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Whydocountriesborrowthewaythey
borrow?∗
Marcos Chamon†
Harvard University, Dept.of Economics Ph.D. Candidate
Ricardo Hausmann‡
Harvard University, Kennedy School of Government
First Draft: July 2002. This Version: November 2002.
Abstract
This paper analyzes the interplay between individual borrowers’
choices for liability denomination and the optimal monetary policy. If
the monetary authority cares about preventing bankruptcy and most
liabilities are denominated in dollars, it will stabilize the exchange rate
at the expense of higher volatility in the interest rate and vice-versa if
most liabilities are denominated in local currency. That can generate
multiplicity of equilibria in the liability composition. If an individual
borrower expects the others to borrow in dollars (pesos) he or she will
expect the monetary policy to be tailored for that liability denomina-
tion and as a result would find it optimal to borrow in dollars (pesos)
as well. If the monetary authority has a strong enough preference for
exchange rate (interest rate) stability the equilibrium becomes unique
with the liabilities denominated in dollars (pesos).
∗The authors are grateful to Roberto Chang, Federico Sturzenegger and seminar par-
ticipants at the Original Sin Pre-Conference and Lacea 2002 for helpful comments. Any
errors are ours.
†chamon@fas.harvard.edu
‡ricardo_hausmann@harvard.edu
1
1 Introduction
Why do companies and households in many emerging markets borrow in
foreign currency? After all, the recent spate of crises in which liability “dol-
larization” has interacted with drastic real depreciations to create massive
bankruptcies and economic havoc suggests that it is an important source of
financial fragility. But what causes this phenomenon? Is it that they do not
want to borrow in domestic currency because their choices are distorted by
moral hazard, or is it that, for other reasons the domestic currency market
does not exist, especially for international lending? A significant part of the
literature has been focused on moral hazard interpretations. If companies ex-
pect to be bailed out by governments, they will not fully internalize the risks
they bear (Dooley (2000), Burnside, Eichenbaum and Rebelo(2002), Schnei-
der and Tornell (2000). But the telltale signs moral hazard in terms of the
pattern of lending have not found much empirical support (Eichengreen and
Hausmann (1999), Fernandez-Arias and Hausmann (2000). A more recent
literature has proposed other models. Caballero and Krishnamurthy (2002)
present a model where excessive dollar debt is the result of domestic financial
constraints that lead firms to undervalue the social benefit of borrowing in
local currency. Jeanne (2002) argues that liability dollarization can be a safe
play when low monetary credibility keeps interest rates in domestic currency
high. Chamon (2001) and Aghion, Bacchetta and Banerjee (2001) present a
model where the correlation of devaluation risk and default risk makes do-
mestic currency lending unattractive. In their setup, it is possible for a firm
to expropriate the claim that domestic currency creditors have on the resid-
ual value of the bankrupt company by increasing their borrowing in foreign
currency, given that in the context of a bankruptcy, the claims of domestic
currency creditors will be automatically written down by the concomitant
depreciation. In anticipation of this, investors refrain from lending in domes-
tic currency. Tirole (2002) proposes a “dual-and-common agency” approach
to the problem, where a foreign investor’s return depends not only on the
behavior of a private borrower but also on that of the borrower’s government
with whom he does not contract.
In this paper we explore the interplay between individual borrower’s
choices for liability denomination and the optimal monetary response of the
Central Bank given those choices. We start from the assumption that the
debt in domestic currency cannot be contracted at long maturities and at
fixed rates. As a result, the terms in which it is rolled over or repriced will
2
depend on changes in the domestic interest rate. In the model presented,
there is a shock to the expected future exchange rate. Since agent’s are for-
ward looking, that shock affects the present interest and exchange rates. The
Central Bank uses monetary policy so as to determine how the absorption of
that shock is divided between changes in the interest rate and in the exchange
rate. If most liabilities are dollarized and the Central Bank cares about pre-
venting bankruptcy, it will stabilize the exchange rate at the expense of larger
movements in the interest rate. Alternatively, if most liabilities are denomi-
nated in pesos the Central Bank will stabilize the interest rate at the expense
of larger movements in the exchange rate. This can generate multiplicity of
equilibria in the liability composition, since if an atomistic agent expects all
other agents to denominate their debt in dollars (pesos) he or she will expect
the monetary policy to be tailored for that particular liability denomination
and as a result may find it optimal to borrow in dollars (pesos) as well. It
is worth noting that the policy maker in our model does not attempt to ex-
propriate foreign investors to the benefitofdomesticresidentsasinTirole
(2002). Instead, it is only trying to make dollar debt safer given that those
contracts have already been written.
