1 / 34 Towards a General Equilibrium Foundation for the Observed Term Structure and Design in Sovereign Bonds K. Wada 1 1 Graduate School of Economics, Hitotsubashi University November 4, 2017 @HIAS. IER, AJRC Joint Workshop
2 / 34 Background Sovereign governments in emerging markets default their external debts. Their defaults involve debt renegotiation between sovereign governments and external creditors since there is no international law about the settlements of defaulted external debts. Surprisingly, the percentage reductions of debts (haircuts) in restructuring vary among debt instruments with different maturities.
3 / 34 Empirical Evidence on Intercreditor Inequality Table : Sturzenegger and Zettelmeyer (JIMF 2008) Corr(Haircut, Maturity Length) Russia -0.56 Ukraine -0.97 Pakistan -0.96 Ecuador -0.90 Argentina +0.30 Urugray -0.46 The sample covers episodes from 1998 to 2005. The short-term debts are reduced more than the long-term debts in debt restructurings.
4 / 34 Research Questions What mechanism does drive this observed haircut term structure? What is the optimal design of the haircut term structure?
5 / 34 Approach I consider the haircut term structure in connection with sovereign defaults and terms of borrowing. I develop a new small open economy model with equilibrium defaults, which incorporates the endogenous determination of the haircut term structures. Two ingredients: - Sovereign defaults with multiple maturities (Arellano and Ramanarayanan, JPE 2012) - Sovereign default with debt restructuring (Yue, JIE 2010) In order to derive the welfare implication, I ask the benevolent sovereign government to make decisions about the allocation within its country.
6 / 34 Model Implications The optimal haircut term structure is determined by terms of borrowing, repayment cost, and the market value of restructured debts. The outcome of the optimal haircut term structure in the model can replicate the observed downward sloping term structure. This indicates that the downward sloping haircut term structures are not necessarily malign phanomena.
7 / 34 Literature Review Quantitative Eaton-Gersovitz sovereign default model: Arellano (2008), Aguiar and Gopinath (2006), Hachondo and Martinez (2008), Chatterjee and Eyigungor (2012), Mendoza and Yue (2012), Gordon and Guerron-Quintana (2017) Analyses on debt restructuring and haircut in general equilibrium: Yue (2010), Bi (2008), Benjamin and Wright (2013) -Only short-term debt or cash settlement Empirical studies on haircut term structure Sturzenegger and Zettelmeyer (2008), Asonuma et al. (2017)
8 / 34 Model Settings Sovereign government behaves as a social planner in a SOE, who maximize the representative household s lifetime utility. Risk neutral foreign creditor supplies the sovereign with the external credit in the form of short-term and long-term debt in non-defaulted periods. Incomplete market: No state-contingent contract available. Time is discrete and one time unit is meant to be a quarter. Endowment economy: No production.
Timing of Events in Good Standing 9 / 34
Timing of Events in Bad Standing 10 / 34
11 / 34 Sovereign: Continuing to repay The value of continuing to repay v c (b s, b l, y): s.t. v c (b S, b L, y) = Max {u(c) + βe yv g (b S, b L, y )} (1) b S,b L,l,c c = y b S [λ+(1 λ)z]b L +q s (b S, b L, y)b S+q L (b S, b L, y)l (2) b s : short term debt at the beginning of this period b l : long term debt at the beginning of this period c: consumption in this period y: output in this period l: long-term debt newly issued in the current period
12 / 34 Long-Term Debt The law of motion of long-term debt (Chatterjee and Eyigungor, AER 2012): b L = (1 λ)b L + l. (3) The fraction λ of the long-term debts matures. A long-term debt delivers a coupon, z.
