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DEWEY
Massachusetts Institute of Technology
Department of Economics
Working Paper Series
A Global Equilibrium Model of Sudden Stops
and
External Liquidity
Ricardo
J.
Management
Caballero
Stavros Panageas
Working Paper 08-05
September 7, 2007
.
RoomE52-251
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This
paper can be downloaded without charge from the
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HB31
.M415
A Global Equilibrium Model
Liquidity
Ricardo
of
Sudden Stops and External
Management
Stavros Panageas
Caballero
J.
MITandNBER
The Wharton School
This draft: September 07, 2007*
Abstract
Emerging market economies, which have much
should run persistent current account
The counterpart
We
of these deficits
make two contributions
model
of
sadden
is
deficits in
their
we use
order to smooth consumption intertemporally.
dependence on capital
in this paper:
stops. Second,
of their growth ahead of them, either run or
First,
inflows, wliich
can suddenly stop.
we develop a quantitative global-equilibrium
this structure to discuss practical
mechanisms to insure
emerging markets against sudden stops, ranging from conventional non-contingent reserves
cumulation to more sophisticated contingent instrument
strategies.
ac-
Depending on the source
of sudden stops, their correlation with world events, and the quality of the hedging instrument
available, the gains
from these strategies can represent a substantial improvement over existing
practices.
JEL
Codes:
Keywords:
E2, E3, F3, F4, .GO, CI.
Capital flows, sudden stops, reserves, international liquidity management, world
capital markets, swaps, insurance, hedging, options, hidden states, Baj'esian methods.
*We
at
are grateful to Lars Hansen, conference participants at
IMF, UPF-CREI
for their
AEA
2007, and seminar participants
comments, and to Herman Bennett, Fernando Duarte and Jose Tessada
research assistance. Caballero thanks the
circulated as
CIRJE, Tokyo,
"A Quantitative Model
of
NSF
for financial support.
A
for excellent
partial equilibrium version of this paper
Sudden Stops and Liquidity Management,"
NBER WP
11293,
May
was
2005.
Introduction
1
Emerging market economies, which have much of
run persistent current account
of these deficits
domestic deficiencies, there
itself,
growth ahead of them, either run or should
smooth consumption intertemporally. The funding
a perennial source of fragility since
is
many
can suddenly stop. While in
the country
deficits in order to
their
is
it
requires ongoing capital inflows which
circumstances the breakdown in capital inflows just amplifies
extensive evidence that in
many
other cases the main culprit
is
not
but the international financial markets' response to shocks only vaguely related
to the country's actions.
The real
costs of this volatility for countries that experience
open
noticed, but at least as important in terms of their pervasiveness
large costs paid by prudent economies. These economies do not
to incur in a variety of costly precautionary measures
crises are
extremely large. Less
and cumulative impact, are the
fall
into
open
crises
but are forced
and build large war-chests of international
reserves, a trend that has only increased in the aftermath of the capital flow crises of the late 1990s.
By
now, emerging markets' reserves often exceed 20 percent of their
GDP, which
contrasts with
How
the 4 to 5 percent held by developed open economies such as Australia or Canada.
are reserves in smoothing the impact of sudden stops unrelated to a country's actions?
of
them should be accumulated? How
to deal with capital flow volatility?
How
are these
fast?
Who
mechanisms limited by
Are there potentially
would be the natural counterpart
How much
mechanisms
mechanisms?
for these
financial constraints?
These are among the most pressing questions
market economies and the international financial
significant conceptual progress over the last
for
policy-makers and researchers in emerging
institutions. Unfortunately, while there has
two decades
in
been
understanding some of the hmitations
of financial contracting with emerging markets, there has been
integral
less costly financial
effective
framework to analyze these questions quantitatively. In
much
this
less
progress in providing an
paper we take one step in this
direction.
Our framework
considers two representative agents, one from emerging markets
from the developed world (W). EM's problem has two main ingredients:
significantly higher
account
deficits.
natural lender
is
than
its
Second,
it
W,
its
but
current income so
it
would
like to
(EM) and one
First, its future
income
is
borrow and run persistent current
has difficulty in pledging future income to finance these
deficits.
willingness to lend varies over time in response to shocks to
its
The
pref-
erences, monitoring technology,
dechne
in capital flows to
tions against these
EM,
sudden
a so called sudden stop}
stops.
EM. At
markets available to
and output. These shocks, when negative, lead to an equilibrium
We
common
EM
sudden
environment,
The
practice by adopting a hedging strategy.
costs
is
and
framework and depend on the particular source
we estimate a regime switching model
of emerging markets, as implied by our theoretical framework.
of sudden stops in a group
As part
of this exercise,
estimate the joint incidence of sudden stops with jumps in the price of assets traded in
of risk
it
stops.
In our quantitative analysis
markets.
EM precau-
can accumulate noncontingent reserves, but
benefits of these strategies are endogenous in our
of
this
discuss a range of options as a function of the risk-sharing
a minimum,
often possible to improve on this
Faced with
A calibrated version of the model
premia and the
size of reserves
matches the extent of capital flow
accumulation.
US
we
also
financial
reversals, the behavior
We then use the model to quantify the potential
gains from adopting hedging strategies and conclude that, depending on the source of sudden stops,
their correlation with world events,
and the quality of the hedging instrument
from these strategies can be substantial.
When compared
available, the gains
to the welfare gains of existing practices,
they represent improvements that range between about 10 and 115 percent.
Our paper
relates to several strands of literature.
sudden stop of capital
and Russian
crises.
inflows.
The work
The main shock that concerns
The hterature on sudden
stops gained
of Calvo (1998) describes the basic
momentum
us here
is
a
since the Asian
mechanics and implications of
sudden stops and Calvo and Reinhart (1999) document the pervasiveness of the phenomenon. The
modelling of these sudden stops as the tightening of a financial constraint
of Caballero
among
is
also present in the
and Krishnamurthy (2001), Arellano and Mendoza (2002) and Broner
et al.
work
(2003),
others.^
As an intermediate
step in developing our substantive argument and quantifying the effects
we
describe,
we model
The view
that reserves can be used to cushion the impact of external shocks exists at least since
Heller (1966),
reserves accumulation as a buffer stock
model against
was enhanced by the work on precautionary savings
'See an earlier version of this paper, Caballero and Panageas (2005),
three agents:
insure with
EM, W, and
W.
specialists.
a
macroeconomics during the
more reduced form model, but with
There, sudden stops are the result of shocks to specialists which
Shocks to the monitoring technology play a similar
^See also Kiyotaki and Moore (1997)
for
in
capital flow reversals.
for
role in the current
EM
tries to
model.
a comprehensive model of how financial constraints are linked with cycles.
1980s and has recently returned to the fore with the large accumulation of reserves exhibited by
emerging markets since the
crises at the
end of the 1990s
Lee (2004)).
(see, e.g.,
Importantly, the main reason for seeking insurance and hedging in our 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 Caballero and Krishnamurthy
(2001) for emerging markets. While the substantive themes developed in those articles differ from
ours, the basic
model
in
our paper
is
in
many
respects a
dynamic global-equilibrium version
of
theirs.
Closely related to our recommendations are those in the sovereign debt literature.
mality of contingent debt and the limitations to
of that literature. In particular, the
Wright (2000), characterize
it
imposed by financial
The
frictions are also a feature
work of Kletzer, Newbery and Wright (1992) and Kletzer and
feasible financial structures consistent with different degrees of
ment by a sovereign borrower and
opti-
its
lenders. In our
model we capture
financial frictions
committhrough
monitoring costs, which capture features similar to those of their richer limited enforcement framework. Our paper reinforces
much
of the message in that literature
and provides a tractable model
that can be estimated and quantified.
The
closed
interaction between precautionary savings and financial constraints
economy framework
of Aiyagari (1994).
He
calibrated such a
is
also present in the
model to estimates
of
US
microeconomic income processes and other parameters and concluded that eventually agents would
save enough to relax
all
a steeper growth rate than
W.
time varying financial frictions
increasing endowment.
and recurrent episodes
There has been a
(which
is
the index
happen because
EM faces
Hence, EM's incentive to save in order to mitigate the
effects of
financial constraints. In our
The
of
is
model
this does not
counterbalanced by the reluctance to save in light of a steeply
interaction of these two effects leads to a stationary level of reserves
sudden stops and consumption drops.
significant rise in volatility trades in financial markets, including the
we use
for
our main example of hedging). The finance literature has studied
the impact of such trades on the performance of hedge funds and other markets participants
e.g.,
Bondarenko
(2004)).
VIX
From a
risk
management
(see,
perspective, the issues faced by these economic
agents are similar to those faced by emerging market policymakers.
We
setup the model in Section 2 and describe global equilibrium under different assumptions on
the risk-sharing options available. In Section 3
we estimate the sudden
stop process and cahbrate
the model. Section 4 presents our main quantitative results and Section 5 concludes.
The paper
also contains several appendices.
A
2
Model of Sudden Stops
In this section
we
in Global Equilibrium
describe a model of optimal global risk-sharing in the presence of financial
constraints in emerging markets.
In the model, systemic crises are caused by events in world
financial markets that lead to abrupt changes in capital flows to
We
emerging markets.
start with a
characterization of a perfect (sudden stop) risk sharing benchmark, and conclude with a discussion
of a
more
realistic
proxy-hedging context where available insurance instruments are only imperfectly
correlated with sudden stops.
Agents and Endowments
2.1
There are two "representative" agents
one from the developed world (W),
Et
t
The parameter p
(it
is
world economy, one from emerging markets (EM) and
who have common
j,
preferences
for j
1-7
>
represents the discount rate, 7
consumption of agent
The world
in the
1
e {W,
EM}
(1)
the coefficient of relative risk aversion, Ci the
and A^ captures a taste shock, whose properties
an endowment economy. EM's current income
is
will
be specified
low relative to
its
future income
has yet to catch up), which we capture through a pre-development phase (which
of the paper)
and a post-development phase. During the former, EM's endowment
AfYt, where At
is
a process that can take either the value
the exact dynamics of these transitions shortly)
,
and
—^^fi,dt +
where
fXf^a
>
and dBt
is
g.
is
or the value A"^
<
1
is
independent of
the focus
described by
(we shall specify
adBt.
(2)
EM
(permanently) transits from the
an exponentially distributed random time
This catching-up transition
is
is
given by
a standard Brownian motion.
pre- to the post-development phase at
hazard
Yt
1
later.
all
r*^,
with constant
other sources of uncertainty in the
model. EM's post-development income
equal to ^Aflt, where k
is
>
1.
W's endowment
is
given by
PAtYt, where
> Ac>
/3
>
1
(3)
for all
t
2.2
Growth-Contingent Debt and Capital FIoavs
0.
Since in the pre-development phase
EM
for
to
EM's expected growth
rate
is
borrow from W. Moreover, given the stochastic nature of catching up
optimal debt contracts are contingent on the transition to development.
contingent) debt contracts establish that
>
t,
development arrives
if
contingent debt contracts
is
is
natural
in our
model,
W provides a flow of resources ftPtYt to EM over the next
in the interval dt
pt while ft
it is
Concretely, (growth-
infinitesimal time interval dt, in exchange for receiving a promise to a stream of
for all s
W,
higher than that of
number
the
and
The
otherwise.
payments ft^Ys
By
of contracts (expressed as a fraction of Yt).
EM can exchange a stream of post-development for pre-development
using these contracts,
growth
price of these
income.
In the absence of frictions, growth-contingent debt contracts are claims that complete the market
with respect to the uncertainty introduced by the random arrival of development, and ensures that
the marginal utilities of
W
and
EM
are always proportional to each other, both pre-and post
development.'^
Since there
assuming that
markets after
only one source of uncertainty in the model in the post-development phase,
is
EM
t'^.^
and
To
W are able to trade
simplify the analysis,
in
W's
we
will
equities implies that
assume
this trade
is
W and EM face complete
feasible (after
''Note that these contingent contracts can be implemented through equity trades.
exist
be
two equities that give the owner an entitlement
Pf^
opment.
and P}^ respectively and their dividends be
To produce the
financed by xtPt''^ /
xtYt
[1
to the
P^
- (Pf^/P^)
(
[1
-
units of
W's
equities.
P] pre-development
{Pfc^i
and
(5Yt
IP%- )
(k/^)] )
^
The
Duffie and
and
W respectively.
Let their prices
pre-development and nVt and pVt post devel-
and receives a capital gain equal to
By
P^a
arbitrage
.
Since Xt
Pfci
is
=
(f^/P)
arbitrary,
^ pBM
Xi.
'iv
Huang
(1985).
£,.
P^+
one can
units of
growth contingent contract. In summary, a growth contingent contract
e.g.
EM
assume that there
EM's equity
holder of such a claim delivers a stream of payments equal to
Hence, by purchasing xt units of EM's shares financed by
"See
of
this,
payoffs of a growth contingent contract, one needs to purchase xi units of
the transition to development takes place.
pressed as £t
Yt
endowment
For
t^) and use
is
(
[Pfc^
- (Pfc^ /P^- ) P^+])
once
and hence the capital gain can be ex-
let xt
W's
=
1/
shares,
(
[l
- {Pf^- IP%-)
(/t//3)] )
one can create one unit of a
equivalent to a carry trade in equities.
