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Massachusetts Institute of Technology
Department of Economics
Working Paper Series
Hedging Sudden Stops
&
Precautionary Contractions
Ricardo
J.
Caballero
Stavros Panageas
Working Paper 03-19
May
Revised:
26,
2003
November
10,
2005
RoomE52-251
50 Memorial Drive
Cambridge,
MA 02142
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j
Hedging Sudden Stops and Precautionary Contractions
Ricardo
MIT
Stavros Panageas
Caballero
J.
and
NBER
University of Pennsylvania
November
10,
2005*
Abstract
Even
well
bulk of which
managed emerging market economies
is
At a moment's
financial.
are exposed to significant external risk, the
notice, these
economies may be required to reverse
the capital inflows that have supported the preceding boom.
sudden nonlinear events (sudden stops),
we address
in the
paper
is
sudden stop jump
itself.
their likelihood fluctuates over time.
how should a country
hedging possibilities the country
faces,
While capital flows
react to these fluctuations.
crises are
The question
Depending on the
the options range from pure self-insurance to hedging the
In between, there
is
fluctuations in the likelihood of sudden stops.
the more likely possibility to hedge the smoother
The main
contribution of the paper
is
to provide
an analytically and empirically tractable model that allows us to characterize and quantify
optimal contingent
liability
management
in a variety of scenarios.
management can
example, that the gains from contingent
liability
of cutting a country's external liabilities
by 10 percent of GDP.
JEL Codes:
Keywords:
We
show, with a concrete
easily exceed the equivalent
E2. E3, F3, F4. GO, CI.
Capital flows, sudden stops, financial constraints, contractions, hedging, insur-
ance, signals.
"We
are grateful to Victor Chernozhukov,
seminar participants at MIT,
UTDT, and
the
Gordon Hanson. Arvind Krishnamurthy, an anonymous
referee,
AEA
for excellent
meetings
for their
comments, to Andrei Levchenko
research assistance, and to Alejandro Izquierdo for providing us the data on sudden stops from Calvo et
Caballero thanks the
NSF
for financial support.
and
al (2004).
Introduction
1
Even
well
managed emerging market economies
capital inflows.
That
is,
moment's notice these economies may be required
at a
capital inflows that supported the preceding
The deep
are subject to an occasional "sudden stop" of net
boom.
contractions triggered by this sudden tightening of external financial constraints have
great costs for these economies, and therefore represent one of the
problems
to reverse the
many
emerging market economies. There are
for
main macroeconomic
policy-
domestic and external factors behind
these sudden stops in net capital inflows. While there
is
adjustments that reduce domestic
consensus on what are the adequate external
liability
management
we analyzed
there
risks,
is less
substantial agreement on the kind of policy
strategies to deal with external shocks.
reserves accumulation
and hedging mechanisms
In Caballero
and Panageas (2005)
to soften the impact of
once these take place. In this paper we focus on a different aspect of the problem:
there are substantial fluctuations in the likelihood of a sudden stop,
and that
sudden stops
We
argue that
many
in
instances
these fluctuations are correlated with persistent external variables which offer hedging possibilities
free of
moral hazard or (emerging market
specific) informational
concerns (see Sections 5 and 6 for
evidence).
The main
technical contribution of the paper
is
sive analytical characterization, but is also flexible
guidance.
The model has two
a model that
and
central features: First
is
realistic
is
stylized
enough
enough
to allow exten-
to generate quantitative
the sudden stop, which
we
characterize as
a probabilistic event that, once triggered, requires the country to reduce the pace of external (net)
borrowing
significantly.
This specification of sudden stops captures well policymakers' concerns not
only with the stock of external
variable).
We
Second
but also with the size of current account
deficits (a flow
a signal, which describes the likelihood of sudden stop at each point in time.
characterize the impact of changes in this signal on optimal precautionary savings, and the
role of different
actions
On
is
is
liabilities
and the
hedging opportunities in reducing the extent of fluctuations due to precautionary
direct cost of
one extreme,
if
sudden
stops.
the country has no access to hedging instruments, the only tool at hand
a sort of precautionary contraction.
worsen, the country slows
down
As external indicators
the pace of borrowing.
Of
of the likelihood of a sudden stop
course, in reality
many
countries choose
to ignore these signals
and tighten monetary and
excessively prudent,
The
first
and experience deeper contractions once sudden stops take
of our goals in this paper
is
policy
fiscal
normative, and
it is
beyond what many
the other extreme
directly
and
is
is
reasonable.
the case where the country can hedge both the (jump) sudden stop
(diffusive) fluctuations in the price of
such hedge as the signal fluctuates. In this case
the country has maximal insurance. However, since our constraint
deficits,
remove the need
that removes
for
precautionary contractions and
jumps
in the value of
reduction in external
for the welfare loss
diffusive
it
minimal
of the latter.
on the current account
can go a long way
What
does do,
it
is
sudden stop to a
to
level
the post-sudden stop stage enough to compensate
from reduced consumption during the sudden stop).
hedging
One
cost.
limits the extent of a
is
is
the more feasible scenario where the
available. In this case the country still
tionary contractions and reduce the cost of sudden stops, but
it
of the quantitative contributions of the paper
in reducing the overall costs of living in a
jump cannot be hedged but
can remove the bulk of precau-
remains exposed to the realization
is
to
show that
this
type of hedging
sudden stop prone environment.
case of Chile, a prudent emerging market economy, provides a useful framework of reference.
Chile's business cycle
so that this price has
investors
is
highly correlated with the price of copper,
become a
and policymakers
signal, the decline in
larger
directly
a marginal dollar at the time of the sudden stop (since the
liabilities raises utility in
In an intermediate range,
The
is
such insurance does not remove sudden stop entirely but only reduces the country's level
of external liabilities at the onset of the sudden stop, at
smooth
see as reasonable.
to take a step in the direction of offering
an empirically based metric to evaluate how much prudence
On
place. Others are
its
main export good;
signal of aggregate Chilean conditions to (foreign
alike.
As a
result of the
many
internal
scenario in Australia, a developed
effect of
the price decline.
economy not exposed
falls
sharply
is
is
to this
many
times
This contrasts with the
to the possibility of a sudden stop,
similar terms-of-trade shocks are fully absorbed by the current account, with almost
domestic activity and consumption. 1 Chile
much
and domestic)
and external reactions
Chilean economic activity when the price of copper
than the annuity value of the wealth
so
where
no impact on
a natural candidate for the type of hedging strategies
we propose.
We
'See
do not attempt to explain why international financial markets treat Chile and Australia
,
e.g.,
Caballero (2001).
so differently, or even
the signal for the sudden stops should be so correlated with the price
we take
Instead,
of copper.
why
characterize the optimal risk
in
the "sudden stops" feature as a description of the environment and
management
strategy under different assumptions about imperfections
hedging markets. -
We estimate
sudden stop as a function of the price of copper and
Chile's probability of facing a
calibrate the parameters to
match observed consumption
hedging strategies and their impact on the
volatility
fluctuations.
and
levels of
We
then describe different
We
consumption.
start
by
showing that in the absence of hedging instruments, Chile should exhibit a significant precautionary
business cycle. For example, more than half of the deep reduction in the current account deficit in
Chile during the 1998/99 crisis can be accounted for by the optimal response to the fear (not the
realization) of a
We
sudden stop.
then go on to show the benefit of adding contingencies to
liability
management. For a wide
range of scenarios the gains associated with hedging are equivalent to reducing Chile's net external
about 50 percent of
liabilities (of
GDP)
by a
we
third. Similarly,
find that the required risk
premia
to justify for a country like Chile to not adopt an active hedging strategy exceeds 800 basis points
for the simplest strategies
Of course the
illustrative.
"We
policy
— as identified by Calvo
strongly correlated with lagged
is
for richer ones.
benefits of these strategies are not limited to the case of Chile, which
As another example, we show
sudden stops
latter
and several thousand basis points
for
only
a panel of emerging markets that the likelihood of
et al (2005)
US
is
— exhibits large time variation,
and that the
high yield spreads.
focus on the aggregate financial problem vis-a-vis the rest of the world, in an environment where domestic
is
managed optimally and
seldom hold
in practice.
Such
decentralization
failures
exposure to sudden stops. See,
e.g.,
compound
is
not a source of problems.
the problems
we
Needless to say, these assumptions
deal with in this paper by exacerbating the country's
Caballero and Krishnamurthy (2001, 2003) and Tirole (2002) for articles dealing
with decentralization problems. The literature on government's excesses
is
very extensive. See,
e.g.,
Burnside
et al
(2003) for a recent incarnation.
J
In his
Nobel
lecture,
Robert Merton (1998) highlights the enormous savings than can be obtained by designing
adequate derivatives and other contracting technologies to deal with risk management. He also argues that emerging-
market economies stand to gain the most. Our findings
in this
paper
fully
support
his views.
See
also, e.g.,
(1988), Froot, Scharfstein, Stein (1989), Haldane (1999), Caballero (2001), for articles advocating
ation of emerging markets debt.
framework and to
constraints.
link the
The
contribution of this paper relative to that literature
hedging need not to commodities per
se,
is
Krugman
commodity index-
to offer a quantitative
but to the signal aspect of much
costlier financial
we describe the environment and characterize the optimization problem once a
In section 2
sudden stop has been triggered
phase." In section 3
The main
— this provides
we study the phase
goal of this section
is
the boundary conditions for the "precautioning
that precedes a sudden stop
when
the country self-insures.
to characterize precautionary contractions.
Section 4 describes
aggregate hedging strategies under different degree of imperfections in these markets.
illustrates our results
through an application to the case of Chile.
Section 5
Along the way, we outline
an econometric approach to gauging the likelihood of a sudden stop and
its
correlation with an
underlying signal. Section 6 provides empirical evidence for a panel of emerging market economies
on the
large time variation in the likelihood of
sudden stops and
exogenous) global indices. Section 7 concludes and
2
is
its
correlation with persistent (and
followed by an extensive technical appendix.
Preliminaries
Intertemporal smoothing implies that emerging economies typically experience substantial needs for
external borrowing from abroad. For a variety of reasons that
on borrowing
is
a source of fragility.
The sudden
we do not model
here, this
dependence
tightening of financial constraints generates large
drops in consumption. In this section we formalize such environment and characterize the value
function of consumers once a sudden stop has occurred. This serves as a boundary condition for
the pre-sudden stop problem faced by the country, which
2.1
is
our main concern in this paper.
Endowment and Preferences
Let the endowment grow at some constant rate, g
yt
=
>
yoe
0,
during
to
t
<
oo:
9t
(l)
.
This specification of the income process excludes feedback
and from output
<
effects
from aggregate demand to output
sudden stops (although we recover some of the
latter channel in the signal
process, see below),
and hence underestimate the impact of the phenomenon we
we adopt
its
it
because
study. Nonetheless
simplicity allows us to isolate the direct effect of sudden stops
more
cleanly.
Reflecting emerging markets' net debtor position, the country starts with financial wealth,
X(0)
= Xq <
0.
However the country's
total wealth
X
where
r
Let
Ct
and
cjT
represent date
t
The
it
=
ct
>
must be positive
at all times:
0,
exceeds the rate of growth of the endowment, r
consumption and
cj"
i
r-g
denotes the riskless interest rate and
I
t
—
>
t
(X + ^-
- Kyt
>
g.
"excess" consumption, respectively, with:
< k <
,
1.
representative consumer maximizes:
/OO
u(c*)e~^'-^ ds
(2)
ith
=
u{($)
c*
1
The parameters
simplicity,
5
1
'
-7
<5>0, 7 >0.
,
and 7 are the discount rate and
we assume
r
=
6 throughout.
The
The
in a
manner
habit term plays two roles:
similar to
First,
is
increasing at the rate of the country's growth
raises the effective risk aversion of the
it
Campbell and Cochrane (1999) and thus brings the demand
to realistic magnitudes.
