GDP-indexed bonds in perpetuity and sovereign default

Transcription

1 GDP-indexed bonds in perpetuity and sovereign default Yasin Kürşat Önder April 4, 2017 Abstract This paper analyzes the gains of introducing GDP-indexed bonds in an equilibrium sovereign default model with long-term debt. The interest in these financial products re-emerges with cycles and has recently peaked with European debt crisis. This paper aims to inform policy discussions with a formal analysis and I find that depending on the indexation method, GDP-indexed bonds may generate both positive and adverse outcomes. This paper shows the gains for the government and investors are amplified if investors fully participate in GDP developments with which periodic coupons are proportionally linked. Remarkably, these instruments yield welfare losses with one-period debt. Keywords: sovereign default, sovereign debt, GDP-indexed bonds, debt dilution, longterm debt JEL Codes: F30, F34 kursatonder@gmail.com, Address: Muhsin Yazicioglu Cad Sok. Cankaya Ankara, Turkiye. Phone: +90 (533) i

2 DISCLOSURE STATEMENT: I have nothing to disclose i

3 1 Introduction As the severity of European debt crisis unfolded, the interest on GDP-indexed bonds re-emerged. The rationale for tying debt repayments to economic activity has a long history and interest in such contracts typically re-emerge with recurring debt crises. It first gained prominence in the aftermath of the debt crisis in the 1980s. A further interest gained momentum with Shiller s 1993 and 2012 proposals which called for a new market for GDPindexed securities. There is now a renewed interest with the recent debt crisis in Europe and debates reinvigorated among policy makers and platforms including G20 which has recently issued a statement we call for further analysis of the technicalities, opportunities, and challenges of state-contingent debt instruments, including GDP-linked bonds (G20 (2016)). 1 Proposals of GDP-indexed bonds are motivated with the mitigated debt-services during economic downturns so as to provide relief for governments to avoid a default. To date, some countries such as Costa-Rica, Bulgaria, Bosnia and Herzegovina, Argentina, Greece and Ukraine issued GDP-indexed bonds as part of their debt restructuring deals. Singapore so far has been the only nation issuing these instruments that is independent of a debt restructuring agreement. 2 There are conflicting views on the likely gains of introducing GDP-indexed bonds. One view conjectures that the government would default less and both investors and the government would benefit from such bonds. These bonds may facilitate as an automatic stabilizer through lower payments during economic downturns and may provide room for a countercyclical policy. 3 Another view stipulates that the gains would be very limited and may lead losses for both parties. This view also stems from the previous unsuccessful 1 Greece issued GDP warrants in 2012, and on top of that it has suggested to replace its existing debt with GDP-indexed bonds in perpetuities in 2015 (see Economist (Feb 7, 2015)). Similarly, Ukraine also reached a debt restructuring agreement with its creditors and swapped its debt payments with GDP warrants (see Eder (Aug 28, 2015)). 2 See Borensztein et al. (2004) for a summary of the established deals that Costa-Rica, Bulgaria and Bosnia and Herzegovina had as part of Brady restructuring deals. 3 This view is more prevalent: Shiller (1993), Shiller (2012) and Borensztein and Mauro (2004) to name just a few. 1

4 GDP-indexed bond issuances of Bulgaria, Bosnia and Costa Rica in the 1990s as well as Argentina in This is partly because of the poor contract design. 4 On a broader level, my results highlight that both the government and investors may either be better off or worse off depending on the undertaken indexation method. To this end, my paper sheds light on current policy discussions with a formal analysis by quantitatively investigating the gains from different indexation methods and highlights the importance of contract design for a successful bond issuance. Formally, this paper presents a quantitative analysis of GDP-indexed bonds using an endogenous sovereign default framework à la Eaton and Gersovitz (1981). The benevolent government maximizes the utility of the representative household which receives stochastic stream of single tradable good. The government and investors observe the government s current debt holdings and income at the beginning of the period and the government then decides to repay its debt or default. It is a typical assumption in the literature that the government is excluded from the credit markets for a stochastic period of time after defaulting and is subject to an income cost during its exclusion. After a repayment decision, the government issues non-contingent debt in the baseline model. In an economy with GDP-indexed bonds, the government s current payment obligations vary with the changes in its income and the payment schedule differs according to the undertaken indexation method. Bonds are modeled as perpetuities and I consider four different widely discussed indexation methods along with alternative forms in order to offer the highest welfare generating contract. The first one considers a floor on the payment so that the investor receives extra coupon if GDP exceeds a certain predetermined threshold. The coupon payment alternates proportionally depending on the income realization. The floor is offered by the issuers with the presumption that investors would not like their coupon payments declining during economic downturns (see Bank of England (2015)). The Greek GDP-indexed securities issued in 2012 can also be considered under this first category. 5 The second alternative considers unfloored payments with which the return 4 Previous GDP-indexed bonds for the listed countries are considered, at best, to have very limited success. 5 See Hellenic Republic Ministry of Finance (2012) for details of such securities. 2

