Digitized by the Internet Archive
in
2011 with funding from
Boston Library Consortium IVIember Libraries
http://www.archive.org/details/quantitativemodeOOcaba
DEWEY
HB31
.M415
Massachusetts Institute of Technology
Department of Economics
Working Paper Series
A Quantitative Model of Sudden Stops and
External Liquidity
Ricardo
J.
Management
Caballero
Stavros Panageas
Working Paper 05-10
April 4, 2005
Room
E52-251
50 Memorial Drive
Cambridge,
This
MA 021 42
paper can be downloaded without charge from the
Paper Collection at
Social Science Research Network
http://ssm.com/abstract=703241
^^"^^TT^EfHUfL'Sir''''
APR
2 6 2005
LIBRARIES
A Quantitative Model of Sudden
Stops and External Liquidity
Management
Ricardo
MIT
J.
Stavros Panageas
Caballero
and
NBER
The Wharton School
April
4,
2005*
Abstract
Emerging market economies, which have much
account
their
deficits in order to
dependence on capital
which can suddenly
inflows,
model of sudden stops and use
quantifiable
against them.
We
first
of their growth
ahead of them, run persistent current
smooth consumption intertemporally. The counterpart
it
stop. In this
to study practical
of these deficits
mechanisms
to insure emerging markets
assess the standard practice of protecting the current account through the accu-
mulation of international reserves and conclude that, even when optimally managed, this mechanism
expensive and incomplete. External insurance, on the other hand,
often
come together with
Thus, one needs
We
distress in
is
of such
these countries portfolios,
it
emerging market investors themselves (the most natural insurers).
an asset based on the
would
significantly
S&P
500's
imphed
volatility index.
sudden stops.
If
enhance their sudden stop risk-management
In our simulations, the median gain in terms of reserves available at the time of sudden stop
30 percent. Moreover, in instances where the level of non-contingent reserves
close to 300 percent.
affiict
We
is
hard to obtain because sudden stops
to find global (non-emerging-market-specific) assets that are correlated to
show an example
is
paper we develop and estimate a
manage
also find that as countries
is
low, the
to reduce the size of the
added to
strategies.
is
around
median gain
is
sudden stops that
them, they should reduce their stock of reserves and significantly increase their share of contingent
reserves.
The main
insights of the paper extend to external liquidity
and
liability
management more
generally.
JEL Codes:
Keywords:
specialists,
*We
E2, E3, F3, F4, GO, CI.
Capital flows, sudden stops, reserves, international Hquidity and liability management,
world capital markets, swaps, insurance, hedging, options, hidden states, Bayesian methods.
are grateful to Lars
Fernando Duarte and,
Hansen and conference participants
particularly, Jose
Tessada
at
CIRJE, Tokyo,
for their
for excellent research assistance.
support.
1
comments, and to Herman Bennett,
Caballero thanks the
NSF
for financial
Introduction
1
Emerging market economies, which have much of
their
growth ahead of them, run persistent current account
deficits in order to
smooth consumption intertemporally. The funding
of fragihty since
requires ongoing capital inflows which can suddenly stop.
it
of these deficits
While
the breakdown in capital inflows simply amplify domestic deficiencies, there
many
others the
main
culprit
is
not the country
itself
is
in
is
a perennial source
many circumstances
extensive evidence that in
but the international financial markets' response to
shocks only vaguely related to the country's actions.
The
real costs of
tliis
volatihty for coimtries that experience
open
crises are
dramatic and well known.
Less noticed, but at least as important in terms of their pervasiveness and cumulative impact, are the large
costs paid
by prudent economies. These economies do not
and build
variety of costly precautionary measures
fall
into
open
crises
but are forced to incur in a
large war-chests of international reserves, a trend that
has only increased in the aftermath of the capital flow crises of the late 1990s.
for
GDP
example, hold close to 20 percent of
developed open economies such as Australia or Canada.
Are there potentially
mechanisms
less costly financial
natural counterpart for these mechanisms?
How
How
effective are reserves in
How much
of sudden stops unrelated to a country's actions?
Chile and South Korea,
which contrasts with the 4 to
in reserves,
of
them should be accumulated? How
mechanisms limited by
by
smoothing the impact
to deal with capital flow volatihty?
are these
5 percent held
fast?
Wlio would be the
financial
and
collateral
constraints?
These are among the most pressing questions
for policy-makers
and researchers
in
emerging market
economies and the international financial institutions. Unfortunately, while there has been significant conceptual progress over the last two decades in understanding
with emerging markets, there has been much
these questions quantitatively. In this paper
some of the
less progress in
we take one
providing an integral framework to analyze
step in this direction.
In a nutshell, our framework considers three type of agents:
investors,
and the world
problem we wish to model has two ingredients:
current income so
difficulty in
it
The
capital markets at large.
would
like
to borrow
and run
An
emerging market country, specialist
essence of an emerging market
First, its future
pledging future income to finance these
limitations of financial contracting
income
is
economy
significantly higher
persistent current account deficits. Second,
deficits. Specialists
it
for the
than
its
has great
can alleviate this problem but they
themselves are subject to shocks that limit their ability to commit to deliver resources. These shocks, which
in our
model are driven by a Poisson process,
beginning of this period
is
the sudden stop stop
or implicit short term commitments, but
rebuild their international collateral.
country would
it
trigger a period of significantly reduced capital inflows.
itself,
when
specialists are
unable to rollover
all
can continue even after specialists recover as countries have to
For simplicity, we refer to the entire episode as a sudden stop.
like to insulate its current
The
their exphcit
account financing from these sudden stops, but
it
The
cannot do so with
its specialists
since they are constrained during these events. Resorting to the world capital markets after
the sudden stop takes place does not work either, since the country has very limited credibility with nonspeciahsts. However, world capital markets can
made
are
One
still
be used ex-ante, as long as contracts and investments
contingent on variables that do not require emerging markets knowledge.
main obstacles
of the
in building
tliis
type of structure for quantitative analysis
is
principle quite complex, involving several layers of potential financiers, contractual problems,
Om
state variables.
We make
of
framework preserves some of
this ricliness
but
it
several stylized assumptions that allow us to represent the
managing the
is
that
it
is
same time manageable.
at the
problem in terms similar to those
an exogenous non-diversifiable "income" diffusion-jump process, and
risk associated to
in
and multiple
is
amenable to estimation.
A key simphfication is that countries' and their speciahsts engage in growth-swaps.
These swaps eliminate
debt-overhang type considerations and focus the analysis on the obtention of new, uncommitted, resources.
This
is
probably not an overly costly simphfication in terms of reahsm, as
of external financing.
renegotiation. It
is
Unhke
existing liabihties,
it
isolates the
In the
is
part of the paper
more
or less
we
for a small
smooth the
First,
in practice,
impaict of sudden stops
we
Calvo, Izquierdo
on the current account.
Since
to estimate the sudden stop processes
panel of emerging market economies with open capital accounts for at least two decades. Our
external funding declines by 10 percent or more, and
estimation procedure
is
its
Calvo
when managed
costly events. In a typical
lasts for over six years.
designed to capture the impact of the sudden stop beyond any
in interest rates.
This
et al (2004)).^
reserves, as central
and
main impact
is
stop,
turmoil or short
sudden stop, which
is
what
is
often measured
Second, we conclude from this part that holding non-contingent international
banks do in practice,
optimally.
initial
sudden
Note that our
important because in practice current account reversals are significantly
persistent (and hence costly) than the initial impact of the
(see, e.g.,
e.g.,
use this structure to estimate and cahbrate the key
we use a Bayesian- hidden-state model
findings indicate that sudden stops are infrequent, but recurrent
more
through
discuss the pure reserves-management problem. Countries hoard (non-
what countries do
parameters of the model.
run spike
in
(2004)).
first
contingent) international reserves to
this
fragile source
perhaps for this reason that in practice sudden capital flow reversals are more closely
associated to the current account deficit (a flow variable) than to the stock of debt (see,
and Mejia
most
uncommitted resources cannot be forced to stay
is
a costly and incomplete sudden-stop insurance mechanism even
Reserves require large consumption sacrifices prior to sudden stops per unit
'See Caballero and Krishiiamurthy (2005) for a model where sudden stops
country's ability to produce financial assets, and hence to attract savings.
come together with a
Moreover, after the
decline below the pre-crisis level, but inflows do not recover. This characterization
SS-identification procedures in this paper.
is
persistent decline in the
initial crash, interest rates
consistent with our empirical findings
and
of protection, especially while their stock
is
being
This
built.
costly for emerging market economies,
is
which experience limited access to external resources even during normal times. Here again, the standard
practice of measuring the cost of hoarding reserves from the spread between US-Treasuries
and country debt
underestimates the actual opportunity cost.
This takes us to the second part of the paper, where we expand the set of investments (or contracts)
countries can make.
country's portfolio.
In particular,
The VIX
is
we
VIX
in the
S&PSOO
firms,
consider the optimal inclusion of digital options on the
an index of imphed
volatilities
extracted from options on
that sirriultaneously satisfies the conditions to appeal to world capital markets and to provide a good hedge
against sudden stop for emerging markets:
sudden
We
stops.
discuss implementation
It is
a developed world variable that
is
higlily correlated
with
and run the economy through simulations with the same sample
paths of sudden stops used in the case with only non-contingent reserves. Relative to the latter case, the
median gain
in
terms of reserves available at the time of sudden stop
instances where the level of non-contingent reserves
stop
is
high, the
median gain
is
is
In terms of dynamics, these observations
is
mean
that a country should
particularly concerned about the worst possible event.
the time of a sudden stop.
is
When
the stock of reserves
is
first
is
more
Moreover, in
likely to take place
if
build the contingent part of the
The reason
That
is,
for this strategy is that
the
to be found without resources at
small, the best chance of avoiding this worst-event
by getting a large contingent payment at the time of the sudden stop.
the worst-event
about 30 percent.
close to 300 percent.
portfolio aggressively, adding non-contingent reserves only gradually.
country
is
low (lower quintile) and hence the cost of a sudden
When
the stock of reserves
the country overcommits to contingent contracts
is
large,
and these do
not deUver, than by holding a large amount of non-contingent reserves.
Extrapolating these dynamic implications to the cross-section,
we
find that as countries
improve along
dimensions that reduce the size of their sudden stops, they should not only reduce their reserves accumulation
but also reallocate their portfolios toward more contingent instruments.
we show
Finally,
liquidity
and
liability
that the insights developed for reserves
management extend more
generally to external
management. Non-contingent debt with world capital markets has the same
deficiencies
that non-contingent reserves have, and hence can be improved by adding contingencies similar to those
discussed for reserves. These contingencies can be obtained by either embedding
by
raising the contingent
Our paper
The
them
in the
debt
itself or
of reserves.
relates to several strands of literature.
stop of capital inflows.
The work
component
literature
The main shock
on sudden stops gained new
of Calvo (1998) describes the basic mechanics
life
that concerns us here
since the Asian
is
a sudden
and Russian
crises.
and implications of sudden stops and Calvo and
Reinheart (1999) document the pervasiveness of the phenomenon. The modelhng of these sudden stops as
the tightening of a Kiyotaki-Moore (1997) style collateral constraint
is
also present in the
work of Caballero
and Krishnamurthy
(2001), Arellano
As an intermediate
we model
and Mendoza (2002) and Broner
step in developing our substantive argument
reserves accumulation as a buffer stock
model against
et al. (2003),
among
and quantifying the
capital flow reversals.
others.
effects
The view
we
describe,
that reserves
can be used to cushion the impact of external shocks exists at least since HeUer (1966), was enhanced by
the work on precautionary savings in macroeconomics during the 1980s and has recently returned to the
fore
with the large accumulation of reserves exhibited by emerging markets since the
1990s
(see, e.g.,
crises at the
end of the
Lee (2004)).
Importantly, the main reason for seeking insurance and hedging in ova context
is
not income fluctuation
per-se but the potential tightening of a financial constraint. This motive parallels that highlighted by Froot
et al (1993) at the level of corporations,
and by CabaUero and Krishnamurthy (2001)
While the substantive themes developed
in those articles differ
many
respects a
dynamic version of
for
contingent debt and the hmitations to
work
of Kletzer,
it
lenders.
