Policy, Planning, and Research
WORKING PAPERS
Debt and Internallonal Finance
International Economics Department
The World Bank
November 1988
WPS 132
Is the Discount on the
Secondary Market a Case
for LDC Debt Relief?
Daniel Cohen
A discount in the secondary market is a case for debt service
relief but not necessarily for a write-off. The author derives a
"maximum repayment" rescheduling program, which trades off
higher current investment for lower current debt service.
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Proposit.on 1: The "maximum repayment" wvile the market price of the debt is stabilized at
program the lenders would like to monitor a constant equilibrium price below par.
involve a fixed investment rate that is smaller
than the socially optimal rate and larger than the (Implication: Observing a discount on the
post-default rate. It involves a transfer of debt does not automatically warrant a write-off.
resources from the debtor that is a fixed fraction The discount implies the possibility of default,
of GDP - a fraction that is smaller than the cost but lenders should not write the debt off until the
of default. possibility materializes. But the service of the
debt shlould always be scaled down by its
Proposition 2: When the debt-to-GDP ratio market value rather than kept in line with its face
is above a floor value (h*), the lenders can value.)
capture the "maximum repayment" value (V*)
by fictitiously splitting the debt into performing Proposition 4: When the lenders reschedule
and nonperfonning components. Each period, the debt on a period-by-period basis, they induce
they should ask the borrower to service the the country to follow a growth pattem that
perfonming component of the debt only, and let exactly mimics the post-default path. The
the performing component grow at a rate equal lenders capture each period the penalty they
to the economy's expected growth rate. Mean- could impose on the defaulting country. As a
while, the nonperforming asset is automatically result, they get more on a period-by-period
capitalized at the riskless rate. When the actual basis, but less on average than under the "maxi-
growth rate of the economy is above (below) its mum repayment" schedule. Under such a ("time
expected level, the performing part of the debt is consistent") rescheduling strategy, a write-off
scaled up (down). When this "maximum and multiyear rescheduling may prove benefi-
repayment" rescheduling strategy is undertaken, cial, but the gains fall short of the strategy
the equilibrium market value of the debt is equal defined in Proposition 2.
to V*.
How relevant is the idea of "debt overhang"
Proposition 3: When the debt-to-GDP ratio (according to which the market value of the debt
is above the hreshold h*, the debt can be written may depend negatively upon its fact value)?
down to h* GDP without impairing the lender's Empirical evidence presented here indicates that,
return. If the write-off is repeated each time the at a 75 percent confidence level, 9 of 33 coun-
economy declines, and if the rescheduling is tries studied may suffer from a debt overhang
undertaken according to Proposition 2, the prblem. At a 90 percent confidence level, only
lenders capture the "maximum repayment" 4 of them may be affected by it.
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Is the Discount on the Secondary Market
a Case for LDC Debt Relief?
by
Daniel Cohen
Table of Contents
Introduction ............................................. 1
I. The Setup ............................................ 5
(a) Production ........................... . 5
(b) Preferences ........................... 7
(c) External Debt ........................... 7
II. The Optimal Growth Rate: The (Totally) Open Case
and the Post-Default Case ................................ 9
(a) The open economy case ............................ 9
(b) The post-default case ............................ 10
III. The "Maximum Repayment" Which can be Extracted
from an Indebted Country ................................ 11
IV. How to Implement the "Maximum Repayment" Scheme ...... 14
V. The "Debt Overhang" Problem Revisited ................ 17
VI. Empirical Evidence of the "Debt Overhang" Problem.... 20
Appendices ............................................... 24
References ............................................... 31
I gratefully thank Plutarchos Sakellaris for his research assis-
tance, Homi Kharas for his insights on the last section of the
paper, and the members of the Debt and International Finance
Division for their comments and their warm hospitality.
Introduction
In 1988, the price on the secondary market of LDC dJebt averaged 50
cents per dollar of face value. This figure is certainly an indication that
the lenders do not expect (on average) to be repaid the full value of their
outstanding claims on LDCs and indeed that they expect perhaps no more than
half the value of these claims to be serviced. 1/ Prom the observation of such
a discount in the secondary markets, can one go one step further and argue
that the debt should be written down in order to account for the discrepancy
between the face and the market value of the debt? It is this question that
this paper tries to shed some light on, both theoretically and empirically.
Theoretically, the rough answer is as follows. A discount in the
secondary market can be the effect of two distinct causes: one is that
past shocks may have impaired the capability of a debtor to service its
;ebt. Another cause is that future shocks may be expected to impair, when
they occur. .he servicing capacity of the country. Por instance, the fall in
the price of oil that occurred in 1986, if viewed as permanent, is a shock
which (at that time) certainly reduced the expected ability of Mexico to
service its external debt, and, to some extent, was translated into a fall of
the market price of Mexico's debt. On the other hand, the prospect of say, a
Middle-Eastern peace settlement, which brings the expectation of increased oil
1/ The secondary market is a thin market in which, until recently, swap
transactions have predominated. Hence secondary market prices may not
accurately reflect market expectations. For the purpose of the analysis here,
we assume that the secondary market price does reflect the expected value of
discounted future debt service.