5 varies depending on the income realization; coupon payments decline during economic downturns and increase during upturns. As in the first option, the coupon payments vary proportionally with changes in income however investors are taking two-sided symmetric risks with the second alternative. When the economy s growth rate is less than its long-run trend, investors receive lower coupons. As one of the very earlier proposals of GDPindexed bonds, Shiller (1993) can essentially be considered under this category. 6 The third alternative considers a ceiling on the payment so that there is a cap on the payments which would ensure the sovereign not paying excessive returns during upturns. For instance GDP warrants which were issued by Bulgaria, Bosnia, Costa Rica, Argentina, Greece and Ukraine have ceilings on their payment structure (see Bank of England (2015), Griffith-Jones and Sharma (2006)). The first three indexation methods were also proposed by Schroder et al. (2004) after conducting interviews with market participants. In the fourth and last alternative, which was also undertaken by Singapore in 2001 and was highlighted by Borensztein and Mauro (2004), investors are guaranteed to receive coupons each quarter and promised to receive extra returns if GDP exceeds its long-run average. Different than the first three alternatives, the extra return does not proportionally vary with coupons, that is, for example if the real-gdp growth is three percent higher than its long-run growth then investors receive extra three percent returns. With the last indexation method the government s payments vary with the outstanding debt, however, in the first three alternatives the government s payments alternate with the coupon payments rather than the outstanding debt. 7 I also further analyze GDP-indexed contracts which encompass the features of both ceiling and floor. By investigating these different indexation methods, this study would help to design an optimal contract for policymakers. 6 Robert Shiller declares these financial products to be one-to-one linked to the developments in GDP and refers to them Trills (a trillionth of a share of GDP). 7 For example, consider a coupon structure such that the sovereign s outstanding long-term debt to GDP ratio is 50 percent and makes periodic coupon payments that are equivalent to 5 percent of its GDP. For the first two indexation methods the sovereign s total debt obligations increase by 0.15 percent and for the last method the total debt obligations increase by 1.5 percent when the economy s growth rate is 3 percent larger than its long-run trend. When the economy s growth rate is 3 percent lower than its long-run trend, the total debt obligations fall by 0.15 percent under unfloored and ceiling methods, remains unchanged under floored method and declines 1.5 percent with the unproportional method. 3

6 This paper finds that the type of GDP-indexation method is vital for a successful implementation. Some contract suggestions that are widely cited and some of which are implemented leave the agents with lower welfare and the government fails to raise funding following an increased default risk. For instance, the first alternative (floored method) generates, at best, very limited gain for the sovereign while the last alternative (unproportional method) generates losses. Nevertheless, this paper features a contract with which both the government and investors are able to benefit. In a contract where investors take two-sided risks (claims are higher (lower) if GDP grows more (less) than the trend) and the returns are proportionally linked to the baseline coupon, I show that the government s ex-ante welfare gains increase by 0.16 percent in terms of compensating variation in permanent consumption when there is no initial debt and investors enjoy higher capital gains. Also, these contracts provide a better hedge for a countercyclical policy than with non-contingent debt. Thus, contracts where investors fully participate in the GDP developments generate the highest welfare. It should also be noted that while unfloored payments provide a better hedge during downturns, floored payments do also provide limited insurance. As they promise a higher payment during upturns, floored payments induce higher prices which lead to a reduction in the sovereign s outstanding debt obligations and default risk. These results stem from the non-linearity of the price schedules. The price of unfloored instruments is higher during economic downturns than with non-state contingent debt because the default risk is mitigated following the declined debt burden. On the other hand, the default risk with such instruments is higher during economic booms since the government now promises to make higher payments. Concurrently, the yield is lower (yield falls when a bond price rises) because the government promises higher coupon payments. So, there are different forces working in opposite directions. With the quantitative exercise, this paper shows that the asymmetric gains during economic downturns is higher than the possible losses during economic booms during which the price schedule is more flat. One of the key contribution of this paper is on the understanding of long-term debt. I evaluate the gains of introducing GDP-indexed bonds with one-period debt and this 4

7 analysis reveals dramatic differences in the implications. With one-period debt, it is puzzling that these contracts generate welfare losses for the sovereign under all different specifications. I will explain the benefit and the rationale of using long-term debt as a modeling choice in more depth under literature review. To this end, this paper documents the potential pitfalls of using short-term debt as a modeling tool which may lead to false policy implications. The discussions on GDP-indexed bonds are often centered on the gains of a sovereign while the gains/losses of investors are not addressed. In this regard, this paper also contributes by computing the gains/losses of investors for all analyzed indexation methods. It is important to note that there are concerns on the feasibility of arranging GDPindexed bonds such as data revisions or a methodology change in GDP calculation. Borensztein and Mauro (2004) and Griffith-Jones and Sharma (2006) address these concerns by proposing to establish independent institutions to ensure that data revisions can be adjusted for the subsequent payments as in the case of inflation indexed bonds issued by the United Kingdom (Council of Economic Advisers). If there is a change in the calculation of GDP series, the old series that the indexed coupons are tied to can be maintained (see Borensztein and Mauro (2004) and Griffith-Jones and Sharma (2006)). Literature Review. The framework provided in Eaton and Gersovitz (1981) is the backbone of the recent quantitative sovereign default models (e.g., Aguiar and Gopinath (2006), Arellano (2008)). This paper belongs to the same class of models. Robert Shiller is a big advocate of such bonds, which he calls Trills because a government would pay one-trillionth of a country s GDP each year in quarterly dividend payments. The payoff, however, varies. If the economy grows on the upside, dividends would increase; if the economy slides, dividends would decline (see Shiller (1993), Shiller (2012) and Borensztein and Mauro (2004)). In addition, there are some studies suggesting to link the payments to commodity prices as in Froot et al. (1989). However, all these models are abstracted from a formal analysis where the default decision of a sovereign is endogenously determined. Despite the growing interest in GDP-indexed bonds, the number of formal studies investigating the effects of such bonds falls short. Athanasoulis and Shiller (2001), Durdu 5

