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FUNDAÇÃO GETULIO VARGAS

ESCOLA de PÓS-GRADUAÇÃO em ECONOMIA

Pedro Vaissman Guinsburg

The Eaton-Gersovitz-Arellano

Environment with Collateral

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Pedro Vaissman Guinsburg

The Eaton-Gersovitz-Arellano

Environment with Collateral

Dissertação para obtenção do grau de mes-tre apresentada à Escola de Pós-Grauação em Economia

Área de concentração: Macroeconomia

Orientador: Ricardo de Oliveira Caval-canti

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Ficha catalográfica elaborada pela Biblioteca Mario Henrique Simonsen/FGV

Guinsburg, Pedro Vaissman

The Eaton-Gersovitz-Arellano environment with collateral / Pedro Vaissman Guinsburg. – 2014.

27 f.

Dissertação (mestrado) - Fundação Getulio Vargas, Escola de Pós-Graduação em Economia.

Orientador: Ricardo de Oliveira Cavalcanti. Inclui bibliografia.

1. Inadimplência (Finanças). 2. Dívida externa. 3. Investimentos estrangeiros. I. Cavalcanti, Ricardo de Oliveira. II. Fundação Getulio Vargas. Escola de Pós- Graduação em Economia. III. Título.

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Resumo

Neste artigo eu introduzo colateralização no ambiente de dívida soberana de Eaton-Gersovitz-Arellano. Esta colateralização pode ser vista como Investimento Estrangeiro Direto. A entrada de recursos colateralizados serve como uma estratégia de comprometimento dos países. Ao abrir a economia para este tipo de aporte de recursos, meu modelo prescreve maior tomada de dívida em equilíbrio pelos países e menos uso de default como instrumento de suavização da trajetória de consumo. Além destas características, eu mostro que limitação de colateral pode gerar mais default em equilíbrio do que um modelo sem Investimento Estrangeiro Direto ou colateral.

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Abstract

In this article I introduce collateralization in the Eaton-Gersovitz-Arellano environment. In my paper, collateral can be understood as Foreign Direct Investment. I find that collateral increases the equilibrium levels of defaultable debt and, at the same time, avoids the use of default. Restrictions of collateral leads to higher use of default in equilibrium.

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List of Figures

2.1 B Bond-Prices and C . . . 10

2.2 B Policy Function. . . 11

2.3 Foreign Direct Investment and External Debt (Credit for Imports) . . . 12

6.1 Large Levels of C . . . 21

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List of Tables

2.1 Parameters . . . 9

2.2 F.D.I. Vs. Foreign Debt . . . 11

2.3 Long Run Statistics 1 - First Moments . . . 13

2.4 Long Run Statistics 2 - Variances and Covariances . . . 14

2.5 Long Run Statistics 3 - First Moments . . . 15

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Contents

1 Introduction 1

2 Model 5

2.1 The Economy . . . 5

2.1.1 Lenders . . . 5

2.1.2 Countries: Government and Agents . . . 5

2.1.3 Financial Instruments . . . 5

2.1.4 The Problem of the Government . . . 6

2.2 Equilibrium . . . 8

2.3 Parametrization . . . 8

2.4 Results. . . 9

2.4.1 Large C . . . 9

2.4.2 Small C . . . 14

3 Conclusion 17

4 References 19

5 Code 20

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Chapter 1

Introduction

Default models based on Eaton-Gersovitz (1981), when calibrated to developing countries economies manage to generate the empirically observed countercyclical interest rate spreads. In these models, default probabilities are increasing with the level of debt and decreasing with the level of growth shocks of the sovereign country. The intuition for this result is straightforward: Sovereign Countries ruled by benevolent planners with the objective of temporal, and inter-state, consumption smoothing, face a very high cost of honouring its debts obligations precisely when the hired debt amounts are large or when the output is low. This happens because, in these states of the world, the country usually cannot have inflows of resources due to the debt obliga-tions and the large discount over the face value of its bonds. The outflows of resources in such periods, at the same time, would make the consumption path of the representative agent more volatile, worsening the social welfare of the country. Therefore, in this situations, the planner might choose to default in order to smooth the consumption in bad periods thereby facing severe punishments, such as exclusion from markets and direct output costs. Because consumption will be too low in this periods, smoothing it becomes a more important issue than having access to financial markets in the near future and the only way to have inflows of resources in such bad states is by defaulting. As a result, default choices can be viewed as a completing markets instrument in response to an unfavourable price menu of debt and the poorness of the asset structure.