Second,
borrowing demand to reasonable
it
For
functional form of the utility function captures an
external habit formation, with a habit level that
rate.
risk aversion coefficient, respectively.
for
country
hedging
introduces a trend in consumption which allows us to reduce
levels
without introducing overlapping generations or gaps between
the interest and discount rates which would complicate the analysis. 4
A
frictionless
It is
benchmark
instructive to pause
and study the solution of problem
(2)
subject to a standard intertem-
poral budget constraint (and absent a sudden stop constraint):
dX
t
= {rX -cl+y* )dt
t
t
(3)
^Yq given
Accordingly, this type of utility function implies that the indebtedness of the country at any point in time will
be larger than
- (1 ~^yt
r-g
with y*
By
ee (1
-
n)y.
standard reasoning we have that in such case:
=
c*
and the
"total resources" of the
r
(x + -^—j
foralli>0,
country remain constant throughout:
r
X
-g
+
Vo
(4)
r-g
so that
lim ——
i^oo y*t
1
=
r-g
and, accordingly:
lim
i->oo
Moreover, for any level of
^
above
-K
r-g
—=
y
1
t
value, the ratio
its limit
^£ decreases monotonically
to
l
r-g'
This case serves as a frictionless benchmark in what follows.
2.1.1
Sudden Stops
We place no
limits
on the country's borrowing
country faces a "sudden stop."
this constraint, or the
stop
is
we look
is
for a realistic
model the sudden stop
up
to the stochastic time r.
we
this time, the
do not model the informational or contractual factors behind
as a
to produce a quantitative assessment of the hedging aspects of
and
fairly
robust (across models) constraint. For
this,
we simply
temporary and severe constraint on the rate of external borrowing. In
particular, since the "natural" aggregator of a country's total wealth
total resources,
At
complex bargaining and restructuring process that follows once the sudden
triggered. Since our goal
the problem,
We
ability
is
the net present value of
its
place the constraint on:
*
XT
We
assume that
at time r, financial
^
<
r-g
markets require the country to increase
c.
e
gT
<
C
<
e
rT
its total
resources by
within
T
periods. Formally:
y
(xT+T+ -^-)>dx
T + ^-).
r-gj
T-gJ
V
V
It is
apparent that this constraint
unconstrained benchmark
show that
debt/gdp
this constraint
at time t
+ T,
I
Xt
+
will
-^-
)
(5)
be always binding because, as we showed above, at the
remains constant at
all
times.
debt/gdp
then straightforward to
maximum
can be expressed as a constraint on the
as a function of the
It is
To
ratio at time r.
amount
allowable
of
see this, simply rearrange
obtain
(5) to
r
^>Ce^+
Vr+T
(Ce-*
T
-D—
r
Vt
Higher levels of £ make the constraint tighter. At
its
minimum,
(6)
9
C,
=
e aT the constraint literally
,
reduces to the requirement that debt/gdp cannot grow any further between r and r
XT +T
>
Vt+t ~
Finally,
it
interesting to note that
is
+
T:°
Xt_
v*
one can think of
(6) as
a constraint on the
T
balance-of-trade surpluses the country has to accumulate over the next
minimum
To
periods.
see this,
budget constraint:
start with the intertemporal
rr+T
e
and replace
(6) into
it,
-rV-T>(
y
*
-^) dt=
e~
rl
X T+T - XT
to obtain:
rT+T
e"
r(i_T)
(y*
i;
~ <)
dt
>
(Ce
_pT -
l)^- e -(
r-g
r ~ 9)T
-
(1
- Ce~ rT )XT
.
This gives us the constraint in terms of the balance-of-trade surpluses required from a country
that
is
faced with a sudden stop. These required surpluses rise with the country's endowment, with
£,
and with the
is
a net debtor).
That
This
is,
level of
we have
debt at the time of the sudden stop (recall that
XT
<
when the country
arrived to a flow constraint rather than to a conventional stock constraint.
specification not only facilitates our analysis but also captures well policymakers' concerns
with current account
°Even
for the £
=
e
deficits regardless of the level of external indebtedness. In practice
gT case the
constraint we consider
the fact that the debt/gdp ratio
is
is
binding at the time of the sudden stop. This follows from
always growing for an unconstrained country as we showed above.
8
we observe
a wide array of debt-to-gdp ratios but warnings emerge quickly as soon as current account deficits
There
(a flow) crosses certain thresholds.
above and beyond stocks: while there
flows
is
much harder
at least
is
one good reason
for this concern
with flows
always some space for stock renegotiation, financing
is
new
to obtain during distress.
As we mentioned above, note that while the
constraint
is
on the
flows, stocks
A
do matter.
higher debt-to-gdp ratio at the outset of the sudden stop requires a larger consumption sacrifice
since debt-servicing
is
more expensive. This
down
countries to slow
,
is
initial
Note
stock of debt
diminished but excess consumption during the sudden stop approaches
and hence the value of even small contributions to the relaxation of the constraint have great
value.
If
7 >
which we assume throughout, the
1,
adjust debt before a sudden stop rises with
latter effect
dominates and the incentive to
£.
The Sudden Stop Value Function (A Boundary Condition)
2.2
We
to reduce this debt-burden cost.
is
upper limit erT the effectiveness of the changes in the
in affecting the constraint
zero,
sudden stop
in anticipation of a
also that as C goes to its
key for our results below, as the main reason for
is
solve the optimization
crisis
problem
in three steps. Starting
backwards, we
first
solve for the post-
period, then for the sudden-stop period, and finally for the period preceding the sudden stop.
In this section
we summarize the
results of the first
V
function at the time of the sudden stop,
(XT
two
y T ), which
,
The
steps.
goal of these
we then
is
to find the value
use to find the solution of the
optimization and hedging problems before the
crisis
with the latter phase, we assume that there
only one sudden stop, which simplifies the analysis
without sacrificing
Lemma
7 >
1.
1 Let
much
is
insight.
V ss (XT ,y*)
denote the value function at the the time of the sudden stop and set
Then:
v ss (xT
where
is
takes place. Since our concern in this paper
V + (XT ,y*)
y ;)
,
=
Kv + (x
,
y ;)
(7)
denotes the value function in the benchmark frictionless case:
V + (XT ,yl)
.
.7(xT
rJ
and the constant
T
K
-i--a1
—7
is
K=
(1
-
e-
rT
)^(l
-
e-
1-7
rT
C)
+
e-
rT 1-7
C
,
(8)
such that
K = K(Q
We
leave the proof for the appendix.
y T independently, but instead
it
A">1;A"'>0.
with
depends on the net present value of resources
is
K
>
to the
and increases with
£.
While the particular simplicity of our formulae
is
more
1
The model
general.
will allow us to
(
XT + £f-
)
.
and
This
reduce the dimensionality of
constant K,
we have
problem without sudden stops.
identical to the one in the
we assume throughout,
is
it
programming problem. Moreover, up
function that
XT
Note that the value function does not depend on
observation will prove convenient later on, because
the dynamic
(9)
arrived at a value
Lastly, for
7>
1,
which
6
due to stylized assumptions, the basic message
be understood as an approximation to a potentially more
also can
complicated specification of constraints that result in higher marginal disutility of debt in the event
of a crisis.
Precautionary Contractions
3
Let us
now
address our main concern and begin to characterize the country's problem prior to
the sudden-stop phase.
If
there are no hedging instruments, the country's only
reducing the cost of a sudden stop
takes place.
We
varies over time.
show that
if
is
to cut
mechanism
for
consumption and borrowing before the sudden stop
sudden stops are somewhat predictable, the amount of self-insurance
This time varying precautionary savings implies that when the signal of a sudden
stop deteriorates, the country
falls into
a sort of "precautionary contraction." That
is,
into a sharp
reduction in consumption to limit the cost of the potential sudden stop.
Let
st
represent a publicly observed sudden stop signal that follows a diffusion process:
Observe that
value function
7
>
1
is
is
for
7
>
1
the function
^3—
=
+ odB
ds
t
is
negative for
jidt
all
t
.
Z >
0.
Thus
one, because the lender does not reduce
is
>
1
reflects that the
lower than the continuation value function for the unconstrained problem.
that the flow aspect of the constraint implies that having a higher
one, this effect
K
its
XT
10
crisis.
The reason we need
does not relax the constraint one
demands by the same amount during the sudden
strong enough to discourage saving for the
constrained
stop.
When
7
is
for
below
This signal
imperfectly related to the sudden stop process.
is
Concretely, the stochastic time r
depends on the realization of a stochastic jump process, the intensity of which
=
At
We
exp(a
-
ais t );
can now write the dynamic programming problem
V(X
t
= maxE f -$
1-7
Jt
,s u yl)
>
+
e-
given by
0.
as:
r{u - l)
e-
<*i
is
du
r
^- V ss (X ,y*)\F
T
t)
t
(10)
~ exp(l).
(11)
where
r
=
inf{£ 6 (0,oo)
/
:
<
X(s u )du
z},
=
X(s t )
e
Q °- QlS(
z
,
Jo
and the evolution of (Xt,
is
st,yt)
given by:
dX
= {rX -c* +y* )dt
(12)
dy*t
=
ylgdt
(13)
ds t
=
[idt
t
From the previous
section
we know
t
t
t
+ adBt.
(14)
that once the sudden stop takes place, the value function
takes the form:
(iy( x^ +
,
,
Hence the
HJB
with
first
1-7
equation for the value function
= max j£—- -
Vxdt
is:
\-rV+ Vx (rX + y") + Vy .y* g + VsM + ^a V
2
ss
In Proposition 5 in the appendix
{Vx )- lh
we show that the
X(s t ) [V
ss
-
V]
.
(15)
v=
b(s t
(16)
.
solution to the
X + ^~9
in
more
detail in
what
follows:
11
takes the simple form:
r
an appropriate function b(s t ). The function b(s t ) captures
it
HJB
1-7
y±-1-7 ^
t
we study
+
order condition:
t=
for
\1-T
rX
.
all
(i7)
the economics of the problem and
Precautionary Savings
3.1
Assume
that sudden stops are totally unpredictable, Qi
first
in this case b(s) is
no longer a function of the signal
yields a simple algebraic equation for
0,
so that X t
replacing the function
s,
=
V
exp(ao)
in the
=
HJB
A.
Since
equation
b:
V
b
- rb+ -A
7
1
=
1
-
1
(18)
where
b
captures the
jump
in
the value function once the country enters into a sudden stop.
relegate the proof of the following
Lemma
It is
lemma
Let us
to the appendix:
2 Equation (18) has a unique solution in \^,b\
now
straightforward to obtain the (excess) consumption function from the envelope theo-
rem:
(x +
-£-
t
<
This result -combined with
larger
of
Lemma
than - yet smaller than
b.
,
sudden stops. Yet,
it is
= ~
r
-^-
(19)
2- demonstrates the presence of precautionary savings: b
Hence consumption
more than what
it
is less
would be
if
than what
it
would be
in the
is
absence
the country experienced a sudden stop.
This reflects the country's desire to accumulate extra savings (compared to the no-sudden stop
case) in order to
By
smooth out the
time-differentiating (19)
extreme events.
effects of these
and using
(1),
(3),
and
(18)
we obtain the country's
(excess)
consumption process:
dr*
dc;
q
That
is,
a
r//>\ 7
!
7
1
dt
>
the (excess) consumption process prior to a sudden stop has an upward
turn reflects a precautionary savings
of
1
i
= -A
sudden stop taking place
with the signal
self-insurance
is
s,
in the
demand
drift:
This in
attributable to the uncertainty generated by the risk
next instant. But, because this probability
is
not correlated
there are no precautionary contractions. In this case, the pattern of saving for
not a source of business cycles.
L2
Time Varying Precautionary Savings
3.2
now return
Let us
the function
V
to the general case,
of equation (17) into the
where a\
HJB
>
0.
After a few simplifications, substituting
equation (15) shows that b(s)
satisfies the following
ordinary differential equation (ODE):
-b
+
b s fi
+ (7-1)
(b s
+b
V+-A(
2
s
^7
St )
The boundary
1
0.
(20)
b.
7
conditions reflect the facts that as the signal becomes extremely positive, the
problem approximates the
frictionless (no
sudden stops) benchmark, while when the signal
is
ex-
tremely negative, the problem converges to that at the sudden stop time. Let
b*
=
lim
s—»oo
K=
where
and
b*
6* are
lim
b(s).
(21)
b(s).
(22)
given by:
b*
=
r
K lh -.
K =
We
can easily obtain the solution to the ordinary
differential
plot this function (multiplied by r) in the top subplot of Figure
with respect to
s.
As the
1.
equation (20) numerically.
