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Massachusetts Institute of Technology
Department of Economics
Working Paper Series
HEDGING SUDDEN STOPS &
PRECAUTIONARY RECESSIONS:
A QUANTITATIVE FRAMEWORK
Ricardo
J.
Caballero
Stavros Panageas
Working Paper 03-1
May 26, 2003
Room
E52-251
50 Memorial Drive
Cambridge,
This
MA 02142
paper can be downloaded without charge from the
Social Science Research Network Paper Collection at
http://ssrn.com/abstract^XXXXXXX
JUN
5 2003
jJBRARJES
Hedging Sudden Stops and Precautionary Recessions:
A
Quantitative Framework
nicardo
MIT
J.
and
Sta\TOS Panageas
Caballero
NBER
MIT
May
-'
Even
which
is
Abstract
managed emerging market economies
well
financial.
are exposed to significant external risk, the bulk of
At a moment's notice, these economies
that have supported the preceding boom. Even
often leads to costly precautionary measures
cycle of an
26, 2003*
economy that on average needs
to
if
and
may be
required to reverse the capital inflows
such a reversal does not take place,
recessions. In this paper,
borrow but
we
its
anticipation
characterize the business
faces stochastic financial constraints.
We
focus
on the optimal financial policy of such an economy under different imperfections and degrees of crowding
hedging opportunities.
out
in its
and
realistic
is
simple enough to be analytically tractable but flexible
enough to provide quantitative guidance.
JEL Codes:
Keywords:
E2, E3, F3, F4, GO, CI.
Capital flows, sudden stops, financial constraints, recessions, hedging, insurance, signals,
contingent credit
*We
The model
lines,
Eisymmetric information.
are grateful to Victor Chcniozliukov, Arvind Krishnamurthy and the participants at the macroeconomics and econo-
metric lunches at
for financial
MIT
support.
for their
comments, and Andrei Levchenko
for excellent research assistance.
Caballero thanks the
NSF
Introduction
1
Most emerging economies need
tunately, even well
moment's
borrow from abroad
to
as they catch
managed emerging economies are subject
up with the developed world. Unfor-
to the "sudden stop" of capital inflows. At a
and with only limited consideration of initial external debt and conditions, these economies
notice,
rnay be required to reverse the capital inflows that supported the preceding boom.
The deep
contractions triggered by this sudden tightening of external financial constraints have great costs
for these economies.
Not
surprisingly, local policymakers struggle to prevent such crises. During the cycle,
the anti-crisis mechanism entails deep "precautionary recessions:" tight monetary and
the
first
sign of
symptoms
of a potential external crunch.
build up of large stabihzation funds
and international
Over the medium
reserves,
fiscal
contractions at
run, the tools of choice are the
and taxation of capital
inflows. All of tliese
precautionary mechanisms are extremely costly.
The
case of Chile illustrates the point well.
of copper, its
investors,
and
Chile's business cycle
main export good: so much so that
this price
is
highly correlated with the price
has become a signal to foreign and domestic
pohcymakers ahke, of aggregate Chilean conditions. As a
to
external reactions to this signal, the decline in Chilean economic activity
sharply
is
many
result of the
when the
and
This contrasts
a developed economy not exposed to the possibility of a sudden stop, where
in Australia,
similar terrns-of-trade shocks are fully absorbed by the current account, with almost
no impact on domestic
and consumption.'
In this paper
we do not attempt
so differently, or even
Instead,
internal
price of copper falls
times larger than the annuity v'alue of the income effect of the price dechne.
with the scenario
activity
many
we
why
to explain
why
international financial markets treat Chile
and Australia
the signal for the sudden stops should be so correlated with the price of copper.
take the "sudden stops" feature as a description of the environment and characterize the optimal
hedging strategies under different assumptions about imperfections
the precautionary business cycle that arises
The main
technical contribution of the paper
analytical characterization, but
model has two
when hedging
is
also flexible
central features: First
is
and
is
in
hedging markets.
We
also characterize
opportunities are very limited.^
a model that
reahstic
enough
the sudden stop, which
we
is
stjdized
enough
to allow extensive
to generate quantitative guidance.
The
characterize as a probabilistic event that,
once triggered, requires the countrj' to reduce the pace of external borrowing
significantly.
Second
is
a signal.
'See
,
-We
focus on the aggregate financial problem vis-a-vis the rest of the world, in an environment where domestic policy
e.g.,
Caballero (2001).
managed optimally and
compound
decentralization
is
we
deal with in this paper by exacerbating the country's exposure to sudden stops. See,
Such
failures
e.g.,
Caballero and Krishnamurthy (2001, 2003) and Tirole (2002) for articles dealing with decentralization problems.
the problems
literature on government's excesses
is
is
not a source of problems. Needless to say, these assumptions seldom hold in practice.
very extensive. See,
e.g.,
Burnside
et al
(2003) for a recent incarnation.
The
Sudden stops have some element of predictability
there
is
a perfect signal
is
We start
by studying a simple environment where
the high yield spread, or the price of an important commodity
(e.g.,
that triggers a sudden stop once
where the threshold
to them.
it
crosses a well defined threshold.
We
for
the country)
then study the more reahstic case
may
blurred and a sudden stop, while increasingly likely as the signal deteriorates,
occur at any time.
Within
that
is,
in
this model,
we develop two
substantive themes. First
we
characterize precautionary recessions:
the absence of perfect hedging, the business cycle of the economy follows the signal even
sudden stop actually takes place. This
is
because as the likelihood of a sudden stop
Consumption
into a precautionary recession.
is
if
no
the country goes
rises,
cut to reduce the extent of the adjustment required
if
a
sudden stop does take place.
The second and main theme
reduces the extent of the
assets
and
for incurring
is
crisis in
aggregate hedging strategies.
An
adequate hedging strategy not only
the case of a sudden stop, but also reduces the need for hoarding scarce
sharp precautionary recessions as the signal deteriorates.
We
study two polar-
opposite types of generic hedging instruments or strategies, as well as their intermediate cases. At one end,
the hedging contract relaxes the sudden stop constraint one-for-one. That
to relax the constraint.
The
other extreme
is
each dollar of hedge can be used
if
the resources obtained from the
motivated by crowding out:
hedge facihtate the withdrawal of other lenders, then the main
effect of
debt in bad states of the world but not to provide fresh resources.
excellent substitute for precautionary recessions, but
The paper concludes with an
is,
it
illustration of our
By
the hedge
its
to reduce the country's
is
nature, this type of hedging
cannot remove the sudden stop
main
is
an
entirely.
We
results for the case of Chile.
estimate the
probability of a sudden stop as a function of the price of copper and calibrate the parameters needed to
obtain sharp consumption drops such as those experienced by Chile in the recent contraction of 1998/99.
We
then describe different hedging strategies and their impact on the volatility and
We
discuss credit hnes
problems. For example,
and
its
copper.
and
their indexation to the signal as
we argue that
le\'els
of consumption.
one way of reducing asymmetric information
Chile could virtually eliminate sudden stops, precautionary recessions,
large accumulation of precautionary assets, with a credit line that rises nonlinearly with the price of
The
cost of this line,
if
fairly priced,
should be around 1-2 percent of GDP. This
compared with the costs of the precautionary measures currently undertaken, including
is
very
little
when
large accumulation
of reserves, limited short-term borrowing, and precautionary recessions.^
Currentl)',
although futures markets exist
ations, the size of the financial
problem
is
for
much
much
of the income-flow effects of
larger than that.
The markets
commodity price
fluctu-
required for these strategies
'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
stand to gain the most. Our findings
in this
paper
fully
risk
support
management. He
his views.
also argues that
emerging-market economies
do not
exist, at least in the
magnitude required. In
titatively the usefulness of these
this sense, our
framework also serves to highlight quan-
markets and allows us to begin gauging the potential
size of
the markets to
*
be developed.
we
In section 2
has been triggered
describe the environment and characterize the optimization problem once a sudden stop
—
this provides the
boundary conditions
for
the "precautioning phase." In section 3 we
The main
study the phase that precedes a sudden stop when the country self-insures.
is
goal of this section
to characterize precautionary recessions. Section 4 describes aggregate hedging strategies under different
imperfections and the degrees of crowding-out in these markets.
an application to the case of Chile.
likelihood of a
sudden stop and
Section 5 illustrates our results through
Along the way, we outline an econometric approach to gauging the
its correlation
with an imderlying signal. Section 6 concludes and
is
followed
by an extensive technical appendix.
The Environment and the Sudden Stop Value Function
2
Intertemporal smoothing implies that, during the catch up process, emerging economies typically experience
substantial needs for borrowing from abroad.
For a variety of reasons that
dependence on borrowing
The sudden
a source of fragility.
is
we do
not model here, this
tightening of financial constraints, or the mere
anticipation of such an event, generates large drops in consumption. In this section,
we formahze such an
environment and characterize the optimization problem once a sudden stop has occurred. The next sections
complete the description by characterizing the phase that precedes the
The Environment
2.1
Endowment and
2.1.1
We assume that
{y{i)A
>
0), is
Preferences
the endowment grows at some constant rate, g, during
dvU)
yO
y[t)
aspects of this process are to be highlighted.
to characterize,
•"See,
e.g.,
<
i
<
oo. Thus, the
income process
described by:
-^^gdt,
Two
crisis.
sudden stops are
significantly
5
=yo.
First,
it
is
>
0.
deterministic.
In the economies we wish
more important than endowment shocks as
Krugnian (1988), Froot, Scharfstein. Stein (1989), Haldane (1999), Caballero (2001),
commodity indexation
of emerging markets debt.
The contribution
of this
for
triggers of
articles
paper relative to that literature
quantitative framework and to link the hedging need not to commodities per se, but simply as a signal of
financial constraints.
is
deep
advocating
to offer a
much
costlier
.
contractions.^
and hence
>
this perspective, the
their potential to trigger
any collateral
since g
Prom
effects
there
0,
Reflecting
is
main importance
We
a sudden stop.
an incentive
and
riskless interest rate
c^ represent
date
t
dimension and group
and
i
r~g
it
= ^o <
0.
times:
all
^
>
X(0)
>
0,
exceeds the rate of growth of the endowment, r
>
g.
consmnption and "excess" consumption, respectively, with:
c\
The
this
net debtor position, the country starts with financial wealth,
Xt
Ct
model along
their "collateral" effect,
country to borrow early on.
for the
However, the country's total wealth must be positive at
Let
simplify the
is
contained in endowment shocks with the signal process (to be described below). Second,
its initial
where r denotes the
endowment shocks
of
=
~
Ct
<
K.yt,
K
<
1
representative consumer maximizes:
POO
c:)e-*('-*' ds
(1)
with
C*l-7
u(cn
The parameters
assume r
=
6
and 7
is
T—
1-7
are the discount rate
6 throughout.
with a habit level that
=
The
and
6>0,7>0.
,
risk aversion coefficient, respectively.
For simplicity,
we
functional form of the utihty function captures an external habit formation,
increasing at the rate of the country's growth rate. This
economies that exhibit strong growth, since future generations are
likely to
is
a natural assumption in
have a higher standard of
living
than the ciirrent ones.®
A
frictionless
It is
benchmark
instructive to pause
and study the solution of problem
(1) subject to
a standard intertemporal
budget constraint (and absent a sudden stop constraint):
=
dXt
{rXt-cl+y;)dt
Xo
"Of course, sudden stops reduce growth as
well,
given
but we capture these effects directly through the decline in consumption.
In this sense, y can be thought of as potential, rather than actual, output.
Accordingly, this type of utility function allows us to impose
indebtedness of the country at any point in time
a simple reduced form) a barrier on the
of:
r
and on saving,
(in
-
g
since:
yt-
Ct
<(l -
K)yt
t
>
0.
amount
of
•
=
with y*
By
—
(I
K)y.
standard methods
can be shown that the solution to this deterministic problem
it
c*
and the
=
"total resources" of the country
r
(Xo +
aU
for
^:^)
f
>
is
given by:
0,
remain constant tliroughout:
— Xo+
-y
r-g
^°
(2)
-y
r-g
I
'
so that
lira
—=
1
and, accordingly:
lim
'-°°
Moreover,
for
any
level of the
—
l-K
=
r-g'
yt
debt-to-income ratio below
its
limit value, the ratio
^ decreases monotonVt
ically
—.
to
This case serves as a
2.1.2
There
frictionless
benchmark
in
what
follows.