Discounts over the face value of debt, on the other hand, are high in this periods precisely be-cause lenders can correctly predict that default probabilities are high during bad economic shocks or periods of debt overhang. In other words, lenders perfectly predict default probabilities. This perfect forecasting feature, at the same time, is a consequence of the rational expectations, per-fect information, characteristic of the Eaton-Gersovitz environment. If the level of the output on one date can predict future output, as it happens in a markovian structure of shocks, for example, low levels of output in a certain time indicates low levels of output in the near future. As mentioned before, the lower the output, the higher the probability of default and lenders, knowing the borrowing policy function of the country, will predict future decisions of default. Consequently, low levels of output in a certain date pushes the price of a sovereign bond down, making the capacity of the country to roll over its debt worse, in a self reinforcing dynamic that ends with default as an important consumption smoothing policy for the governments in equilibrium.

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The cost of relaxing commitment and proxiing for realistic assets in the EGA environment, however, is precisely that the model looses all its ex-ante commitment. In this models, repay-ments are not due to contractual obligations but rather because of the benefits of financial inclu-sion and the cost of direct utility (output) costs. In particular, there are no sort of collateralized assets in this environment simply because there are no collateralized assets being traded in the sovereign debt market. On the other hand, even though countries do not use REPO agreements for funding, for example, collateral can be a metaphor for another important feature of sovereign debt: endogenous guarantees. Although countries cannot underwrite mortgage contracts on its bridges or buildings, it is interesting to think on a wide range of commitment contracts proxiing variables such as fiscal responsabilities acts, commitment to international financial institutions (IMF, World Bank) or international reserves accumulation. All of these instruments have the important common feature that they serve as guarantees given to lenders. In other words they can be viewed as some sort of collateral.

In terms of a characterization of different levels of collaterallized assets coexisting and play-ing part on a general equilibrium approach, Geanakoplos and Zame (2007) is the benchmark model. In this paper, there is a new definition of equilibrium that also departs from the clas-sical Arrow-Debreu approach and permits default. Because of simplifying assumptions, there is no default penalty in this model and default is only a decision taken by individuals inasmuch they choose their type of borrowing assets. Default will happen as a consequence of different collateral requirements of the assets used and equilibrium prices of the collateral good. Indi-viduals can sell certain asset as long as it meets its collateral specification. If the future state of the world is such that the repayment promises specified in the asset are higher than the value of collateral required, no rational individual will choose to pay more than less and, as a result, default will happen and the collateral will be executed. Remember that this automatic response to equilibrium prices happens only because there are no default costs associated in this model. With default costs it is perfectly possible that the value of the collateral requirement is lower than promises and, still, the agent decides not to default because of default costs associated.

Finally, the researcher finds that default may coexist with collateral for an asset structure where the promises are constant and so the only source of heterogeneity is collateral requirements. As an asset pay-off will be the minimum between its promise and its collateral requirement value in a specific state, there will only be a more complex structure of asset if prices of the collat-eral good are so low in certain states that default happens on certain riskier, low collatcollat-eralized, assets. If for example, there is an agent that wants to smooth inter-state consumption in the future, he can use the default option to enhance the asset span. Competitive equilibrium, on the other hand, can guarantee the presence of lenders because the prices of this bonds will be lower (higher yields). As a result, collateral enhances the complexity of the asset span. Note however that complexity of the asset span does not necessarily means Pareto Efficiency once there are distortions associated with the need for collateral in the first period constraint.

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the minimum present costs. And the least first period consumption distortive way of borrowing money is to use the minimum collateral requirement asset to borrow.

The problem, however, is that this poor agent will not offer any other kind of bonds to borrow nor buy other bond, therefore making the active financial structure poor. To sum up, almost certain default happens when there are agents facing serious illiquidity problems, desperately needing to bring wealth to the first period but trying to do this without changes in the first period optimal consumption basket. They, then, have to face lower prices on their debt, which means higher yields.

In an effort to study collateral in the sovereign country debt environment, I introduce collat-eralization choices in the Eaton-Gersovitz environment. This is done simply by introducing new types of assets in the economy: collateralized bonds. Once the country defaults, it has to pay the collateral requirements of the assets used in the last period. For a large set of parameters, I find that countries manage to turn risky assets into risk-free ones by choosing large levels of full collateralized assets in order to show to the market it will not default at any chance. The idea is that the country deliberately chooses to increase the cost of default - engaging with large obligations in case of this extreme event - to make the market believe it will not default at all. This effort made by the country, at the same time, happens because, in my model, the set of risky debt is wider than the set of full collateralized debt. Consequently, the consumption smoothing process is better managed with the use of this larger set instead of using the more scarce collateralized bond.