8 (2009) and Barr et al. (2014) study the effects of GDP-indexed bonds in frameworks of one-period debt and they are all abstracted from an endogenous default decision. To this end, the framework provided in this study contributes to this literature by highlighting the differences generated by different indexation methods. Sandleris et al. (2011) and Hatchondo and Martinez (2012) present the effects of such an arrangement with strategic defaults. Both studies model the debt with one-period maturity. The former one assumes that the payments are constant across states and time. The latter study makes an assumption such that with the GDP-indexed bonds the government is abstracted from choosing a debt level for which it would default. This is achieved by allowing the sovereign selecting a payment schedule after the endowment realization; therefore the government is not committed to any indexation method ex-ante. With that assumption in place, the government never defaults. In stark contrast, this paper documents that these financial instruments generate welfare losses with one-period debt when endogenous sovereign default risk is accounted. Standard quantitative models of endogenous sovereign default with one-period debt typically fail to generate key stylized facts of sovereign debt moments such as debt-to-gdp ratio and sovereign spreads. Using long-term debt as a modeling tool features matching such moments observed in the data, and Hatchondo and Martinez (2009), Arellano and Ramanarayanan (2012) and Chatterjee and Eyigungor (2012b) document the superiority of using long-term debt over short-term debt in order to generate plausible debt and business cycle moments. Another feature of long-term debt is its functionality for exploring alternative forms of indexation methods. To this end, the framework provided in this analysis is more amenable for quantitative analysis and also for the evaluation of the effects of different indexation methods. To the author s knowledge, the analysis in this paper is the first to formally evaluate the impact of GDP-indexed bonds with long-term debt on sovereign spreads, default risk, debt-to-gdp ratio and the government s welfare as well as investors gain in a framework with endogenous sovereign default risk. There are many more alternatives of GDP-index methods that are currently discussed among the policy circles such as conditionality, callable bonds and trigger mechanisms. However, we are bounded by modeling choice and it is not technically feasible to evaluate each of them. This study covers significant 6

9 alternatives of indexation methods that are commonly discussed and it is not to say that any other mechanism cannot generate significantly higher gains or they are all welfare reducing. The analysis of GDP-indexed bonds is also related to the literature on debt dilution problem which refers to the adverse impact of a sovereign s current borrowing on outstanding debt that is induced by a future government s borrowing behavior (see Hatchondo et al. (2016) and Chatterjee and Eyigungor (2012a)). The probability of a future government s high debt issuance and thus the risk of defaulting, leads the current creditors to lend at a discounted rate so that they can cover their future losses in expectation. This makes borrowing with long-term debt costly and may eventually leave the sovereign with welfare losses. To this end, some indexation methods that are covered in this paper may help tackle this issue for the sovereign by limiting the sovereign s ability to issue debt during economic upturns. This paper also relates to recent growing literature on equity based contracts and risk sharing contracts to resolve the debt crises. Önder (2016a) shows that if the sovereign debt contracts are linked to tangible assets, such as building or land, where the usufruct of the asset is transferred to the contract holder, the risk of default falls, the government raises higher funds and enjoys higher welfare gains than the economy with non-contingent debt. These types of sovereign contracts are prevalent in Islamic finance and GDP indexed bonds is considered to be promoting risk sharing between the borrowers and investors. The remainder of the paper proceeds as follows; Section 2 presents the benchmark model and subsequently introduces GDP-indexed bonds; Section 3 explains the benchmark calibration and Section 4 compares the outcome of the GDP-indexed model with the one obtained with benchmark model. Section 5 documents the dramatic differences in the implications of the model with one-period debt; Section 6 concludes and Appendix A discusses the numerical algorithm. 7

10 2 Model with non-contingent debt This section features a dynamic small open economy model with non-contingent defaultable long-term debt. The economy is endowed with single tradable stochastic consumption good. The shocks to the economy are uninsured and time is discrete. 2.1 Environment Preferences and endowment. This paper considers a small open economy in which benevolent government maximizes the utility of the infinitely lived representative household and has preferences given by: E 0 t=0 β t u(c t ) (1) where E denotes the expectation operator, 0 < β < 1 is the intertemporal discount factor and c t denotes aggregate consumption at time t. The utility function u(.) is an increasing, continuous and strictly concave function which belongs to the class of CRRA utility functions and can be written as: u(c) = c1 γ 1 γ (2) with a risk aversion parameter γ. Each period households receive a stochastic income y Y R ++ which follows a Markov process: log(y t ) = (1 ρ) µ + ρ log(y t 1 ) + ε t, with ρ < 1, and ε t N ( 0, σ 2 ɛ ). Asset space. It is assumed that the government issues a perpetuity with geometrically decreasing coupon κ at a rate δ (0, 1]. 8 A bond issued in period t promises to pay 8 If default is avoided, price of the debt will be risk free and average one-period bond price is equivalent to the present value of the non-contingent long-term debt. Coupon payment κ is calculated such that default-free price of the debt is equal to the average one-period debt price and is formally derived in Section

