Download Sovereign Default: The Role of Expectations.

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
"
Ã
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

Ã
!
"
1
1 0
1−
; =


Ã
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  () +  E0 ⎣
00 ⎩
+(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  () +  E0 ⎣
0
0

00 ⎩
+(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.