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supplies, is also part of Mexico's debt value, but its implications for debt
relief are dramatically different. As I will indicate in the theoretical part
of this paper, only the shocks of the first kind--the "backward shocks"--are
open to debt relief., As for the others--"forward shocks"--writing off and
forgiveness is only optimal after the shocks occur but not beforehand.
Even though it may not always be a good thing to write off the debt
in order to account for the discrepancy between its face and its market value,
I will show that the service of the debt should always be scaled down by its
market value rather than kept in line with its face value.
Specifically, I will show that the optimal rescheduling of the debt
should proceed as follows. The lenders should split the debt into two
components: a performing and a non-performing part. They should act "as if"
the debt amounted to the performing component and scale how much money the
borrower should pay in debt service on that part only (while the non-
performing part is automatically capitalized at the riskless rate). The
performing component of the debt should reflect the market value of the debt
but it is important that the lenders calculate it themselves. If they were to
rely on the market estimate, the borrower would have an incentive to bring
down the market price through poor policies or through confrontations with
creditors. Nevertheless, at equilibrium, the lenders' and the market's
evaluation should coincide. 2/
On the other hand, the non-performing asset should not necessarily be
written down. Good outcomes can occur (or expected bad outcomes may fail to
materialize) and, conditionally on the good news, the size of the performing
2/ See the caveat in footnote 1.
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asset may be scaled up. In brief, the answer to the question in the title of
this paper is as follows: (1) Yes, a discount on the secondary market implies
that the service of the debt should be scaled down in line with its market
values (2) No, such a strategy does not necessarily imply that the face value
should be written down. A discount in the secondary market is a case for debt
service relief, not necessarily for a write-off.
Obviously, there may come a point in time when the non-performing
asset becomes so big as to warrant a write-ofi. However, if creditors operate
optimally (along the lines sketched above), the write-off does not modify the
market value of the debt; it simply raises the market price. This should come
as no surprise. If they behave optimally, the lenders should not lose by
having more nominal claims than less.
How does this result relate to the "debt overhang" idea, according to
which the market value may depend negatively upon the face value of the
debt? The link may come as follows. In order to achieve their first best
outcomep I will show that the lenders must reschedule the performing component
of the debt generously enough to allow for the country's investment needs and,
on average, they should let the performing asset grow along with CDP. While a
strategy along these lines is shown to be optimal for the lenders, it is not
however the case that such a strategy is "time-consistent" (as initially
defined by Calvo and Kydland and Prescott). It is not a strategy which can be
implmented on a period-by-period basis.
Indeed, if dealt with on a period-by-period basis (without setting
out the rules of future reschedulings) I will show that a self-fulfulling
downward spiral is bound to appear: one in which the fear that the lender
will not acknowledge the investment needs of the debtor immediately raises the
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cost of capital in the debtor country, reducing investment immediately and
making it ex post-optimal for the lenders to tighten their rescheduling
strategy.
In the previous literature on the "debt overhang" problem (originated
by and Krugman (1987) Sachs (1988)), only these second best equilibria have
been examined (for the technical reason that most of the models examined in
this literature are two-period models in which the commitment issue could not
be addressed). In such a case, when the lenders fail to commit themselves to
their first best strategy, a write-off may indeed prove beneficial, by helping
the lenders to commit themselves to let the country invest. In no case,
however, can it help them get the first best repayment stream.
Whether the "debt overhang" problem is empirically relevant has
remained an open question (see Claessens (1988) for a review.). Following an
idea in Krugman (1988), one may check whether the elasticity of the market
price of the debt with respect to its face value is larger than one in
absolute value. In the last section of the paper, I will indicate that, at
the 75 per nt degree of confidence, 9 countries (in a sample of 33) may
suffer (accord.ug to Krugman's test) from a debt overhang problem, while at
the 90 percent degree of confidence, only 4 of them may be affected by it.
Before closing this introduction, I should emphasize that the case
for a write-off which is explored in this paper only rests upon the question
of knowing whether the private lenders may find it in their best interest to
do so. This is obviously only a narrow way of dealing with the overall
question. A write-off may prove beneficial to the lending countries as a
whole, when all the relevant spillovers are taken into account, and not to the
private lenders themselves. (For such a broader viewpoint see Dornbusch
(1988).) In any case, if the industrialized countries wanted to help the
debtors (for whatever reason: altraism or educated selfishness), then a
partial forgiveness of the debt may be . crucial preliminary step that the
industrialized countries would want to encourage--perhaps through regulatory
or tax measures--on the part of private lenders. Without partial forgiveness,
the lenders may tend to be the main beneficiaries of the public funds poured
into the debtor country.) All these crucial issues are outside the scope of
this paper but should be kept in mind before any policy conclusion is drawn.
Section 1 spells out the model. It is a stochastic version of the
model examined in Cohen and Sachs (1986). Section 2 calculates the socially
efficient and the post-default growth rates of the economy. Section 3 shows
that the lenders, if they were to monitor the investment and the consumption
strategy of the borrower (in order to maximize their return) would choose a
lower investment strategy than the socially efficient one. Section 4 shows
how an optimal rescheduling (based upon the distinction between performing and
non-performing assets) can achieve the equilibrium described in Section 3.