The interaction of liability denomination and monetary policy has re-
ceived recent attention. Calvo and Reinhart (2000) and Hausmann, Panizza
and Stein (2001) show that emerging markets that formally float their cur-
rency tend to limit the movement of the exchange rate vis a vis the interest
rate and to accumulate significantly larger stocks of international reserves.
Hausmann et al show that this behavior is strongly correlated with measures
of the ability of a country to borrow internationally in its own currency.
Hence, the title of their paper “Why do countries float the way they float?”
receives implicitly the answer “because they borrow the way they borrow”.
There is also a recent literature relating the structure of liabilities to the
choice of monetary policy. Aghion, Banerjee and Bacchetta (2001) show
how balance sheet effects can make devaluations contractionary and opti-
mal monetary policy apparently pro-cyclical. Cespedes, Chang and Velasco
(2000) present a model where the effectiveness of monetary policy although
still positive is significantly compromised in its ability to dampen cyclical
fluctuations by the presence of liability dollarization. These works focus on
how liability dollarization can affect the optimal monetary policy. There is
also a literature that studies how monetary policy can affect the currency
composition of corporate debt (for example, Jeanne 2000). In this paper,
we emphasize how the choice of an individual entrepreneur’s liability com-
3
position can be affected by the choice of the other entrepreneurs through the
effect of their choice on the resulting monetary policy.
2 The basic environment
Consider a small open economy subject to shocks to its expected future
exchange rate. These shocks are assumed to be out of the control of the
economy’s policy maker (for example terms of trade shocks) and are not
modeled explicitly. We assume the resulting exchange rate expectations are
distributed according to a random variable zt.
The focus of the model is on a small segment of the nontradable sector
which consists of atomistic entrepreneurs. Those entrepreneurs have access
to a production function that requires an initial investment of one unit of
the local currency (henceforth pesos) and whose output is worth Apesos.
Two types of debt are considered. The first, which we refer to as dollar debt
is denominated in the foreign currency. The second type which we refer to
as peso contracts are denominated in the home local currency. We assume
one cannot write peso contracts in terms of a fixed domestic interest rate.
Instead, peso debt contracts must pay the ex post domestic interest rate for
the period the loan was made. This assumption aims at capturing the fact
that in practice the maturities of local currency contracts are much smaller
than those of foreign currency ones in emerging markets, and that by bor-
rowing through short-term local currency debt the borrowers are vulnerable
to shocks to the interest rate at which that debt is rolled over.
The debt contracts must be written in the beginning of time twhen the
exchange rate is given by e0
t. The expectation at time tfor the exchange rate
in period t+1 is given by the random variable ztwhose realization occurs
at the beginning of period t, but only after the debt contracts have been
written. The higher zt, the larger the expected depreciation (which can be
interpreted as the result of an adverse shock). Thus, the expected exchange
rate in period t+1 at the beginning of period tis given by E0
t[et+1]=E[zt],
and is updated to Et[et+1 ]=ztfollowing the realization of that uncertainty.
The monetary authority decides how to accommodate the shock zt−E[zt]
between changes in the interest rate and in the exchange rate.We assume an
uncovered interest parity condition must be satisfied:
1+it=(1+i∗)zt
et
(1)
4
where itis the domestic risk-free rate, i∗the constant world interest rate and
etisthevalueoftheexchangeratefollowingtherealizationofthatshock.
The value of e0
tis given by arbitrage between peso and dollar instruments at
the beginning of time t:
E0
t[1 + it]=(1+i∗)E0
t[et]
e0
t
We assume that entrepreneurs are risk-neutral, but face a non-pecuniary
cost associated with defaults. As a result, when deciding whether to borrow
in pesos or dollars they seek to minimize the probability of a default occur-
ring. An entrepreneur that has borrowed through peso debt, paying a risk
premium rpeso defaults if:
A<(1 + it)(1 + rpeso)(2)
while an entrepreneur that borrowed through dollar debt paying a risk pre-
mium r$defaults if:
A<(1 + i∗)(1 + r$)et/e0
t(3)
In principle an entrepreneur could mix the two denominations, but we
show that this is not optimal in this model.