The
function b(s)
is
We
decreasing
signal deteriorates, this function rises rapidly, reflecting the increasing
marginal value of wealth when a sudden stop
is likely
in the near future.
This setup allows for an explicit characterization of the optimal consumption policy in feedback
form. Given the value function,
it
follows
from (16) that the optimal consumption policy takes the
form:
X +
t
Vt
r-g
(23)
Kst)
The
effect of
the signal can be seen clearly in this expression.
its frictionless limit
when the
and hence consumption increases
signal worsens,
consumption
falls for
for
any given
contraction.
13
any given
level of
As
s rises, b(s) falls
level of wealth.
toward
Conversely,
wealth causing a precautionary
-i.
1.3
-20
1
2
.o
\
1.6
1.4
1.3 "
\
1.2
1.2
\
"~
-
^
0.8
11
1
0.4
i
12
-10
i
'
i-
i
'
i
'
3
Standard deviations of the state Variable
Figure
1:
The
top subplot depicts rb(s t ) for a wide range of values of s t normalized by
its (yearly)
standard deviation. Similarly, the bottom panel focuses on a more narrow range of values
(normalized by the yearly standard deviation), and plots rb(st) (dashed line
axis)
and the hazard rate
(solid line
-
measured on the
14
right axis).
-
for s t
measured on the
left
Finally, applying Ito's
lemma
hand
to the right
side of (23) using (20), (3),
and
(1)
we obtain
the country's (excess) consumption process before the sudden stop:
dc?
The
case a\
drift
=
0,
+
i
(ob s
y
k^
i
TYT)
1
i
{st)
(24)
b
-A(sj)
I
1
b,
,
which we
also
found in the
and captures precautioning against the Poisson jump event that can occur
now
at
any
there are two additional terms associated with the sensitivity of the likelihood
sudden stop to the
signal.
For the reasons explained above,
deteriorating signal increases the need to accumulate resources
respond more to a one-standard-deviation increase
also
—odBt
dt
7
term includes a precautionary term
time. However,
of a
7
an extra precautionary savings term
^
is
strictly negative, so that
a
and accordingly makes consumption
in the signal
by a factor of —
^
There
c-
is
in the drift in response to the additional uncertainty in
consumption created by precautionary contractions. In the application section of the paper we
quantify these effects.
3.3
Ignoring time varying hazards
Before we turn to hedging,
we
shall develop a
simply ignoring time-variation in the hazard
framework to measure the costs associated with
In order to quantify these costs,
rate.
a country chose to ignore time variation in the hazard rate and determines
hazard rate was given by a constant
A.
assume that
its policies
as
if
the
In that case, the country's "optimal" policy would be given
by:
(**
+
#
d
where d would be a solution to equation
(18) with A replacing A. If the hazard rate
varying, this policy will be clearly suboptimal.
with this suboptimal policy by plugging
it
We can evaluate the expected utility
V'
is
truly time
associated
(instead of the optimal policy (23)) into the
Bellman
equation (15). This results in the partial differential equation:
•V
o
i
An
sub
+ X(s
t )
(V ss - V sub )+V^ b (rX +
t
y*
t
-
c*}
7
informed guess about the solution to
this
equation
is:
l-T
V sub
[/(*)]'
1
15
7
Vy ub gy; + Vr
s
.
b
H+^V T
s
b
(25)
Plugging back this conjecture into (25) and simplifying, shows that
/
V sub
solves (25)
if
and only
if
solves:
r^-4
1
7
2
loses of ignoring
we
+
/
2
f'2
+/> + y (7 In the application section
rf
"f + y/ss
1)
and evaluate the welfare
solve this differential equation numerically
time variation in the hazard
rate.
Aggregate Hedging
4
Precautionary savings and contractions are costly and imperfect self-insurance mechanisms for
smoothing the impact of a sudden stop. In
allow
it
to
this section
we
enlarge the options of the country and
hedge using insurance contracts and derivatives.
Of
course, the effectiveness of the
hedging strategy depends on the contracts and instruments that are available to the country, and
how accurate the
crisis-signal
In this section
is.
we provide a framework
to explore a variety of
scenarios in a unified framework.
A
4.1
unified
model of hedging
There are two fundamental sources of
associated with variations in the signal
The second
as "diffusive" risks.
of the
s.
Since
s follows
The
first
type are risks that are
a diffusion process,
we
shall refer to those
source of risks are those that are associated with the actual arrival
random sudden stop time
r.
We
of the country to mitigate the effects of
of markets to
our framework.
risk in
hedge against these
risks
shall refer to this type of risk as
sudden stops
and the
will critically
"jump"
risk.
depend on both the
The
ability
availability
costs of such strategies.
Let us assume that diffusive risks can be hedged in futures market. That
is,
a country can enter
into contracts with payoffs given by:
dF
t
—=-
=
(fiydt
+ adBt
Ft
where
<p x
is
a risk premium, that could depend on
global risks, then
cp
1
=
and the futures
price
is
st
.
If
the diffusive risks are not associated with
a martingale. In the presence of futures contracts,
16
the dynamic budget constraint becomes:
dX = (rX t
where
7r t
denotes the
number
the portfolio process as pt
=
Tv t
dF
{rX -
t
t
risks are
c*
t
+ y* + pt4>i)dt + p odB
t
stop over the next interval dt, then the insurer pays
Allowing
for
If
t
N
It
t
is
actuarially
is
fair,
+ y ; + ptfa -
c*t
t
then
(26)
t
+
when
1
the country enters into a sudden
amount equal
(j>
=
2
2
0.
to A(s t )(l
should be viewed as a
Else the insurer charges
<t>
2 )It)dt
+ Pt adB + I dN
t
t
t
=
presence of insurers, then It
and
(3) as special cases.
(27) reduces to (26).
presence of futures markets to hedge against diffusive
4.2
The
to the
markup
2
>
0-
(27)
the country enters into a sudden stop, and
denotes the amount of insurance contracts purchased (in units of the numeraire).
budget constraint (27) includes equations (26) and
+ (j)2)dt
the evolution of assets becomes:
risks,
A(s t )(l
and jumps to
identically equal to
If
the parameter
1;
markets in both diffusive and jump
dX = {rX where
Similar to
insurance
constraint:
unit of consumption to the country, else
1
nothing. In exchange for this contract the country pays an
insurer per contract purchased.
Defining
hedged through "insurers" who are willing to enter into an
(instantaneous) contract of duration dt in the following terms:
or a risk premium.
t
we obtain the new dynamic budget
TttFt,
jump
+ yl)dt +
of futures contracts that the country wishes to purchase.
dX =
Let us assume that
c*
t
t
risk,
Furthermore,
The dynamic
If
we assume away the
if
we assume away the
then (26) becomes
(3).
general case
Let us solve the country's problem subject to the dynamic budget constraint (27), and then turn to
study important subcases.
for the
If
the budget constraint
is
modified to (27), then the Bellman equation
country becomes:
= max
i
+A(.s ( )
^
Vx c
I
max {V ss (X +
t
(28)
I)
- Vx (l +
<f>
2
)l}
2
+ max <
p
{
a
—
VxxP
„2
t/
„ 2 +pV\<p
+V\S pa
,
2
x
2
+VX (rX + yl) + piV +
t
s
^V
17
ss
+ Vy -gy* t
[r
+
X(s t )}
V
i
The
first
order conditions for this problem are:
(cT 1
(29)
v
vxss
The
first
optimality condition
is
Vx (l +
(30)
<p 2 )
Vx
4>\
VXi
VX x
v2
Vxx
(31)
the familiar Euler equation.
The second equation captures
the country's attempt to equate the marginal value of a dollar pre- and post- sudden stop, which
is
fully
accomplished
if
2
=
0.
Finally, the last condition
optimal portfolio of futures consists of two parts:
and captures the willingness
some
to accept
The
first
is
part
identical to Merton's formula.
is
positive (since
volatility in return for a risk-premium.
simple intuitive argument shows that the second term captures a hedging
The reason
is
that the marginal value of wealth will
that a sudden stop
interesting case
is
is <p
l
imminent. This happens as
=
0, i.e.
when the
Vxx <
signal
is
demand and
become higher the more
s decreases
By
and hence Vx s <
likely
0.
A
0,
The
Vx >
0)
contrast, a
is
negative.
it
becomes
particularly
uncorrelated with the world economy. Then p
<
0,
since only the hedging motive prevails.
To
solve (28)
we proceed
as in Section 3
and conjecture that the value function has a separable
form:
i—
v=
(*+£)
d( St y
(32)
i
i
With
this conjecture, equations (29)
through (31) become, respectively:
/~
i
y*t
d{s t ) V
r
{l
1
(
I
1^1
ya
It will
be useful to
l
+
4>2
-
K
rd \
+
.
(33)
~9
d.
X
d
X +
r-g
Vi
1
t
(34)
Vi
(35)
t
define:
t
Plugging back (32) into
(28), using the
r-
above four equations and performing several simplifications,
L8
shows that
V
satisfies (28) if
1
-
and only
+ fid s +
rd
1-7
70-'~
—d
/
1
if
d(st) satisfies the equation:
ss
+
(36)
ds
0J
y
jv 2
i
1-7
\
;
(l
1-7
7
is
A'
7
A(s t
+
1
it
stiU characterize several properties of the solution.
consumption,
Lemma
for
i
cps)
+
{l
+
6 2 )d-
a non-linear second order differential equation which
the boundary conditions (21) and (22). Even though
can
2
d
2
7
Equation (36)
(d s )
is
to be solved under
does not have an analytical solution, we
Of
particular interest
the behavior of
is
which we have:
3 The excess consumption process evolves according
dc*
-rd+
A(s t )(l
+
-f <p 2 )id
[
fi
to:
+
—
)
ds
+ %-d s
dt
(37)
~~c*
7 a
Let us
now
dBt
analyze a few special cases to understand the different component of the consumption
process.
No
4.3
The
risks
restrictions
on hedging
natural starting point
and the
this case
is
when
<p l
risks that threaten the
=
cp
2
=
0.
In other words, markets exist for both types of
country are diversifiable. The following
Lemma
shows that in
pre-sudden stop consumption wiU be constant:
Proposition 1
If
4>i
=
<p 2
=
0,
then
dc[
=
<t <T.
0:
This result asserts that consumption will be constant prior to a sudden stop
hedge perfectly
in
markets
for
both diffusive and jump
risks.
country can costlessly trade in contingent claims, then
it
This result
will
is
if
a country can
hardly surprising.
be able to stabilize
its
If
the
consumption
stream pre-sudden stop. Precautionary savings ceases to be necessary as a mitigation mechanism.
19
Let us pause and note that both markets need to be costelessly accessible in order to obtain
the above result.
may seem
This
surprising at
for diffusive risks as long as the country
desires
upon the
all,
is
is
time varying.
The
resolution of this puzzle
is
that the
Since the contracts with the insurers are instantaneous, there
uncertainty about the terms under which these contracts can be rolled over in the future.
s decreases,
needed
To
then the actuarially
in order to
insurance will become more costly.
fair
hedge against such variations
see this, let us momentarily set
In the appendix
we show that
it
variations in s are only useful in predicting
the likelihood of a jump, and are otherwise useless.
cost of insurance
should there be a need for a market
can costelessly contract with insurers on the amounts
sudden stop? After
arrival of the
Why
first.
</>
2
=
0,
If
Therefore, futures are
in the price of insurance.
but
in this case the (excess)
consumption process has the following
properties:
Proposition 2 Assume
that
cp
2
=
and
=
4> Y
dc*
1
c*
d
—jcr 2 ^f
(7+1)
Then
dt
d
2
c Is
T^dBt
i
for d(s t ) that solves (36) with
That
zero
is,
<f>
2
=
and fa
the consumption process
"dB" term). This shows that
is
= - 7°" 2 ^f
a non-degenarate diffusion (in the sense that
in the presence of
stop both markets (for diffusive and
•
jump
time variation
risk) will contribute
in
it
has a non-
the likelihood of a sudden
towards hedging. Neither
will
be
sufficient in isolation.
Let us not return to the case with
is
to use the martingale
This methodology
is
(t>
1
=
q> 2
=
0.
An
alternative
way
to arrive at Proposition
methods developed by Cox and Huang (1989) and Karatzas
particularly attractive
when dynamic trading
stochastic payoffs.