Our
imposed by
paper
is
in
The optimahty
of
financial frictions are also a feature of that literature.
Newbery and Wright
(1992)
and Kletzer and Wright (2000), characterize
with different degrees of commitment by a sovereign borrower and
feasible financial structures consistent
its
in our
theirs.
Closely related to our recommendations are those in the sovereign debt hterature.
In particular, the
emergiag markets.
from oms, the basic model
collateral constraints capture features similar to those of their richer limited enforcement
framework. Our paper reinforces
much
of the message in that hterature
and provides a tractable model that
can be estimated and quantified.
The
interaction between precautionary savings
economy framework
of Aiyagari (1994).
He
and
financial constraints
is
also present in the closed
calibrated such a model to estimates of
US microeconomic
income processes and other parameters and concluded that eventually agents would save enough to relax
aU financial constraints. Ova model has similar ingredients,
over time
and
in that there
is
However, in sharp contrast with his quantitative
gotten close to that level of savings.
for these countries,
in that countries
tend to accumulate resources
a level after which sudden stops would have no consequences on consumption.
which face two
results, ours suggests that liistorically countries
The main reason
is
financial constraints.
On
enough resources from the post- to the pre-development phase.
one hand, countries are imable to transfer
On
the other, within the pre-development
phase the country experiences sudden stops. The problem equivalent to that in Aiyagari
built to
smooth the sudden
stop.
The
have not
the high opportunity cost of accumulating reserves
is
the buffer stock
long run constraint, on the other hand, raises the opportunity cost of
such buffering.
Over recent years there has been a
significant rise in the volatihty trades, including the
VIX. The finance
hterature has studied the impact of such trades on the performance of hedge funds and other markets
participants (see, e.g., Bondarenko (2004)).
From a
risk
management
perspective, the issues faced by these
economic agents are similar to those faced by emerging market policymakers.
We
setup the model in Section 2 while constraining the precautionary options of the country to the
accumulation of non-contingent reserves. In Section 3 we estimate the sudden stop process, quantify the
basic model,
and
assess the effectiveness of reserves as a precautionary
comrtry purchase digital options from world capital markets.
using the
VIX
as the contingent indicator
mechanism. In section 4 we
let
the
Section 5 implements the extended model
on which options are written. Section 6 extends the
adding debt and discuss contingent liabihty management more broadly. Section 7 concludes and
results
is
by
followed
by several appendices.
Sudden Stops
2
Eind
While there are many important
Non-contingent Protection
issues that arise
from decentralization
in
economies with poor institutional
development, we leave these aside and focus on the problems between the country as a whole and international
investors.
We
study a representative agent economy with a benevolent government that seeks to maximize
the expected present value of utihtj' from consumption:
r
E
with r being both the discount and
U{Cs)e-''^'-'Us
riskless interest rate.
While
it is
not essential, assuming a
CRRA
utihty
of consumption also simplifies the exposition:
^<^) = rr:7
2.1
Emerging Market Economies and World Capital Markets
There are two featmres of an emerging market economy that are important
income
is
low relative to
its
future income
current account deficits. Second,
it
(it
for
has yet to catch up), and thus
it
our analysis. First,
would
like to
has difficulty jaledging future income to finance these
its
current
borrow and run
deficits.
Let Yt represent the country's income in the pre-development phase, and follow the geometric Brownian
motion:
dYt
with
<
/iy
<
=
fiYYtdt
+ crYYtdBt
r.
Income post-development, on the other hand,
is
KYt
equal
to:
K>
1.
A
from the pre- to the post-development phase
transition
The
T^.
focus of the paper
assume that
r'^ is
is
irreversible
on the former phase. In order
is
and takes place
to eliminate inessential time dependency,
governed by a Poisson process with constant hazard g and independent of
Note that
of uncertainty in the model.
g{K,
—
1)
>
is
random time
at a
all
we
other soiu-ces
the difference in the expected rate of growth of a
developing and a developed economy.
The
swift transition
hke to borrow against
at large
(WCM,
and
clear
demarkation ensures that a country in the pre-development phase would
post-development income.
its
for short),
and speciahsts.
We
We
spHt potential financiers into world capital markets
focus on the former in this section
and
discuss specialists in
the next one.
World
capital markets have limited connections
and understanding of emerging market economies, hence
they do not accept contracts contingent on variables endogenous to these economies. For now,
let
them
just
accept plain- \'anilla debt. Moreover, they have only limited wilhngness to invest resources in any particular
emerging market economy.
at
some
finite \'alue
The country can
unit of time.
The
We
shall
assume that a country's debt capacity with respect to
WCM
is
capped
Wt.
accumulate international
also
constraint with respect to
Both
assets.
assets
and habihties pay a return of
WCM appUes to the country's
r per
net assets, Xt.
Xt > -Wt.
Let us define reserves, Rt, to be the difference between debt capacity with respect to
WCM
and actual
net assets:
Rt
Note that
reserves
if
the country
is
at its debt capacity,
and net uncontingent
assets coincide.
= Wt + Xt
then Xt
By
= —Wt
and reserves are
construction Rt
>
0.
0.
Similarly,
if
Wt =
then
For now, we shall focus our attention
on the case
Wt=0
and
relax this simplification in Section
will
be introduced shortly
in the
6.
Until then,
all
borrowing
will
be done through
specialists,
which
income process, a stable
WCM
next section.
Note that given oui assumptions of a continuous sample path
for the
and, most importantly, a high expected growth, the country has no incentive to accumulate reserves.
We
estabhsh this as our benchmfirk result.
Lemma
1
Assume
that
Wt =
and
M.-^^>0
Then, starting from Rt
=
0, the
optimal solution
is to
keep Rt
=
(1)
throughout and set Ct —Yt.
That
is,
(^^),
as long as income growth {fiy)
is
sufficiently strong to
outweigh the precautionary savings motive
then there are no reserves accumulation in the post-development phase. This result can only be
strengthened during the pre-development phase since there the expected growth rate
Simply put, hoarding reserves during the pre-development phase
primarily because the country as a whole
is
is
is
Hy + g{i^ —
1)
> iiy
particularly costly for an emerging market,
already constrained in
its ability
to transfer resources from the
post-development phase. This means that any accumulation of reserves in aggregate carries with
it
a costly
reduction in current consumption.
Henceforth we shall assume that condition
1
holds.
have a source different from the ingredients described up
Tliis
means that any accumulation
We now turn
now.
to
of reserves will
and
to the role of specialists
their risks.
2.2
Specialists
and Sudden Stops
Speciahsts are investors that have developed some expertise and connections in the country and can engage
in
more information-intensive
contracts.
Concretely,
we capture
this feature
by allowing countries to sign
we
contracts with specialists that are contingent on country-specific variables. In particular,
and countries write contracts contingent on the country's transition to development. With
let specialists
this
we capture
the idea that specialists big payoffs are closely tied to the country's success. In practice, these contracts
represent equity investment, FDI, the riskiest tranches of
GDP-indexed bonds,
or toxic-waste
more
may
generally.
Later on we also consider a richer set of contingent contracts that hedge the diffusive component of income,
but this we do for technical reasons and does not change the substantive message we isolate here.
Countries' can pledge
—that
after netting
is
up
At each time
time interval
dt, in
t,
—
> t^
may
.
sign
In aggregate
and are not
engage in "swap-like" contracts with the
risk neutral speciahsts optimally
the specialists commit to provide resources at a rate Ft over the next infinitesimal
exchange
(t'^) arrives in the interval dt
The maximum
all s
out the multiple type of financial contracts that individuals
of our concern in this paper
country.
to a share z of post-development output, kYs, for
for receiving a
and
promise to a stream of payments znYt forever
otherwise.
By
risk neutrality tliis
stream
commit
flow that competitive speciahsts are willing to
if
development
valued at ZKYt/{r
is
for the
next dt
is
—
/ly)-
then:'^
Fi^jYt
-Formally, this expression can be derived from the pricing equation:
rPt
and noting that a swap
compensate
at the time of its inception has value Pt
specialists for the resources spent in
significant way.
= -Ft+g{zKYt/{r -
becoming one, but
=
fXy)
0.
this
-
Pt)
Note that
would not
it
would be
alter the
trivial to
add a markup to
formulae or main results
in
any
with
^'
7-
My
Assumption
1 (Limited Smoothing) z
<
My
^^ ^g-Hy
This ensures that the funds provided by speciahsts before development are
after development.
as "normal" times, an emerging market
than the unpledged income
economy engages
in as
much swaps
as
it
can:
smooth pre- and post-development consumption.
in order to
now
Let us
less
Thus, during non-sudden stop times, abbreviated as ''NSS" and sometimes referred to
introduce the sudden stops, which are our main concern in this paper.
completing financial contracts and markets, but they can also experience shocks.
are episodes
when
speciahsts' abihty to pledge resources
is
significantly reduced.
We
We
Specialists help
assume that there
neither specify the
particular agency problem or capital constraints that afflict speciahsts dining sudden stops, nor let
take precautionary measures against these.
and focus on the country's
Assumption
We simply state
them
the aggregate constraint implied by these shocks
side of the problem:
2 (Sudden Stops) During sudden stops, specialists (collectively) can commit at most
ft
<
ft
resources to the country. Specialists transit from the normal to the sudden stop stage with hazard X and do
the reverse with hazard X, both of which are independent of the transition to development.
That
is,
the
maximmn
flow of resources received from the specialists before development drops to ftYt
during sudden stops, with
/</
This tightening of the aggregate financial constraint can be interpreted as a drop in the share of swaps rolled
over by the speciahsts, which are clearly binding:
For simplicity, we shall assume that
if
the country transits from the sudden stop to development,
the effectively pledged resources fnz/g per unit time to specialists.
transfer of fKz/g, or anything in between, in which case not only the
rate
would
analysis.
rise
during sudden stops. This modification
More importantly,
in the empirical section
reduction in capital inflows; in practice this
we
is
it
transfers
Note that we could have assumed a
shadow but
also the observed interest
straightforward but not a fhst order issue in our
associate the sudden stop to a period of prolonged
may come from
a combination of an
initial
shock to speciahsts
and a longer
lasting decline in the country's collateral
This
z.
is
straightforward to
accommodate
model, with results analogous to attributing the entire episode to speciahsts' problems, which
in the
what we do
is
throughout.'^
Letting At denote the smxi of income and contingent flows from specialists,
ically
we can write
in a
mathemat-
compact form:
At
= (9^^^1{NSS} + e'^^liSS} +
e'^^^^l{G\SS}
+
e°^'^^'^l{G\NSS}) Yt
(2)
with
qNSS
0^^
0G\NSS
^°'^^
= 1+J
=
1
+
it is
/
(4)
(5)
=
(6)
/c-/(r-/x,.)/g
behind the external
crises
normal times ("NSS"), or
in
is
we wish
in a
sudden stop
to capture,
nNSS
qSS
Finally,
(3)
^ ^_J^^^^^yg
where 1{NSS} and Ij^^} indicate whether the country
("SS"). Importantly, for
.
1{G|55}, 1{G|A''55} indicate that the comitry
is
currently developed and the transition to
development took place from SS or NSS, respectively. Note that.
nNSS ^ nG\NSS
The
first
of these inequalities
is
important since
it
gG\SS_
captures the constraint to transferring resources to the
emerging market (pre-development) phase. The second one
is
inessential, except for implicitly stating that
the former constraint, binds regardless of whether the particular transition into development takes place from
NSS
or SS.
Taking stock, we have that net assets accumulation
is
now
dXt^{rXt-Ct +
Let us conclude this section with two remarks.
First,
described by:
At)dt.
by introducing
specialists
we have
effectively
introduced short term borrowing whose availability can be reversed instantaneously in a manner that
largely unrelated to coiuitry specific state variables, such as the level of debt. This captures
is
an important
'Note, nonetheless, that we do need that specialists are a significant part of the problem, otherwise we have to consider
additional insurance contracts between the country and
its specialists.