Thetimingofeventsissummarizedbelow:
1. Debt contracts are written
2. Theshocktotheexpectationofthenextperiodexchangerateis
realized.
3. Given that shock and the currency composition of the debt contracts,
the monetary authority sets itand et.
The monetary authority (henceforth the Central Bank) takes that cost of
default into account when choosing itand et.It also seeks to minimize the
gap between the economy’s output and an ideal target and to minimize the
inflation rate. Output and inflation are a function of the interest rate and
of the exchange rate, given by the equations below where a sans-serif font
indicates a variable is expressed in log terms:
yt=y+αet−βit
πt=π+γet−δit
where ytis the log of the output, πtthe inflation rate, yand πare constants,
and α,β,γand λare positive constants. Since the entrepreneurial sector
5
that borrows from abroad is assumed to be small, the effect of its liability
composition on these parameters is ignored. The Central Bank’s inflation
versus output trade-offis given by the loss function:
`=(
e
y−yt)2+χπ2
t(4)
where e
yis the log of the ideal output target and χis a constant.In addition
to that trade-offbetween output and inflation, the Central Bank’s welfare
function is also affected by the share of entrepreneurs that would default
given its choice for itand et. The Central Bank’s loss function taking this
effect in consideration is given by:
L=`+s(it,e
t)C(5)
where s(it,e
t)is the share of bankrupt entrepreneurs given the monetary
policy and Cis the cost of that default to the Central Bank’s welfare. That
cost differs from the private one incurred by the entrepreneurs depending on
the extent to which the Central Bank internalizes their welfare and on the
externalities that their bankruptcy can impose on the rest of the economy.
It is useful to initially consider the case where the Central Bank does not
take into account those entrepreneurs when setting its monetary policy (i.e.
C=0). The Central Bank’s loss function becomes L=`. Minimizing (4)
subject to (1) yields the following policy rules:
et=ψ+ζzt(6)
it=i∗−ψ+(1 −ζ)zt(7)
where:
ψ=(α+β)(e
y−y−βi∗)−χ(γ+δ)(π−δi∗)
(α+β)2+χ(γ+δ)2(8)
ζ=(α+β)β+χ(γ+δ)δ
(α+β)2+χ(γ+δ)2(9)
and 0<ζ<1. Thus, the Central Bank accommodates the shock in a
way that both the elasticities of itand etwith respect to ztare positive but
smaller than one and together they add to unity.
Consider the case where parameters are such that ζ=1/2(i.e. the
Central Bank distributes the shock to ztbetween itand etwith the same
6
elasticity). We solve the recursive equilibrium and show that if everyone
else borrows in dollars, an atomistic entrepreneur is better offborrowing in
dollars as well since the monetary authority will stabilize the exchange rate
at the expense of higher volatility in the interest rate. The opposite is true
if everyone were to borrow in pesos.
Suppose that in the firststageofthegameallentrepreneurschooseto
borrow in dollars. The monetary authority knows that in order to prevent a
default from occurring, it must set etbelow a critical level ec
tgiven by:
ec
t=Ae0
t
(1 + i∗)(1 + r$)
There are three regions of interest for the problem solved by the monetary
authority in stage 3:
Region 1: The condition et≤ec
tis not binding:
In this region, the realization of ztis such that the monetary authority
does not need to worry about defaults occurring. As a result, it sets etand
itaccording to (6) and (7):
et=ψ+1
2zt(10)
it=i∗−ψ+1
2zt(11)
In this region both etand itincrease on the square root of zt.
Region 2: The realization of ztis such that the Central Bank chooses to
set et=ec
tin order to avoid defaults and accommodates the remaining part
of the shock through it.
In this region the Central Bank sets:
et=ec
t
it=i∗+zt−ec
t
Note that instead of increasing on the square root of zt,itis linear on zt
in this range.