Signal
is
a pubhcly observable signal,
S(,
correlated with the sudden stop and, for simplicity, uncorrelated
with world endowments. This signal follows a diffusion process:
dst
In our basic model,
threshold,
s,
We
In our second model, the signal
only
its
likelihood that
+
adBi.
crisis is triggered
is
is
=
mi{t e (0,00)
S(
:
time the signal reaches a
<s}.
(smoothly) influenced by the signal
A(
We
first
imperfect. In this case, a sudden stop can happen at any point in time.
stochastic time, t, depends on the reahzation of a stochastic
2.1.3
the
associate with this event the stochastic time, t:
T
It is
fJ-dt
a perfect signal and a
St is
from above.
=
jump
5j.
In this case, the specification of the
process, the intensity of which
is
given by
=exp(Qo -a\St)-
Sudden Stops
place no hinits on the country's borrowing ability
faces a "sudden stop."
We
up
to the stochastic time t.
do not model the informational
or contractual factors
At
this time, the country
behind this constraint, or
the complex bargaining and restructuring process that follows once the sudden stop
to produce a quantitative assessment of the hedging aspects of the problem,
goal
is
and
fairly robust (across
models) constraint.
the net present value of
is
its
assume that
within
It is
T
a realistic
as a temporary
we
place the constraint on:
~g
>
e^'^
periods. Formally:
obvious that this constraint will be always binding because as
(
Xt +
-^
remains constant at
)
that this constraint can be expressed as a constraint on the
+
for
at time t, financial markets require the country to increase its total resources by ^
the unconstrained benchmark
T
we look
particular, since the "natural" aggregator of a
total resources,
r
We
triggered. Since oiu-
we simply model the sudden stop
For this,
and severe constraint on the rate of external borrowing. In
country's total wealth
is
r, as a function of the debt/gdp ratio at time
^
=
e^^
r.
+
To
0(1
-
we showed
times. It
all
maximum
is
in a previous subsection, at
then straightforward to show
allowable
amount
of
debt/gdp at time
see this, redefine
e-<'-»'^)
(4)
and observe that the constraint becomes:
X.,r > X.
k^ + 0(1 - e-(^-)-)l
L
^Ml-^-'"-^^")
r - g
+
J
(5)
or
X.+r ^ X^
" Vt
[e^^
+
levels of
imply higher
that for the special case
=
- e'C-g)^)
0(1
^
e9T{r-g)
esT
y;+r
Higher
- e-(-^)^)]
0(1
levels of
(^
and thus make the constraint
this constraint literally reduces to the
grow any further between r and t
tighter. It is also trivial to verify
requirement that debt/gdp cannot
-\-T-J
Xt+T ^ Xt
Vr+T " Vr
Finally,
it is
interesting to note that
one can think of
surpluses the country has to accumulate over the next
(5) as a constraint
T
periods.
To
on the minimum balance-of- trade
see this, start with the intertemporal
budget constraint:
Xr+ I
e-^^-'-^^y;
dt=
e-'-f'-'^c* dt
I
+
e""^ X.
or:
r^+^
Xr - e-'-^X^+T = j
4>
=
the debt/gdp ratio
is
Even
for
t.lie
case the constraint
we consider
,
^
!;*fl-e-('-s'^^
e-"<*-">< dt
r
is
-g
binding at the time of the sudden stop. This
always growing for an unconstrained country as we showed
in section 2.1.1.
is
due to the
fact that
.
and replace
(5) into
to obtain:
it,
rT + T
j-r+T
or, finally:
fT + T
rT + T
-
e-''"-^> (y;
/
cl) dt
> X^lle-f'-s)^ +
This gives us the constraint
in
9!.e-'-^(l
-
e-<'-5>^))
-
1)
+
<t>e-'''^
/
e-'"*'-^'!/; dt
terms of the balance-of-trade surpluses required from a country that
faced with a sudden stop. These required surpluses rise with the level of debt (recall that ,Yt
endowment
the
<
0),
(p,
is
and
of the country.
The Optimization Problem
2.2
Let us assume for
indexed to s or
t.
now
that the only financial instrument
f
Jo
riskless debt, so there are
maximizes the expected
In this case, the country
V(X,,y;)=maxE
is
utility
no hedging instruments
of a representative consumer:
<
7
1
s.t.
=
dXt
[rXi —
cl
+
yl)dt
X^+T > Xr+T = Xr
-^ =
[e^^
+
- e-f-'"^)] + ^ylO^_£ll:^^
0(1
<
<
e'-^
gdt
Vt
dst
t
=
hm
—
iidt
+
cfdBt
X(;)e-'''
=0,
a.s.
'OO
and either
Perfect signal: r
=
inf{i
6 (0,oo)
:
Imperfect Signal: T
=
inf{?
e (0,c»)
:
St
/
<
or
s]
\[st)dt<z},
At
=
e"°~°'*',
z
~exp(l)
Jo
independent of the standard Brownian Ft —Filtration.
where
z
2.3
The Sudden Stop Value Function and Amplification
We
is
solve the optimization problem in three steps.
period, then for the sudden-stop period,
we present the
first
the sudden stop,
two,
and
V^^[Xt,
problems before the
crisis
J/t).
and
trivial, steps.
Starting backwards,
we
first
solve for the post-crisis
finally for the period
preceding the sudden stop. In this .section
The goal
to find the value function at the time of
of these
is
which then can be used to find the solution of the optimization and hedging
takes place.
8
Post Sudden Stop:
2.3.1
Since
we made the
problem
for
i
>
r
t>T + T
simplifying assumption that the country suffers only one financial crisis, the maximization
+T
is
simply:
y(X.+r,J/;+T)=max/
< Jt+t
^^
1
-7
e-(*-(-+^)) di
(6)
s.t.
Jt+t
Jt+t
This problem has the
trivial solution:
c*
= rW^
T
+ T <t<oo
(8)
^^.
(9)
with
VV'|-X.+r +
r
-g
This constant can be interpreted as the "excess" wealth at date t
which
is
2.3.2
needed to cover the reservation-consumption
Sudden Stop:
t
for the
(that
is,
the wealth in excess of that
level kj/j).
<t<T + T
With the continuation value
problem
+T
function,
V{Xr+Tiy*+T)^ from above, we can now write the maximization
sudden stop phase:
y^^(X,,
y;)
= max T^ ^e--(t-) dt + e-'^V{X^+T,y;+T)
S.t.
rT+T
j-T
Cj*e-^(*-^)
/
dt=
+T
Xr+
y;e-''('-^> dt
-
e"''^ Xr+T
Xr+T > Xr+TSince the country
allows, the
is
(10)
growing and wishes to expand
sudden stop constraint,
(10), is
its
consumption at a faster rate than the constraint
always binding (see the appendix for a formal proof):
Xr+T = Xt+tGiven
this result, the optimization
subject to a final wealth condition.
A
q*
problem
is
straightforward.
(11)
It is
a deterministic consumption problem
few steps of algebra show that:
= ^_[_rT ^Vl,
T<t<T + T
(12)
with
Wi = Ar
It is
r
are
now ready
crisis in
Ar+T-
e
-
g
Xt+t
easy to see from these expressions that as
consumption during the
We
H
The country has
c^ falls.
rises,
to cut back
order to satisfy a tighter sudden-stop constraint.
to determine the sudden-stop value function:
1-7
Jr
1—
7
1-7
Jr+T
r
\^
1~7
J
(14)
It is
easy to verify that our careful choice of the constraint pays off at this stage.
The value
function simphfies
to:
=
where V{XT,y*) denotes the vahie fimction
KV(Xr.y;)
in the absence of
(15)
a sudden-stop constraint and
ii" is
a constant
given by:
K=
That
{1-
is,
e-'"^)'>(l
-
e-'"^,^(l
-
e-('-5)^)
-
e-''"-*)^)'-^
the dimension of the state space effectively
Moreover, up
to the constant
K
,
we have arrived
is
+
e"*"^ (<^(1
-
+
e-'^-^)^)
e^^)^"^
reduced from two, (Xr.yr), to one, (Xr
at a value function that
is
(16)
.
+
-^f-z )•
identical to the one in the problem
without sudden stops. This problem corresponds to the trivial case of an infinite horizon consumption-savings
problem under
certainty.
Note that
for
7 >
1,
which we assume throughout,
While the particular simplicity of our formulae
more
general.
The model
also can
is
K>
1
and increases with
0.^
due to styhzed assumptions, the basic message
is
be vmderstood as an approximation to a potentially more complicated
specification of constraints that result in higher marginal disutihty of debt in the event of a crisis.
'Observe that
is
for
7 >
1
the function
-^-^
—
negative for
all
Z >
0.
Thus
lower than the continuation value function for the unconstrained problem.
of the constraint implies that ha\'ing a higher
its
is
Xt
K
>
1
The reason we need 7 >
does not relax the constraint one
demands by the same amount during the sudden
stop.
When 7
for the crisis.
10
is
reflects that the constrained value function
below one,
for
1
is
that the flow aspect
one, because the lender does not reduce
this effect
is
strong enough to discoXirage saving
—
Precautioriciry Recessions
3
Let us
now
address our main concern and begin to characterize the country's optimization problem prior
to the sudden-stop phase.
s or T, the country's only
borrowing before
it
we have eissumed
Since
mechanism
We
takes place.
that there are no hedging instruments contingent on
reducing the cost of a sudden stop
for
show that
when
some elements
the signal of a sudden stop deteriorates, the country
a "precautionary recession:"
sudden stop. Such behavior
that
is
is,
a sharp reduction
in
of predictability in them.
falls into
what can be
First,
we study
signal hits a
stops stiU are
3.1
minimum
is
complex
for
a case where there
threshold,
more hkely as the
Perfect Signal:
In
labelled as
consumption to hmit the cost of the potential
many
risk.
reasons,
one of the most important being the uncertainty
that surrounds the factors that trigger such crises. In order to isolate the main issues,
steps.
amount
widely observed in emerging market economies, where private decisions and
macroeconomic policy tighten on the face of external
In practice, such a problem
to cut consumption and
in addition to a precautionary savings result, the
of self-insurance varies over time, because sudden stops have
particular,
is
s,
is
a perfect (stochastic) signal:
for the first time.
A
we proceed
in
two
sudden stop occurs when
this
Second, we add (local) uncertainty: while sudden
signal deteriorates, they can occur at
any time.
The Threshold Model
In the threshold model, the dynamic
programming problem
*l
rT
V{Xt,st,yt)
= max£:
is:
t
(17)
1-7
Jt
where
=
T
and the evolution of {Xt,St,yt)
The boundaries
is
inf{(i
:
s
<
s)
Ago}
(18)
given by:
dXt
=
{rXt-c*+y;)dt
(19)
—
=
9df
(20)
dst
=
/.idt
of the value function,
V{Xt,
+ adBf
Sj, ?/<),
can be found readily.
(21)
On one
end,
we showed
in the
previous section that:
ViX.,,,y;)
where
K
>
1
measures the intensity of the
=
crisis
K{l}l'i^^0—
and
is
11
given in (16).
(22)
'
On
we show
the other end,
the appendix that
in
goes to
St
eis
infinity,
the value function converges to
the value function of the deterministic problem with no sudden stops:
£n^y(X.,.,y;) =
That
,_;^
(23)
the value function becomes independent of the signal as the crisis event becomes less and less
is,
The
^
(;j
V
value function
satisfies
the following Hamilton-Jacobi-Bellman (henceforth
f.*i-T
= max
<!
-^ - Vxcl
I
- rV + Vx {rX +
\
yl)
+
Vy-yy',
+
Km +
HJB)
likely.
equation:
1
^Vsst'
(24)
subject to the boundary conditions (22) and (23).
In the appendix,
we show that the
solution has the form:
1-7
V{XuSt,y:)
=
^)
^^
+
(-^-
a{str^
for
some twice
The
difTerentiable function a{st).
Mm
latter satisfies the
=
a{s)
s—*oo
and
solves
mization
an ordinary
in (24)
differential
with respect to
=
(24).
Cj
exists
function a{s)
is
and
is
unique.
Ma,
We
This setup allows
for
Given the value function,
(27)
= {Vx)-"^
(28)
.