P^
to denote the price of
and
W respectively, with 5f^ + S^ ^
Normal Times,
2.3
W's
shares and
Crises,
Sf'^,S^
to denote the holdings of such shares by
1.
and Sudden Stops
During the pre-development phase, there are two states of nature
(Sf
=
respectively.
all t
The
1) times.
>
We
EM
st:
"normal"
transition hazards from the former to the latter
(st
=
0)
and
"crisis"
and vice versa are A and
For simplicity, there are no crises after the time of development
r'^.
(i.e.,
st
—
A,
for
r'^).
assume that
for
each growth contingent contract
ft,
W pays a monitoring cost equal to
itYt,
where
'z
^t
if Sf
shall
assume throughout that
growth contingent debt
i
>
and
g
(4)
if
i
We
=
={;,..::::
+q
St = 1
>
These monitoring costs capture the idea that
0.
effectively represents uncollateralized lending to
subject to a series of incentive problems that are costly to overcome.
EM, which
in practice
is
For our purposes, these
monitoring costs inhibit intertemporal consumption smoothing between pre-and post-development
times.
Moreover, we assume that during normal times Af
Hence, crises are also times when world output
To produce sudden
Assumption
stops within our framework,
1 Either q
Assumption
1
may
>
or
A'^
<
fall
1,
while during crises Af
below the (stochastic) trend
in
monitoring
its
trend, so that
crisis:
it
becomes
costlier to
activities.
two extra layers of
f^o
we assume that the
1.
Either the cost
generality.
First, the
growth rate
/Xj
+
St (Mi
-
^^o)
if t
<
r"^
preference shock process At in equation (1)
7
is
it is
also
could take different
values during crises, normal times, and development:
Second,
<
Yt-
For the purposes of calibrating the model and performing robustness checks later on,
useful to allow for
A''
(or both)
I
requires that at least one of two events happens during a
consumption
=
we assume that
of monitoring increases, or world output declines below
sacrifice
=
given by
where
^ >
happened
expected
and
1
until
utility
A'';
time
number
transitions from normal times to crises that have have
is
the
t.
In the special case where
of
all
A —
During normal times
Benchmark
EM precautions against sudden stops. We assume that there are two potential
The
contracts that can be used for that purpose.
which are zero net supply claims that pay an
kind are non-contingent bonds (reserves),
first
interest rate rt per unit of time dt. Second, agents also
have access to (sudden stop) hedging contracts. For each of these contacts,
of TTtYt to
W per unit
stays at
over the interval dt, then
However,
if St
W until
from
jumps
Sj
of time dt while
to
transits
EM
happens.
delivers the
We
=
Sj
0,
where nt
over the next time interval
1
back to
delivers a
payment
to
EM
EM
and the contract
is
If st
terminated.
receives a flow of Yj (per contract)
0.
W
payments to
W has
assume that
dt,
EM
the "price" of such a contract.
is
W makes no transfers
Hedging contracts are hence similar
case
becomes a standard
(1)
maximization problem.
Perfect (Sudden Stop) Risk Sharing
2.4
the objective
1,
in
and
first,
full
nature to contingent debt contracts, although in this
W has to repay when a transition to a sudden stop
commitment
to such contracts
and hence there
is
no
risk of
W defaulting on EM.
In summary, sudden stop contracts represent forms of insurance that can help
EM hedge against
both the occurrence and the duration of sudden stops.
2.4.1
We
Net Asset processes
can now describe the dynamic budget constraints
(reserves), ft the
amount
of contingent debt as a share of Yt,
contracts as a share of Yt,
1.
(Pre-development,
st
we
=
rt
Letting
X^ denote
j's
assets
and nt the number of sudden stop
have:
0) If
i
dXf ^ =
with
EM and W.
for
<
r'^
and
[rtXf ^
sj
=
[nXJ!^
-
EM's
- Cf ^ +
being the prevailing interest rate, and
dX^ =
0,
Cf +
8
(1
W's
{0-ft
+
net asset process
ftpt
cisset
{Pt
-
ntnt)
process
+ i)+
Y]
is
dt
(5)
is
ntnt)
Y]
dt
(6)
2.
(Pre-development,
=
Sj
1) If
<
i
EM
time at which
last
and
=
s*
EM's
1,
rEM riEM
+
rtXr-Cr
dX.
with r being the
r*^
{^'
jumped from
St
T
=
=
SUp{s„
net asset process
+
ftPt
to
+ nAY,
is
dt
(7)
1
0}.
u<t
The corresponding W's
asset process
w
dXl
3.
(Post-development)
If t
is
ru
>
dt
t*^
then
dXf ^ = [nX^"^ _ cf ^ +
ciXf
^
[nXf" - Cf'
+
Let us now summarize our setup. Equations
to development
"borrows"
[t
<
r*^)
whenever
St
noncontingent bonds. The processes
Equations
St
—
I
(7)
and
and hence the
0.
development,
EM has
pt,7rt
(8) are similar to (5)
n,-
the promised payoffs to
-
+
(/3
(5)
/,g) KYt] dt
«/,g)
and
and
EM.
(6),
we assume that
if
EM
rit
(10)
the evolution of assets prior
has access to three markets:
it
hedging contracts, and also invests in
except that the economy
EM
Finally, equations (9)
to deliver payoffs to
M^uW
+ ^.'^dP,
yt] dt
EM
(9)
are the respective prices in these markets.
rt
and
+ 5f ^dFf
(6) describe
In this regime
hedging contracts that
the time of transition into development
Finally,
=
(1
units of growth-contingent debt, invests in
ft
(8)
entered
and
when
St
is
jumped
(10) state that after
in a regime
to
1
EM's
where
are delivering
transition into
W as specified by the outstanding growth contracts at
(/t-g).
enters short positions in non-contingent bonds, then
itYtm.in\Xf^ ,0] monitoring costs per unit of time
dt.
W pays
This assumption implies that growth con-
tingent contracts and short positions in non-contingent bonds are subject to the same monitoring
costs, since
they both present forms of borrowing.
2.4.2
Global Equilibrium
We
now
are
able to define an equilibrium in the global economy:
An
Definition 1
nt,
TTt,
equilibrium
a collection of (progressively measurable) processes C^^
is
pu ru Xf^\ Xi^, 5f^,
maximize
(1) (for j
— EM)
and
subject to (5), (7)
(1) (for j
Markets
3.
= W)
subject to (6), (8)
_
rEM.r^
SfM^gW ^
1
ft,
and
Cf^,Xf^,ft,
nt
and
Sf^
,
X^
,
ft,
nt
and St^ maximize
(10).
clear, i.e.:
Xf^ + X^ =
Remark
,
(9).
Given the price processes rt,ni,pt,P^ the quantity processes Cj^
2.
Cp^
5j^, Pt^ such that
Given the price processes rt,7Tt,pt,Pt^ the quantity processes
1.
,
Note that
all
for
all t
(11)
+
(yt[{l+l3){l
St{A^-l))-itft] tft<T(^
iforallt>T°
(13)
remaining financial markets clear by construction, since equations
(10) recognize the zero net supply nature of the markets for hedging contracts
(5)-
and growth contingent
contracts.
The
prices
definition of equilibrium
and that
all
Before proceeding,
markets
it
clear.
is
It
requires that
and
Proposition
1
prices.
= ^,
Yt
where
i
=
{W,
properties.
EM].
some properties
of pre-development
(Post-development allocations and prices are given in the appendix).
For appropriate constants c^'^ /°, f^ there
,
properties:
Xp^ = X^ =
its
= ^, wherei-{W^,£;M},
following proposition constructs an equilibrium and outlines
allocations
actions should be optimal given
wih be useful to define
cj
1.
all
Next we construct an equilibrium and characterize
x\
The
standard.
for
all
t.
10
exists
an equilibrium with
the following
2.
W
The consumption process for
is
given by
r.Q,W
ifst^O
cY^
C^'^
if St
(14)
=
I
where
(15)
+ 1 -'iP
and
3.
the
consumption process for
w
Let the constants
and u
EM
is
given by (12) after plugging in for
be defined as
/9
+ A-(l-7)(Mi-7f
p- (1-7) (mg-ty)
and assume
<1. When
that v
st
—
= ,+ (,.-!-),'A'^(/3
XwA
TV
K fp
while in the state
J
st
1
(2£)
-A
-7
0\ -7
-
,o,w
1
(18)
-7
—
(19)
1
the prices rt
and
pt are given by the constants
,0,W\ -7
—
a+
Kp1\
.l.W
Ac
P
V
There are two
there
is
(/?
EM's income post development, any form
all
1
(20)
(21)
no accumulation of noncontingent bonds
resources for
-
{c^y/)-^
results in Proposition 1 that are
against a sudden stop
-7
+ K/^)"^
worth highlighting:
{Xf^ =
contracts to precaution against the arrival of the state
in
(17)
rO,W
+ l)-(i + g)/i'
(3 + 1 -if
. fO \
+ Kf
=
(16)
•
G the prices rt^pt^'n't «re given by the constants
a,vi/\
..
C^
is
costly.
sj
=
X^
1.
—
First, in general equilibrium
0).
EM
only uses contingent
Given the anticipated steep increase
of pre-development
consumption
sacrifice to
Since reserves are noncontingent, they imply that
states of the world - even states of the world
11
EM
is
where no sudden stop takes
hedge
saving
place.
This makes reserves too costly as a hedging instrument compared to claims that only deliver payoffs
exclusively in states of the world where such payoffs are needed
Second, Proposition
1
(i.e.
states that unconstrained trading in hedging contracts completes markets
with respect to the arrival of state
st
—
1.
To
see this, note that (15) together with (12) implies
that
"'
(CX\
—
St-
0,
refers to
and s^+
_
A^(/3
/^
~[
[C}^J
where r~
+ l)-(i +
/3 + l-^/°
an instant before the arrival of
=
1).
The
simple limit case
(/?
—
>
st
—
g)/^ \
'
)
and
1
r"^
(C^\ -'
_
^''>
"l^Cf.^J
to the instant thereafter. (Formally,
equality in (22) follows from (15), while the second equality follows
first
from the goods market clearing condition
A
during sudden stops).
W becomes very large relative to EM, we can sharpen the
When
oo)
(12).
results further:
Lemma
1
As
13
^f oo the solution to the system (S0)-(82) satisfies c
p^=g-{A'^]
i,
f^
is
—
(i
+
q)
'
/ (3 -^ 1
and
andn = \wA {-')
(23)
given by
-1/7
p
A'^-
l+P^TT^
=
f
K
+
p°
1
?
(24)
-1/7
pO
I+ttA''
-]-
p^Tl
(
^
"^
^
and
(25)
Furthermore /°
> /\ and
since both f^
in equation (22) converges to 1
and f^ have
finite limits, the
term
and hence
qEM
A^
(jEM
Lemma
1
implies that
when
W
is
large
compared to EM, the only shocks that can make EM's
consumption drop when entering a sudden stop are world-wide
drop
in available resources to
EM
that
is
cyclical shocks.
caused by capital flow reversals
12
is
Hence, the extra
smoothed
out.
To
see
most
this
clearly,
by jumps
monitoring costs
in the
Then Lemma
1
instructive to focus on the special case
it is
implies that
and there are no
it,
and
(7)
=
cyclical variations in output, so that A'^
EM's consumption experiences no change upon
In order to understand this result, note that since
(5)
where sudden stops are caused purely
=
A''
^^
1,
=
entering a sudden stop.
and
1
1.
Xf^ =
equations
0,
imply that
(/V-/V)
EM
Equation (26) states that
-
decline in capital flows (/°p°
equation (19) implies that
p°
> p^
Since /°
>
it
f^
p'''
buys an amount of hedging contracts that
f^p^)
-^ g^
upon entry
- i and
n >
follows that
sudden
into a
The
0.
n >
fact that
P''
^ 5p -
when consumption
suspect that
insurance must be actuarially
A>
1.
from
fair.
This
is
In this case the value (in state s
to
1
is
of
=
0) of obtaining
^ ^_
/ A'^(/3
V
fi
+^)
,
*
oo,
so that
non-zero trading.^
The
easiest
way
then sudden stop
to see this
is
to allow
a unit of consumption once s switches
given by
^
When
is
EM becomes perfectly smooth,
not true however.
(i
^ —
implies that adjustments of prices
are not enough by themselves to attain consumption smoothing, and there
One might
proportional to the
Additionally, as
stop.
equation (21) implies that
is
—^ oo, and A'^
not actuarially
fair.
=
1,
+ l)-(I + g)/^ \ "^
H + l-'iP
I
the above term becomes
Nevertheless, consumption
is
XA >
A.
Hence sudden stop insurance
is
not affected by a sudden stop (although marginal
utility is).