Region 3: The realization of ztis so large that the Central Bank gives up
accommodating the change in etand lets the entrepreneurs go bankrupt:
The Central Bank decides to “throw in the towel” if the interest rate hike
necessary to keep the exchange rate at ec
tis so high that the loss function
7
is actually larger than the one where it lets them go bankrupt and accom-
modates the shock between the two instruments. The Central Bank’s loss
function given a realization of ztwhen ignoring bankruptcy issues can be
defined as a function of et.Thedifference between the value of that function
obtained by setting et=ec
tas opposed to the level ψ+1
2ztit would choose
in the absence of bankruptcy considerations is obtained by taking a Taylor-
series expansion1:
`(ec
t)−`µψ+1
2zt¶=`0µψ+1
2zt¶µec
t−µψ+1
2zt¶¶
+`00 ¡ψ+1
2zt¢
2µec
t−µψ+1
2zt¶¶2
(12)
=`00 ¡ψ+1
2zt¢
2µec
t−µψ+1
2zt¶¶2
(13)
>Cfor large enough zt(14)
The above expression is increasing on the difference between ec
tand ¡ψ+1
2zt¢.
Thus, for large enough ztthat loss will dominate the one associated with the
cost Cof letting the entrepreneurs go bankrupt. Once that level is reached,
the Central Bank’s response is given by (10) and (11). Note that there is
a discontinuity around that critical value of ztwithadiscreteincreaseinet
and a discrete decline in it.
So far, we have shown that given dollarization of liabilities the Central
Bank will “float with a life-jacket”, letting the exchange rate float over some
range but aggressively intervening if a certain threshold is reached. But for
ahighenoughrealizationofztit will give up on that intervention and let
it float again. It remains to show that given that the Central Bank will act
this way, agents would indeed prefer to borrow in dollars.
Since the entrepreneurs are risk neutral, when choosing the composition
of their liabilities they only care about which of them decreases the likelihood
of a default.
Let zc
$and zc
peso denote the critical values of the realization of the zt
above for which an entrepreneur would default given dollar and peso liabilities
respectively.
Since from arbitrage both types of liabilities must yield the same expected
1Note that since `(et)is quadratic, `(n)(et)=0for n>2.
8
return to the lenders, we have:
Zzc
$
z
(1 + i∗)(1 + r$)et
e0
t
Pr(z)dz =Zzc
peso
z
(1 + rpeso)(1 + it)Pr(z)dz (15)
where both etand itare increasing functions of z,andtheriskpremiumsr$
and rpeso are decreasing on zc
$and zc
peso. Arbitrage between risk-free short-
term peso and dollar instruments implies:
Zz
z
(1 + i∗)et
e0
t
Pr(z)dz =Zz
z
(1 + it)Pr(z)dz (16)
The solution of the Central Bank’s problem implies that there exists zA
and zBsuch that zA>z
c
$>z
Band:
et(zA)−et(zB)≤1
2(zA−zB)≤it(zA)−it(zB),with strict inequalities for some zB
(17)
Proposition 1 zc
$>z
c
peso
Proof. Suppose zc
$≤zc
peso.
Equation (16) can be rewritten as:
(1 + i∗)
e0
tÃZzc
peso
z
etPr(z)dz +Zz
zc
peso
etPr(z)dz!=
Zzc
peso
z
(1 + it)Pr(z)dz +Zz
zc
peso
(1 + it)Pr(z)dz
The equation above and inequality (17) imply:
(1 + i∗)
e0
tZzc
peso
z
etPr(z)dz < Zzc
peso
z
(1 + it)Pr(z)dz
which implies
(1 + i∗)
e0
tZz0
z
(1 + rpeso)etPr(z)dz =Zzc
peso
z
(1 + rpeso)(1 + it)Pr(z)dz
9
for some z0>z
c
peso.That implies r$<r
peso, which in turn implies zc
$>z
c
peso,
a contradiction.
Thus if all other entrepreneurs borrow in dollars the resulting monetary
policy is such that dollar debt is safer. Note that since etand itare perfectly
correlated in the range where dollar debt holders would default, there are
no benefits from mixing the two debt denominations (unless the borrowers
could short the peso instrument which we do not allow in our analysis).
The problem presented is completely symmetric between etand it.Asa
result, if all liabilities were in short-term pesos the resulting monetary policy
would be such that short-term peso instruments would be safer2.There-
fore, if an atomistic agent expects all others to borrow through dollar (peso)
debt, he or she will choose to borrow through dollar (peso) debt as well and
multiplicity of equilibria in the debt composition occurs.
