Why Prices Don’t Respond Sooner to a Prospective Sovereign Debt Crisis.∗ R. Anton Braun†
Tomoyuki Nakajima‡
Federal Reserve Bank of Atlanta
Kyoto University
July 11, 2014
Abstract Since 2008 actions have been taken in Europe and elsewhere that increase the cost of short-selling sovereign debt. We show that such actions can have a profound effect on the timing and magnitude of price responses to bad news in periods leading up to a sovereign default. When financial markets are frictionless, prices drop instantly in response to bad news even if the prospect of a crisis is very remote. Imposing costs on short-selling disrupts this dynamic. Government bond prices exhibit no response to bad news when the prospects are remote. Instead price declines only occur immediately prior to a sovereign default and then in a nonlinear way. Keywords: sovereign debt crisis; bond prices; inflation; leverage; heterogenous beliefs. JEL Classification numbers: E31, E62, H60.
∗
These are our personal views and not those of the Federal Reserve System. For helpful comments and
suggestions, we thank Kosuke Aoki, Fumio Hayashi, Atsushi Kajii, Keiichiro Kobayashi, Felix Kubler, Masao Ogaki, Kozo Ueda, and seminar participants at the Bank of Japan, Canon Institute for Global Studies, Federal Reserve Bank of Atlanta, Hitotsubashi University (ICS), Indiana University, Kyoto University, NBER Summer Institute, North Carolina State University and the University of Tokyo. A part of this research was conducted while Braun was a visiting scholar at CIGS and while Nakajima was a visiting scholar at the Bank of Japan and the Federal Reserve Bank of Atlanta. Nakajima also thanks the financial support from the JSPS. †
Federal Reserve Bank of Atlanta. Email:
[email protected].
‡
Institute of Economic Research, Kyoto University, and the Canon Institute for Global Studies. Email:
[email protected].
1
1
Introduction
Recently, many developed countries have experienced large increases in government debt-GDP ratios against a background sharp and persistent economic contractions. Figure 1 reports debt-GDP ratios for Greece, Japan, Spain and the U.S. This figure has three noteworthy properties. First, Debt-GDP ratios are high by historical standards in all four countries. Second, the growth rate of the debt-GDP ratio accelerates in all four countries after 2008. The third and most interesting feature of this figure is that Greece’s debt-GDP ratio is lower than Japan’s and Spain’s debt-GDP ratio in Spain is lower than the U.S. Recent developments in these countries have demonstrated that it is a politically difficult task to increase taxes and lower purchases sufficiently to stabilize the debt-GDP ratio. As these challenges become more apparent credit ratings on sovereign debt were downgraded in all four countries. The response of markets to these events has been quite different. Both Greece and Spain saw yields on their sovereign debt rise suddenly and sharply after 2008. Yields on 10 year government bonds in Greece rose from 4.5% on September 1 2009 to 37.1% on March 1 2012 shortly before it renegotiated the terms of its debt. Yields on 10 year Spanish bonds rose from 4.8% to 7.2% between March 1 2012 and June 18 2012.1 Bond yields in Japan and the U.S., in contrast, have fallen since 2008. The Japanese 10 year bond yield fell from 1.9% on June 19 2008 to 0.8% on June 18 2012 and the U.S. 10 year bond yield fell from 4.14% on June 19 2008 to 1.6% on June 18 2012. Most sovereign debt is nominally denominated and high inflation reduces the real value of outstanding nominally denominated government debt. So one might expect to see inflation rates rise in the U.S. and Japan since they are not bound to a monetary union. However, inflationary pressure has also remained low in the U.S. and Japan. Why did bond yields rise so sharply and suddenly in Spain and Greece? Why haven’t bond yields and/or inflation responded to the elevated risk of a sovereign default in the U.S. and Japan? It is not uncommon for policy makers to attribute sharp unfavorable changes in the price of government liabilities to the actions of short-sellers. Short-selling activities are subject to special regulations in many countries (see Angel (2004) for a description of regulatory restrictions on short-selling in the U.S., Europe and Asia.). These regulations make it more costly for investors to take short-positions as compared to long positions. It is also not uncommon for sovereigns to increase the costs of short-selling when the price of government obligations including debt and/or currency falls. Germany banned naked short-sales of sovereign CDSs in 1
These price quotations are taken from www.Bloomberg.com
2
Figure 1: Public Debt-GDP ratios in Greece, Japan, Spain and the United States Gross Debt−GDP Ratios
240 220
Greece
200
Japan Spain
180
USA
%
160 140 120 100 80 60 40 1996
1998
2000
2002
2004 Year
2006
2008
2010
2012
*General government gross financial liabilities as a percentage of GDP from OECD.
2010. In November of 2012 this ban was extended to the entire Euro area. Governments also take actions to increase the costs of shot-sellers when their currencies are threatened. Some of the more extreme measures include splitting onshore and off-shore currency markets (Spain in 1992 and Thailand in 1997), imposing capital controls (Malaysia in 1998), or undertaking large interventions in equity markets (Hongkong in 1998).2 In this paper we use a general equilibrium model to analyze the effects of costly shortselling on prices and trading activity along a path leading to sovereign default. We find that costly short-selling disrupts the channel through which bad news about future events, gets reflected in prices today. A standard result in forward looking models is that bad news about future payoffs affects bond prices today as traders act on this information. We find that when short-sales are costly both potential short-sellers of government debt and those wishing to take long positions are pushed to the sidelines of government bond markets. This results in significant disruptions in bond price and/or inflation dynamics. In one example we provide the response of bond yields to bad news about the prospects of a future sovereign-debt crisis falls from 5% when short-selling is costless to only 15 basis points when short-selling positions 2
In August 1998 the Hongkong Monetary Authority purchased domestic stocks amounting to about 7% of
the Hongkong Stock Exchange’s total market capitalization and 30% of its free float in an effort to fend off short-sellers. See the discussion in Corsetti, Pesenti and Roubini (2001) for more details.
3
are subject to an annual transactions fee of 4%. In this situation bond yields are a poor leading indicator of a sovereign default. Thus, according to our model, it could be a mistake to infer from the fact that bond yields and inflation are low in the U.S. and Japan that the risk of a sovereign default in these countries is also low. Instead it is possible that bond yields and/or inflation will only respond to the prospect of a sovereign default one to two years in advance of that event and then in a dramatic way. We start by considering a situation where a sovereign debt crisis is resolved by a high inflation rate. This scenario appears to be particularly relevant for Japan and the United States. Most government debt is nominally denominated in the local currency in these countries and resolving a sovereign debt crisis via inflation is a viable and possibly more attractive option to these governments as compared to an explicit default. Individuals in our model have heterogenous beliefs about the probability of a sovereign debt crisis. Agents who are optimistic can borrow to purchase government debt, and agents who are pessimistic can short government debt. Both agents are subject to collateral constraints that restrict the sizes of their positions. Those taking long-positions hold government bonds as collateral and those taking short positions hold cash as collateral. The leverage rates are determined endogenously as in Geanakoplos (2003, 2010). Under these assumptions and the further assumption that all government debt is one period debt, no short-selling occurs in equilibrium when the fiscal crisis is resolved by a high inflation rate. We find that the trajectory of the inflation rate along a path leading to default is very different in our “leverage equilibrium” as compared to an Arrow-Debreu reference point. When financial frictions are absent the inflation rate jumps in response to bad news about the possibility of a future debt crisis, even when this event is distant. In our leverage equilibrium the inflation rate only increases in states that occur immediately prior to the crisis event. When it does respond the size of the movement in the inflation rate is very large as compared to our Arrow-Debreu reference equilibrium. In this model short-selling is too costly and pessimists choose not to participate in the government bond market. Instead they make safe loans to the optimists. Optimists stochastic discount factors have a dominant effect in moulding the dynamics of the price of the nominal bond and the inflation rate. We then consider an environment with explicit default and multiple period government debt that is meant to reflect the situations of Spain and Greece. Both countries are bound to the Euro and an explicit default is the most likely type of default. In this version of our model there is short-selling in equilibrium and we can analyze how government bond yields
4
and the pattern of short-selling changes as the costs of taking short positions are increased from zero. In our model increasing the cost of short-selling is an effective way to reduce government bond yields. Higher costs of short-sales reduces participation in bond markets and this lowers bond-yields in all periods leading along a path that eventually results in a sovereign default. Higher costs of short-selling also have a big impact on the dynamic properties of the model. When short-selling is sufficiently costly, trading in government bonds experiences a burst in participation shortly prior to default. Interestingly participation by those taking long positions increases even more than participation by those taking short positions. This burst in participation is so large that it can exceed participation rates in the frictionless benchmark economy. Associated with this increase in participation is a sudden sharp decrease in oneperiod holding returns on government debt. We also investigate the welfare properties of costly short-selling. Welfare comparisons are more subtle in our model because there is no objective truth and agents heterogenous beliefs about the prospects of a sovereign-default are all equally valid. We find that the imposition of costs on short-selling can be justified by a Rawlsian welfare criterion. Agents in our model are risk neutral and the fraction of agents that go bankrupt is lower when short-selling is costly. Our model of costly short-selling is related to models with heterogeneous beliefs and financial frictions considered by Geanakoplos (2003, 2010). He investigates the role of ruling out short-sales on asset pricing. Our model with explicit default extends the work of Geanakoplos by allowing for both leveraged short and long-sales and differs in other respects due to our interest in sovereign default. The combination of heterogenous beliefs and an exogenous ban on short sales has also been used by Harrison and Kreps (1978), Scheinkman and Xiong (2003) and Hong and Sraer (2011) to account for bubbly phenomena in asset prices. For instance Scheinkman and Xiong (2003) show that agents are willing to purchase an asset even when it exceeds their evaluation of its fundamental value because they expect to be able to sell it in the future at a higher price. The difference between their setup and ours is clearest in our model where the sovereign debt crisis is resolved by inflation. Only one period bonds trade in that setting and there is no bubble. In our model with explicit default there are multiple period bonds but the subjective evaluation of cash flows for optimistic agents who purchase these bonds exceeds the equilibrium price. It follows that this type of bubble does not arise in our model with explicit default either. Our research is also complementary to previous research by Bi (2011) and Bi, Leeper and Leith (2012). These papers also produce nonlinear movements in bond rates leading up to
5
a sovereign default in representative agent dynamic general equilibrium models. The source of the nonlinearity in bond rates in their setup is nonlinearities in the objective probability of default. We also generate nonlinearities in the dynamics of bond yields. In our model the nonlinearities are jointly determined by the initial distribution of beliefs, the market structure and the resulting patterns of trade in the bond market. The principal message of our analysis is that the micro-structure of the bond market is important. Financial frictions and asymmetries in the cost of short-selling government debt creates nonlinearities and magnifies any nonlinearities that might arise in frictionless financial markets. The remainder of the paper is organized as follows. Section 2 describes the model where a sovereign debt crisis is resolved by high inflation. default. Section 3 describes the model with explicit default and Section 4 contains our concluding remarks.
2
Implicit Sovereign Default
We start by discussing a version of the model where a sovereign debut crisis is resolved by creating inflation in the final period. Inflation erodes the real value of nominally denominated government debt but, is not treated as a default event by private ratings agencies. We thus refer to this as an implicit sovereign default. The equilibria that we consider here have the property that there is no short-selling in equilibrium. This result follows from our assumptions that short-sellers have to post cash as collateral and that the government only issues one period nominal bonds. Under these assumptions government bonds and cash face the same exposure to inflation risk but, cash is dominated in rate of return. These assumptions facilitate the exposition and solution of the model. More importantly, these assumptions capture in a parsimonious way the fact that it is difficult to take short positions at multiple year horizons on implicit default by large economies like the U.S. and Japan. Sovereign debt credit default swaps (CDS) do not pay out when default is implicit and betting on default by taking a short position on the Dollar or Yen in foreign exchange rate markets exposes the investor to a bundle of other risks that make this an expensive way to place a pure bet on implicit default. Those wishing to take short-positions have the option of making loans at a real risk-free rate and this is their preferred investment strategy.
