A model of strategic default of sovereign debt

Journal of Economic Dynamics and Control 11 (1987) 483-498. North-Holland

A MODEL OF STRATEGIC DEFAULT OF SOVEREIGN DEBT

Nalin K U L A T I L A K A and Alan J. M A R C U S *

Boston University, Boston, MA 02215, USA

Received May 1986, final version received March 1987

This paper presents a continuous-time stochastic model to study the timing decision of strategic default of sovereign debt. The debtor country precommits to an investment plan, finances part of it with a foreign loan, and maximizes the present value (PV) of utility from an infinite stream of consumption. Default risk arises from uncertainty surrounding the evolution of GDP over time. As the debt ratio increases, it introduces an increasing drag on the growth of GDP and also increases the risk premium on the loans, At each moment the nation must decide whether to service the loan over the next period (infinitesimally small) or to default, thereby gaining the PV of the loan, but correspondingly suffering a default penalty, and forgoing the option to default in the future. This choice is cast as a first-passage problem: default occurs (if and) when the PV of consumption under default first exceeds the PV given continuance of debt service. The actual values of the debt ratio for several debt-ridden countries is found to be close to the theoretically derived critical level. This framework also enables the study of austerity programs, rescheduling and other policy alternatives.

I. Introduction

The growing magni tude of foreign debt of L D C s and, more important ly, the p ropor t i on of debtor ' s export earnings devoted to servicing this debt in recent years has raised considerable concern in the international financial communi - ty. Table 1 lists various measures of the debt burden for several of the countr ies with the most severe debt obligations. Perhaps the most striking result of this paper is that the debt ratios of several countries are quite close to our theoret ical ly derived ratios at which it is opt imal to default.

In much of the previous analysis of potential debt repudiation, researchers have focused on high debt-service to export-earnings (or G D P ) ratio as the dr iving force towards default. In effect, the cause of a default is a type of shor t - run l iquidity problem: al though the debtor count ry might over time be able to repay its debt, it currently is unable to meet even accruing interest p a y m e n t s wi thout domestic austerity programs of unacceptable severity. The mos t f requent ly discussed solutions to the debt crisis a t tempt to reduce current debt-service obligations, for example, via a rescheduling that replaces short- term loans with loans of longer duration.

*We thank Thierry Bollier, Alex Kane, and David McClain for discussions and comments on an earlier draft. Any remaining errors are our own.

0165-1889/87/$3.50©1987, Elsevier Science Publishers B.V. (North-Holland)

484 N. Kulatilaka and A. J. Marcus, Strategic default of sovereign debt

Table 1

Debt ratios, a

Country

Foreign Debt service Debt/f low Debt/s tock debt to exports GDP GDP

(billions $) (percent) (percent) (percent)

Argentina 45.3 58.1 70.6 8.5 Brazil 93.1 51.7 41.1 4.9 Chile 18.6 55.9 89.1 10.7 France 94.5 9.6 18.3 2.2 Indonesia 29.5 17.3 37.3 4.5 Israel 29.3 32.7 116.1 13.9 Ivory Coast 9.4 44.5 116.9 13.9 Mexico 9.8 56.7 60.5 7.3 Nigeria 17.0 21.7 24.0 2.9 Peru 12.5 31.6 77.2 9.5 Phillipines 26.4 31.6 77.2 9.3 Poland 27 80.8 29.3 3.5 South Korea 40.1 19.1 53.1 6.4 Turkey 23.9 31.8 44.4 5.3 Yugoslavia 19.5 28.0 41.6 5.0

~Source: 'External debt of developing countries: 1983 survey' (OECD, Paris).

In a series of recent papers, Eaton and Gersovitz (1981a, b) and Sachs and Cohen (1982) have studied the possibility of default for reasons other than liquidity considerations. They model a country which uses international lending markets to offset fluctuations in stochastic domestic income. The country's objective is to maximize the expected utility derived from two or three periods of consumption. Upon repudiation of debt the country faces a permanent embargo on future international loans, and in addition, incurs a penalty stemming from expulsion from world capital markets, trade sanctions, and/or the cutoff of aid. However, for sufficiently indebted nations, repudiation still might be welfare-maximizing. These models thus introduce and analyze the possibility of strategic or voluntary default.