3 Preference towards exchange rate or inter-
est rate adjustment
The previous section focused on the case where the elasticities of the ex-
change rate and the interest rate with respect to ztwere the same. But if
parameters are such that the resulting optimal monetary policy exhibits a
strong preference towards exchange rate or interest rate stability, the prob-
lem changes quite significantly. While for some range of parameters there is
still multiplicity of equilibria, welfare is higher in the equilibrium where firms
choose to borrow in the instrument whose return the central bank is trying to
stabilize. Hence, if the Central Bank is more concerned with exchange rate
stability, social welfare is higher if entrepreneurs borrow in dollars. Some
ability to coordinate would allow them to choose the better equilibrium.
This ability may be provided either by a few large borrowers or by the fiscal
authority. If the government were to denominate its debt so at to minimize
the risk of debt service to the fiscal accounts, it would choose dollar debts
and firms would just follow suit. Moreover, once a large enough asymmetry
is introduced, there is a unique equilibrium for the debt composition.
2In theory there could be a third equilibrium where half the liabilities are denominated
in dollars and half in pesos. The Central Bank would not be able to help both groups of
creditors, and would randomize which group to help. But that equilibrium is not robust
to small perturbations since if one agent was to switch from peso to dollar debt, every
agent would prefer to borrow only through dollars and vice-versa.
10
Recall equation (9) which defines the elasticity ζof the exchange rate
with respect to ztaccording to the parameters that determine the effects
of the exchange rate and of the interest rate in the output and inflation
of that economy (in the region where the Central Bank ignores bankruptcy
considerations). If ζis small, most of the shock will tend to be accommo-
dated through changes in the interest rates. That elasticity is small when
the expansionary effect of exchange rates on output is low (αis small), the
exchange rate pass-through to inflation is high (γis large) and interest rates
have little impact on aggregate demand and inflation (βand δare small).
These assumptions seem particularly relevant to emerging markets. Because
of that we focus on the case where ζis small. But just like in the previous
section, the actual realizations of etand itare influenced by the composition
of debt liabilities.
Figure 2 illustrates the case where ζis small and liabilities are denomi-
natedindollars. Therangeofztin which a default occurs is smaller than in
the basic scenario of the previous section since the Central Bank is now more
willing to stabilize etat the expense of it. But if liabilities are denominated
in pesos, the Central Bank’s greater willingness to stabilize the exchange
rate will conflict with its willingness to prevent bankruptcy. That case is
illustrated in Figure 3. For some range of ztthe Central Bank will refrain
from raising itbeyond a certain threshold in order not to bankrupt the en-
trepreneurs, accommodating the rest of the shock through the exchange rate.
In that range itis held constant while etincreases linearly in zt. But just like
in the analysis of the previous section, this deviation from the Central Bank’s
ideal rule for accommodating the shock becomes too costly for a large enough
realization of zt. Beyond that critical point, the Central Bank gives up trying
to save the entrepreneurs and is again willing to stabilize the exchange rate
at the expense of larger movements in the interest rate. Therefore the set
of values of ztfor which a default under dollar debt occurs can be disjoint
since etis not monotonic in zt. For example, if an entrepreneur borrows in
dollars she may be bankrupt for a given realization zAif that realization lies
intheregionwheretheCentralBankletstheexchangeratedepreciatemore
in order to keep interest rates low. But she may be solvent for realizations
zB>z
Aif zBis in the range where the Central Bank would have given up
trying to save the peso borrowers and e(zB)<e(zA).Whether or not the
resulting distributions of etand itcan sustain an equilibrium where the debt
is denominated in pesos depends on parameter values. If the costs of default
are low (or if they are high but the Central Bank does not internalize them
11
much) or if the Central Bank is much more concerned about etthan about it,
then one would prefer to borrow in dollars even if everyone else were to bor-
row in pesos. As a result, there would only be a single equilibrium where all
debt is denominated in dollars. But again, if parameters are such that multi-
plicity of equilibria still occur, welfare is higher in the equilibrium where the
debt is denominated in the instrument whose movements the Central Bank
would rather stabilize. It seems reasonable to assume that if large players
are involved (such as the government), the economy will eventually manage
to coordinate on the preferred of the two equilibrium.
4 Discussion
The model presented in this paper argues that the interplay between indi-
vidual borrower’s choices for liability composition and the optimal monetary
policy can lead to an outcome where liability dollarization is widespread.
That result was obtained under a policy maker that is fairly benign towards
foreign investors in the sense that it is not attempting to expropriate them
to the benefit of domestic borrowers. Instead, all that policy maker is trying
to do is to make dollar debt safer given that those contracts have already
been written.