10
aspect of reality, especially for sudden stops that stem from contagion effects (see,
Second,
we have managed
savings
model apphed to the
to reduce the
jump nature
specific
e.g.,
problem to one where the setup resembles
Calvo
et al (2004)).
closely a precautionary
Thus we can use standard and weU
of sudden stops.
developed intuitions to analyze most of our results.
The Problem
2.3
We
now ready
are
its specialists
economy
to study the reserves accumulation problem for an
facing temporary shocks to
(sudden stops). Let us now coUect the ingredients into the country's optimization problem:
ViXt,Yt)
maxE
V°^^^{Xt,Yt)
= maxE
yGlNSS^j^^^Yt)
maxE
r
r(s-t)
u{Cs)ds
s.t.
ViXr,Yr)
= V°\^^^{Xr,Yr)l{T =
=
T^}
+
l{T
=
T^^}V^^{Xr,Yr)
V^'^{Xr,Yr)
=
V°^^^{Xr,Yr)l{T
dXt
=
[rXt-Ct + At]dt
At
=
(9^^^1{NSS} + 0^^1{SS} + e^^^^l{G\SS} + 0°^^^^l{G\NSS}')Yt
dYt
=
^yYtdt
Xt
>
e-^'^Xt
=
hm
T^}
+ V{Xr,Yr)l{T=T'^^^}
(7)
+ ayYtdBt
for all
i
>
0.
where V, V^'^ and V'^ denote the value functions
in the states
NSS, SS and G,
respectively.
In words, in post-development (in the state G) the country faces a standard precautionary savings
problem. This state
The
first
(and
final)
is
absorbing, so that there can be no further transitions to the states "SS" or "NSS".
time that development arrives
to development, the country can be either in a
is
denoted by
r'-',
sudden stop "SS" or
in
from normal times to sudden stops occur with a constant hazard rate of
reverse transitions (from SS to
NSS)
which occurs with hazard
occm' at the rate A, at stochastic times
of resources available differs.
11
Prior
normal times "NSS". Transitions
A, at
the stochastic times
t^^^
.
r'^'^.
The
Aside from the different
potential transitions, the practical implication of being in any of the three regimes
amount
g.
is
that the
maximum
many growth-swaps
Aside from the decision of how
discussed and solved out of the problem, the country
to engage with specialists, which
how much
faced with the decision of
is
we have already
to
consume
(Ct).
Reserves play the role of providing the country with resources during sudden stops. However, accumulating
them
is
costly because the country
extent against
While a
here
its
full
already constrained in normal times, since
is
of the solution analytically.
Let us start backwards, from the post-development phase, where the problem
When
fluctuation problem.
^
full
characterization of the solution to this problem has to wait until the quantitative section,
we gauge the nature
yG\NSS
cannot borrow to the
it
post-development income.
not leading to confusion,
since they both satisfy the
= max
<;
use the generic notation
a conventional income-
V'-'
for
both V'^^^^ and
Belhnan equation:
nV,^
^^ - CtV^
where 9° should be taken to be
we
is
+ V^ (rX
[rXt +
j>
^^>tj
-
rV^'
+
+yy^My>'t
+
^c7l-Y~V^y
(8)
I
e^^^'^ for V^l^'^ while
shoiUd be taken to be ^g'Ia'ss ^^^ yG\NSS_ Defining
it
noticing that
and plugging
back into
this expression
+My
The
first
[(1
(8)
- l)y° -
and
simplifying'*,
xtv^]
T^^Y (-7(1
obtain:
- l)v° +
"i-ixtv^
order condition for consumption (normahzed by potential income)
Ci
An immediate
is
+
we
implication of this
first
+ x1v°^)
is:
= {v?y^
order condition
is
(10)
that the consumption to (potential) income ratio
a function of the reserves to (potential) income ratio only.
Following similar steps in NSS,
we
obtain:
= niax/^-c,<^^4+P^''"^^^(^t)
='
+
[ 1
+
{rxt
+t.y
'For details see Duffie et
[(1
al.
7
1)
v^^^ -
(11)
J
(r
+ A + p)i;^^^ +
- 7)-^^^ - xtv^''] +
Xv^^
\a\. (-7(1
(1997)
12
-
7)-^^"^
+ 27.xt.f ^^ + :.?<Y")
Once
again, the
order condition for consumption
first
is
ct^{v^'')-'
Finally,
we obtain
(12)
the Bellman equations for the SS region:
= msx\^^-Ctv^A+gv^^^^{xt)
='
+My
which has a
first
[ 1
-7
[(1
- l)v^^ -
+
xtv^^]
and compare the
instructive to study
neighborhood of
=
Xj
0.
It
The
NSS
7)1;^^
+ 27Xt^f ^ + x\vll)
region:
(.f^)-^.
first
(14)
order conditions across the regions, in the (positive)
foUows from the condition
«f l^^(O) <
-
\a\- (-7(1
order condition similar to that in the
C=
It is
(13)
J
B'^^^^
v^^'^^^iO)
<
>
g^^W^^^
t;^^^(0)
<
>
e'^^^
>
6^^ that:
«f^(0)
inequahty imphes that consumption drops at the sudden stop, while the next to
last
the country
is
not smoothing pre- and post-development consumption.
It
last
turns out that, as
imphes that
we show
in the
next section, these features of the solution carry over to most of the empirically relevant range of reserves.
Quantitative Analysis
3
In this section
we estimate the main parameters
of the model, cahbrate others,
and
assess the optimal
reserves strategy quantitatively.
3.1
The
The Sudden Stop Process
core of our analysis
process: A, A,
(see the
The
and
9
Appendix
first
step
and SS. For
this,
flows, {dt
—
l)5^t-
is
/9
is
the sudden stop process. In this section
we estimate
the main parameters of such
using data from 1983 to 2003 for six representative emerging market economies
for selection criterion): Chile,
Colombia, Indonesia, Malaysia, Mexico, and Thailand.
to find empirical counterparts for the processes describing available resources during
we note that
in the
model these resources can be decomposed
into income, Yt,
and
NSS
financial
In practice, there are several additional complexities in doing such a decomposition. These
stem from the existence of multiple goods whose
relative prices
13
change during the transitions between
NSS
and SS and viceversa, the presence
declines during sudden stops.
a nutshell,
we approximate
sum
with the
of
temporary terms of trade shocks, and endogenous domestic output
The Data Appendix
describes our
of capital flows in terms of imported goods
of trade effects.
Our main
methodology to deal with these
Yt with the permanent component of domestic national income, and
left
hand
side variable
issues. In
{6t
—
l)Yt
and the transitory component of exports and terms
the ratio of these two, which can be loosely interpreted
is
as external financing over "normal" pre-development income:
Vzt
_
=
l^it
-
i-
-
[eu
_
—
i)Yu
•
T7
'it
In our model,
ip
—
can take only two values: {9
characterization for actual data,
we assume
that that
i/'
1)
is
and
—
{9
1).
Since this this
is
too stark a
observed with (state contingent) noise,
with
eiMt) - N{0, alisu)),
Then
ipu follows a process described
su € {SS,
by a standard regime-switching model a
with parameters: {i>^^^ ,ipf^ ,a^f^ ,af^ ,pi{NSS
-^ SS),p^{SS -^
NSS)}, such
= Pr{si^t+A=SS\si^t=NSS) =
la
Hamilton (1989, 1990),
that:
P^{NSS
-^
SS)
p^{SS
-^
NSS)^Pi{s,^t+A = NSS\si^t=SS)^l-e-'^'^
where we have approximated the continuous time model by
at
NSS}.
most one transition
in
its
l-e~^^'^
(15)
(16)
discrete analog assimiing that there can be
a time interval of A.
For the calibration exercises we convert aniiual transition probabilities into annual frequencies by setting:
A
= - log [1 - P{NSS
A
= - log
[1
- P{SS
Given the limited number of SS observations we have
the hidden states model
and Nelson 1999).
•
We
fix
We
we
are estimating,
for
is
each country and the highly nonlinear natiu-e of
(see
Kim
describe the precise procedure in the Appendix, but the main steps are as follows:
on these parameters, we
draw a sample path from
in a
NSS)]
we use a Bayesian approach based on a Gibbs Sampler
filter
the data using Hamilton's algorithm to find the posterior
probability that at a given point in time each country
We
-^
arbitrary starting parameters.
• Conditional
•
-^ SS)]
SS and
this joint distribution.
otherwise.
14
is
in
a SS or in NSS.
Such a path takes the value
1
if
a given country
Average
Dev
Std
5%
25%
50%
75%
95%
A
0.19
0.050141
0.11337
0.15434
0.1868
0.22153
0.27772
A
0.14769
0.046718
0.079083
0.11403
0.14322
0.17685
0.23111
Table
•
We
Pooled Estimates using 1983-2003: Poisson Parameters.
1:
condition on these paths (one for each country) and derive the joint posterior distribution of the
parameters.
We
require only that parameters related to
-^ SS), pi{SS
pi{NSS
— NSS)
>
are the
same
across aU countries, while the rest of the parameters can vary across countries. These parameters are
key
for the quantitative exercises that
on time
series observations alone.
foUow, while they seem to be the hardest ones to measure based
Hence we exploit the cross sectional dimension of the panel
Throughout we use uninformative
to iucrease the precision of these estimates.
we draw a new
• Finally,
thousand times.
several
set of
By
in order
priors.
parameters from this posterior distribution and iterate the procedure
standard results in Bayesian computation, the stationary distribution of
the sampled parameters coincides in law with the posterior distribution of the parameters.
Tables
i/jj
,
1
and 2 present the estimates we obtain form
Note that from the
Ai, Xi.^
first
this
procedure for the parameters
two parameters we can recover jws^
rp^
^Vi
—
'4^i^~
since:
nSS
= 1+
Vi
Based on the medians, we conclude that sudden stops are
typically
beyond 10 percent,
after exiting the previous
sudden
stop).
As we
main
after the
event.
dechnes in available resomces
said La the introduction, oui measure of sudden stop
to captm'e not only the initial spike in interest rates
remain even
large, leading to
about 6.5 years years and occur about every 12 years (about
last for
and turmoil but
Our estimates suggest that these
also the effects of
effects
5.5 years
is
meant
sudden stops that
can be quite large: Countries
take a long time in resuming significant borrowing after experiencing a severe capital flow reversals.
Finally, Figure 2
illustrates the
summarizes the output of the corresponding Gibbs sampler run
diuing a given period.
want
it
°We
to capture
It is
apparent from
when studying sudden
also estimated for
(if
we trim the
last
tlris
our economies.
figure that our procedure does capture the events
half.
It
0.5
one would
stops.
a sub-sample from the 1990s using quarterly data
very similar except for A, which were cut in
sample
for
path of the share of economies with posterior probabihties of being in sudden stop above
However,
this estimator
is
for a
broader set of countries. The results were
too affected by end of sample condition in a short
two years, the estimates become very similar to those of the 1983-2003 sample that we report).
15
5%
25%
50%
75%
95%
0.066
0.024
0.031
0.049
0.063
0.081
0.108
-0.091
0.030
-0.132
-0.109
-0.094
-0.076
-0.041
Average
Chile
^NSS
Std
Dev
Colombia
^NSS
0.059
0.008
0.044
0.055
0.060
0.065
0.070
-0.051
0.009
-0.064
-0.057
-0.052
-0.047
-0.036
Mexico
^NSS
0.078
0.010
0.061
0.072
0.079
0.085
0.094
-0.084
0.015
-0.106
-0.093
-0.085
-0.075
-0.059
0.043
0.006
0.033
0.039
0.043
0.047
0.053
-0.066
0.009
-0.081
-0.072
-0.067
-0.061
-0.049
0.114
0.020
0.081
0.101
0.114
0.127
0.146
-0.164
0.023
-0.199
-0.178
-0.164
-0.149
-0.126
0.098
0.019
0.064
0.085
0.099
0.111
0.127
-0.146
0.023
-0.179
-0.161
-0.147
-0.133
-0.107
Indonesia
V
Malaysia
^NSS
V
Thailand
^NSS
Table
2:
Country
Specific
Parameters 1983-2003: Normal Level of Resources and Sudden Stops.