2.1
1-Period Model with Implicit Default
We begin by considering a one-period version of the model. Then in Section 2.2 we generalize the model to allow for an arbitrary number of time periods. The 1-period model has two instants of time that are indexed by t = 0, 1. There are two states of nature in period 1, U
6
and D. As discussed below, they are distinguished by the amount of taxes collected by the government. In this 1-period model the event D is associated with low taxes and a debt crisis and the event U is associated with high taxes and no debt crisis. We let st denote the state of nature in period t, where s0 = 0 and s1 ∈ S ≡ {U, D}. Individuals:
There is a continuum of agents indexed by h ∈ [0, 1]. Each agent receives an
identical endowment of yt units of the consumption good in period t = 0, 1. Note that the
endowment does not depend on the state in period 1. Agents are also endowed with equal ¯ > 0, in period zero. All agents have access to a amounts of nominal government debt, B risk-free storage technology that offers a gross rate of return denoted by R > 1. Agents are identical except for their beliefs about the probability that state U occurs in period 1. Specifically, we assume that agent h believes that s1 = U with probability h. Thus, agents with high h assign less probability to the debt crisis event. All agents have linear preferences of the form: c0 +
X
γ h (s1 )c(s1 )
(1)
s1 ∈S
where γ h (s1 ) denotes the subjective probability of agent h that state U occurs in period 1: ( h, for s1 = U , γ h (s1 ) = 1 − h, for s1 = D. Government:
¯ > 0 nominal liabilities to the private sector, The government starts off with B
collects taxes and issues one-period bonds. We abstract from government consumption and assume that taxes are lump sum and identical across agents. Neither of these assumptions are essential to our arguments. Let T0 and T (s1 ) denote the real amount of taxes in period 0 and in state s1 ∈ S of period 1, respectively. Let B0 be the nominal amount of bonds issued by the government in period 0 and q0 ∈ (0, 1] be the price of those bonds. Given these definitions
the “flow budget constraints” of the government in period 0 and in state s1 ∈ S of period 1
are given by3
¯ = P0 T0 + q0 B0 , B B0 = P (s1 )T (s1 ), 3
(2) s1 ∈ S,
(3)
Here we refer to (2)-(3) as the flow budget constraints of the government. But, as is well known, in the
standard formulation of the fiscal theory there is ambiguity about exactly what constraints the government faces. Bassetto (2002) posits a game theoretic version of the fiscal theory that removes this ambiguity. It is straightforward to rewrite our model in the same way as Bassetto (2002).
7
where P0 and P (s1 ) are the price levels in period 0 and in state s1 , respectively. The government’s fiscal policy is given by its choice of {T0 , {T (s1 )}s1 ∈S }, and its monetary
policy is given by its choice of the one-period nominal interest rate 1/q0 . We do not model the optimization problem of the government and instead set fiscal policy and monetary policy exogenously. Fiscal policy is set in the following way: T0 = 0, ( T (s1 ) =
(4) TH ,
if s1 = U ,
TL ,
if s1 = D,
(5)
where TH TL > 0. Taxes collected by the government are assumed to be very small in
state D in period 1 to justify our use of the expression “debt crisis” to refer to this state. It
follows that a government policy is given by (q0 , TH , TL ) ∈ (0, 1] × R2++ . Given a government policy, the amount of government bonds issued in period 0, B0 , is determined endogenously so as to satisfy demand from the private sector. We consider a market structure where agents can borrow and lend to each other. Short sales of government debt do not occur in equilibrium but some agents will choose to take leveraged long positions on government debt. We refer to this specification as the leverage specification. It follows from our assumption of risk neutrality that optimistic agents, who believe that the rate of return on government bonds is greater than the borrowing rate, will want to borrow as much as possible and use the proceeds to purchase government bonds. Their total positions are limited by the requirement that they post government bonds as collateral in order to obtain a loan. How much can an agent borrow with one unit of government bonds as collateral? One way to proceed would be to impose an exogenous ad hoc constraint as in e.g. Kiyotaki and Moore (1997). We pursue an alternative avenue that determines the collateral constraint endogenously. Geanakoplos (2003, 2010) posits a broad array of loan/default contracts and determines which ones trade in equilibrium. Applying this approach to our model yields a “no-default constraint,” that requires that the amount of repayments not exceed the value of the collateral in any state. We simplify the ensuing exposition of the model by directly imposing the no-default constraint. Since there is no default on loans, loans are risk-free. Thus the interest rate on loans is equal to R in equilibrium (as long as the storage technology is used). Consider an agent who borrows φ0 and purchases government bonds b0 in period 0. She must repay Rφ0 in period 1. The no-default constraint requires that Rφ0 ≤
8
b0 P (s1 ) ,
for all s1 ∈ S. Thus the budget
constraints for each agents become4 ¯ B b0 ≤ + y0 + φ0 , P0 P0 b0 c(s1 ) ≤ Rk0 + + y1 − T (s1 ) − Rφ0 , P (s1 ) b0 Rφ0 ≤ , s1 ∈ S, P (s1 )
(6)
c0 + k0 + q0
c0 , k0 , b0 , c(s1 ) ≥ 0,
s1 ∈ S,
(7) (8)
s1 ∈ S.
(9)
Definition 1 (Leverage Competitive Equilibrium). Given a government policy (q0 , TH , TL ), a leverage competitive equilibrium consists of an allocation {ch0 , [ch (s1 )]s1 ∈S , k0h , bh0 , φh0 }h∈[0,1] , supply of government bonds B0 , and prices {P0 , [P (s1 )]s1 ∈S } such that (i) for each agent h ∈ [0, 1], {ch0 , [ch (s1 )]s1 ∈S , k0h , bh0 , φh0 } solves her utility maximization problem; (ii) the
government flow budget constraints (2)-(3) are satisfied, where taxes {T0 , [T (s1 )]s1 ∈S } are given by (4)-(5); and (iii) all markets clear: Z 1 (ch0 + k0h ) dh = y0 , 0 Z 1 Z 1 h c (s1 ) dh = y1 + R k0h dh, 0 0 Z 1 bh0 dh = B0 , 0 Z 1 φh0 dh = 0.
(10) s1 ∈ S,
(11) (12) (13)
0
To characterize this equilibrium start with the utility maximization problem. No one consumes in period 0: ch0 = 0 for all h ∈ [0, 1]. Observe that P (U ) < P (D) in equilibrium and thus the collateral constraint (8) can be expressed as φ0 ≤
b0 RP (D) .
In this setting there
will be a marginal purchaser h0 who is indifferent between storage and borrowing and using the proceeds to purchase government debt. This indifference relationship is given by 1 1 P (U ) − P (D) h0 q0 1 P0 − RP (D)
= R.
(14)
Optimistic agents h > h0 want to borrow as much as possible to purchase government bonds. On the other hand, pessimistic agents h < h0 do not want to hold government bonds. However, they are perfectly willing to lend to optimistic agents at the interest rate R. Thus there will be leverage in equilibrium. 4
Rather than allowing for short-selling and then showing that it does not occur in equilibrium, we simplify
the exposition here by ruling out short-selling. The more general specification of the budget constraints with short-selling can be found in Section 3.
9
Given these results the solution to the utility maximization problem is summarized by: ch0 = 0, bh0 = k0h − φh0 = h c (s1 ) =
h ∈ [0, 1], −1 ¯ q0 B 1 − + y , 0 P0 P0 RP (D)
0, − RP1(D) ¯ B P0 + y0 ,
q0 P0
1 1 − P (D) P (s1 ) q0 1 − RP (D) P0
R
¯ B P0
−
1 RP (D)
−1
¯ B P0
(15) h > h0 ,
(16)
h ≤ h0 , + y0 , h > h0 ,
(17)
h ≤ h0 ,
¯ B P0
+ y0 + y1 − T (s1 ),
+ y0 + y1 − T (s1 ),
h > h0 ,
(18)
h ≤ h0 .
Since storage and lending are perfect substitutes, only k0h − φh0 is determined for agents with h ≤ h0 . Agents with h > h0 set storage to zero, k0h = 0, and borrow as much as they can: φh0 =
bh 0 RP (D) .
Next consider the government. Equations (2) and (4) imply B0 =
¯ B q0
(19)
and it follows that the price level in period 1 is P (s1 ) =
¯ 1 B , T (s1 ) q0
s1 ∈ S.
(20)
Then it follows from (16) that the market clearing condition for government bonds is expressed as −1 ¯ ¯ B q0 1 B = (1 − h0 ) − + y0 . q0 P0 RP (D) P0
(21)
Given P (s1 ), s1 ∈ S, the initial price level P0 and the marginal agent h0 are determined as the solution to (14) and (21). The following proposition summarizes these results.
Proposition 1. In the 1-period leverage equilibrium, (i) the equilibrium consumption allocation/portfolio, {ch0 , k0h − φh0 , bh0 , [ch (s1 )]s1 ∈S }h∈[0,1] , is given by (15)-(18); (ii) the marginal buyer of government bonds, h0 , is given by (14); and (iii) the equilibrium price levels {P0 ,
[P (s1 )]s1 ∈S } are given by (20) and (21).
Numerical Example We now turn to consider a particular thought exercise that is aimed to help us understand how the inflation rate evolves in the leverage specification when agents begin to anticipate the possibility of a debt crisis. As a reference point we also report the evolution of the inflation rate in a frictionless (Arrow-Debreu) market structure that is described in the Appendix. 10
Table 1: Inflation rates (%) and marginal buyers in the two-period model with implicit default
Market Structure
π−1
π0
π(D)
h0
(1) Arrow-Debreu Equilibrium
-1.96
30.72
47.06
0.5
(2) Leverage Equilibrium
-1.96
9.46
75.62
0.79
¯ = 1, y0 = y1 = 1, q0 = 1, R = 1.02, TH = R2 , and We set the parameter values as B TL = TH /2. The choice of a gross nominal interest rate of one is chosen to reflect the current situations in Japan and the United States. These choices are based on our assumption that a model period is one year. Suppose that in all periods prior to period zero the gross nominal interest rate is one and everyone believes that s1 = U with probability one. That is, prior to period 0, no one believes that a debt crisis will occur. Under these assumptions it follows that P−1 = 1. Since q−1 = 1 the inflation rate in period -1 is given by π−1 = q−1 /R − 1 = −1.96%. At the beginning
of period zero news arrives that a debt crisis may occur in period one and individual agents interpret this news in different ways. Table 1 shows the inflation rate at t = −1, t = 0 and at s1 = D for the “Arrow-Debreu”
and “leverage” equilibria, respectively. In the the Arrow-Debreu equilibrium when the news about a future crisis arrives in period zero these beliefs are instantly reflected in the price level and the inflation rate jumps from -1.96% to 30.72%. The inflation rate also jumps up in the leverage equilibrium in period zero. However, the size of the response at 9.46% is much smaller than the the Arrow-Debreu equilibrium. Instead there is a much larger jump in the inflation rate in the leverage equilibrium in the debt crisis state D. It is 75.62% as compared to 47.06% in the Arrow-Debreu equilibrium. There is simple intuition behind these results. In the model with the Arrow-Debreu market structure, every agent can freely bet on her beliefs: optimistic agents buy the Arrow security U and sell Arrow security D, and pessimistic agents do the opposite. In the leverage specification, optimistic agents can undertake collateralized borrowing which makes it possible for them to promise to deliver the return on government bonds in period one and thereby mimic a purchase of Arrow security U . However, they are not able to mimic a sale of Arrow security D. For pessimistic agents the problem is more severe as there is no way for them to mimic either selling the Arrow security U or purchasing the Arrow security D. In this
11
sense, there is an asymmetry in the portfolios held by optimistic and pessimistic agents. This asymmetry results in a higher value of h0 in the leverage equilibrium and thus a lower inflation rate π0 as compared to the Arrow-Debreu equilibrium.