Section 5 shows the dynamic inconsistency of the optimal strategy spelled out
in Section 4, and shows the link with the "debt overhang" literature.
Section 6 investigates the empirical relevance of the "debt overhang."
I. The Setup
(a) Production
I will consider a one-good economy, in which the same good can be
used for export, consumption or investment. In each period, the available
stock of capital is a pre-determined variable. The production, Qt' is a
linear function of existing capital:
-6-
(1) Qt ' Kt
Capital can be increased through investment, and investment itself is a costly
process. Let us assume that an increase It of capital costs Jt:
(2) Jt I ( 2 t
t t 2 Q
The investment decision It, while taken at time t, increases the capital stock
at time t + 1, according a stochastic law of motion:
(3) K t+l [ Kt(l-d) 4 It (l*+t,i)
in which d is the rate of depreciation of installed capital and 3t is an iid
stochastic variable which is worth:
(4) et= u with probability p
3,= v with probability 1-p.
Here, the investment decision It must be taken before its productivity (o t,)
is known. One can think of Et as a stochastic shock which exogenieously
increases (or decreases) the productivity of installed capital stock. I shall
refer to the event of probability p (where the rate of growth of the
productivity of capital is u) as of the "good state" and to the event of
probability 1 - p (when productivity experiences a slower, possibly negative,
growth rate) as the "bad state." I will call 8 the expected rate of growth of
the productivity of capital:
(1+0) - p (1+u) + (1-p) (liv).
(See Gennotte, Kharas and Sadeq (1987) for a model with a similar structure.)
(b) Preferences
I will assume that the country is managed by a social planner who can
impose on the country an investment and consumption decision. The planner's
preferences are represented by an intertemporal expected utility function:
(5) UO a E 0 B u (C)
0 0 t
in which C. is the aggregate consumption of the country at time t; and
u (C) - 1 Cy 1 y < 1 and Y * o, or u (C) Log C when y = �.
(c) External Debt
In order to focus on the question raised in the title, I will simply
assume that the country inherits an initial debt Do (assumed to be short-term)
which is large enough to be quoted below par on secondary markets; and I will
investigate the optimal rescheduling strategy for the lenders. It is not
difficult to show how the framework which is used here could imply that the
optimal borrowing strategy does involve such a risk. But some technical
issues (such as that of calculating the optimal maturity of the debt) would
take this paper too far afield. (See e.g. Cohen (1988) for an analysis, in a
- 8 -
three period model, of the difficulties at hand.)
Following the Eaton and Gersovitz (1981) approach and my earlier work
with Sachs, I shall assume that the country always has the ability to
repudiate its stock of outstanding debt while the lenders can retaliate and
impose on the borrower the following two sanctions:
(a) A defaulting country is forced to financial autarky forever
after it has defaulted.
(b) The productivity of capital of the defaulting country is reduced
by a factor A so that the post-default technology of production
is:
(6) Q (1 - A) t
In all that follows, I will assume-that the len.ders are risk-neutral, act
competitively, and have access to a riskless rate of interest which stays
constant all along. I will assume that 8 is low enough to insure that the
country will be constrained on the borrowing side. Furthermore, I will leave
aside all bargaining issues and assume that the lenders can credibly make (at
each point in time, but not necessarily for the entire future) a take-it-or-
leave-it offer to the debtor.
- 9 -
II. The Optimal Grovth Rate: The (Totally) Open Case and the Post-Default
Case
In this section, I would like to calculate the optimal investment
strategy in the two extreme cases when the country has a free access to the
world financial markets en the one hand and when it is forced to a post-
default path on the other hand.
(a) The open economy case
Assume in this sub-section that A I in equation (6) above, i.e.
assume that the country cannot repudiate its external debt (because it is too
costly). With that assumption, the model boils down to the standard Fisherian
case where the investment decision can be separated from the consumption
decision. I will simply solve, here, the optimal investment decision. The
country wants to maximize its productive wealth when the return on its
investment is taken to be the world riskleas rate of interest.
Kathematically, this amounts to solving the following program:
(7) Wo - Max Eo { 1 [ IK - I (1.1i It)
(I)t > � (I + r) t|Kt I(1 20t)
The solution to this program is given in Appendix 1. Given the linearities in
the model, Wo is shown to be a linear function of initial output:
(8) w0 w QO
and is obtained by picking up a fixed investment rate:
- 10 -
i It
(9) x
associated to a fixed rate of gross investment
(iO) y t i +
(All the technical conditions for the equilibrium to exist are spelled out in
appendix). The equilibrium growth rate of the economy oscillates. It is high
in the good state of nature [ (l*u) 'l4x-d) ] and low in the bad state of
nature t (l+v) (l-d) ].
From here on, I will refer to this equilibrium as the socially efficient
equilibrium.
(b) The post-default case
Assume now, as another extreme case, that the country has defaulted
upon its external debt. In that case, the social planner must choose its
investment decision so as to allocate consumption optimally over time.