The most recent model employing this framework is presented by Sachs and Cohen (1982), who construct a model of international lending and macroeco- nomic equilibria in debtor countries. They find several interesting implications of default risk. Among these are an incentive for pre-commitment to an investment program, if such pre-commitment is feasible, and an upward-slop- ing supply schedule of loans until a loan ceiling is reached.

In this paper we extend their analysis in a continuous-time framework in order to explicitly study the timing decision of strategic default. In contrast to the above models, this framework allows us to analyze the optimal timing of the default decision as a first-passage problem. Default occurs (if and) when the present value of consumption under default first exceeds the present value giyen continuance of debt service. The stochastic dynamics (given by a partial

N. Kulatilaka and A. J. Marcus, Strategic default of sovereign debt 485

differential equation) and the appropriate boundary conditions of this present value function are solved to obtain the critical debt ratio at which default is optimal. The actual values of the debt ratio for several debt-ridden countries are found to be close to the theoretically derived critical level for reasonable parameter values.

To gain insight into the timing decision, consider a case in which there is no penalty for defaulting. Then a default simply adds the value of the loan to the net worth of the country. However, if default results in expulsion from capital markets, then repudiation becomes a one-time-only option. In this case, it may be optimal to postpone the default decision: defaulting now gains the value of the loan, but sacrifices the option to default later at possibly greater gain. Hence, the decision to default requires that the benefits from immediate default exceed the value of a 'live' option to default later; in the certainty case, postponement of an already-profitable repudiation requires only that indebtedness be rising more rapidly than the rate of interest. The default decision is more complex and interesting under uncertainty. When outstanding debt obligations and national income evolve stochastically, we show that the incentive to postpone default increases. The same logic holds true in the case when there is a penalty to defaulting. Now the net benefit to default is the difference between the loan's face value and the penalty, and the penalty will affect the optimal strategic-default rule. However, increased uncertainty still increases the incentive to wait to default.

The debtor country's objective is to maximize present value of the utility from an infinitely-lived stream of consumption. This is achieved with the choice of investments, consumption, and foreign borrowing decisions. We do not attempt to solve the full intertemporal consumption-borrowing-investment problem. Instead, we will focus on the default decision per se, and assume that the country already has pre-committed to an investment plan which has been partly financed with foreign loans. The ability to both maintain the investment plan and service the debt is dependent on the level of GDP. If the investment plan could be fully realized, the country would in turn realize its full potential growth rate of national output. In practice, however, the actual growth rate of output will fall short of this potential growth rate by an amount that reflects the burden of debt service. This drag on GDP growth is due to interruptions in the investment plan that arise from greater than anticipated debt-service obligations (in units of the borrowing country's output). As creditor nations become aware of impending difficulties in the repayment of debt, credit lines or debt-rollover provisions may be restricted, resulting in disruptions to home capital markets.

Our formulation provides richness by allowing for the study of the strategic-default option under various debt-rescheduling scenarios. For example, recent measures taken by Western banks have included the extension of the maturities of previously short-term obligations. The clear motive here is to

486 N. Kulatilaka and A. J. Marcus, Strategic default of sooereign debt

alleviate the pressure on liquidity, but there also are effects on the strategic decision to default. Even though the face value of the loans is not affected by rescheduling, a smaller drag on GDP growth may be achieved by freeing part of national income that otherwise would be diverted from the planned investment program. Our framework also facilitates the study of austerity programs such as those recommended by the IMF. Such programs increase the debtor's ability to meet its investment plan and can result in an increase in the growth rate of output.