20
et.al.
allows one to span
all
1
(1987).
possible
Formally, these methods start by solving an inherently static problem. Namely:
t
max
c
*l-7
+ e -r(T-t)ySS\xT ,y*T )\F
3S__ e -r(«-*) du
E
(38)
t
subject to the intertemporal budget constraint:
-Hv-t)
This
(39)
an inherently
is
and maximizing
c
*
du
+
static problem.
E(e^ T -
at least
of Proposition
decomposed
1.
into
is
>
1
T,
t
r{u - t]
y* du
(39)
u
(40)
y*
'
(41)
r-g
times between
all
First,
<u<
t
Second, observe that the optimal level of
observe that the (excess)
r.
This
is
just a restatement
financial wealth at time
r can be
two components:
r-g
r
the second term
is
in
and
-
- 1
captures the
it
=
(A'-
-
This formula shows that optimal (excess) consumption
sudden stops compared to an economy that
for insurance.
Com-
>
k
r(xt +
+
demand
_i
k'y
1
(A>
Extra insurance needed to cover SS
the absence of SS
solve for c*
=
i
r
clearly positive
we can
cT
-k
-i
i
i
1
bining (39), (40) and (41)
to
-
<U<T
K!
-
equal to a constant for
Optimal Wealth
K
+E
two observations that deserve comment.
XT =
Since
t
(38) over c*, A"T gives:
consumption process
<X
KXT )
Adjoining a lagrange multiplier k to the budget constraint
k
There are
t
1)
£
Ee-^-V
will
be lower in an economy that
is
exposed
not exposed to such events by the factor:'
is
1
(42)
Ki -l)Ee'
It is
for the
is
interesting to cross-check that
consumption process. To see
given by
r(T
-V
dynamic programming and the martingale methods produce the same answer
this note that the martingale
methods
in conjunction with (33)
:
d( s
,)=*+-(/w
21
-
l)
£e- r(T
- £)
imply that d(s t )
The reason
to
pay
intuitive:
is
Even when
such insurance, and hence
for
hedging possibilities are unconstrained, the country
all
it
have to cut back consumption somewhat
will
be able to finance this insurance. However, this insurance
An
consumption process subsequently.
volatility of its
will allow
still
has
order to
in
the country to eliminate the
interesting feature of (42)
us to determine the cost of "hedging" as a percentage of excess consumption.
is
That
that
cost
it
is
allows
given as
i
*•(* +
£)-<?
1+1/0-1 )£e- r
<
T -'>
= (K$ -
l)
Ee-rW
This formula allows a simple "back of the envelope" computation. Note that
moment generating
random time
function of the
Ee -r(r-t) =
2
r.
Ee~ r
(
T~
'"'
is
the
Accordingly:
_ rE T _
(
tj
+
£(,.2)
Ignoring the last term in the above equation, the cost of insurance as a percentage of excess
consumption
given approximately
is
as:
(K$ -l)(l-r£(r-t))
This formula
ity of the
t))
high) the less
To
see that
and the
quite intuitive.
sudden stop
— rE(r —
(1
is
K
captured by
1
'
1
=
(f>i
<Pn
=
will
and the constant of proportionality
)
1
is
equal
d(st)
fid,
yd
+
33
+ \(s
t )
=E
(,-t)
e
/
ds
+ Ee
J
I
—
ODE
(T-i)
is
given by:
K
to:
=E
r(a-0
ds
+
H--'"-"
^ -f
and therefore
d(s t )
This
is
given by
produce the same answer, note that according to the Bellman equation
- rd +
d(s t )
turn
is
the expected distance to the sudden stop (say because s
is
According to the Feynman Kac Theorem the solution to this linear
in
proportional to the sever-
:
=
which
is
is
the cost of insurance.
dynamic programming
fact that
says that the cost of hedging
It
Hence, the longer
.
is
(as
(43)
exactly the
same expression
=
-+E
r
1
-r(T-l)
as the one produced by the martingale approach.
22
>-'-' ds
(36)
Restricted insurance
4.4
One could reasonably argue
that the hedging of
jump
may be
risks
sudden stops directly probably requires to overcome a long
concerns.
ballero
One
possibility
infeasible in practice. Insuring
of moral hazard and informational
list
to do proxy-hedging using exogenous variables, as described
is
and Panageas (2005), but there
by Ca-
always be an important residual that probably can only
will
be insured (and to a limited amount) by international financial institutions rather than by private
markets.
However,
if
one can find tradeable variables (such as the price of copper) that are useful in pre-
dicting the arrival of a
sudden stop, then hedging
"diffusive" risk alone
can
provide significant
still
benefits.
The
easiest
way
assumption that
4> 1
to see this
=
0.
is
to
impose the constraint 7
Let us enforce the constraint I
fc-o^"
If
4>2 is
risk
given by this process, then clearly /
premium but
result
in this case the
Proposition 3 Assume
Then
the optimal
that
<p 2
A key observation about
variation. This
with equation
is
<p 2
is
by assuming that
is
<p 2
satisfies:
(44)
(34). It
consumption process
is
most
useful to think of
is
=
less volatile
0.
=
than
0.
not as a
<^ 2
The
following
in Section 3.
Furthermore assume that
cp^
=
0.
given by:
given by (44) an d
<t>\
=
the above proposition
in contrast to Section 3
(37)).
but continue to maintain the
1
given by (44) so that 1
consumption process
where d solves (36) with
by
0,
Lagrange multiplier associated with the constraint 1
as the
shows that even
=
=
=
is
0.
that the consumption process has no quadratic
where we allowed precautionary savings only (compare
This means that hedging with futures achieves the goal of mitigating pre-
cautionary contractions, even in the complete absence of markets that insure against the arrival of
sudden stops (jump
risk).
Note, however, that relative to the
<^ 2
=
ing the presence of precautionary savings
case, the
consumption process
still
has a
due to the presence of unhedged jump
23
drift reflect-
risk.
Moreover,
there
on
st
(smooth) variation in the consumption rate pre-sudden stop since both A and d depend
is still
.
Pure Precautionary savings as a
4.5
In the previous section
we
In this section
special case
we postulated equation
(44) for
shall additionally postulate that
=
p
hedging possibilities altogether. From equation (35) we
That
is,
is
(f>i
the appropriate "shadow" risk
0,
set
impose the constraint
so that the country
p
=
is
1
=
0.
excluded from
by setting:
-^ ~
2
=
4>i
in order to
4>2
(45)
premium
that ensures that p
=
throughout.
The
next proposition shows that under the joint assumptions (44) and (45) we recover the result in
section 3.2
.
Proposition 4 Assume
section 3.2
and
that (44)
and (45)
the consumption process
dc*
1
c*
d
'
Then
d(st) solves the
same equation
as b(st) in
given by
4^y. 1+(1+1)'2d
'1
iW
J
d*
is
hold.
\
\
di
rd
,„
-^CrdBt
a
which
is
The
identical to equation (24).
value of this proposition
one that we used in
"shadow"
4.6
risk
is
to
this section and,
premia
embed
the results in Section 3 in the
more importantly,
in order to justify
same framework as the
to identify the necessary
assumptions on
no hedging.
Discussion
We can now view the results
that
we have obtained thus
far in a unified
eliminated every hedging possibility. Therefore, the country was
its sole
protection against sudden stops.
by both variations
in the
The
with precautionary savings as
country's consumption pre-sudden stop was affected
hazard rate (diffusive
conditional on the hazard rate (jump risk).
left
framework. In Section 3 we
risk)
and the random
arrival of the
At the polar opposite, Section
24
sudden stop
4.3 allowed for the
presence of an insurer and a market to hedge against diffusive
effects of
both sources of
risk
risk.
This helped eliminate the
on consumption. Hence consumption pre-sudden stop was constant.
Subsection 4.4 helped demonstrate that even in the absence of an insurer, the presence of hedging
markets can help eliminate the
process will
still
on consumption. However, the consumption
effects of diffusive risk
be affected by the non-insurability of jump
which leaves a random
risk
the
drift in
consumption process.
Importantly, Sections 4.4 and 4.5 helped to demonstrate that the assumptions on the presence or
absence of markets are equivalent to making appropriate assumptions on the prevailing "shadow"
risk
premia on these markets.
This
will facilitate robustness
checks after
we
calculate optimal
hedging strategies.
5
An
The Case of Chile
Illustration:
In this section
we
good case study
illustrate
since Chile
our main results through an application to the case of Chile. This
is
an open economy, with most of
to capital flows' volatility. Moreover, the price of copper
(its
source of income fluctuations but, more importantly for us,
attitude toward Chile.
aggregate
demand and
its
a
recent business cycle attributable
main export)
it is
is
is
not only an important
an excellent indicator of investors
In fact, Caballero (2001) shows that the typical contraction in Chilean
net capital flows after a sharp decline in the price of copper
larger than the corresponding annuity value of the decline in Chilean wealth
is
many
times
due to the drop
in
copper prices.
5.1
Calibration
While our purpose here
we spend some time
5.1.1
Estimating
Sudden stops
However, we
is
only illustrative, and hence our search for parameters
implementation.
rather informal,
describing our estimation/calibration of the key function A(s).
A(s)
are severe but rare events, which
still
is
makes
highlight our procedure because
Similarly, a key issue
is
it
it
hard to estimate A(s) with any precision.
provides a good starting point for an actual
determining the components of the signal,
25
s.
Given
we use only the logarithm
the limited goal of this section,
implementation
also
it
of the price of copper.
would be worth including some global
U.S. high yield spread, the
EMBI+
VIX
or the
In a true
risk-financial indicator, such as the
(see Section 6 below),
and removing slow moving
8
trends from the price of copper.
With
we use
these caveats behind us,
us describe the reduced-form
Markov regime switching model
to estimate A(s). Unlike the standard regime-switching model, ours has time-varying tran-
sition probabilities.
GDP
let
data but
sudden stop
Our
left-hand side variable
does not
it
make any
at each point in time)
Y=
t
The growth
Mo
+
aggregate demand, Yt (which we will proxy with
is
difference for our purpose of identifying the likelihood of a
and
is
generated by the process:
+c*e,
!{::,=:i}Mi
e
~
N(0,l),n <
i
and variance
is 0,
the regime
Oq. Otherwise, the
is
"normal" and growth
regime
is
is
=
0|ai
= 0,X SiS£(M+1) = atp"/"
=
1
—
1
*(*">*•
)
Pr(z(+ i
=
0|z (
POO, PiO
=
=
1,
= poo
X^m+i)) = Pio
1 ~~
<?i
just a
assumed
is
Prfo+i
Pr(^t+i
1.
When
=
p.
+
p. 1
l\= t
= 0,Xs s6(M+1) =
)
GDP
l|zf
=
1,
X
for the regressors that enter the
data starting from 1976 to 1999 and monthly copper data
(source, International Financial Statistics).
s
s s e(t,t+i))
=
hazard function
Few
-*'
copper, only discrete points. However,
to
improve the
j
s
fit
unobservable, since
we can obtain copper data
exp
is
same period
but also
is
a key ingredient in designing
investors/ insurers are likely to be willing to hold long-term risk on a
t
ignore this issue and calculate the integral by a
'"There
for the
we
may
turn out to be non-stationary.
D
1 ")"*"
Strictly speaking the quantity exp
variable that
Q
10
Removing the slow moving trend not only seems
long run insurance and hedging contracts.
and
m
(the logarithm of the price of copper in this application). 9 For the purpose of estimating A(s),
use annual
/i
to be the following:
,
=
,
,
and Xt stands
the
normal variable with mean
"abnormal" and growth has a lower mean,
a variance a\. The transition matrix between the two states
where poi
and
rate of Yt depends on an unobserved regime z t that takes values
value of the regime
Pr(c (+1
0.
"
Rieman sum
v
'
a
at reasonable high frequencies so that
we can
of
safely
as
exp
^
a caveat here. Our extended-sample starts from 1972, but
26
we cannot observe the continuous path
all
the years up to 1976 are part of a deep
For estimation, we use a Bayesian approach that
we
Bayesian inference seems natural
have.
do not require asymptotic
justification.
we use a multimove Gibbs Sampler
in this context, since
we
fix
is
to
as described in
augment the parameter
set
Kim and
we
Nelson (1998,1999) and
based on
is
The
refer the reader to these references.