Kathematically, the planner must solve the following programs
()u d(Qo) Mix I u [Qo 11 - (1 + + O)
+ B p Ud[QO(lOu) (l*x-d)J + 8(l-p) Ud [QO(lIv)(l*x-d) }
in which Ud is the utility level that the planner can reach when the available
output is QO at the initial time.
- 11 -
The solution is spelled out in Appendix I where it is shown that the
solution Ud (QO) can be written:
(12) Ud (Q) C QY if y * o and I log Q+ CO if y - o
Yo ~~~~1-s
in which CY is a constant. The solution is also shown to involve a fixed
investment rate:
It
which is smaller than the socially efficient investment rate (obtained in the
open economy case).
III. The "Maximum Repayment" Which Can Be Extracted from an Indebted Country
In this section, I will consider the following simple problem. I
will assume that the lenders can monitor both the invPstment and the repayment
strategy of the debtor in such a way as to maximize the value of the transfers
made abroad by the country. While the borrower will be assumed to give up its
sovereignty over its consumption and investment decision, it will nevertheless
keep its sovereignty over the matter of defaulting: at any point in time, the
borrower will stay free to break the lenders' rule and to follow afterwards
the post-default path defined by equation (12). In other words, the rules of
the game in this section are as follows: the lenders monitor the debtor's
economy so as to maximize the value of the transfers channelled abroad by the
debtor, subject to the constraint that the program is never expected (neither
today nor later on) to be dominated by a post-default path. Clearly, under
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this set of hypotheses, the value of the transfers channelled abroad by the
debtor will provide an upper bound to the market value of any debt accumulated
by the country.
Formally, the problem can be written as follows. Call Pt the amount
of transfers abroad made by the debtor, Yt the gross investment rate
(inclusive of the cost of installation) achieved by the country, and Ct the
consumption left to the country. One has:
Ct Qt (I-y) -p
Call:
(14) Ut E t I !St atu (C")}
the level of utility which the lenders' program is expected to deliver to the
country. With this notation, the program that the lenders must solve is as
follows.
(15) Maximize E 0 o 0 t
subject to Ut ! Ud (Qd) for all t.
in which Ud (Qt) is the post-default level of utility (as defined in equation
(12)).
This problem is solved in Appendix 2. Given the many linearities
built in this model, the problem boils down to finding a fixed (gross)
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investment rate y and a fixed debt service ration Pt/Qt which solves the
problem (15). The solution is shown to involve an investment rate which lies
between the socially efficient rate and the post-default rate. One can state:
Proposition 1: The "maximum repayment" program which the lenders would like
to monitor involves a fixed investment rate which is smaller
than the socially optimum one and larger than the post-
default one. It involves a transfer of resources from the
debtor which is a fixed fraction of GDP, a fraction which is
smaller than the cost of default.
From Proposition 1, we therefore see that the idea according to which
the debt may have a "pro-incentive" effect is not granted in the context of
the exercise which is carried through here. (For another approach see also
Corden (1988) or Helpman (1988).) Even when it is the banks themselves that
design the investment and consumption policy of the borrower, they will choose
a lower investment rate than is socially desirable. The reason is that the
banks must take care to avoid a situation in which the country may one day
choose to default. A too rapid path of capital accumulation, even while
socially desirable, will raise the post-default utility of the country and, if
not carefully balanced, can be counterproductive to the banks.
Prom here on, I will call V the "maximum repayment" that the lenders
can expect to receive from the debtor. Due to the linearities
involved, Vt can be written as a linear function of current output:
(16) V* = Z Q
- 14 -
In appendix 2, I also show that the fraction of CDP which is channelled abroad
can be written:
(17) b = z [(1 + r) - (1 + 0) (1 + x- d)J
in which z is the net investment rate that is described in Proposition 1.
IV. How to Implement the "Maximum Repayment" Scheme
I will now indicate how the lenders can indeed capture the "maximum
repayunt" even when they do not monitor the investment and consumption choice
of the borrower. Consider the following decomposition of the debt:
(18) Dta V + R=
in which Dt is the face value of the debt, Vt is the maximum value calculated
above, and Rt is the residual. Assume that the lenders fictitiously regard Rt
as a non-performing asset and only insist on V being serviced (while Rt is
automatically capitalized). Furthermore, assume that, each period, they ask
the borrower to transfer an amount Pt which is the amount necessary to keep
v* growing at the expected rate of growth of the economy.
Under these assumptions Pt must solve:
- 15 -
(19) V* = (+r) V P = (1+0) (1+x-d) V
t+l t t t
in which (1+) (l+x-d) = p (l+u) (l+x-d) * (l-p) (l+u) (l+x-d) is the expected
growth rate of the economy when the investment rate x has been selected by the
debtor. Pt is then given by:
Pt = l(lr) - (1+0) (lex-d)] Vt
Pt = [ (l+r) - (1+e) (lex-d) ] Q
t ~~~~~~~~~~~t
and the optimum investment decision chosen by the country will coincide with
the "maximum repayment" strategy designed in equation (17). (See Appendix 2,
And Portes (1987) for a suggestion in the same spirit.)