11 (κ, (1 δ)κ, (1 δ) 2 κ,..., 0) units of the tradable good in subsequent periods. This is a typical formulation for long-term bonds to avoid keeping track of the entire distribution of different bond maturities (see Hatchondo and Martinez (2009), Chatterjee and Eyigungor (2012b) and Arellano and Ramanarayanan (2012)). Hence, long-term debt dynamics can be represented as follows: b t+1 = (1 δ)b t + l t, where b t and b t+1 are the total number of outstanding coupon claims at the beginning of time t and t + 1, and l t is the number of long-term bonds issued in period t. Notice that δ = 1 is a special framework of long-term bonds and corresponds to the economy with one-period debt. The government chooses how many bonds b to sell from a finite set B = {b 0, b 1,...b I 1, b I } where b I > b I 1 >... > b 1 > b 0. It is common in the literature to assume that the government is a net issuer, so cannot accumulate asset. 9 Throughout the text, x and x denote any variable x at period t and t + 1, respectively. GDP-indexed payments function. GDP-indexed payments function can be represented as follows: ( y y ) Γ(y) = 1 + θ k y (3) where y is mean trend-income and θ k {θ L, θ H } with subscript L = low and H = high. 10 This flexible representation of GDP-indexed payment function will enable a researcher studying a wide variety of different indexing methods and will help us sharpening our understanding of the mechanics of the long-term debt as well. The function encompasses several alternatives: θ k = 0 is the non-contingent economy with no GDP-indexed bonds (benchmark economy), θ k = 1 is an economy with GDP-indexed bonds promising unfloored payments, θ L = 0 for y < y and θ H = 1 for y y is an economy with GDP- 9 Önder (2016b) shows that allowing the government to have a saving technology generates same results for the class of Eaton-Gersovitz models. 10 Typically, these indexation methods depend on a threshold GDP level to tie the coupon payments (see Bank of England (2015) and Government of Singapore (2001), Borensztein and Mauro (2004) and Durdu (2009)). 9

12 indexed bonds promising floored payments and many more variants of this indexing function is going to be explored in depth in subsequent sections. Timing. The timing of events within each period can be summarized as follows. The government starts a period with total debt b and income y which are both public information. If the government repays its debt, then investors offer a menu of prices depending on the government s amount of debt issuance and current income. The government subsequently chooses how much debt to issue this period. 11 If the government reneges on the contract, it will be excluded from the credit markets and its income becomes y φ(y) for each period it is excluded. The government comes back next period with constant probability ψ. It is common in sovereign default studies that the recovery rate is zero when the government regains access to the credit markets. 2.2 Recursive formulation Let V R (b, y) and V D (y) denote the value of repayment and value of default functions, respectively. The latter is not a function of debt because the government cannot issue debt during a default episode. Let V(b, y) be the value function for the government that has option to default with outstanding debt holdings b and income y. For any price function q, V(b, y) satisfies the following functional equation: V(b, y) = Max {V R (b, y), V D (y)}, (4) where the government s value of repayment is given by { } V R (b, y) = max u (c) + βe y yv(b, y ), (5) b B subject to c = y κbγ(y) + q(b, y) [ b (1 δ)b ]. (6) where Γ(y) is a flexible GDP-indexed payments function. 11 This is identical with the government offering investors a menu of prices and investors choosing how much to lend in equilibrium. Majority of the studies in the quantitative sovereign default literature describe similar game set ups. 10

13 When the government defaults, it is barred from credit markets for a stochastic period of time and there is an income cost during the time it is excluded. The government regains access to the credit markets with constant probability ψ [0, 1] and has zero debt at the time of market re-entry. The value of defaulting satisfies: [ V D (y) = u (c) + βe y y (1 ψ)vd (y ) + ψv(0, y ) ], (7) subject to c = y φ(y). During the exclusion period, the government faces a default cost of φ(y) = d 0 y + d 1 y 2 and it is assumed that the government repays when value of repayment is equal to the value of default. The solution to this problem yields a default decision rule ˆ d(b, y) {0, 1}, 1 if the government defaults, 0 otherwise; and borrowing decision rule ˆb(b, y) when repayment is optimal. In equilibrium, defined in Section 2.4, investors use these decision rules to price contracts. 2.3 Investors problem Bond market is competitive and investors are assumed to be risk-neutral so they take the price schedule q(b, y) as given. The opportunity cost of funds is given by the exogenous risk free interest rate r > 0. With the zero-profit assumption and no arbitrage condition in place, price function solves the following functional equation: ( 1 ˆ ) d (b, y ) κγ(y) q(b, y) = E y y = E y y b = ˆb(b (y ), y ). 1 + r [( 1 d ˆ ) (b, y ) + ( 1 d ˆ ) ( (b, y ) 1 d ˆ ) (b, y ) κ(1 δ)γ(y ) (1 + r) 2,... ] [κγ(y) + (1 δ)q (b, y )] 1 + r, (8) 11