More importantly
reserves,
EM
for
our purposes, note that
no matter how large
to purchase insurance in
premium and those that do
world where
it
all
not.
needs insurance
in these states of the world
is
is
A
(or
V^
EM
for that matter).
states of the world
By
prefers to use hedging rather than hold
-
The reason
is
any
that reserves "force"
both the states that command a high
risk
EM to isolate the states of the
Since EM only needs insurance
contrast, hedging always allows
(i.e.
when a sudden
and wants to avoid
stop occurs).
sacrificing
pre-development consumption, hedging
always a preferred precautionary measure.
^This
is
in contrast to
adjust in such a
way
many examples
of
endowment economies
in
macroeconomics and asset
that agents are happy to just hold their endowments without trading.
trading in both growth-contingent debt and sudden stop hedging contracts.
13
pricing,
where prices
Here there
is
active
More
generally,
when
/3 is
finite,
even though sudden stop contracts cannot eliminate the pres-
ence of sudden stops, they can substantially mitigate their effects on consumption.
Imperfect (Sudden Stop) Risk Shciring
2.5
In practice there
and
is
a variety of impediments to perfect risk sharing, including agency, behavioral,
liquidity problems.
risk sharing
is
In this section
we capture some
incomplete along dimensions that seem
hold their reserves entirely in
US and Euro
of these
realistic.
by assuming that sudden stop
For the most part, central banks
treasuries rather than investing in hedging instruments.
Thus, later on we calibrate our model using
extreme noncontingent scenario.
this
our analysis in this section we also introduce assets whose payoffs are correlated
perfectly
— with sudden stop
arrivals.
However,
in
— although not
This extension allows us to discuss practical improvements
to current central banks' practices in emerging market economies.
Concretely,
we preserve the environment developed up
sharing instruments no longer exist but instead there
correlated with the occurrence of sudden stops.
now
the country will
factors, the particular
to now, with one exception: Perfect risk
an
asset with a payoff that
is
imperfectly
As one would expect, the optimal
portfolio of
is
include both, shares of this asset and noncontingent reserves.
Among
other
composition of the portfolio depends on the degree of correlation of the asset
with sudden stops, on the
level of reserves,
and on the sources
of
sudden stops and
their general
equilibrium implications.
2.5.1
Danger zones and net
asset processes
In order to introduce the hedging asset
First, there is a Poisson process
we decompose the
with intensity x that puts
danger zones take place at random times rf,
instead of r^.
1)
i
=
arrival of
EM
1,2,3....,
at
—
1)
preserve the earlier environment
and avoids
we
which we generically denote by r^
is
with probability P{s^d
(i.e.
the state s^d becomes
=
=
0)
1
- P{s^d =
1).
To
require throughout that
A
Let us now assume that there
it
steps.
danger of a sudden stop. These
At these times r^, the country enters a sudden stop
with probability P{s^d
sudden stops into two
=
xP{s,o
=
1).
(27)
a financial asset with payoff Ft, that also has the potential to
14
jump
exhibit a
Using J to denote a jump, we have that Ft exhibits jumps
at danger times r^.
according to a Poisson process with intensity:
= xP{J =
\j
The
assumption
critical
in this section
between sudden stops and jumps
1)
that there
is
(28)
P{J =
in the payoff Ft:
some
is
=
1,s^d
Letting $j denote a counting process that increases by
correlation, although imperfect,
times
1 at
6
1)
(0, 1).
r^ and
,
A^ denote a required
rate of return (in excess of the interest rate) for holding Ft, the price of asset Ft follows the process
—
dFf
^
=
ndt
+
- Kdt)
(Jd^t
(29)
i't
Without
(J
—
we can condition on those times r^ where we observe
loss of generality,
and/or a transition into SSfs^o
1)
X
which
=
is
1,
Hence from now on
-'-/^
= X{1-
=
Pisro
the hazard rate for observing either a
is
corollary
(s^D
~
jump
in
=
J
0,
J
let
either a
jump
us define:
(30)
0))
=
or a transition into s^d
1.
An
obvious
that there are only tluree possible outcomes that can take place at those times, namely:
J=
1
j
,
or (s^D
=
0,
J
=
1
j
,
or (s^d
^
J
1,
=
Oj
.
Let pj=i,s=i, Pj=i,s=o and pj=o,s=i
denote the respective (conditional) probabilities of these three events.
We
will
not consider the possibility of insuring the duration of the sudden stop, and simplify
the analysis by assuming that, conditional on entering a sudden stop, the transition out of
independent of any jumps in J and happens with intensity
The presence
dXf^ =
^^ is
Xf^
{nXf'''
-
\;^,Ft
- Cf ^ +
=
0.
[l
+
(A'^
-
l)st
+
ftPt]
Yt} dt
the dollar amount invested in the risky asset Ft. Since the asset Ft
= [nXf +
^The reason
J
A.
becomes:
the equivalent dynamic evolution equation for
dXf"
is
is
of a risky asset with the above properties modifies the analysis in only two respects.
First, the evolution of
where
it
XlitFt
- Cf"
+[p+
{A'
-
X"^
is
l)st
-
that the value function of neither agent
Hence the times when s^o
=
and J
=
(W
ftPt] Yt
or
EM)
-
ftYt (i
+
+ itPtJd^t
is
in zero net supply,
siq) } dt
- ^tFtJd^t
(32)
=
and
experiences any change
are irrelevant for equilibrium allocations, prices
15
(31)
when s^d
and welfare.
As with
that
and
we require that
<^f
>
so that
0,
EM
has to pay upfront for the Ft contracts
purchases.^ Finahy, also note that while equations (31) and (32) hold in both states
it
St
reserves,
—
I,
in equilibrium
it
turns out that
=
Ct
when
—
St
I,
since the asset Ft has
st
=
no hedging
value in that state.
Global equilibrium with imperfect risk sharing
2.5.2
The
equilibrium definition in the presence of asset Ft
modification that
brevity.
now
is
identical to definition 1 with the obvious
the consumers optimally choose ^t instead of nt-
The next proposition shows how
We
do not repeat
it
here for
to construct an equilibrium in the presence of imperfect
hedging contracts
Proposition 2 Define
a^t
=
= ^7-
(33)
K^ (xt) K^ (xt)
given in the appendix, there exists an
-^r-'Cf
Then, for a system of differential equations
,
equilibrium with the following properties.
1.
Pre-development, the consumption process for
EM
K°
{xt)
ifst
=
\ K'ixt)
tfst
=
Cf^ _
^*
and
the respective
f
is
given by
(34)
W
consumption process for
is
l
given by
(35)
^i
I
^^{Xt) z/st
=
l
where
^An
?P(xt)
-
{\-r^-~ift-K\xt))
(36)
VL\xt)
=
Ul+l3)k'-{l+q)ft-K\xt))
(37)
alternative assumption that will leave our results unaffected
the same monitoring costs as growth contingent contracts.
16
is
that short positions in asset Ft are subject to
The equilibrium values of p{xt) and f{xt) are given
2.
P
as
.fO
+ Kf
P
1/
+ Kp
= 9- P
=
if St
i
V
(38)
'^
\
[xt)
(i
n^{xt)
where the quantities f°{xt), f^{xt) in the state
=
5f
+
^
q) if St
0, 1 respectively
I
(39)
are given as the solution to
the equations
+
K '0
V
[
Kf° - UXt
ni (xt)
and
K /?
9-
.{l-f')+i^xt
I
K°
-
+
k/1 -VXt'
^'
(xt)
K
9-
(i+q)
K
(1
(40)
{xt)
-
/^)
+
UXt
(41)
K'
J
The equilibrium value of A*
3.
—
(xt)
given by
is
-7
'n^xt+l)
K=x
and
+ Pj=i,s=o
Pj=i,s=iA
(42)
fi°(Xt)
the equilibrium value of
n^ixt)
solves the equation
S,t
QHxt+iii
Pj=i,s=iA
(43)
Pj=i,s=q
QO{x,.)
\
\
)
K°(xt)
Proposition 2 asserts that in the absence of perfect hedging instruments
construct an equilibrium where
to Proposition
some
1,
reserves in
all
it
is
the quantities depend exclusively on xt (and
still
possible to
St).
In contrast
the absence of a perfect hedging instrument implies that in general
its
portfolio to insure against the possibility that
it
EM
enters a sudden stop
holds
and the
imperfect hedge does not deliver any payoffs.
For our purposes, the most illuminating equation in Proposition 2
tion implies that
EM
The
to see this
easiest
way
=
1 is
^To
equation (43). This equa-
always wants to hold some amount of imperfect hedges as long as pj=i,5=i
is
to note that ^j
>
Otherwise, some amount of ^j
St
is
smaller than
EM's
see this, note that the
,
i.e.
is
in
is
a root of equation (43) only
when pj^ig^i
0.
=
0.
optimal as long as W's consumption drop upon entrance into
as long as
denominator on the
the numerator needs to be negative
=
>
order to
n'(x.+C,)
left
n°{xt)
hand
make the
K'i^fH,.)
K"(xt)
side of (43)
left
17
^
<
hand
side
is
8
negative as long as
i^-bZ
<
)
nffjf)-
Henee
and the right hand side equal. However,
this
Estimation
3
Before we can proceed with a quantitative analysis of the model and
its
imphcations,
we need
to
obtain estimates for some of the parameters of the model. In particular we are interested in the
and departure
arrival
sudden
rates of
the benefits of hedging,
we
stops,
and
sudden stops and jumps
also estimate the joint incidence of
appropriate financial instrument. The next section explains
how we obtain
The
core of our analysis
of such process:
A and
E{9°jel), where
Ot is
the sudden stop process. In this section
is
A.
We
also estimate the
these estimates.
we drop the
simplify notation
magnitude
of the average
=
i
+ fipi,
=
j
or less as described by the
on the selection
The
during
first
superscript from Oj
when
there
NSS and
reversal,
no place
We
for confusion.
we had complete data and behave
(see the
Appendix
for details
Colombia, Indonesia, Malaysia, Mexico, and Thailand.
to find empirical counterparts for the processes describing available resources
is
SS. For this,
we note
that in the model these resources can be
regular income, Yt, and financial flows, {9t
plexities in doing such a decomposition.
relative prices
is
model with no contingent instrument
criterion): Chile,
step
sudden stop
{0,l}.
use post 1983 data for six emerging market economies for which
more
we estimate the main parameters
defined as
el
— l)^^
decomposed
into
In practice, there are several additional com-
These stem from the existence of multiple goods whose
change during the transitions between NSS and SS and vice versa, the presence of
temporary terms
of trade shocks,
The Data Appendix
and endogenous domestic output declines during sudden
describes our methodology to deal with those issues. In a nutshell,
imate Yt with the permanent component of domestic national income, and
of capital flows in terms of imported goods
of trade effects.
Our main
can only happen when
is
some
in
The Sudden Stop Process
3.1
To
Furthermore, in order to assess
their severity.
^j
>
left
hand
since Q,^
is
{9t
—
stops.
we approx-
l)Yt with the
sum
and the transitory component of exports and terms
side variable
is
decreasing in xt (since
increasing in Xt.
18
the ratio of these two, which can be loosely
W's holdings
of
bonds
is
equal to —Xt), whereas
K°
interpreted as external financing over "normal" pre-development income:
where
i
the country index.
is
Since the measurement procedure for On can potentially produce measurement error and the
data are time aggregated as opposed to continuous, we assume that Vit
is
observed with (state
dependent) noise,
with
e^t{s^t)-^N{Q,al{s^t)),
Since
stops
{st
t/ij^
=
1)
will exhibit
economy
as the
6 {0,1}.
from tranquil times
transits
we use a standard regime-switching model a
the average value of
ipj^i
jumps
Sit
ipji
during sudden stops
during tranquil times (that we
(t/)j
.
)
As part
of this procedure,
probability of transition from tranquil times
^
^
Pi{NSS
Pi{SS
55)
=
iV55)
[st
=
Pr(s,,t+A
=
=
Pr(s,,t+A
l|s,,t
-
-
Ois,,t
sudden
ipf^^) and the equivalent value of
we can
also obtain estimates for the
sudden stops
0) to
0) to
Hamilton (1989, 1990) to estimate
la
call
=
(sf
0)
=
=
1)
1
-
=
1
[st
=
1):''
e-^'^
(44)
-
(45)
e-^''^
For the calibration exercises we convert annual transition probabilities into annual frequencies by
setting:
Given the limited number
A
= - log [1 - P(Ar55
A
= - log [1 - P(55
of
-»
SS observations we have
^ 55)] /A
(46)
N55)] /A
(47)
for
each country and the highly nonlinear
nature of the hidden states model we are estimating, we use a Bayesian approach (with
based on a Gibbs Sampler
Appendix.
Pi{SS —>
(see
To obtain more
NSS)
are the
Kim and
Nelson
precise estimates,
same
across
all
[1999]).
in
describe the precise procedure in the
we assume that the parameters pi{NSS
-^ SS),
countries, whereas i'i'^^, ipf^ are allowed to differ. Tables
'These equations approximate the continuous time model by
can be at most one transition
We
flat priors)
its
discrete analog,
by imphcitly assuming that there
a time interval of A. This approximation becomes exact as
19
A—
>
0.
Average
Std
Dev
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
1:
Posterior distribution of A, A.