2. The market for international loans: Institutional background

An analysis of international loans must first consider their special institutional features and contrast these with those of corporate debt. The most striking aspect of LDC loans is that they are made with little or no collateral. The borrower tends to be a central government of a sovereign nation or a foreign firm with a guarantee from its government. The lenders are almost exclusively banks in the U.S. and Western Europe. This institutional structure does not provide an enforceable legal compulsion to repay. Instead, the incentives to repay stem from a host of other factors.

First, reliable debt servicing will maintain the credit worthiness of the debtor country, thereby enabling future trips to the loan market to finance exports, pay for imports, and finance long-term development projects. How- ever, the competitive Western banking industry and East-West political competition tend to reduce the penalties which a country would incur in the event of debt repudiation. In fact, some argue that the country could re-enter a different segment of the loan market almost immediately following a default. In contrast, corporate-bond default results in bankruptcy.

A second incentive to repay is due to the potential ability of lending banks to seize the debtor country's assets in the West. Of course, in the event of such seizures, one would also need to consider the exposed Western-country assets in the debtor country.

Perhaps the most effective disincentive to default is the threat of trade sanctions. In particular, Latin American countries with debt problems depend largely on the United States as a market for their exports and, thus, a trade boycott would cause substantial losses. The counter-arguments are that (1) there are significant trade barriers, motivated by protectionist policies already in place, (2) in many cases other markets can be found for export products, (3) a halting of bilateral trade would also be detrimental to the U.S., and (4) political instability created by such sanctions may not be congruent with U.S. foreign policy.

We will treat all foreign loans as homogeneous. In reality, a country might have an implicit ranking based on the lender. For example, loans from private banks might be treated as junior to loans from the World Bank or the IMF.


Therefore, it is likely that a country would repudiate one part of the foreign debt while servicing the more senior parts. Such an action could alleviate a liquidity crisis and thereby reduce default probabilities. Our model simplifies by treating the case of a single creditor; in this case default is an all-or-nothing decision. Our treatment is equivalent to assuming that all loans contain cross-default provisions that place all loans into technical default if any loan is defaulted.

3. A model of strategic default

3.1. M o d e l specif ication

In order to focus on the default decision per se, we will consider a country that has already committed itself to a given investment program and has financed part of that program with a foreign loan in the amount L. L is defined in terms of the currency of the lending country and might conveniently be considered as measured in U.S. dollars. The loan is short-term, so that in the absence of repayments, principal would grow at the loan rate, r L. Because of the possibility of default, r r exceeds the risk-free rate, r 0, by a risk premium '71".

National output measures the capacity of the debtor nation to repay the loan. We will denote by G the present value of the country's stream of national output and refer to this notion as stock GDP. Flow GDP can be interpreted as the 'earnings' on G. If the capitalization rate for flow GDP is denoted by k and the growth rate of flow GDP is called g, then the rate of flow GDP at any moment is simply ( k - g ) G .

Default risk arises primarily from uncertainty surrounding the evolution of G through time. x Because the loan is denominated in the currency of the lending country, it is convenient to denominate G in that currency. G (in dollars) can fluctuate substantially as exchange rates shift; for example, a decline in the price of oil would be expected to cause a depreciation of an oil-exporting nation's currency. The value of G measured in dollars declines, and the ratio of the loan to (stock) GDP increases. The burden of the debt measured in the output units of the debtor nation has increased. This rea- soning suggests that the risk premium charged on the loan should be an increasing function of L / G : r L = r o + ~r( L / G ) .