-^^
—
^_''
.
to (26)
and
(27).
We
show
in
(29)
the appendix that the solution to
s.
As the
signal deteriorates
toward
s,
a[st)
12
The
in
likely.
feedback form.
from (28) that the optimal consumption policy takes the form:
.-Ml.
1.
this function rises rapidly,
wealth (and marginal wealth) as the sudden stop becomes more
follows
yield:
l^K)^
^2 ( ,..
=0
+ -CTM(7-l)^-^+a,J
an exphcit characterization of the optimal consumption policy
it
These steps
plot this function (multiplied by r) in the left panel of Figure
decreasing with respect to
reflecting the increasing \'alue of
(-\
K^'-^
Third, di\'ide the resulting expression by
which can be solved numerically subject
ODE
(26)
/
to get:
l-ra +
this
r
equation that can be obtained in three steps. First, carry out the maxi-
c*
Second, substitute this into
boundary conditions:
(-]
V
a{s)
(25)
-7
1
The
effect of
limit. Conversely,
when the
mechanism: as the
and hence exposure
to reduce external debt
Ito's
consumption
signal worsens,
precisely the precautionary recession
Applying
As
the signal can be seen clearly in this expression.
Lemma
to the right
to the
hand
falls for
crisis
s rises, a{s) falls
any given
level of
toward
its frictionless
income and debt. This
becomes imminent, consumption
falls in
is
order
sudden stop.
we
side of (30),
obtain the country's (excess) consumption
process:
^ = 0^ + ()
dt-^adBt.
1
2
Cl
\
a
(31)
a
J
This process reinforces the above conclusion and shows that consumption has a nontrivial diffusion term.
Without hedging, consumption "picks up" the
Up
to the functions
^, which can be
(— ^cr)
savings, whereas the second term
signal news. Positive
is
news about the
and
fulfills
the function of a hedging strategy.
evaluated numerically, the rest of the terms in this expression
The
have a straightforward interpretation.
volatility
first
is
term
is
positive
also positive
signal increases
and captures the
and captures the
effect of precautionary
sensitivity of
consumption
consumption whereas negative news decreases
it.
to
This
the outcome of the country's attempt to accumulate resources as the sudden stop becomes imminent. As
can be seen in the right panel of Figure
deteriorates.
3.2
That
is,
1,
which plots ^,
this effect
becomes more intense as the
the sensitivity of consumption to news about the signal increases during downturns.
Imperfect Signal: The
jump model
Countries do not have a perfect signal for their sudden stops.
Variables such as terms-of-trade and the
conditions of international financial markets raise the probability of a sudden stop, but
as in the threshold model. In this section
we capture
sudden stop a probabihstic function of the underlying
The setup and optimization problem
T, in (18).
Now
the sudden stop
is
this
Thus, we replace (18)
=
it is
never as stark
dimension of reality by making the trigger of a
signal.
are exactly as in (17)-(21), with the exception of the stopping time,
a Poisson Jump>-Event with intensity:
A(5t)=e°''-"»^S
T
signal
a,
>0.
for:
inf {i
e
(0,
go)
:
/
\{st)dt
<
z},
A(s,)
=
e""""'^'
,
2
~
exp(l).
Jo
Once the sudden stop takes
value function
still
place, events unfold exactly as in the threshold model.
takes the form:
rj
13
1-7
In particular, the
Function a
Function ra(s)
/a
1.25
-0.15
-0,2
-0.25
105
-
2
2
4
Level o( the state Variable
Figure
1:
Functions ra{s)
s=-0.6,so =0,r
(left
= 0.09,T=
of the state variable
is
1,
panel) and
=
4
Level of the state Variable
^
0.35e''^,a
(right panel).
=
0.23,7
set to 0.
14
=
7,a
The parameters used
=
0.03,
k
=
0.8,
yo
=
example
are:
-O.S.The
drift
for this
I.Xq
=
The only
difference
In this case the
that
is
HJB
V^^ now can be
reached from any
equation for the value function
which A(s)
s for
Vxc;} - rV
With
still
rather than from just
0,
s.
is:
r*l-T
- inax{^^
>
+ Vx{rX + j/*) + Vy-y'g +
essentially identical steps as in the previous subsection,
V^^i
+
1
^-a^V^s
+
[V^^
X{st)
we can show that our
stylized
V
framework
yields a simple solution of the form:
1-7
V=
(^'
bistY
+
^)
1-7
Precautionary Savings
3.2.1
Let us pause and focus on the case where sudden stops are totally unpredictable, aj
6(s) is
no longer a function of the
algebraic equation for
signal s, replacing the function
V
in the
HJB
=
0.
Since in this case
equation yields a simple
b:
l-rb+-\
0.
7
where
RIt is
now
straightforward to obtain the (excess) consumption function from the envelope theorem:
b
Applying simple differentiation to
this expression,
we obtain
the country's (excess) consumption process:
dc;
-A
dt
1
7
Relative to the threshold model, there
reflects
sudden stop
correlated with the signal
self-insurance
new
drift
term
in the excess
consumption process. This term
is
s,
tsiking place in the
next instant. But, because this probability
is
not
there are no precautionary recessions. In this case, the pattern of saving for
not a soruce of business cycles.
Precautionary Recessions
3.2.2
Let us
in
a
the additional precautionary savings attributable to the local uncertainty introduced by the strictly
positive probabihty of a
V
is
now
the
>
return to the general case, where ai
HJB
equation yields the following
1
—
r6
+
bsi-i
+
0.
After a few simplifications, substituting the function
ODE:
(7-1)^+6.
15
0,
2
7
(32)
which
differs
from (29) only
The boundary
probabiUty of a
in
the last term.
conditions are also different. First,
crisis.
when
Second, as conditions worsen, there
goes to infinity there
s
is
is still
a strictly positive
no equivalent to the tlu-eshold * where a
crisis
happens with probability one. Let
b*
bt
b*
and
6, are
=
=
lim
b{s).
lim
b{s)
given by:
=
b*
r
=
b,
Again,
it is
K'/-'-.
straightforward to obtain the (excess) consumption function from the envelope theorem:
(^"^)
,.
Finally, applying Ito's
lemma
,33,
hand side of
to the right
we obtain
(33),
the country's (excess) consumption
process:^
7
dc*
+ 1 fabsY
1
1
b
7
b'
.
(It
b
J
This case integrates the insights of the pre\'ious models.
tributable to local uncertainty, but the
section,
^
is
amount
- jadBt
There
is
ongoing precautionary savings at-
same reasons
varies with the signal. For the
more to a one-standard-deviation increase
in the signal
new precautionary term (-A(S() (|)
precautioning against the Poisson
paper we quantify these
4
of the previous
strictly negative, so that the conclusions of the pre^aous section also carry tlirough here.
deteriorating signal increases the need to accumulate resources and accordingly
a
(34)
'
jump
—
1
),
by a factor of —^f^-
which we
e\'ent that
can occur
also
at
found
A
makes consumption respond
Finally, the drift
in the
case a\
=
0,
term includes
and captures
any time. In the apphcation section of the
effects in the context of Chile.
Aggregate Hedging
Precautionary savings and recessions are very costly and imperfect self-insurance mechanisms for smoothing
the impact of a sudden stop.
using deri\'atives
'We
In this section,
and insurance
contracts.
are obviously focusing on a ca.se where the
Of
we enlarge the options
of the country
and allow
it
to
hedge
course, the effectiveness of the hedging strategy depends on
jump
lias
not yet taken place so that dq
16
=
0,
where q
is
the poisson event.
how
the contracts and instruments that are available to the country,
stop constraint, and
The
issue of
how
how accurate
the crisis-signal
is.
the fresh funds relax the country's constraint
optimal assistance mechanisms and
is
We
not well understood.
first case,
is
at the core of the current debate
do not
characterize optimal hedging strategies under different scenarios.
In the
these contracts enter into the sudden
We
about
try to solve this debate but rather
begin by exploring two polar cases:
the hedge cannot relax the constraint directly but only improves the initial conditions of
the country once
hits the constraint.
it
This would be the case, for example, when the resources from the
hedge are used entirely to pay other lenders.'" In the second
That
relaxes the sudden stop constraint.
case,
the country can buy hedging that directly
each dollar received from the hedge can be used to
is,
fulfill
the
sudden stop constraint.
In this section,
we
isolate the
for analytical tractabihty,
closed-form solutions.
signal case,
4.1
for the
constraint.
this
do
this
model that allow us to obtain
use these results as an approximation benchmark for the more reahstic imperfect
the focus of the empirical section.
moment
that the country
litis
no mechanism of injecting net resomces into the sudden stop
However, the country faces complete hedging markets before the sudden stop
we can
only with
An
is
because we can use complete-markets tools in
We
Hedging Precautionary Recessions
Assume
sense,
which
We
above distinction and focus on the simpler threshold model.
its
interpret the
sudden stop as a time when
resources at the outset of the
alternative interpretation
is
financial markets close
all
arises.
In this
and the country
is left
Xr-
crisis:
that the hedge
is
used (crowded out) by existing lenders and the sudden-
stop constraint remains unchanged, except for the positive effect of a reduced
initial
debt (hence, there
is
a
reduction in the required balance of trade surplus).
Let us re-write the dynamic budget constraint for
dXt
where dFt denotes the
=
{rXt
-
c;
t
>
+
profit/loss in the futures position
Since under the assumption of zero risk
as:
y;)dt
and
+
tTj
ntdFt
is
the number of future contracts.
premium and any constant convenience
yield for copper, d,
have:
Ft
we can apply
Ito's
lemma
=
to obtain:
dFt
Recall that our
5tef'--'^«^-*)
model has no straight
= oFtdBf
default. Implicitly, however,
17
it
does allow
for limited
rescheduling.
we
Defining the portfolio process as pt
=
dXt
tt;
=
Ft,
we obtain
-
{rXt
+
cl
y;)dt
Thus we modify the dynamic programming problem
spondingly,
we
refer to this
hedging portfoUo as
+ ptodBf
(35)
in (17) to allow for a
hedging portfolio
pt (corre-
p— hedging):
^»l-7
rT
V{Xt.St,yl)
the new dynamic budget constraint:
= msxE
(36)
1-7
Jt
where
r
and the evolution of {Xt,St,yt)
is
=
inf{(i
now given
dXt
=
<
s)
c*
+
y^) dt
A
oo}
by:
-
{rXt
-^ =
s
:
+ ptcrdBt
(37)
gdt
(38)
y't
dst
It
=
+ adBt.
j-idt
turns out that the solution to this problem
(39)
simpler than the no-hedging problem because
is
we can use
weU-knowTQ techniques from the complete markets case." Following a derivation similar to the no-hedging
case,
one
verifies that the value function of
the problem in the presence of complete hedging
is
given by (see
the appendix):
V{Xuy:,St) =
"'
(x
a^{s^)^
^
I
'
-'>V""'
1-7
with
a''(st)
= -
(l
+
e-^'(^-^)(K^ - 1))
Correspondingly, the consumption policy in feedback form
Wliile
it
case, this is
may appear from
not
so.
To
this expression that not
see this, apply Ito's
lemma
(40)
is:
much
to the right
has changed with respect to the no-hedging
hand
side
and simphfy
(see the appendix), to
obtain:
dc*
That
is,
excess consumption
is
=
0.
constant throughout the pre-sudden stop phase.
precautionary recessions, and the signal does not affect consumption.
"This
is
possible because markets are complete at
all
dates but r.
18
There are no more
>
How
can this be reconciled with the consumption expression
Xt was simply made
Xt- While in the no-hedging case
component,
which once replaced
imphes that
all
+
e-^^i^-s.)(K~
=
{rXt
the variation in precautioning
portfolio, p, is always negative.
when
Its
the signal goes to
of this investment position
is
now
the behavior of
includes a hedging portfolio
it is
-l)J
r-g^
\
is
c;
+
y;)dt
+ ptadBt
absorbed by Xt rather than by consumption.^^ The hedging
is
largest
when the
This means that the coimtry
and the amount
signal
is
is
at the trigger point s
and
always shorting the asset that
of shorting rises as the signal worsens.
The
is
counterpart
a reduction in the countries' external debt as the signal deteriorates.
