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Sovereign Default: The Role of Expectations. Gaston Navarro. NYU Juan Pablo Nicolini, Minneapolis Fed and Universidad Di Tella Pedro Teles, Banco de Portugal and UCP • Can sovereign debt crisis can be the result of self-fulfilling equilibria? (Calvo 1988) • High interest rates imply higher probability of default which in turn feeds-back into high interest rates. • Argentina 2001? Currency board. Average rate was 10%. (93-98) Average debt to GDP ratio was 30%. • Same interest payment than a 3& rate (the US bond) and 100 % of debt to GDP ratio. • IMF withdrew support in September 2001. Default in December 2001. • Southern Europe 2012? No default with debt to GDP ratios of 80% for Spain and over 100% for Italy and Portugal. • ECB intervened in September 2012: Outright Monetary Transactions (OTM). European Bond Spreads Basis points, 10-year bond spread to German bonds 700 600 500 400 300 200 100 0 Jan-10 Apr-10 Source: Global Financial Data Jul-10 Oct-10 Jan-11 Apr-11 Jul-11 France Oct-11 Italy Jan-12 Apr-12 Spain Jul-12 Oct-12 United Kingdom Jan-13 Apr-13 Jul-13 Oct-13 Jan-14 • This paper 1. Shows Arellano (2008) exhibits multiplicity studied in Calvo (1988). Characterize equilibria (in a two-period model). 2. Evaluates the effect of policy by an agent with deep pockets (IMF or ECB?). 3. Shows (in an infinite period model) how a sunspot can be used to replicate behavior of spreads in Italy and Spain during a sovereign debt crisis. Plan • Two period model. • The price of the bond (or the interest rate) as a function of the ”debt”. • Discussion of equilibria. Multiplicity. • A sunspot in the infinite period model: engineering a sovereign bond crisis. • Skepticism. A two period model • Two-period, endowment economy populated by a representative agent (1) + 1[ (2)] • The endowment is 1 in the first period. • Second period endowment 0 ∈ [1 ] with density ( 0) and cdf ( 0) • In period one, the agent can borrow in a non contingent bond in international financial markets. • The risk neutral gross international interest rate is ∗ (the price of the bond ∗ = 1∗). • At = 2 after observing 0, the representative agent decides either to pay the debt or default. • If default, consumption in second period is 1. • The representative agent takes into account the effect of the level of debt on interest rates. • Lenders offer a schedule of interest rates as a function of the level of debt. • There are two classes of schedules consistent with zero profits. • In one, we guess that depends on the resources obtained in the first period, 0. (Calvo schedules) • In this case, the representative agent will default if and only if ( 0 − 0 1 ) ≤ (1), or 1 0 0 ≤1+ h i 1 and the probability of default is given by 1 + 0 • In the other class, we guess depends on the resources paid (if no default) in the second period that we denote by 0. (Arellano schedules). • Expressed in terms of 0, the condition for the threshold is written as 0 ≤ 1 + 0 £ ¤ and the probability of default is given by 1 + 0 • Investors are risk neutral, so arbitrage in international capital markets implies " à 1 1 1 0 = 1− 1+ ∗ 0 ( ) (0) ¡ 0¢ for the schedule , or ∗ = ¡ ¢ for the schedule 0 . !# ³ ´i 1 h 0 1− 1+ 0 ( ) (1) (2) • This verifies that the schedule depends on 0 in the first case and on 0 on the second. 1 for ∈ {0 0} so the choice between or • Note that () = () is irrelevant. • What matters, is if the schedule is defined over the variable 0 (the value of the debt at maturity) or 0 (the value of debt when issued). • This depends on which one foreign lenders coordinate upon, not on any decision taken by the small open economy. • The optimal choice of debt depends on which schedule the representative agent faces. We will discuss this in the context of a specific example below. ¡ 0¢ Equilibrium An equilibrium is a schedule (or (0)) and a point in the schedule (0∗ ∗) (or (0∗ ∗)) such that: 1. Given the schedule, 0 (or 0) maximizes utility 2. The schedule solves the functional equation (1) (or (2)) 3. ∗ = (0∗) (or ∗ = (0∗)). ¡ ¢ Calvo schedules 0 • Let à ! " 1 1 0 1− ; = à 1 + 0 1 !# . ¡ ¢ 1 0 • For = 0, 0; = 0. With a bounded support, for 1 such that ³ ´ 1 1 0 0 1 + ≥ , then ; = 0. ³ ´ 1 0 • For standard distributions, the function ; is concave, so that ³ ´ 1 1 ∗ 0 there are at most two solutions of = ; . h(R) 1.5 b =2.2, γ = 0 b =2.6, γ = 0 R∗ =1.04 1 0.5 0 0 1 2 3 4 R 5 6 7 8 10 9 8 7 1/q 6 5 4 3 2 1 0 0.2 0.4 0.6 b′ 0.8 1 Arellano Schedules • The model is also consistent with an alternative class of schedules that have the contrasting feature that the equilibrium is unique. • These are the schedules that Arellano (08) considers. ³ ´ 0 • A schedule now is , so that the price of a bond in the first period, , is a function of the bonds to be paid in the second period, 0. • It must satisfy ³ ´i 1 1 h 0 = 1− 1+ ∗ 0 ( ) • This schedule is increasing in 0 in all the support. • But this is the same schedule as before, with only a change of variables. ³ ´ • So while this schedule 0 is increasing, it includes the high rate, ¡ ¢ decreasing schedule for 1 0 • In particular high values of 0 can be associated with low values of 0, and high values of 1 . 