If the loans are traded in a competitive secondary market, the price and default premium would be determined endogenously. The equilibrium condition for a fairly priced loan would set the value of the creditor's claim equal to

t The other major source of uncertainty arises from stochastic changes in interest rates. Modeling interest-rate dynamics in a satisfying manner is, however, extremely difficult [Marsh and Rosenfeld (1983)] and adds little to our analysis of the default decision since the uncertain path of G already makes the default decision an optimal-timing problem.


the amount lent. If the loan did not affect the dynamics for the country's growth rate, this condition would be easy to impose [Merton (1974)]. However, in this model, foreign debt affects the domestic growth rate, which in turn affects the fair risk premium. The resulting simultaneity is exceedingly complex [Bollier (1986)]. Moreover, in practice, these loans are institutionally administered, and risk premia are assigned using negotiated rules such as LIBOR rate plus points. See Cline (1984) for examples of this practice. 2 Therefore, in our numerical simulations below, we will stylize the risk premium using a specific functional form that makes the risk premium directly proportional to indebtedness: 3

,r ( L / G ) = aL/G, (1)

where a is a known parameter. We will assume that the debtor nation and the lenders (banks) have agreed

that loan repayments will depend on the level of the country's indebtedness. At low levels of L / G the country is considered a good credit risk and has the option of borrowing additional funds simply to cover interest payments. As the ratio of L / G increases, however, the debtor is expected to devote an increasing fraction of flow GDP to debt service. Ultimately, however, there might be a cap on the fraction of output that the debtor is wilhng or able to devote to debt service. This description seems to capture the de facto nature of many outstanding loan agreements. In fact, these arrangements occasionally have been made explicit, as in the case of Peru's decision to peg payments to a specified fraction of export earnings. The recent debt crisis seems to support the view that banks will abstain from declaring a default if the debtor nation is acting in good faith, that is, devoting the 'maximum possible' resources to debt service.

We will denote the rate of payment (in dollars) made on the loan as the function pmt(L/G) . In numerical simulations, below, we will consider the specification

p m t ( L / G ) = [rmn( CAP, L/2G)] ( k - g)G. (2)

This function captures the stylized nature of loan agreements discussed above. At L = 0, no resources are devoted to debt services, and for small L/G, a small fraction of output is so devoted. As L / G increases, the fraction of output [which equals (k - g)G] dedicated to debt service increases with slope one-half, but finally, if L / G exceeds 2CAP, the fraction of output directed to

"-Since international loans are not auctioned in an open market and market values are not observable, the use of such second-best administrative rates is common practice.

3 This formulat ion also abstracts from the possibility of credit rationing.


debt repayment is limited to CAP. This allows us to study the effects of austerity programs and debt rescheduling as variations in CAP. In our numerical simulations we set the CAP at 20 percent.

Eq. (2) implies that the evolution over time of the outstanding loan can be described by the equation

d L / d t = rLL - pmt( L/G). (3)

At very low levels of debt, it is likely that the loan will grow [rL > pint(L/G)]. As L increases relative to G a more strenuous repayment plan is implemented which is meant to keep debt within manageable limits. However, adverse movements in G can preclude such 'containment', and L/G can grow to a point at which default may be optimal for the debtor.

We will model the dynamics for the evolution of G as a diffusion process. The instantaneous growth rate of G in the absence of debt obligations would equal g, which is taken to already impound the effect on growth of the planned investment program. The debt burden places a drag on growth, however: the drag is attributable to the flow of resources leaving the country for debt service. As L/G increases, disruptions to the economy arise as additional credit becomes difficult to obtain and the investment program must be curtailed. Ideally, one would like to model the effect of these disruptions endogenously. However, this is not feasible without explicitly modeling the production side of the economy. This task would take us far afield. Therefore, in order to focus on the problem at hand, we assume that the instantaneous change in G is described by

dG/G = [ g - drag( L/G)]dt + odz (4)

The o dz term is a Gauss-Weiner process, which may be considered as the limit in continuous time of a random walk, with zero drift and unit standard deviation.

The drag term should be increasing in L/G as debt obligations increase. Further, it should be a convex function of L/G since the burdens and dislocations attributable to debt service increase more than proportionally with L/G. For the reasons already discussed, we will consider the admittedly ad hoc specification

d rag(L/G) = 8( L /G) 2, (5)

and will examine the sensitivity of our results to the magnitude of the coefficient 3.