2.2
T -period Model with Implicit Default
From the 1-period model we know that the inflation rate in the leverage specification undershoots the Arrow-Debreu inflation rate. There are other interesting differences between the two specifications that become more apparent when the time horizon is extended. We turn to discuss these dynamics next. In the T -period model there are T +1 instants of time indexed by t = 0, 1, . . . , T . In periods t = 1, . . . , T , a shock st ∈ S = {U, D} is realized. For t = 1, . . . , T , let st = (s1 , . . . , st ) ∈ S t
denote the history of shocks, and also let s0 ≡ 0 and S 0 ≡ {0}. In each period t, regardless
of the history st , agent h believes that st+1 = U with probability h and st+1 = D with probability 1 − h. In other words each realization of st alters agents’ subjective assessments
of the conditional probability of default in the final period. However, each agent’s subjective
belief about the probability of st+1 is not affected by the history of shocks st .5 Then her expected utility is expressed as T X X
γ h (st )c(st ),
(22)
t=0 st ∈S t
where γ h (st ) denotes the subjective probability that agent h assigns to history st in period 0. That is, ( h
t
γ (s ) =
hγ h (st−1 ),
if st = U ,
(1 − h)γ h (st−1 ),
if st = D,
with γ h (s0 ) ≡ 1.
In each period t, all agents are endowed with equal amounts of the consumption good,
y(st )
for st ∈ S t . For simplicity, we assume that for t = 0, y0 , t y(s ) = 0, for all st with t = 1, . . . , T − 1, y , for all sT . T
In addition, at the beginning of period 0, all agents are endowed with equal amounts of ¯ > 0. government debt B 5
These assumptions, which can be justified by assuming that each realization of st is drawn from a different
urn, allow us to sidestep an inference problem that significantly increases the complexity of the model.
12
Figure 2: The event tree when T = 2. The history D2 results in low taxes and implicit default in period 2. All other histories result in higher taxes in period 2.
U h
T1 = 0 U h
D 1−h
D 1−h
U h
T2 = TH
T2 = TH
T0 = 0
D 1−h
T1 = 0
T2 = TH
T2 = TL
As in the two-period model, the government specifies a fiscal policy which consists of a state-contingent path of lump-sum taxes {T (st ) : st ∈ S t , t = 0, . . . , T }. To be specific, fiscal
policy takes the following form: 0, T (st ) =
TL , T , H
for all st with t = 0, . . . , T − 1, for sT = DT ,
(23)
for all sT 6= DT .
The government also chooses a monetary policy which is a state-contingent path of the price of government bonds, {q(st ) : st ∈ S t , t = 0, . . . , T }. Given these government policies, the government debt B(st ) evolves as
¯ = q0 B 0 , B
(24)
B(st−1 ) = q(st )B(st ),
st ∈ S t , t = 1, . . . , T − 1,
B(sT −1 ) = P (sT )T (sT ),
sT ∈ S T .
(25) (26)
Figure 2 illustrates the implications of these assumptions using an event tree for the special case where T = 2. A debt crisis only occurs when the government collects TL taxes in the final period. Under our assumption (23), it occurs only if st = D for all t = 1, . . . , T . In the case of T = 2 illustrated in the figure this corresponds to the bottom outcome where the 13
history is {D, D}. In the other three histories in the last period there is no crisis. For general
T there will only be one history in the final period that produces a crisis. From this it follows that extending the horizon acts to lower the perceived probability of a crisis. The flow budget constraints for agents in the T-period model with implicit default are ¯ B b0 ≤ + y0 + φ 0 , P0 P0 b(st ) c(st ) + k(st ) + q(st ) P (st ) t−1 b(s ) ≤ + Rk(st−1 ) − Rφ(st−1 ) + φ(st ), t = 1, . . . , T − 1, st ∈ S t , P (st ) b(sT −1 ) c(sT ) ≤ + Rk(sT −1 ) + yT − T (sT ) − Rφ(sT −1 ), sT ∈ S T , P (sT ) b(st ) Rφ(st ) ≤ , t = 0, . . . , T − 1, st ∈ S t , st+1 ∈ S, P (st , st+1 ) c0 + k0 + q0
c(st ), k(st ), b(st ) ≥ 0,
st ∈ S t , t = 0, . . . , T,
(27)
(28) (29) (30) (31)
Numerical Example We now discuss the dynamics of inflation leading up to an implicit sovereign default using a model with six periods. ¯ = 1, y0 = yT = 1, We use the same parameterization of the model as before. Let B q(st ) = 1 for all st and t, R = 1.02, TH = RT +1 , and TL = TH /2. Also, assume that in period -1, everyone believes that Pr(sT = DT ) = 0, that is, everyone believes that the government collects taxes of amount TH in period T for sure. It follows that P−1 = 1. Recall also that π−1 = −1.96%.
Table 2 contains results for the Arrow-Debreu and Leverage equilibria when T = 5. The
first five rows report values of the inflation rate in periods t = −1, 0, . . . , 5 along the path leading to default. The final five rows report the identity of the marginal purchaser.
Consider first the results for the Arrow-Debreu market structure. For this parameterization of the model the inflation rate jumps from -1.96% to 6.95% in period zero when individuals realize that there is a possibility of a debt crisis. The size of the jump in the inflation rate in period zero is much smaller as compared to the 1-period model which saw the inflation rate jump to 30.7%. This difference is due to the fact the probability of a fiscal crisis is lower when the model has 5-periods. It is now (1 − h)5 as compared to (1 − h) before. Still, a jump in the inflation rate of nearly 7% is very large and inconsistent with the experience of Japan and the
United States who have seen their debt-GDP ratios rise with no discernible response of the inflation rate to events such as ratings downgrades in their debt issues. Notice also that the jump in the inflation rate in the crisis state, π(D5 ), is smaller in the model with five periods.
14
Table 2: The inflation rate and identity of the marginal buyer in the 5-period model along the path leading to implicit default.
Period
Arrow-Debreu
Leverage
Inflation (Percent) -1
π−1
-1.96
-1.96
0
π0
6.95
-1.96
1
π(D)
7.84
-1.96
2
π(D2 )
8.93
-1.96
3
π(D3 )
10.29
-1.01
4
π(D4 )
12.04
10.97
5
π(D5 )
14.38
71.57
Marginal Buyer of Bonds ¯0 0 h 0.5 ¯ 1 h(D) 0.33
1.00 1.00
3
¯ 2) h(D
0.25
1.00
4
¯ 3) h(D ¯ 4) h(D
0.20
0.94
0.17
0.75
5
15
From this we see that extending the horizon acts to smooth out the inflation response in the the Arrow-Debreu market structure along the path leading to implicit default. Subtle changes in participation in the market for government bonds underly these price dynamics. As proved in the Appendix, h(Dt ) = 1/(t + 2) in the Arrow-Debreu market structure. In the 1-period model the marginal buyer is h0 = 1/2 or the midpoint of the interval. This continues to be the case in the 5-period model. In period 1 when the state D is realized the net worth of the optimistic individuals with h > h0 falls to zero. The remaining individuals with positive net worth enter into new agreements and the new marginal purchaser is h(D) = 1/3. This process repeats itself as successive values of D are realized. The leverage market equilibrium has very different properties. Perhaps the most notably difference is the lack of response to the bad news in period zero. Its response is zero two decimal places. It follows that this model can account for the lack of a response in the inflation rate to either the downgrade of Japanese sovereign debt by Fitch in 2012 or the downgrade of U.S. sovereign debt by Standard and Poors in 2011. We can also see a very clear delay in the response of the inflation rate. As additional bad news arrives in period 1 and period 2, there continues to be no discernible response in the inflation rate. Even when the inflation rate begins to respond in period 3, its overall level is still very low at -1.01%. It is only in period four that we see a substantial increase in the inflation rate to 10.97%. The biggest increase in the inflation rate is concentrated in the final period when the crisis state is realized (71.57%). In fact, the size of the inflation rate in period 5 in the 5-period model is about the same as it’s final value in the 1-period model (75.62%). This last point can be related to the identity of the marginal buyer in the final period before the crisis. For the 1-period model her identify was 0.79. Her identity is also very high at 0.75 in the six period model. This is why the final period inflation rate is so similar in the two simulations. It is clear from these results that the standard channels whereby bad news about future outcomes gets translated into prices today is severely disrupted when short-sellers are absent. A final interesting observation is that the fraction of the population taking long positions on government debt is very concentrated in the leverage market-structure in early periods. In period 0 all government debt is held by less than 0.01% of the agents in the leverage market structure as compared to 50% of the population with Arrow-Debreu markets. As a sovereign default approaches the picture is reversed. In period 4, for instance, government debt is held by 19% of the population in the leverage market structure and by only 3% of the population when markets are frictionless. Previous work by Scheinkman and Xiong (2003) and Hong and Sraer (2011) has empha-
16
sized the role of the resale option in generating an asset price bubble. We wish to emphasize that their mechanism is not operative here. The only security that is traded in the leverage specification is a one period bond which by definition has no resale value. In other words there is no asset price bubble in this economy. To summarize our findings, when short-selling is not possible and leverage is available to optimists, in early periods virtually all agents are pessimistic and choose to lend to a tiny fraction of very optimistic agents. The extreme optimism of these agents determines the price of government debt and the inflation rate. Bad news wipes out these agents. But, they are replaced by other agents who are nearly as optimistic. It follows that at horizons of three to five years in advance of a sovereign default bad news has no discernible effect on the equilibrium inflation rate. Inflation does eventually respond but only shortly prior to the implicit default. When the inflation rate responds the size of the response is sudden and very large.
3
Explicit Default
3.1
1-Period Model with Explicit Default
We now turn to discuss a sovereign debt crisis that is resolved by an explicit default. Implicit default is not possible for countries that are credibly bound by currency unions and explicit default is the more relevant option. We will see that short-selling occurs in equilibrium in this setting and we are thus able to illustrate how the dynamics of asset prices and participation in government bond markets change as the cost of taking short positions is altered. We once again start by considering a 1-period version of the model. In Section 3.3 we generalize the model to allow for an arbitrary number of time periods. The sequence of events is the same. Prior to time zero all individuals agree that the probability of an explicit default is zero. At the beginning of time zero before any trade takes place news arrives. Agents interpret the news in different ways and this induces a nondegenerate distribution of beliefs h ∈ [0, 1] over the probability of an explicit default in period
1. Periods are indexed by t = 0, 1 and there are two states of nature in period 1, U and D that are distinguished by whether the government defaults on its debt. Government:
¯ > 0 nominal liabilities to the private sector The government starts off with B
that mature in period 1. It then raises revenue to pay off the debt in period 1 with lump sum taxes that are identical across agents. Let T (s1 ) denote the real amount of taxes in state s1 ∈ S of period 1. Then the “flow budget constraint” of the government in state s1 ∈ S of 17
period 1 is given by ¯ B = T (s1 ), s1 ∈ S (32) P1 where α(s1 ) is the fraction of debt repaid in state s1 and P1 is the price level in period 1. In α(s1 )
this version of the model the price level is exogenous and evolves according to P1 = Π = P0 where Π = 1 + π is the gross inflation rate. Default occurs in state D: α(D) < 1. In state U we have α(U ) = 1. Thus a government policy is given by (α(s1 ), T (s1 )) ∈ (0, 1]2 × R2++ . We assume that government policy is exogenous.
Individuals
There is a continuum of agents indexed by h ∈ [0, 1]. Each agent receives an
identical endowment of yt units of the consumption good in period t = 0, 1. Note that the
endowment does not depend on the state in period 1. Agents are also endowed with equal ¯ > 0, in period zero. All agents have access to a amounts of nominal government debt, B risk-free storage technology that offers a gross real rate of return denoted by R > 1. Agents are identical except for their beliefs about the probability that state U occurs in period 1. Specifically, we assume that agent h believes that s1 = U with probability h. Thus, agents with high h assign less probability to the debt crisis event. All agents have linear preferences of the form: c0 +
X
γ h (s1 )c(s1 )
(33)
s1 ∈S
where γ h (s1 ) denotes the subjective probability of agent h that state U occurs in period 1: ( h, for s1 = U , γ h (s1 ) = 1 − h, for s1 = D. Let k0h denote the amount of safe storage by agent h, mh0 the amount of money held by agent h, bh0 the amount of government debt held by agent h and φh0 the amount of loans obtained by agent h. Given these definitions the budget constraints in period 0 and 1 for agent h are ¯ bh B mh0 + q0 0 = q0 + y0 + φh0 − χh0 P0 P0 P0 mh bh ch (s1 ) = α(s1 ) 0 + Rk0h − Rφh0 + 0 + y1 − T (s1 ). P1 P1 ch0 + k0h +
(34) (35)
where q0 is the price of government bonds, and χh0 is a proportionate fee associated with short sales of government bonds: χh0
bh = χ max 0, −q0 0 P0
18
.