However, this form of insurance
the coimtry can do
is
-
absolute value
infinity.
perfectly correlated with the signal,
surpluses
it
is in
in
dXt
AU
of riskless debt,
The answer
pti
l
goes to
in (41)?
will
to arrive at the
not entirely remove the consumption drop during the sudden stop.
sudden stop with
required to have during the
crisis.
less debt,
Because of
and borrowing throughout the pre-sudden stop phase
for
and hence to reduce the
this drop, the
country
precautionary reasons.
It is
still
size of the trade
cuts consumption
simply no longer state
(signal)-dependent.
Optimal consumption before the sudden stop
is
constant at (see the appendix):
V
_l_
^0
E[j^e-rtdt +
Since
K
>
1, it
clearly is lower
than the
4.2
Hedging Sudden Stops
Let us
now assume
that there
level of
e-rrl<l!l
consumption
in
a framework without sudden stops:
a second asset, H. with the property that
is
it
can be excluded from the
sudden stop constraint and hence can be used directly to overcome the forced savings problem. This can
be thought of as a credit
line
that does not crowd out alternatiT.'e funding options (we refer to this case as
/i— hedging).
Formally, the problem becomes:
.1—
V{Xt,St,yt)
=
E
max
it
'^The appendix provides a formal proof of
and return to a
fuller characterization in
this
1
(43)
-7
statement, which follows steps similar to those in Karatzas and
the implementation section.
19
Wang
(2000),
where
=
T
and the evolution of {Xt, Ht,st,yt)
inf{(t
now given
is
:
s
<
Aoo}
s)
(44)
by:
dXt
=
{rXt
dHt
=
vHt+ptadBt+dNt
vHt+ptodBt+dNt
(46)
gdt
(47)
-
c*
+
dst
=
ndt
+ adSt
Ht
>
Q,
i>l
ij;)
dt
+ ptodBt -
idNt
(45)
(48)
.
(49)
Observe that we now have a second portfoho process pi and an increasing process dNt representing
Moreover, every addition into Ht from Xt
additions into the second type of asset.
markup
^
fee, C
to imply that
Ht
Since
1-
To be more
Xt+t ^
X
+
,
the presence of a
that at time t the holdings of the asset
we explain
we can
This
is
=
X;~,
<
</>< e"-^.
follows that the constraint will be non-binding
Ht >
h*
(50)
because the country can use
its
write:
Xt^=Xt-^Ht.
It
will
as:
rather than Xt-
to relax the constraint directly, so that
Ht
below.
e9^) +<?i^^(l-e-('"-»>^)
now reads Xr~
markup would seem
not the case precisely because Ht has the ability to
is
sudden stop constraint
precise, let us re-write the
('?^(l-e-''"-s>^)
the form of
we assume
^t- In other words
Notice that the constraint
Ht
this
in order to relax the constraints as
^r+T>^r-
holdings of
arri\'e in
dominated by Xt- However,
is
relax the constraint:
be added to X^.-
income flows
tlie
accompanied by a
is
(51)
if
(Xt- +
^^]
(52)
where
h'
To
see
why
this
Xt+
is
so,
Xt- + Ht
=
Xt-
('0(1
-
(<A(1
xr;
e-f'-s'^)
h*
[Xt- +
e-(^-')^)
+
e^^)
)
V
=
-
Ht =
note that for
=
=
y'r+T
T
—
g
_
—
-^
)
at time
+ <t>^^{l —
T
Vt
r
e^^ - l)
+
g
20
Q
(53)
.
t+
total resources
e-<'-s)^)
+
(e^^
^
-
l)
'
become:
^
yr_
T
—
9
However, as we established in equation
the unconstrained solution automatically satisfies the relation:
(2),
A^+
H
r
which implies that
A
Xt+t ^ Xr
,
i.e.
-
—
A.T-+T H
r
g
the constraint
is
1
g
Ht >
satisfied automatically for
modification of the martingale approach of Karatzas and
Wang
>
and characterizing the solution < Cf,pt,pt,Xt, Ht,dNt
finding
-
(2000)
any feasible consumption-portfolio plan we
Applying
in this context.
=
^^ + '^(1 [y*t-cl)e-^'dt-i
E{e-^^Hr)
=
Ho +
Eie-^^ {Xr
We
will
assume that at time
+
e(
f e-'-'dNt^
^Hr))
it
can structure as
a "markup,"
^,
on
it
the country starts with
desires. I.e.,
this credit line.
Then
-^f-
)
•
Ito's
is finite
lemma
to
almost surely,
e-''dNt]
j
.
=Xo + iHo + Eff
interesting interpretation of this equation
line that
+
get:
E{e-'^{Xr+(Hr)) = Xo +
An
X^.-
get:
Eie-'-^X^)
Combining the above two equations we
I
particularly well suited to
is
e~^*Xt and e~^* Ht, and because we have a bounded portfolio and a stopping time that
for
h*
is
it
Hq =
E(j
as follows:
can choose
0,
{yt
{yl
-
c*) e"''' dt\
so that
-
we end up
cl) e-^'
dt\
Suppose the country
Hr
"state by state."
the price of the credit line
with:
(54)
.
offered a contingent credit
is
But assume too that
it
pays
is:
Pcci=^E{e-^^H^).
With
this definition
we can
rewrite (54) as:
E(e— XO =
which
is
the standard budget constraint
(Xo - Pcc)
(e.g.
+ E (j^
[y;
of Section 4.1.)
-
c,*)
e'^' dt\
after subtracting the initial cost of the
contingent credit line from Xq.
Prom now
on,
it
will
prove useful to express the amount of the credit
line as a fraction of
\Xt-
+
;rf^
)
so that:
hr
For
It is
T.'alues
of h
<
h* the constraint will be binding and consumption will drop during the sudden stop.
not difficult to show that in this case consumption after the sudden stop
21
(i
>
r
+ T)
is
equal
to:
—
which follows from
Given Xras long as h
{t
<
t
<
T
,
<
this
(8).
the same quantity as in the no-hedging case because the sudden stop constraint binds
is
However, the
h*.
+ T)
and
(4)
since
Ht
<. =
entirely used during this period
is
for
=
h
.
Consumption during the
crisis is
now
equal
to:
-^^^
(X^- + -^) (l+h- e--^<P{l - e-(-^)^) - e-t-^)^)
— e~^^
r — q
1
which
consumption during the sudden stop phase
effect of the credit line is to raise
\
h* rises to
+ h')[X^-+-^^]=c%r-
r[l
Accordingly, the value function at time r
is:
vss ^
c=
(55)
1-7
+ h'y-n^^^,.y +
[(1
(^^
-W
c^
i^ii„<,,.j
}J
where: ''^
Ki = {l-
Now
e-'-^)^(l
+ h-
+
e""-^!!
+ e"'-^ (1 +
/i*))^-^
/i*)^"^
< K.
the optimization problem has the same form as before except that the intertemporal constraint
takes into account the fact that the consumer essentially faces two types of
Adopting the Cox-Huang (1989) methodology and
Wang
ping time in Karatzas and
min
k
max E
is
X
and H.
application to problems involving a
its
random
stop-
are able to reduce the problem to the following static problem:
e-'-'dt + e-'-'V^^(Xr,Hr)-k( f Cte'^Ut +
f V
1-7
V^o
Ct,Xr,Hr
e'"'^
{Xr
+ e.Hr
Jo
s.t.
where V^^
we
(2000),
sissets,
e(
f
c'e-'-' dt
+
e-'-"
{Xr
+ ^Hr)) =Xo +
e[
f yle-"' dt\
given by (55).
The presence
of dynamically complete markets allows us to reduce the problem to
an essentially
static
problem. Parties can contract on the payoffs to be transferred "state by state." In other words, the objective
inside the brackets
It is
is
maximized state by state and the optimal
''It might
seem puzzling why
time r+ are now given by
in the ab.sence of
Xr
rephcated dynamically.
easy to show that in this framework one can derive optimal consumption to be:
Cj*
in the
payoflF is
{l
a markup,
C=
(^)'' (1
-I-
/i')'"''
+
h")
[X^- + -^^
it
will
be the case that
)
.
=
yt-i,iG (O.r).
when h = h' and not
Actually one can
(l
^
+
h')/
(x^T
\^
complete absence of sudden stops.
22
C=
ea.sily show^ in
=
+ -^^1
T-g
J
(-)"'
•
The
rea.son is that total resources at
the framework of the threshold model that
X:^""
^
+
-^^. where Xi"""
denotes the
^
1
g
level of
where
fc
a constant that
is
is
determined
a way that the intertemporal budget constraint
in such
is
satisfied.
The
crucial step
In the appendix
is
the maximization of the problem involving the continuation value function at time r.
we show
that the solution to this problem
.
Xr-
=
Ifpt
^
T{l
h*
is:
—J-
+
-
Vt
r-v h*)
[\
-
+
e-'"^){\
h*)
with
K-i)
o<r
—
Notice that as ^
>
1,
/i
^>
/i*.
As might be expected
<i.
in this case,
the constraint, the country will optimally choose h
=
consumption adjustment during the sudden stop
be required.
will
h*
>
li ^
.
1 it will
Because of the homotheticity of the problem, the optimal credit
initial
wealth.
To complete the
solution to the overall problem,
and combine everything to solve
+
costs nothing to avoid
be true that h
us return to the time
let
<
h* and
does not depend on the
line ratio
+ /i''p*or"
il + h*
£[e-''^] (i
Jo
-
it
some
level of
budget constraint
for c*:
dt
1
where
1
e-^(^-^) (1
e-^(^-^>
+
T
-1
/i°P*Or"
(l
+
Vo
Xo
Xo +
Vo
ft*)
where as usual:
A
The
credit line reduces the
consumption. However, since
=
(56)
magnitude of the sudden
/i°p'
<
h*
,
there
is still
the remaining need for a precautionary recession.
significantly less
4.3
stop,
scope
which translates into higher pre-sudden stop
for
The economy
hedging; pt takes the slack and eliminates
still
suffers
through the sudden stop, but
than without hedging.
Imperfect Signal
In general, crises will
be correlated with a signal but not
but the nature of the solution
can occur for any
to contrast
it
s,
is
perfectly.
This complicates the p-insurance case
similar to that of the threshold model.
On
the other hand, the fact that r
considerably complicates the /i-hedging case. In this section
with the perfect-signal scenario.
We return
23
we develop
the former case
to the ft— hedging case in the empirical section.
The
steps of the derivation are similar to those in the threshold model,
appendix.
The optimal
(excess)
consumption and portfolio
6f(..)(A',
policies
and we relegate them to the
become:
+ ;^)
Unfortunately the function b^ {s) can no longer be characterized in closed fonn, but
it
can be computed
numerically.
Applying
Ito's
lemma
to the right
hand
dc*
side of (57),
we obtain the
process:
/ A(s,
dt.
1
(.59)
c;
Comparing
unhedged case
this expression to that of the
eUminates the diffusion term.
in (34),
However, unlike the hedged case
shows that the
in the threshold
possibility of hedging
model, there
is still
pre-
sudden-stop precautionary savings and these fluctuate over time as the signal changes. This happens because
the sudden-stop
An
5
In
tliis
jump
is
not directly contractible.
Illustration:
we
section
illustrate
case study since Chile
flows' volatility.
is
The Case
of Chile
our main results througli an application to the case of Chile.
an open econom>\ with most of
Moreover, the price of copper
(its
its
This
is
a good
recent business cycle attributable to capital
main export)
is
an exceUent indicator of investor attitude
toward Chile.
5.1
Calibration
While our purpose here
is
only illustrative, and hence our search for parameters
some time describing our estimation/calibration
Estimating
5.1.1
we
still
we use only the
we spend
of the key function A(s).
rare events. This
highlight our procedure because
Similarly, a key issue
rather informal,
X{s)
Sudden stops are very severe but
ever,
is
is
it
makes
it
hard to estimate
X{s:)
with any precision. How-
provides a good starting point for an actual implementation.
determining the components of the signal,
logaritluri of the price of copper. In
s.
Given the hmited goal of
a true implementation
24
it
also
this section,
would be worth including
some
global risk-financial indicator, such as the U.S. high yield spread or the
moving trends from the
EMBI+, and removing
slow
price of copper.^''
With these caveats behind
us, let
us describe the reduced-form Markov regime switching model
we
use to
estimate A(s). Unlike the standard regime-switching model, ours has time-varying transition probabihties.