At each moment, the debtor nation must decide whether to service the loan over the next interval of duration dt or whether to default. The goal of the

490 N. Kulatilaka and A. J. Marcus. Strategic default of sovereign debt

nation is taken to be the maximization of the present value of consumption. The consequence of default is expulsion from world capital markets and possible appropriation of assets in creditor nations. The expulsion causes a disruption to the economy and causes the stock value of GDP to fall by d percent. We assume that the expulsion from capital markets is permanent - default is a one-time-only option. This represents a severe version of the reputation effect resulting from debt repudiation. Thus, the time t present value of consumption given a default at time t equals Gt(1 - d ) . 4

If the nation chooses not to default at t, the present value of consumption will be denoted by P(G, L, t). 5 Note that P(G, L, t) must exceed G, - L,, the present value of consumption assuming repayment, by the value of the option to default profitably once in the future. Strategic default occurs, therefore, at the first passage of P(G, L, t) below Gt(1 - d) , that is, when the present value of consumption under a default exceeds the present value under continuance of debt service. The debtor nation's optimal default decision therefore can be solved by the characterization of the function P(G, L, t).

3.2. Solution of the model

In order to solve for P(-), we will use techniques from the contingent-claims literature. We have assumed that the evolution of G over time can be described by a diffusion process. Ito's lemma then implies that the dynamics for P(G, L, t) are given by

dP = P, dt + Pc dG + PLdL + ½PGc dG2, (6)

which becomes, upon substitution from (2) and (3),

1 2 2 d P = [P ,+ iPccG a + [rL-pmt(L/G)]Pt .

+ [g - drag(L/G)] PcG] dt + aPcGdz. (7)

4The present values of consumption and GDP are equal. Consider the certainty case in which the rate of return on new projects (and the discount rate) equal k. Upon default the debtor would operate under autarky, and thus, there will be no drag on consumption. Hence, under autarky g = k - c where c is the ratio of consumption to output. The present value of consumption for any value of c is

fo ogre- 'dt=cfo G°e(k-()te-ktdl=G°'

where time 0 is normalized to be the instant of repudiation.

~The problem is defined within a state-space consisting of the state variables G, L, and r Hence. P(-) must be a function only of these variables.

N. Kulatilaka and A. J. Marcus, Strategic default of sovereign debt 4 9 1

The last term in (7) is the stochastic component of the evolution of P and determines the risk premium that a security with the same risk characteristics as P would be priced to earn in a competitive security market [Constantinides (1978)]. If a security that is perfectly correlated with G would command a risk premium of 0, then comparison of the stochastic components of (4) and (7) implies that the risk premium for P (over the next interval dt) would be pPcG/P. This is because the stochastic terms describing the 'rates of return' on the pseudo-securities P and G differ only by the (known at time t) scale factor PeG~P, which is the elasticity of P with respect to G. Thus, adding the risk premium that the capital market would deem appropriate for P to the riskless rate, we obtain the equilibrium expected rate of return on P,

d P ) pPcG E ~ = r o + P (8)

Equating E(dP) from (7) and (8) implies that

Pt + ½Pcc G202 + [rLL - pmt( L/G)] PL

+ [g - p - drag(L/G)] PeG - roP = O. (9)

Eq. (9) can be further simplified by setting x = L / G and letting Gp(x, t) = P(G, L, t). [This substitution is aUowable, since, as we shall see, the boundary conditions for (9) are homogeneous in G and L.] Thus, using the specific functional forms for pmt(.), the risk premium, and drag(-) from (1), (2), and (5), (9) becomes

pt + 1/2p:,xx2a 2 + [r L - f ( x ) / x - g* + 8x 2] pxx

where

+ ( g * - r o - S X z ) p = O ,

f ( x ) = min( CAP, x /2)( k - g),

g* = g - p, i.e., the certainty-equivalent drift in G.