(36)
The collateral constraints are bh0 − Rφh0 ≥ 0, P1 bh mh α(s1 ) 0 + 0 ≥ 0, P1 P1
α(s1 )
s1 ∈ S,
(37)
s1 ∈ S.
(38)
Constraint (37) imposes the restriction that loans received to acquire bonds do not exceed the value of bonds in any state and constraint (38) imposes the condition that agents who wish to short-bonds hold sufficient collateral in the form of money to deliver bonds in any state. Even though we are imposing these constraints exogenously as we noted above they can be derived as endogenous constraints by proceeding in the same fashion as Geanakoplos (2003, 2010). It turns out to be convenient to consider equilibrium for a small open economy in which demand for loans is allowed to differ from the domestic supply of loans.6 This reduces the number of endogenous variables and significantly reduces the computational complexity of computing an equilibrium when we allow for more than two periods.7 Definition 2 (Competitive Equilibrium with Explicit Default). Given a government policy (α(s1 ), T (s1 )) a competitive equilibrium with explicit default consists of an allocation {ch0 ,
[ch (s1 )]s1 ∈S , k0h , mh0 , bh0 , φh0 }h∈[0,1] and a price of bonds q0 , such that (i) each agent h ∈ [0, 1], {ch0 , [ch (s1 )]s1 ∈S , k0h , mh0 , bh0 , φh0 } solves her utility maximization problem; (ii) the government
budget constraint (32) is satisfied; and (iii) the market for government bonds clears ¯= B
Z
1
0
bh0 dh.
(39)
Characterization of Equilibrium Agents hold one of three portfolios in an equilibrium with explicit default. The most optimistic agents borrow and use the proceeds to purchase government bonds. The most pessimistic agents short-sell government bonds and hold cash. The remaining agents lend to the optimists and/or save using the safe storage technology. The range of individuals who hold each of these portfolios is determined by the following 6
Even though we are not explicitly modeling money supply here, a second implicit assumption we are
making is that demand for cash by short-sellers of government bonds is much smaller than the domestic supply of money. 7
We have also computed a closed economy equilibrium for the 1-period model and found that the results
are qualitatively similar to the small open economy results that we present here.
19
indifference relations α(U ) − α(D) ¯ h0 Πq0 − α(D)/R α(U ) − α(D) (1 − h0 ). R= Πα(U ) − (1 − χ)Πq0 R=
(40) (41)
Equation (40) is the indifference relationship for a marginal purchaser of government bonds. ¯ 0 the return from safe storage is equal to the return of borrowing at the rate For individual h R and using the proceeds to purchase government bonds. Equation (41) is the corresponding relationship for individual h0 who is indifferent between safe storage and holding money as collateral in order to make short sales of government bonds. Given these definitions we can now describe the optimal portfolios of each type of investor. We will assume throughout that R > 1. Under this assumption all agents choose to defer ¯ 0 will take leveraged long positions consumption until period 1. It follows that agents h > h in government debt. Their optimal portfolio is k0h = mh0 = 0 α(D) h b RP1 0 P1 q0 ¯ y0 + B . bh0 = Πq0 − α(D)/R P0 φh0 =
(42)
P1 Πq0 −α(D)/R
is the amount of leverage by those taking long positions in government ¯ 0 ] the optimal portfolio is debt. For agents h ∈ [h0 , h where
bh0 = mh0 = 0
(43)
q0 ¯ B k0h − φh0 = y0 + P0 and the remaining agents h ∈ [0, h0 ) are short-sellers of government bonds and with portfolio k0h − φh0 = 0
mh0 = −α(U )bh0 bh0
P0 =− α(U ) − (1 − χ)q0
where the amount of leverage by short-sellers is
q0 ¯ y0 + B . P0
P0 α(U )−(1−χ)q0 .
Using equations (42), (43) and
(44) we can now express the government bonds market clearing condition as ¯ B Π q0 ¯ 1 q0 ¯ ¯ = (1 − h0 ) y0 + B − h0 y0 + B . P0 Πq0 − α(D)/R P0 α(U ) − (1 − χ)q0 P0 20
(44)
(45)
Finally, observe that equation (45) in conjunction with (40) and (41) allows us to solve ¯ 0 , h and q0 . for h 0 It will prove useful in what follows to compare the price of government bonds, q0 , in the model with short-selling to its price when markets are frictionless. The Arrow-Debreu price provides a reference point for discussing over-shooting or under-shooting. We show in the Appendix that the price of bonds in this market structure is: 1 1 1 AD q0 = α(U ) + α(D) . ΠR 2 2
3.2
(46)
Analysis of the 1-period Model with Explicit Default
We now use analytical and numerical methods to analyze how the size of the response of bond price to the same news changes as we alter the initial level of government debt, the costs of selling government debt short and the inflation rate. To anticipate our results we find that a higher debt-output ratio results in overshooting of the bond price in the short-selling equilibrium as compared to the Arrow-Debreu market structure, but that higher transactions costs of short-selling and higher inflation result in undershooting of the bond price. A Case Where Financial Frictions are Irrelevant We start by considering a shortselling equilibrium with no government debt. Even though there is no trade in bonds we can still price government debt and agents depending on their beliefs will choose to take long or short positions on the payoff of government debt in period 2. This situation is of interest because the response of bond prices in the short-selling equilibrium is the same as in the Arrow-Debreu equilibrium when ΠR = 1 and χ = 0. This later assumption implies that cash earns the safe interest rate and thus that the collateral posted by short sellers earns R. ¯ = 0, χ = 0 and ΠR = 1 then q0 = q AD . Lemma 2. Suppose B 0 ¯ = 0 then (45) becomes: Proof. Set B ¯0 h0 1−h = . ¯ 1 − h0 h0
(47)
Next setting χ = 0 implies (40) – (41) become 1 ¯ 0 + α(D)(1 − h ¯ 0) α(U )h ΠR 1 q0 = {α(U )(ΠR − 1 + h0 ) + α(D)(1 − h0 )} . ΠR
q0 =
21
(48) (49)
If ΠR = 1 it follows from the previous three equations that ¯0 = h = 1 h 0 2 1 1 1 α(U ) + α(D) q0 = ΠR 2 2
(50) (51)
Even though the price of bonds is the same as the allocations in the short-selling equilibrium and the Arrow-Debreu equilibrium are different. This difference stems from a difference in what can be pledged in time zero. In the short-selling equilibrium agents cannot borrow against their future endowment. If this restriction is relaxed the allocations are also identical in the two market structures. Government Debt. This equivalence breaks down when χ = 0, ΠR = 1 but government debt is positive. On the one hand, in the Arrow-Debreu market structure the Ricardian equivalence proposition obtains. Increasing the initial government-debt-output ratio from zero has no effect on q0 . This result follows from the observation that government debt is a redundant security. In the short-selling equilibrium, on the other hand, q0 < q0AD when the government debt-output ratio is positive and the degree of overshooting in the bond price increases monotonically with the amount of government debt. ¯ ≥ 0. Then Lemma 3. Suppose χ = 0, ΠR = 1, and B ¯ = 0; 1. q0 ≤ q0AD with equality when B ¯0 = h , 2. h 0
¯0 dh ¯ dB
=
dh0 ¯ dB
< 0 and
dq0 ¯ dB
< 0.
The proof can be found in the Appendix. Figure 3 illustrates these various points. The top panel shows that q0 falls monotonically as the debt-output ratio is increased. The second panel displays the identity of the marginal purchaser of government bonds and the marginal short-seller of government bonds at alternative debt-output levels. There is only one line because the identity of each agent is identical. When the debt-output ratio is positive the identity of the marginal purchaser of government bonds is less optimistic as compared to the Arrow-Debreu equilibrium. This finding follows from the fact that purchasers of government debt have more access to leverage when government debt is in positive net supply. This can be seen in the third panel of Figure 3. Leverage by those taking long-positions ¯ and leverage by those taking short positions falls. Intuitively, q0 is the cost increases with B for those taking long positions and a lower value of q0 lowers this cost. For short-sellers q0 is 22
Figure 3: One-period model with explicit default. Government bond price, (q0 ), marginal purchasers and short-sellers and leverage at alternative government debt levels: π = 1/R − 1 and χ = 0.
q0 (χ = 0.00, π=−0.02 )
0.76
Leverage Arrow−Debreu
0.755 0.75 0.745 0.74 0.735 0.73 0.725 0.72
0
0.2
0.4
0.6
0.8
1 1.2 debt−output ratio
1.4
1.6
1.8
2
marginal buyers (χ =0.00, π=−0.02 )
0.5
hU hL
0.49 0.48 0.47 0.46 0.45 0.44
0
0.2
0.4
0.6
0.8
1 1.2 debt−output ratio
1.4
1.6
1.8
2
leverage (χ = 0.00, π=−0.02 )
4.5
long short
4
3.5
0
0.2
0.4
0.6
0.8
1 1.2 debt−output ratio
23
1.4
1.6
1.8
2
the benefit and this benefit is smaller when q0 is low. More formally, recall that leverage for those taking long positions is given by: Levlong =
P1 Πq0 − α(D)/R
(52)
and that leverage for those taking short-positions is: Levshort =
P0 . α(U ) − (1 − χ)q0
(53)
¯ reduces q0 . Then observe that increasing B Our definition of leverage is different from how this term is used when referring to leverage in exchange traded funds (ETFs). These funds offer a return that is generally two or three times the daily inverse return of a long-term government bond index. This notion of leverage is defined as: ET F Levshort
q0 = q0 − α(D)
α(U ) − α(D) −1 . Πα(U ) − (1 − χ)Πq0
(54)
According to this definition leverage of short-sellers is about 3 when government debt is zero and is about 2.9 when government debt is one. For purposes of comparison the Direxion Daily 20 Year Plus Treasury Bear 3X (TMV) ETF is designed to offer short-sellers leverage of 3. Our model has two mechanisms that produce undershooting in the price of government bonds as compared to the Arrow-Debreu benchmark. We describe each of these mechanisms in turn. Transactions costs on short-sales of government debt.
Introducing a proportionate
cost on short-sales results in undershooting of the bond price in the short-selling equilibrium ¯ = 0 and ΠR = 1. when B ¯ = 0, ΠR = 1, and χ ≥ 0. Then Lemma 4. Suppose B 1. q0 ≥ q0AD with equality when χ = 0; 2.
d q0 dχ
> 0,
¯0 dh dχ
> 0, and
d h0 dχ
< 0.
