36
0.46
Germany
29.24
91.38
18.23
0.68
Greece
259.50
1015.24
263.89
1.46
Ireland
152.11
486.69
101.84
0.50
Italy
102.61
249.73
56.27
0.41
Netherlands
39.26
127.83
28.24
1.18
Portugal
112.36
461.22
96.65
1.49
Spain
100.04
273.62
64.09
0.95

4.72
2.44
4.64
2.07
3.71
3.85
3.01
2.09
4.22
4.22
3.06

48

49

3

3

3

−

−

−

−

+

Real GDP growth

Exports/GDP

Real GDP/Pop.

Gov. surplus/GDP

Debt/GDP

Legend:
(+) ⇒ βij > 0
(−) ⇒ βij < 0
3⇒ βij ’s p-val < .1

3

+

UE rate

−

−

−

−

−

+

3

3

3

3

3

+

+

−

−

−

+

+

−

+

Reserves/GDP

−

+

+

+

M3/GDP

3

3

Belgi
um

3

3

3

3

3

3

+

−

−

−

−

+

+

+

−

−

Finla

nd

3

3

3

3

3

3

+

−

−

−

−

+

−

−

−

−

Fran
ce

3

3

3

3

3

3

−

+

−

−

−

+

+

+

−

−

Germ
any

3

3

3

3

3

+

−

−

−

−

+

−

−

+

−

Gree

3

3

3

3

3

ce

+

−

−

+

−

+

+

+

−

−

ηi ∼ N (0, Ri )

3

3

3

3

3

3

nd
Irela

M CIi,t+1 = ρi M CIi,t + ηi,t+1 ,

+

−

−

Inﬂation

3

+

−

EU

Pol. stability

Aust
ria

yi,t = βi M CIi,t + Γi Wi,t + εi,t , εi ∼ N (0, Qi )

−

−

−

−

−

+

+

+

−

+

3

3

3

3

3

3

3

+

−

−

−

−

+

+

+

−

+

N
e
t
h
erlan
ds

3

3

3

+

−

−

−

−

+

−

+

−

−

3

3

3

3

3

3

Port
ug

al

Italy

+

−

−

−

−

+

+

+

−

−

3

3

3

3

3

3

Table 2: Economic/political risk indicators and the MCIs. For each Euro zone country i, the table shows the sign and statistical signiﬁcance of
the loading βij on the MCIi index for the economic/political risk indicator yij . The sample period is 01/03/2001-09/30/2010. The state-space model
is

Spain

50

i

∆CDSi = b1,i ∆MCIi + b2,i ∆MCIEU⊥ + b3,i ∆log VIX + b4,i ∆BB-BBB spread + b5,i ∆PC1CDS−i + εi .

i

CDSi = b0,i + b1,i MCIi + b2,i MCIEU⊥ + b3,i log VIX + b4,i BB-BBB spread + b5,i PC1CDS−i + εi

i

Adj. R2
mean(abs(ε))
max(abs(ε))

1
P CCDS
−i

BB-BBB spread

log V IX

M CIEU ⊥

M CIi

0.21
15.09
86.08

1.02
( 6.67)

(1.1)

0.22
15.49
82.45

1.02
6.84)
-0.80
( -1.45)

(

(1.2)

0.46
12.15
63.15

37.97
( 10.26)

(1.3)

(

(

0.48
11.66
66.21

33.97
6.81)

0.30
1.79)

(1.4)
0.13
0.79)

0.51
10.92
67.56

16.15
2.35)
9.35
( 3.22)
(

(

0.13
( 23.65)
0.88
6.40
33.17

27.58
( 20.18)

0.79
( 12.88)

Panel A: France
(1.5)
(1.6)

0.01
4.36
46.33

-0.30
( -1.02)

(2.1)

0.04
4.40
43.64

-0.42
( -1.39)
-0.62
( -2.29)

(2.2)

0.11
4.78
49.97

14.37
( 4.79)

(2.3)

(

0.14
4.97
42.60

14.79
4.64)

-0.35
( -1.30)

(2.4)

0.24
4.51
38.47

6.68
2.17)
7.05
( 3.36)
(

-0.18
( -0.74)

(2.5)

0.07
( 7.14)
0.50
3.87
35.14

16.56
( 6.75)

-0.06
( -0.41)

(2.6)

The dependent CDSi variable is the daily 5-year credit default swap basis-point spread for country i. MCIi is the macroeconomic conditions index
for country i. MCIEU⊥ is the residual of an OLS regression of the daily Euro zone MCIEU against the daily country-i MCIi and a constant. The log
i
VIX variable is the logarithmic daily S&P 500 percentage implied volatility index published by the Chicago Board Option Exchange. The BB-BBB
spread variable is the diﬀerence between Bank of America Merrill Lynch corporate bond eﬀective percentage yield indices, sampled at the daily
frequency. To compute the PC1CDS−i variable, for each Euro zone country j ̸= i we regress CDSj against the MCIj and the logarithmic VIX. PC1CDS−i
is the ﬁrst principal component extracted from the panel of the regression residuals (we exclude Finland, Ireland, and the Netherlands due to data
availability). Compared to Model 1, Model 2 has both left- and right-hand side variables in diﬀerences rather than in levels, e.g., we measure ∆CDSi
with the series of daily overlapping changes in the 5-year CDS spreads for country i relative to the prior month, ∆CDSi (t) = CDSi (t)−CDSi (t−21).
The sample period goes from 02/12/2004 to 09/30/2010. The results for Austria, Belgium, Finland, Ireland, the Netherlands, Portugal, and Spain
are in the Online Appendix. The coeﬃcient t-ratios in brackets are based on (Newey-West) heteroskedasticity and autocorrelation robust standard
errors.