(/i
,
n1
,
ao,
<j\, q,
an, a\)
parameters we draw from the posterior distribution of the states
and conditional on these
in a single step as described in
Nelson (1999). Given the draw from the posterior distribution of the states we sample
means
the conditional
jj.
q
,
/^ using a conjugate normal / truncated normal prior which leads to
11
a normal / truncated normal posterior.
se
of interest (namely (ao,ai))
by treating the unobserved states as parameters. Then
a draw from the posterior distribution of
Kim and
allows exact statements that
it
To estimate the parameters
the filtering algorithm of Hamilton (1989). For details
basic idea
suitable for the very few datapoints that
is
Conditional on the states and the draw from
sample from the posterior distribution of
posteriors in the
same
(o"o, cr i)
class. Similarly, fixing the states
using inverse
gamma
and the draw from
q using a conjugate beta prior leading to conjugate beta posterior.
priors
(/i n
,
(/x
,
fi^)
which lead to
// 1; crn,
a\)
The sampling
we draw
of (an,a:i)
presents the difficulty that there does not seem to be a natural conjugate prior to use and thus
we use a Metropolis Hastings-Random
normal
we sample from
a bivariate
centered at the previous iterations' estimate as described in Robert and Casella (1999).
This provides us with a new draw form
we can
walk-accept-reject step where
iterate the algorithm
standard result
in
by
(fi
,
fj.^,
ao, a\, q, ao, a\)
filtering the states again,
and conditional on
based on the new draw
this
new draw
etc. It is
then a
Bayesian computation that the stationary distribution of the parameters sampled
with these procedure (treating the unobserved states as parameters too) coincides with the posterior
distribution of the parameters.
12
contraction due to political turmoil combined with extremely weak external conditions. However, since the extended-
sample starts during abnormal years these do not influence the estimation of X(s) (which
transitions from
normal
to
abnormal
states).
It is in this
is
estimated from the
sense that the sample relevant for the estimation of the
latter starts in 1976. For details see below.
"For
details see Albert
and Chib (1993).
'"To reduce computational time and satisfy the technical conditions required for the applicability of the Gibbs
Sampler, we used proper priors for
proper prior
for
(/j
,^j) was
{fJ.
Normal
/
,p-i)
and (qo.Qi) and improper
Truncated normal with means
priors for the rest of the parameters.
(0,0)
The
and a diagonal covariance matrix. The
standard deviations where chosen to be roughly 5 times the range of the sample, so that the priors had effectively no
influence on the estimation. Similarly for (qq,Qi)
we used independent normal
27
priors centered at
with a standard
A
first
state. The
output of this procedure
is
Figure 2 which plots the probability of being in an abnormal
model recognizes roughly three years out
and the recent episode following the Asian/Russian
a devastating debt-crisis, and was significantly
episode appears to be a
An
mix
of the 24 as
The
crises.
more
abnormal
years:
the early 1980s
early 1980s episode corresponds to
severe than the recent one. In fact, the recent
sudden stop and a precautionary recession.
of a milder
interesting observation about these probabilities
that they allow us to identify the abnor-
is
mal regimes with great confidence. To improve the tightness
of the posterior confidence intervals
concerning the parameters of interest (an,a:i) we observe that conditional on the states the (log-)
likelihood function becomes:
Yl
'
Xl lo g(Poo(ao, a\
))
+
(1
-
Xi) log(poi ("o, "i
))]
2,=0
where Xi takes the value
1 if
there
is
no transition
to state
words
all
our purpose of estimating
states. In other
words
if
and takes the value
other parameters of the model including the
(cvo,c*i)
we were
otherwise.
depends on the data only through the
interesting to notice that the likelihood for (arj,ai)
states. In other
1,
GDP-data
It is
filtered
are relevant for
only to the extent that they influence the identification of the
to condition directly on the states
we would be
able to get rid of
the noise introduced by filtering.
Given the few data points and the quite clear identification of the abnormal states we report
directly the posterior distribution of (an,a:i) conditional
abnormal
states,
we report
on the early 80's and 1999 being the
the posterior distributions of the parameters (ao,ari) conditional on
the states that are identified as abnormal. 13 Table
Correspondingly, in what follows
we use an
=
1
reports these results.
2.5,
a\
=
5.2 as our
benchmark values
for the
function A(s).
deviation of 10.
Even when we experimented with more
diffuse priors the algorithm
produced virtually identical results
but computational time was significantly increased, because convergence was significantly slower. Most importantly
though, for the results that we report
in
Table
1
and that we use
in
the calibration exercise
we used completely
flat
priors.
'
I.e.,
states that have a posterior probability of being
abnormal above
0.75.
The
conditioning
is
done
increase the precision of the estimates and seems to be warranted just by a visual inspection of Figure
28
2.
in
order to
Probability of
abnormal regime
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
1975
1980
Figure
1985
2:
2000
1995
1990
Probability of abnormal regime
Posterior Distributions
mean
Oil
Table
1:
Deviation
10
25
50
75
90
-2.522
1.067
-3.826
-3.027
-2.316
-1.789
-1.420
-5.170
3.387
-9.259
-6.802
-4.708
-3.040
-1.515
std.
Posterior Distributions of the Parameters (ao,ai) Conditional on the States that are
Identified as
Abnormal.
10,
25
etc. refer to
the respective quantiles.
29
Other parameters
5.1.2
We
calibrate £
and
T
to generate the average cumulative
stops in the sample, which
es
T
+ 0.35(e rr
is
consumption drop caused by the sudden
about 8-10% of GDP. One such configuration
is
T =
and
1
Q
=
-eff T ).
The parameters k and j
are calibrated to generate reasonable
amounts of steady-state debt
(and hence reasonable amounts of insurance needs) together with significant precautionary fluctu-
—but not — of the precautionary contraction
much
ations (see below, in particular, to explain
all
experienced by Chile at the early stages of the 1998-9 episode). For
The
interest rate r
is
set to 0.09 to
emerging market and the growth rate g
is
set to 0.03.
path that grows significantly faster than that
up,
and then decelerates below that
debt-to-GDP
ratio to 0.5
X =
The
latter
level for a
level forever.
(i.e.,
we
set
k
=
0.8
and 7
=
7.
capture the high cost of borrowing faced by the typical
to a
initial
this,
We
is
a constant-rate approximation
few years, while the country
normalize
initial
GDP
(j/q)
to
1,
is
catching
and
set the
-0.5).
FinaUy, we approximate the process for copper by a driftless Brownian motion with constant
volatility a,
which we estimate using monthly data from 1972 to 1999 (source, IFS).
value of roughly
a
=
0.23
and normalize the
initial
value of
We
find a
s to 0.
Aggregate Hedging
5.2
Given these parameters, we now turn to a quantitative evaluation of the effectiveness of the various
strategies described in Sections 3.2
5.2.1
and
4.
Precautionary Savings and Hedging
Let us illustrate the impact of the different hedging strategies in steps: In the
first
one we
illustrate
the gains from going from a pure precautionary savings strategy to one that incorporates diffusion
risk hedging. In the
We
8 years
start
second one we enrich the strategy further by adding jump
by drawing a random path
and no sudden stop takes place
for s t
in
it.
,
plotted in panel (a) of Figure
The main
3.
risk hedging.
The path runs
features of this path are not too different
from the realization of the price of copper during the 1990s. In particular, the large
the middle of the path followed by a sharp decline toward the end of the period
30
for
is
rise in s
near
reminiscent of
The path
the path of the price of copper in the second half of the 1990s.
corresponds to the empirical
The two lines in
mean
of
Sj
=
0.
which
in the data.
subplot (b) illustrate the corresponding paths of (excess) consumption generated
for the precautionary savings only case
The former
starts at so
(dashed
and the
line)
tracks the signal closely while the latter
diffusive risk hedging case (solid line).
manages
to avoid precautionary contractions.
Note, however, that both paths have positive slope, reflecting precautionary savings against the
jump
risk (both)
Subplot
risk (the case
shows the the paths of the currents account
(c)
scenarios. In
and precautionary contractions
both cases the current account
worsens the current account
deficit
Of
deficit shrinks.
with the signal but the point we are isolating
a tightening of credit conditions.
The
is
is
without diffusion hedging)
deficits
course, in our context
much
both
in
income does not change
the sensitivity of consumption to news signifying
analysis would not be changed in any substantive
deterministic process, in order to illustrate in the simplest possible
of exported commodities can have an effect above and
effectively "normalized" to
differ
"procyclical," in the sense that as the signal
by adding some correlation between such news and income shocks.
have
do not
.
beyond
We
way
way
chose to keep income a
that variations in the price
their direct
income
effect
(which we
for simplicity), as long as they are useful in predicting the arrival
of sudden stops.
Importantly, the procyclical behavior of the current account deficit does not reflect the
same
behavior of consumption in the two scenarios.
In the absence of diffusion risk hedging,
When
a point in case:
than increased
its
it is
consumption that does the adjustment.
Chile
is
the price of copper collapsed at the end of the 1990s, Chile reduced rather
current account deficit - exactly the opposite behavior of that of Australia, a
developed economy which was facing similar terms of trade shocks. 14 Moreover, the sharp decline
in the current account deficit
from 6 to about
3.5 percent of
GDP
toward the end of the sample,
suggests that about half of the current account adjustment observed in Chile during the 1998-9
crisis
can be accounted by optimal precautionary behavior in the absence of hedging. 10
While the current account does follow a similar path when
H See
1,1
The
Caballero,
rest of the
Cowan and Kearns
diffusive risk
is
hedged, the adjust-
(2005).
adjustment could be accounted
for
by excess adjustment (some have argued that the central bank
contracted monetary policy excessively during this episode) and by the partial sudden stop
spread tripled during this episode).
31
itself
(the "sovereign''
excess consumption
Path of state variable
0.24
1a.
°- 23
/
| 0.22
c
2 0.21
in
0.2
I
K
if
'
V
'
i
0.19
4
2
time -
6
(b)
Path of Debt/GDP
Path of current account/GDP
-0.02
- prec. savings
—
-0.03
-0.04
k
hedging
diff.
o
0.8
A
I
L_
1
-0.05
U
"Li
''
,'
'
ft
/f
M,
§0.6
''.
.-
-0.06
<D
Q
-0.07
2
2
3:
0.4
V
-0.08
Figure
Subplot
4
time -
(a) depicts
6
4
2
time -
(c)
and k
=
0.8.
The dashed
The
solid line depicts the path of excess
line depicts the
savings exclusively. Subplot
(d) plots the
its
yearly standard deviation) as a
when income
at time
is
consumption when the country uses
normalized to
diffusive hedging.
path of excess consumption when the country uses exclusively precautionary
(c)
plots the current account
path of the debt/gdp ratio
hedging and the dashed
6
(d)
the path of the state variable (normalized by
function of time. Subplot (b) depicts the path of excess consumption
1
-
,
n
line to
for the
rXt —
c[
+yl
for the
same two
two scenarios. Throughout the
precautionary savings.
32
scenarios. Subplot
solid line refers to diffusive
ment
variable
account
t
behavior of rXt
X
t
t
-
c*
+ y*
t
quantity remains practically unchanged across the two scenarios,
If this
is
see this note that the current
given by:
is
CA = rX
of c*
To
not consumption but the net asset position.
is
— c%
is
it
means that the
roughly similar across them. However, as we have seen already, the volatility
dramatically different across the two alternatives. This must
mean
that the net asset position
has to become more volatile as we move from the precautionary savings case to diffusive hedging.
Subplot (d) confirms this conjecture. In the absence of hedging, the country's net asset position Xt
is
smooth, while
debt to
is
GDP
it
fluctuates
when the country
is
hedging.