14 In the absence of default risk, price of the non-contingent long-term debt (notice that θ = 0 for non-contingent debt and thus Γ(y) = 1) equals so, one can write κ = r+δ 1+r. q = κ r + δ = r The expected value of investing in a bond should be equal to the expected return of investing in a risk-free asset with a return of r. If an investor purchases non-contingent debt today, he will receive κ units of goods plus the remaining value of its receivables which worth (1 δ)q (b, y ) tomorrow conditional on government s repayment. In an economy with GDP-indexed bonds, the investor receives κγ(y) units of goods tomorrow instead of receiving κ. 2.4 Definition of equilibrium This paper focuses on Markov Perfect Equilibrium (MPE). Hence, equilibrium conditions on the current state variables, not on the entire history. Definition 1 A Markov Perfect Equilibrium is characterized by 1. a collection of value functions V, V R and V D ; 2. rules for default ˆ d and borrowing ˆb; and 3. a debt price function q; such that: { i. given a price function q; V, V R, V D, d, ˆ ˆb } solve the Bellman equations (4), (5), and (7). { } ii. given policy rules d, ˆ ˆb, the price function q satisfies condition (8). 2.5 Model with GDP-indexed bonds This section investigates the gains from different GDP indexation methods (to be described below). The government now offers periodic payments which are tied to the developments in income growth. 12

15 2.5.1 Proportional changes in coupon payments Coupon payments vary proportionally with the realized income. For instance if the realized income at time t is three percent higher than mean trend-income, coupon payments would be three percent higher than the model with non-contingent debt. Under this subsection, I will study three different indexation methods: floor, unfloor and ceiling on payments Floored payments The rationale behind such type of an indexation method was to assure investors by promising a coupon amount that at least pays a lower bound which is similar to what they would have received with non-state contingent debt. Thus, this type of GDP-indexed bonds provides a partial insurance for investors because periodic coupon payments are not negatively affected with the adverse developments in GDP. This is similar to what Greece issued in 2012 and is widely discussed among policy circles (see Griffith-Jones and Sharma (2006), Borensztein and Mauro (2004) and the references therein). Upon income realization at the beginning of time t, total coupon payments become κbγ(y) conditional on the government s repayment decision. The parameter θ L = 0 and θ H = 1 in the GDP-indexed function Γ(y), thus investors would at least receive κb and are insured from adverse developments of country s income. The expected payment the investors receive depends on the fractional income growth Γ(y). In order to better understand the dynamics of the model and the behavior of long-term debt as well as other variants of floored payments, this paper also considers alternative values for θ H Unfloored payments A criticism of the floored payments is that they do not carry out the underlying motivation of risk sharing during economic downturns. The novelty of such instruments that is discussed in Shiller (1993) and Shiller (2012) is to have a market that shares the macroeconomic risks. Also, this is expected to help the government recovering from the economic slowdown without a default by lowering the government s debt burden. 13

16 In this variant of GDP-indexed bonds, investors fully participate to the developments in a country s GDP. There is no floor and thus investors fully participate both to positive and adverse income realizations. To this end, θ k = 0 in the GDP-indexed payment function Γ(y). With this way of indexing, investors now share the cost of economic downturn. Note that this way of linking coupons with GDP realizations ensure a non-negative payment and thus monotonicity in the pricing function. For the baseline analysis, investors take symmetrical risk sharing for both the upside and downside. To better evaluate these unfloored payments, alternative values of θ k will be utilized. When we are discussing the results, it will turn out that sharing the cost of a crisis will provide the highest gain both for the government and investors. Proponents of floored type of payments argue that investors would be worse off with unfloored payments and may not participate. Relegating the details of the mechanism in Section 4, our analysis shows that such bonds would reduce the default risk and thus increase the price of the asset during downturns which leaves investors better off Ceiling on payments This section now investigates the impact of ceiling on payments. Such types of clauses were considered by sovereigns so that during high growth states payment obligations do not rise substantially (Griffith-Jones and Sharma (2006)). Although investors do not like the idea of ceilings (see Schroder et al. (2004)), there is still a merit analyzing these bonds to better understand the long-term debt dynamics and to inform the policy discussions considering such indexation method. Under this method, GDP-indexing function Γ(y) becomes Min ( y y, y +θ c σ y y ) where σ y denotes one standard deviation of aggregate income and θ c is a constant indicating the degree of ceiling on the payment. For instance when θ c = 2 and the current aggregate income is three standard deviations above the mean, the coupon payments will proportionally increase by two standard deviations of aggregate income. In the results section, I consider different values of θ c and I also consider a contract with both ceiling and floor on payments to investigate the price and welfare dynamics. 14