Model 2 :

Model 1 :

Table 3: Linear Model Regressions. For each Euro zone country, the table shows OLS regressions for the models,

51

i

i

Adj. R2
mean(abs(ε))
max(abs(ε))

1
P CCDS
−i

BB-BBB spread

log V IX

M CIEU ⊥

M CIi

Adj. R2
mean(abs(ε))
max(abs(ε))

1
P CCDS
−i

BB-BBB spread

log V IX

M CIEU ⊥

M CIi

Table 3, continued

0.73
81.49
481.63

0.82
65.67
368.16

10.62
( 17.31)
-5.19
( -6.16)

(1.2)

(1.1)

10.62
( 10.94)

0.47
8.60
62.14

0.81
4.31)
1.81
( 7.85)

(

(1.2)

0.15
12.19
64.27

0.81
( 3.72)

(1.1)

0.17
111.00
817.56

199.49
( 5.82)

(1.3)

0.54
8.70
51.53

29.40
( 9.21)

(1.3)

0.73
81.66
482.66

-8.42
( -0.42)

10.73
( 9.92)

(1.4)

0.54
8.54
50.25

28.08
( 8.22)

0.15
( 0.94)

(1.4)
0.02
0.14)

0.73
81.81
478.67

3.52
0.11)
-5.68
( -0.34)

170.83
6.45)

1.61
( 11.45)
0.91
44.12
273.40

(

(

1.48
1.65)

-0.03
24.78
530.28

0.95
( 1.01)

(2.1)

Panel C: Greece
(1.5)
(1.6)

-0.27
( -1.07)

0.02
3.44
41.38

10.74
( 9.83)

(

0.83
7.87)

22.00
( 15.02)

(

(2.1)

0.08
( 11.95)
0.80
5.76
30.99

0.61
7.53
43.62

7.75
( 1.66)
9.99
( 4.10)

(

Panel B: Germany
(1.5)
(1.6)

0.01
26.04
524.57

0.81
( 1.01)
-2.51
( -1.84)

(2.2)

0.03
3.39
41.76

-0.42
( -1.26)
-0.39
( -1.13)

(2.2)

62.61
2.36)

0.02
27.13
510.57

(

(2.3)

0.09
3.68
43.02

10.43
( 4.15)

(2.3)

61.87
2.35)

0.02
27.46
514.43

(

(

0.85
0.92)

(2.4)

0.12
3.83
40.66

10.83
( 4.12)

-0.31
( -1.31)

(2.4)

0.04
26.64
521.18

38.96
1.21)
20.28
( 1.93)
(

1.04
( 1.15)

(2.5)

0.23
3.46
35.85

4.13
( 1.50)
5.87
( 2.77)

-0.22
( -1.07)

(2.5)

1.22
( 7.33)
0.64
17.37
280.36

215.50
( 7.18)

0.73
( 1.47)

(2.6)

0.05
( 5.32)
0.38
3.39
32.54

12.16
( 5.59)

-0.14
( -0.93)

(2.6)

52

i

0.11
40.43
223.23

2.27
( 3.49)

(1.1)

0.24
36.07
211.05

2.27
3.51)
3.97
( 5.10)

(

(1.2)

0.51
29.05
154.89

95.48
( 11.48)

(1.3)

0.52
30.10
144.19

107.58
( 8.08)

-1.00
( -1.55)

(1.4)

0.61
25.87
139.07

36.14
( 2.32)
38.22
( 5.17)

-1.87
( -2.61)

1.57
4.04)

0.29
( 15.82)
0.80
19.79
104.57

76.43
( 11.24)

(

Panel D: Italy
(1.5)
(1.6)

-0.02
9.69
105.49

-0.08
( -0.13)

(2.1)

0.02
10.20
104.49

-0.66
( -1.00)
-1.36
( -2.15)

(2.2)

0.10
10.70
93.58

30.32
( 4.63)

(2.3)

0.10
10.82
94.95

31.32
( 4.40)

-0.33
( -0.63)

(2.4)

0.28
9.67
93.34

9.02
( 1.13)
20.10
( 3.97)

-0.41
( -0.96)

(2.5)

0.16
( 6.27)
0.46
8.68
88.07

30.67
( 5.93)

0.48
( 1.42)

(2.6)

5.72
13.94

55.68
109.21

Contagion Model
Linear OLS ﬁt

Contagion Model
Linear OLS ﬁt

Austria

23.67
55.82

2.05
11.41

Belgium

19.76
39.83

1.95
4.99

Finland

Germany

178.34
183.41

max absolute error
9.36
12.55
201.23
33.17
30.99
273.40

Ireland

14.83
19.24

Greece

16.04
44.12

mean absolute error
1.27
1.70
6.40
5.76

France

45.29
104.57

4.78
19.79

Italy

28.87
40.22

2.54
8.09

Netherlands

118.47
231.08

6.84
38.23

Portugal

38.17
102.51

4.66
24.25

Spain

Table 4: CDS Pricing Errors: Model vs. OLS Regressions. The table compares 5-year CDS spread pricing errors from the model to those
from the OLS regressions. We estimate the sovereign CDS spreads model via quasi maximum likelihood in combination with the unscented Kalman
ﬁlter. For each country, the OLS linear regressions explain the sovereign CDS spreads with the country-speciﬁc MCI, the logarithmic VIX, and a
proxy for a latent factor (this is the regression in column 6 of Table 3). The sample period is 02/12/2004-09/30/2010.

Adj. R2
mean(abs(ε))
max(abs(ε))

1
P CCDS
−i

BB-BBB spread

log V IX

M CIEU ⊥

M CIi

Table 3, continued

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Full text of Working Papers (Federal Reserve Bank of Chicago) : Modeling Credit Contagion Via the Updating of Fragile Beliefs, Working Paper 2012-04

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