Our
left-hand side variable
is
=
Yt
The growth
the regime
o-Q-
tJ.o
+
the regime
Otherwise, the regime
"normal" and growth
is
is
=
=
Q\zt
0,Xs,seit,t+i))
Pr(24+i =0\zt
where poi
=
1
—
Poo, Pio
=
=
following:
= exp"/.'^' ^(-")<^- = poo
Pr(2t+i
1
=Pio
l,Xs,s€{t,t+i))
—
g,
and Xt stands
and
1.
When
the value of
normal variable with mean Hq and variance
just a
assumed to be the
is
by the process:^^
0.
that takes values
Zt
"abnormal" and growth has a lower mean,
transition matrix between the two states
Pr{zt+i
is
to be generated
£-7^(0,1),^! <
l[^^=i}^ii+a^_^s,
rate of Yi depends on an unobserved regime
is 0,
we assume
aggregate demand, Yj, which
+
^Zg
=
Pr(2t+i
Mi>
l\zt
=
=
l|-2t
^nd a variance
0,X,^se(t,t+i))
=
= Poi
l,Xs^se{t,t+i))
for the regressors that enter the
The
a\.
=
9
hazard function (the
logarithm of the price of copper in this application).'^ For the purpose of estimating A(s), we use annual
GDP
data starting from 1976 to 1999 and monthly copper data for the same period (source, International
Financial Statistics).'''
For estimation,
we
use a Bayesian approach that
is
Bayesian inference seems natural in this context, since
suitable for the very few datapoints that
it
we
have.
allows exact statements that do not require as-
ymptotic justification. To estimate the parameters of interest (namely (ao,ai)) we use a multimove Gibbs
Sampler as described in
(1989).
Kim and
Nelson (1998,1999) and
is
For details we refer the reader to these references.
'''Removing the slow moving trend not only seems to improve the
based on the
The
fit
filtering
basic idea
but also
is
is
to
algorithm of Hamilton
augment the parameter
a key ingredient in designing long run
insurance and hedging contracts. Few investors/insurers are likely to be willing to hold long-term risk on a variable that
may
turn out to be non-stationary.
'*A11 aggregate quantity data are highly correlated during large crises; hence the particular series used
is
immaterial for our
purposes.
'^Strictly speaking the quantity
exp~
J«
.^(^v)<'u
jg
unobservable, since we cannot observe the continuous path of copper,
only discrete points. However, w"e can obtain copper data at reasonable high frequencies so that w"e can safely ignore this issue
and calculate the
integral
by a Rieman sum as
exp-/.'^'^<-")''"=^exp-£^(-'')^"
'^There
due
is
a caveat here.
to political turmoil
Our extended-sample
combined
w-ith
starts
abnormal years these do not influence the estimation
states). ]t is in this sense that the
from 1972, but
all
the years up to 1976 are part of a deep contraction
extremely weak external conditions. However, since the extended-sample
sample relevant
of X{s) (w-hich
for
is
estimated from the transitions from norma! to abnormal
the estimation of the latter starts
25
starts during
in 1976.
For details see below-.
by treating the unobserved states as parameters. Then we
set
(/ig, ^ij
,
ctq.
f1
,
<?,
oo,
cti
we sample
Kim and Nelson
the conditional
means
a'o>/^i
leads to a normal / trucated normal posterior.'*
(1999). Given the
same
Conditional on the states and the draw from
and the draw from
Similarly, fixing the states
class.
beta prior leading to conjugate beta posterior.
The samphng
gamma
(i-Iq,
where we sample from a bivariate normal
priors
(/^q, /.ij)
se
which lead to posteriors
in
we draw
iii,C(j.Oi)
new draw we can
using a conjugate
use a Metropolis Hastings- Random walk-
centered at the previous iterations' estimate
Robert and Casella (1999). This provides us wdth a new draw form
as described in
q
of (qq.oi) presents the difficulty that there
we
does not seem to be a natural conjugate prior to use and thus
accept-reject step
draw from the posterior distribution
using a conjugate normal / trucated normal prior which
sample from the posterior chstribution of {oQ,ai) using inverse
the
a draw from the posterior distribution of
and conditional on these parameters we draw from the posterior chstribution of the
)
states in a sigle step as described in
of the states
fix
(/Iq!
Mi-f^o.
'^i-
(/,
«i)
<^o.
the states again, b;ised on the
and conditional on
this
new draw
then a standard result in Bayesian computation that the stationary distribution of the
etc.
It is
iterate the algorithm
by
filtering
parameters sampled with these procedure (treating the unobserved states as parameters too) coincides with
the posterior distribution of the parameters.'^
A first
output of this procedure
Figure 2 which plots the probabihty of being in an abinormal state.The
is
model recognizes roughly three years out of the 24 as abnormal
The
following the Asian/Russian crises.
was
significantly
more
years: the early 1980s
and the recent episode
early 1980s episode corresponds to a devastating debt-crisis,
severe than the recent one. In fact, the recent episode appears to be a
and
mix of a milder
sudden stop and a precautionary recession.
An
interesting observation about these probabihties
is
that they allow us to identify the abnormal regimes
with great confidence. To improve the tightness of the posterior confidence intervals concerning the parameters of interest (ao,Q:i)
we observe
X]
where x, takes the
\'alue
1 if
that conditional
[Xtlog(poo(«o-a:i))
there
is
on the
+
(1
-Xi)log{poi(Qo,ai))]
no transition to state
to notice that the hkelihood for (qq, oi
)
states the (log-) likelihood function becomes:
1,
and takes the value
depends on the data only through the
otherwise.
It is
interesting
filtered states. In other
words
'*For details see Albert and Chib (1993).
To reduce computational time and
used proper priors
was Normal
/
for (/jqiMi)
^"d
satisfy the technical conditions required for the applicabibity of the
{o'0,o>l)
and improper priors
Truncated normal with means
(0,0)
for the rest of the
we used independent normal
Gibbs Sampler, no
The proper
prior for (aiqi/'i)
and a diagonal covariance matrix. The standard deviations where
to be roughly 5 times the range of the sample, so that the priors
for (ao,o;i)
parameters.
priors centered at
had effectively no influence on the estimation.
with a standartl <leviation of
10.
cho.sen
Similarly
Even when we experimented
with more diffuse priors the algorithm jjroduced virtually identical results but computational time was significantly increased.
because convergence was significantly slower. Most importantly though,
u.se in
the calibration exercise we used completely
flat jjriors.
26
for
the results that
we
report in Table
1
and that we
Probability of
0.6
abnormal regime
-
0.3
1975
1980
Figure
all
1985
2:
1995
1990
Probability of abnormal regime
other parameters of the model including the
GDP-data
are relevant for our purpose of estimating (qq. ch)
only to the extent that they influence the identification of the states. In other words
directly
on the states we would be able to get
Given the few data points and the quite
rid of the noise
introduced by
clear identification of the
if
we were to condition
filtering.
abnormal
states
we
report directly
the posterior distribution of (ao.«i) conditional on the early 80's and 1999 being the abnormal states,
we
report the posterior distributions of the parameters (ao.cn) conditional on the states that are identified as
abnormal.^" Table
reports these results.
1
Correspondingly, in what follows we use qq
=
2.5,
qi
=
5.2 as
our benchmark values for the function
X{s).
5.1.2
We
Other parameters
calibrate
4>
and
the sample, which
T
is
to generate the average cumulative consumption drop caused by the sudden stops in
about 8-10% of GDP. One such configuration
The parameters k and 7
^"^I.e.,
states that
h.i\'c
are calibrated to generate reasonable
is
T=
1
amounts
and
<p
= 0.35e''-^.
of steady-state debt (and
a postfrior probability of being abnormal above 0.75. Tlie conditioning
the precision of the e.stimates and seems to be warranted just by a visual inspection of Figure
27
2.
is
done
in
hence
order to increase
Posterior Distributions
mean
"1
Table
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
1:
Abnormal.
10,
25
etc. refer to
the respective quantiles.
reasonable amounts of insurance needs) together with significant precautionary fluctuations (see below, in
particular, to explain
1998-9 episode). For
The
to a
much
of the precautionary recession experienced by Chile at the early stages of the
we
tliis,
set
=
k
interest rate r is set to 0.09
0.8
and 7
=
and the growth rate
path that grows significantly faster than that
Xo, to
Finally,
to 0.03.
level for a
The
latter is
(yo) to 1,
and
is
catching up, and
set the initial
debt-to-GDP
0.5.
we approximate the process
for
copper by a
tlriftless
Brownian motion with constant volatihty
which we estimate using monthly data from 1972 to 1999 (source, IPS).
and normalize the
5.2
a constant-rate approximation
few years, while the country
We normahze initial GDP
then decelerates below that level forever.
ratio,
7.
initial
value of s to
We
find a value of roughly
a
=
a,
0.23
0.
Aggregate Hedging
Recall that our concern in
tliis
economy
is
to stabilize both precautionary recessions
and sudden
stops. Let
us start with the former.
5.2.1
Hedging Precautionary Recessions
Evaluating at
in
^
as
1
,
and
s
=
in the calculation of
^
,
we can approximate the
volatility of log
consumption
our model before the sudden stop takes place by:
-(l-K)--^rr==0.01.
That
is.
despite the absence of income fluctuations in our model, fluctuations in precautionary behavior
can account for about a fourth of consumption volatility in the regimes that are characterized as normal by
our algorithm. More importantly, the contribution of precautionary fluctuations
sudden
stops, as
Panel
shown by Figure
(a) in this figure
a sudden stop takes place.
is
particularly relevant near
3.
shows a random realization of the path
The main
s
that runs for nearly eight years before
features of this path are not too different from the realization of the
28
0.24
— ^ 023
No Hedge
Hedge
y
^\\
g
"•^'
I 0.22
C
«
<D
;j:/!L
,../>
--
'V:yr
•\i^
0.21
..-'^'^M
0.2
;'.;/"
/
0.19
4
time
Figure
3:
p-Hedging
for the Chilean case
price of copper during the 1990s. In particular, the large rise in s near the middle of the path followed by
a sharp dechne toward the end of the period
is
reminiscent of the path of the price of copper from 1996 to
2000.
The dashed hne
in the
bottom
left
panel (b) illustrates the corresponding path of (excess) consumption
generated by our model without hedging. The
More
hne
interestingly,
drift in the process
is
due to average precautionary savings.
one can clearly see the precautionary recessions caused by the decline
in the top right panel (c)
shows the impact of the
latter
in
s.
The dashed
on the current account. In particular, the sharp
decline in the deficit in the current account from 6 to about 3.5 percent of
GDP toward the end of the sample.
This suggests that about half of the current account adjustment observed
in Cliile
during the 1998-9
crisis
can be accounted by optimal precautionary behavior in the absence of hedging. The rest of the adjustment
could be accounted for by excess adjustment (some have argued that the central bank contracted monetarj'
policy excessively during this episode)
and by the
partial
sudden stop
itself
(the "sovereign" spread tripled
during this episode).
Now,
let
the country hedge by shorting copper futures, which do not relax the sudden stop directly but
29
Xr
only through their effect on
to the
(p-hedging).
The
dashed paths of the unhedged economy. Panel (b) clearly shows that hedging, even of this very hmited
kind, virtually eliminates precautionary fluctuations.
consumption from precautionary cycles
improvement
in the current
latter are reflected in a
As a
practical matter,
are very large.
The
is
Interestingly, panel (c)
account as
size of
s deteriorates
The
difference
comes from the
comes from hedge-transfers
sharp dechne in external debt as s worsens (panel
it is
shows that the insulation of
not done through a smoothing of the current account, which looks
hedged and unhedged economies.
virtually identical in the
The
show the paths corresponding
solid lines in panels (b) to (d)
fact that the
in favor of the coimtry.
(d)).
important to point out that the hedge-ratios required to achieve this success
the imphed portfolios can be calculated as (evaluating at s
=
0):
^(Xo + (1-k)-^)«-0.56.
r
b
That
the notional
is,
very large
amount
g
would have to exceed 50 percent of GDP. This
in the futures position
when compared with the
existing futures
markets
for copper, pointing to the
or related markets for sudden-stop-insurance purposes. However,
it is
is
need to develop these
not a disproportionate
number when
compared with the very large and much costher practice of accumiilating international reserves (around 30
GDP).
percent of
and
Finally,
in
as indicated in section 4, this strategy
smoothing the sudden stop
hedging of
this sort
—
if
Our estimates show
itself.
subject to
full
crowchng out
—
is
not very effective
that in case of a sudden stop, consumption with
exceeds consumption in the unhedged case only by about
2%
for
consumption drops of
roughly 8-10%.