As
the state variable deteriorates, the
under hedging. Hence the increased
ratio improves
just a natural consequence of
what hedging
transfer resources between states of the world
is
meant
volatility of the net asset position
to achieve in the
where they are
less
place:
first
Namely, to
needed to states of the world
where they are needed the most. As we increase hedging, a substantial portion of consumption
volatility is "absorbed"
Let us
Section
now turn to
4,
stop. This
with
full
by the procyclical
variability of the net asset position.
the second step and add the possibility of jump risk hedging.
hedging
possibilities excess
means that the country
As we showed
consumption becomes constant prior to the sudden
insulates itself
from the sudden stop signal
—
it
uses
jump
hedging to reduce the decline in consumption at the sudden stop, and diffusive hedging to
fluctuations in the price of
jump
risk
in
hedging as the signal changes. The top
risk
offset
subplot in Figure
left
4 reproduces the signal path in the previous figure, while the top right subplot illustrates the paths
of the country's net foreign liabihties:
The
solid line
shows the path of debt/gdp which fluctuates
with the signal as the price of jump-risk insurance fluctuates. The important concept, however,
is
the dashed line which corresponds to the debt/gdp ratio the country wiU have
takes place.
That
and trending
it is, it
plots
To generate
risk
levels at the
and the diffusive-hedging only
smooth
hedging over and above diffusive hedging at
these panels
we ran 500 simulations
kept only those that yielded a sudden stop during that interval.
consumption
is
model would.
show the gains from jump
the time of the sudden stop.
of difference in
a sudden stop
includes the return from the jump- risk hedges. This dashed line
as the frictionless
The bottom
if
case.
The
The
time of the sudden stop
right panel does the
33
same
left
for
ten years, and
panel shows the histogram
between the
for the
full
hedging case
consumption drops
at
the time of the sudden stop. Both panels have the
common message
that the possibility of
jump
hedging can significantly improve the outcome during a sudden stop.
risk
Welfare calculations
5.2.2
We now summarize
the different gains by evaluating the value functions of a country under different
hedging scenarios. For
this, let
VH
and
to markets for diffusive risk only
V HH
and
denote the value function of a country that has access
and jump
to diffusive
risk, respectively.
As a benchmark,
V
denotes the value function of a country that only accumulates savings without accessing hedging
We now
markets of any kind.
compensate a country
How much
ask the question:
resources would be required to
hedging possibilities? Formally, when contrasting
for its lack of access to
with the diffusive risk hedging only scenario, we solve for the amount
VH
We
do the same
X ,y* =
V(s u
X
for other scenarios as well (with
V HH
instead of
(.s t
,
t )
t
t
W such that:
+ W, y*t )
V
(46)
,
and so on)
.
It will
be useful
to define:
W
(47)
r—g
1
Then, using
(17), (32), the definition of
w
and
<P
where
b solves the differential
=
(46) yields:
(1
+ w)P
equation (20) and d
on whether we want to allow jump
risk
is
(48)
determined as in sections 4.4 or 4.3 depending
hedging or not. Finally equation (48) leads to
u
from which we can determine
Figure 5 presents
only,
W by using
W (normalized by gdp), when the country
both diffusive and jump
risks
and when
(Clearly in the third case the welfare gain
loss
(47).
when the country
is
it is
is
allowed to hedge diffusive risks
only allowed to accumulate precautionary savings.
zero by construction) Finally
ignores predictability altogether, as
we described
we
also report the welfare
in section 3.3
and hence
adopts suboptimal precautionary savings. 16
b
To determine the constant hazard
A,
we ran a simulation
34
to determine the probability that the country enters
Debt/GDP
Path of state variable
at
SS
0.8
Q_
n
0.6
L
/
tl
^
-TV
ft/
I'
.Q
WW'
(1)
Q
0.4
r
0.2
Hedge
Pert.
Prec. Savings
4
6
time
Consumption Differences
-0.04
-0.02
Diff. in
Figure
The top
4:
normalized by
time.
The
its
left
at
SS
0.02
(excess) consumption at
subplot
Cons. Drop Differences
is
0.04
SS
identical to the top left plot of figure
yearly standard deviation.
The top
various lines capture the level of debt to
stop at time
t.
The bottom
left
0.01
Diff. in
It
bottom
left
at
0.04
SS
depicts the path of the signal
GDP
as a function of
assuming the country were to go into a sudden
from a simulation. Conditional on so
500 paths for a duration of 10 years. Keeping those paths where the country enters
years the
SS
0.03
consumption drop
right subplot depicts debt to
GDP
figure depicts results
3.
0.02
at
in to a
=
0,
we draw
SS within 10
subplot shows the differences in the level of consumption at the time of the sudden
stop between the perfect hedging scenario and the diffusive hedging scenario. Similarly, the bottom right
figure
shows the differences
in
consumption drop upon entering the sudden stop between the two scenarios.
35
The
figure
shows that
for a
range of values of
corresponding to the range of the (log) price
s
of copper in our sample, the gains from hedging are substantial.
cautionary savings typically results
hedging
To put
W between 8 and 10% of GDP,
in values of
used and are potentially up to
is
18% when both
this in perspective, notice that the
Using hedging in place of pre-
debt/gdp
GDP
ratio of the country
from 50%
to a
Another important observation about the
ratio for the country
becoming
strategies start
to
irrelevant.
and the expected time
"very far away"
.
to the next
become
essentially 0.
reassuring feature of Figure 5
equivalent to reducing the
is
state variable approaches
intuitive. If s t
—
»
is
it
By
infinity.
a similar reasoning, as
explicit insurance against
jump
risk.
from the lowest
the sudden stop hazard
not too costly
Sj
—
>
is
when the
good
This
is
line (with crosses) that ignoring
at
it
sudden stop
will
for
is
in
there
is
no
jump (sudden
many
cases.
at a constant rate
—
is
high as long as the country has the right
crisis:
neither precautionary savings only
handling the sudden stop. However the cost of a non-contingent precautionary
non-contingent resources
that
when
hedging and the variability of
—essentially accumulating precautionary savings
likelihood of a sudden stop
is
to perfect hedging, irrespective of the level
important since developing markets
savings strategy takes places during good times (high
into a
is
-co the sudden stop
that the welfare benefits of only hedging diffusive risks
sense of the unconditional (on the signal) likelihood of a
strategy
hedging
Hence, the sudden stop
substantially harder than for diffusive risk, which are readily available in
Finally, note
all
too expensive).
and covers more than half of the distance
is
±oo
oo then the hazard rate goes
of the state variable. This suggests that the benefits of hedging will accrue even
stop) risk
is
that most of the benefits of hedging accrue
sudden stop approaches
becomes imminent (and insurance against
significant
under consideration
Therefore any comparative benefits of one strategy over the other get so heavily
discounted, that they
A
is
As the
The reason
is
used.
is
number between 32% and 40%.
figure
for intermediate values of the state variable.
only "diffusive"
"jump'' and "diffusive" hedging
50%. This implies that using hedging instead of precautionary savings
debt to
if
when the low
risk of a
s),
when the country overaccumulates
sudden stop does not
the next 10 years conditional on so
produce exactly the same probability.
36
=
0.
We
justify
costly
it.
then determined the constant hazard rate A so
Equivalent variation
20
15
Q.
Q
&
10
c
o
CD
•f?f?"fM-" ? —
>
" "
" °
"
"""----
_CD
CD
>
Diff.
CT
LLI
Hedging
Perfect Hedging
-10
%3
Optimal Prec Savings
"%.
Suboptimal prec. savings
-15
12
-10
3
Standard deviations of the state variable
Figure
5:
Equivalent variation for a "typical" range of values of
St
(normalized by
its
yearly standard
deviation). For a precise definition of the notion of equivalent variation see the text.
37
Decomposing the Gains
5.2.3
Our model produces a
Let us not decompose the gains from different hedging strategies.
clear
demarcation of time into three stages: Before a sudden stop, during a sudden stop, and after a
sudden stop. The marginal value of a dollar
the phase that precedes
welfare gains
if it
and lowest
it
manages
in
is
somewhat lower
highest during the sudden stop,
in
the post-sudden stop phase. Hence, a strategy produces
to "shift" resources into the
sudden stop and
also into the
phase before
the sudden stop.
Table 2 reports the results of a simple simulation exercise:
a number of paths for s
t
for 10 years.
We
We
start with so
also simulate the arrival of
=
and simulate
0,
sudden stops by drawing
independent exponential variables and using formula (11) to determine the time of arrival of sudden
stops. In
about 67% of the simulated paths
then compute the behavior of
stop
c*
c£,
X
t
we observe a sudden stop within 10
for st
+ and the associated consumption drop upon entering the sudden stop. Results are reported
from worst to best: suboptimal precautionary savings,
optimal precautionary savings, diffusive hedging only, and perfect hedging.
initial (excess)
hedging.
consumption, which
The reason
is
The
for this is that perfect
columns that capture the most important
consumption
hedging does not require precautionary savings, while
results,
column shows that the drop
lower in the case of
diffusive
of
consumption
and jump
in all the cases
rises over
but the
is
jump
reflected in the last
the next three
risk hedging.
sudden stop
The reason why the
is
hedged
important: In
all
strategies; the
is
difference in drops
but the
full
risk
hedge case, the economy
is
when the country has not had
column of the
table,
relatively vulnerable
if
no correlation. With
hedging can also reduce
it
is
insurance
all
the sudden
the time to accumulate resources.
which shows the correlation between the time
takes for the sudden stop to arrive and the level of consumption at the sudden stop.
risk hedge, there is
significantly
time as precautionary savings accumulate. But this also means that
stop happens in the near future,
This
for the
at the time of the
greater than that of levels at the time of sudden stop
consumption
It is
however. The second column shows the level of
time of the sudden stop, with a clear advantage
at the
column shows
first
highest for perfect hedging and second highest for diffusive
that of diffusive hedging requires less precautionary savings than the other two.
case,
We
along the path, consumption immediately following the sudden
for the four strategies of the previous section
third
years.
the rest, this correlation
significantly.
38
is
it
With jump
positive, although diffusive
c*(0)
mean c*(r+)
mean Ac*
cov(c* + ,t)
Suboptimal Precautionary Savings
19.99
12.00
9.39
0.99
Optimal Precautionary Savings
19.83
12.04
8.37
0.98
Only Diffusive hedging
20.24
12.66
9.00
0.67
Diffusive+Jump hedging
20.64
13.02
7.60
Table
2: Initial
(excess)
consumption
c*(0), (excess)
consumption
at the time of the
sudden stop
c* +
,
standard deviation of the consumption path leading to the sudden stop (std(c*)) and instantaneous
consumption drop upon entering the sudden stop (Ac*)
to
we go
hedging scenarios.
Introducing risk premia
5.2.4
Up
for various
now we have assumed no
to the opposite extreme
to give
risk-premia on either diffusive or
and
ask:
how
large
would the
risk
jump
risk hedging. In this section
premia have to be
for the
country
up hedging?
The framework we developed
equation (45) shows that the
in Section 6 is well suited to
"critical" risk
answer
this question. In particular,
premium that would render
diffusive
hedging pointless
is:
fa
and equation
hedging
(44)
shows that the
critical
d
overestimation of the hazard (risk premium) for "jump"
is:
K
™
Since d
~1°~2"s
=
= b if p =
with the function
1.
(rd) 1
(no diffusive risk hedging) and 1
b
from Section
=
(no
jump
risk
hedging) we can replace d
3.
Figure 6 reports the results when one computes the "critical risk premiums" fa an d fa as
implied by the above formula, for the range of (log) price of copper observed in our sample.
top panel shows that the critical risk
shows that the
critical
critical risk
premium
premium
for
for diffusive
jump
risk
8%
is
very large
is
about 8%. The bottom panel
hedging typically exceeds 100%. These large
premia confirm the benefits of hedging. For
have up-and-running markets,
hedging
diffusive risks,
which
in
many
cases already
when compared with reasonable premia
39
The
in liquid
critical risk
premium -
diff
hedging
0.08
i 0.06
To
0.04
12
-10
3
Standard deviations of the state variable
critical risk premium - jump hedging
12
-10
3
Standard deviations of the state variable
Figure
6:
Critical Risk
the critical risk
altogether.
Premia
premium
The bottom
<£j
for diffusive
= — -ycr
line depicts
-f
2
hedging and jump hedging.
which would make
1, i.e.
{rdp
it
plot depicts
optimal to avoid diffusive hedging
the percentage by which the true hazard
rate would have to be "over-estimated" by insurers in order to
1(1
The top
make jump hedging
too expensive.
markets. 1
For
'
jump
risk hedging, a
premium above 100%
reflects the large potential gains of
developing such markets.