Provided that the non-performing asset is initially large enough,
which amounts to assuming that D/Q > h with h a given threshold, this
scheme can be shown to be repeated for ever and indeed deliver the "maximum
repayment" scheme (see Appendix 2 for further details). If D/Q is below h ,
then the non-performing asset should be charged a larger interest rate until
the face value of the debt reaches the h Q ceiling.
It is crucial to note that this fictitious decomposition of the debt
into a performing and a non-performing part is updated each period. Indeed,
along equation (19) V* is only teft to grow at a rate (1+0) (l+s-d) which is
the average growth rate of the economy. If things go well the actual growth
rate will be larger and Vt+l must be scaled up; conversely, Vt+l will be
scaled down if the bad state occurs.
The second crucial remark to make is the following: the performing
- 16 -
asset is not calculated from the observation of the market value of the debt
but from the theoretical computation of the maximum repayment scheme. Even
though they do coincide at the equilibrium, it is crucial that the lenders do
not let Pt depend upon the observed market value of Dt. Indeed, if they were
to do so, they would ask to be repaid:
Pt = 8 (X) [ (1*r) - (1+8) (1+x-d) t
and tL. country would be induced to bring down the market value of the debt.
Theme results can be summarized as follows:
Proposition 2: When the debt to GDP ratio is above a floor value h*, the
lenders can capture the "maximum repayment"' value V* by
proceeding as follows. They should fictitiously split the
debt into a performing and a non-performing component, the
performing component being equal to V*. Each period, they
should ask the borrower to service the performing component
of the debt only, and let the performing component grow at a
rate equal to the expected growth rate of the economy.
Meanwhile the non-performing asset is automatically
capitalized at the riskless rate. When the actual growth
rate of the economy is above (below) its expected level, the
performing part of the debt is scaled up (scaled down). When
this rescheduling strategy is undertaken, the equilibrium
market value of the debt is equal to V .
- 17 -
Now obviously, as time passes, the size of the non-performing asset
grows relative to the performing one, and some write-off of the debt may
become possible without impairiag the lenders' ability to capture V*t. One
can actually show:
Proposition 3: When the debt-to-GDP ratio is above the threshold h*, the
debt can be written-down to h* CDP without impairing the
lenders' return. If the write-off is repeated each time the
economy goes into the bad state and if the rescheduling is
undertaken according to Proposition 2, the lenders capture
the "maximum repayment" while the market price of the debt is
stabilized at a constant equilibrium price below par.
One important implication of Proposition 3 is that it is not enough
to observe a discount on the debt to warrant a write-off. The intuition is
that hinted at in the introduction: the discount on the debt takes into
account the possibility that the economy may go into a bad state. But lenders
have no reason to write-off the debt before that prediction materializes. It
is only in the deterministic case when u = v that the optimal strategy is
indeed to write-off the debt "once and for all" (in order to erase whatever
backward shocks may have lifted the debt-to-CDP ratio above h*) and let the
debt be quoted at par.
V. The "Debt Overhang" Problem Revisited
In view of Proposition 2, it appears that the face value of the debt
is of little importance in assessing the optimal rescheduling strategy of the
- 18 -
debt. This should come as no surprise: when they behave optimally, lenders
get as much as the country can transfer and more nominal claim cannot imply
less actual payments. (See also Bulow and Rogoff (1988).) This result,
however, contradicts the "debt overhang" argument according to which too large
a nominal claim may excessively discourage investment and reduce the market
value of the debt. I would now like to indicate how these two conflicting
views can be reconciled.
A key feature of the optimal rescheduling strategy described in
Proposition 2 is that lenders should let the performing asset grow at the
expected growth rate of this economy. As apparent from equation (17) this
implies that the service of the debt is negatively correlated with the
investment decision of the borrower. Even though such behavior is in the
lenders' self-interest, I now want to show that this is not a "time-
consistent" decision, that is: it is a decision which is an optimal one to
take only if the lenders can commit themselves (in whatever way: sophisticated
contracting or a built-in reputation) to implement it later on. In order to
see why such a commitment is necessary, assume instead that the lenders
operate on a period-by-period basis and simply reschedule the debt each period
to the best of their ability, taking for granted that they will do the same
(and will be expected to do so) later on. Such a policy can be characterized
as a "time-consistent" policy: it -is one which is found to be optimal to
implement today, when it is expected to be implemented in the future. Since
the work of Kydland and Prescott (1977) and Calvo (1978), it is well known
that such a policy maybe intertemporally sub-optimal (even though it is
pointwise optimal). Let us see what the outcome of such a "time-consistent"
rescheduling strategy would be.
- 19 -
As shown in Cohen and Michel (1988) calculating a time-consistent
policy simply amounts to finding a feed-back decision rule which, here, can be
writtent
( 19 ) Pt , bQt
in which b is the largest amount that the lenders can ask at time t #hen it is
expected that future payments will be set according to another rule:
(20) Pt+ b~ Qt.s
which they take as given. A time consistent strategy is one for which, at the
equilibrium, b - b.