13 / 34 Stochastic Process The log of output process follows AR(1). log(y ) = ρlog(y) + ϵ, (4) where ϵ is an exogenous standard normal shock to the output. E[ϵ 2 ] = η 2. (5)
14 / 34 Sovereign: Bad Financial Standing The value of being in bad financial standing, v b (b S, b L, y): v b (b S, b L, y) = u(c) + βθe y v g (b S, b L, y ) + β(1 θ)e y v b (b S, b L, y ), (6) where c = y L(y). (7) θ: exogenous probability of reentering the international financial market. v g : value of good financial standing (defined later). L(y): Output loss in bad financial standing.
15 / 34 Cost of Default Two default costs: 1. The direct output loss L(y). 2. Exclusion from the international financial market.
Sovereign: Good Financial Standing The value of being in good financial standing, v g (b S, b L, y): v g (b S, b L, y) = { Max (1 d)v c (b S, b L, y) d {0,1} + dv b (b r S (b S, b L, y), b r L (b S, b L, y).y) }, (8) where b r S (b S, b L, y) and b r L (b S, b L, y) are determined in debt renegotiation. 16 / 34
17 / 34 Default Indicator Function A default indicator function: d(b S, b L, y) = { 1 (v c < v b ) 0 (otherwise) (9) r: constant world real interest rate.
18 / 34 Debt Restructuring After a default event, the external creditors unite and send a representative external creditor to the debt exchange with the sovereign government. The representative external creditor aims to maximize the total market value of restructured debts. After the restructuring, each external creditor receives the debts newly issued with the same maturity he had before the default event. The sovereign wants to maximize the difference between the utilities of joining the restructuring and staying in autarky forever.
19 / 34 Debt Restructuring The Nash bargaining game is employed. Two parties maximize the weighted average of the debtor s and creditor s surplus: [( Max v b (b bs r S r, br L, y) v a (y) ) α,br L ( qs b (br S, br L, y)br S + qb L (br S, br L, ) 1 α ] y)br L (10) s.t. b r S b S, b r L b L (11) qs b, qb L : prices of short-term and long-term bond in bad financial standing respectively. v a : value of autarky (= E u(y L(y))).
20 / 34 Asset Prices The expected one-period return from sovereign bond must be equalized to the world interest rate (actuarially fair pricing). Expected returns are composed of the expected repayment and the recovery rate after default events.
21 / 34 Asset Prices The price of the short term debt is, in good financial standing: q S (b S, b L, y) = 1 1 + r { E y [1 d(b S, b L, y )] } + E y [d(b S, b L, y )(bs r /b S)qS b (br S, br L, y )], (12) where r is a world interest rate.
22 / 34 Asset Prices The price of the long-term bond: q L (b S, b L, y) = 1 1 + r { E y [(1 d(b S, b L, y )) (λ + (1 λ)(z + q L (b S (b S, b L, y ), b L (b S, b L, y ), y ))] } + E y [d(b S, b L, y )(bl r /b L)qL b (br S, br L, y )]. (13)
23 / 34 Asset Prices The short-term bond price in bad financial standing: qs b (b S, b L, y) = 1 { (1 θ)e y [qs b 1 + r (b S, b L, y )] + θe y [(1 d(b S, b L, y ))] } + θe y [d(b S, b L, y )(bs r /b S)qS b (br S, br L, y )] (14)
24 / 34 Asset Prices The long-term bond price in bad financial standing: q b L (b S, b L, y) = 1 1 + r { (1 θ)e y [q b L (b S, b L, y )] + θe y [(1 d(b S, b L, y )) (λ + (1 λ)(z + q L (b S (b S, b L, y ), b L (b S, b L, y ), y )))] } + θe y [d(b S, b L, y )(bl r /b L)qL b (br S, br L, y )] (15)
25 / 34 Recursive Markov Equilibrium 1. Given the asset prices and haircuts, the debt policy functions b S and b L solve the Bellman equation (1). 2. Given the asset prices, haircuts, and policy functions, the default indicator function d(b S, b L, y) solves the Bellman equation (9). 3. Given the asset prices, policy functions, and the default indicator function, haircuts (b S b r S (b S, b L, y))/b S and (b L b r L (b S, b L, y))/b L are the solution to the Nash bargaining game (10). 4. Given the debt policy functions, the default indicator function, and the haircuts, the asset prices q S, q L, qs b, and ql b satisfy the actuarilly fair pricing from (13) to (16).