While the model presented only predicts corner solutions, a richer model
with different types of shocks is likely to generate equilibria with internal
solutions for the share of dollarized liabilities (with that share depending on
how the monetary policy responds to different shocks). The preferences of
the central bank towards exchange rate or interest rate adjustments can play
averysignificant role in terms of focusing the market on a type of borrowing
and of monetary policy. In this sense, countries that exhibit original sin are
countries where the central bank cares more about exchange rate movements
than about interest rate movements. This is often the case in emerging mar-
kets, which can be explained by a number of reasons. In those countries, the
exchange rate pass-through into prices tends to be higher than in developed
ones, and the evidence for devaluations being expansionary is at best mixed.
Moreover, a low level of financial development weakens the link between in-
terest rates and aggregate demand, and as a result domestic interest rates
have a lower impact on inflation and employment. All these elements will
bias the choice towards more stable exchange rates at the expense of higher
volatility in interest rates. Finally, emerging markets have more imperfect
12
and incomplete financial markets, so the costs of bankruptcy are likely to be
larger. As a result, that bias towards exchange rate stability is amplified,
with the monetary authority being more willing to use interest rates aggres-
sively in order to stabilize the exchange rate in the presence of dollarized
liabilities. In fact, the volatility of interest rates tends to be much higher in
emerging markets than in developed economies.3
Why countries borrow the way they borrow? Why do countries differ in
their borrowing behavior? The most likely candidates would be countries
where the central bank has a preference for exchange rate stability and that
suffer from high volatility and bankruptcy costs.
Finally, while the model has focused on the borrower’s choice for liability
denomination, some of the insights can shed light into the related problem
of denomination of savings. If households are risk averse and their income
is not correlated with the shock to the expected future exchange rate, then
they would rather just save in whatever instrument makes the value of their
savings more stable. For example, if the debt composition is such that the
Central Bank stabilizes the exchange rate at the expense of the interest rate
and the parameters are such that the resulting distribution of peso savings is
riskier than that of dollar ones, the households would rather save in dollars.
If however, the realization of the shock to the expected future exchange rate
is correlated with household income (if for example it reflects productivity
shocks that affect the marginal product of labor and as a result the house-
hold’s labor income) then matters become more complicated. On the one
hand households dislike uncertainty on the return to their savings. But on
the other hand they want that return to covary negatively with their labor
income. As a result, they will be willing to hold some peso denominated
instruments, since those instruments do better than dollar denominated ones
over some range of “bad” realizations of the shock4. The share of their sav-
ings held in peso instruments will depend on the distribution of returns and
3Hausmann (2002) estimates the volatility of changes in 12-month real interest rates
in a sample of Latin American countries in the period 1994-1999. Using a sample that
excludes observations when inflation exceeded 40% the average volatility was 10.5%, while
the corresponding figure for the United States was only .9%
4The Central Bank will give up trying to stabilize the exchange rate for very bad
realizations of the shock, and dollar savings would provide higher returns than peso ones
in those states. But for intermediate levels of a bad shock, the exchange rate is stabilized
at the expense of an interest rate hike and peso savings will have a larger return than
dollar ones.
13
on how that shock to the expected future exchange rate covaries with their
income.
14
Figure 1: Exchange rate and interest rate if ζ=1/2and liabilities are
denominated in dollars.
0.8
z
0.764
0.779
0.793
z
e
1+i
15
Figure 2: Exchange rate and interest rate if ζis small and liabilities are
denominated in dollars.
0.8
z
0.7625
0.775
0.7875
0.8
0.8125
0.825
0.8375
0.85
0.8625
0.875
0.8875
0.9
0.9125
0.925
0.9375
0.95
0.9625
0.975
0.9875
1
1.0125
1.025
1.0375
1.05
1.0625
1.075
1.0875
1.1
1.1125
1.125
1.1375
1.15
1.1625
z
e
1+i
16
Figure 3: Exchange rate and interest rate if ζis small and liabilities are
denominated in pesos.
0.8
z
0.764
0.779
0.793
0.808
0.822
0.837
0.851
0.866
0.880
0.895
0.909
0.924
0.938
0.953
0.967
0.982
0.996
1.011
1.025
1.04
1.054
1.069
1.083
1.098
1.112
1.127
1.141
1.156
1.170
1.185
1.199
1.214
1.228
1.243
z
e
1+i
17
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