The appropriate boundary conditions for (8) are that

(10)

a p ( x , t ) 2. lim - O, ( l lb)

x ~ 1 OX

3. p ( x , t ) = l - d , at a repudiation of debt. (11c)

1. lim p(x , t) = 1, (11a) . v ~ O


Condition ( l la) states that with very low external debt, the present value of consumption simply equals G: at x = 0, Gp(x, t) =p(G, L, t) = G. (See footnote 4.) The value of the option to repudiate is zero, since as debt approaches zero the probability that default will ever be optimal also goes to zero. With zero debt the full value of stock GDP is available to the home economy. Eq. ( l lb) states that at very large debt levels, relative to the value of stock GDP, the present value of resources available to the home economy becomes insensitive to x = L/G. For very large x, repudiation becomes certain, so that further increases in x no longer affect the net resources of the home economy. Finally, ( l lc) is the free boundary condition. Default occurs at the moment that the present value of consumption assuming no default falls to the present value of consumption if default occurs.

If all loans are ultimately called in at some terminal date T, then at that time

o r

P(G, L, T) = max[G- L , G ( 1 - d ) ] ,

p ( x , T ) = max(1 - x , 1 - d ) . (12)

At T, the default decision is easy:6 repudiate if the loan exceeds the default penalty, that is, if L >Gd. For t < T, however, the repudiation criterion is stricter, since repudiation now imposes an additional implicit cost which is the loss of the opportunity to repudiate later. For times prior to T, the value of the loan must be greater than Gd by at least some critical value to induce a default. 7

3.3. Numerical results

We numerically solved the partial differential eq. (10) subject to the boundary conditions (11) to obtain the present value of consumption, p.8 Prior to obtaining comparative-static results we investigate the behavior of p and the 'consumption gain', denoted V, from deferring immediate repayment of the loan under base-case parameter values. The value of V can be obtained as the difference between the actual present value of consumption, P(G, L, t),

6Although ideally we should use an infinite-time horizon the numerical techniques required finite-maturity loans. In the numerical computations we used a value of 75 years for T. We experimented with larger values of T but found that the solutions barely changed. Thus, our choice of T = 75 closely approximates the infinite-time case.

7This is precisely analogous to the result that at times prior to maturity an American put option should not necessarily be exercised even if it is in the money. Only at maturity is being in the money a sufficient condition to induce exercise.

SWe used a finite difference method and solved the p.d.e, backwards, following Brennan and Schwartz [1978].


.9C 0 .04 .08 .12 36 ,20 .24

1.0

i= 0 :,: .98 Q.

E

8 . 9 6

-~ .94

c

~ .92 Q.

Debt Ra t i o : x = L/G

Fig. 1. The present value of consumption, p.

and the present value of consumption assuming repayment, G - L. Hence, V = P - ( G - L ) or, expressed as a fraction of stock GDP, v = V/G = p - (1 - x ) .

Notice that our loan-repayment scheduled disallows immediate repayment of the entire loan. This constraint means that the country can incur a GDP growth-rate drag for a substantial period of time. Hence, V captures the consumption gain from an optimal default policy conditional on the no- immediate-payback constraint. This consumption gain therefore equals the value of the option to default less the capitalized value of the consumption loss due to the drag term.

Figs. 1 and 2 plot p and o against the debt ratio, x, for the following base-case parameter values (comparative statics appear in later figures):

d = 0 . 1

8 = 0.05

a = 0.10

o = 0.10

The debtor country will be subject to a penalty of 10 percent of stock GDP if it defaults.

Drag coefficient [see eq. (5)].

Risk premium on debt is 10 percent of indebtedness [see eq. (1)].

Annual standard deviation of G (valued in domestic currency).

CAP = 0.20 Maximum fraction of GDP devoted to debt service.

At extremely low values of debt, the debtor country can devote all of its flow GDP to consumption and/or investment, and hence, p = 1 (see footnote

494 N. Kulatilaka and A. J. Marcus, Strategic default of sooereign debt

0.1!

0 0

o ( .9

c O.1( o

E

o c~ 0 .0

I I I I I I

• 04 .08 32 36 .20 .24

Debt Ratio: x -- L/G

Fig. 2. Consumpt ion gain, o = p ( 1 - x).