Hedging Sudden Stops
5.2.2
now analyze the opposite extreme and assume
Let us
(/i-hedging)
.
Full insurance in
(Xo +
case requires a transfer in the case of a sudden stop
~^]
(1
h'
tliis
-K)yo\ _
=
/ ,.,
(0(1
Importantly, note that 15 percent of
tion to
be
in case of
in the case of
„-(r-s)^^
e-(^-^)^)
+
,
„oT
e^^
-
of:
,(l-«)yo'
Afv
l) (Xo +
r
—
0.15.
g
GDP exceeds the 8-10 percent we calibrated the dechne in consumpis
because 15 percent covers not only the drop in consumption
a sudden stop but also the precautionary sa\dngs and recession that precedes the sudden stop.
depends on the
initial
NPV
of such
a claim, we must multiply the number above by
E[e~'"'^] for different
Using the benchmark case, we see that
GDP
E[e~'"^]. Since the
value of the signal s and on the estimates of a's, which are very imprecise,
report in Table 2 the value of
percent of
-
a sudden stop. This
In order to calculate the
latter
that hedging directly relaixes the sudden stop constraint
(0.15 times 0.536)
if
we
combinations of these parameters.
"fairly" priced full
insurance would cost the country about 8
contracted when the price of copper
30
is
at "normal" levels
and about
5
—
—
Table
percent (0.15 times 0.323)
cost of
2:
s
=
s
=
(2.5,5.2)
(5,5.2)
(2.3,0)
0.536
0.247
0.517
0.323
0.139
0.523
1.5(7
Values of E[e~'"^] for Different Combinations of (qq, oi).
when
While these amounts are small when compared with the
at very high levels.
sudden stops and precautionary recessions, they
for these countries to undertake,
Fortunately,
insurance. In
much
what
even
still
amounts which are probably too large
involve
such markets existed.
if
amounts of
of the advantage of insurance can be obtained with significantly smaller
follows,
we take h
and
as given
n
maxE
solve:
r.*i
f.*i
^
1
r
-7
s.t.
dXt
= {rXt-cl + y;)dt
Xq+
= Xq- — Po
Po
In other words
costing Po which
is
we assume
e-'-^hiXr-
+
^^
r-g
that the country starts with an initial debt
priced fairly
common knowledge
is
=E
and denominated
to both the borrower
as a fraction h
and the
<
Xq-
,
and purchases insurance
h* of the quantity X^--
tools
Po-^^
type of contingent claim results
we developed
The
The
first
in a
problem that we can
in the previous sections, plus a fixed point
results are
summarized
row shows
Fqi the
in
Table
second
cj,
problem
stop.
The next
three rows
while the last three columns do the
same
for sq
There are three main lessons to be learned from
insurance
is
easily solve numerically
maximum
show consumption
=
The
first
with the
The
right before the
final
row describes
three columns present values for
1-5(t.
this table.
First, clearly
much
of the cost of a lack of
paid for with the large precautionary behavior that the economy needs to undertake without
^'Eveii though the contingent claim signed by the borrower and the lender seems complicated,
not varying
Using
value attained by excess
after the crisis, respectively.
the standard deviation of c*+ for a sample of 1000 simulations.
0,
-^f-.
for the initial price of the insurance,
while the third row shows the
sudden stop takes place, during the sudden stop, and
=
X^- +
3.
consumption along the path before the sudden
So
setup
lender, so the lender prices this claim understanding the
subsequent optimization problem of the borrower and accordingly the resulting path of
this particular
+ ^^. The
ver>'
much and
can be approximated reasonably well by
31
Xq + -^,
it
isn't since
especially for large h.
-V^- H
^^
is
.
1.5rT
So
h/h':
0.4
0.8
0.000
0.019
0.039
0.252
0.213
0.247
0.252
0.248
0.253
0.243
0.252
0.254
0.212
0.244
0.252
0.224
0.247
0.253
0.010
0.002
0.002
0.011
0.002
0.001
4+
0.125
0.175
0.229
0.131
0.177
0.230
<+T
0.300
0.269
0.2.59
0.313
0.271
0.260
Standard Deviation of c*^
0.013
0.002
0.001
0.014
0.002
0.001
Initial
Cost
c5
C'r-
Standard Deviation of
c*^^^.
-c;_
Table
insurance.
be seen
in
The
level
dimension of
— c*^
.
less
itself is significantly
0.000
0.032
0.063
0.198
0.243
0.226
/i- Hedging
3:
cj,
while the cyclical component can
Second, much of the precautionary costs can be removed with
very limited amounts of insurance; a value of h
stop
0.8
can be seen in the value of
this effect
the difference between c*max
0.4
=
harder to smooth, but
0.4 does nearly the
same
as one of 0.8. Third, the
possible to
make
a difference with h significantly
it is still
sudden
than h"
5.2.3
Asymmetric
Beliefs
and Contingent Credit Lines
Before concluding, we discuss an important practical
country's actions and there
is
significant
issue.
Sudden stops are not
entirely
exogenous to the
asymmetric information about these actions between borrowers (the
country) and lenders. Suppose tlien that the financial markets overstate (from the \'iewpoint of the country)
the constant part of the hazard, qq. In particular,
The borrower on
the other hand takes this
borrower and the lender be
E""
we assume
number
to be
Oq
that the lender takes this constant to be Qq
<<
Qq
.
.
Let the expectation operators of the
and E^, respectively, so that the problem becomes:
"-""'
^-ds
1-7
+ e-^^V^^ {Xr^)
s.t.
dXt
-=
[rXi
-c*+yl)dt
Xq+ = Xq- — Pq
Po
= E'
e-'-^h{X°1'
+
In this case the country obviously will find the price of the insurance "unfairly" high.
32
One way
to reduce
u;=3
uncontingent
0.4
hlh":
Cost
0.8
0.032
0.063
0.009
0.020
c5
0.219
0.247
0.250
0.238
0.246
<-
0.219
0.246
0.251
0.245
0.255
<+
0.128
0.175
0.228
0.155
0.187
Standard Deviation of c*+
0.009
0.002
0.001
0.016
0.026
Table
the extent of this problem
this CEise the country
fiscal deficits
is
for the
ft— Hedging with
4;
asymmetric Beliefs
country to add another contingency to the credit hne.
would benefit from making the hne contingent on the value of
crises not related to s as
this feature
0.4
0.000
Initial
run up
0.8
—
much
it is
less likely
optimal for
than the markets - for example,
it
clear that in
Since the coxmtry sees
s.
may be
not to pay for insurance in states when s
it
It is
certain that
is
it
will
not
high. Let us capture
by assuming that the borrower and the lender enter an agreement whereby the lender agrees to
pay:
Vr
/(sO/iLy!? +
to the borrower and the fraction of h paid out depends on St
fiSr)
exp{—w{s-r
=
1
Obviously
<
/(s,-)
and only when
Table
4,
s,-
is
<
1.
Taking Qq
~
2.5,
+
2a))
+ exp(— u;(st- + 2a))
a^
=
5
and
w=
very low (2 standard deviations below
3, i.e.
0),
a claim that pays quite steeply
the results are reported in the
when
first half
of
and contrasted with the case without the additional contingency.
By adding
the additional state contingency, a country that
is
confident of
its
"good behavior"
is
able to
lower the price of the claim without incurring a very significant rise in risk exposure.
6
Final
In this paper
Remarks
we
characterized
many
aggregate hedging strategies. For
the key issues but
realistic
aspects of sudden stops, precautionary recessions, and the corresponding
this,
we
built a
model simple enough
to shed analytical hght
on some of
enough to provide some quantitative guidance.
We showed that even after removing aU other sources of uncertainty,
stops and the mere anticipation of
them has the
the combination of infrequent sudden
potential to explain a large share of the volatile business
33
cycle experienced by emerging
market economies. This important source of
volatility could
be overcome
with suitable aggregate hedging strategies, but these would require developing large new financial markets.
In our application,
we hinted
at
one aspect of the insurance arrangements that could faciUtate the
development of such markets by reducing the inherent asymmetric information problems.
credit hues
for
and
financial instruments being contingent
example, Chile could remove
many
its line to
the price of
oil
of the signaling
problems
and US GDP, and so
There are several aspects of an aggregate
we
did not
risk
this
discuss accumulation of international reserves, as these are
these issues properly,
signal worsens.
Of
until the crisis
is
we
also
need to
course, a country
imminent, but
may
for
it
fears from the
IMF's Contingent Credit
Mexico could do the same by
on.
management
model the maturity structure of debt and how
reserves financed with long-term debt
argued
on indicators that are exogenous to the country. Thus,
Lines by adding a clause linking the size of the line to the price of copper.
indexing
We
strategy that
we
left
unexplored. In particular,
changes with the signal,
dominated by oiu credit
s.
Similarly,
lines.
we
did not
However, building
partially substitute for overpriced credit lines. In order to address
ha\'e a better
understanding of the behavior of the supply side as the
would want to postpone accumulating reserves and borrowing long term
it is
highly urdikely that the lenders will go along with this strategy.
intend to explore these dimensions in future work.
34
We
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S.
Bayes inference via gibbs sampling of autoregressive time
markov mean and variance
[2]
C. Burnside,
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Ricardo
J.
shifts.
J.
Bus. Econ.
Eichenbaum M., and Rebelo
Ricardo
J.
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J.
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Caballero and Arvind Krishnamurthy. International and domestic collateral constraints in a
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Scharfstein,
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Michael Harrison.
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Statistics.
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AM J.
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Krugman
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36
Universitext. Springer-
Appendix
7
Propositions and Proofs for section
7.1
Proposition
The optimal solution of the r period problem
1
C;°P*
X-r+T
and
Moreover the maximum in (61)
the above statements,
we have
Proof. To establish the
=
and thus
j/i-e*-^
rT
/
= yx
= max{X,
—
r
r
-g
(60)
(61)
}
g
r
1-7
X
always given by
is
That
.
is,
the constraint
is
always binding. Combining
that the optimal consumption-wealth pair is given by (12)
first
assertion
we need
to
show that the proposed pohcy
and
(11).
(we normalize
satisfies
without loss of generahty)
+ e-^^V{X°'" + ^2^)>
c*
,
X^,
for
e-'-'^vldu
1-7
Jo
< u <
>
By
T.
f
feasibility
„*
+ e'^^ViXr +
du
^^)
r-g
we have
+ e^'-^Xr
e-'-'c^du
Jo
Xt
where we can focus without
e-^^^
r-g
any other admissible policy pair
^*l-7
i-T
t
au
Xo+ f
Jo
By
T<t<T+T
(fcl)-^,
/*opt\ 1-7
g-..iS^J
Jo
for
=
is:
ki is determined as the (unique) solution to:
r
T
2.3.
loss of generality
X
>
on deviations that
satisfy the first equation as
concavity of the utility function and the terminal Value function
1-7
L
it
an
equality.
follows that:
r-g
^
1-7
\Jo
^
/
^-ruc_u
l^H_J
<
f
I
€-""
e-^- (c*°P'^
(C')"'^
Jo
Now we
constant
fci
du
1-7
Jo
r-g
+ e-^^[V{XT +
V
+ ^^^)
^^)-V{X^'
'r-g
r-g
- X°P'
X
-^)(Xt
« - CP*)du + e-^^V'iX^"' + -^^^^MXt
r-g
fc*.
c*°P')du
e~^'^V'{X°J"
need to distinguish between two cases:
such that the budget constraint (62)
is
a)
if
X^^ > X
satisfied
and the
then V'°p*
result follows
last expression:
^-1
I
/
(/
J
e-^-{c:
-
cl°P')du
+ e-^^(Xr -
37
X'^p')
1
=
=
ki
=
(C^*)"^
upon substituting
for
a
in the
since both policies satisfy:
7o
Jo
b)When Xj^ = X,
is
feasible,
it
it is still
the case that
Xt > X = X^^
also satisfies
These two arguments can be combined to
(c^"''')
^
—
and because the alternative candidate policy
kx,
Because of concavity of
.