9 9 8 8 7 7 6 6 1/q 10 1/q 10 5 5 4 4 3 3 2 2 1 0 1 0 0.2 0.4 0.6 b′ 0.8 1 1 2 bq′ 3 4 ¡ ¢ ¡ ¢ • The two schedules 0 and 1 0 are solutions of the same functional equation with just a change of variables. • But it makes a big difference whether the representative agent in the small open economy faces one schedule or the other. ¡ 0¢ • Once offered the increasing schedule, , the representative agent will be able to pick a point on the schedule. • By choosing 0, the probability of default will be pinned down. • That way the country can avoid the default probabilities associated with the high rate, high probability outcome. Fragility of the decreasing Calvo schedule b • Consider a perturbation of a point ( 1 b) in the the schedule 1 ( −1 1 ∗ 2) 4 that consists of the same interest rate, but a slightly lower value for b the debt ( 1 b − ). b b • At the point ( 1 b − ) the interest rate is the same as in ( 1 b) but the debt lower, so the probability of default is also lower. b • Thus, profits for the lenders are higher than at ( 1 b) where profits are zero: Individual investor would cut down rates. • A reasonable refinement would restore uniqueness? A distribution with normal and disaster times • Consider two independent random variables, 1 and 2, both normal with different mean, 1 and 2, respectively, and the same standard deviation, . 0 • Now, let the endowment in the second period be equal to 1 with probability and equal to 2 with probability 1 − . 0 • If the two means, 1 and 2, are sufficiently apart, then ( 1 ) has four solutions, for some values of the debt. h(R) 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 b =2.0556 b =2.5859 b =2.9394 R∗ =1.04 0.2 0 1 1.5 2 2.5 R 3 3.5 4 Perturbing the uniform distribution • Is that bimodal distribution empirically plausible? No. • But the answer can be yes if the debt level is high enough. • Consider a perturbation ( 0) of the uniform distribution, so that the density would be 1 ( 0) = + ( 0) with −1 Z 1 ( 0) 0 = 0 • In particular let ( 0) = sin 0, with = 2 −1 , where is a natural number. • If = 0 the distribution is uniform, so there is a single increasing schedule. • If = 1 there is a single full cycle added to the uniform distribution. • The amplitude of the cycle (relative to the uniform distribution) is controlled by the parameter • The number of full cycles of the sin 0 function added to the uniform, is given by . h(R) 1.6 1.4 1.2 b =2.2, γ > 0 b =2.2, γ = 0 b =2.9, γ > 0 b =2.9, γ = 0 R∗ =1.04 1 0.8 0.6 0.4 0.2 0 0 1 2 3 4 R 5 6 7 8 • As → 0, so does the perturbation • Given a value for , the closer the debt to its maximum value, the larger the degree of multiplicity. • The function ⎡ 1 + 0 1 ⎤ 1 ⎣ 1⎦ 0 1 − 1 ∗ 1 − − 1 − sin = 0 1 has more than two zeros for 1 , for that can be made arbitrarily small, as long as 0 is close enough to . • if is small, it may take a very long series to identify it in the data. Policy • Consider an agent with deep pockets. ³ ´ • Assume it offers to lend to the country, at a policy rate 1 , any amount lower than or equal to a maximum level . • Then, given any schedule offered by the foreign lenders to the country b 1 (0 ), the schedule faced by the representative agent is given by b 1 1 1 0 ( ) = min{ (0 )} so there cannot be an equilibrium with an interest rate larger than 1 . • If well designed, the amount borrowed from the large lender is zero. • The level of lending offered has to be limited. The infinite period model: Numerical exploration • The endowment has bounded support, given by [min max] ⊂ R+ and follows a Markov process with distribution ( 0|). • We let the value after default be ( ) = 1− . • In here we explore interest rate schedules for the depend on 0 (Calvo) • They will also depend on , and , which is a sunspot variable that selects one of the multiple schedules for the interest rate. • Our exploration will be based on the bimodal distribution we studied above. (at most two increasing solutions?) • Thus, = 1 2, with transition probabilities, 11 = 22 = • The value for the representative agent, after deciding not to default, is given by value functions and schedules 1 , satisfying ⎧ ⎨ ⎤⎫ ⎬ max (0 0 1) n o ⎦ ( 1) = max () + E0 ⎣ 00 ⎩ +(1 − ) max ( 0 0 2) | ⎭ subject to ≤ + 0 1 0 = 0 − 0 (0 1) 0 ≤ ⎡ n o and ⎧ ⎨ o ⎤⎫ 0 0 ⎬ max ( 2) n o ⎦ ( 2) = max () + E0 ⎣ 0 0 00 ⎩ +(1 − ) max ( 1) | ⎭ subject to ≤ + 0 1 0 0 0 0 = − ( 2) 0 ≤ ⎡ n Conditions the schedules must satisfy • In state 1 investors offer the schedule 1 (0 1). • The following period the state is 1 with probability , and 2 with probability 1 − . • If in state , the threshold for default is defined by = ³ ´ ( ) Then 1 0 1 0 0 1)|)] + (1 − )[1 − (( 0 0 2)|)]] = 1)[[1 − (( ( ∗ • Similarly, in state 2, investors offer the schedule 1 (0 2) • Then, the arbitrage condition must be written as 1 1 0 0 0 2)|)] + (1 − )[1 − (( 0 0 1)|)]] = 2)[[1 − (( ( ∗ Equilibrium An equilibrium is given by functions 1 ( ) ( ) 0 ( ) (0 ( ) ) ( ) such that, 1. given ( ) ( ) solves the threshold equation 2. given 1 (0 ( ) ) ( ) ( ) 0 ( ) solve the DP problems. 3. The arbitrage conditions and the law of motion for are satisfied.