As in Araujo, Kubler and Schommer, here, scarcity of collateral will increase the use of de-fault in the economy. However, in AKS or GZ scarcity of collateral is just exogenous scarcity of the good that serve as collateral (houses, for example) whereas in the model of this paper, as this sovereign economy has only one good, scarcity of collateral is the result of disastrous states of the world. When an economy have a positive probability of entering in a very bad outcome, the levels of collateralized bond have to account for that and can no longer afford large levels of debt relative to output - otherwise the country would have committed to repay levels of debt it could not have. The levels of this more secure debt, then, have to be smaller than this disaster output. As a result, the use of the self imposed commitment strategy by the country is weakened when serious low levels of output can happen because there is scarcity of the collateralized bond. Consequently, when there is scarcity of collateral, default is used frequently as a consumption smoothing instrument and this economy approximates the Eaton-Gersovitz-Arellano environ-ment.

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available for payments.

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Chapter 2

Model

2.1

The Economy

The economy consists of three types of agents and one consumption good. In addition there is a competitive financial market where assets can be traded among all agents.

2.1.1 Lenders

In one side of the financial market, there is a continuum of risk-neutral lenders with infinite resources available. To allow for rational expectations and perfect information, I will assume that lenders also have access to the countries value function and will anticipate future govern-ment policies in their actions. That is, this is a rational expectations with perfect information environment.

2.1.2 Countries: Government and Agents

There is a single small open economy, that admits a representative agent and is ruled by a benevolent social planner.

The agent is risk-averse with a Bernoulli utility function on the unique consumption good. I will assume that the utility of the agent have all the standard features, such as the INADA properties.

Every period, the agent receives an output shock that follows a markov process with transition function𝑓(𝑦t+1,𝑦t). Households also receive a lump sum transfer of goods from the government.

As a result, this formulation assumes that agents are passive in the economy, in the sense that they do not invest in capital goods or financial assets.

The objective of the country is to use the financial market to smooth the representative agent consumption process with only two instruments of debt/equity, B or C. The third instrument of consumption smoothing that this country can use is the option to default on its obligations.

2.1.3 Financial Instruments

There are two financial assets in this economy and they both have a structure of a bond, or fixed income. I will call them B and C. If the country decides to repay its debts, it can build a portfolio of two types of assets, 𝐵t and 𝐶t. While asset B entails no obligations in case of default, asset

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This feature will be called collateralization due to the enforcement in case of default. It is as if the country was forced to deliver a unit of the consumption good in the future even in the case of default and that is the reason I call it, a collateralized bond.

One could classify this debt instrument as foreign direct investment, for example. When a country fails to meet its debt obligations not necessarily the capital and industry obligations enter in a stage of default because multinational firms may still bring their dividends back from their subsidiaries in a defaulted country. Even if the dividends and profits are shut-off in an balance of payments crises, it is widely known that capital flows circumvent legal restrictions using the black market. Therefore even in a more severe break of contract from the countries, lenders, or investors in a more broad definition, manage to have their dividends sent to their homeland. For this reason, we like to think of C as foreign direct investment.

If it decides to default, the country is excluded from financial markets (even for the lend-ing markets) for a probabilistic amount of time. The probability of re-enterlend-ing in the financial markets by the country will be given by 𝜃 and it does not evolve over time. In addition, if the government decides to default, it suffers a direct output penalty. This feature tries to follow a stylized fact of output costs associated with default on government debt that countries suffer when they choose to default.

After defining the environment we can now turn to the problems of the government and the problem of the lenders. I will focus directly on the recursive formulation.

2.1.4 The Problem of the Government

The consumtpion of the governement in case of default will be

𝑐deft =𝑦deft +𝐶t (2.1)

Where 𝑦def = (𝑦) 𝑦 is an increasing function and represents the direct output costs

associated with default.

In case of repayments, on the other hand, the consumption of the country will be

𝑐t=𝑦t+𝐵t+𝐶t−𝑞b(𝐵t+1,𝐶t+1,𝑦t)𝐵t+1−𝑞c(𝐵t+1,𝐶t+1,𝑦t)𝐶t+1 (2.2)

Where the prices of the assets may depend on the policy functions of the government. These will be clearer when we set the problem of the lenders.

Every period the representative agent decides if she will default or not. If she defaults, she will be excluded from markets in that period and for a probabilistic amount of time. However, the agent will still have to pay the amount contracted in the previous period of the type of asset C.

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If the government decides to repay, on the other hand, she will have to pay all her obligations but will have access to financial markets and, as a result, may borrow. The equations below characterize the value function of this option, 𝑉o, defined by the value function of repayment,

𝑉r, and the value function of default,𝑉d. The value function of the country will, therefore be:

𝑉o(𝐵, 𝐶,𝑦) =𝑚𝑎𝑥r,d{𝑉r(𝐵,𝐶,𝑦),𝑉d(𝐶,𝑦)} (2.3)

Where the value of repayment will be

𝑉r(𝐵, 𝐶, 𝑦) =𝑚𝑎𝑥{B,C}{𝑢(𝑦+𝐶+𝐵−𝑞(𝐵′,𝐶′,𝑦)𝐵′−

𝐶′

1 +𝑟f

)

+

∫︁

y′

𝑉o(𝐵′,𝐶′,𝑦′)𝑓(𝑦′,𝑦)𝑑𝑦′} (2.4)

Note that the price of the security C is the same as the riskless asset because it has the same pay-off structure of a risk-free asset traded among lenders. I will address further this issue of pricing in the lenders section.