Sudden
6
stops:
Time
variation in hazard rates
and traded
vari-
ables
In this section
this,
we argue that Chile
we document two important
is
not an exception in terms of the usefulness of our analysis. For
ingredients for our framework: that there
of predictability in sudden stops, and that this predictability
and tradable
indices which are largely
(hence moral hazard
is
beyond the control
is
is
a significant element
correlated with readily available
of individual emerging
market economies
Moreover, we conduct this analysis in a simple reduced
of limited concern).
form framework, so that our conclusions here are not intertwined with our
specific
methodology
in
the rest of the paper.
6.1
Time
We
identify the
Variation and Predictability of the Likelihood of Sudden Stops
sudden stop hazard using the data and methodology from Calvo, Izquierdo and
Mejia (2004). They identify sudden stops based on a number of cutoffs and indicators, which they
then use to run (probit) regressions against country specific variables (dollarization, openness
and time
effects.
etc.)
Their data are yearly and span the period 1990-2001 for 32 countries, of which
15 are emerging markets (Argentina, Brazil, Chile, Colombia, Czech Republic, Ecuador, Indonesia,
Mexico, Nigeria, Peru, Philippines, South Africa, South Korea, Thailand and Turkey).
exclusively
We
focus
on the 15 emerging markets.
We essentially rephcate the
Calvo
et al steps
y lt
We use the most significant
by running
= Pt(SS =
1)
=
logit
of the form:
F{x'it P)
variables reported in their paper,
openness, and terms of trade, as well as time fixed
and probit regressions
effects.
namely the degree
of dollarization,
After running the regressions,
we
isolate
the predicted probabilities F{x' ft) and the actual regressor values x' (3.
''For example, the corresponding Sharpe ratio
of the
US
stock market (see,
e.g.
is
0.35 which
Campbell and Cochrane
is
(1999)).
41
nearly twice the Sharpe ratio for the long sample
Assuming that the conditional
probabilities F(x'fi) capture well the variation in the likelihood of
running into a sudden stop, we investigate whether there
Columns
rates.
and
1
2 of
is
significant time variation in the
Table 3 show that indeed the bulk of the variation
is
hazard
in the within (across
time) dimension as opposed to the between (across country) variation. Also, the standard deviation
of 0.24 indicates that the conditional probabilities of sudden stops vary substantially across time.
Time
If
there
variability in the conditional probabilities
is little
is
however not synonymous with predictability.
or no serial correlation in the conditional probabilities, then there
from time varying precautionary accumulation of reserves, hedging
effective
if
there
is
some
etc.
is little
benefit
All these measures are
persistence in the process that generates the conditional probabilities.
The
next four rows in Table 3 report results from a dynamic panel data regression of the form:
Vit
Rows
2
and 3 are simple
known
persistence
biases of
and hence
Ot x
+ fitfit-l + P2Vtt-2 + Eit
fixed effects regressions with robust standard errors.
report the equivalent results
the well
=
when one
uses the estimator of Arellano and
dynamic panel data models. Uniformly, the
The next two rows
Bond
(1999), to avoid
results indicate substantial
predictability of the conditional probabilities.
Traded Variables and the variation of the conditional likelihood of Sudden
6.2
Stops
Sofar
we have only examined whether there appears
to be
enough variation and persistence
conditional probability of sudden stops to justify time varying precautionary measures.
question
is
whether traded instruments can be used to hedge away these
this question is to investigate
whether there
risks.
exist traded instruments that
A
One way
in the
separate
to answer
can "predict" the time
variation in the conditional probabilities. Table 4 addresses this question.
We
use
HY -i
t
to denote the difference between
Aaa and Baa bonds
(high yield spread) as
reported in the historical series H.15 of the Federal Reserve. Columns (1) and (2) use the predicted
values
x'/3
and F(x'/3) obtained using a probit model similar to Calvo
regressions:
42
et al.
(2004) to run the
Between /Within
Logit
Probit
Logit
Probit
(1)
(2)
(3)
(4)
0.06/0.24
0.07/0.23
0.51
0.53
(5.32)**
(5.45)**
-0.48
-0.48
(-5.89)**
(-5.89)**
0.46
0.46
(4.64)**
(4.65)**
-0.67
-0.68
(-5.12)**
(-5.30)* *
0.37
0.37
Prob (Lagl)
Prob (Lag2)
Prob (Lagl) Arellano -Bond
Prob (Lag2) Arellano
-
Bond
R-squared
Table
3:
Columns
(1)
and
(2):
Ratio of variation (reported as ratio of standard deviations) attribut-
able to the within (accross time) dimension as opposed to the between (accross panel) dimension.
Columns
(3)
and
(4):
Regressions of predicted probabilities on their lagged values using regular
regession and the Arellano-Bond estimator.
43
(1)
(2)
(3)
(4)
D.regressors
D.lprob
D.regressors
D.lprob
-16.90
-83.30
-71.62
-70.20
(-4.54)**
(-4.30)**
(-4.53)**
(-4.40)**
Observations
135
135
135
135
R-squared
0.20
0.17
0.19
0.19
L.HY
Table
Column
4:
Regressions of
(1):
(2004) on the lagged difference between
(1)
differences in the regressors x'it /3 used in Calvo et
first
Baa and Aaa bonds (L.HY). Column
with the sole exception that we use differences in the
in place of x'it 0.
For
Columns
regressions
all
(3)-(4):
Same
as
columns
A
The
Calvo
first
et.
al.
regression
is
is
column
predicted probability log. F(x'lt j3)
model.
and con-
included but not reported.
a + aiHYt-i +
a + a
A\og(F(x'J))
as
errors allowing for heteroskedasticity
constant
A(4£)
Same
(l)-(2) using a logit instead of a probit
we use panel corrected standard
temporaneous correlation across panels.
(log)
(2)
al.
x
e lt
HY -\ + s
t
lt
a projection of the change in the regressors entering the probit model of
(2004) on the high yield spread at
exercise for the predicted probabilities.
If,
t
—
The second
1.
regression performs a similar
for instance, increases in the
high yield spread coincide
with periods of high risk and risk aversion, then emerging markets might be exposed to contagion
risk.
Hence, in a manner similar to copper
for Chile, variations in
the
HY
can be used to form a
hedging strategy.
As Table
4 shows, the
R-squared of these regressions
is
close to
20%. Hence, 20% of the inno-
vations in the conditional probabilities are predicted by a variable that belongs to the information
set of agents at
time
t
—
1
and has a liquid market. To put
this in perspective, in
predictable variation in the conditional probabilities was about
Comparing the two numbers
37%
Table 3 the total
of their variation in the sample.
reveals a substantial co-variation between the predicted hazard
and
HYt-L
Note that the
coefficient
has the expected sign since
44
HY
is
mean
reverting, thus a high degree
of
HY
today predicts a lower
more
this point
AHY
and
t
,
HY
tomorrow and hence a lower likelihood of a sudden
suppose that there
clearly,
positive
is
stop.
To
contemporaneous correlation between Ax'lt /3
suppose that:
i.e.
E(Axit P) = a + a!E(AHYt ) with 5i >
Now assume
see
that
HY
is
t
(49)
a simple AR(1) process:
HY = pHY -i + Ev,t
t
t
<
with p
1
so that:
AHY = HY
t
and
t
- HYt-! = {p- l)HYt -i + e v
,t
also:
E{AHY = {p-l)HY .
t )
t
(50)
l
Plugging (50) into (49) shows that:
E(Ax'J) = a + a 1 (p-l)HY _ 1
t
Hence the regressions
in
The main reason why
common
Table 4 produce a negative sign on lagged
HY
HY performs well in terms of predictive power
variation of the arrival of
sudden stops across countries. To
since a\(p
is
that
it
—
1)
<
0.
captures well the
illustrate this,
we computed
the average yearly difference in the probability of a sudden stop as implied by the probit model.
Then we
plotted the (negative of) the high yield (lagged once) against
it.
Figure 7 illustrates the
strong comovement between the two series.
7
Final
In this paper
Remarks
we characterized
several aspects of
sudden
the corresponding aggregate hedging strategies. For
analytical light on
We showed
gate
That
demand
is
some
of the key issues but realistic
that even after removing
in
all
we
built a
model simple enough
and
to shed
enough to provide some quantitative guidance.
other sources of uncertainty, the optimal path of aggre-
an unhedged emerging market economy exhibits large precautionary contractions.
contractions that precede sudden stops.
remove much
this,
stops, precautionary contractions,
We then argued that smooth diffusive hedging could
of these fluctuations even in the absence of
45
more complex jump hedging mechanisms.
CO
f
r
0)
c
I
1992
1994
1998
1996
2000
2002
year
neg_L_hy
Figure
7:
The
(D_probs)
D_probs2
joint variation in the average yearly difference in the probability of a
as implied
by a probit model and the (negative
(neg_l_hy).
46
of) the high yield
sudden stop
spread (lagged once)
The
gains from these hedging strategies are large, which raises the question of
countries already doing this in a significant manner.
problem
is
Our somewhat informed
belief
why
is
aren't
that the
mostly one of coordination, which could be reduced significantly with the involvement
of the international financial institutions (IFIs).
On
liability
one end, hedging along the
management: Any
lines
we have suggested
sensible policymaker
is
not standard practice for external
knows that the
political
something going wrong while implementing an unorthodox strategy
of failing while following the standard recipe.
The
is
many
(and personal) cost of
times larger than that
IFIs can help by modernizing their advice on
standard practices.
On
the other, while one can imagine
many
indicators that could be used to start improving
hedging strategies immediately, such as the price of commodities, the high yield spread (HY), the
volatility
index (VIX), indicators of
US and
other developed regions activity, and so on, some of
these markets are not nearly as large as they would need to be (recaU that the issue
hedge the income
lager,
effect of
and probably not
commodity
price fluctuations but the
sudden
just as a multiplier but also as a bellwether).
stop,
which
is
is
not to
significantly
Market creation requires
coordination, and the development of a few general indices that can ensure adequate liquidity.
Again, the IFIs can help develop these markets. For example, the
credit lines to index
them
to indicators that could eventually
traded in open markets.
47
IMF
could modify
its
contingent
be adopted by the private sector and
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Control Optim., 39(1), 306-329.
KlM, C.
and
J.,
and C. R. Nelson
tests of duration
(1998):
"Business cycle turning points, a
new
coincident index,
dependence based on a dynamic factor model with regime switching," Rev.
Econ. Statist, 80(1), 188-201.
49
Kim, C.-J., and C. R. Nelson
State-space models with regime switching Classical and
(1999):
Gibbs- sampling approaches with applications.
KRUGMAN,
P. (1988): "Financing
vs.
MIT
Press,
Cambridge
MA
and London.
Forgiving a Debt Overhang," Journal of Development Eco-
nomics, 29, 253-68.
ROBERT, C.
P.,
and G. CASELLA
(1999):
Monte Carlo
statistical methods. Springer- Verlag,
New
York.
TlROLE,
J.
(2002):
"Inefficient Foreign
Borrowing:
A Dual-and-Common
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Mimeo, Toulouse.
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(1996):
Nonlinear differential equations and dynamical systems, Universitext.
Springer- Verlag, Berlin, second edn.
50
—
A
Appendix
A.l
Proofs for section 2.2
Proof. (Lemma
1)
we made the
Since
simplifying assumption that the country suffers only one
maximization problem post sudden stop
financial crisis, the
*l
POO
V + (XT+T ,y*T+T )=max
=t
J
Jt+T
(t
>
r
+ T)
simply:
is
i
^—e^-^+^dt
-
(51)
7
1
S.t.
POO
/CO
c*
t
/
e-^-( T+T »
dt
= XT+T +
Jt+T
This problem has the
^e-^-( T+T »
/
dt
(52)
Jt+T
trivial solution:
q*
= rW£
t
+T <
t
<
oo
(53)
with
W| = XT+T + ^±Z.
(54)
This constant can be interpreted as the "excess" wealth at date r
of that which
(r
<
t
<
t
is
needed to cover the reservation-consumption
+ T) we
can use the continuation value function,
+T
level
(that
ny
t
is,
the wealth in excess
During the sudden stop
).
V(Xt+ t, y* + x),
from above, to write
the maximization problem for the sudden stop phase:
ySS (X
rr+T „*l-7
T
,
y* )
r
= max
f
±JL e -r(t-r) dt + e -rTy+ {x
.j
s.t.
rr+T
T
f
4e~ r{t
Xt+t >
- T)
X T+T
i-t+T
XT +
dt=
r
T
rT
y;e- ^- Ut-e-
f
XT+T
(55)
-
where
y?(i
X T+T = C
by
(5).