Average
Std Dev
5%
25%
50%
75%
95%
Chile
-0.091
0.030
-0.132
-0.109
-0.094
-0.076
-0.041
Colombia
-0.051
0.009
-0.064
-0.057
-0.052
-0.047
-0.036
Mexico
-0.084
0.015
-0.106
-0.093
-0.085
-0.075
-0.059
Indonesia
-0.066
0.009
-0.081
-0.072
-0.067
-0.061
-0.049
Malaysia
-0.164
0.023
-0.199
-0.178
-0.164
-0.149
-0.126
Thailand
-0.146
0.023
-0.179
-0.161
-0.147
-0.133
-0.107
Table
1
2:
Posterior distribution of capital flow reversal
during sudden stops
(77)
and 2 present these estimates. Based on the posterior medians, we conclude that sudden stops are
large, leading to
last for
about
dechnes in available resources sometimes beyond 10 percent of
6.5 years
previous sudden stop)
in interest rates
.
and they occur about every 12 years (about
Our measure
and turmoil but
of the capital flow reversal.
of
sudden stop
also the effects of
is
meant
GDP,
5.5 years after exiting the
to capture not only the initial spike
sudden stops that remain well
Our estimates suggest that
their effects
after the onset
these effects can be quite large: Countries
take a long time in resuming significant borrowing after experiencing severe capital flow reversals.
Finally, Figure 2
It illustrates
summarizes the output
of the corresponding
Gibbs sampler
It is
apparent from this figure that our procedure does capture
the transitions into sudden stops that occur around major financial
The VIX and the
To evaluate
our economies.
the path of the share of economies with posterior probabilities of being in sudden stop
above 0.5 during a given period.
3.2
for
joint incidence of
the usefulness of hedging,
when a country
transits into a
sudden
we need
sudden stops
crises.
zind
VIX jumps
to find an instrument that exhibits
stop. If such
20
an asset
exists,
then
it is
jumps
at times
straightforward to use
Chile
0.1
h-
0.05
Y
^v'^«^
5-
l\
-0.05
-0.1
Colombia
W^
'
1
'
\
1990
1995
Mexico
1985
1990
1995
2000
Indonesia
0.05
V
2
CO
0.5 w.
\^
^
)
-0.05
1985
1990
1995
2000
1990
1985
Malaysia
0.2
0.1
:'"
\
-
to
1985
Figtixe
a combination of
detail
how
Our
to
1990
1:
tp'^
sliort
dollar
2000
1995
1985
and posterior probability
dated put and
when
the
jump
call
is
of being in a
call
and
otherwise.
(We explain
in
options shortly.)
Rather, we seek to show that there exist assets with such properties and
Moreover, by finding an asset that delivers payoffs
stops,
2000
for 6 different countries.
in the underlying asset occurs
obtain an order of magnitude of the benefits of expanding
sudden
SS
1995
not to conduct a thorough search for the optimal risky instrument for specific
countries' portfohos.
of
1990
options on the asset in order to synthetically approximate
approximate such a claim with
goal here
^;
Va
^u.
.
0,1
1
2000
^.
0.5
a payoff of
1995
Thailand
we can use
risk
EM's
strategies to include such an asset.
in states of the
premia on that asset to
world with increased incidence
infer the price of obtaining payoffs in
such
states.
With
this
purpose
in
mind, we chose the
formed from quoted put and
call
CBOE
options on the
Volatility
S&P
21
Index (VIX). This
is
a traded index
500, available since the mid-1980s.
The VIX
Sudden Stops
0.9
0.8
LTCIVI/Russia-*
I Crises
in
Asia
Asian Crisis
S«
0.4
Tequila Crisis
4-
0.1
1983
1989
1987
1985
Figure
1991
2:
1993
Year
jumps
in the
calls
on the
VIX, making
it
S&P
1997
1999
2001
2003
Fraction of Economies in SS
captures traders' anticipations of volatility in the
by various puts and
1995
500.^"
US
stock market over the next
month
as implied
As we show below, sudden stops often coincide with
^^
a good candidate for our purposes.
'"For more details on the construction of the VIX, sec http://www.cboe.com/micro/vix/vixwhite.pdf
^'See also the recent work
CDS'
prices
of,
Pan and Singleton
(2007) documenting the tight correlation between
and the VIX.
22
EM's
sovereign
The VIX Process
3.2.1
Let us postulate a continuous time process of the VIX, described by an Ornstein-Uhlenbeck
(i.e.
a
continuous time analog of an AR(1) process) with jumps
dlog{VIX)
a
(log)
is
= -a
(\og{VIX) - log(y/X))
a parameter that controls the speed of
VIX, uvix
to a standard
mean
second and third terms
and then
to
is
Xj^^dt]
(48)
.
the average level of
remove
\og{VIX) and
and standard deviation
acj).
is
Finally,
(f>
assumed to be drawn
Given these assumptions, the
discrete time counterpart. Since (48)
its
discrete time observations follow an
its
this predictable
isolating the residuals.
in
intensity Xj respectively.
martingale differences.
in (48) are
an Ornstein Uhlenbeck process,
is
fi^
jump
we approximate the above process by
In estimation,
-
[(f>d^t
log(y/X)
reversion,
Brownian motion and a Poisson process with
from a normal distribution with mean
step
+ avixdZt +
the instantaneous volatility of the process, dZt and d^t capture the increments
is
captures the (potentially random) magnitude of the
first
dt
AR(1)
is
process. Hence, the
component by estimating an AR(1) process
for
log(V^/X)
For small time intervals, these residuals are characterized (ap-
proximately) by a mixture of normals:^^
£t
where pvix
=
(1
=1-
-py/x)iV(My/A'A>^WA'A) +pvixN{iiyjxA
e"-^-''^,
fiyj^
=
+ i^^,a^vix^ +
(49)
^l)
-Xjfi^A.
In principle, estimation of this process can proceed from this point on along standard estimation
techniques
(see, e.g.,
Caballero and Panageas [2004]).
Such estimation
delivers estimates of the
underlying parameters of the process for \og{VIX). Given these parameters, we can also estimate
the posterior probabilities that the
VIX
also estimate the joint incidence of
jumps
has exhibited a jump in a given month, and we can then
in the
VIX and
the arrival of sudden stops.
Unfortunately, these estimates are not directly useful in practice, because in real markets one
can only find contracts whose payoffs are contingent on the
the month). However, there are no contracts on whether the
'^The source of the approximation to the continuous time
possibility of
more than one jump
and infrequent jumps and
in
hrait
is
level of the
VIX
A.
23
(say, at
has exhibited a
the end of
jump
over the
that the discrete approximation excludes the
the interval A, which seems reasonable
relatively small time intervals
VIX
if
we want
to focus
on
relatively large
course of a month, because such jumps are unobservable (and impossible to define) with discrete
time trades.
To measure the
strategy,
correlation between the arrival of sudden stops
we adopt an
alternative procedure. ^'^ In particular,
We
monthly data.
of (49) using
we
let
first
>
=
e|J
0)
<
and consider a contract that
interval of
lif
log(F/Xt+A)>log(V^/A7^J
is
above
small A, the payoff
a payoff that
it is
more
is
1 if
1 if
+e
the unexpected change in the \og{VIX) over an
also important to note that in the limit
e. It is
small, this contract pays off
and only
Qt+A approaches
if
there
is
a
jump
in the
where
VIX
A
and
becomes
from
arbitrarily
otherwise. Hence, for
the contract described in section 2.5.1.^^ Given that (50)
measurable with respect to the market participants' information
useful
be defined as
otherwise
this contract delivers a payoff of
A
A
= {1- aA) \og{VIXt) - aA\og[VIX)
\0
is,
such that
delivers the following payoffs:
Q^^^^i
That
e,
0.01.
the expected level of ]og{VIX) after a time interval of
log(y/Xi%A)
feasible
perform a Bayesian estimation
then use the obtained estimates and set a cutoff value
Pr{et
In a next step
we
and the payoffs of a
Qt+A =
this point to think directly of the event
set at
time
t
is
+ A,
as a "jump".
1
Furthermore, figure 3 illustrates how to approximate the payoff Qt+A by combining a standard
call
option with strike price ]og{VIX^_^_^)
\og{VIX^_^_^)
+ e + 6. The
+
e
f
e
+
5
\ogiVIXt+A) < log(y/Xf+^)
+
e
6if\og{VIXt+A)>^og{VIX^^^) +
(51)
if
As was shown by Breeden and Litzenberger
Given
'^If anything, this
this
option with strike price
<
[
a-e.
call
resulting payoff is^^
Qt+A =
Qt+A
together with a short
approximation
result,
(1978),
when
^
5
and the existence
we obtain
that /im^^o
^"^
=
of a rich set of options of various
procedure biases the results toward finding a lower "correlation", because
it
introduces the
potential of "over-identifying" jumps, and hence understating the benefits of hedging.
^''When
A =
course of that
^^When
1/12
month
(i.e.
is
monthly data) the probability that Qt+A
1%. This
log{VIXt+A)
e
is
so by the construction of
[\og{VIX^+^)
will
be
24
when no jump has occured
over the
e.
+ e,logiVIXt+A) + e + 5]
(log(V/X.%^)+e).
1
then
Qt+A
=
\og{VIXt+A)
-
Payoffs
Payoff of Long Call
Option
Sum of the
Payoffs
I
\ogiVIX „^)
Payoff of Stio rt Call
Option
Figure
3:
Payoffs of a long call option with strike price
B=
with strike price
strike prices,
\og{VIX^_^^)
we assume
+e+
S.
The
A=
\og{VIXf_^^)
solid line presents the
directly that the payoff given
by
is
(50)
+e
and a short
sum
of the
call
option
two payoffs.
a feasible payoff given existing
securities.
Importantly
Qt+A
for
for small 6
is
our purposes, equation (51) implies that the arbitrage
free price of the payoff
given (approximately) as the price difference of two standard
strike prices \og{VIX^_^_^)
+
e
and \og{VIX^_^^^)
+e+
5 respectively, divided by
options with
call
5.
Denoting that
price as P'^t-^^^ a feasible strategy to replicate a payoff such as (29) in section 2.5.1
follows:
Every time interval
payoff such as (50).
there
is
If
A
(say every one or two months) an
the (unexpected) change in the
is
less
than
can pay P<5«+a and obtain a
e
,
then Qt+A
a discontinuity that produces an (unexpected) change larger than
the risk compensation A* in equation (29) as'^ A*
details
VIX
EM
on how we used the prices
'^In the limit
where 5
—
>
and
A—
»
of existing
this
VIX
~
call
e,
—
0. If
then Qt+A
P^'+^/A. In the appendix we
=
however,
1-
Hence,
give further
options to obtain an estimate for the average
approximation becomes exact.
25
given as
is
Average
Std Dev
5%
25%
50%
75%
95%
Pr{SS\J)
0.236
0.096
0.095
0.165
0.226
0.296
0.413
Pr{J)
0.159
0.032
0.109
0.137
0.158
0.180
0.215
Table
3:
Posterior distribution of
Pr {SS\J) and
Pr( J) based on quarterly data,
value of P'^t+A and hence A*. Based on that analysis our baseline value for A*
3.2.2
It is
is
0.94.
Joint jumps, risk premia and implied preference shocks
now
straightforward to measure the joint incidence of sudden stops and positive payoffs to a
claim such as the one defined in equation (50). Figure 4 gives a
incidence. It shows the residuals of an
those instances
when
AR(1) model
log(y/X). The shaded areas represents
these residuals are above e and hence the times
constructed in equation (50) would pay
off 1 dollar.
onset of the gulf war) in 1997 (around the Asian
after 9/11/2001,
for
visual depiction of this joint
first
and around the beginning
The shaded
crisis), in
when a claim
like
the one
areas occur in the early 90's (at the
1998 (around the Russian/LTCM
of the U.S. corporate scandals
crisis),
and the Argentinean
default.
These large movements
in the
VIX
at times of crisis, together with the increased incidence of
sudden stops during such times, suggest that there
can be used
as
for
hedging purposes.
"jump" times and we
will
We
is
some
will refer to the
times
denote such events with J
Conditioning on the times where J
=
1, it is
joint incidence of the
=
1
when
two events, which
the payoff of equation (50)
is 1
in analogy to section 2.5.1.
straightforward to use a Bayesian procedure similar
to section 3.1 in order to estimate the parameters
P(SS\J -
1)
=
Pl^h£^
and PiJ)
=
1
-
e"^^^
PJ = 1,5 = +PJ=1,S=1
where pj^i,;^i,pj^i^s^Q
etc.
were defined
in section 2.5.1.
The
exact procedure
appendix. The resulting posterior distributions are given in table
With
the estimates of P{J) from table 3
,
namely Xj
the estimates for A (from section 3.1) and also Aj, and
26
of the arrival intensity
— - log [1 -
P{SS\J =
described in the
3.
we can obtain estimates
a manner analogous to equations (46) and (47)
is
1)
Xj
in
P{J)] /A. In turn, given
from table
3,
we can
identify
VIX Residuals and Jumps
Jan92
Figure
4:
VIX
residuals.