16
Chile
0.5
«
1985
1990
1995
0.5
a
0.5
w
0.5
52.
2000
Indonesia
w
0.5 52^
1985
1990
1995
2000
Thailand
1985
Figure
1990
1:
ip'^
1995
1985
2000
and posterior probability of being
17
in a
SS
1990
1995
2000
for 6 different countries.
Sudden Stops
7
LTCM/Russia^
-
J-
Crises
in
/
Asia
Asian Crisis -^
w
0.6
1
/A
-
I
/
Debt Crisis/
\
/
Tequila Crisis!
_
/
\_
1
1983
1985
1
1987
1
1
U
Figure
1991
2:
\
1993
Year
/
1
1995
V
1
1997
Fraction of Economies in SS
18
1
1999
1
2001
2003
Other parameters
3.2
In addition to A, A and
^^^s^^ss
^^^^ ^^ determine
^^^
Hy, ay,
r,
g, a, 7,
before simulating the model. Within reasonable ranges, the choice of
and the discount
rate
^y
is
rate, is not of first order
set to 0.018,
which
importance; we simply set
it
r,
^^iss^^ss ^^^ qG\nss jqNSS
which represents both the interest
to 0.04.
The post-development growth
approximately the rate of growth of per-capita consumption in the
is
the very long sample considered in Campbell and Cochrane (1999).
We
set
ay =
0.05,
which
is
US
over
higher than
the developed economies volatihty in national income because emerging markets also have significant terms
of trade fluctuations.
The parameter g
We
Barro and Sala-i-Martin (2003).
in
is
set
an emerging market economy during normal times
liy+g log
which
is
Q
follows
16
'
2,
--^-^/o
qNss
I
so that the expected rate of growth of income
is:
from the previous ones
qSS
g
y
used 7 to match a reasonable stationary
=
£ilog(a;t)
^l°gf ^Tvsl^j
- (My -
=
—
(
r
Y
X*
Beyond
=
I
8,
we obtain an x*
—
0.2,
since:
IJ
+ qNSS
level of central
= (^-(^y- ^Tvss^
ct
bank
lj
reserves. Consider the
dynamics of
Y)-^ + ^^) '^^~^^-^« =
[~^(r) +
+
gSS
~^^ )
" "^^* =
^*
dt- adBt
as:
min{(r
this level, the process for Xt
=
)
qSS
- (^y - —)xt -
Let us define the "steadj' state" level of x
setting 7
^^
on the conservative end of an emerging market economy's expected growth during normal times.
The parameter
We
which matches the speed of convergence estimated by
set to 0.025,
q<^\'^ss jqNSS
-
^))x -
(/zy
has a negative
which
is
drift
Ct
and
+
1
=
0}
reserves start to dechne on average.
a reasonable estimate for the
maximum
By
desired level of
reserves for countries not attempting to use reserves accumulation with goals other than smoothing sudden
stops (such as maintaining an undervalued currency, and so on).
Finally,
to the
we use
77
=
—0.1,
tp
=
0.08 as a base case scenario in this section. These correspond roughly
median values of these parameters across the countries that we
consider.
The
values for A, A are
taken from the estimations of the previous section. Table 3 summarizes the values of the parameters used.
19
.
7
8
A
0.211
5
0.04
A
0.158
r
0.04
77
-0.1
jji
0.018
g
0.025
a
0.05
ijj
0.08
Table
3.3
Parameters used in Simulations.
3:
Implications
Figure 3 depicts the policy functions.
The upper curve
represents consumption during
The
represents consumption during SS, both as a function of the level of reserves.
NSS and the
lower one
vertical distance
between
these two curves illustrates the instantaneous drop in consumption once the country transits into SS. In the
neighborhood of
this
drop
when
is
largest
is
which
reflected in the difference
very large
declines as the level of reserves rises. Because
it
the country has no reserves, precautionary savings
the drop
is
and around 7.5%, and
Note that the policy functions c^^^{x) and c^^{x) do not converge. This
the calibration section, once the country reaches x*
this level, the country rather
costs of reserves
is
maximum
is
also
is
because, as
a
tliis
point,
between one and c^^^
—
0.2, it
stops accumulating reserves.
consume them than use them to further smooth sudden
simply to great to purse
full
we discussed
If
stops.
in
reserves exceed
The opportunity
insurance. Despite the large costs associated to sudden stops,
the strength of the other financial constraint (the inabihty to pledge a significant share of future, and higher,
income) raises the opportunity cost of storing reserves enough to make
somewhat unprotected against these sudden
Panels
(a)
and
(b) in
the country starts with
sudden stop
Figure
lasts exactly its
accumulates reserves, which
is
consumption and reserves, respectively,
sudden stop takes place exactly at
expected time, 1/A. For
clarity,
we
Early on, the consumption path
it
(constrained) optimal to remain
stops.
plot the paths of
reserves, then a
value and policy functions).
reserves
4,
it
is
also set
dBt
increasing
then uses during the sudden stop.
The
its
a case where
for
expected time, 1/A, and the
=
(in
the path but not the
and below
one, as the country
country's incentive to accumulate
very steep initially and less intense once some reserves have been accumulated.
even though the sudden stop in this example takes place exactly at
consumption drop at the time of the sudden stop,
its
expected time, there
Importantly,
is
reflecting the imperfect nature of the reserves
a significant
accumulation
self-insurance mechanism.
The next two
to a large
figures contain the distributions of the
number
transitions in
of paths in our economy.
and out of sudden
stops.
We
main
quantities generated by the model, associated
Recall that randomness stems from Yt as well as from the
start with a country that has
20
reserves initially
and simulate
Consumption as a function
of
Reserves
0.96
0.94-
0.25
Figure
3:
Consumption functions
21
in
NSS and SS
A and consumption
-
0.99
0.98
y
X
-
0.97
-
-
0.96
-
-
-
0.95
\
0.94
-
0.93
-
-
\
-
\
1
0.92
-
0.91
^---A,
-
L
_ _ _ _
Jr~-:=r-
"=,
n n
,
,
1
40
20
20
quarters
Figure
4:
40
quarters
Typical path of consumption and reserves.
22
Density of Reserves (no hedging)
700
-
600
1
400
-
300
-
200
-
100
0.02
0.04
Figure
5000 paths for
T=
5:
0.06
0.08
0.14
0.12
0.1
0.16
O.lf
Distribution of Reserves immediately prior to SS.
80 years, without a transition into development.
Figure 5 displays the histogram of reserves available at the point in which the country enters a sudden
stop.
The
possibihty of a sudden stop leads to an accumulation of reserves well above
resources during NSS. However
low
levels of reserves.
it is critical to
of available
This mass corresponds to those cases where the country has not had the time to
accumulate reserves since the previous sudden stop. These early
reserves are particularly inefficient in
deahng
crises are
an important source of
risk,
that
with.''
Figure 6 displays average consumption during
is large,
5%
observe that there a significant mass concentrated at very
NSS and SS.
The average
difference between
consumption
around 7 percent. Moreover, often the country simply rims out of reserves during sudden stops and
consumption
falls
by the
full
extent of the capital flow reversal.
In summary, these results raise the question of whether there are other ways to
^In practice, countries often deal with this risk
manage
reserves that
by not using reserves very aggressively during sudden stops, which
suboptiraal strategy.
23
is
a clearly
Average Consumption during
NSS
Average Consumption during SS
120
120
Figure
6:
-
Average consumption during SS and NSS.
24
could increase their efEciency and avoid the risk that
We
is
associated with the unpredictability of sudden stops.
turn to this question in the next section.
Contingent Instruments
4
In practice,
ously, the
financial
how much
answer to
do by holding contingent instruments
better could countries
depends on the
this question
markets available to hedge those
Our
specific factors.
to illustrate, in a conservative scenario in terms of information
significant potential gains
goal here
and
is less
financial
ambitious.
exclusively related to emerging markets (and hence not
We
simply want
development requirements, the
appeahng to world
more
generally, of factors
capital markets).
Portfolio decision
4.1
Before introducing additional assets,
sudden
stop.
intensity
x
It is
let
us develop in more detail the events behind a transition into a
useful to think of this transition in two steps.
that puts the
economy
First, there is
in a "danger zone" at stochastic times r-°.
enters a sudden stop with probabihty
P{SS ~
1)
and avoids
it
with probabihty
a Poisson process with
Second, at r^, the comitry
P{SS =
0)
= 1 — P{SS =
1).
evident that,
A
and that nothing
Let us
in our analysis
now assume
that there
jump
in "danger zones"
while
J
—
up
is
to
now
is
= xP{SS -
(17)
1)
modified by this decomposition of events.
a financial asset with payoff Ft
When J =
r^, which we denote by J.
denotes the absence of a
(conditional on t^) are denoted by
also
Obvi-
from improving current sudden-stop-risk-management practices by adding con-
tingent instruments that are largely independent of the country's actions and,
It is
in their reserves?
behind a country's sudden stops and the
specific factors
jump
P{J =
1)
at such time.
and P{J
—
1
,
that also has the potential to exhibit a
the asset's payoff exhibit a
jump
at r^,
Correspondingly, the probability of each event
0), respectively. It follows
that this
jump
process
foUows a Poisson process with intensity:
Xj=xP{J=l)
Note that
sofar
(18)
we have not excluded the possibihty that the jump
pendent from the transition into the SS (P(J
=
1, SS =
1)
= 0)),
in the asset
with
payoff' Ft is inde-
although clearly our interest
is
in the case
where:
P(J =
We
can now write the risky asset's
1,
55 = 1)>0
payoff" process as:
dFt
=
rFtdt
+
Ft {JdNt
25
- xJdt)
(19)
where dNt
is
the
is
mean
a
jump
process that takes the value of one at time
of J. Correspondingly, the asset has a return
we
In what follows
is
without
zero otherwise, and
r^ where we observe
condition on those times
transition into SS. This
t^ and
loss of generality, since (as
we show
either a
jump (J
shortly) the
=
BeUman
depends on those events and not on situations where neither takes place. Hence from now on
which
is
is
the hazard rate for observing either a
jump
in
J
An
or a transition into SS.
=
0,
J=
1)
,
{SS
or
To complete the
transition out of
The
=
1,
J=
0)
To
.
and/or a
1)
equation only
us define:
let
—
1,
J
=
1),
or
simplify notation let us defuie:
Pj=i,S5=i
=
^(55=1
and J
=
1|5S'
=
1
or
J
=
l)
P5S=i,j=o
=
P{SS =
and J
=
Q\SS
=
1
or
J
=
1)
pj=j^ss=o
=
P(55'
= 0and J =
l|5'S'
=
lor J
=
l)
1
description of the data generating process
which happens with intensity
it,
1)
obvious corollary
{SS
that there are only three possible outcomes that can take place at those times
{SS
= P[J ~
J
^.
A, is
we assume
that once in a sudden stop, the
independent of the jumps in J.
addition of a risky asset with the above properties modifies the analysis in only two respects. First
the evolution of reserves (7) becomes
dXt
where
^^ is
the
amount invested
=
{r{Xt
-
itFt)
-Ct+ At) dt + ^,dFt
in the risky asset Ft.
It is
important to note that the constraint Xt
>
0,
imphes the "short selUng" constraint
?t
Else, there could
If
inax(J)
—
1
be
(as
case,
where the jump
is
> -Xt/max(J).
sufficiently large to violate the non-negativity constraint
we have assumed) we have the
portfolio constraint: ii/Xt
risky asset cannot be used to circumvent the borrowing constraint
Second, the optimization problem
now
(i.e., it
>
must post
full
involves a portfolio decision. This decision
the post-development and sudden stops regions since investing in the asset Ft only
on Xf.