The proof can be found in the Appendix. The effect of alternative settings of χ on the government bond price are reported in the top panel of Figure 4. The gap in the bond price across the two equilibria is moderate when short-selling costs are less than 10%. When χ = 0.05 prices fall to 0.77 in the short-selling equilibrium and to 0.79 when χ = 0.1 However, at larger but finite values of χ, the gap is quite substantial. For instance, the bond price only falls to 0.96 in period 0 when χ = 0.4. 24
Figure 4: One-period model with explicit default. Government bond price, (q0 ), marginal purchasers and short-sellers and leverage at alternative costs of short-selling (χ): π = 1/R − 1 and no government debt.
q0 (debt−output ratio = 0.0, π= −0.02)
1
Leverage Arrow−Debreu
0.95 0.9 0.85 0.8 0.75 0.7
0
0.05
0.1
0.15
0.2 cost of short sales
0.25
0.3
0.35
0.4
marginal buyers (debt−output ratio = 0.0, π= −0.02)
1
hU hL
0.8 0.6 0.4 0.2 0
0
0.05
0.1
0.15
0.2 cost of short sales
0.25
0.3
0.35
0.4
leverage (debt−output ratio = 0.0, π= −0.02)
4
long short
3.5
3
2.5
2
0
0.05
0.1
0.15
0.2 cost of short sales
25
0.25
0.3
0.35
0.4
In order to understand why the bond price undershoots consider the second and third panels of Figure 4. As χ is increased, short-selling becomes more costly and the identity of the marginal short-seller, h0 , declines. A smaller fraction of the population taking short positions reduces the supply of government bonds to those wishing to take long-positions and the price of government bonds falls by less in response to the news. It follows that leverage available to those wishing to take long positions falls because in the D event bond prices will fall by more. Thus, the identity of the marginal purchaser of government debt must rise. Those taking long and short positions have the same access to leverage when transactions costs on short-selling are positive. With zero government debt, clearing in the bond market implies that the net supply of government bonds is also zero. This can only occur if shortsellers and long-sellers have the same amount of leverage. Now there are three types of portfolios in the short-selling equilibrium. Some mildly pessimistic individuals choose to lend at the safe rate of R as compared to taking short positions on government debt. Once the transactions costs of taking short-positions are factored in they receive a higher expected return from lending as compared to short-selling government debt. Combining these various responses we see that costly short-sales depresses participation in the market for government bonds. As the costs are increased, the identify of the marginal purchaser increases, the identity of the marginal short-seller falls and an increasing fraction of the population chooses to stay on the side-lines. Inflation
Inflation also produces undershoting in the short-selling equilibrium. To see why ¯ = 0, χ = 0 and ΠR > 1. In this situation cash is dominated this is the case suppose that B in rate of return by safe storage and only short sellers hold cash. They hold cash because it is required as collateral for their short-positions. It follows that one effect of inflation is that it acts as a type of transactions tax on short-sellers. A second effect of inflation arises from the fact q0 is the nominal price of bonds. Higher inflation reduces the price of bonds for the standard reasons (it raises the nominal interest rate). This second mechanism is operative in both the Arrow-Debreu and the short-selling market structures. ¯ = 0, χ = 0, and Π ≥ R−1 . Then Lemma 5. Suppose that B 1. q0 ≥ q0AD with equality when Π = R−1 ; 2.
dq0 dΠ
< 0,
d(q0 −q0AD ) dΠ
> 0,
¯0 dh dΠ
> 0 and
dh0 dΠ
< 0.
The proof is provided in the Appendix. These various effects are illustrated in Figure 5. The most noteworthy feature fo the figure is that higher inflation is associated with 26
Figure 5: One-period model with explicit default. Government bond price, (q0 ), marginal purchasers and short-sellers and leverage at alternative inflation rates: χ = 0 and no government debt.
q0 (debt−output ratio = 0.0, χ = 0.00)
0.8
Leverage Arrow−Debreu
0.75
0.7
0.65
0.98
1
1.02
1.04
1.06 inflation rate
1.08
1.1
1.12
1.14
marginal buyers (debt−output ratio = 0.0, χ = 0.00)
1
hU hL
0.8 0.6 0.4 0.2 0 0.98
1
1.02
1.04
1.06 inflation rate
1.08
1.1
1.12
1.14
leverage (debt−output ratio = 0.0, χ = 0.00)
4
long short
3.9 3.8 3.7 3.6 3.5 3.4 0.98
1
1.02
1.04
1.06 inflation rate
27
1.08
1.1
1.12
1.14
a lower value of q0 in both market structures. In other respects the results are simular to those associated with higher values of χ. The short-selling equilibrium price of bonds is higher (undershoots) than the Arrow-Debreu bond price and that the degree of undershooting increases with the level of the inflation rate and a higher inflation rate reduces participation on the market for government bonds. An applying the model to Greece
We have seen that positive government debt results in
overshooting of government bond prices in the short-selling equilibrium but that proportionate transactions costs on short sales and inflation result in undershooting. Which of these effects are most important? To answer this question we consider the combined effects of these three factors on government bond prices. It turns out that transactions costs on short sales have much stronger effects on government bond prices as compared to the size of the debt-output ratio. Figure 6 reports government bond prices for the short-selling and Arrow-Debreu equilibria using a parameterization that is based on Greece. The inflation rate is set to 2%, χ is alternatively set to 0.01 or 0.05 and the debt-output ratio varies from 0 to 3. For purposes of comparison, the annualized CPI inflation rate in Greece was 2.2% in December 2011 and the debt-GDP ratio in Greece was 150% in 2010 and 178% in 2011. Results reported in this figure show that government bond prices in the short-selling equilibrium undershoot the Arrow-Debreu prices when χ exceeds one and the debt-GDP ratio is close to 1.5. When the value of χ is 1%, the price response in the short-selling equilibrium is about the same as the Arrow-Debreu prices at a debt-output ratio of 1.5. Larger values of χ imply that bond prices in the short-selling equilibrium are undershooting the Arrow-Debreu prices. For instance, when χ = 0.05 bond prices in the short-selling equilibrium undershoot the Arrow-Debreu prices for all debt-GDP ratios between 0 and 3. The results in Figure 6 raise the possibility that costs associated with selling Greek debt short acted to attenuate the overall decline in bond-prices in the period leading up Greece’s debt workout with the EU and IMF in March of 2012. Although it is difficult to measure the overall costs of short-selling government debt, indirect evidence suggests that these costs were significant and that they increased as Greece moved towards default. Credit default swaps were traded in Greece well prior its sovereign debt crisis and they did payout when Greek reached an agreement with the EU and IMF to reschedule its debt payments in March 2012. However the net size of the CDS positions was small. At the time of default the net notional amount of CDS contracts was estimated to be about $3 billion. This constituted less than one percent of total outstanding Greek government debt which was about 360 billion euros. One
28
Figure 6: Price of government debt, (q0 ), in the 1-period model: Greece Scenario (inflation rate is 2% and explicit cost of short-selling is either 1% or 5%). q0 (χ1 = 0.01, χ2 = 0.05)
0.76
χ1 = 0.01 χ2 = 0.05 Arrow−Debreu
0.75
0.74
0.73
0.72
0.71
0.7
0
0.5
1
1.5 debt−output ratio
2
2.5
3
reason the size of CDS positions may have been so small was uncertainty about whether the CDS contracts would pay out at all. In the weeks leading up to the workout efforts were made to structure it in a way so that the workout would not trigger a payout of credit default swaps. Another reason why the volume of CDS positions was low is that European governments had taken previous measures to make short-selling more costly. Germany imposed a ban on naked short sales of foreign sovereign CDSs and financial stocks on May 18, 2010. As conditions worsened in August of 2011, stock prices of Greek banks and other companies plummeted. Greece responded by banning short-sales on all stocks on Aug. 8 2011.8 France, Italy, Spain and Belgium followed suit shortly thereafter banning short-sales in financial service sector stocks on August 11, 2011. Then in November 2011 the European parliament voted to ban naked CDS on sovereigns. This legislation, however, only came into force in November 2012 well after the Greek CDS credit event. 8
Greek banks hold large amounts of Greek government debt and one way to take a short position on a
sovereign default is to short Greek banks.
29
3.3
The T-Period Model with Explicit Default
We now turn to consider the T-period model with explicit default. Extending the number of periods allows us to analyze the dynamics of government bond yield movements leading up to a sovereign default. We assume that a model period corresponds to a year and when we calibrate the model we calibrate it using annualized interest rates. Generalizing the model in this way makes it possible to analyze the dynamics of bond price movements leading up to a sovereign debt crisis and in particular to illustrate situations where the initial response of bond prices to bad news about sovereign default is very small. ¯ be the face value of government debt in period 0 and suppose that the government Let B does not issue new debt in any other period. All government debt is long-term and matures in the final period, T . Suppose also that the government only collects taxes in the last period. ¯ in all but Under these assumptions the nominal outstanding value of government debt is B the last period. As before, a shock st ∈ {U, D} is realized in periods t = 1, . . . , T . The government defaults ¯ otherwise. When the government in period T only if sT = DT ; it repays the full amount of B ¯ It follows that the amount of taxes collected defaults, it repays only a fraction α ∈ (0, 1) of B. in the last period, T (sT ), is given by
T (sT ) = α(sT )
¯ B , P (sT )
where α(sT ) is defined as ( T
α(s ) =
α,
if sT = DT ,
1,
otherwise.
It is convenient to write q(sT ) = α(sT ). The flow budget constraints for agent h are given by q0 h mh0 q0 ¯ b0 + ≤ B + y0 + φh0 − χh0 , P0 P0 P0 q(st ) h t 1 ch (st ) + k h (st ) + b (s ) + mh (st ) Pt Pt q(st ) h t−1 1 ≤ b (s ) + R[k h (st−1 ) − φh (st−1 )] + mh (st−1 ) + φh (st ) − χh (st ), t = 1, . . . , T − 1, st ∈ S t , Pt Pt T) q(s 1 h T −1 ch (sT ) = bh (sT −1 ) + R[k h (sT −1 ) − φh (sT −1 )] + m (s ) + yT − T (sT ), sT ∈ S T , PT PT ch0 + k0h +
where the short-selling fees are q(st ) h t χ (s ) = χ max 0, − b (s ) . Pt h
t
30
Figure 7: Yields and one-period holding returns on government bonds in the 4-period model with explicit default at alternative costs of short-selling, χ. Greece scenario: inflation rate is 2% and debt-GDP ratio is 1.5. Yield to Maturity
60 χ = 0.01 50
One−Period Holding Returns
5 0
χ = 0.04 χ = 0.08
−5
No Short Sales 40
−10
Arrow−Debreu
%
%
−15
30
−20 −25
20
0
1 Period
2
χ = 0.04
−30
χ = 0.08 No Short Sales
−35
Arrow−Debreu
10
0 −1
χ = 0.01
−40 −1
3
0
1
2
3
4
Period
(a) Yield to maturity of a government bond
(b) One period holding returns on govern-
that matures in period 4.
ment debt.
This structure has the property that the short-selling fees are paid every period. We believe that this is a reasonable assumption. ETFs that are used to short-sell U.S. treasuries, for instance, reset to one hundred either every day or every month. An investor who uses this security to take a short-position at a longer horizon must readjust his portfolio on a daily or monthly basis which incurs transactions costs. Similarly, open positions in options and futures contracts are concentrated at very short-horizons of less than six months. Rolling over these contracts to take short-positions are longer horizons also incurs transactions costs. The collateral constraints are: Rφh (st ) ≤
q(st+1 ) t b(s ), Pt+1
t = 0, . . . , T − 1, st+1 ∈ S t+1 , st+1 ∈ S,
mh (st ) ≥ −q(st+1 )bh (st ) The non-negativity constraints: c(st ), k(st ), m(st ) ≥ 0 for all t and st .
3.4
Analysis of the T-period model
The T-period model is sufficiently complex that we have no alternative but to rely entirely on computational methods in characterizing the equilibrium. Finding the marginal purchasers and sellers of government debt is a rather subtle numerical problem that quickly becomes intractable as the number of periods increases and the number of potential trading strategies for each individual increases.
31
Price Dynamics We start by considering how costs on short-sales affect the dynamics of bond yields and 1-period holding returns along the path to default in our model with explicit default. Figure 7a reports yields to maturity of a government bond that matures in the final period in a 4-period version of the model.9 Results are reported for the Arrow-Debreu market structure, the leverage market structure with no short sales and three intermediate settings of χ. All of the results use our previous Greece scenario with a debt-output ratio of 1.5 and an inflation rate of 2% . Our first finding is that costly short-selling results in lower bond-yields than the ArrowDebreu benchmark. This occurs for all values of χ > 0 and in all periods leading up to default. In the 1-period model with explicit default we saw that a higher debt-GDP ratio in isolation acted to produce overshooting and that the size of this effect could be quite significant. The range of model parameters that produce undershooting of government bond yields is much larger in the multi period model. For instance, using a two percent inflation rate and χ = 0.01, overshooting occurs in the 1-period model when the debt-GDP ratio is larger than 1.5 with χ = 0.01. Using the same values of inflation and χ, the debt-output ratio has to be in excess of 30 to produce overshooting when T = 4. This finding is due to the fact that adding more periods reduces the overall level of short-selling activity and this limits the scope for overshooting. Observe next that the extent of overshooting as measured by the vertical distance between the yield in the Arrow-Debreu market structure and the short-selling market structure increases along the path leading to default. When χ = 0.01. the extent of undershooting varies from a low of 71 basis points in period 0 to a high of 346 basis points in the period immediately prior to default. As χ is increased the magnitude of undershooting relative to the Arrow-Debreu benchmark increase. For instance, the gap between the two market structures is 958 basis points in period three when χ = 0.04 and 3596 basis points in period three when short-sales are not allowed. Note next that as χ is increased the initial response of bond yields drops to zero. For instance, when χ = 0.04 the yield of the government bond rises by only 15 basis points in period zero and when χ = 0.08 there is no response in the bond yield in either period zero or period one. Instead, responses get delayed and concentrated into states that are close to a sovereign default. The fact that costly short-selling is associated with undershooting in government bond yields in all periods might appear to be at odds with our previous results for the model 9
To be more precise the figure reports 1/(T − t) ln(1/q(st )) with T = 4 and t = −1, 0, . . . , 3.