The value of default, on the other hand, will be:

𝑉d(𝐶,𝑦) =𝑢(𝑦+𝐶)

+𝛽

∫︁

y′

(𝜃𝑉o(0,0,𝑦′) + (1−𝜃)𝑉d(0,𝑦′))𝑓(𝑦,𝑦′)𝑑𝑦′ (2.5)

The Problem of the Lenders

The most important consequence of the assumption that there exists a continuum of lenders with plenty of resources is that prices of bonds will be solely given by the lending Euler Equation.

Once lenders are risk-neutral, their first order condition for positive, and finite, lending on any asset,𝑎, will be

𝑞a=𝛽lenders𝐸[𝛾] (2.6)

Where𝛾 stands for the pay-off of the asset, taking account for possible default situations as well. In the case of a bond that pays one unit of the consumtpion good,𝛾 is the indicator variable of repayment. As a result, in our asset structure of bonds, it is straightforward that

𝑞C =𝛽lenders ≡

1 1 +𝑟f

(2.7)

because the bond C is always repaid by the country. That is the only price that makes lenders want to lend the amount asked by the country C. It is important to notice that this price is independent of the amount asked by the country C.

In the case of the bond B

𝑞B =𝛽lenders𝐸[𝐼Vr≥Vd] (2.8)

Where I stands for the indicator random variable.

Moreover, as lenders know the policy functions of the country and, specifically, in what regions of output and debt the country decides to default, they can calculate regions of default,

𝐷conditional on the amount lent by them of the bond B and C. These regions will then be:

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By knowing these regions, lenders may then calculate, in period 𝑡, the probability of default in period 𝑡+ 1. This given by

𝑔(𝐵′,𝐶′, 𝑦) =

∫︁

D(B′,C)

𝑓(𝑦′,𝑦)𝑑𝑦, ∀𝐵′,∀𝐶′ (2.10)

As a result, the only price of the defaultable bond that balance offer and demand by satisfying the euler equation of the lenders will be

𝑞(𝐵′,𝐶′,𝑦) = 1−𝑔(𝐵

,𝐶,𝑦)

1 +𝑟f

(2.11)

In the competitive equilibrium then, there will be a menu a cost for every amount asked by the country B.

2.2

Equilibrium

A rational expectations recursive competitive equilibrium in this economy will be a list{𝑉r,𝑉d,𝐵′, 𝐶′} and a competitive price function, q, such that

1. Given the price function𝑞, the value functions𝑉r,𝑉d and the policy functions𝐵′ and𝐶′

solve problems (2.4) and (2.5).

2. The price function 𝑞 satisfies the break-even condition for the lenders. In other words,

𝑞( ˜𝐵,𝐶,𝑦˜ ) satisfies (2.11) ∀𝐵˜ ∈B,∀𝐶˜ ∈C,∀𝑦∈𝑌.

Proposition 1Suppose the set of assets B,B, and the set of assets C,Care intervals. Then, there exists a rational expectations recursive equilibrium in this model.

Proof:

2.3

Parametrization

I will assume that the agent has the C.R.R.A. on the unique consumption good𝑐t. The Bernoulli

utility function is therefore:

𝑢(𝑐) = 𝑐

1−σ

1−𝜎 (2.12)

Following Arellano (2008), the de-trended output of the country in period t, 𝑦t, follows a

log-AR(1) process. That is:

𝑙𝑛(𝑦t) =𝜌𝑙𝑛(𝑦t−1) +𝜖t (2.13)

Where

𝜖t∼𝑁(0,𝜂2) (2.14)

Note that 𝑦t, is not the current output but rather the ratio of the current output over the

trend output. As a result, typical values of t y will be 1.05 or 1.07 or 0.975 and they will be readed as an output of 5% above the trend, 7% above the trend or 2.75% below the trend and this way of reading the variables will carry on to all other debt variables. That is, by read-ing the output in that manner all debt/equity variables will be in percentages of the trend output.

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As for the rest, I have used the same set of parameters than that of Arellano (2008).