The equilibrium is calculated in Appendix 3. It is shown that the equilibrium
growth rate is nothing else but the post-defauLt path and that b ) X In
other words the "time-consistent" policy is simply one in which the lenders
take every period the costs that the borrower would incur by defaulting and,
as a result, their rescheduling strategy simply mimics the post-default path
that the country could follow on its own.
As apparent from equation (19) a time consistent rescheduling
strategy act as a tax on outputs the borrower expects that the lenders will
ask for as much as it can pay and this is an amount which, it can foresee,
will be proportional to how much output it can generate. These expectations
increase the shadow cost of capital in the debtor country and reduce
investment immediately, making it optimal for the lenders to do what they are
expected to: disregard the incentive to invest and ask for as much as they
can.
- 20 -
It is this downward spiral that most people (I think) have in mind
when discussing the debt overhang problem: debt acts as a tax which
inefficiently discourages investment and less annual payment from the debtor
would imply more overall income to the lenders. Under these circumstances, a
write-off may help the lenders. In fact, a write-off cum a multi-year
rescheduling can perform even better inasmuch as it helps the lenders commit
theuselves to put an explicit ceiling on how much money they will ask for each
period to come. It should be clear, however, that neither a write-off nor a
multiyear rescheduling can help the lenders get the first best, unless, the
rescheduling is made contingent upon the investment decision of the borrower.
(See Appendix 4 for a formal proof of these statements).
To summarize, one can state:
Propomition 4: When the lenders reschedule the debt on a period-by-period
basis, they induce the country to follow a growth pattern
which exactly mimics the post-default path. The lenders
capture each period the penalty that they could impose on the
defaulting country. As a result they get more on a period-
by-period basis, but less on average than under the "maximum
repayment" scheme. Under such a ("time-consistent")
rescheduling strategy, a write-off and a multi-year
rescheduling may prove beneficial, but the gains necessarily
fall short of the optimal strategy defined in Proposition 2.
VI. Empirical Relevance of the "Debt Overhang" Problem
Let us now investigate whether the "debt-overhang" problem is or not
empirically relevant.
- 21 -
Krugman (1987) has suggested that we regard the "debt-overhang" as a
"Debt Laffer Curve" problem, the question at hand being: does more nominal
debt imply a lower market value for this debt? A test of the "debt overhang,"
according to this formulation, therefore amounts to deciding whether the
elasticity of the market price of the debt with respect to its face value is
strictly larger than one (in absolute value). Certainly if this elasticity is
larger than one, then one can make the case that the lenders operate
inefficiently. However, an elasticity equal to or smaller than one is not in
itself sufficient to accept the hypothesis that the lenders reschedule the
debt efficiently. In this section, we shall stick to Krugman's test, but
certainly more work is needed in order to investigate the efficiency of the
rescheduling process which has been undertaken since 1982.
Previous attempts to measure the elasticity of the price of the debt
with respect to its nominal value systematically found a low estimate. A
study by Purcell and Orlanki, following a previous estimate by Sachs and
Huizinga, reported an elasticity of 0.34. We have estimated an equation,
representative of these earlier studies, as follows:
(21) Log p = 5.06 - 0.653 log D/X - 2.231 A/D - 1.016 R/D
(0.152) (0.603) (0.373)
- 0.274 Dummy 1987.12
(0.132)
B2 * 0.560 pooled equations for 1986.12 and 1987.12 data; 60 degrees of
freedom. (Standard errors in parenthesis).
p: price of the debt (cents on the dollar).
D: debt; X: exports; A: arrears; RB amount of rescheduling
since 1982.
- 22 -
From this equation, one would tend to reject at the 95 percent level
of confidence that the elasticity of the debt was larger than one. Before
comnenting on the insufficiency of such an equation, it is interesting to
report that the price of the debt seems to be very poorly correlated to
macroeconomic data related to the country. For instance, the most important
of these macroeconomic data (one would guess), such as the non-interest
current account or the domestic inflation rate, never appeared to be
significantly correlated with the price. On the other hand, arrears or
rescheduling data (as we can see from equation (21)) alurays perform extremely
well.
These rasults are summarized in diagrams 1 to 3. They tend to
indicate that the market is extremely sensitive to the "punctuality" of
payments and pay little attention to overall macroeconomic performance.
Finally, one also sees from equation (21) that a dummy separating the 1986 and
1987 data appears to be significant. This may be a reflection of Citibank's
decision to build up $3 billion of reserves against developing country
exposure, a move which significantly influenced the market.