26 / 34 Optimal Haircut Term Structure, Default, and Terms of Borrowing Consider the FOC of the objective in debt restructuring w.r.t. b r S and br L. With several simplifying assumptions, the FOCs are reduced to qs b b b r S r S + qb S + qb L b b r S r L v b b r S v b b r L = qs b b b r L r S + qb L + qb L b b r L r L. (16) The LHS is the relative cost for the sovereign of issuing an additional short-term debt to long-term debt. The RHS is the relative benefit for the external creditor of receiving an additional short-term debt to long-term debt.
27 / 34 Optimal Haircut Term Structure, Default, and Terms of Borrowing The LHS can be rewritten as { E y,r [u (c ) 1 + { E y,r [u (c ) λ (1 λ)(z + q L ) + q S b r S (y)b S }] }] q L bl r (y)(b L (1 λ)br L (y)) (17) where the expectation is conditional on continuing to repay in the next period, R.
28 / 34 Optimal Haircut Term Structure, Default, and Terms of Borrowing The numerator is expressed as: ( E y u(c )E y 1 q ) ( S Cov u (c ), q S bs r b S } {{ } Cost of Repayment bs r b S ) } {{ } Consumption Insurance (18) The second term arises since the terms of borrowing changes.
29 / 34 Optimal Haircut Term Structure, Default, and Terms of Borrowing The denominator is described as: ( E y u(c )E y λ + (1 λ)(z + q L ) q ) L bl r (b L (1 λ)br L ) }{{} ( Cov u (c ), q L b r L Cost of Repayment (b L (1 λ)br L ) ) } {{ } Consumption Insurance + Cov(u (c ), (1 λ)q L ) }{{} Cost of Repayment (19) The final term is typically negative since the higher default risk associated with higher marginal utility of consumption decreases the price of the long-term debt.
30 / 34 Quantitative Analysis The slope of the haircut term structure is a quantitative problem. The parameter values and functional forms largely follow Chatterjee and Eyigungor (2012), who consider default episodes in Argentina. Exceptions are β (the discount factor) = 0.93, targeted for the debt-to-output ratio in Argentina (100%). α (bargaining power of the sovereign gov.) = 0.7, targeted for the overall haircut rates (70%). This choice of parameter values is not based on the solid structural estimation.
Model Prediction Table : Theoretical Moments Stats Model Data Debt-to-Output Ratio (Targeted) 88% 100% Mean Overall Haircut Rates (Targeted) 62% 70% Consumption-to-output-Ratio 100% 109% Mean Haircut of Short-Term Debt 74% Mean Haircut of Long-Term Debt 57% The model can capture the high consumption-to-output ratio observed in emerging countries. This model prediction corresponds to the observed haircut downward sloping term structure. 31 / 34
32 / 34 Conclusions The optimal haircut term structure in my model coincides with the observed haircut term structures. Therefore, the observed downward sloping haircut term structures are not always malign phenomena. The future research Sovereign default models incorporating intra-creditors confliction, which can lead to the delay of debt restructurings.
33 / 34 Functional Form The utility function is CRRA: u(c) = c1 γ 1 γ (20) The output loss function is nonlinear: L(y) = max{a 0 y + a 1 y 2, 0} (21)
34 / 34 Parameter Values Parameter Value Description σ 2 RRA β 0.93 Yearly discount factor r 0.01 World Interest Rate θ 0.0385 Probability of reentry a 0-0.18 Parameter of output loss function a 1 0.24 Parameter of output loss function ρ 0.95 Serial correlation of ln(y t ) η 0.027 Std. dev. of innovation ϵ t α 0.7 The bargaining power of the debtor λ 0.05 Reciprocal of Average Maturity z 0.03 Coupon Repayments