It is important to notice that, as in Arellano (2008), I evaluate the utility with respect to rel-ative deviations of the output from the trend. As a result, the debt or equity levels are measured in terms of percent of trend G.D.P..The rest of the parameters are summarized in the table below:

Table 2.1: Parameters

Parameters

𝛽 0.953

𝜎 2

𝜃 0.282

𝜂 0.025

Output Costs 𝑦ˆ=0.969E[y]

2.4

Results

2.4.1 Large C

The first exercise I conduct is with the output dynamic close to that of Arellano (2008). In this environment, the lowest output is 0.77, which means the lowest output that can be reached in this markovian structure is 20% less than the trend output. In addition, I have made exercises with and without direct output costs in case of default. Because I get very few default cases in equilibrium with direct output costs, I have decided to analyse the model without this structure to generate default in equilibrium. The Arellano (2008) environment uses direct output costs. This model without costs make prices so low that there is almost no debt in equilibrium.

In terms of portfolio choice the country is faced with two types of debt/asset instruments. First, the country can be a net borrower or lender and this is done by letting B be positive 1

. Note first, that the country would never default on a positive claim it has because this decision would take resources from the present and be costly in terms of exclusion. Consequently, if the country chooses to be a lender, the price of the positive amount lent by the country, B, would be trivially the risk-free interest rate,𝑟f, in equilibrium. So let’s turn to debt analysis.

Obviously, the debt -C has a better pricing, once it is a full commitment debt instrument and, as so, the interest rate of this debt is trivially 𝑟f. As a consequence, this debt is good for

bringing resources from the future to the present. However, the country face potentially serious disadvantages in using this type of debt in its portfolio. Because this debt is of full commitment, if the country decides to default, it will face the same payments as it would if it did not entered in a default state.

The debt -B, on the other hand, has the advantage that, once the country chooses to default, it does not have to pay anything from the principal borrowed. Therefore, the debt B is good for completing markets, once default can be viewed as a completing markets instrument. To state this in other words, the debt B is good for inter-state consumption smoothing because of its discretion. The problem with this instrument, however, is that, precisely because of the lack of commitment, the competitive equilibrium prices set by the market in equilibrium will have

1

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to account for default risk in the next period and, as a result, the interest rate will have to compensate for that risk and so will higher than the risk-free rate 𝑟f in equilibrium. The price

of this debt must, then, be lower therefore making debt obligations harsher.

Importantly, however, the price of the bond B’, q(B’,y,C’) depends positively on the amount of the debt -C’. When the country chooses to make the consumption smoothing process by us-ing high levels of debt -B and low levels of debt -C, the price of the debt -B becomes lower. This happens because the proportion stolen by the country from international creditors in case of default is lower when the debt -C is higher, ceteris paribus. Consequently, default is more attractive for the country when the proportion of debt B/C, is high because there is more to steal. This relationship yields higher interest rates when the debt -C is lower. Once the price of the risky bond fully reflects default probabilities, the higher the amount -C’, the higher the price q(B’,y,C’). This is depicted in figure (2.1). The pattern shown is reproduced in all states of C and y.

Figure 2.1: B Bond-Prices and C

−0.50 −0.45 −0.4 −0.35 −0.3 −0.25 −0.2 −0.15 −0.1 −0.05 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 B q(B,C,1.03)

Bond Prices − y=3% Above the Trend

If we use the analogy of C as foreign direct investment, for example, we can conclude that F.D.I. serve as an endogenous commitment policy for the governments. The more intense is the investment/"lending", the more committed with repayments the country is. The benefit in terms of prices from the F.D.I. comes from the fact that when the country enters in a default state, it still has to pay the C obligations (dividends or profits). As C can be large, the amount stolen from international lenders B, responsible for improving the levels of consumption, might be small relative to the C obligations. The debt -C, as a result, allows the country to choose the relative cost/benefit of default by choosing the proportion B/C and, in doing so, the country can now choose a commitment strategy every period by letting more or less F.D.I. enter the country.

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Figure 2.2: B Policy Function

−2.5 −2 −1.5 −1 −0.5 0 0.5

−2 −1.5 −1 −0.5 0 0.5

Use of B − Trend Output

B’(B,C,1)

B C=−80% of the Trend Output C=−30% of the Trend Output C=0

Each of the coloured lines represent risky debt policy functions, B’, with one level of current collateralized debt level, -C, kept fixed. In the figure, it can be viewed that the higher the level of guaranteed present payments by the country, C, the more explosive the risky debt policy func-tion is. This can be seen comparing the blue line, where C is -40% of the trend GDP, with the red and green lines - 0 and -20 % of the trend GDP, respectively. This increasing levels of debt happen because low levels of C (high debt) means high present payments and resources leav-ing the country which makes the consumption smoothleav-ing effort of the planner demand more debt.