17 2.6 Unproportional changes in coupon payments In this subsection, the government makes additional payments on top of κ. This type of GDP-indexed bond was issued by Singapore in 2001 and also commonly studied in papers without endogenous default risk (see Borensztein and Mauro (2004), Bank of England (2015) and Schroder et al. (2004)). 12 In contrast to Section 2.5.1, coupon payments do not vary proportionally. For example if the realized income at time t is three percent higher than the mean trend-income, then the government s total debt obligations increase by three percent. Whereas in Section 2.5.1, the current coupon payments go up by three percent. To illustrate, consider a sovereign with 50 percent outstanding debt-to-gdp ratio promises 5 percent periodic coupon payments. With this indexing method, the sovereign s current payment would rise 1.5 percent while increasing 0.15 percent under proportional changes in coupon payments for an income realization three percent higher than the mean-trend income. In this section, I only study the floored payments because with unfloored payments a researcher would have a non-monotonic price function with respect to income y. Details of this non-monotonicity are discussed in Section Floored payments With this type of indexation method, investors are guaranteed to receive at least coupon κ conditional on government s repayment decision. If the government s income realization is bigger than its mean income, investors also receive additional coupon payment that is equal to the income growth rate. In this specification, budget constraint in equation (6) becomes c = y (κ + Γ(y))b + q(b, y) [ b (1 δ)b ], (9) 12 Singapore issued the New Singapore Shares which pay a fixed three percent dividend payments per annum plus an extra dividend if the real growth rate is higher than the predetermined baseline GDP (see Government of Singapore (2001)). Singapore so far has been the single country issuing these instruments out of sovereign debt restructuring deal. 15

18 with θ L = 0 and θ H = 1 in the GDP-indexed function Γ(y). Note that GDP-indexing function is added to coupon payments κ in contrast to equation (6). Thus, for an income realization bigger than mean trend-income, the sovereign has to make Γ(y)b additional payments to the investors in contrast to additional payment of κb(γ(y) 1) in Section Price function in equation (8) now reads as q(b, y) = E y y [( 1 d ˆ ) ] (b, y ) [κ + Γ(y) + (1 δ)q (b, y )]. (10) 1 + r 3 Calibration It is typical in the literature to have quadratic loss function with which the cost of default increases more than proportionally with income. Mendoza and Yue (2012) provide a framework where defaults endogenously lead higher output declines when income is higher. This assumption is also empirically documented by Chuhan and Sturzenegger (2003) and Arteta and Hale (2008) and also allows the equilibrium default models to match the debt-to-gdp ratios as well as the sovereign bond yield observed in the data. Following Chatterjee and Eyigungor (2012b) I assume the following functional form for cost of defaulting: φ d (y) = d 0 y + d 1 y 2. Table 1 presents the model parameters used in the paper. A period in the paper corresponds to a quarter. Risk aversion rate γ and risk free rate r are common in sovereign default literature. Probability of market re-entry parameter ψ assumes an average exclusion of one-year from the market which is also the value used by Hatchondo and Martinez (2009) and Arellano (2008). Average exclusion period is also within the range of average exclusion period documented in Cruces and Trebesch (2013) and Gelos et al. (2011). It is common to calibrate the economy to Argentina in the literature. The income process parameters are estimated using the detrended real GDP series of Argentina between the first quarter of 1980 and the last quarter of The auto-correlation coefficient for the output process as well as the standard deviation of innovations is standard estimates. 16

19 To match the duration of debt, δ is set to be With this value, average debt duration is around 4 years in the simulations, which is roughly the average debt duration for Argentina reported in Broner et al. (2007). Remaining parameters to explain are discount factor β and income cost of defaulting parameters d 0 and d 1. These parameters are used to match the average debt-to-gdp ratio as well as the mean spreads. Cost parameters d 0 and d 1 are calibrated to be and 1.076, respectively and β is set to be 0.96 which is a standard value in the literature. The debt statistics and the business cycle statistics in the data are obtained from Hatchondo and Martinez (2009). Macaulay (1938) definition of duration is used to compute the duration of the long-term debt. Duration D is the weighted average maturity of future cash flows. A bond issued at time t promising to make periodic payments κ for the subsequent periods at time 1, 2,..., J years into the future with the final price of zero has duration D Quantitative results I initially present the simulation results of the baseline model (economy without the GDP-indexed bonds) and match the target statistics. I then show the model results of GDPindexed bonds with different indexation methods one by one and compare the outcomes with the one obtained in the baseline model. I start presenting the results with floored and unfloored payments in Sections 4.2 and 4.3. Then I analyze the impact of ceiling and unproportional methods in Sections 4.4 and 4.5. Subsequently I investigate the welfare 13 D = 1+i i+δ where i is the periodic yield an investor would earn if the bond is held to maturity with no default and it satisfies κ(1 δ) q = j 1 j=1 (1 + i) j. The sovereign spread r s is computed as the difference between yield i and the risk free rate r. Annualized spread reported in the tables is computed as 1 + r s = ( ) 1 + i r The debt levels obtained from the simulations are equivalent to the present value of future debt obligations and computed as b δ+r. 17