Jan94
The grey
JanOO
Jan98
Jan96
Jan02
areas correspond to instances where the
VIX
residual
is
above
the jump-cutofF.
all
the parameters of section 2.5.1 by solving the following linear system of equations
Aj
X(PJ = 1,5 = 1 +PJ = l,S=0)
(52)
A
X(PJ=l,s=l +PJ=0,s=l)
(53)
PJ = 1,S=1
Fv{SS\J)
(54)
PJ=l,s=0 +PJ=l,s=l
I
for the four
for A, Xj,
unknowns pj^j
and P{SS\J
=
1),
of equations (52)-(55) are
x
PJ=l,s=\
+ PJ=l,s=0 + Ps=l.J=0
^^j, Ps=i,j=o, Pj=i,s=o
and
x-
Based on the median posterior values
the values of pj=i,5=i, Ps=i,J=o, Pj=i,s=o and
=
0-74, pj^i^s=i
=
0.23,
27
(55)
ps=i,j=o
=
0.06,
x that
ps^ij=o
=
solve the system
0.71.
Calibration and Results
4
In this section we input the estimates obtained in the previous section into a cahbration exercise,
and then use the resulting quantitative model to evaluate the gains from
different precautionary
Since the standard practice of central banks in emerging markets
strategies.
is
to hold reserves
almost exclusively in the form of noncontingent assets, we calibrate the model of section 2.5.1
assuming that a country does not use any hedging instruments. This standard practice seems to
be more the
result of inertia
how much would
to ask
and domestic pohtical economy
welfare increase
if
and hence
factors,
is
it
reasonable
countries adopted optimal hedging strategies.
All the
counterfactual exercises are done in general equilibrium, and hence take into account the endogenous
reaction of prices.
Parcuneters
4.1
We
calibrated the model following two different approaches. In the
CRRA
=
preferences with no global preference shock (A
this kind, this
approach
is
able to
match quantity data
in actual markets, especially during times of turmoil.
1
first
for all
approach we use standard
As with
t).
all
well but not the large risk
Within
calibrations of
premia observed
approach, we allow for two
this first
subcases -one where cyclical components in world output are small (case la) and hence sudden
stops are caused almost exclusively by changes in monitoring costs, and one where sudden stops
coincide with significant global downturns (case lb). In the second approach (case 2)
jumps
in the preference shock
price of risk.
helps us
fit
We
As
in order to calibrate interest rates
also allow in this case for a
EM
common
across
all
cases.
for the three cases
The parameter g
difference
between these cases
is
we
income
of a developed economy.
The parameter
in
The
consider.
set to 0.03 to
is
that
W's
We
first
twelve parameters
approximately match the speed of
set
k
=
3,
an emerging market economy g\og{K)
of (logarithmic) growth of
we
in the global
growth rates during sudden stops, which
convergence estimated by Barro and Sala-i-Martin (2003).
Since
for
varies across them.
Table 4 reports the parameters
are
in
The main
the volatility of interest rates.
willingness to insure
drop
and fluctuations
we allow
controls the relative size of
so that the expected rate
—
EM
3.2% higher than that
and
W
in the
model.
are interested in estimating the benefits of hedging for a large part of the world, such
28
Case la
Case lb
Case 2
g
0.03
0.03
0.03
K
3
3
3
13
7
7
7
A
0.14
0.14
X
0.74
0.74
0.74
Ps=i,j=i
0.23
0.23
0.23
Ps=o,J=i
0.71
0.71
0.71
Ps=i,j=o
0.06
0.06
0.06
i
0.01
0.01
0.01
Q
0.35
0.35
0.35
7
7
7
7
P
0.01
0.01
0.01
a
0.027
0.02
0.02
Mo
0.018
0.018
0.03
Ml
0.018
0.018
Mg
0.018
0.018
0.018
1
1.8
0.96
0.96
A
1
A'i
.
0.99
Table
4:
Parameters used
in the calibration exercises.
29
.
0.14
GDP
we added the (PPP adjusted)
as Latin America,
GDP
compared that to the
American
of Latin
US,
of
EU
countries and Japan
The parameters
countries.-'^
and then
p^^u^Q,
pj=i,s=i,
Pj=i,s=o and X are those that solve the system of equations (52)-(55) as determined in the previous
section.
9%
GDP
of
flows
A was estimated and reported
in tranquil times.
become
The
in table
value of q
We
1.
set high
is
We
sudden stops.
practically zero during
chose
so as to
i
chose 7 to generate reserves accumulation in the 20
— 30%
growth rates of
(see
Y
fiQ
=
fii
—
fXQ
=
but assume some variation
in the
riskless interest rates. ^^ In case 2
we
2
Y
A
calibrate
premia (assuming that originally only
in
set the
developed economies
match the observed
volatility in real
match the observed VIX's
to approximately
in this market).
Finally, in all cases
case la produces a standard deviation of the cyclical component corresponding to what
The drop
output. ^^
in cases
risk
we
set
The drop
At to capture the standard deviation of the cyclical components in output in the US.
US
we
For cases la and lb
growth rates
in order to
W participates
from a Beveridge-Nelson decomposition of
we
preferences. Finally,
preserve (approximately) the average growth rate
growth rate of
we
three cases capital
range.
0.018, so as to capture
Campbell and Cochrane 1999). In case
CRRA
uses
parameters varies across the different cases.
to
all
p to a low number in order to reduce
set
when one
set of
capital flows of about
enough, so that in
the high risk-free rates that are typically obtained,
The next
match
is
in
obtained
lb and 2 produces a
model-implied standard deviation of cyclical component that corresponds to the one estimated from
a standard
H-P
volatility of
post-war
We
US
filter
decomposition
for the US.^''
output produced by the model
We
set
ay
so that in
all
three cases the annual
about 0.028, corresponding approximately to annual
is
data.^^
also estimated the
model by matching the
relative size of
US,
and East Asian economies (Indonesia, Korea, Singapore, Thailand and
'^The presence
EU
and Japan to Latin American countries
Malaysia.) obtaining similar results.
of preference shocks necessarily increases the volatility of the interest rate.
By
allowing for
some
small time variation in output growth that volatility can be reduced within reasonable levels. Since in case 2 we are
interested in matching asset prices,
we allowed such time
variation in the stochastic trend. However,
we
also simulated
the model allowing for constant growth rates and preference shocks. This did not affect the results substantively.
^'See Nelson (2006),
p.
12.
^°See Nelson (2006),
p.
12.
^'Note that the total variance of output
is
given as
E [Vart
(Yt+i
—
is
the variance of the predictable component Var[Et (Vt + i
total volatility stays the
somewhat the
same compared
total volatility of
to case lb.
Yt)]
—
Yt)]
+ Var
smaller,
we assign a
In case 2, the variation in
output compared to case
16,
30
but this
effect
[Et (V/.+i
—
Yi)]
.
Since in case la)
larger value to
growth rates
fJ,Q,fJ.i
ay, so that
tends to reduce
has a negligible effect on total volatility
Model
Data
Case la
Case lb
Case 2
Reserves
0.31
0.32
0.2
0.17
Mean Resource Drop
0.091
0.091
0.091
0.1
1.01
1.04
1.33
1.39
0.14
0.14
0.01
0.020
0.008
0.04
0.04
[0-0.05]
Mean
Jump
Risk Intensity ratio (A*/Aj)
Mean
Interest
Rate
Volatility of the interest rate
Table
Table
5:
Comparison between model and data, assuming no hedging.
5 reports several statistics
accumulated and the corresponding values
column are based on the
countries that
The jump
as
^
we consider
as reported in Table 9 in the appendix.
risk intensity ratio
is
GDP)
(In the
Mean
W participates
resource drop refers to the
in this market).
The data on the average
real interest
volatility of the real interest rate is
from
data one can only obtain estimates about the volatility of the realized
anticipated real interest rate,
as an interval.)
data
obtained by our estimates of \j and A* in the previous section
on bonds. These estimates provide an upper bound on the
real return
in the
during the sudden stop as reported in table
Campbell and Cochrane (1999), while the
(Jermann 1998).
reserves are
following: Reserves refers to the historical average levels of reserves for the
(assuming that only
rate are from
The numbers
our sample of countries. ^^
in
average reversal in capital flows (as a fraction of
2.
when only noncontingent
generated by the model
and hence we report the
volatility of the
unobserved
volatility of the anticipated real interest rate
The models produce reasonable numbers
accumulation and the
for reserve
the sudden stops. However, as explained above, only case 2
is
size of
designed to match asset prices by
introducing an additional source of shocks to the marginal utility of consumption. In particular,
the
jump
premium"
in
As
is
needed to generate jumps
implicit in the
they generate a positive
(about 20
beisis
VIX
during
drift in the
in
crises.
marginal
marginal
Jumps
utility consistent
in ^4^ also help
utility of
with the
rise in
the "risk
reduce the interest rate, since
consumption.
points reduction in annual volatility).
"*To obtain the stationary quantities we simulated 20000
development.
31
artificial
yearly data, conditioning on no transition to
Hedging
4.2
Case la
Case lb
Portfolio (imperfect correlation)
0.87
0.86
0.78
Portfolio (perfect correlation)
1.00
1.00
1.00
Proportional reduction in stationary reserves (imperfect correlation)
0.16
0.16
0.23
Proportional reduction in stationary reserves (perfect correlation)
0.42
0.47
0.62
Table
Case
2
Portfolio allocated to hedging and proportional reduction in stationary non-contingent
6:
reserves (reserves prior to hedging divided by reserves after hedging).
In this section
we study
focuses on the former.
spent in hedging, that
The
the effect of hedging on portfolio decisions and welfare.
first
available (imperfect correlation)
occurrence of a sudden stop
is
and
is
+
for
OfYt
—
for the case
Cf),
The
where only options on VIX are
an upper bound where an asset perfectly correlated with the
available (perfect correlation).
invested in hedging instruments.
two rows
two rows report the share of the flow of precautionary savings
X*^^l{rtXt
is
Table 6
We refer to
—
portfolios are evaluated at xt
clear in all cases: the bulk of the precautionary resources,
this ratio as the "portfolio"
The message
0.^'^
of these
even when the correlation
is
imperfect, should be allocated to hedging instruments rather than reserves accumulation.
We
already
knew from Proposition
sudden stop insurance that
EM
is
is
1
that this "portfolio" notion
is
always
1
when
there exists
both perfectly correlated with the sudden stop and also insures
against the duration of th& sudden stop ("complete risk sharing").
Once the
price of risk
correctly assessed, a country always prefers a hedging instrument to non-contingent reserves,
irrespective of the
magnitude
of preference shocks
that this mechanism remains intact even
stops (second row). This
hedging
The
is
limited to
mechanism
VIX
is
when
A
there
and the associated
is
weakened but
risk premia.
Table 6 shows
no insurance against the duration of sudden
still
remains dominant
(first
row) even when
instruments.
third and fourth rows of 6 describe a stock rather than a flow concept.
^''Evaluating the portfolio at xt
=
is
conservative.
When we
solve the
model
the portfolio at levels of xt that are higher than zero, the denominator rtXt
numerator X'^,, making the portfolio
rise
toward
1.
32
+
They
for all three cases
BtYt
—
are based
on
and we compute
Ct decreases faster than the
a simulated run of the model with and without hedging. ^^
decline in average non-contingent reserves that
different cases that
reserves
when
What
is
it
we
consider.
The
is
The two rows
report the proportional
achieved with various forms of hedging in the
table shows that
EM
holds between 16 and 62 percent less
can hedge.
We
the impact of these changes on welfare?
by computing the equivalent variations
particular form of insurance. For this,
from hedging strategies
income required to compensate
in
let
assess the gains
for the
absence of a
us define the no-insurance case as one in which
EM only
trades in growth contingent contracts without any access to either noncontingent bonds or any
form of hedging. In such a world EM's
utility
is:
(A.
fsPsy-'Y,
1-7
Jq
Jo
where
oo
1-7
JtO
Correspondingly,
country
(in
let
V'^*(0, Yq) and
V"*"''(0,
Yq) denote the
EM's
utility in
a world in which the
allowed to use reserves only to insure against sudden stops and hedging instrument ins
is
addition to reserves), respectively, both evaluated at zero initial reserves.
variation
is
defined as the additional pre-development income that needs to be given to an
with no access to insurance or reserves, in order to achieve
Since utility
Then, the income
is
homogeneous
of degree
1
—7
=
(l
+
fc^^*)^-^Eo /
Jo
yi"-
=
(1
+
fc^"^)'^^
where the
fc's
Yo)
Eo
and F*"*(0, Yq)
respectively.
with respect to the country's income, we have that:
y-=^(0,yo)
(0,
y^'^*(0, Yo)
EM
A / '^'^^
^
1-7
r
Jo
A^
i^^
+
fsPs)^~'"^s
1-7
e'P'ds
\
.ps
^^
+
V^^"
^
ydev
represent the proportional income variation. Since our goal in this section
is
to
evaluate the improvement over the standard practice of accumulating noncontingent reserves, we
^""See footnote
22 for a description of the simulation design. Importantly, when we compute average reserves, we
also take into account states of the world
stop (as in Proposition
1),
it
when
Si
=
1.