Despite its appeal and its simplicity, an equation such as (21) is
extremely misleading. First, it leads us to reject the hypothesis that the
elasticity of the price with respect to debt is larger than one for the entire
sample. But it may very well be the case that only a sub-group of countries
was hit by the debt-overhang problem. Running, for instance, the same
regression for the sub-sample of countries for which the debt-to-export ratio
is larger than 3 (a sub-sample of 16 countries) would yield a larger
elasticity, which we estimated to be at 1.183 (with a standard error of
0.339). Second, and perhaps more importantly, an equation such as (21) takes
- 23 -
the arrears and the rescheduling variables as exogeneous, while these
variables obviously depend upon debt and perhaps upon the price itself. In
order to overcome these two difficulties (to which one should also add a more
technical one which is that the price being smaller than one hundred, log p
cannot be normally distributad), we have estimated a reduced form equation in
which the dependent variable has the logistic form log (p /100 - p), so as to
let the elasticity depend upon the level of the price. The result comes as
follows:
(22) Log P u 2.152 - 1.509 log D/X
lOO-P (0.318) (0.305)
-0.048 X growth - 0.583 Dummy 87.12
(0.024) (0.288)
*2 u 0.389; pooled equation for 1986.12 and 1987.12 data;
60 degrees of freedom; X growth: rate of growth of
exports.
According to this equation the elasticity of the pricw with respect
to debt (100 - p) is 1.509 (with a standard error of 0.305). This indicates
that the debt overhang problem could not be rejected at the 95 percent level
degree of confidence for these countries in the sample for which the price was
almost zero (such as Sudan). More generally, Table 1 indicates the countries
for which the debt-overhang problem could not be rejected at various degrees
of confidence. At the 90 percent level of confidence, only 4 countr-es pass
the test.
- 24 -
Table 1
Countries with a Potential Debt
Overhang Problem
(as of 1987.12)
At the 501 level of Argentina (34)
confidence: p < 34 Jamaica (33)
1 Nigeria (29)
At the 75X level of Dominican Republic (23)
confidence: p c 23 Congo (23)
Zaire (19)
Zambia (17)
Costa Rica (15)
At the 90X level of Bolivia (11)
confidence: p ! 11 Peru (7)
Nicaragua (4)
At the 951 level of
confidence: p - o Sudan (2)
(The numbers in parenthesis are secondary market prices in cents
per dollar).
Appendix 1. Optimal Growth in the (Totally) Open and In the Post-Default
Economy Cases.
A) The open economy case
Prom equation (7), doubling QO would also double 0 so that one can look for
* such as in equation (8). w is the solution to the following Bellman
equation underlying the definition of W0 in equation (7):
(Al.l) c Mnax { 1 - x (I + x)
+ @ r p(l + u) + (1 - p)(l * v)+ (1 * x - d))
- 25 -
The equilibrium value of x is
;a1 (1*o --
(A1.2) s * I(i+@)+r , with 1 + 0 a p (lou) + (1-p)(l+v).
We shall assume * to be positive.
Equation (Al.I) yields that x is a solution to:
(Al.3) 1 20
(A1.3) X x2 _ X Ir @_ + d) + O(l - t _ 0+e
The solution that is socially efficient is:
(AY.4)( + d) [I - VI (1-+- --i ) I
which exists and is positive if:
(Aoi5) in2(1 rc s ) /(hl as- s t o); l+ o
a condition which we shall assume to hold.
- 26 -
B) The Post-Default Case
Let us "guess" that the solution to equation (11) can indeed be
written:
(Al.6) Ud (QO) a C Q�
Then the "guess" will prove to be the right one if
(Al.7)
C - Miz { [1-k-yly+ B p 1(14u) (lex-d))YCy+ B [ (l-p) (lv) (l+x-d) IYCYi
By the envelope theorem, the derivative of the right - hand side is
maller than one when B is small enough to induce the country to be in the
borrowing side.
Appendix 2: The "Maximum Repayment Scheme"
Because of the linear structure of the model, one has to find b and
s such that
-b Q*
b, x (1c+ r) *0
subject to to (IBt u [(1 - b - y ) Q )> Ud(Qo)
t = o
(The statinarity of the problem implies that this inequality, if it holds at
time o, will also hold at later times).
- 27 -
CSll w (x) the solution to:
(A2.1) a c(x)Qo Q * ,.,t 3lr)-lO)(1z^d) QO
when the investment rate is x. The problem at hand is therefore simply that
of finding:
(A2.2) z* Max b w(x)
b;x
subject to:
A2.3) 80 0 t (1 - b - y)Q Ud (Qo)
Let
A2.4) ut- B E I 8o 4 Ui)I
(A2.3) can be written:
(A2.5) �t p(I - b - y)Y (I + s - d) ty > U d(Qg
By duality, maximizing z* in (A2.2) subject to (A2.5) amounts to finding
z which is a solution to:
ud ( QO) - Hex ! ut( 1 - Z* y)Y (I + s - ) tY
- 28 -
From the definition of w (x) - (1 + r) - (1 + e) (1 + x - d) in equation
(A2.1), this amounts to asking the country to transfer:
Pt abQt a Z (1 + r) - (1 + 8) (1 + s - d) Q
in which x is freely chosen by the country so as to maximize its utility.
Since Ud (Qo) = 6 Pt (1 - A - Yd)Y(l d5
one can see from (A2.5) that
bt< A and x > x
The investment rate is larger under the optimal scheme than under default.