20 Table 1: Parameter Values Parameter Value Risk aversion γ 2 Risk-free rate r 1% Probability of reentry after default ψ Income autocorrelation coefficient ρ 0.9 Standard deviation of innovations σ ɛ 2.7% Mean log income µ (-1/2)σɛ 2 Debt duration δ Calibrated Discount factor β 0.96 Income cost of defaulting d Income cost of defaulting d dynamics for the sovereign and document the highest welfare generating contract as well as the ones featuring welfare losses in Section 4.6. In order to fill the gap in the literature that remains silent on the likely gains of investors in the event of a hypothetical voluntary debt exchange, I compute the capital gains for each method in Section 4.7. Lastly, in order to better understand the intuition of the results we obtained in these sections I study the price dynamics in Section Results with non-contingent debt Table 2 presents the results with non-contingent debt as well as the moments in the data. The simulation results of the baseline economy (θ k = 0) match both the long-run sovereign debt moments and the business cycle moments reasonably well. To date, business cycle moments were matched by previous studies. 14 For the next subsections, I will discuss the role GDP-indexed bonds play and compare the outcome of different indexation methods with the one obtained here. 14 See Aguiar and Gopinath (2007) and Neumeyer and Perri (2005). 18

21 Table 2: Long-run statistics: effects of introducing GDP-indexed bonds Floored, θl = 0 Unfloored Data Bmark θh = 1 θh = 2 θh = 3 θ k = 1 θ k = 2 θ k = 5 Unprop Debt Statistics Mean debt-to-gdp Mean rs Mean rs downturns σ(rs) Default rate Resource inflows Debt duration (yrs) Business Cycle Statistics σ(c)/σ(y) σ(tb/y) ρ(c, y) ρ(rs, tb/y) ρ(rs, y) ρ(tb/y, y) The second column reports the data moments and the third column presents the simulation results with noncontingent debt. Columns 4-6 show the results of GDP-indexed bonds with floored coupon payments. Columns 7-9 show the results of GDP-indexed bonds with unfloored coupon payments. The last column presents the results of GDP-indexed bonds where the coupon payments do not proportionally vary with the developments in income. Standard deviation of a variable h is denoted by σ(h) and the coefficient correlation between variables h and m is denoted by ρ(h, m). Consumption and income are reported by natural logs. For GDP-indexed bonds, debt stock is computed using expected payments; the yield and duration as well as spreads are computed by generating a debt covenant that it is never defaulted on. To this end, the simulations incorporate the adjustment stemming from different GDP-indexation methods. In the simulations, details of which are explained in Appendix, I compute the yield (and the spreads) such that yield of the debt covenant is equal to the price of the GDP-indexed bond. Resource inflows are measured by 100*q(b, y) [b (1 δ)b]. 19

22 Table 3: Long-run statistics: effects of introducing GDP-indexed bonds Ceiling and Unfloored Ceiling and Floored Data Bmark Unfloored θc = 1 θc = 2 θc, θ f = 1 θc, θ f = 2 Debt Statistics Mean debt-to-gdp Mean rs Mean rs downturns σ(rs) Default rate Resource inflows Debt duration (yrs) Business Cycle Statistics σ(c)/σ(y) σ(tb/y) ρ(c, y) ρ(rs, tb/y) ρ(rs, y) ρ(tb/y, y) The second column reports the data moments,the third column presents the simulation results with noncontingent debt and the fourth column displays the results with unfloored coupon payments. Columns 5-6 show the results of GDP-indexed bonds with a ceiling on coupon payments. Columns 7-8 show the results of GDP-indexed bonds with both floor and cap on coupon payments. Standard deviation of a variable h is denoted by σ(h) and the coefficient correlation between variables h and m is denoted by ρ(h, m). Consumption and income are reported by natural logs. For GDP-indexed bonds, debt stock is computed using expected payments; the yield and duration as well as spreads are computed by generating a debt covenant that it is never defaulted on. To this end, the simulations incorporate the adjustment stemming from different GDP-indexation methods. In the simulations, details of which are explained in Appendix, I compute the yield (and the spreads) such that yield of the debt covenant is equal to the price of the GDP-indexed bond. Resource inflows are measured by 100*q(b, y) [b (1 δ)b]. 20

23 4.2 Results with floored payments The outcome of this exercise is reported in Table 2 under floored column and I compare the outcome of this model with the one obtained with non-contingent debt. Proponents of such type of indexation method have argued that such bonds would have significant effects on reducing the borrowing costs and increasing the amount of fund the government can raise. As anticipated by the detractors of such bonds, floored payments fail in mitigating cost of debt services during downturns. Mean spreads during downturns, which is measured as two-standard deviation below the mean trend income, do not change compared to the economy with non-contingent debt. With floored payments these bonds also lead to higher defaults in equilibrium for the states when the government is holding excessive debt (see upper panel of Figure 2). 15 This is because the coupon payment burden is multiplied by the existing debt and realized income (κbγ(y)) for the income levels that are higher than mean income. For each increase in outstanding debt holdings b, the government has to pay additional κγ (y) amount of goods. This increases the debt burden and thus the default states. However, in simulations the government s mean debt-to-income ratio is similar to the one obtained with non-contingent debt indicating that the government avoids holding such high levels of debt. Table 2 in summary shows that this indexation method with θ H = 1 does not generate much different moments than the one obtained with non-contingent debt. In order to better understand the dynamics of GDP-indexed bonds with long-term debt, alternative values of θ H is engineered and following important observations stand out from Table 2. Floored payments provide a limited insurance; as the sovereign promises higher payments during economic upturns, say θ H = 2, price of the asset increases following an increase in κγ(y) term in equation (8). This induces the sovereign to borrow less and a reduction in the sovereign s outstanding debt obligations mitigates default risk. This shows that such type of indexation method helps mitigating the debt dilution problem in the presence of long-term debt through avoiding higher debt issuance during upturns. 15 Part of the increase in debt-stock presented in Table 2 is because of the adjustment made to account for the developments in GDP. 21