Our prescription in terms of macro variables would then be that the higher the levels of foreign direct investment, the higher the level of foreign debt demaded. This is precisely what I have found for this relationship in Brazil2

. As can be seen from the table and the graph below, foreign direct investment and foreign debt seems to have similar dynamics.

Table 2.2: F.D.I. Vs. Foreign Debt

Correlation (F.D.I. Vs. Foreign Debt - Credit for Imports) F.D.I.t−1 Vs. Credit for Importst 0.72

F.D.I.t−1 Vs. Credit for Importst(Trend) 0.89

F.D.I.t Vs. Credit for Importst 0.69

F.D.I.t Vs. Credit for Importst(Trend) 0.86

2

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Figure 2.3: Foreign Direct Investment and External Debt (Credit for Imports)

0 10 20 30 40 50 60

0 0.5 1 1.5 2 2.5

3x 10

4 Foreign Direct Investment and External Debt

Time (quarterly, beginning in 4th quarter of 1999, end in 4th quarter of 2012)

Reals

F.D.I. External Debt F.D.I. Trend

Therefore a way of analysing F.D.I. is to view it as an investment that ties the country within good behaviour policies. This then leads to better prices that facilitate increasing levels of debt. Overall, we have found that the country chooses large quantities of both types of debt and their trajectories are mostly increasing. The country mostly chooses the highest quantity of debt -C, therefore increasing current consumption on the expense of future large obligations. This debt is mostly kept fixed in its higher level. This happens for two reasons: first, because

𝛽lenders > 𝛽country, the model prescribes natural tendency for leverage by countries, if they have

good pricing of bonds. The price of the debt -C is the one that fully reflects 𝛽lenders and,

con-sequently, taking a lot of this debt is an arbitrage opportunity. Second, the debt -C serves as a commitment strategy, as already mentioned. In doing so, it helps to increase the price of the debt -B, approximating that price to that of the -C debt.

The country then uses this commitment strategy to increase the debt -B, taking advantage of that arbitrage opportunity between betas, until the level of approximately 1.5 of debt (150% of Trend Output). 3

. After reaching this level, the country chooses to smooth consumption using -B. The debt -C, on the other hand, is mostly kept fixed at its higher level4

.

As this debt -B lacks commitment and is increasing with the debt -C, one might think the sovereign risk of default increases with increases in the level of current debt -C. However, in this environment, where the lowest relative output is 0.77, I find that the majority of the debt demanded in equilibrium is free of risk - 𝑞 = 1

1 +𝑟 - , meaning that there is almost no default

on the equilibrium path. The biggest probability of default in equilibrium is 1.36%5

. In other words, although there is considerable use of the risky debt, -B, the country almost never defaults in equilibrium and this debt is free of risk. The reason for this risk-free environment is that as C can go as low as -70% of the trend output, the country can deliberately control the cost of default using the collateralized payments. In particular, the country can make default costs so

3

Details of increasing Policy Functions are in the Appendix section

4

It is important to notice that a large part of the high levels of debt encountered in this model are due to the innate disequilibrium of betas between lenders and countries. This is also present in Arellano (2008)

5

Given a state (B,C,y) I define the probability of default in equilibrium, given a state (B,C,y), as

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13

large that is (almost) never worthy to default in equilibrium no matter what y realizes in the future.

Table 2.3 show that debt levels are much higher in our model than in Arellano and default events are much rarer (almost non-existent). These statistics are driven by the presence of the Debt -C and its effects on prices and on the capacity of leverage on the defaultable debt. Even if we treat -C not as debt, our model prescribes higher defaultable debt levels than in Arellano. In other words openness in form of F.D.I. increases the equilibrium levels of (defaultable) foreign debt and, at the same time, eliminates default events. These characteristics are shown in the table below.

Table 2.3: Long Run Statistics 1 - First Moments

Long Run Statistics6

Variable Arellano (2008) BC Model BC Model D.O.C.7

Gross Debt (B+C) -0.04 -2.18 -2.76

Defaultable Debt -0.04 -1.48 -2.06

Probability of Default 0.01 0.0009 0

Table (2.4) show that the debt B is used for consumption smoothing purposes. Here the positive correlation shows that the country uses more debt -B when the output is lower. The high standard deviation of this debt shows that this instrument escape substantially from the mean level, therefore indicating an intense use of this debt. The reason for using B for con-sumption smoothing is also straightforward. Because the set of possible resources that can be brought to present by the debt -B is larger when its pricing is good, and having -C kept large is very important,8

the consumption smoothing effort must be done with the defaultable debt instrument, B. Last but not least, the debt -B allows for default. Consequently when the output is lower it is good to have a defaultable debt for consumption smoothing reasons.

As already mentioned, the Debt -C is mostly kept at its maximum level (0.7 of the Trend Output). This is due to the arbitrage opportunity and the commitment use of this debt.