4). As the debt ratio increases, p declines and v increases monotonically until default occurs at about a 20 percent debt ratio. 9 Upon default, the debtor gains the face value of debt and faces a penalty equal to 10 percent of G. The consumption gain increases monotonically with the 'debt ratio', x, until default, at which point it is worth G ( 1 - d) - (G-L)= (L - Gd), i.e., the value of the loan repudiated less the costs of repudiation.

We now turn to the behavior of o (consumption gain as a fraction of G) for different parameter values. Fig. 3 presents the value of the default option as a function of the debt ratio for different risk premia rules, a. The most striking characteristic of the figure is the appearance of an 'internal' optimal debt ratio for sufficiently sensitive risk-premium rules, i.e., for alpha greater than ap- proximately 0.2. For smaller values of alpha, however, the borrowing country optimizes at a corner solution, with an unbounded debt ratio, and eventual default. For the internal-optimum cases, consumption gain is positive and increasing for small values of debt. As x grows, the value of o reaches a maximum, starts to decline, and sometimes reaches negative values. Finally, for even larger values of x, the consumption gain reaches a minimum and begins to increase.

In order to interpret these results we must first consider the meaning of a negative value for the consumption gain. This results from a constraint in our model. We provide only two alternatives to the debtor country: default on its loans or continue to service them at a prespecified payment schedule. We omit the alternative of immediately paying back the loans when the costs of

"The relationship need not be monotonic for other parameter values; see below.


0.15

o.~o

E 0.05

I I f I I I

a=O

~ a :0 .2

I " - [ I I I I .04 .08 32 .16 .20 .24 28

Debt Ratio: x = L / G

Fig. 3. Responsiveness of consumption gain, v, to changes in a.

servicing them outweigh the benefits. Such a policy is probably not available to any nation facing significant debt-servicing problems. Hence, the present value of the consumption stream, P, might be less than G - L. Although G - L might appear to be a logical lower bound on the value of the consumption stream, it would be so only if the debtor country could repay the principal in one lump-sum without incurring a GDP drag from the sacrifice of resources. A negative consumption gain tells us that ex post the loan turned out to be detrimental to the debtor country. However, ex ante the loan might have appeared welfare-increasing. Of course, P must always exceed G(1 - d ) because of the option to default.

Thus, in fig. 3, consumption gain starts positive. As debt increases, the drag increases since debt-service requirements respond both to the size of the loan and to the risk premium. Ultimately, however, because we impose a ceiling on debt payments for higher debt ratios via eq. (2), the drag effect is limited and eventually is dominated by the increased value of the option to default. Hence the upturn in D.

A second, and more direct way to observe the effect of drag on the option value is depicted in fig. 4 which reports a comparative-static analysis with respect to variations in the term 8. As the above intuition suggests, higher values of the drag parameter result, ceteris paribus, in reductions in the option value. Higher values also increase the incentive to default once debt is incurred. Each graph in fig. 4 ends at the debt ratio that sets off a default. That ratio is lower for higher values of 8.

Other unreported simulations consider the effects of uncertainty and the default penalty on the consumption gain. As expected, increases in g increase


c o

(.9

8

( J

0.:30

0.24

0,18

0.12

0,06

0 0

I I I I I I I I

- - 8 - 0 .

~ 8 =0.1

" I I I I I I I I .04 .06 .12 .16 . ~ .24 ,28 .32

Debt Ratio: x =LIG

Fig. 4. Responsiveness of consumption gain, o, to changes in drag.

the value of the consumption gain. This result is in accordance with the conventional wisdom from option-valuation models and is due to the down-side protection provided by the default option. Although the debtor country can benefit from the positive swings in GDP, it is somewhat protected against negative swings by the option to default, which allows it to transfer some of the misfortune to the creditor country. Also as expected, an increase in the default penalty lowers the consumption gain as the option to default carries a higher 'exercise price'.