32
with implicit default. In that model the inflation rate exhibited undershooting relative to the same Arrow-Debreu benchmark in early periods and overshooting shortly before default. This difference between the two models has to do with the fact that bond yields and inflation are different prices. Inflation in the model with implicit default corresponds to one period holding returns in our model with explicit default. Figure 7b displays one-period holding returns which are given by q(st+1 )/q(st ). From this figure we see that one-period holding returns exhibit the same pattern of under-shooting in early periods and over-shooting in states close to default that we found in the model with implicit default. All of the leverage market structures undershoot the Arrow-Debreu returns in the first two periods and overshoot in the final period. In our model with implicit default no short-selling occurred in equilibrium and we were thus not able to document how the dynamics of the model change as the costs of short-selling are altered. It is very clear from the results in Figures 7a-7b that even moderate transactions costs of a magnitude from 4-8% significantly alter the price dynamics in a way that makes them resemble the dynamics of the model when no short-sales are banned. Overall, the dynamics of prices in the multi-period model with explicit default are surprisingly similar to the previous model with implicit default. When χ ≥ 0.04 bond yields and
one-period holding returns exhibit no discernible response to the initial bad news in period
0. The responses of both variables also exhibit a discernible delayed as compared to ArrowDebreu. Bond yields undershoot the Arrow-Debreu reference point in all periods leading up to default but, one-period holding returns overshoot in states immediately prior to default as inflation did in our previous model. Trading Dynamics
The multi-period model with explicit default also has a rich set of
implications for the dynamics of trade. Consider Table 3 which reports the percentage of the population taking short and long positions on government debt and the total percentage of the population that is participating in the government bond market in each period for the history that results in a sovereign default. We saw above that transactions costs on shortselling reduced participation of short-sellers in the government bond market in the 1-period model. This same phenomenon occurs in the T-period model. When short-selling constraints are absent 50% of the population takes short-positions along the path leading to a sovereign default (see row 5 of Table 3).10 This percentage drops to 34.3% when χ = 0.0 and is due to the fact that inflation also increases the costs of taking-short positions as we documented in 10
The “total” percentage of the population that takes a short position is the maximum percentage of short-
sellers in the given column.
33
the 1-period model above. As χ is increased, the percentage of the population taking short positions falls to 29.6% when χ = 0.01 and to 21.5% when χ = 0.04. The effect of the transactions cost on short-selling is most pronounced in early periods when the prospect of a default is distant. No agents choose to take short positions in period zero when χ = 0.04, and no agents take short positions in either period 0 or period 1 when χ = 0.08. As default approaches short-selling activity increases. For instance, short-selling is monotonically increasing as default approches when χ = 0.08. This pattern is the mirror opposite of the Arrow-Debreu equilibrium which exhibits a monotonic declines in short-sales activity along the path to default. Transactions costs on short-sales also have a depressing effect on the participation of purchasers of government debt. In the Arrow-Debreu market structure of 80% of the population takes a long position on government bonds at some point along this path.11 In the leverage market structure participation of those taking long positions falls to 76.5% when χ = 0, 67.9% when χ = 0.04 and 28.1% when short-sales are banned entirely. The reason for this decline in long positions was discussed above in the 1-period model. As χ increases, the supply of bonds available to those taking long positions falls and leverage available to those wishing to take long positions declines. This in turn increases the identity of the marginal purchaser of government debt. Using our 4-period model we can document the dynamics of this general equilibrium effect of costly short sales on long bond trading. The depressing effect of costly short-selling long bond trading is most pronounced in early periods. For instance, when χ = 0.04 only 4.3% of the population takes long positions in government debt on period 0 as compared to 50% in the Arrow-Debreu equilibrium. Costly short-sales also disrupt the timing of long-trading. When χ ≥ 0.08. the fraction of agents taking long positions increases monotonically as default approaches. Whereas in the Arrow-Debreu equilibrium the fraction falls monotonically.
The final four rows of Table 3 reports the total number of active traders in the government bond market in each period. As the costs of short-selling rise, an increasing fraction of the population chooses to stay on the sidelines and hold their assets in the form of safe storage or equivalently safe loans to traders taking long-positions. The percentage of agents that are on the sidelines is largest in early periods. Only 4.3% of the population has an active trading position in period zero when χ = 0.04 and only 0.71% is active when χ = 0.08. However, as the economy moves closer to a sovereign default, the returns from betting on this event increase and the percentage of the population taking 11
The “ total” percentage of the population taking long positions is the sum of long-purchasers in each row
in a given column .
34
Table 3: Traders in government bonds in the 4-period model with explicit default for the history that results in a sovereign default.
Period
History
Arrow-Debreu
χ = 0.0
χ = 0.01
χ = 0.04
χ = 0.08
No short sales
Short in government bonds 0
0
50.00
34.34
29.58
0.00
0.00
0.00
1
D
33.33
20.31
19.46
21.49
0.00
0.00
2
D2
25.00
15.98
15.47
19.12
20.13
0.00
3
D3
20.00
15.43
14.90
18.70
22.80
0.00
Total Short
50.00
34.34
29.58
21.49
20.13
0.00
Long in government bonds 0
0
50.00
31.52
28.71
4.30
0.71
0.01
1
D
16.67
18.45
18.40
23.10
6.10
0.77
2
D2
8.33
15.09
15.01
19.89
23.89
7.10
3
D3
5.00
11.43
13.66
20.65
27.03
20.19
80.00
76.49
.75.78
67.94
57.73
28.07
Total Long Total active traders 0
0
100.00
65.86
58.29
4.30
0.71
0.01
1
D
50.00
38.76
37.86
44.59
6.10
0.77
2
D2
33.33
31.07
30.48
39.01
44.02
7.10
3
D3
25.00
26.86
28.56
39.35
49.83
20.19
All numbers are percentages of the total population. Total active traders refers to all agents who participate in the government bond market and is the sum of those taking long and short positions. Total Short (Long) refers to the total percentage of the population that takes a short (long) position along this history.
35
both short and long positions increases. For the case of χ = 0.04, 39.4% of the population participates in the government bond market in period 3 (state D3 ) with 20.7% taking long positions and the remaning 18.7% taking short positions. Higher transactions costs act to delay and concentrate trading activity into states closer to the sovereign-default. However, this effect is not monotonic. Trading activity in period 3 is higher, for instance, when χ = 0.08 as compared to short-sales are ruled out. Note also that when this burst trading activity occurs its scale can be so large as to exceed the overall level of trading activity in the Arrow-Debreu market structure. This can be seen either when χ = 0.04 or 0.08. Higher participation is in turn is associated with the sudden and large movements in bond yields and one-period holding returns in periods two and three that we documented above. Short-selling activity is often attributed to the large price swings that occur shortly before sovereign debt crises and also exchange rate crisis. When chi ≥ 0.04, participation of short-
sellers is increasing but participation of those taking long-positions is increasing by even more in every period along this path. Discussion
We next consider the effects of increasing the costs of short-selling as was done
by Europe when it banned naked short-sales of sovereign debt in 2011-2012. Short-sellers face a range of costs even in normal times (see Angel (2004)). Using our short-selling specification. It thus makes sense to recognize this fact and start from a baseline where the transactions costs of short-selling are positive. We have already seen that bond yields are lower when the costs of short-selling are positive. The results in Figure 7a also indicate that increasing the costs of short-selling as bond yields start to rise in anticipation of a possible sovereign debt crisis substantially reduces the magnitude of the increase in bond yields. For instance, imposing restrictions on short-selling that are equivalent to increasing the transactions cost χ from 1% to 4% reduces government bonds yields by 180 basis points in period 0 if the restrictions are imposed at that juncture. The differences are even larger if the same measure is taken in subsequent periods. In our model a sovereign-default is an exogenous event that does not depend on funding costs. Still, these results suggest that the savings to the fiscal authority from this type of measure could be substantial. The fact that a small change in the cost of short-selling has such a large impact on bond yields suggests that raising this cost could also have a large impact on welfare. Conducting welfare comparisons in our model is subtle because agents have heterogenous beliefs and it is not clear how one should aggregate these beliefs. We start by considering a Rawlsian notion of welfare and document the ex post percentage of the population that goes bankrupt. Up to now we have only considered the single history that results in default. In order to investigate
36
Table 4: Percentage of the population that goes bankrupt in each history in the 4-period model with explicit default.
Period
History
χ = 0.0
χ = 0.01
χ = 0.04
χ = 0.08
No short sales
1
U
34.34
29.58
0.00
0.00
0.00
2
D, U
51.83
48.17
25.79
0.71
0.01
3
D, D, U
65.95
62.58
46.52
26.94
0.78
4
D, D, D, U
80.49
77.02
65.99
53.5
7.88
4
D, D, D, D
76.49
75.78
67.94
57.73
28.07
*All numbers are percentages of the total population. A sovereign debt crisis does not occur in any history in which the event U occurs. We thus terminate the history at the point that the U event is realized.
ex post welfare we also need to document outcomes for all other histories. Table 4 reports the percentage of the population that is bankrupt for each history. Recall that a realization of D results in bankruptcy of all agents who took long positions at the previous stage. This follows from the fact that agents are risk neutral. A realization of U results in bankruptcy for all agents who took short positions in the previous stage. It is also useful to keep in mind that we have assumed that any history in which a realization of U results in higher taxes in the final period. The results reported in Table 4 indicate that there is a strong rationale for imposing short-selling constraints under the Rawlsian welfare criterion. The percentage of the bankrupt population falls monotonically as χ is increased from zero. This pattern occurs in all histories leading to higher taxes and also the single history that results in a sovereign default. As the costs of short-selling rise, a larger percentage of the population stays on the side-lines in all histories. Instead of participating in the bond market, they choose to make safe loans instead. Brunnermeier, Simsek and Xiong (2014) propose an alternative Bergsonian welfare criterion for aggregating individual utilities when agents have heterogenous beliefs.12 our model increasing χ from zero results in a loss in resources due to the fact that every short-seller faces a loss in goods whenever short-selling occurs in equilibrium. When inflation exceeds the Friedman rate, short-sellers also face a cost when χ = 0 because they have to hold their collateral in the form of cash. Only a complete ban on short-selling avoids this cost and it 12
See Definition 1 in Brunnermeier et al. (2014).
37
Figure 8: Ex ante welfare by type of individual (h) in the 1-period model with π = −0.02 and
a debt-output level of 1.5.
Ex Ante Utility 5 χ = 0.00 No Short−Sales
4.5
4
3.5
3
2.5
2
0
0.1
0.2
0.3
0.4
0.5 Individual (h)
0.6
0.7
0.8
0.9
1
follows that the Brunnermeier et al. (2014) welfare criterion will select the no-short-selling specification which is the same specification chosen using a Rawlsian criterion. If the inflation rate is set according to the Friedman rule instead, both the χ = 0 and the no short-selling specifications are selected by the Brunnermeier et al. (2014) welfare criterion. Utility is linear in our model and aggregate payouts are identical both when transactions costs on short-selling are infinite and when there are no costs of short-selling. It is interesting that this welfare criterion gives the same ranking to an allocation in which the minimum percentage of the population that goes bankrupt is 57.5% (no costs of short-selling) and to an allocation where the maximum percentage of the population that goes bankrupt is (28.07%) (a total ban on short-selling).13 We conclude by documenting properties of ex ante welfare for each individual in our economy. We limit attention to the no-short-selling scenario and the scenario with (χ = 0 and Π = 1/R) and a debt-GDP ratio of 1.5. Agents with h sufficiently low always prefer the specification with no costs on short-sales. We have explained above that costs on short-sales also increase costs for those wishing to take long positions and it follows that those with h sufficiently high will also prefer the economy with costless short-selling. However, for h close 13
The former percentage is based on a simulation not reported in Table 4.