Since the country
constraint allows,
it
is
I
A' T
+
r
-
growing and wishes to expand
g
its
consumption
at a faster rate
can be easily shown 16 that the sudden stop constraint,
XT+T =
'For details see Caballero and Panageas (2003)
51
Xr+T-
(55), is
than the
always binding
(56)
Given
problem
this result, the optimization
problem subject to a
final
wealth condition.
'
7
4=
1
_
e -rT
a deterministic consumption
is
straightforward.
A
few steps of algebra show that:
WU
It is
T<t<T + T
(57)
with
W — XT H
1
It is
crisis in
A-t+t-
g
X t+ t
easy to see from these expressions that as
consumption during the
e
-
r
rises,
cjT
falls.
The country has
order to satisfy a tighter sudden-stop constraint.
to cut back
We are now ready
to determine the sudden-stop value function:
V ss (XT ,y*T
)
=
^~\ -^-r)
r T {rW1-7
dt
+
e
1
(rWf) -'
1— 7
It is
(I-
e~' T \
\
r
r
-rT
Jt+t
Jt
+
1
J
^
(rWj) 1
(rWtf-> _ r{t _ {T+T))
e
1 - 7
dt
/e^ T \
—7
easy to verify that our careful choice of the constraint pays
off at this stage.
The
value function
simplifies to:
V ss {XT ,y'T
K
)
KV
where
V(XT ,y*)
rJ
+ (X ,y*
T
T)
the fact that the value function (which
A. 2
(60)
denotes the value function in the absence of a sudden-stop constraint and
the constant given in the statement of the proposition.
in the
—7
1
absence of the constraint (55)
A
is
negative
if
7 >
K
1)
>
1 is
K
is
a straightforward consequence of
has to be lower than the value function
simple differentiation shows that
Kq >
as long as 7
>
1.
Proofs for section 3.1
Proof. (Lemma
2)
We
give a brief sketch of the proof. Differentiate the
respect to b to obtain:
-r + (l-7)^( = )'<0
52
left
hand
side of (18) with
=
=
since 7
>
than
This establishes existence and uniqueness of the solution to (18) by the intermediate value
0.
1.
Moreover, at
b
£ the left
hand
side of (18)
greater than
is
while at
b
b
less
it is
Theorem.
Propositions and Proofs for section 3.2
A. 3
Proposition 5
(15) and
is
If there exists a function b(s) solving (20) subject to (21)
and (22) then (17) solves
the value function of the problem as long as:
1-7"
lim
t
Proof. Substituting
The reason why
b(st)
—>oo
E
f
satisfied
The above
(61)
1-7
simplifications leads to (20).
6).
Since (17) satisfies (15) for b(st) solving (20), 'a classical verification
it
must be the value fimction as long
as the "tail" condition
(Fleming and Soner (1993) p.172).
proposition
to establish that (17)
to b are given
—
and performing some obvious
(17) into (15)
theorem can be invoked to show that
is
t}6(s) 7
has to satisfy the boundary conditions (21) and (22) follows from the next
proposition (proposition
(61)
>
l{r
by
(21)
is
is
a "place-holder" for
the properties that need to be verified in order
all
the value function of the problem.
and
We start
by showing that the boundaries
(22).
Proposition 6 The value function of
the problem with sudden stops for s
>
s satisfies:
1-7
M
i
Proof.
We
focus on the 7
(62)
5J
V(XT ,s,y*T )=K[
>
r-g
'
\
(
lim
V
1 case.
1
|
(xT + ^-
1-7
rJ
1
The proof proceeds
—7
in
two
steps.
upper bound on the value function of the problem with sudden stops, which
given by the value function of the problem in the absence of sudden stops:
53
First one obtains
in this case
is
an
naturally
It
tuns out that this expression
The lower bound
sudden stops
<
,
sudden stops (since
<
t
r
satisfies
it
also a lower
bound
to the value function
when
is still
goes to
St
upon observing that the consumption policy
established
is
c*
is
infinity.
the absence of
in
a feasible consumption /portfolio plan in the presence of
>
the intertemporal budget constraint and also Xt
—^-
for all
t
).
Accordingly we have the two inequalities:
rr
v(xu s
t
,
(r
L
-r(u-t)
y;)>Et
\
*NSS\ l -1
)
-du
V
s as
s
—
>
+ e-^^hir
1-7
I
However by standard arguments
goes to
7
U
oo
.
To
it is
lim
— OQ
E
e
see that, note that Pr(r
-r{u-t)\Pu
)
+ e -r(r-t) K1
r(u-t)
r—a
<.oc}A"
< Q) —
1-7
for all finite
>
\
Cu
)
,
T < oo }
VL
1-7
i
f
-\F
t
"
Vr
-
1
V»^{X u y*
du\F
1-7
Q, and thus
X NSS +
/,
du
1-7
\l-7"
...
X.
easy to show that the right hand side of the above equation
(*NSS\*NSS\ l -1
S
t-„
BO
NSS
t
t )
1-7
1-7
Since the upper and the lower
similar claim can be used to
Showing formally that
(21)
and
(22)
is
a
bound coincide
show the second
to
as s
—
»
claim
oo, the
is
established.
A
result in the proposition.
b(s t ) possesses a unique solution subject to the
hard mathematical task that
fairly
V NSS
is
well
beyond the scope of the present paper.
how one would proceed
The next Proposition provides
a sketch of
substantive results of the paper,
we were always
two boundary conditions
to obtain a proof.
For the
able to obtain numerically a solution to this
ODE
at arbitrary precision levels.
Proposition 7 There
Proof.
exists a solution to the
(Sketch) First
we compute the steady
(20) subject to (21)
we transform the second order
states at s
= +oo
eigenvalues. Using the stable manifold
establishes the result. For
ODE
more
and
s
into a
= — oo,
linearize the
=
twice (as s
theorem around
details see Caballero
54
ODE
s
and
(22).
Then
system of ODE's.
system and compute
— +oo and
»
as s
and Panageas (2003) Proposition
3.
its
— — oo)
>
The
(61)
last step in order to verify that (17) is
indeed the Value function
is
To show
to verify (61).
we need two Lemmas:
Lemma
4 Consider a consumption policy that has the feedback form:
c\
Then
X
t
=
X^ ss
>
(x + -l£- J
A(s t )
,
t
where Xj^ ss
is
K~^r
,
result
One
is trivial.
Vs t G (-co, +oo)
when one
=
uses the optimal
stops:
r
(
x nss + _Vt_
T
r-,
- 9,
\
Proof. This
<r
the asset process that results
consumption policy in the absence of sudden
c?ss
< A(st )
solves for the asset process that results
from the two
policies
to find that
X NSS = X
t
The
equivalent calculation for the optimal policy
d(e~( rt ~foM^)du) x
= _^ _ A
\
=
Now
integrating both sides
e
*°
and rearranging
result
Lemma
5
Proof.
to the
this
local
now
b(s) is
means that
if
if
r-A(s )-g
t
e
-(rt-f< A(s u )du)
dXt
gt
gives:
( [\-((r- 3 )*-£Ms»)du) {r
_
A{Si)
_ g)
A
=
J
<
Vs t 6
r
bounded between
(
— oo, +co).
-K
1
'
1
and
-.
by means of a contradiction. Suppose
(21), (22).
there exists a b
max. Similarly
,
^+
- e (rt-ftAMdu)\
will derive the result
boundary conditions
-( rt -f< A(s u )du)
\
follows since A(st)
The function
We
(e st
e
St gives:
r-g
t
V
XQ - rg
^
1)
conditional on a path of
ct
-{rt-fZMsv)du)
X = X + AeO-t-Jo'^H
The
-^^
(e* r-g
Since b(s) asymptotes to £ as s
>b then
there exists a b
<
—
»
b(s) solves (20) subject
oo and to 6 as
there must be a s* such that b(s*)
- then there
55
must be a
s*
>b and
such that b(s*)
<
^
s
— -co
>
b(s*)
is
is
a
a local
Both
min.
We
are impossible though.
similarly. Indeed,
establish the
suppose that there exists one point
such that b s (s*)
s*
The second
contradiction.
first
=
and
is
obtained
>
b(s*)
b.
Then
(20) becomes:
—
1
rb(s
A(s
H
)
-1
)
-bss-^cr
6(5*)
which
in
turn implies that
Hence every extremum
Similarly
local
min
if
b(s*)
-,
since the left
>
in the area b(s*)
b
hand
must be a
side of the
above equation
minimum, and cannot be
local
a symmetric argument can be used, to show that
is
a
is
negative.
maximum.
impossible to have a
in this region.
Now, we are
Lemma
<
>
b ss (s*)
in a position to establish the "tail condition" (61):
6 Assume (±)
7
t
Proof. For any
<
6(s t
lim
E
—»oo
t it is
p <
K (A)
>
'i{t
7
Then
Vsj e (-oo, +oo).
t}b( St y
(63)
1-7
the case that
(
e~
/
e
- rt
rt
l{T >t}b(s t
e~
J-
l{T>t}K
ri
y
•
N
1
—
>
'
>
1-7
(x?
A opt +
-
\l-7
vL
>
1-7
/
-,
.
KE
Y ^
yNSS
\{T>t}
1-1
i
V't
1-7
1-7
The
number.
first
inequality follows from the assumption
The second from
the previous
and the
lemma and
fact that
'
-^
1
is
a negative
the monotonicity of the value function and
the last limit follows from the fact that one can trivially show that in the standard model without
sudden stops
e"
-- 0.
1-1
To
establish the result observe also that
1-7
e-
irrespective of
t,
since
7 >
1.
rt
^t
l{r
Hence,
it
>
t}b(s t
y
"+"
r-a
1-7
must be the case that
56
<0
(63) holds.
A.4
Proofs for section 4
For
the value functions in this section one can apply similar verification arguments to the ones
all
we
given in the previous section. Hence,
on the substantive proofs of the Lemmas and
just focus
the Propositions in the text.
Proof. (Lemma
The proof
3)
is
a straightforward application of Ito's
Lemma
in conjunction
with (27)
dc*
^C
c
=
Cv
—dX
c
Cs
P?0""Cvv
Cv*
*
+ —dy? + ^- —dt + —ds +
,
,
t
\
-A 1
d
n /
,
A(s t )(l
O C
oCXs
——dt
+ p a —dt
2 c
c
ss
,
ds
-
2
,
o~
2
+ <&,>-- -f/z+-
some obvious
follows directly after
Proof. (Proposition
1).
=
For fa
dc* _
<£ 2
1
1
c*
Notice also that for fa
=
fa
=
d
"
=
1
0,
=
cancellations
I
equation (37) becomes:
— rd +
a"
\(st)id
+ nd + —d„
dt
s
equation (36) becomes:
- rd +
fid s
+
°— d
ss
+
X(s t )
I
f
—1
-d
J
so that:
dc*
c*
Note
finally that
i
is
,
t
c
c
dBt
7 a
The claim now
,
2
c
_
X(s t )
K'-
d
'
-d
given by (34) as
fi
K \h
+
so that
A=
1<lh
A
d
id
and therefore
dc*
X(s t )
{id
c*
57
-
id} dt
=
dt
/
^^l
jff2
,
4\
ds
d J d
dt
Proof. (Proposition
2) Similar to Propositions 3
Proof. (Proposition
3)
=
Suppose fa
0,
Then the Bellman equation
(see below)
and that
rd
K
and 4
h =
T7r
l
,
(l
+ ^2
._i
-'
)
(36) becomes:
=
and the consumption evolution
(37)
1
- rd +
1
— rd + d s
jj.d s
+
—d
+
—d
ss
+
is:
dc*
f.i
dt
s
=
c*
dt
7
-X(s t )
Proof. (Proposition
4)
1
1
- rd +
-(1-7)
fxd s
-X(s t )d'
then follows upon using
Lemma
+
—d
O 2 (dsf
2
result
\dt
rd
Using (44) and (45) the Bellman equation (36) becomes:
(I
The
=
rd
I
3.
58
d
rd
ss
+
1*28**
23
2 4 ?nnc
JAR I
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