V^'this also
impMes V'{X
+ -^) <
fci-
get:
/ e-'-"(CP')"^«-CP')du + e-''^V"(X + ^^)(Xr-X°'")
r - g
=
- X""*)
f e-'""A:i« - CP')du + e-^'^V'(X + -^^^){Xt
'^-9
Jo
<
Jo
/
+
e-'-"fci(c:-c°P')du
e-''^fci(,Yr-X°P')
'
=
Jo
This
verifies the optimality of the
proposed
policy.
For the second assertion, that the constraint
otherwise.
Then
is
always binding, the argument runs as follows. Suppose
k\ is given as:
r
r
-
g
Under our (counterfactual) assumption we should have:
X^
{4>{\
-
e"''"-''*^)
+
+ 0^^(1 -
e^^)
r
or
-
e-(''-9>^)
< Xr +
r
-g
g
—
Xr <
-
g
e-*'"-^'^)
-
r
(63)
since
(1
But
=
-
es^)
<
consumption when combined with the transverality condition
0.
Propositions and Proofs for section 3.1
In this section
the
(/.(I
(63) contradicts non-negativity of
lim,_ooe-'-'.Y,
7.2
-
same
we proof
all
the steps of the perfect signal case.
steps but has a few steps which are
much harder
numerical simulations.
38
The proofs
to proof, hence
of the imperfect signal case follow
we only
\^lidate
them though our
)
,
The proof
consists of the following steps. First
value function should satisfy.
text subject to the required
that
we use
Then we prove the
we
boundary conditions that the
establish rigorously the
ODE
existence of a unique solution to the
boundary conditions. Then we
in order to apply a (classical) verification
verify the
"tail''
Theorem along the
presented in the
condition on the value function
lines of
Fleming and Soner (1993)
p. 172.
Proposition 2 The value function of
the
problem with sudden stops for
\imV{Xr,s,y;)=
Proof.
we
We
7 >
focus on the
allow hedging or not
(i.e.,
IV {Xr^^)
{in
s satisfies:
1-7
(64)
two steps and does not depend on whether
whether we require that the portfolio p
of the optimization problem). First one obtains an upper
sudden stops, which in this case
>
1-7
The proof proceeds
1 case.
s
=
or whether
p
determined as part
is
bound on the value function
of the problem with
naturally given by the value function of the problem in the absence of
is
sudden stops:
1-7
V NSS
1
J-
To
see
why
this is
—7
an upper bound notice that the following inequahty holds
for
(Xr^ +
V
\-Hu-t){<LJLdu
>• -'Al-I ^
e >" -'^^
au + 1^7-<oo^e
1{t < oo}e-^(^-"K
f-
I
1-7
any
^
feasible policy:
1-7
r-c,
'-
\Ft
:
1-7
Vr
l-->
<
E
^
1-7
^
\Ft
1-7
\rj
1-7
-r(u-t)
V'-n)
1-7
since for
7 >
when one
-du
+
1{t < oo}e-^^^-*^V^^^{Xr,y;)\Ft
XxH
1
1-7
<
0.
This holds true for any feasible consumption /portfolio policies and thus.
evaluates this inequality at the optimal plan c*°P',p*:
V{XuSt,y:)
/
op(\l-7
„-r(u—t) V-u'—^J
1-7
/IN
du
+
1{t
<
l^*opt \ 1 -7
r
-r{
1-7
7
(xT
oo}e-'"(^-*^A'
1-7
/
[{t
<
oo}e
39
+^y~''
1
N 7
(X°P'
+ ^^)
1-7
-\Ft
1-7
l^(
=
-
(
< max E
* X^"'!'
1-7
+
du
1{t
<
V^^^iXuVl)
oo}e-''("-''l/(X^,y;)|Ft
1-7
1-7
rj
The
last equality follows
true as
from the principle of dynamic programming. Since
upper bound. The lower bound
—> oo. This establishes the
st
consumption /portfolio policy
in the
Xt > —-^ti
for
^U
^
)•
It is also witliin
it
satisfies the
the class of pohcies that
also
it is
upon observing that the
established
absence of sudden stops c^ss pNSS
/portfoho plan in the presence of sudden stops (since
also
is
this holds for all Sj
a
(=^0) is still
consumption
feasible
intertemporal budget constraint and
mandate Pu
=
0.
Accordingly
we have
the two inequalities:
(x;NSS
V^''-^=\XuSuVl)>E
l-l
Kt
,
r—a
\Ft
1-7
1-7
-r(.-t)i5^^
VSs.v=P°-\x^,suy;)>E
^^
1-7
i:
However by standard arguments
it is
^
e-'-(^-"l{r
—
easy to show that as s
oo
>
< oo}K
,
Pr(T
-)
\rj
^{xr' + ^^)
1
< Q) —
»
\Ft
- 7
for all finite
Q, and
thus
1-7
„*NSS\'^-'^
lim
s—•oo
E
/
e-(.-<)ifl^
Jt
1
1-7
du + e-'-(^-')A-l{T < oo}
(
1-7
V^
1-7
du\Ft
\Ft
V^'^^C^t.yr)
1--I
1 — 7
yi^ss
Since the upper and the lower bound coincide to
as s
^r J
Before
following
we can
invoke a standard verification
Theorem we
—
»
oo, the
claim
is
estabhshed.
establish the existence of a solution to the
ODE:
-yC
Even though
this
>
-^rC + ^iCs +
second order non-linear
hard to establish that
it
ODE
has a unique solution that
ifsso"^
=
(65)
does not seem to have any closed form solution
satisfies the required
in the following proposition:
40
boundary conditions.
it is
This
is
not
done
ODE (65)
variables a"^ = C
Proposition 3 The
change in
has a unique solution satisfying the boundary conditions (26), (27), with the
.
Proof. The steady state of
this
2nd order nonhnear
ODE
can be obtained at once
as:
7
C^""
The Theorem now
a system of two
first
-
(66)
basically follows from the stable manifold
Theorem upon reformulating the system
as
order ODE's, realizing that the system has one positive and one negative eigenvalue and
applying the stable manifold Theorem (see
thus
=
we do not provide any
details apart
Verhulst(2000)
e.g.
p. 33.)
The argument
is
particularly easy
and
from the fact that the linearized system has one positive and one
negative eigenvalue. Indeed, one can approximate the solution by means of a hnear approximation around
the steady state value to get:
7(C^^)^+(7-l)(C^^)"MC^-C^^)-r7C + Ma + ^a.a2 =
The two
(67)
eigenvalues of the characteristic polynomial of this equation are:
^ ,±^,^-,2o-r
Obviously one eigenvalue
is
positive,
one
is
negative and this establishes the claim after a few straight-
forward steps and the use of the stable manifold Theorem.
Remark 4 A
ODE
simple reformulation of the
(65), yields the function a{s) that
we use
in the text
and
is
obtained by defining:
and rewriting the
ODE
7aT-i
as:
^ _^^^7 + ^^a^-ifl^ +
1^2 (7(7 - l)a^-2(a,)2 + 70^-ia,,)
=
or:
l-ra +
;ia3
+
^a2U7-l)i^ + a,, =0
(69)
)
subject to the boundary constraints:
a{s)
lim a{s)
5
The
last step in verifying the fact
the problem at hand
is
—too
= A-^f-l
(70)
=
(71)
(
)
V ^ /
that the conjectmed Value function
to invoke a verification
Before we do that we prove an almost obvious
is
indeed the Value function for
Theorem along
the lines of Fleming
Lemma
of independent interest:
41
that
is
and Soner (1993)
p. 172.
Lemma
1
Consider a consumption policy that has the feedback form:
c*
=
A{st)
(Xt +
Then Xt > Xj^^^ where Xj^^^
^"j
the asset process that results
is
,
K-^'^r < A{st) <
,
r
Vst e {-s,
+00)
when one uses
the optimal
consumption
policy in the absence of sudden stops:
c^ss
^
(,^Nss ^ _yL_
^
r-g
\
Proof. This result
is trivial.
One
solves for the asset process that results
from the two
policies to find
that
^NSS
The
Xo -
equivalent calculation for the optimal policy
d (e-(^'-/o ^(«.)d-)x,)
Now
integrating both sides
X,
=
^^
-5
yp
Ct
=
-(r
=
,-frt-L- A(s,Adu)
-
-
1)
conditional on a path of
A(s.)du)
yl(si))e-('-'--^o
and rearranging
Xo + ^e('-*-/o
f^gt
(e^'
r
^^ ^ e'i^'-fi
A{st)
[
r-g
-
\. r -
result
now foUows
To invoke a
Lemma
since A{st)
verification
<
Theorem we
2 Assume (i)^ < C{st)
<
gives:
Ms.)du) f
r Vst €
f'
^-Hr-g)^-fi .4{.„)d.)(^
{5,
_
+00).
need to show
finally
K {^Y
Vs,
that:
€ {s,+oo). Then
-
lim
Proof. For any
t it is
E
e-'-'l{r
>
t}C(s]
N
1-7
=
1-7
the case that
e-'-'l{T
>
t}C{s)
>
1-7
1--)
E
[xr
e-''*l{T
>^}A-
+
^
1-7
1-1
KE
e-'''l{T>^}
^^'-^''")dXt
g ^^;
Xo-^2_('eSt_e('*-/,M(^..)<'.)^
r - g \
J
The
S( gives:
\V {xr' +
^)
1-7
42
^(^.,)
_
^)^.\
^
—
The
inequality follows from the assumption
first
The second from the previous lemma and
and the
fact that
'-
-^^
—iz:~
is
the monotonicity of the value function and the last limit follows from
the fact that one can tri^dally show that in the standard model without sudden stops e
The
in a rather straightforward
— C~
a{s)
(If it
increasing).
To
establish that
maxima
extremum. Then asis")
=
— ra <
it
1) 1
to
be a
local
0.
Then
the region where
2)
1
— ra >
0.
>
1
s*
—
Then
To
or
we
establish the claim
it
we
decreasing,
would have
just need to
to
<
show that
Now
<
minima. So suppose indeed that there
would have to be
0.
exists
ODE)
it
one point s* that
that a^^
maximum, which
local
<
<
would be
is
a local
>
0,
so that s* would have
impossible since
and a^sis**)
>
which
is
it
would
still
would have to be a
local
maximum. But
minimum
Lemma
Lemma
impossible.
consumption process:
derives the properties of the
3 The optimal consumption process
^=
del
c^
Proof. Straightforward apphcation of
1
satisfies:
,
Ito's
•
/
oa,\'^
,
a,
,„
+ l)(—^) dt^^adB,
^
o(7
2
\
a
Lemma
~
/
to:
«(»,)
along with equation (69.)
43
in
s** to the right
one can apply a standard verification Theorem and verify that indeed the proposed function
following
be
a{s**)).
so that 5*
0,
is
the Value function for the problem.
The
everywhere
cases:
(Since -necessarily- a{s*)
has to be true that Uss
a{s*)
is
have at least one section where
order to satisfy the boundary condition at infinity there would have to be a local
of s* satisfying a{s**)
a(s)
order for a{s) to satisfy the Boundary condition at infinity there would have
in
that
ra
-.
are going to establish 2 contradictions by studying points
has to be true (in order to satisfy the
,
it
it is
and
a decreasing function that stays between -K^'~'
is
and accordingly we have 2
minimum. But
to exist another s**
in
,
exited the "band" [^, ^K^^'']
that could be local
^'
by means of two contradictions. Suppose a{s) solves (69) subject to
will derive the result
the boundary conditions (70), (71).
decreasing.
^J
manner:
Proposition 5 The function
We
^
\j:)
prove that any solution to a{s) (respectively C{s)) will be bounded. This can be done
final step is to
Proof.
a negative number.
a
V
is
Propositions and Proofs for section 4.1.
7.3
We
first
first
show that the excess consumption process, Cj,
one
is
to apply Ito's
lemma
to
Cj
is
There are two ways
constant.
and observe that both parts of the
Wang
for distributions of hitting times of
tliis.
The
resulting Ito process (informally
speaking the dt and the dBt terms) are identically zero. Another more direct way
methodology developed by Karatzas and
to see
is
to use the martingale
(2000) to deal with this prot>lem along with standard formulas
Brownian Motion. This approach
particularly appealing in the context
is
of this section because of the presence of complete markets (up to the point where the sudden stop takes
place).