When EM
has to hold non-contingent bonds
has no way to insure the duration of the sudden
in this
regime.
Because of
this,
even when there
is
perfect correlation between the arrival of the sudden stop and the payoffs of the contingent instrument, the stationary
reserves will always be non-zero, unlike in Proposition
1.
33
report the welfare gains as the proportional gain over
fc'"^*.^^
Table 7 shows that the extra benefits
Case la
Case lb
Case 2
Imperfect Correlation
20
15
8
Perfect Correlation
91
60
44
Complete Markets
114
81
60
Table
7:
Percent increase in the relative effectiveness of hedging
(fc^"''
—
for various
/c^*^^) /fc'"**,
cases.
from enriching the precautionary strategy with hedging instruments ranges between
when only VIX
of
k^"^^
to
114%
results
we move
as
is
used, from 44 to
91%
to the complete markets case. Importantly,
=
1
utilitj'
others,
5
all
"hurts"
W more than
making the
in the price of risk.
The
W's income
for risk
falls
sharing
is
and
in case 2
W's marginal
larger in case la
than
in the
benefits of hedging larger.
Remarks
Emerging market economies hold
by developed economies
latter, the
and from 60
that in cases lb and 2 the transition into state
(In case lb
This means that the potential
increases).
Fined
in case la.
is
and 20%
these are general equilibrium
and therefore do take into consideration the endogenous changes
reason for the difference between the three cases
St
in the perfect correlation scenario,
8
levels of international reserves that greatly
(relative to their size).
This would seem paradoxical given that, unlike the
former face significant financial constraints with
The paradox disappears once
exceed the levels held
much
of their
these greater financial constraints also
of volatility, which countries seek to smooth. This
is
growth ahead of them.
become an important source
the context we have modelled, analyzed, and
assessed quantitatively.
^"The absolute value
of k^"" itself
instance in cases la), lb) and 2)
magnitude
and any
k''"^ is
rather small in line with
0.26, 0.32
all
welfare calculations in the Lucas tradition.
and 0.23 percent of
GDP
as Lucas's estimates of the costs of business cycles (0.1 percent of
links
welfare gains,
problems.
is
respectively. This
GDP).
is
in
For
the same order of
Since the absence of heterogeneity,
between temporary and permanent consumption components may be understating the magnitude of
we use a concept
An added
advantage
of proportional increases in welfare benefits that should
is
that
it
makes our
results less sensitive to
34
be more immune to these
assumptions about
risk aversion etc.
The main
contributions of this paper are twofold:
equilibrium model of sudden stops. Second,
to insure emerging markets against
sudden
we use
First,
we develop a
quantitative global-
this structure to discuss practical
stops, ranging
mechanisms
from conventional non-contingent reserves
accumulation to more sophisticated contingent strategies. Depending on the source of sudden stops,
their correlation with world events,
and the quality of the hedging instrument
available, the gains
from these strategies can be substantial.
On
the
first
contribution, the model
is
matching the extent of capital flow
successful in
the behavior of risk premia and the size of reserves accumulation.
On
the second one,
reversals,
we estimate
that the potential gains from adopting hedging strategies are, in a worst case scenario, about 10
percent more
than conventional reserves management, but the gains can also reach up to
efficient
115 percent as the quality of hedging instruments improves and the correlation between sudden stop
arrivals
and marginal
investors' risk attenuates. All these strategies imply a substantial reduction
accumulation and
in the rate of reserves
productive investments.
in
many
For example, in our case
practical instances
up reserves
free
for
(which calibrates quantities and prices), we
2
compute that our representative country would need
may
to hold between
23-62%
less reserves
than
it
does without hedging (depending on the quality of the hedging instrument).
There are several natural extensions
frictionless trading in (imperfect)
to our work.
Our model assumes
hedging instruments to explain observed risk premia. This mech-
anism allows us to reproduce variations
in the
marginal
utility of
consumption of the representative
agent in a spirit similar to Campbell and Cochrane (1999). As a
ful since it
preference shocks and
first
pass, this
approach
safeguards that welfare computations are consistent with asset prices.
An
is
use-
interesting
extension of our model would be to assume some form of segmentation between bond markets
and hedging instruments and derive the observed
in these markets.
It is
risk
premia from the heterogenous participants
reasonable to conjecture that in such a world, increasing participation in
the markets for (imperfect) hedging instruments by agents in the developed world would
outcomes resemble the cases
la)
and lb)
in
make
the
our model.
Furthermore, we have assumed a representative agent within each of the regions of the world.
It is
quite apparent that in reality there are plenty of financial frictions within each of these regions
that clearly interact with the aggregate risk
By
the
same
token,
it
management and sharing problem we have
discussed.
seems important to endogenize the production side of the economy, as
35
in
practice real investment and
employment are severely
36
affected
by sudden stops.
Appendix
6
A
Proofs
Proof of Proposition
Shreve 1998), Ciiapter
{
^w
W's endowment post development
= pEt
The absence of arbitrage
is
f
J-
so that
results in (Karatzas
and
dynamically complete and hence there exists a
for all
for
any
> r2
(56)
i
>
r*-"
is
then
(57)
Ht
implies that the value of the payouts that
W's intertemporal budget
t
H-.G
^nds.
I
t
_ dW
/t-gP^,
is
(jEM
p,'^
«
on development. By the
first
Ht such that
p-p{t~r1)
value of a claim to
by focusing
start
market post development
4, tlie
stochastic discount factor
The
We
1.
W
is
collecting
constraint post-development
from
EM post development
is
dt
'
+
'
(58)
+
Using (56) inside (58) and simplifying, leads to
i+^fro)prc+x!:^a = c:
The market
IT
J^-^o
+
w
-<w
Ht Cf
—
/
/
J^a H^G
—rrrdt
w
C
C%
E^a
e-K*-?)
I
4
1-7
dt
(59)
clearing condition (12) together with (56) implies that
^t
=
e-p(t-T?)
H-G
^t
(
for all
t
> T°
(60)
+
Using (60) inside (59) and (57) allows us to compute the expectation of the integral on the right hand side
of (59) exphcitly.
at time
r^
Using the
fact that Yt
is
lognormally distributed, allows us to compute
W's consumption
as
Cw
i
{l3
w
+ Kf,c)+yx'^G
(61)
y:>
For future reference,
of
W
at time
t%
as
it
will also
be convenient to combine (61) with (56) and
(1) to
obtain the value function
A'^roy^ {X^G,Y^a) where
1
—7
'
;y
37
'
(62)
The
rest of the proof
We
equilibrium.
save space.
A
is
V^{Xt,
the processes for
straightforward and
is
C^ ^Cf^ ,it, ft,nt
and hence W's value function can be expressed
'
when
Ft)
if
1
We
are optimal.
constitute an
we omit
to
it
by showing
start
Since the preference shocks enter multiplicatively in the objective, the value function
A
in
markets clear
shall verify this directly. Verifying that
W.
optimality for
'
devoted to showing that the prices and quantities of proposition
remains to check
It
homogenous
is
=
Sj
1.
Specifically, the
A
as
''V°{Xt,Yt)
when
Hamilton Jacobi Bellman (HJB) equation
=
St
and as
W when
for
is
sj
=
given by
=
(i2LLl + v°{nXr-Cr +
ma^
+^yi>o + \v?WYy- -{p +
where V'^
when
=
Sj
=
is
+ 9)V°
X
(63)
the value function in development.
+ Vk {nXr - Cr +
^7 J
+V^-Yfio
(64)
+
and
\vi-yo\y''
-
(p
+
A
+
ff)
[PA' -h{pt+-i
first
is
+ q)-
W
order conditions for
=
V?
,-
/^ in
n.) Y,)
=
Vl, fori
Y,;
h) +
=
the states j
0, 1
0,1
~XV'
(X^, Y,)
(65)
can be written as
K,(^+«/r''
+ n) = 9-——^-=^
.
(j
The
equation in (66) follows by re-arranging the
first
,
C^^
5:^ -
given by (16).
+ gV^ [X^
(64)
(63) imply the following first order conditions for
P^
where v
for
V^
(Cn"^ =
Similarly the
Hamilton Jacobi Bellman equation
Similarly, the
given by
1 is
inp
Equations
{P-ft{pt+i)+nt7T,)Y,)+gV^{xr,YtJt)+XAV^Xr,Yun,)]
(^
+ 3i)
for j
=
0,
first
(66)
order condition for
/t
and the second equality follows by combining (65) with (62) and noting that in the postulated allocation
Xf^'^
the
= Xf^ =
same
for all
t.
Note that equation
(66)
is
identical to the equations (19)
steps as in equations (63), (64), (65) and (66) for
.
Pt=9
To determine the optimal
choice of Ut for
EM
i-^fl-Z/V^
) .^J-,
W
in state 0,
denote the time of entry and t the time of exit from state
respect to
rir
=
0,1
shall first
St
Repeating
(21).
gives
for J
we
and
=
1,
compute V^
differentiating
(67)
[X^ ,Yt;nt)
both sides
.
Letting r
of (64)
with
and using the envelope theorem gives
= -V^Yt+Vl^ (nAT - cr +
(/3A'
-ft{Pi
+ + q)i
38
nr)
Yt)+VlyY^i,+^Vlyy<rlY^--[p + A + g)
V^^^
The Feynman-Kac Theorem
above
to the
differential
equation
is
and Shreve 1991) and (Oksendal 1998)) imphes that a solution
given by
= -E n^-^t'-^Vi-ytdil = -E
Vn.
The second
f/^'e^"'*-^' {c'''^)-'Yl-'d^
= -A
=
\^n^r
V°Yt
(63)
and
-1—
{c°'^r^
^^
(69)
^
W nor EM, the analog of equation (69)
'
holds
as well, so that
nt
Finally since
Equations
=
—
AtJ?
Since there are no constraints in the choice of Ut for neither
for j
(68)
order conditions with respect to n^ to obtain
first
EM
{Y^c'-'^f-'vo
Having computed Vn^ we can now return to
explicitly the associated expectation.
TTt
for
=-
equality in (68) follows from (65) whereas the third equality follows by using lognormality of Yt
and computing
take
(see (Kaxatzas
Xf^ = X^ =
^^
'—-.
in the postulated equilibrium,
we
(70)
also
know that
cO.i^^
=
/3_(pO_^-)^o^^„
^i,w
^
A^/3-(p^+i + g)/i-n
(72)
^^.BM
^
i+pOfO_^^
(73)
^i,BM
^
A-^+pV^+n
(74)
(66), (67), (69), (70), (71)-(74)
0, 1.
= Aro—
Plugging (73)-(74) into
(71)
form a system of ten equations
(69),
in ten
unknowns
{p>
,
f^ ,n,n,c^'^ jC^'^'^)
combining (69) with (70) and using the market clearing condition
(12) leads to
^i.w
Ad(^+i)_(^ + g)/
1
^i,iv
(75)
Equation
(75) implies that
ci,tv_ Ad(/3+l)-(i
+
g)/l
(76)
p + i^lfo
cO.iv
Using (76) inside (69) we arrive at equation
which
is
precisely equation (18). Using (66)
and
(77) inside equation (71) leads to
-7
.0,w,,_..,;),.„„,,_,;(jLt3QZ,.,,„^f l^^')A'-gr'^'
39
|
»
(78)
Solving for n from equation (72), and using (76) and (66) gives
+ ^f n-^
{P
n^A"
A'^{P+l)-{i + q)P
f
rO,W
(79)
P+l-lf°
J
Using (79) inside (78) and rearranging, we arrive
P
+
1
at
-gH^^n^iP + ^fV /O.A.^ig^/^
X^AA^(^I^1^,±^
„o,w
(80)
1-7
Combining
(66)
and
=
(67) for j
leads to
0, 1
niP+^^fy
K
(/?
" {{P
+
-
1) A"^
+
(i
1
+
- if —
IM-^
q)
«
\.
,
(81)
(cO,W)-T
1/
[P
+ nf n-7
-p+g)
"
-
P
= g-
^
rO,W\
(82)
vd>'^^)
where
+ l)A''-(7+g)/i
P + l-~if°
(/3
By
solving the system of equations (80), (81) and (82) for
obtain n, into (77) to obtain
tt,
into (66) to obtain p°
f^,P we can
c°''^',
then substitute into (79) to
and p^ and then inside (71)-(74) to obtain the
rest of
the allocations.
To complete the
verification that the postulated allocation
the consumption processes Cf^
optimality for
W
it
C^^
and
X^^ X^
(cry = E,
-''"')
\
(34)"'^
e(^''"-"
°
'^"
*"
"
{C'^y
^- Nt
=Et{e
(^)
(/o'('-.-o)'i")
f-T\Nt
(a)
'
(^w\-i
(Cf^)
,l,M/\
r°
-
p
+
A
M,W
is
a martingale. Applying
-T
+
p+
g
nfo\
r.O,W
40
for all
{Ct)
Hence, an equivalent way to test whether the Euler equation holds
_
and
W
respectively.