38
to 1/2, these benefits are smaller and they in fact prefer the economy with no short-sales in some situations. For instance, in the 1-period model agents with 0.4 ≤ h ≤ 0.5 have higher
expected utility in the economy with no short-sales (see Figure 8). However, when the number
of periods is extended to four, all agents prefer the economy with χ = 0 to the economy with no-short sales.
4
Conclusion
The world has recently witnessed a number measures taken by governments that have increased the cost of short-selling or even banned short-selling entirely. We have found that these restrictions disrupt a basic price-revelation mechanism associated with forward looking behavior. In frictionless markets bad news about future outcomes gets reflected in prices today as individuals trade on the news. Our findings suggest that the action of short-sellers plays an essential role in this price-revelation mechanism. Small transactions costs on shortsellers of a magnitude ranging from 4 to 8% severely disrupt this mechanism. An outright ban on short-sales of government debt has an even more pronounced effect on bond price and inflation dynamics. In the context of our model there are two justifications for a government to impose costs on short-selling. First, higher costs on short-selling government debt reduces downward price pressure on government debt in the short-run. This short-run benefit has a cost. When prices do move, the movements are more sudden and large. Second, higher costs of short-selling reduce participation in government bond markets and this in turn reduces the fraction of agents that go bankrupt.
References [1] Angell, James. 2004. “Short selling around the world.” Unpublished Manuscript. [2] Bassetto, Marco. 2002. “A game-theoretic view of the fiscal theory of the price level.” Econometrica, 70, 2167-2195. [3] Bi, Huixin. 2011. “Sovereign default risk premia, fiscal limits and fiscal policy.” Unpublished Manuscript. [4] Braun, R. Anton and Tomoyuki Nakajima. 2011 “Why prices don’t respond sooner to a prospective sovereign debt crisis.” Federal Reserve Bank of Atlanta Working Paper, 2011-13.
39
[5] Brunnermeier, Markus, Alp Simsek and Wei Xiong. 2014. “A welfare criterion for models with distorted beliefs.” Forthcoming “Quarterly Journal of Economics.” [6] Corsetti, Giancarlo, Paolo Pesenti and Nouriel Roubini, 2001. “The role of large players in currency crises.” NBER Working Paper number 8303. [7] Davig, Troy, Eric M. Leeper and Todd B. Walker. 2011. “Inflation and the fiscal limit.” European Economic Review, 55, 31-47. [8] Fostel, Ana, and John Geanakoplos. 2008. “Leverage cycles and the anxious economy.” American Economic Review, 98, 1211-1244. [9] Geanakoplos, John. 2003. “Liquidity, default, and crashes: Endogenous contracts in general equilibrium.” In Advances in Economics and Econometrics: Theory and Applications, Eighth World Conference, Vol. 2, 170-205. Econometric Society Monographs. [10] Geanakoplos, John. 2010. “The leverage cycle.” In NBER Macroeconomics Annual 2009, ed. Daron Acemoglu, Kenneth Rogoff, and Michael Woodford, 1-65. University of Chicago Press. [11] Harrison, J. Michael and David M. Kreps. 1978. “Speculative investor behavior in a stock market with heterogeneous expectations.” Quarterly Journal of Economics, 92, 323-336. [12] Hong, Harrison and David Sraer. 2011. “Quiet bubbles.” Unpublished manuscript. [13] Kiyotaki, Nobuhiro and John Moore. 1997. “Credit Cycles.” Journal of Political Economy 105, 211-48. [14] Scheinkman, Jos´e A. and Wei Xiong. 2003. “Overconfidence and speculative bubbles.” Journal of Political Economy, 111, 1183-1219. [15] Simsek, Alp 2011. “Belief disagreements and collateral constraints.” unpublished manuscript.
40
A
Appendix
A.1
Arrow-Debreu Market Structure
Consider a setting with Arrow-Debreu markets and no trading frictions. Here we only describe the model with implicit default. The model with explicit default is very similar and thus omitted. A.1.1
1-Period Model
We start with the 1-period model. Suppose that a complete set of one period contingent claims (Arrow securities) are traded in period 0. Let q(s1 ) denote the price of an Arrow security that pays off one unit of account in period 1 if and only if s1 occurs. Each agent h ∈ [0, 1] maximizes utility (1) subject to the budget constraints: c0 + k0 +
X
q(s1 )
s1 ∈S
c(s1 ) ≤ Rk0 +
¯ b0 B b(s1 ) + q0 ≤ + y0 , P0 P0 P0
b(s1 ) b0 + + y1 − T (s1 ), P (s1 ) P (s1 )
c0 , k0 , b0 , c(s1 ) ≥ 0,
s1 ∈ S,
(55) s1 ∈ S,
(56) (57)
where k0 is the amount goods stored in period 0, b0 is the amount of government bonds purchased in period 0, and {b(s1 )}s1 ∈S are the amounts of Arrow securities purchased in period 0. Notice that (57) imposes a short-selling restriction on government debt. Given that
agents have access to a complete set of one period Arrow securities this restriction doesn’t matter in this setting. Having described the agent’s problem we can now define a competitive equilibrium. Definition 3 (Arrow-Debreu Competitive Equilibrium). Given a government policy (q0 , TH , TL ), an Arrow-Debreu competitive equilibrium consists of an allocation {ch0 , [ch (s1 )]s1 ∈S , k0h , bh0 ,
[bh (s1 )]s1 ∈S }h∈[0,1] , supply of government bonds B0 , and prices {P0 , [P (s1 )]s1 ∈S , [q(s1 )]s1 ∈S }
such that (i) for each agent h ∈ [0, 1], the allocation {ch0 , [ch (s1 )]s1 ∈S , k0h , bh0 , [bh (s1 )]s1 ∈S }
solves her utility maximization problem; (ii) the government flow budget constraints (2)-(3)
41
are satisfied, where taxes {T0 , [T (s1 )]s1 ∈S } are given by (4)-(5); and (iii) all markets clear: Z 1 (ch0 + k0h ) dh = y0 , (58) 0 Z 1 Z 1 h k0h dh, s1 ∈ S, (59) c (s1 ) dh = y1 + R 0 0 Z 1 bh0 dh = B0 , (60) 0 Z 1 bh (s1 ) dh = 0, s1 ∈ S. (61) 0
We now turn to provide a characterization of the the Arrow-Debreu equilibrium. In this setting both storage and the government bond are redundant assets. Since the government bond pays off one unit of account irrespective of the state in period 1, no-arbitrage requires that X
q(s1 ) = q0 .
(62)
s1 ∈S
Similarly, since one unit of goods put into storage in period 0 yields R units of goods in all states in period 1, no-arbitrage also implies X q(s1 )P (s1 ) R = 1, P0
(63)
s1 ∈S
when storage occurs in equilibrium. It follows from these observations that the flow budget constraints (55)-(56) can be combined to obtain the lifetime budget constraint: ¯ X X q(s1 )P (s1 ) B q(s1 )P (s1 ) Rc0 + Rc(s1 ) ≤ R + y0 + R y1 − T (s1 ) P0 P0 P0 s1 ∈S
(64)
s1 ∈S
and the agent’s problem can be restated as maximizing (1) subject to (64). Since R > 1 our preference structure implies that no one consumes in period 0: ch0 = 0, Hence
R1 0
for all h ∈ [0, 1].
(65)
k0h dh = y0 > 0 and thus there is storage in equilibrium. Define h0 as h0 =
q(U )P (U ) R. P0
It follows from (63) that h0 is between 0 and 1, and that 1 − h0 =
q(D)P (D) R. P0 42
(66)
Then we see that h0 is the marginal agent in the sense that agents h > h0 make different choices than agents with h < h0 . Agents with h > h0 are optimistic and choose ch (U ) = 0. Agents with h < h0 are pessimistic and choose ch (D) = 0. Next consider the government. Equations (2) and (4) imply B0 =
¯ B q0
(67)
and it follows that the price level in period 1 is P (s1 ) =
¯ 1 B , T (s1 ) q0
s1 ∈ S.
(68)
Using the no-arbitrage condition (62), we can combine the flow budget constraints (2)-(3) to derive the government’s present value budget constraint X q(s1 ) ¯ B = P (s1 )T (s1 ). P0 s ∈s P0 1
Then use (66) to write this expression as ¯ B TH TL = h0 + (1 − h0 ) . P0 R R
(69)
Note that this equation determines P0 given h0 . Next we use the market clearing restrictions to determine the identity of the marginal agent. Starting with the agent’s lifetime budget constraint (64) and using (69), its right hand side becomes X ¯ B q(s1 )P (s1 ) + y0 + R y1 − T (s1 ) = Ry0 + y1 . R P0 P0 s1 ∈S
Thus (64) can be rewritten as Rc0 + h0 c(U ) + (1 − h0 )c(D) ≤ Ry0 + y1 . It follows that the consumption allocation is given by: ( 1 for h > h0 , h h0 (Ry0 + y1 ), c (U ) = 0, for h ≤ h0 , ( 0, for h > h0 , ch (D) = 1 for h ≤ h0 . 1−h0 (Ry0 + y1 ), Market clearing implies Z Ry0 + y1 =
1
ch (U ) dh =
0
43
1 − h0 (Ry0 + y1 ), h0
(70)
(71)
which yields the identity of the marginal agent 1 h0 = . 2 Finally, the Arrow security prices q(s1 ) are determined by solving
q(s1 )P (s1 ) R P0
=
1 2
for each
s1 ∈ S. The next proposition summarizes the result. Proposition 6. In the 1-period model with complete asset markets, (i) the equilibrium consumption allocation {ch0 , [ch (s1 )]s1 ∈S }h∈[0,1] is given by (65), (70), and (71), where the marginal agent h0 = 21 ; and (ii) the equilibrium price levels {P0 , [P (s1 )]s1 ∈S } are given by (68) and (69).
One implication of this proposition is that the consumption allocation is independent of government policy or in other words Ricardian equivalence obtains. In the other two market structures we consider Ricardian equivalence fails. A.1.2
T -Period Model
Now consider the model with a more general time horizon. Let q(st+1 |st ) denote the price of
the Arrow security traded at st that pays off one unit of account if and only if st+1 occurs in the next period. The price of one-period government debt is q(st ), which is determined by monetary policy (i.e, the inverse of the gross nominal interest rate). With a complete set of Arrow securities, the flow budget constraints for each agent are given by c0 + k0 +
X s1 ∈S
q(s1 |s0 )
c(st ) + k(st ) +
X st+1 ∈S
¯ b0 B b(s1 |s0 ) + q0 ≤ + y0 , P0 P0 P0
q(st+1 |st )
t b(st+1 |st ) t b(s ) + q(s ) P (st ) P (st )
b(st |st−1 ) b(st−1 ) + + Rk(st−1 ), t = 1, . . . , T − 1, st ∈ S t , P (st ) P (st ) b(sT |sT −1 ) b(sT −1 ) c(sT ) ≤ + + Rk(sT −1 ) + yT − T (sT ), sT ∈ S T , P (sT ) P (sT )
(72) (73)
≤
c(st ), k(st ), b(st ) ≥ 0,
st ∈ S t , t = 0, . . . , T,
(74) (75)
where b(st+1 |st ) denotes the quantities of the Arrow securities purchased at st , and b(st ) denotes the quantities of government bonds purchased at st .
Given a policy ({q(st )}, TH , TL ), a competitive equilibrium is defined as in the two-period case. Since the government collects taxes only in the last period, T , the government debt, B(st ), evolves as q(st )B(st ) = B(st−1 ), 44
¯ With complete asset markets, given the initial debt B. T X B(st−1 ) T t P (s ) q(s |s ) = T (sT ), t) P (st ) P (s T t s |s
for all st and t, where q(sT |st ) are defined as q(sT |tt ) ≡ q(st+1 |st ) × · · · × q(sT |sT −1 ). Given R and q(st ), the Arrow prices, q(st+1 |st ), satisfy two no-arbitrage conditions: X q(st+1 |st )P (st , st+1 ) = R−1 , P (st ) st+1 ∈S X q(st+1 |st ) = q(st ).