—1. This means that the
is
margin).
straightforward in
means adding
risk to the
country's reserves holdings, without reward in terms of hedging value. Accordingly, the country picks it
in these regions
and the associated Bellman equations
Thus the portfoho
where the country
is
decision
preparing
is
—
^
of section 2.3 remain unchanged.
interesting only in the pre-development non-sudden-stop region (NSS),
itself for
a potential sudden stop. Letting:
26
the Bellman equation in the
r
Cj
I
1
max
NSS
region
is
now'':
'
-
<
Ct^''^^
^t-F'txVi.i
i,je{o,i}
_(,
+ ^.+5yvss
+Mk
[(1
- 7)^^^^ - v^'^t] +
^4 (-7(1 - 7)-^^^ + 2jv^''x + .^^/^x^)
where:
„55i{ss=,}
^
^^^ if 55 = 1
^wss if 55 =
/
and
J = E{J\SS=1
The
additional
first
order condition
Y^
or
J
=
l)
is:
+ FtCMJ = ^}-^^v^''
pj=,,ss=,v!''^''='Hx,
(20)
i,je{OA}
is
a standard Euler equation. Defining the rate of return on asset Ft
first
order condition for consumption given in (12) we can rewrite (20)
Notice that this
as:
J
and using the
=
-RISS
1
or
=
J
as:
1
1
_u'{CrD-)'
where the expectation
of
J and SS
is
is
taken at time t^ but without knowledge of which of the three possible combinations
about to materialize.
In the extreme case where pj=i,ss=i
Euler equation collapses
wliich
imphes CtD+
=
—
1,
the country has an "ideal" instrument at
its
disposal and the
to:
CtD- and hence there
is
no consumption drop upon entering a SS. Moreover, one
can show that in this case the country has no incentive to accmnulate reserves and instead just purchases
enough contingent instruments to ensure that CtD+
More
generally,
=
both pj=o,ss=i ^nd pj=\^ss=Q are
injections of resources that
Ct-d-.
positive,
which implies that there can be
do not come with sudden stops, as well as sudden stops with
sumption drops since the contingent strategy does not yield the resources
the Euler equation simply balances the risks of these events.
non-contingent and contingent reserves, an issue
''In
the second line pj=o,ss=o
=
since
we return
significant con-
manner. In
tliis
case,
Doing so generally requires accumulating
to in the empirical implementation below.
we condition on events where
27
in a timely
significant
either
J
=
1
or
SS =
1.
Implementation
4.2
The
strategy described above assumes that the country and the world capital markets can write a contract
that
is
jump
contingent on a
a contract.
exhibits a
jump and
in the price of a traded asset. It
we
In this section
otherwise.
Assume
—
Wt
+
ayj^/A
+ o"^„ \/A + e + ei
+ e. Now
ji^dt
suppose that
.
Such a scheme pays
it
A
assume that we take
1
if
Wt+— Wt' >
to
1 if
to write such
a traded asset
+ JwdNt
this state variable.
1 if
ei
N
can be repeated
if
+
wj+a > wt
cr^.x/A
and
if
bound then taking
a jump and
otherwise.^
The
wt exhibits a discontinuity larger than
e to
and
-^
if
otherwise. Furthermore
e. If
the distribution of
be that lower bound yields a contract that has a payoff of
and the analysis
important to note that the above argument
that delivers a payoff conditional on a
jump
is
an approximation argument.
We
as the limit of trading strategies with existing securities.
the limit as
A
—>
0.
A
fixed
amount
5
Quantitative Analysis
Our
goal here
is
1 if
Prom
Wj+a — Wt
is
we
The more
therefore
larger than a
taken to be a month.
is
not to conduct a thorough search for the optimal risky instrument for specific countries'
portfolios. Rather,
imphcations.
is
calls is relatively
In the econometric estimations that follow
estimate the "correlation" between a contract that would have a payoff of
and
has a
there
obtain a contract
straightforward. Investment banks are often willing to provide quotes for such contracts directly.
is
J^,
1 if
of the previous section follows.
a practical perspective, creating a payoff that resembles a digital option from puts and
e,
wj+a <
Then, one
1.
payoff of such a strategy would then be equivalent to the payoff of an asset
following the process in equation (19)
subtle part of the argument
and
short a call option
»
+ (Tu,-s/A + £,
>
A,e,ei
first
+ <^w V^ + e + e\
— and Nei
times, so that £i
wt+A > wt
Fix
+ e and
by considering contracts of very small maturity. Then we have a contract that
0,
£, i.e.
positive lower
It is
how
not immediately obvious
option with exercise price wt
call
obtains in the limit a "digital" option that pays
pays
(j^dBt
-\-
and put options that are written on
exist call
assume that the country goes long a
with strike price wt
is
to create a contract that pays
that there exists a state variable with dynamics:
dwt
and that there
way
describe a practical
With
we seek
to illustrate the kind of properties that such instrrunents ought to have
this purpose,
from quoted put and
call
we chose
options on the
the
S&P
CBOE
Volatihty Index (VIX). This
500, available since the late 1980s.
is
and
their
a traded index formed
This index extracts the
"implied" volatility from the underlying options. Traders then can take positions in this index to implement
If
the distribution of the
probability as long as
we
jump has no
lower
set the "cutoff" e low
bound then we can
enough and Pr(Cu,
28
still
>
obtain a payoff similar to the above with high enough
^) 'S close to
1.
)
hedges or speculate. In
events. Moreover, as
line
with the model, this index
we show below, sudden stops
is
US and
primarily driven by
are liighly correlated with
jumps
not emerging market
in the
VIX. These two
properties are key for useful and potentially hquid SS-hedging instruments.
The VIX Process
5.1
Let us postulate a continuous time process of the VIX, described by an Ornstein-Uhlenbeck with jumps:
dlog{VIX)
Notice that this process
component
is
of the form considered in section 4.2.
of the process, which plays
A, B, cTvix to be constant and
jump
^A{B- log{VIX)) dt + avixdBt +
<?!>
process dNf takes the value
hazard rate
for
the
jump
no interesting
when a jump
takes place and
process. Finally, note that the second
The
first
-
mean
Xjfi^dt)
term captures the predictable
what foUows.
role in
to be a normal distribution with
1
{4>dNt
For simphcity
we
shall take
and standard deviation a^. The
/x^
otherwise.
The parameter \j denotes
and third terms
the
in the process are martingale
differences.
In estimation,
we
shall
approximate the above process by
to remove the predictable component, for which
its
discrete time counterpart.
we estimate an AR(1)
The
first
step
is
process for log{yiX) and focus on
the residuals. These residuals are characterized by a mixture of normals:
et
with
fJ-Yjx
between
~
~Xjl^,p^ and 1{J
and
t
i
+ A,
and
=
/Uy^^fA
=
+ o-v/xVAet+A + '?!'t+Al{-^=
1} denotes
{^et+A,'Pt+A)
^.re i.i.d.
K't't+Aj
The
an indicator that takes the value
relatively large
Note that
M^
=
in the interval
and infrequent jumps and
1 if
a jump takes place
al
is
that the discrete approximation excludes
A, which seems reasonable
relatively small time intervals
if
we want
to focus
on
A.
(21) can be rewritten as:
et
with pvix
jump
(21)
Normal:
source of the approximation to the continuous time limit,
the possibihty of more than one
1}
1
^
=
(1
- pvix)N{fiyjxA, aljxA) + pvixN^fXyj^A + /x^, alj^A +
a^
(22)
e"^-''^.
In principle, estimation can proceed ,from this point on along conventional jump-diffusion estimation
(see, e.g.,
Caballero and Panageas (2004)). Instead of following this path,
29
we adopt a
strategy that
is
more
directly linked to the implementation strategy described in section 4.2.^
obtain contracts that pay off
is
1
unit
if
the residual from the log{VIX)
the cutoff that ensures that the type
1%
I
We
assume that the country can
ARl-regression
is
error of wrongly accepting an observation as a
above 0.259. This
jump
is less
than
(see the appendix):
Pr{et
Note that
this rule clearly selects the
component of
diffusion
>
0.2591 J
=
jumps, but there
<
0)
0.01
some
is
residual
Figure 7 shows the residuals of an an AR(1) model for log(y/X).
instances
when
Type
error associated with the
I
ej.
this statistic is
VIX
above 0.259. The
onset of the gulf war) in 1997 (around the Asian
The shaded areas
exhibits significant
crisis), in
jumps
represents those
in the early 90's (at the
1998 (around the Russian/LTCM
crisis), after
9/11/2001, and around the beginning of the U.S. corporate scandals and the Argentinean default.
VIX
This procedure gives us estimates of the times when the (ARl residuals of the)
With
these estimates
we turn next
to determining the joint occurrence of
jumps and
experience jumps.
transitions into
SS
for
the various countries.
Conditional Probabilities
5.2
The
final
step to be able to assess the benefits of hedging,
between jumps
We
in the
VIX and
an estimate of the degree of "correlation"
use a Bayesian procedure to estimate the parameter
P{SS\J)
PJ=i,ss=i
=
Pj=i,ss=o
,
to find
is
transitions into SS.
we
as described in detail in the appendix. In essence,
VIX
in the
as determined above.
Effectively
we
+ Pj=i,ss=i
start
by marking the dates associated with jmnps
are conditioning on this information.
We
then use the
paths associated with the repetitions of the Gibbs sampler to identify the dates in which individual countries
entered into a SS (applying the parameters estimated for the long annual sample on quarterly data for the
1990s).
Second,
VIX jumps and
we update
the (uniform
-
uninformative) prior by counting the
the country transits from an
NSS
state into
in section 3.1 in order to exploit the cross sectional
states
and count the
total
number
of
jumps
probability that within each quarter the
in the
VIX
VIX jumps
for
P{SS\J) and P{J) from which we take a separate draw
I
error
compared
number
pool
all
of times that the
countries together as
for
we obtain a
30
We
also count
(beta) posterior distribution
each repetition of the sampler.
biasing the results towards finding a lower "correlation", because
to Bayesian filtering.
NSS
during such states. This allows us to update the
of times that a given country transits into SS. Fourth,
is
We
(we denote this probability as P{J))-
number
'if anything, this procedure
state.
dimension of the panel. Third, we condition on the
the
of type
an SS
it
To check
introduces the potential
VIX Residuals and Jumps
>
0.1
Jan92
Figure
7:
VIX
residuals.
The grey
Jan94
Jan96
Jan98
JanOO
areas correspond to instances where the
jiunp-cutofF.
31
Jan02
VIX
residual
is
above the
Average
Std Dev
5%
25%
50%
75%
95%
Beta
0.159
0.032
0.109
0.137
0.158
0.180
0.215
Empirical
0.155
0.006
0.145
0.151
0.155
0.159
0.163
Table
4:
Pr
(J) Pooled Estimator, quarterly data 1990-2003. Bayesian
Average
Updating and Empirical Distribution.
Std Dev
5%
25%
50%
75%
95%
Beta
0.236
0.096
0.095
0.165
0.226
0.296
0.413
Empirical
0.211
0.040
0.150
0.190
0.200
0.238
0.286
Table
5:
Pr{SS\J) Pooled Estimator, quarterly data 1990-2003. Bayesian Updating and Empirical
Distri-
bution.
we
the influence of this prior on our estimates
occurrences
{NSS
SS =
-^
1
and J
=
1) to total
also report directly the distribution of the ratio of joint
occurrences
report the distribution of the ratio of total occurrences of
event{NSS
—
>
In tables 4
and
5*5
=
and
1
5
or
we
=
J
1)
.
We
{NSS
SS =
-^
jumps only
(
J=
1) to total
Similarly,
we
also
occurrences of either
table 5 contains estimates for P{SS\J).
The top
line in
Table 4 contains the estimates
both tables includes
distribution obtained through the formal Bayesian procedure described above.
is
1).
label this the "empirical" distribution.
report the results of this estimation.
the "empirica" distribution, which
J =
or
1
for
P{J)
statistics for the posterior
The bottom
line contains
only given for an intuitive "robustness" check on the influence of the
uniform prior on the posterior location parameters hke the mean and the median.