Agpendix 3: The Time Consistent Path
For equation (17), the lenders want to induce the country to repay in
each period:
(A3.1) Pt m Z ((l + r) -(1 + )(1 + * - d)] Qt
in which z and x are the optimal choice defined in Proposition 1. Let us
show that Proposition 2 solves this problem when h* is defined as
(A3.2) h* = ^* (1 + r) - (1 * x - d) (1 + e)
(1 + r) - (1 + x - d) (1 + u)
- 29 -
and when the price of the debt is
(A3.3) q* ,( + r) -(I + zs M )l + u)
(1 + r) - (1 + x - d)(l + 0)
If (A3.1) and (A3.3) are satisfied, the market value of the debt is Z Qt (the
maximum value) and Proposition 2 indicates that the borrower should repay Pt
so that, when measured in market terms, the debt grows at the rate
(1 + ) (1 + x - d). This implies that Pt must be such that
q D5 1 (I + r) q Dt Pt a q Dt (1 + )( + X - d)
so that
? *[(l + r) - (1 + e)( + *x - d)| Qt
Given this rule of the game, the country must:
Naxi idze ' ( 1 + x J)t Y { (1 - z ((1 + r) - (1 + x - d)] - y P
0
which is exactly the problem at hand in Appendix 2.
Appendix 4: MYRAs and Write-Offs a
In order to see how a multi-year rescheduling agreement associated
with a write-off can help time-consistent lenders, let us restrict the
analysis to the deterministic case when u = v. Assume that the lenders are
trapped into the tim-consistent strategy by which they are expected to (and
indeed do) levy Pt-b t each period. Assume that they reschedule the debt on
- 30 -
a long run basis so as to let each period's payment falling due equal:
t
with g being some exogenous growth rate. The country now must:
(A-l) Max B u [Qt- J Pt]
and one sees that the disincentive to growth is eliminated (inasmuch as the
borrwer takes Pt as not contingent upon Qt). Clearly there exists a value of
Po and g for which the equilibrium growth rate of the economy coincides with g
and for which the borrower is exactly indifferent between servicing the debt
and defaulting. For this equilibrium one finds that lenders raise the value
of their claim above the time-consistent pay-off (to the extent that the
disencentive to grow has been eliminated) but fall short of the first best
strategy (to the extent that the incentive to grow has not been optimally
designed). In order to require (A4.1), the lenders must therefore write off
part of the debt below the "maximum repayment" value.
- 31 -
References
Bulow, J. and K. Rogoff "A Constant Reconstracting Model of Sovereign
Debt", forthcoming Journal of International
Economics.
Calvo, G. (1988) On the Time Consistences of Optimal Policy in a
Monetary Economy," Econmetrica no. 6.
Claeesens, S. (1988) "The Debt-Laffer Curve: Some Estimates" World
Bank, mimeo.
Cohen, D. (1988) "The Inefficiency of Private Lending to Sovereign
States" forthcoming in a volume in honor of E.
Malin+ajid edited by Crandmont, Laroque and
Monfort.
Cohen, D. and P. Michel (1988) "How Should Control Theory be used to Calculate a
Time Consistent Policy" Review of Economic
Studies, March.
Cohen, D. and J. Sachs (1986) "Growth and External Debt Under Risk of Debt
Repudiation," European Economic Review, vol 30,
pp. 529-550.
Corden, Max (1988) "Debt Relief and Adjustment Incentives: A
Theoretical Exploration," Mimeo, IMF.
Dooley, M. (1987) "Market Discounts and the Valuation of
Alternative Structures for External Debt," IMP,
Mimeo.
Dornbusch, R. "Our LDC Debts," in The United States and the
World EconomY edited by M. Feldstein Universy
of Chicago Press.
Eaton, J. and M. Cersovitz "Debt with Potential Repudiation: Theoretical
(1981) and Empirical Analysis" Review of Economic
Studies, March.
Cenotte, C., H. Kharas "A Valuation Model for Developing - Country Debt
and S. Sadeq (1987) with Endogenous Rescheduling," World Bank
Egonomic Review, January.
Helpman, E. (1988) "Debt Relief: Incentives and Welfare," Mimeo.
Kruguan, Paul (1987) "Financing vs Forgiving a Debt Overhang: Some
Analytical Rates," Mimeo, MIT.
Kydland, P. and E. Prescott "Rules rather than Discretion: The Inconsistency
(1977) of Optimal Plans," Journal of Political Economy,
no 3.
- 32 -
Portes, R. "Debt and the Market," Centre for Economic Policy
Research, London.
Purcell, J. and D. Orlanki "Developing Countries Loans: A New Valuation Model
(1968) for Secondary Market Trading," June. Corporate
Bond Research, Solomon Brother Inc.
SAchs, Jeffrey (1988) "The Debt Overhang of Developing Countries," in de
Macedo and Findlay (ed.), Debt, Growth and
Stabilization: Essays in Memory of Carlos Diaz
Alejandro. Oxford, Blackwell, 1988.
Sachs, J. and H. Huizinga (1987) "US Comercial Banks and the Developing - Country
Debt Crisis," Brooking Papers on Economic Activity,
no 2.
PPR Working Paper Series
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1988-89 edition My T. Vu October 1988 S. Ainsworth
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