(76) (77)
st+1 ∈S
In each state, two Arrow securities corresponding to U and D in the next period are traded. Let h(st ) denote the marginal buyer of the Arrow securities traded at st . That is, she is indifferent between Arrow securities U and D there. The expected real rate of return for individual h of Arrow security U is h
1/P (st , U ) ; q(U |st )/P (st )
and that of Arrow security D is (1 − h)
1/P (st , D) . q(D|st )/P (st )
For the marginal buyer h(st ), these two rates are equal, which implies that h(st ) q(U |st )P (st , U )/P (st ) = 1 − h(st ) q(D|st )P (st , D)/P (st )
(78)
Using no-arbitrage condition (77), it follows that q(U |st ) = q(st )h(st ), q(D|st ) = q(st ) 1 − h(st ) .
(79) (80)
As before, the last period prices, P (sT ), are determined by the condition: B(sT −1 ) = T (sT ). P (sT )
(81)
Then, starting from the last period price levels P (sT ), the equilibrium price levels P (st ) are determined recursively by using (76): P (st ) = R
X st+1 ∈S
q(st+1 |st )P (st , st+1 ).
45
(82)
Thus, given the marginal buyers {h(st )}, equations (79), (80), (81), and (82) determine the equilibrium prices, {P (st )} and {q(st+1 |st )}.
The next lemma describes how the marginal buyers evolve along the path leading to the
crisis: st = Dt , t = 0, 1, . . . , T (D0 is interpreted as the initial state 0). Lemma 7. Consider the Arrow-Debreu market structure. Then, the marginal buyer in the initial state is h0 = 12 , independent of time horizon T . Along the path that results in a crisis, the identity of the marginal buyer evolves as: h(Dt ) =
1 , t+2
t = 1, 2, . . . , T.
(83)
Proof. Consider the present-value budget constraint for each individual h at date-event st : T X X j=t sj |st
q(sj |st )
P (sj ) h j c (s ) ≤ wh (st ), P (st )
where wh (st ) is the lifetime wealth of individual h at the beginning of st : wh (st ) ≡
T X bh (st |st−1 ) bh (st−1 ) h t−1 T t P (s ) T + + Rk (s ) + q(s |s ) y − T (s ) . T P (st ) P (st ) P (st ) T t s |s
Notice that X sT |st
X sT |st
q(sT |st )
q(sT |st )
P (sT ) yT = R−(T −t) yT , P (st )
P (sT ) B(st−1 ) T (sT ) = , t P (s ) P (st )
where the first equation follows from (76) and the second one from the government budget R1 constraint. In equilibrium, we have ch (st ) = 0 for all st and t = 0, ..., T −1, 0 bh (st |st−1 ) dh = R1 R1 0, 0 bh (st−1 ) dh = B(st−1 ), 0 Rk h (st−1 ) = Rt y0 . It follows that 1
Z 0
wh (st ) dh = Rt y0 + R−(T −t) yT ≡ Wt ,
(84)
where Wt denotes the aggregate wealth, which depends on t but not st . Using the lifetime wealth wh (st ), we can rewrite the flow budget constraint for each individual as ch (st ) +
X q(st+1 |st )P (st , st+1 ) wh (st , st+1 ) ≤ wh (st ). t) P (s s t+1
46
In equilibrium, ch (st ) = 0 and the budget constraint is satisfied with equality so that X q(st+1 |st )P (st , st+1 ) wh (st , st+1 ) = wh (st ). t) P (s s t+1
Let h(st ) be the marginal buyer at st . Then the portfolio of each individual at st is expressed as ( h
t
w (s , U ) = ( wh (st , D) =
P (st ) h t q(U |st )P (st ,U ) w (s ),
for h > h(st ),
0,
for h < h(st ), for h > h(st ),
0, P (st )
q(D|st )P (st ,D) w
h (st ),
for h < h(st ),
(85)
(86)
We first show h0 = 21 . Notice that all individuals start with the same lifetime wealth in the initial state: w0h = w ¯0 ≡ y0 + R−T yT ,
∀h ∈ [0, 1].
Given h0 , the portfolio chosen by each individual is given by ( 0, for h > h0 , h w (D) = w(D), ¯ for h < h0 , ( w(U ¯ ), for h > h0 , wh (U ) = 0, for h < h0 , where q(D|0)P (D) −1 w(D) ¯ = w ¯0 , P0 q(U |0)P (U ) −1 w ¯0 w(U ¯ )= P0
As shown in (84), the aggregate wealth depends only on t but not on st so that W1 = h0 w(D) ¯ = (1 − h0 )w(U ¯ ). That is, 1 − h0 q(U |0)P (U )/P0 = . h0 q(D|0)P (D)/P0
(87)
On the other hand, (78) implies that h0 must satisfy h0 q(U |0)P (U )/P0 = . 1 − h0 q(D|0)P (D)/P0 47
(88)
Equations (87)-(88) imply that h0 = 12 . Now consider the path leading to the crisis st = Dt , t = 0, 1, . . . , T . Given h(Dt−1 ), the marginal buyer h(Dt ) is determined as follows. Starting from w ¯0 , define w(D ¯ t , U ) and w(D ¯ t+1 ) recursively as −1 q(U |Dt )P (Dt , U ) w(D ¯ t ), w(D ¯ , U) = P (Dt ) −1 q(D|Dt )P (Dt , D) t w(D ¯ t ). w(D ¯ , D) = P (Dt ) t
Then, at st , individuals choose portfolios so that ( 0, for h > h(Dt ), h t w (D , D) = w(D ¯ t , D), for h < h(Dt ), ( 0, for h ∈ / [h(Dt ), h(Dt−1 )], h t w (D , U ) = w(D ¯ t , U ), for h ∈ [h(Dt ), h(Dt−1 )]. It follows that h(Dt )w(D ¯ t , D) = h(Dt−1 ) − h(Dt ) w(D ¯ t , U ) = Wt+1 . Therefore, h(Dt−1 ) − h(Dt ) q(U |Dt )P (Dt , U )/P (Dt ) = . h(Dt ) q(D|Dt )P (Dt , D)/P (Dt ) Combined with equation (78), this equation implies that h(Dt ) h(Dt−1 ) − h(Dt ) = , 1 − h(Dt ) h(Dt ) which can be rewritten as h(Dt ) =
h(Dt−1 ) . 1 + h(Dt−1 )
Given the initial condition h0 = 12 , the solution for this difference equation is h(Dt ) =
48
1 . t+2
(89)
A.2
Proof of Lemma 3
When χ = 0 and ΠR = 1, equations (40)-(41) become α(U ) − α(D) ¯ h0 , q0 − α(D) α(U ) − α(D) (1 − h0 ). 1= α(U ) − q0 1=
They imply that ¯0 = h = h 0 ¯ 0 = h ≡ h0 . Clearly, h0 = Thus, write as h 0
1 2
q0 − α(D) . α(U ) − α(D) ¯ = 0, and h0 < when B
1 2
¯ > 0. when B
It is convenient to express the bond price, q0 , as a function of h0 : q0 (h0 ) = α(U )h0 + (1 − h0 )α(D).
¯ = 0. Since h0 ≤ 21 , q0 ≤ q0AD and q0 = q0AD when B
Then equation (45) is rewritten as ¯ B 1 − h0 h0 1 q0 ¯ = − y0 + B . P0 h0 1 − h0 α(U ) − α(D) P0
That is, 1 = f (h0 )
1 ¯ + q0 (h0 ) , x(B) α(U ) − α(D)
where h0 1 − h0 − ≥ 0, h0 1 − h0 ¯ ≡ P0 y0 > 0. x(B) ¯ B
f (h0 ) ≡
Note that 1 1 − < 0, 2 h0 (1 − h0 )2 ¯ = − P0 y0 < 0, x0 (B) ¯2 B q00 (h0 ) = α(U ) − α(D) > 0. f 0 (h0 ) = −
Differentiating (90), we obtain n o−1 d h0 0 ¯ ¯ = − f (h ) x( B) + q (h ) + f (h ) α(U ) − α(D) f (x0 )x0 (B). 0 0 0 0 ¯ dB 49
(90)
¯ < 0, in order to prove Since f (h0 ) > 0 and x0 (B)
d h0 ¯ dB
< 0, it suffices to show that
¯ + q0 (h0 ) + f (h0 ) α(U ) − α(D) < 0. f 0 (h0 ) x(B) To see this, ¯ + q0 (h0 ) + f (h0 ) α(U ) − α(D) f 0 (h0 ) x(B) f (h0 ) < f 0 (h0 )q0 + h0 α(U ) − α(D) h0 f (h 0) < f 0 (h0 )q0 + q0 h0 1 1 1 = − − q0 − (1 − h0 )2 h0 1 − h0 < 0. ¯ < 0; the second inequality Here, the first inequality follows from the fact that f 0 (h0 )x(B) from the fact that q0 > h0 α(U ) − α(D) .
A.3
Proof of Lemma 4
¯ = 0, equations (40), (41), and (45) become: When πR = 1 and B α(U ) − α(D) ¯ h0 , q0 − α(D) α(U ) − α(D) 1= (1 − h0 ), α(U ) − (1 − χ)q0 ¯0 1−h h0 0= − . q0 − α(D) α(U ) − (1 − χ)q0 1=
(91) (92) (93)
Using (91)-(92), equation (93)can be rewritten as 0=
¯0 1−h h0 − , ¯0 1 − h0 h
that is, ¯ 0. h0 = 1 − h Rewrite equations (91)-(92) as ¯ 0, q0 − α(D) = α(U ) − α(D) h α(U ) − (1 − χ)q0 = α(U ) − α(D) (1 − h0 ).
50
(94)
Using (94), these equations can be solved for q0 as q0 =
α(U ) − α(D) . 2−χ
Thus, q0 ≥ q0AD with equality when χ = 0, and ¯ 0 is determined as Given q0 , h ¯0 = h and thus
¯0 dh d dχ
> 0.
q0 − α(D) , α(U ) − α(D)
(96)
> 0.
Finally, h0 is given by (94), and therefore,
A.4
d q0 dχ
(95)
d h0 dχ
< 0.
Proof of Lemma 5
¯ = χ = 0, equations (40), (41), and (45) become: When B α(U ) − α(D) ¯ h0 , RΠq0 − α(D) α(U ) − α(D) 1= (1 − h0 ), RΠα(U ) − RΠq0 ¯ 0 )RΠ h0 RΠ (1 − h − . 0= q0 − α(D) RΠα(U ) − RΠq0 1=
(97) (98) (99)
Using (97)-(98), equation (99) again implies that 0=
¯0 1−h h0 − , ¯ 1 − h0 h0
that is, ¯ 0. h0 = 1 − h Rewrite equations (97)-(98) as o 1 n ¯ 0 + α(D)(1 − h ¯ 0) , q0 = α(U )h RΠ o 1 n q0 = α(U )(RΠ − 1 + h0 + α(D)(1 − h0 ) . RΠ
(100)
(101)
It follows that ¯ 0 + α(D)(1 − h ¯ 0 ) = α(U )(RΠ − 1) + α(U )h + α(D)(1 − h ) α(U )h 0 0 ¯ 0 as Using (100), this equation is solved for h α(U ) ¯0 = 1 + h (RΠ − 1). 2 2[α(U ) − α(D)] 51
(102)
Substituting (102) into (101), we obtain 1 1 α(U ) q0 = [α(U ) + α(D)] + (RΠ − 1) , RΠ 2 2
(103)
and thus q0 − q0AD =
RΠ − 1 α(U ) . RΠ 2
(104)
It follows from (103)-(104) that q0 ≥ q0AD with equality when RΠ = 1,
d (q0 −q0AD ) dΠ
> 0. Also, (100) and (102) imply that
52
¯0 dh dΠ
> 0 and
d h0 dΠ
< 0.
d q0 dΠ
< 0, and