Prom
this table
one can obtain the imphed intensity of a jump in the
\j
VIX
as:
= -41og(l-P(J))=0.69
where P{J) denotes the mean of the posterior distribution of P{J).
Now we
can identify
all
the key parameters of the joint
VIX and SS
process, which can be recovered
from
the following four relations:
^J
=
X* {pj=i,ss=i+PJ=i,ss=o)
A
^
=
X* (pj=i,ss=i +Pj=o,ss=i)
Pr(5S|J)
=
(23)
PJ=1,SS=1
Pj=i,ss=o +Pj=i,ss=i
1
Since
we have obtained estimates
=
PJ=i,ss=i +Pj=i,ss=o+Pss^i,j=o
for Xj,
(24)
A (from section 3.1) and Pr(5'S'|J) from the above table, we
32
7
8
A
0.211
Pr {SS =
1
and J
=
1)
0.221
1
andj =
0)
0.063
=
1)
0.716
X*
0.743
Aj
0.696
S
0.04
A
0.148
Pt {SS =
r
0.04
V
(0.07,0.1,0.14)
Pr {SS =
IJ'
0.018
9
0.025
a
0.05
Table
and J
Parameters used in Simulations.
6:
Average
Std Dev
5%
25%
50%
75%
95%
Beta
0.636
0.16122
0.354
0.526
0.645
0.759
0.883
Empirical
0.672
0.088
0.556
0.625
0.667
0.714
0.857
Table
Pr {J\SS) Pooled
7:
Estimator, quarterly data 1990-2003. Bayesian Updating and Empirical Distri-
bution.
can solve
for the four
unknowns pj=i,ss=i, Pss=i,J=o, Pj=i,ss=o and x* Table
6 smxnnarizes the resulting
parameters that we use in the simulation in the next section.
Finally,
we
which report
also obtain estimates Pr( 7155),
of observing a "jtmip" conditional on a transition to SS
is
in
Table
7.
They
indicate that the probability
about 70 percent. Wliich means that with very
high probability the contract that we consider delivers payoffs exactly in the states where inflows are needed
the most.
5.2.1
Implications
In this section
we cahbrate
a base case scenario
5.1
we can determine
the model with the same parameters as the ones used in section 3.2.
we use the median
pj^ss- Using the
x*, Pss=i,J=o a^nd pj=i,ss=o
same A
as in section 3.2
To provide
and the Xj obtained
in
from the procedure described above and solve the model
numerically.
For the same realizations in section
3.2, plus
the corresponding realizations of simulated
VIX
processes,
panel (a) in Figure 8 reports the difference in reserves at the time of the sudden stop between the contingent and non-contingent strategies, normahzed by the average of the latter.
distribution
is
close to 30 percent.
This
non-contingent reserve accinnulation.
transfer
more resources towards the
states
However, the above distribution and
it is
much worse to have
is
By
less reserves
The
empirical
mean
of this
a direct consequence of the efficiency of hedging compared to
adopting the optimal hedging strategy the country manages to
concerned with, namely the onset of a SS.
it is
its
when
mean underestimates
there
is little
of
the benefits of hedging.
them
to start with (which
strategy typically dominates by a wide margin) than to have fewer reserves
,
33
when
is
It is
apparent that
when the hedging
the country has had plenty
Density of absolute Reserve Gain
Density of proportional Reserve Gain
1400
1200
h^Ji
0.2
-2
4
2
(X!;SS-XSS)/E(XSS)
T
T
Figure
8:
'
^
(x
4
ss-X ss)/x ss
T
Reserves Gain
(in
absolute and proportional terms)
34
6
Difference
n
in
consumption drop
:
.
I
1
-0.04
-0.03
-0.02
0.01
0.02
0.03
|AcV|Ac|
Figure
9:
Difference in the magnitude of consumption drop
35
upon entering the SS
0.1
7^
Mean
Level of Reserves
Median Level
0.1
7]
of Reserves
Reserves
Mean
Reserves (fourth quartile)
quartile)
Median (normalized)
reserves gain
Median (normalized)
reserves gain
Median (normalized)
reserves gain (fourth quartile)
(first
quartile)
=
0.14
77
=
0.137
0.065
Mean
(first
=
0.07
0.019
0.064
0.132
0.019
0.011
0.022
0.0034
0.122
0.263
0.0356
0.293
0.047
1.658
2.797
1.265
9.422
-0.058
-0.130
0.801
Absolute value of Median difference in consumption drop
0.010
0.010
0.022
Absolute Value of Median difference (75th Percentile)
0.015
0.013
0.027
Absolute Value of Median difference (25th Percentile)
0.007
0.006
0.015
Median
0.700
0.562
0.776
Portfolio
Table
8:
Performance of hedging strategies
of time to prepare for the sudden stop (which
is
when
the unhedged strategy
important asymmetry, panel (b) in figure 8 plots the histogram
this
may do
better).
To capture
for:
Ax =-^
This expression gives the relative reserves gain. As might be expected the distribution of Ax^
The hedge performs
reserves.
An
especially well
alternative
way
of
when
making the same point
the row labelled "median (normalized) gain
contingent strategy
when
reserves are low
is
right skewed.
the country has not had the time to accumulate non- contingent
(first
is
is
seen in the
quartile),"
close to
first
column
{rj
=
0.1), for
which shows that the median gain from the
300 percent (since the
implies an absolute gain in reserves around 3 percent of 9
of Table 8
mean
of reserves
is
0.011, this
Y).
In terms of the drop of consumption at the instant of the sudden stop, the above difference translate into
a median difference that exceeds
significant
mass
1
percent of
maximum NSS-consumption
and, as shown in Figure
9,
has
at drop-differentials four times as large as that.
Figure 10 displays the notional amount of contingent contracts normalized by reserves as a function of
xj.
Two observations emerge
from
this figure: First, as a percentage of xt, the
contracts declines. Second, the country enters a large
number
the country enters contracts that equal total reserves.
amount invested
of these contracts. For instance
And when x
is
in contingent
when x —
as large as 0.15, the contracts
0.05,
still
exceed 30 percent of reserves.
In terms of dynamics, these two observations
mean
that the country should
part of the portfoho aggressively, adding non-contingent reserves only gradually.
36
first
build the contingent
The reason
for this strategy
Portfolio
Figure 10: Notional amount invested in contingent instruments normalized by reserves.
37
is
that the country
is
particularly concerned about the worst possible event.
resources at the time of a sudden
this worst-event is
of reserves
stop.
When
the stock of reserves
by getting a large contingent payment
is large,
the worst-event
is
more
contracts and these do not deliver, than by holding a large
These features also have implications across
8,
at the
likely to take place
is
That
is, is
to be found without
small, the best chance of avoiding
time of the sudden stop.
if
amount
When
the stock
the country over commits to contingent
of non-contingent reserves.
Going back to Table
different configurations of parameters.
the columns report results for three scenarios that only vary in the size of the sudden stop. In addition
to the
benchmark
scenario,
we
report results for the 25 and 75 percentiles of our estimated
suggest that as countries improve along dimensions that reduce the size of sudden
stop,^'^
t;s.
The
results
they should not only
reduce their reserves accumulation but also reallocate their portfolios toward more contingent instruments.
Debt and Contingent
6
Let us
now
W=
relax the constraint
Management
Liability
and allow the country
Xt>-Wt
for
some process Wt >
mean
This
0.
turn negative. In principle
Wt
to borrow from
WCM,
so that:
a.s.
WCM as
that net assets with
could be any process. However,
it is
can potentially
the counterparty (Xt)
most reasonable to assume that Wt
is
proportional to income:
Xt > -wYt
The
point of this short section
is
(25)
that the essence of our analysis of reserves remains vahd here,
now
for
net assets (or habilities) management.
In fact, without further modifications the analysis
country would behave as
in
if
w=
0.
The reason
for this
is
income that would lead to negative consumption.
impUcation
identical to that in the previous sections since the
is
that for any Xt
to "complete" the market by introducing a
is
<
0,
there
The standard approach
new
is
to
a path of the diffusion
remove
this unpleasant
asset St with payoffs:
—1 = rdt + asdBt
where dBt
is
the
By adding
willing to
same Browirian motion that
this asset to its portfolio, the
drives the
income process. ^^
country removes the possibility of a perverse path and
is
now
borrow from WCM.^^
For example, the estimate of
77
we report
for Chile
corresponds to an average of 0.11 and 0.08
for the
1980s and 1990s,
respectively.
I'See Duffie and
^"We view
Huang
(1986).
the introduction of this asset as a technical device to
38
make
the substantive point
we wish
to address here rather
Rt=Xt+ wYt >
The presence
5
of
income process but
stops,
and
if
w
sudden stops,
in the coiintry's portfolio allows
this changes in
it
to effectively remove the diffusion
no substantive way our
In particular,
analysis.
component
there are no sudden
if
not too large, the coimtry never wants to increase X. But, more importantly,
is
it is
of the
if
there are
optimal for the country to rely on contingent instruments correlated to the sudden stops
rather than just on ex-ante slack in the constraint 25.
In summary, the imphcations and recommendations stemming from oui analysis carry beyond the problem
of reserves
management
to that of external assets
sudden stops benefits from indexing
stops.
Pubhc
its
assets
and
and
Hability
debt, for example, should have coupons
and
A
management.
liabilities to variables
country that experiences
that are correlated with these sudden
principal that
fall
as the
VIX
(or similar variables)
crosses certain critical thresholds.
7
Final
Remarks
Emerging market economies hold
levels of international reserves that greatly
developed economies (relative to their
size).
This would seem paradoxical given that, unUke the
latter, the
much of their growth ahead of them. The paradox
disappears
former face significant financial constraints with
once these greater financial constraints also become an important source of
to smooth. This
is
the context
Once such perspective
is
we have modelled,
volatility,
analyzed, and began to assess quantitatively.
smoothing the impact of sudden stops unrelated to a country's actions?
accumulated?
volatility?
How
fast?
Are there potentially
Who would be the
by financial and
less costly financial
than
natural counterpart for these mechanisms?
markets" solution
How much
How
aspects of an answer to each of these questions:
is
of
them should be
are these
mechanisms Umited
is
is
a lower bound for Y.
One
An
alternative
of the
mechanism
main reasons the
to
Even
if
optimally managed,
and used more aggressively during
being done by prudent emerging market economies.
than US a claim that such assets exist in reahty.
assume that there
effective are reserves
mechanisms to deal with capital flow
reserves offer hmited insurance, should be accumulated at a slower pace
stops,
How
collateral constraints?
Our framework provides
sudden
which countries seek
adopted, one must ask whether current practices, consisting primarily in accu-
mulating non- contingent reserves, are the best countries can or should aim to do.
in
exceed the levels held by
However, the most important
make the same substantive
literature prefers the
somewhat
point,
is
artificial
simply to
"complete
that in such case one can use the very general methods of El Karoui and Jeanblanc Pique (1998), and
He and
Pages (1993) to solve the country's post-development problem. As El Karoui and Jeanblanc Pique (1998) demonstrate, solving
problems of this kind amounts to solving the complete markets problem and subtracting the value of a perpetual American
option on the difference between the constraint-free net asset process and the constraint.
39
message of the paper
is
that there are potential insurance contracts, credit hnes
that could significantly reduce the cost
The natural counterpart
and improve the
of these instruments
and useful instrument
is
The much touted GDP-indexed and peso-bonds,
to trade with specialists (in fact, they
and should form the
and contractible global
do
in oui
its
strategies these countries can engage in. In particular, since
side.
in
specialist
risks.
hedges are unhkely to be
It is clear,
nonetheless, that
sudden
Importantly, the very same financial
and amount of insurance and hedging
sudden stops are mostly times when
must be such that require
stops.
mind when
variables that are significantly correlated with
constraints that are behind these countries' troubles, limit the type
country
smooth sudden
model) are unlikely to appeal to the
own
basis for a better contingent strategy.
are constrained as well, the strategies
and hedging markets,
example, while a natural
for
well, since in practice
perfect, and overcommitting to an imperfect hedge comes with
stops
to
an important consideration to have
broad markets. Non-contingent reserves have a place as
there are enough verifiable
mechanisms
and contracts are not the regular emerging market
investors but the world capital markets at large. This
designing such instruments.
efficiency of
little
credibihty
specialists
and commitment on the
This means, essentially, policies and investments that are paid (or collateralized) up-front
rather than simple swaps of future contingencies.