Finally, we consider the impact of austerity programs, which we model as increases in CAP, that is, the maximum fraction of flow GDP that the debtor country is required to devote to debt service. Fig. 5 examines the critical value of indebtedness at which the country will default for base-case parameter values and varying values of CAP. We see that the resources that austerity programs can coax from the debtor country are limited. As CAP increases, so too does the critical value of L/G. However, beyond a value of roughly 0.15, further increases in CAP no longer affect (L/G)*. At that point, the payment CAP is no longer binding: if the country reaches a debt level at which required debt service is dictated by such large values of CAP, the country will choose instead to default.

(L/G)* can be plotted as a function of other parameters. Unreported simulations show that, as one would expect, the critical debt ratio falls with the risk premium on the loan and the drag term 8, and increases with the penalty for default and the volatility of the GDP process.

Finally, we highlight the practical importance of our findings with some rough comparison of x* to actual debt ratios. Column 4 of table 1 gives the


0.27

0.25

0.23

0.21 . J

0.19

0,17

0.15

0,13

I ! I I I 1 I I

I I I I I I .04 .06 .08 .10 32 .14

Payment Cap

I .16 .18

Fig. 5. (L/G)* as a function of payment CAP.

I • 2 0 .22

actual-debt ratios in 1984 for several countries based on the assumption of a 15 percent capitalization rate for stock GDP (k) and 3 percent growth rate of GDP (g). The theoretically computed critical-debt ratios in fig. 5, which range from about 13 to 27 percent, approach the actual-debt ratios reported for some of the more severely indebted countries. These values highlight the gravity of the debt crisis and indicate that some countries might be perilously close to the point at which default might be welfare-increasing.

4. Concluding remarks

This paper develops a continuous-time model of the strategic decision to default on sovereign debt. We specify dynamics for the level of debt and the stock GDP of the debtor nation, where debt-service payments impose a drag on the GDP growth rate. At any time the country has two alternatives: (a) default, in which case it gains the face value of the loan, foregoes the opportunity to default in the future, and incurs a penalty, or (b) continue to service the debt, thereby maintaining the option to default at some future time.

Strategic default occurs when the debt ratio reaches a critical value. We solve for this value and obtain the value of the option to default. We study the present value of the consumption gain that the borrowing nation derives from the loan under an optimal default policy. We find that that gain increases with the variance of the GDP process and decreases with the default penalty and GDP drag. We are also able to study the effect of austerity programs, such as those prescribed by the IMF, by varying the payment schedule. We find that


increased austerity measures will increase the incentive to default; the option to default thus limits the efficacy of such austerity programs.

References

BoUier, Thierry, 1986, International debt-repudiation and opportunistic behavior in an uncertain environment, Unpublished dissertation (Sloan School, M.I.T., Cambridge, MA).

Brerman, M. and E. Schwartz, 1978, Finite difference method and jump processes arising in the pricing of contingent claim: A synthesis, Journal of Financial and Quantitative Analysis 13, 461-474.

Cline, William, R., 1984, International debt: Systematic risk and policy response (Institute for International Economics, Washington, DC).

Constantinides, G., 1978, Market risk adjustment in project valuation, Journal of Finance 33, 603-616.

Eaton, J. and M. Gersovitz, 1981a, Debt with potential repudiation: Theoretical and empirical analysis, Review of Economic Studies 48, 288-309.

Eaton, J. and M. Gersovitz, 1981b, Poor country borrowing in private financial markets and the repudiation issue, International finance section (Princeton University, Princeton, NJ).

Marsh, T. and E. Rosenfeld, 1983, Stochastic processes for interest rates and equilibrium bond prices, Working paper no. 83-45 (Harvard Business School, Cambridge, MA).

Merton, Robert, 1974, On the pricing of corporate debt: The risk structure of interest rates, Journal of Finance 29, 449-470.

Sachs, J. and D. Cohen, 1982, LDC borrowing with default risk, NBER working paper no. 925.

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A model of strategic default of sovereign debt