24 In our quantitative exercise we show that the sovereign cannot generate much revenue from the debt issuance. That is, the surge in price q(b, y) is outweighed by the decline in net issuance (b (1 δ)b). Thus, the sovereign s inflow of resources measured by q(b, y) [b (1 δ)b]. In addition, Table 2 shows that as the degree of floored payments θ H amplifies, the standard deviations of consumption to income falls and trade balance (tb = y c) becomes procyclical with income or in another words borrowing policy becomes countercyclical. To account for these results and to better understand the intuition, I plot the sovereign s borrowing rule with zero initial debt in Figure 1 which shows that the sovereign avoids issuing debt for high income states and borrows during low income states for θ H = 2. So the government smooths consumption by borrowing more during low income states and this also explains the decline in inflow of resources. In addition, the figure shows how the debt dilution problem is mitigated with floored payments. The sovereign avoids issuing debt for good income states and borrows during low income states. 4.3 Results with unfloored payments Table 2 under unfloored column reports the simulation results with unfloored indexation method. Proponents of this indexation method are right in their conjecture that more funds can be raised and the spreads are mitigated during economic downturns. Middle panel of Figure 2 also confirms the proponents of this method: defaults are less likely during economic downturns. The figure shows that the necessary adverse shock for a government to default is larger in an economy with unfloored GDP-indexed bonds than in an economy with non-contingent debt. In this formulation, debt obligations would be mitigated if current income realization is below the mean trend-income and these contracts act as an automatic stabilizer by facilitating countercyclical policies. The government is able to generate higher (lower) debt during economic downturns (upturns). I further undertake additional exercises for different values of θ k in order to sharpen our understanding of such unfloored payments and the long-term debt. In the table, 22

25 default frequency and the spreads decline while debt-to-gdp as well as the inflow of resources (q(b, y) [b (1 δ)b]) surge. This stems from the fact that default risk declines in economic downturns while slightly increasing during upturns following the promised coupon swings. As θ k increases, unfloored payments facilitate a better hedge against adverse income shocks and lower consumption volatility. Unfloored payments allow the sovereign to better smooth its consumption during economic downturns by carrying tomorrow s resources to the present with reduced debt burden while limiting its ability to over-borrow during upturns (see Figure 1). The non-linearity of the pricing function features different asset price elasticity with respect to income which is further discussed in Section 4.8. Saving/Income benchmark floored (θ H = 2) unfloored (θ k = 2) unfloored (θ k = 5) Income Figure 1: Borrowing decision rule with zero initial debt for unfloored GDP indexing function with θ k = 5 and θ k = 2, floored indexing function with θ H = 2 and the benchmark economy with no GDP linked bonds. 4.4 Results with ceiling on payments This exercise considers ceiling on payments with and without floor. For payments with ( ( ) ) ceiling and floor, GDP-indexed function Γ(y) y becomes Max Min y, y +θ c σ y y, y θ f σ y y with θ c and θ f > 0 which are both constants and determine the degree of ceiling and floor, respectively. If σ y, which denotes one standard deviation of aggregate income, is greater (less) than y, then there is a ceiling (floor) on the payment. Recall that for contracts ( without a floor Γ(y) y = Min y, y +θ c σ y y ). 23

26 Results of this section are displayed in Table 3. Even though the moments are not significantly different than the unfloored economy, following important observations stand out and improve our understanding of the model dynamics with GDP-indexed bonds and long-term debt. When the payments are not floored but capped (Columns 5 and 6), we observe that spreads resemble unfloored economy and when we introduce floor, the spreads look similar to the benchmark economy. Payments on ceiling feature higher resource inflows than the benchmark economy but they are not as high as the unfloored economy. The ceiling and floored columns show us that it is the degree of floor that is driving down the spreads and amplifying the revenues generated by debt issuance (resource inflows). The intuition stems with the fact that declined claims during downturns are vital to avoid a default and lead an increase in the price. Also as θ f increases, floor falls and gets closer to the unfloored economy and we will discuss in capital gains section that investors would prefer switching to an unfloored economy. 4.5 Results with unproportional payments These bonds were issued by the Government of Singapore in My analysis in this section shows that the government s ability to raise funds drops when the government guarantees a fixed coupon rate plus a promise of extra return if the realized income is higher than a threshold income. Table 2 under unproportional column shows that following a decline in total debt-to- GDP ratio, total borrowing costs as well as the mean default rates decline. Table also shows that such an indexation method generates procyclical policies while non-contingent debt economy features countercyclical policies. Lower panel of Figure 2 illustrates why the government fails to raise higher funds clearly. Figure shows that for an income level that is higher than the mean-trend income, equilibrium default states are higher than the one with non-contingent debt. This is not surprising. Let s rewrite equation (9) as y + σ y ( ( y )) + σ y κ + Max y, 1 b + q(b, y) [ b (1 δ)b ] 24