40
Source
Series
Nominal
GDP (GDP)
World Development Indicators
(quarterly
and annual)
CPI (P)
IFS
Nominal Exports {PxX)
World Development Indicators and IFS
(quarterly
(local currency)
(quarterly
Nominal Imports
(PmM)
and annual)
and annual)
World Development Indicators and IFS
and annual)
(local cm-rency)
(quarterly
Real Exports (X)
World Development Indicators
(local cm-rency)
(annual)
Real Imports (M)
World Development Indicators
(local cm-rency)
(annual)
Nominal Capital Flows (CF)
IFS
(dollars)
(quarterly
Nominal Exchange Rate {E)
World Development Indicators and IFS
Net Factor Payments (NFP)
IFS
(doUars)
(annual)
(quarterly
Table
8
A
9:
Data used
in the construction of
and annual)
and annual)
tp.
Appendix
Data
The Data
The VIX
is
publicly available from the
of this index, see
CBOE
(2003).
For the construction of
V'it,
We
CBOE on a daily frequency.
used a monthly frequency
we used data from
in order to
the World Bank's
While
in the
smooth spurious daily
volatihty.
World Development Indicators Database,
and from the International Monetary Fund's International Financial
of the variables
For an introduction to the construction
Statistics (IPS). Table 9 presents
a
list
and corresponding sources.
model
ip is
straightfor-ward, in the data its
computation
is
more cmnbersome
since there are
multiple goods, exchange rate fluctuations, intermediate goods, and so on. All our steps below are aimed at
isolating in
V
the component of external resomces and income which
41
is
transitory in nature. For this,
we
let:
,Cycl
{Qt
where
N
and
X
[\-p^-^-^^^
correspond to real nontradables and exports;
and
local currency;
-f^^
-\)Y _
E
and
CF
Px and Pm
to export
and import prices
in
to the nominal exchange rate and capital flows.
Real nontradables are constructed from:
Nt
=
^
{GDPt -
iPx,t
-
0.5PM,t)Xt)
PN,t
where
GDP
is
the country's
GDP,
PN,t
and the term 0.5PM,t removes a proxy
(
p''
We
as
—
'
]
the price of nontradables approximated by the local CPI,
captures the terms of trade
as
we could
The expression
effect.
decompose between trends and cycles using a standard Hodrick-Prescott
much
filter
0.5Xt
is
for intermediate inputs in export- production.
filter;
extending the series
We
in order to reduce the effect of the end-of-series bias in this procedure.
applied the
the log of the corresponding variable.
In siunmary, the denominator in equation (26) measures the average (trend level) of total income and
resources, while the
numerator attempts to capture the cychcal component of external resources.
The Sample
We
work with a sample of
six developing
countries/emerging markets:
Malaysia, Mexico, Thailand, and Turkey.
We
and Mejia (2004) plus Malaysia. However,
since their
time
series
chose them from the
sample
is
list
Chile, Colombia, Indonesia,
of countries in Calvo, Izquierdo
for the 1990s only
dimension (we used data from 1983 to 2003), we dropped
all
andwe needed a longer
the countries that either were closed
during the 1980s, or had primarily domestic macroeconomic problems, or did not have complete data. Our
marginal drop was Korea, for which we did not have good deflators. However, when we re-estimated our
model with Korea included (using only capital flows data divided by nominal
GDP
for
Korea
),
our results
remaiiied essentially unchanged.
We made
one adjustment to oui construction of
crisis is significantly larger
t/ijj.
Thus
than that of the 1990s.
In the case of Chile, the cycle around the debt
in a first stage
we standardized
1990s in order to estimate the hidden-states and the transition probabihties.
After that,
the 1980s and
we
inverted the
standardization to recover the rneans of each state in each sub-sample.
Finally,
we constructed
capital flows.
We
quarterly series for
i/'it
using a related series approach with quarterly data on
restrict the average of the quarterly values to
directly using equation (26). Unfortunately,
the 1990-2003 period.
To
solve
tliis
we
be equal to the aimual figure we computed
lack quarterly data for the capital flows for
problem we use a
linear (or a quadratic) interpolation
42
all
countries in
method
to obtain
}
quarterly series for the years
much
we only
if
when we
lack quarterly data.
Although the point estimates do not change
use interpolation methods, the timing of the event matters for the synchronization between
sudden stops and the VIX, so we use as much information from the quarterly capital flows as possible
We
order to preserve the timing of the joint events.
shown
(dates are
in parenthesis): Chile (1990),
in
use interpolation methods for the following countries
Colombia (1990-1995), and Malaysia (1990-1998).
Reserves
As a
reference, the next table
shows
reserves, reserves net of short term,
and
total external liabilities for
the six economies we study.
B
on the econometric procedure of Sections
Detedls
To estimate the process described
Section 3.1:
Gibbs Sampler. The Gibbs Sampler
states (See
Kim and
Nelson (1999)
is
in this section
We
all
an introductory treatment). The basic idea
for
We
precise estimates
allow
all
transition into a
The
p{NSS
of
i/jj;
first
we assume
sudden stop
p{SS
-^
a time (fixing
all
to exploit knowledge
the others) to construct the joint
Nelson (1999) present by poohng
Hamilton (1989,1990) type
all
the countries into a
However, in order to obtain
that the joint probability of a jtunp in the
common
is
NSS)
was drawn from the
differ across countries.
that the transition probabiUties into and out of a sudden stop are the
step of the procedure
-^ 55),
Kim and
parameters of the model to
we assume
across countries. Moreover,
at
is
parameters.
modify the basic model that
single sample.
we apply a Bayesian methodology, by using a
by now a standard methodology in estimating models involving hidden
about the conditional distribution of one parameter
posterior distribution of
3.1, 5.1, 5.2.
is
VIX and
same
a simultaneous
across countries.
to fix a set of initial parameters
$=
{ip^^^, ipf^
-
tpf^^^, CTef^^ «^f,f
>
and then determine the posterior probabiUties that a particular reahzation
}
first
To do that we run a standard
(7V55)or the second (SS) distribution.
as described in
filter
Kim and
Nelson (1999).
This allows us to determine
a sequence of posterior probabilities that a given realization of the data was drawn from the second (SS)
normal
We
distribution.
We repeat this process for each country separately and obtain one sequence per country.
shall denote this as
Pr(55 =
these posterior probabihties.
In the next step
we
whether each economy
l\'ipi;
We
take these
is
in
SS
I's
and
O's as given.
first:
To
We
1
and
to
(artificial)
sample of
Effectively this allows us to proceed as if
or not at a given point in time.
the posterior distributions of the elements of $.
Nelson (1999).
I's
corresponds to a Sudden Stop and
we draw an
$). In the next step
use the convention that
Then we
Once again we do
we
this in steps as described in
use conjugate priors: a) a beta prior for
43
from
we knew
use this information to determine
start with determining the posterior distribution of {^/jf ^^, ipf^
facihtate the updating
O's
NSS.
p{NSS
-
Kim and
tp^^^ cr^f ^.
-^ SS),
,
crf.f
p(SS — NSS)
>
Res
Res
-
Chile
Colombia
Mexico
Indonesia
Malaysia
Thailand
ST Debt
Res
+
-
ST Debt
Total Ext
Priv Res
Debt
Avg
20.6%
15.3%
15.0%
46.5%
Median
21.5%
16.3%
12.6%
45.7%
Max
22.7%
21.0%
29.1%
63.4%
Min
15.4%
4.3%
-5.0%
30.3%
Avg
11.3%
6.1%
1.5%
36.6%
Median
10.0%
6.5%
2.1%
39.1%
Max
16.6%
10.8%
6.4%
44.9%
Min
8.4%
2.4%
-3.2%
27.1%
Avg
5.5%
0.1%
-6.5%
28.8%
Median
5.6%
0.2%
-6.7%
25.6%
Max
7.2%
3.6%
2.3%
43.4%
Min
2.7%
-4.9%
-18.6%
20.4%
Avg
11.3%
7.1%
6.9%
22.5%
Median
7.4%
4.8%
4.7%
17.4%
Max
19.8%
15.8%
17.4%
44.4%
Min
4.7%
1.9%
-0.7%
13.3%
Avg
29.4%
-4.8%
-9.1%
187.7%
Median
28.6%
-7.7%
-12.1%
186.2%
Max
42.3%
24.3%
24.6%
237.2%
Min
19.6%
-25.6%
-35.9%
147.4%
Avg
21.3%
3.6%
-7.8%
56.4%
Median
20.8%
3.3%
-3.7%
58.2%
Max
28.0%
18.6%
21.9%
93.8%
Min
14.1%
-6.8%
-36.1%
32.9%
Table 10: Debt and Reserves: 1990-2002 (% of
GDP)
Res: Total Reserves minus Gold;
External Debt; Priv Res: Net Private Reserves.
44
ST
Debt: Short-term
—
with a
=
(3
which coincides with a miiform prior on
1
and an (improper)
Kim and
(see
for
—
ipi
gamma
inverse
Nelson (1999))
V'f
c)
a trmicated (improper) normal and an inverse (improper)
^^ explained in
i'^ef)
,
an (improper) normal prior
b)
[0, 1]
Kim and
for
ip^
that lead to posteriors that depend only on the data
prior for (o^gf^)
Nelson (1999).
we assume
Finally,
that
gamma
all
prior
priors are
independent of each other. By well known results in Bayesian Statistics the posterior distributions are in the
same
class as these conjugate priors.
of the posterior distributions. This
each country separately. In the
at a time
keeping
(i.e.
Moreover there are simple closed form expressions
spirit of
distribution of each parameter,
when we update
the Gibbs Sampler,
parameters except one fixed).
all
for
the parameters
aUows us to determine the posterior distribution of the parameters for
Finally,
we make a random draw from that
the posterior probabilities one
once we have determined the posterior
distribution
and keep that number
fixed
imtil the next iteration of the sampler.
The updating
of
p{NSS
-^ 55), and
countries. So, for each country
number
p{SS — NSS)
>
we count the number
and
of periods in normal times
in
sudden
done by poohng the observations
is
of transitions to
stops.
We
and out of sudden
then add
all
stops,
the episodes for
all
for all the
and the
total
countries and
find the posterior distributions as follows
p{NSS
where af^
is
number
We
shall refer to this as a transition to a
that a normal year
this
(l +J2''f^^
the
sudden stop.
then we add
^ SS) ~ beta
all
is
of observations
marked
^
+ E"''^^^
as normal years that are followed
sudden
stop.
Conversely,
followed by another normal year (NSS). This count
them up
into a single niunber
)
which
is
is
by a year marked
as
a^^^ counts the times
done country by country, but
used in the updating process. Let us emphasize that
approach exploits the panel dimension of the data in order to obtain precise estimates of the transition
probabilities.
Similarly, the posterior for the other
p{SS
-^
parameter in the transition matrix
NSS) ~
beta
+ X! ''^^'^^ ^ + X^ ^f ^
1
(
where bf^^
is
the
number
of observations
marked
is
as
given by the following formula
)
sudden stops that are followed by a year marked as
"normal". Conversely, bf^ represents the other case, which
is
when a
transition out of a
sudden stop does
not occur.
We
conclude by making a random draw from these distributions too.
We
record the
random draws
parameters {ip^^^, ipf^
NSS)
.
-
Then we repeat
V-f
^•^,
of a) the paths of I's
^^f ^,
o-f,f }
and
c)
and
O's for
each country, b) the country specific
the "pooled estimates"
o{p{NSS
the above procedure several times and at each time
the paths of I's and O's for each country and the parameters.