From Table 2.4 it can also be noted that the model with direct output costs present much lower correlation of the debt -B with the output than the BC Model. This is so because with this formulation, default in the good outputs is much rarer which leads to better prices in states of the world. This good prices, then, allows the country to use the same arbitrage opportunity of the -C debt. That is the reason this formulation has even more average defaultable debt (Table

2.3) than the standard formulation without direct output costs.

7

This statistics are built using simulations. I have created 500 time series of 103 quarters of the process of y (starting with the trend output). This allowed me building debt and default trajectories. When a country entered in default, I did not exclude it from markets. Because in my model default is rarer than in Arellano (2008) this would only increase the gap between my levels of debt and that of the author.

7

D.O.C. stands for Direct Output Cost

8

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14

Table 2.4: Long Run Statistics 2 - Variances and Covariances

Long Run Statistics9

Variable Arellano (2008) BC Model BC Model D.O.C.10

Corr(B’,y) -0.62 0.37 0.01

Corr(C’,y) 0 -0.01 0.00

Std. Dev. (B’) 0.04 0.40 0.49

Std. Dev. (C’) 0 0.01 0.02

To sum up, I have found in this first exercise that the country chooses large levels of both types of debt. The large levels of the debt -C have two motives. Firstly, the debt -C serves as a commitment strategy policy because large levels of this debt instrument increase the relative cost of defaulting. In addition, the high price on this debt instrument allows for an arbitrage opportunity due to the difference between the betas of the country and the lenders. The country, then, chooses, and constantly maintains, large levels of this debt .

In terms of default and the B debt, the country chooses high levels of defaultable debt and, at the same time, it almost never defaults. This is so because as -C reaches extreme large values, the country increases sufficiently the cost of default. As a result, no matter what y realizes in the future, default will practically never be optimal for the country, which then makes the B debt a risk-free instrument for large levels of debt11

Things change when the country cannot increase sufficiently the cost of default. If there is a harsh constraint in the level of collateralized debt, for example, default costs might not be high enough to guarantee no default in equilibrium for all states of future output. This is analysed in the next section.

2.4.2 Small C

In this new exercise I keep the distribution of y fixed, but I expand the computational trunca-tion to lower levels of output. Whereas in the last sectrunca-tion my markovian structure covered 3 standard deviations of the output process (where 𝑠.𝑑. = √︀ 𝜂

1−𝜌2). Now, I am going to cover

10 standard deviations of the output realization for both sides in the markovian structure and preserve the distribution. With this change the truncation will be less intense and the output can reach extreme, but rare, low values.

Whereas in the first exercise the range of the set Cwas from -0.7 to 0, now the range of the grid for C will be only from -0.2 to 0 because, now, the lowest possible output is 0.2356. As a result, the country can no longer increase substantially the cost of default, thereby impeding the commitment policy of the previous exercise for many outputs realizations. As this commitment policy can no longer be fully used, equilibrium bond prices are less favourable and the use of default increases. The general equilibrium forces therefore, makes default significantly present as a policy function in this environment. This can be seen in table 2.5.

Interestinly, however, now there is more default in equilibrium than in Arellano (2008). Be-cause there is still some sort of endogenous commitment strategy, the country still choose large levels of debt, compared to the Arellano environment - this is shown in Table 2.5. However this

11

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15

larger debt expose the country to the risk of output reversal. If the output suddenly changes to worse states of income, the country will inevitably default. This would not happen so intensely in Arellano because high levels cannot be visited by the country in general. Therefore, in Arel-lano there is less debt exposure than in this environment. This new environment facilitates debt accumulation, relative to Arellano, without substantially improving the commitment possibili-ties of the country. Moreover, now extreme low output events may happen. This forces the use of default. This model than prescribes that limiting the amount of F.D.I. may have actually increase the use of default because of the increased debt exposure.

Comparing to the previous formulation, with large levels of the C debt, now the country uses less debt in general. Because the commitment policy is less effective and, as a result, default is more present, prices are less favourable. Consequently, the country cannot visit regions of large levels of debt without incurring in high interest rates. In equilibrium is not worthy to reach this levels because lower levels generate less obligation and better prices. When direct output costs (D.O.C.) are introduced default becomes less favourable, prices get better and, as a result, more debt is used. In equilibrium there is still more default than in Arellano, though. Again, this is due to more debt exposure in conjunction with extreme low outputs.

Table 2.5: Long Run Statistics 3 - First Moments

Long Run Statistics

Variable Arellano (2008) BC Model BC Model D.O.C.

Gross Debt (B+C) -0.04 -0.08 -0.45

Defaultable Debt -0.04 -0.06 -0.27

Probability of Default 0.01 0.08 0.07

Table 2.6: Long Run Statistics 4 - Variances and Covariances

Long Run Statistics

Variable Arellano (2008) BC Model BC Model D.O.C.