To show
i
T>
and
i
leads to
{]J'(r„-p)du)
(Cr)
[A)
EM
are optimal for
,
Multiplying both sides of this equation with e
e
an equilibrium, we need to show that
check that the asserted allocations satisfy the familiar Euler equation;
suffices to
(A)"'
,
is
\
is
Lemma
Ito's
for all
to check
in state
i
and
T>
t.
whether Zj, defined
cis
gives
—y
-
1
dt
+ dMt
(83)
where dMt represents martingale increments. Using equation
A
hence the Euler equation holds.
when
St
=
and
driftless
argument can be applied to show that the Euler equation holds
to
EM,
our assumption that short positions in bonds are subject to the same monitoring costs
growth contingent contracts implies that
EM.
is
1.
Turning
as
similar
(17) inside (83) implies that Zt
In particular,
it
suffices to
^(;„V„_„,.„)
Equation
^^;v,
borrowing at the rate
rt
+
EM
it
is
sufficient to
check that
X^'^ =
check the following pair of Euler inequalities for
^^^^y, ^
(84) asserts that
it is
Hjf <..-.,..)
^^
^^;v.
not optimal either. To show equation (84),
development and r^ denote the instant
after
all t
^c^Myl\ ^^
does not want to invest in bonds at the rate
a "corner" solution
is
<
^jj ^
and
t'^
^^d
all Sj
T>
6
for
{0, 1}
t.
(84)
vt,
whereas (85) asserts that
let r't
denote the instant before
development. Then for states j
=
0, 1
equations (81) and (82)
imply that
^I;em
C'}^
J^)=\^\
+-i'
g^'^ + ''"
C^f
\
'
j
For times prior to development
(r'EM\
where the
follows
is
first line
from the
{t
<
t'^)
1
'
and any
T>
—
i
I
\
C'-j""'
I
/
C
M 7^
<
I
(86)
I
-G
we obtain
!r'EM\~''i
follows from the fact that
TOT
lr'W\
(
W's Euler equation
—
''
holds as an equality and the inequality
(if
T<
r*^) or is larger
than
1
(if
T>
r"^)
by equation
(86).
To show
(85),
apply
Ito'
use (17), (20) to obtain
_k
^
'
^
e-.*(l)'^'(Cf-)where dMt
=^
/
fact that
either equal to
Lemma and
\
'
is
^
=-rtdt + g\
^
^
^
U^.-)
-
^ A
K-)
^
^
dt
+ dMt
(87)
a martingale increment. Equations (81) and (82) imply that
-7
(88)
41
Combining
Assuming
and
(88)
tliat
z/
(87)
<
1,
if
we obtain
holds. This concludes the verification that the postulated
Xf^ =
are optimal for
Lemma
Proof of
Then equation
c^,wi^Q,w
=
=
for j
=
fr
>
^
)
/3
and
letting
Using these facts inside
it
1,
(89) with (25), (73)
follows that /°
and (81) yields
Proof of Proposition
W
is
-^ oo shows that c°-^^ /P
/3
(19), (21)
2.
>
(l
/i.
+
and
—
>•
1.
(18) gives (23). Since
^^^>
^A-^)
and using
•
=
A'^
q}-'^^ jcP^^^^ implies that
1-/1
/p°\^
Rearranging the above equation yields
EM
and
combining
Verifying that the proposed allocations and prices constitute an equi-
W are the same as
in proposition 1
given by equation (62), while the value function of
1/°
(25). Finally,
but tedious manipulations.
(24) after straightforward,
librium pre-development follows similar steps to the proof of proposition
function of both
the asset process
(/V-/V)A^
0, 1
1
(
Cf^, and
c''^^/c°'^^', equations (73). (74) imply that
Furthermore, applying equation (67)
t^z
consumption process
Dividing both sides of (80) by
1.
_
"-
Since
a submartingale and hence (85)
is
EM.
(76) implies that c^'^' /P —* A"^.
A-*
{Cf^y
follows that e(^o('-"+'"-p)'^«) (a)'^'
(X,c,nc) =
^i
1
—
7
EM
\k{\
Post development, the value
1.
This implies that the value function of agent
.
is
-
/,g)
+
vx,a\^-^
(90)
;y
vEM
Since in the proposed equilibrium
all
—=
the pre-development equilibrium prices depend exclusively on '-^
—
5^, the value function of both agents can
j
= {EM, W}.
=
V
of agent
Yi,
W
Si,
and Xl where
in state Sj
=
Hcn^\ yO^^^^w_cW^(^0_f^(^p^^j^^^^^\^'^Y,)+gV^{xr,Yuf,)
[pj^i.s=iAV'
+V°Yf,y^o
Computing the
be expressed as a function of
For instance, the Bellman equation for the value function
max
+X
still
first
(^X^-lYuYt) +pj=i,s=oV° (xr
(91)
+PJ=o,s^iAV° {X^ ,Yt)]
+ ^V°yalY'-{p + X + g)V°^
order condition for
V°X:Yt
-Ct^t.i^t)
is
^^
leads to
= X [pj=i,s=iAVk (Xf - lYt, Yt) + pj=i,s=oV^ [x^ - ItYu Yt)]
42
Yt
(92)
By combining
jjj'BM
_ -x}^
condition for
(92) with the first order condition for
and using the definition of Xt
EM
By combining
(42)
when pj=i,s=i
=
arrival of
St
We
=
and
0,
(Cl^)"^), using (35), noting that
equation (33) we arrive at (42).
in
then
zr^
PJ^l,S=lA
we
(93)
jumps
there are no joint
when
=
(V^,
The analogous
first
order
is
K=x
and the
consumption
^j
=
(93)
also explain in the text, equation (43) implies that
Hence, unless there are joint jumps in the marginal
0.
in the
As we
arrive at (43).
in the asset Ft, the
jumps
+PJ=1,5=0
marginal
optimal holdings of Ft are
utility of
=
S,t
0.
By
consumption and the jumps
utility of
consumption
sj
=
^j
=
assumption, in state
in Ft
and therefore
1
1.
next turn to the determination of the optimal /° and the equilibrium price p". Differentiating (91)
with respect to /°, using (35) and (34) and repeating the same steps as
order conditions (38) and (39). Deriving the analogous
with (38) and (39) leads to (40) when
=
St
and
first
to (41)
as functions of
K°
{xt)
^.
Sofar
we have derived
when
EM
St
=
1.
To
We
1,
leads to the first
and combining them
simplify notation,
will also
we
shall
denote
apply the same shorthand
the quantities of interest (/°, /'^,p°,p\ A* and
and K^{xt).
To complete the equilibrium
verification,
it
remains to derive these functions K°,K^.
utilizing the Euler equation. Formally, define tq to
To
Using similar steps as
all
Proposition
order conditions for
the solution to (40) as f° instead of the more lengthy f'^(K°{xt)).
notation to /"^,p°,p\A* and
in
in
Proposition
1,
=
be the smallest time after time
inf {X,
=
s>t
we obtain that
for
t
We
do
such that Xto
by
this
=
:
0}
any time
t
<T < tq, the following Euler equation
must characterize EM's consumption
A''^^Ct'^My'r^E^^^(r,-p)(T^t)^'^T^^EMy-r^^
An
identical equation needs to hold for
W
Following identical steps to the proof of Proposition
St
=
imply that
Applying
Ito's
e''"'"''^*^'^'
Lemma
{C^y
(94)
and
1,
e<'''-'')*34'^'
one can show that equations (94) and (95) when
(Cf
^)"'^ are both (local) martingales when
Xt>
0.
to Xt gives
dxt
=
d
(^^\
=^+Xtd (y]
43
=
a{xi)dt
-
XtadBt
(96)
where a[xt)
is
defined a^
a{xt)
Using (96) and applying
Ito's
=
-
(r°(xt)
Lemma
Mo
+ ^')
to e''"'"'''*^
'
Xt
-
- XUt +
K'^ixt)
when
{C^^')~"^
1
=
s*
+
f?P°
(97)
gives
d[e(-.-p)*A'^'(Cf^)^^]
l"^"' (nEM\~y
eCr.-P)*^'"'
(C,
=
<!'-"(a:t)-p-7(Mo-
+X
Pj=i,s=i^
d.K"
dxi
I
[
where dM^'^
is
+^+5
y)
+ Pj=i,s=o
i^° (xt)
9
f
f
2\
\
(98)
^
'^
2
,
,
Since
a (local) martingale.
7
(d,T[p
"'"^t
,
of K°,K'^,
^, ^^
to determine
i<"°,
and Xf
We
> \
/
I
Mo
-
y
>
2
Kf-^G
+-^- + 5
1
nUx, + i,
dit
I
dM^
we can
(98)
r
(k°,K\
Tit
no
is
2
2
I
I
\
y'^^^t)
~
2\
y^t'^)
+
,
'^
Xi
a local martingale. Since
set
-^
with the
must
VX^a
-
also hold for
.
Finally
W, namely
(99)
1
(xt)
,
,
n^{xt)Y''
0°
[xt]
"
(dxi)'
(7+1)
no
e'^'~''^*yl
'
The terms
them equal and use equations
[97])
^, ^^,xt)
fn°{xt+~Q
brackets in equation (99) must be zero.
so
=
fi° [Xt)
\
where
a (equation
+ Pj=i,s=o
Pj=i,s=iA
f dn°
-y
definition of
zero allows us to solve for r°[xt) as a function
+P—
n°
a local martingale, the term inside
is
denote this function as r°[xt)
shall
+ dMf ^
dt
'''
[Cf^^
is
2
s
dTt
Combining the
brackets of (98)
curl}'
/ dA"
K^, we observe that an expression analogous to
r''(xt)-p-j
+X
'
9
Xf
T\ '^
e^'''"'''^*A
the curly brackets in equation (98) must be zero.
observation that the term inside the
i^° [xt]
{Cf)
is
and
a local martingale, the term inside the curly
(42), (93) to
'^^[xt)^
no
+ dM;w
inside the curly brackets of (98)
(40), (41),
gi- +xPj=o,s=iA
dt
[xt)
44
and
(99) are
both zero,
obtain
[K^xtY
K°
(100)
{xt)
dK°
=
dxt
7
d-n°
dx.
{oi{xt)
A'o
0.°
Using equation
-jxta-) +
2
QP
\
we obtain
(36)
axt
i^t
and
By
^
=
^^^
ti^
^*' "^ similar line of derivations allows
"
dn'
7
ni
{a{xt)
A'l
5(a;t)
=
-
-yxtCJ
'n°{xt
ni (xi
)
(103)
{r'^{xt)
-
Mi
+
o"^)
Xt
=
-
'—
-^
H
'
2
non-contingent bonds when Xj
B
+ q)-+X
d^n'
d/<'
di,
dx.
where
us to obtain a differential equation in state
namely
I,
g{i
=
(102)
using (36), (101) and (102) inside (100) allows us to obtain an ordinary differential equation involving^^
^°'-^^' "d^'
St
--/-l^ -(+:«- /J.)) ^-/;.
- K^{xt) +
A''
-
A'l
fii
I
+
'
I
'
^'
2
EM
flp\. Proving that
f^W
\
1
\
A'^
not choose to borrow in
will
follows similar steps to the proof of Proposition
1
.
Data
For the construction of
0^j,
we used data from
the World Bank's World Development Indicators Database,
and from the International Monetary Fund's International Financial
of the variables
While
and corresponding
in the
model there
is
Statistics (IFS).
Table 8 presents a
list
sources.
a single good,
in the
data the computation of
more cumbersome
ipt is
since
there are multiple goods, exchange rate fluctuations, intermediate goods, and so on. All our steps below are
aimed at
this,
we
isolating in
?/)(
the component of external resources and income which
transitory in nature. For
let:
^'
where
is
A^
and
X
in local currency;
Y
-r
r/D
^
correspond to real nontradables and exports;
and
E
and
CF
x-.
V
\^'^^>
Trend'
[Ar,r^"'^+[(^;^-0.5X,)]
Px and Pm
to export
to the nominal exchange rate and capital flows.
and import prices
Real nontradables are
constructed from:
A^t
= TT- iGDPt J
{Px,t
-
0.5PM,t)Xt,)
N,t
'Note also that a{xt) can be expressed as a function of A'°, K^,
is
a function of A'°, A'\
'
'
^, ^^C
dxi
'
(dxt)
and Xf
45
^-,
f^ ^.;
and
xt since the interest rate
Source
Series
Nominal
GDP [GDP)
World Development Indicators
(quarterly
CPI (P)
and annual)
IFS
(quarterly
and annual)
Nominal Exports {PxX)
World Development Indicators and IFS
(local currency)
(quarterly
Nominal Imports
{P^M)
and annual)
World Development Indicators and IFS
and annual)
(local currency)
(quarterly
Real Exports {X)
World Development Indicators
(local currency)
(annual)
Real Imports (M)
World Development Indicators
(local currency)
(annual)
Nominal Capital Flows (CF)
IFS
(dollars)
(quarterly
Nominal Exchange Rate {E)
World Development Indicators and IFS
(quarterly
Net Factor Payments
(NFP)
and annual)
and annual)
IFS
(annual)
(dollars)
Table
8:
Data used
in the construction of
46
ip.