45
By
p{SS
-^
new draw
of
-^ 55),and
we record
the
properties of the Gibbs sampler, the
posterior distribution of tliese
random draws
coincides in law with the posterior (joint) distribution of
all
the parameters (and the hidden states).
we proceeded
Section 5.1: In order to choose the cutoff point x
process for log{VIX) and focus on the residuals
as follows. First,
mixture of normals. Second we estimate a mixture of normals distribution
Pvix,
IJ-vix^ '^vix^
/^<*' '^rf>>
^^ then proceed
to determine the cutoff in the
Namely, given our estimate of Mv/Xi
in hypothesis testing.
we estimate an AR(1)
which -according to the model- follow approximately a
Zt,
""v/.y
^^
for zj.
Having an estimate of the
same way that one would proceed
^^^ the threshold
x high enough so
that:
P{zt
>x\J = Q)<l%
In statistical terms this would correspond to the "size" of a test that would (wrongly) accept the hypothesis of
when
a jump with probability
the
VIX
less
than 1%. Also, with the value of x in hand, we can identify the months
actually crossed the threshold
and we
label
them
as jumps.
Let us reiterate that we adopt this procedure in order to be able to assess the performance of realistic
We
hedging contracts.
also tried
an alternative approach in order to determine \j and pj^ss
ran a Gibbs Sampler to determine pvjx, l^vix^ ^viX'
Zi-
M<^'
'^'4,
^^'^ ^^^^ posterior probability of a
Subsequently we used the estimate oi pvix to determine Xj. In a next step we sampled
the posterior distribution of the states of
VIX
to determine pj^ss-
because
Under
ip^
in section 3.1
and the posterior
we obtained
either procedure
the more conservative of the two and
it is
is
closer to
jointly:
similar results.
how such
We
adopt the
jump
jump
first
in
from
(jointly)
probabilities of a
We
in the
procedure
contracts would be implemented in
practice.
Section 5.2:
jump
in the
By
using the cutoff value for the
VIX. By
that
exceeded the cutoff value
To determine a
VIX
we mean months when
we
also determine the
months
in
which we observed a
x.
distribution oip{SS\J)
we proceed
as follows. First, for each country
we draw paths from
the posterior distribution of the states {NSS, SS)^, as provided by the Gibbs Sampler.
the only relevant observations for the conditional probabihty are the ones
or has just transitioned to a SS.
discard
all
Then we
jump
look at
VIX
all
the times
when
procedure
for
in
there was a
when the country
i
is
in
NSS
SS, except the one that marks the beginning of each SS.
those times where the states switch from
jump
numbers of observations when country
this
is
either in that quarter or the quarter before.
possibility of delayed data reporting etc.
all
Given the model,
Hence, in accordance with the data generating process of the model, we
the quarters in which the country
in the
VIX
the residuals of the estimated AR(1) process of the
to
SS and
VIX.
either in
Let this number be
rij.
46
window
j. Similarly,
we
is
a
to allow for the
also determine
Finally, define ?^5vss+i ^^ ^^^
NSS or just moved to a SS
each coimtrj^ separately.
simultaneously there
allow this short
Let this number be given by n^g
in the
i is
NSS
We
(a transition, T).
We repeat
After completing the above procedure for
all
with
flat priors for
we updated the
p{J\NSS
both
or T)
p{NSS
posteriors for
and p{ T\{NSS or T) and
or T)
p{SS
+ E.n^, 1 + ^lU^ss or
p {T\{NSS or T) and J)
-
beta (l
+
in the
all
i.e.
countries have the
VIX. The benefit
As a robustness check we
prior,
in section 3.1
beta (l
or
also
is
EiU^ss^j^ 1
+
Then we pool
J),
or T)
posterior
which are given by:
t)
of times
joint probabihty distributions
when a jump
in the
between transitions
estimates.
and p {T\{NSS or T) and J) without imposing
of:
{J\NSS
or T)
=
^i''''-NSS or
p(fc)
we obtain the
across countries. This effectively
we can obtain more accm'ate
computed p {J\NSS
distribution
same way that
^i^];)
we determine the munber
same
that
by just recording the random draws
p^"'
across countries.
j,
In exactly the
J).
-
imposes the constraint that
any
NSS)
-^
coincided with a transition into a SS for each country.
SS and jumps
ti)^^^ ^^
and p{J and r|(A''55 or T) and
In a nutshell, for each iteration of the Gibbs sampler,
into a
and
T)
p {J\NSS
VIX
j, n^j,
we have been using throughout, we use a beta
-^ 55), and
p{J\NSS
distributions for the parameters
we sum n^^
countries
In accordance with the Bayesian methodology that
(r (7V55 or r) and J)
=
T
-^-IM^
I
at each iteration k of the
distribution.
Gibbs Sampler and computing the empirical mean of the corresponding stationary
The two approaches
by the assumption of a uniform
C
To
delivered very similar results, suggesting that our results are not influenced
prior.
Details on the numericgd procedure
solve the
model numerically, we proceeded as
in great detail in Section 5.2 of that
the derivatives in
all
in
Kushner and Dupuis (2001). The procedure
book and hence we only provide a sketch. The basic idea
Bellman equations by
discretizations. This
way one can
is
to
is
explained
approximate
reexpress the value function at
each point as an appropriately probability weighted average of the value function evaluated at neighboring
points in the state space.
That way the
solution to a particularly simple
discretized version of the
dynamic programming equation
only transit to neighboring states.
Bellman equation coincides with the
in discrete time,
where the processes can
For the exact formulas of the transition probabihties as well as the
treatment of jumps we refer the reader to Kushner and Dupuis (2001) Ch. 13.2. Once this simple Markov
chain has been determined, determination of the Value function can proceed by the standard value function
iteration procedure.
47
D
Proofs
Proof,
(of
Lemma
1) Post
up
ian motion which
development the country's total available resources At follow a geometric Brown-
to a constant coincides with Yj. Let us use the normalization:
~
^
Ct
Ct
As we
where At denotes post development resources.
establish in section 2.3 the country's
problem
the developed phase) can be reduced to the solution of an essentiaDy one dimensional problem with
(in
HJB
equation
m_ax
This
HJB
is
\
^—-
[
r - My(l - t) + 7(1 - 7)^4] v° + {{r -
satisfied irrespective of the presence of
A^y-
+ 7^)
^t
+ \-
any constraint. Assume now that, at 5
=
Ct)
v^
it is
(27)
optimal
to set
=
2^
the optimal level of consumption
i.e.
is
set to
1
be equal to income, since
1
=
Ct
c^
At
and accordingly:
dxt
=
'dt
and
similarly:
du'(c* )
~ dt
By
equation (3.8) in Yong and Zhou (1999)
it
'-
=0
follows that the above equation can only
fr-/iy(l-7)+7(l-7)-a^-j =
7
If
+
1
2
(r
-
yiiy
be
satisfied
if:
+70-^)
n
the parameters satisfy this restriction, then the Euler equation will be satisfied with equality, and
starting from
Xt
=
0,
Cj
=
volatility,
Xt
the policy that sets
For stronger growth or smaller
1 will
—
be optimal.
will
48
be binding as a constraint, starting from Xt
~
0.
References
AlYAGARl,
R. (1994): "Uninsured Idiosyncratic Risk and Aggregate Saving," Quarterly Journal of Eco-
S.
nomics, 109(3), 659-684.
Arellano, C, and E. G. Mendoza
An Equilibrium Business
Economies:
"Credit Frictions and 'Sudden Stops' in Small
(2002):
Cycle Framework for Emerging Markets Crises,"
Open
NBER Working
Papers: 8880.
Broner,
G. Lorenzoni, and
F.,
Term?," mimeo.
Caballero, R.
in a
J.,
Model
of
MIT
S.
Schmuckler
(2003):
"Why Do Emerging Economies Borrow
Department of Economics.
AND A. Krishnamurthy
Emerging Market
(2001):
"International
Crises," Journal of
and Domestic Collateral Constraints
Monetary Economics,
48(3), 513-548.
"Bubbles and Capital Flow Volatility: Causes and Risk Management," mimeo.
(2005):
Short
MIT
De-
partment of Economics.
Caballero, R.
NBER
J.,
and
S.
Panaceas
(2004): "Contingent Reserves
Management:
An Applied Framework,"
Working Papers: 10786.
(1998): "Capital Flows
Calvo, G. a.
and Capital-Market
Crises:
The Simple Economics
of
Sudden
Stops,"
Journal of Applied Economics, 1(1), 35-54.
Calvo, G. A., A. Izquierdo, and L.-F. Mejia
of Balance-Sheet Effects,"
NBER Working
Calvo, G. A., and C. M. Reinhart
Dollarization," Finance
Campbell,
J.
Y., and J. H.
of Aggregate Stock
DUFFIE, D.,
W.
of
J.
and Development,
Sudden Stops: The Relevance
36(3), 13-15.
(1999):
"By Force of Habit:
Market Behavior," Journal of
HARA
D., AND C. FU
Few Long-lived
El Karoui,
of
Political
A
Consumption-Based Explanation
Economy, 107(2), 205-251.
H. Fleminc, H. M. Soner, and T. Zariphopulou (1997):
Markets with
DUFFIE,
"On the Empirics
"Capital Flow Reversals, the Exchange Rate Debate, and
(1999):
Cochrane
(2004):
Papers: 10520.
N., and
Utihty," Journal of
HuANG
Economic Dynamics and
(1985): "Implementing
"Hedging in Incomplete
Control, 21(4-5), 753-782.
Arrow-Debreu Equilibria by Continuous Trading
Securities," Econometrica, 53(6), 1337-1356.
M. Jeanblanc-Picque
(1998): "Optimization of
Consumption with Labor Income,"
Finance and Stochastics, 2(4), 409-440.
Froot, K.
a., D. S. Scharfstein, and J. C. Stein (1993): "Risk Management: Coordinating Corporate
Investment and Financing Policies," Journal of Finance, 48(5), 1629-1658.
49
Hamilton,
D. (1989): "A
J.
New Approach
to the
Economic Analysis of Nonstationary Time
Series
and
the Business Cycle," Econometrica, 57(2), 357-384.
(1990): "Analysis of
Time
Series Subject to
Changes
in
Regime," Journal of Econometrics, 45{l-2),
39-70.
He, H., and H. F. Pages
Economic Theory,
Heller, H. R.
(1970);
(1993): "Labor Income,
Borrowing Constraints, and Equihbrium Asset Prices,"
3(4), 663-696.
"Wealth and International Reserves," Review of Economics and
Statistics, 52(2),
212-214.
Kim, C.-J., and C. R. Nelson
(1999):
State-space models with regime switching:
sampling approaches with applications.
KlYOTAKl, N., AND
MoORE
J.
(1997):
MIT
Press,
Classical
and Gibbs-
Cambridge and London.
"Credit Cycles," Journal of Political Economy, 105(2), 211-248.
Kletzer, K., D. M. Newbery, and B. D. Wright
(1992):
"Smoothing Primary Exporters' Price Risks:
Bonds, Futures, Options and Insurance," Oxford Economic Papers, 44(4), 641-671.
Kletzer, K. M., and B. D. Wright
nomic Review,
KUSHNER,
H., AND P.
J. (2004):
YONG,
J.,
New
"Sovereign Debt as Intertemporal Barter," American Eco-
90(3), 621-639.
Time. Springer,
Lee,
(2000):
DuPUIS
New
(2001):
Numerical Methods for Stochastic Control Problems in Continuous
York.
"Insurance Value of International Reserves," mimeo. IMF.
AND X. Y. Zhou
(1999): Stochastic Controls:
Hamiltonian Systems and
HJB Equations.
Springer,
York.
Zellner, a. (1997
[1988]):
Bayesian Analysis in Econometricspp. 444-467, Statistical foundations
for
econometrics. Elgar; distributed by American International Distribution Corporation, Williston, Vt.,
Elgar Reference Collection. Foundations of Probability, Econometrics and Economic Games,
Cheltenliam, U.K. and Lyme, N.H.
4836 030
50
vol. 5.
Date Due
3 9080 02618 3860