We start
Lemma
with the
first
approach.
We
have the following result:
4 Under perfect hedging "excess" consumption,
Cj
,
is
constant for
t
<
t
Proof. To simplify notation somewhat we define
aP(st)
=
(e-^'("-^(Ki -l) +
J
and observe that
The consumption pohcy can then be
aP{st)
r-g
V
as:
a
Lemma
(73)
f^^^ydi-^)
1
and the portfoho policy
l=0
reexpressed by means of (41) and (72) as
'
Ito's
(72)
this function solves the ordinary differential equation:
Ja2<, + ^iaP-ra^ +
Applying
l)
to the right
hand
r-g
\
side of this expression gives:
dc'
=
g
) \aP
Cidt
+ C2dBt
where
aP \
a? \
oP \
(aPfV
r
-
aP \
r
-
r-g
T
g
\ aP J
—
aP
J \ aP
A2
"
44
'
^
J
T
)
9
2
2
—g
and
T- g
aP \
J \ aP J
aP
r
\
-
g
J \ aP
'
In other words perfect hedging leads to complete smoothing in our example, despite the possibility of a
fall in
consumption when the state variable crosses the
critical level.
Let us prove the same result with the martingale methodology developed by Cox and
Karatzas, Lechoszky and Shreve (1987), as
be using the results
Lemma
5
in
Wang
Karatzas and
it
facilitate the proofs later on.
(1989) and
For oui exposition
we
will
(2000).
Without loss of generality, take
=
t
=
ci
wiU
Huang
The optimal
0.
policy
is
given as:
(fc-i/^)l{«<r}
\r\K)
r-(
where k solves the intertemporal budget constraint:
Xo +
-^ =
fc-i/^
(
E
e-''"du
/
+
1{t < oo}
e"''"
J
or
k
Xo +
=
(Ej^e'-'-^du
Proof.
We
^1
+ e-'--^l{r <
provide only a sketch of the proof.
The
reader
is
oo})
referred to Karatzas
and Wang (2000)
for
details.^^
The
objective
is:
maxEf
"I'P"
[ ^^^^e-''''du +
\Jo
1-7
e-''''V{Xr,y*r)l{T
<
c»}
s.t.
dXt
and the
We
(2000).
=
rest of the evolution equations
{rXt
-
c*
+
y*)dt
+ aptdBt
remain unchanged.
proceed to show formally that the proposed strategy
is
optimal borrowing from Karatzas and
Wang
Let us denote for an alternative admissible strategy {c*,X*) satisfying the intertemporal budget
constraint with equality the Value of the objective:
Jicp) =
e(
+
[ ^^e-'-^du
-7
\yo
Karatzas and
Even though
it
Wang
(2000) do not strictly cover the case
seems easy to extend
their
e-'-"T/(X,,y:)l(r
<
cxd}
1
we
are considering here, since our problem
is
on an
infinite horizon.
approach to cover our case too, we refrain from doing so and just note that the
techniques used for the limited hedging case at the beginning of section 4 along with the observation about the optimal policy
and the
differentiability of the
a
—function are enough to apply a standard verification Theorem.
45
where
Let / denote the Fenchel
Legendre Transform of the function
-
= max f{x) —
f{k)
/:
kx
so that this maximization e.g. for
-
1
—7
e
~
= max e
"
—
kc„
\^1— 7
X
yields:
and
similarly for X-r-
Then
it is
the case that:
J{c,p)
=
El
<
£
(If^-^
+k (e
<
E
=
E
/
+k
I
e-^'^'cldu
-^
Xo +
r
El
This
verifies optimality of
using well
/
the strategy.
the
e-''^.Y^l{T
-
9/
^-^^—^
known formulas about
+
e-'-'^du
-^
-
first
<
+ e-"K(X^,y;)l{T<oo}
Y^e-^"du + e-^^V{Xr,y*)l{T <
/
j
e-'-''dK
\
+
<
oo}
j
+
oo
e-^^y(X;P',y;)l{T
<
oo}
+
Xo+^^
r -
e-'-''du
g
+ e-'-^V{X°P\y;)l{T<oo}\
We will now
use the optimal strategy to obtain the Value function
hitting times of
Browmian Motions.
V{Xo + ^2-^ r — g
46
;,-(l-7)/7
-
[-. -7
E
-e-^"du
+
1-7
jt-(i—r)/7 / K^/-i\^ fK'^f
e-
1{t
1-7
<
oo}
1-7
(^»
^)
+
{e
1-7
e-^'dif
+
l{T<oo}j
e-'-"
=
1-7
(^0
^)
+
1-7
1-7
-\
—7
1
(l_e-^'(^-^)+e-^><=-^)/^^)''
yr
-Ai(s-s)
1-7
The only important
Vry
(A--
-
+
1)
1
V
step in this straightforward derivation
This involves a well known formula which can be found
is
going from the second to the third hne.
Harrison (1985)
e.g. in
p.
47
Proofs and Propositions for section 4.2
7.4
Proposition 6 For
1
<
^
<
^* ,where:
c*
the solution to the
A
problem (74)
is
given by:
c*
=
/c-7,t e (0,r)
-i
Xr-
l^opt
r
is
r{\
^
/j*
+ h*)
Vl
-
r
_(l_e-rT)(j_^;j
defined by
1-^
e-'-T'il+h']
and h'
is
k~^
given by:
IS
given in the
+
e
text.
E\e~''^] (l
f e-'^'du
Jo
and
g
+
fe°P'C)r"
'*'
r
(l
since:
47
+
/i*)
Vo
Xo +
T
-g
where A
is
given in (56), we get finally that
+
(1
+
h*)
Proof. Adopting the Cox-Huang (1989) methodology and
stopping time Karatzas-Wang (2000)
max
k
c-^.XrMr
where V^^
is
its
-9
apphcation to problems involving a random
are able to reduce the problem to the following static problem:
^^du
e"-'V^^{Xr,R-r) -k( [
1-7 +
E
min
we
T
e-^V-,
+
e""" {Xr
+ ^Hr
\Jo
IJo
given by (55).
By maximizing
the objective inside the integral one can derive optimal con-
sumption to be:
=
c;
The
crucial step
is
k-^,t e (0,t)
the maximization of the problem involving the continuation value fimction at time
max [V^^iXr. Hr) - k{Xr +
where V^^
is
given above.
Assuming
for the
moment
Hr<C[
we can
^Hr)]
r:
(74)
that
Xr- +
r- g
solve the problem (74) using (55), to get:
1
- e-'-^rB
{B(Xr-+ -^^
]+Hr]
r -
+ e-'-^
(1
+
h')''^
Xr- +
T
g
[l
- e-^^r
{
B
{
Xr- +
r
where:
5=
l-e-'"''(l-h/z*)
Let us define
fCi
so that
we
—
k
arrive at:
1-7
Vr
k.^B + e-^' {l-\-h*y-^ (Xr- +
T
Xr- +
y'r
-g
k,il-iB)
Y^
,\l—l
T-gJ
e-"-^ (1 -h /i*)
48
=
k,
-
-
g
+ Hr
g
k^
or
fci(l-^-B)
Vr
X,- +
T
-g
e-'-T (1
/i*)i-^
+
fci(l-^(l-e-^(l +
e-'-^(l
(1
It is
an interesting observation {that we
+
will
+
1-e
+
+
e-^-^Cl
< F <
Now
1,
F
'
>
+
-
+
e-'-^(l
hi
Hr
(1
-
as:
+ H^
kxi
+ Hr
hi
-rr\-y
(l-e-^)
/i*)'-'^
fci(l-^g)
B
hji-JB)
B
e-rT)
*^i-^
e-'-^(l+/i*)
get the ratio
Hr
Xr-+^
we combine
the preceding equations to get:
h
We first
=
{l-e-'-'){l
verify that indeed h
(1
<
h*.
.*.f(
+ h*)
This
is
(l-e-'-^d +
(l-e-^^(l +
-
(1
+
r - e-'-^(l + hn) + h*
+
h*)
h*
Kh*
smce:
>
F
e--'''(l
(since ^
>
1)
and thus
(1
-
e-''^)(l
+
/(*)
/i*))
true since:
.J^
- e-^^){l + h*)[^\
(f)'
49
+ ft-)
<0.
1
we
get:
(75)
C
1.
kijl-^B)
B
e-'-^)'^
> l,^B <
Zi*)
the holdings of the second (hedging) asset are determined
(1
To
/i.)
use later) to note that since ^
1-^
and we have that
+C
e-rT(l
/1*)
/i*)))
+ /i*)-^
/i*)(l
+
1
1
>
.
Finally
we want
to provide conditions in order to exclude that h
becomes negative and
this
tvill
be the
case whenever:
/
£
1
h^
h'
<e-'-^(l + /r)
{l
+
+ h'){l-e-^]
1
or
i>e
1
=
(76)
Accordingly we only focus on cases where ^
As might be expected due
on the
<
c^*
to the homotheticity of the problem, the optimal reserve ratio does not
To complete
level of initial wealth.
the solution to the overall problem
budget constraint and combine everything together to
Xo +
Vo
r
k-^E
-g
+
du
Jo
A-
k-^<
can be determined
^
(1
>
The
1.
+
(1
+
-— +
/J-)
/1-)
as:
r
-'
-;
•
r
e-'^'du
and we have that F < l,r
k
J
r
which shows that
us return to the time
let
get:
e
/
depend
(1
final step of
+
A'o
h')
the proof
is
+
r
-
g
to use the formula for
E
[e
'"^]
from
section 4.2.
Remark
7 The optimal portfolios can
much smaller
7.5
We
be derived in a
manner
similar to sectioii ^.2.
Their magnitude
is
in this case.
Propositions and Proofs for section 4.3
give a sketch of the claims.
section 3.1.
We
=
Formal proofs would proceed along the
lines of
the respective proofs in
focus only on pt insurance for simplicity. In this case the Bellman Equation becomes:
(c*)^-^
max{—
c"
1
—
1
Vxc'} +
max{-a
p
-y
p'V'xx
2
+ ^Xs^p}
1—
-rV + VxirX
+
y')
+
Vyv' g
Once again the optimal consumption and
+
K/z
+
Ja^^/,,
+
A(,.,
portfolio policies are given
c*
= V^
50
^
C
1-7
by (assuming a
V
C^— Value function):
and
-
p=
Notice that the portfoho strategy
to risk
is
Vx?
a "pure" hedging strategy, in the sense that there are no demands due
premia (we assumed them away by positing that the sources of
consumption growth). In
variance motives.
where C'^{st)
this
way we
risk are imcorrelated
are able to distill out the pure hedging component, excluding
As before we conjecture a value function
mean-
of the form:
the same boundary conditions as in the no-hedging case (section
satisfies
with aggregate
3.2.).
Under
this
conjecture the optimal portfolio process becomes:
Notice that the sign and magnitude of hedging
One can show
i.e.
that this
they involve short
non-hnear
ODE
influenced by the derivative of C(sj) with respect to
Plugging back into the value function and simphfying we get a second order
for C{st).
_ r^C" + Cf ,< +
Cif ia^
+
ia^illl^^ + A(.,) [C - C"]
^
defining
CH
we
St-
a decreasing function and thus not surprisingly the hedging demands are negative,
is
sales.
7C"^
Now
is
.H\l
[b")
get
r^{b"v + ^^i{b"y-'h^ + -:^i{b"y-'
1
,(l-7)7^(6f)M^^]''"'
+
X{st) [A
(7
+ H
i)(S
b"
b:
- {b"r]
-r{b")-y
or after simplifying:
C
b"
2
Notice that once again this
ODE
is
[{b"V
7
very similar to the
ODE
with no hedging, namely:
\2
Q
= l-rb +
bsii
+
up to the absence of the term ^0-^(7 —
portfolio pohcies
a
(7-l)%^0.
2
l)-^-|^. In
7
1
b~<
terms of the 6— function the optimal consumption and
become:
51
Vt
Lemma
6 Under perfect hedging "excess'' consumption,
ct
-
Kyt
=
i.e.
c*.,
t
<
r
follows the process:
dcT
^
c
/ A(5t)
dt
C{st
Proof. The proof proceeds
in
a virtually identical
way
52
as in section 4.2.
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^f
.J ij
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