Mean
Median
Min
Max
Chile
20.6
21.5
15.4
22.7
Colombia
11.3
10
8.4
16.6
5.5
5.6
2.7
7.2
Indonesia
11.3
7.4
4.7
19.8
Malaysia
29.4
28.6
19.6
42.3
Thailand
21.3
20.8
14.1
28
Mexico
Table
GDP
where
Reserves for various countries as a percent of
9:
is
the country's
GDP,
and the term 0.5PM,t removes a proxy
(
—
p"^
-
—
0.5A'tj
is
Pa'.j
filter,
for intermediate inputs
mary, the denominator
We
effect.
extending the series as
We
end-of-series bias in this procedure.
applied the
in
export-production.
The
expression
decompose between trends and cycles using a
much
as
we could
in order to reduce the effect of the
to the log of the corresponding variable. In
filter
equation (104) measures the average (trend
in
for the years 1990-2003.
the price of nontradables approximated by the local CPI,
captures the terms of trade
standard Hodrick-Prescott
GDP
level) of total
sum-
income and resources,
while the numerator attempts to capture the cyclical component of external resources.
We
work with a sample
of the following developing countries/emerging markets;
donesia, Malaysia, Mexico, Thailand.
We chose them from
(2004) plus Malaysia. However, since their sample
for the
is
dimension (wc used data from 1983 to 2003), we dropped
with domestic macroeconomic
which we did not have good
issues, or did
deflators.
flows.
we constructed
We
all
list
of countries in Calvo, Izquierdo
quarterly series for
the countries that either were closed economies
Our marginal drop was Korea,
GDP
il'it
for
Korea
),
solve this
quarterly series for the years
using a related scries approach with quarterly data on capital
be equal to the annual figure we computed directly
countries (dates are
As a
shown
reference, table 9
problem we use a linear
when we
for
our results remained essentially unchanged.
using equation (104). Unfortunately, we lack quarterly data for some components of
To
and Mejia
1990s only, and we needed a longer time series
not have complete data.
restrict the average of the quarterly values to
the 1990-2003 period.
Colombia, In-
However, when we re-estimated our model with Korea included (using
only capital flows data divided by nominal
Finally,
the
Chile,
lack quarterly data.
in parenthesis): Chile (1990),
shows reserves
for the six
(or a quadratic) interpolation
We
some countries
method
in
to obtain
used interpolation methods for the following
Colombia (1990-1995), and Malaj'sia (1990-1998).
economies we study.
47
V't for
Details on the econometric procedure of Sections 3.1, 3.2.1, 3.2.2.
C^
Section 3.1:
To estimate the process described
Gibbs Sampler. The Gibbs Sampler
states (See
Kim and
is
in this section
by now a standard methodology
posterior distribution of
all
However,
the
same
first
SS), p{SS -^
of
country
>
for
we run
i
We
allow
we assume
NSS)
at time
is
differ across countries.
that the transition probabilities into and out of a
we assume that the
common
is
joint probability of a
Fx{SS =
l\ip^;'^).
As
was drawn from the
In the next step
in the text,
we use
In the next step we take these
whether each economy
is
in
SS
[NSS)
first
"f
=
{i/'t
,
~
V'f
We
first;
To
Kim and
with
a =
(3
=
we draw an
the convention that
I's
and
O's as given.
Kim and
ipi
"
gamma
Nelson (1999))
''Pi
1
I
i'^ef)
sample of
I's
and
O's
c)
By
We
i
priors: a) a
from these posterior
Then we use
well
known
NSS.
to
this information to
beta prior
[0, 1]
b)
for
p{NSS —
>
we knew
determine
this in steps as described in
Kim and
"P^^^ o-^p, o-f|}
,
SS), p{SS
an (improper) normal prior
— NSS)
»
for
ip^
that lead to posteriors that depend only on the data
a truncated (improper) normal and an inverse (improper)
Kim and
shall denote
Effectively this allows us to proceed as if
Once again we do
prior for [cr^f^)
^^ explained in
independent of each other.
^
To do that
corresponds to a Sudden Stop and
or not at a given point in time.
updating we use conjugate
and an (improper) inverse
<^^f^ o'ff
a sudden stop at a specific point in time.
in
(artificial)
which coincides with a uniform prior on
1
in the
Nelson (1999) to determine a
start with determining the posterior distribution of {ip^^^, ipf^
facilitate the
,
''Pi
or the second (SS) regime.
as described in
filter
the posterior distributions of the parameters in $.
Nelson (1999).
jump
across countries.
to fix a set of initial parameters
a standard Hamilton (1989,1990) type
probabilities.
for
Nelson (1999) present by
repeat this process for each country separately and obtain one sequence per country.
this as
(see
Kim and
and then determine the posterior probabilities that a particular realization
}
t
to exploit knowledge
is
parameters of the model to
all
sequence of posterior probabilities that a given countrj' was
We
models involving hidden
the others) to construct the joint
all
modify the basic model that
across countries. Moreover,
step of the procedure
time (fixing
at a
a simultaneous transition into a sudden stop
p{NSS —
tpit
We
in order to obtain precise estimates
VIX and
The
parameters.
all
the countries into a single sample.
sudden stop are
in estimating
Nelson (1999) for an introductory treatment). The basic idea
about the conditional distribution of one parameter
pooling
we apply a Bayesian methodology, by using a
Nelson (1999).
results in
Bayesian
Finally,
we assume that
gamma
all
prior
priors are
statistics the posterior distributions are in
the same class as these conjugate priors and there are simple closed form expressions for the parameters
of the posterior distributions.
observations for
all
The updating
oi
p{NSS
the countries. So, for each country
48
-^ 55), and
p{SS
-^
we count the number
NSS)
is
done by pooling the
of transitions into
and out of
sudden stops, and the
episodes for
number
total
normal times and
of periods in
^
~
SS)
beta
+
1
(
^ap,
\
sudden
We
stop.
a normal year
we add
all
number
the
is
We
stops.
then add
all
the
countries and find the posterior distributions as follows
all
piNSS
where af^
sudden
in
marked
of observations
+
1
^ ^NSS
1
I
normal years that are followed by a year marked as
as
shall refer to this as a transition to a
sudden
stop. Conversely,
followed by another normal 3'ear (NSS). This count
is
them up
same
probabilities to be the
precise estimates.
number which
into a single
for all countries,
counts the times that
done country by country, but then
used in the updating process.
is
we
is
af ^^
By
restricting the transition
exploit the panel dimension of the data in order to obtain
Similarly, the posterior for the other
parameter
in the transition
^
+
matrix
given by the
is
following formula
^ NSS)
p(SS
where b^^^
number
the
is
of observations
~
beta
marked
+
1
(
as
^^^^^
^
E
^f^
)
sudden stops that are followed by a year marked
eis
"normal". Conversely, bf^ represents the other case, namely when a transition out of a sudden stop does
We
not occur.
specific
record the
random draws
-
parameters {ip^^^, ipf^
NSS)
p{SS
-^
draw
of the paths of I's
of a) the paths of I's
tp^^^, a^f^, o-f|} and
Then we repeat the above procedure
.
and
O's for
the posterior distribution of these
c)
and
each country, b) the country
the "pooled estimates" of
several times
each country and the parameters.
random draws
O's for
and
By
p{NSS
-* SS),and
we record the new
at each time
properties of the Gibbs sampler,
coincides in law with the posterior (joint) distribution of
the parameters.
all
Section 3.2.2:
"jump"
in the
cutoff value
e.
By
VIX,
using the cutoff value e for the
i.e.
months when the
VIX
we determine the months
residuals of the estimated
To determine a distribution oip{SS\J) we proceed
AR(1) process
in
which we observed a
of the
as follows. First, for
VIX
exceeded the
each country we draw
paths from the posterior distribution of the states (NSS,SS)^, as provided by the Gibbs Sampler. Given
the model, the only relevant observations for the conditional probability are the ones
is
in
NSS
or has just transitioned to a SS.
model, we discard
of each SS.
there
is
a
all
Then we
jump
in the
VIX
all
is
in
also determine
all
those times where the states switch from
either in that quarter or the quarter before.
the times
when
n}v55 as the numbers of observations
i
SS, except the one that marks the beginning
allow for the possibility of delayed data reporting etc. Let this
we
the country
Hence, in accordance with the data generating process of the
the quarters in which the country
look at
when
there was a
jump
when country
i
in the
is
49
NSS
We
to
this
^''^^ss—ss
number be
cither in A'^55 or just
simultaneously
allow this short
number be given by
VIX. Let
55 and
moved
ny
J-
window
to
Similarly,
Finally, define
to a SS.
We
repeat
procedure
this
Tiggj,
for
and
n'-j,
each country separately. After completing the above procedure for
n^f^gg
all
countries
In accordance with the Bayesian methodology that
across countries.
we sum
we have been
using throughout, we use a beta distribution with uniform priors to obtain the posterior distributions
p{J)
-
beta (1
+
E,:nj,, 1
p {SS\ J)
-
beta (1
+
T-m^Nss-^ss.j,
+
i:in%ss)
1
+
In a nutshell, for each iteration of the Gibbs sampler, we determine the
VIX
coincided with a transition into a SS for each country.
imposes the constraint that
all
VIX. The benefit
into a
SS and jumps
U
Estimation of Risk Premia
To estimate
the risk
in the
premium
same
countries have the
we
A*
is
that
number
Then we pool
if
P""
{VIX, K,
t)
is
^
We
^^
'^^
'
.
To estimate
used quotes^^ from the
Letting k
= ^7^, we
is
^\l''^'^''^^
VIX,
estimates^''.
Huang
K
(1985).
As explained
we
on VIX
fit first
call
in the
and r days to expiration, when
then the (arbitrage free) price of a payoff such as (50)
a non-parametric function
P'-^
[VIX, K,
is
simply
r) to the data.
options since the beginning of this market in early 2006.
=
0.0306isr-1.115/c-|-0.34/c2-0.0535/c^
-0.0002r/C - 0.0304TA:
...
random draws
+
0.0243rfc2
-
+
0.0024fc4
0.008tA:^
+
(105)
O.OOlrfc*
+
..
additional controls
a robustness check we also computed
recording the
between transitions
then estimated the following regression:
^
"As
in the
followed an approach similar to (Ait-Sahalia and Lo 2000), which itself
this quantity
CBOE
when a jump
across countries. This effectively
we can obtain more accurate
the price of a call option with strike price
the underlying value of the index
of times
joint probability distributions
builds on the ideas of (Breeden and Litzenberger 1978) and Duffie and
text,
S»"!/)
p[J
)
and p{SS\ J) without imposing any
prior,
i.e.
by just
of:
^in'j
p^^>
at each iteration k of the
distribution.
(SS\ J)
Gibbs Sampler and computing the empirical mean of the corresponding stationary
The two approaches
delivered very similar results, suggesting that our results are not influenced
by the assumption of a uniform prior.
^^In particular, we used the midpoint between bid and ask
50
where additional controls included a l/VIX, a constant, VIX,
To avoid micro-structural problems
t,t'^.
with very short maturities,^^ we focused attention on values of r e [30,90]. Furthermore, since we are
interested in isolating the price of risk for
upward jumps
in the
VIX we
focused on
was estimated on 2476 date-price combinations. The resulting E? of the regression was
that the regression specification accounts well for the observed variation in
differentiated the estimated
P^ {VIX,K,t)
in equation (105)
on short-dated options but with expiry dates
and
at the average value of
regression (105).
To be
VIX
first
term
in
over an interval r-
k
is
+ 0.17^ {\og{VIX) -
set so that the probability of
.01.
The second term accounts
2.88 (based on data since 1986).
The
reversion in monthly data. Plugging in these numbers,
As was shown
0.105.
in the text, this
{VIX))
having a jump in the
the distance between the average (log) underlying price in the
mean \og{VIX) =
log
we used
call
for
Since
.
is
'^^
^^^'"^''"^ at
r
days a large fraction of the out of the money
of bid
This
is
and the next highest increment
why we
of ask.
40,
we
set
residuals-assuming no
reversion in
(VIX) =
VIX
jump
by comparing
2.54 to
its
long run
speed of
mean
that the value of a claim such as (50)
at very short intervals.
call
=
.
options log
number imphes that A*(r/360) =
problem
then
an interior value to the
is
0.105. Solving for A* gives 0.94.
^^The data are quoted at prices with discrete increments. For the out of the money
consider, this discreteness presents a
We
we focus attention
coefficient 0.17 accounts for the
we obtain
regression
which suggests
K
in the text
VIX
mean
0.83,
The
option prices.
sure that r
consistent with the definition of "jumps" that
approximately
is
make
1.
call
we evaluated —
larger than 30 days,
over the sample in order to
r
The
with respect to
>
/c
call
As maturity nears
options that
we
to less than thirty
options start to converge to the lowest possible increment
This discreteness
leave out the very short maturities.
51
is
less of
a.
problem
for longer maturities.
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