Corr(B’,y) -0.62 0.03 -0.83

Corr(C’,y) 0 0.00 0.52

Std. Dev. (B’) 0.04 0.05 0.20

Std. Dev. (C’) 0 0.00 0.05

In terms of correlation, it can be seen from table 2.6 that with D.O.C. the use of default-able debt is positively correlated with the output variations. This happens because leverage conditions are better in good outputs. When the C debt was large, in the previous exercise, the consumption smoothing purpose of the defaultable debt was more prominent. As here the commitment strategy is limited, the consumption smoothing purpose of the B debt is limited and so there will be leverage because of the beta disparity.

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16

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Chapter 3

Conclusion

Default models based in rational expectations use lack of commitment and default punishments in the sovereign debt contracts to model default choices. This effort showed that a self reinforc-ing dynamics operate and for sufficiently low outputs and high levels of debt, default is a better choice than repayments because the price of the bond is pro-cyclical and blocks the capacity of the country to roll-over its debt. As payments when output is low are very utility costly, default arises as a policy. However, none of these models account for the (subjective and objective) guarantees given by countries such as fiscal responsibilities acts, foreign reserves accumulation or openness to foreign direct investment. This guarantees, then, can be considered commitment policies. In other words, countries choose not only if it defaults or not or the levels of sovereign debt it will take but also the extent at which these obligations is ex-ante guaranteed.

To account for this additional governmental choice, in this paper, I have introduced collat-eralization in the sovereign debt environment of Eaton-Gersovitz (1981) - later sophisticated by Arellano (2008) - to study commitment policies by the country. I find that, with the usual cal-ibration of Arellano (2008), countries avoid default by choosing the largest level of guaranteed debt therefore making forced payments in case of default sufficiently large. Large payments, at the same time, make defaulting extremely costly. On the equilibrium path, there is no default, which makes the more abundant risky debt a risk-free bond. Because the risky debt is more abundant, the consumption smoothing effort made by the country is accomplished using this bond. Therefore, the guaranteed debt is used for commitment purposes whereas the risky debt is used for consumption smoothing purposes.

Things change when I extend the output structure to cover large, but still rare, deviations from the trend, preserving the output stochastic process. As extremely low outputs can be reached, the range of the possible values for the collateralized debt diminishes. Consequently, the representative country can no longer commit to never default in the equilibrium choices of debt. Prices of the uncollateralized bond reflect, then, this risk. The more the range of the collateralized debt diminishes the more intense is the use of default. As a result, when collateral-ization gets limited this environment produces results similar to Arellano (2008), where default is a present policy of countries.

My results show that the reason for sovereign default might be the lack of collateral. If countries could commit never to default, then pro-ciclicality of prices in relation to output and the counter-ciclicality of prices in relation to present debt levels would be eliminated. As a re-sult, the country would have access to a risk-free bond all the time and default would be avoided.

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18

prices of defaultable sovereign debt. This higher prices allows the country to increase the level of observed debt and better smooth the consumption path of the representative agent. High levels of debt arise because the country is more impatient than lenders and prices do not collapse. We find that in this environment default is mainly avoided.

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Chapter 4

References

Araujo, A. and Kubler, F. and Schommer, S. (2009) "Regulating Collateral Requirements When Markets Are Incomplete ",Journal of Economic Theory

Arellano, C. (2008), "Default Risk and Income Fluctuations in Emerging Economies",American Economic

Conesa, J.C. and T.J. Kehoe (2012) "Gambling for Redemption and Self Fulfilling Debt Crises",Federal Reserve Bank of Mineapolis Staff Report

Eaton, J and M. Gersovitz (1981), "Debt with Potential Repudiation: Theoretical and Em-pirical Analysis"Review of Economic Studies

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Chapter 5

Code

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Chapter 6

Appendix

In the figures below it is shown that mostly the country chooses the lowest level of C - -0.7- or the largest level of debt. This is due to the betas difference between lenders and the country which gives the country an arbitrage oportunity.

Figure (6.1) show that even for the highest possible output and the lowest possible debt (which is to be a creditor of 0.3) the country chooses to have large levels of C debt. At the same time, figure (6.2)shows that when the outptu is in its trend levels of indebtedness increase even more. These are the cases where the C policy function are mildest, In most cases I have found that the C policy function is the highest possible.

Figure 6.1: Large Levels of C

−0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0

−0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0

C

C’

C Debt Increasing

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22

Figure 6.2: Large Levels of C

−0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0

−0.7 −0.65 −0.6 −0.55 −0.5 −0.45 −0.4 −0.35 −0.3

C Debt Increasing

C’

Referências

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