Financial Innovation, Collateral and Investment. A. Fostel (UVA) J. Geanakoplos (Yale)

Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Financial Innovation, Collateral and Investment.
A. Fostel (UVA)
J. Geanakoplos (Yale)
Chicago, October 2015
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Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Outline
1
Introduction
2
Model
3
Leverage
4
CDS
5
Over Investment
6
Conclusion
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Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Financial innovation was at the center of the recent financial crisis.
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Introduction
Model
Leverage
CDS
Over Investment
Conclusion
200.00 0.0% 180.00 2.0% 160.00 4.0% 140.00 6.0% Case Shiller 120.00 8.0% 100.00 10.0% 80.00 12.0% 60.00 14.0% 40.00 Case Shiller Na9onal Home Price Index 2009 Q2 2009 Q1 2008 Q4 2008 Q3 2008 Q2 2008 Q1 2007 Q4 2007 Q3 2007 Q2 2007 Q1 2006 Q4 2006 Q3 2006 Q2 2006 Q1 2005 Q4 2005 Q3 2005 Q2 2005 Q1 2004 Q4 2004 Q3 2004 Q2 2004 Q1 2003 Q4 2003 Q3 2003 Q2 2003 Q1 2002 Q4 2002 Q3 2002 Q2 2002 Q1 2001 Q4 2001 Q3 2001 Q2 2001 Q1 2000 Q4 18.0% 2000 Q3 0.00 2000 Q2 16.0% 2000 Q1 20.00 Down payment for Mortgages-­‐Reverse Scale Leverage and Prices
Avg Down Payment for 50% Lowest Down Payment Subprime/AltA Borrowers Note: Observe that the Down Payment axis has been reversed, because lower down payment requirements are correlated with higher home prices.
For every AltA or Subprime first loan originated from Q1 2000 to Q1 2008, down payment percentage was calculated as appraised
value (or sale price if available) minus total mortgage debt, divided by appraised value. For each quarter, the down payment
percentages were ranked from highest to lowest, and the average of the bottom half of the list is shown in the diagram. This number is
an indicator of down payment required: clearly many homeowners put down more than they had to, and that is why the top half is
dropped from the average. A 13% down payment in Q1 2000 corresponds to leverage of about 7.7, and 2.7% down payment in Q2
2006 corresponds to leverage of about 37. Note Subprime/AltA Issuance Stopped in Q1 2008. Source: Geanakoplos (2010).
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Introduction
Model
Leverage
CDS
Over Investment
Conclusion
2000 0.0% 1800 2.0% 1600 4.0% Investment in Thousands 1400 6.0% 1200 8.0% 1000 10.0% 800 12.0% 600 14.0% 400 Investment 2009 Q2 2009 Q1 2008 Q4 2008 Q3 2008 Q2 2008 Q1 2007 Q4 2007 Q3 2007 Q2 2007 Q1 2006 Q4 2006 Q3 2006 Q2 2006 Q1 2005 Q4 2005 Q3 2005 Q2 2005 Q1 2004 Q4 2004 Q3 2004 Q2 2004 Q1 2003 Q4 2003 Q3 2003 Q2 2003 Q1 2002 Q4 2002 Q3 2002 Q2 2002 Q1 2001 Q4 2001 Q3 2001 Q2 2001 Q1 2000 Q4 2000 Q3 18.0% 2000 Q2 16.0% 0 2000 Q1 200 Down payment in Mortgages-­‐Reverse Scale Leverage and Investment
Avg Down Payment for 50% Lowest Down Payment Subprime/AltA Borrowers Note: Observe that the Down Payment axis has been reversed, because lower down payment requirements are correlated with higher home prices.
For every AltA or Subprime first loan originated from Q1 2000 to Q1 2008, down payment percentage was calculated as appraised
value (or sale price if available) minus total mortgage debt, divided by appraised value. For each quarter, the down payment
percentages were ranked from highest to lowest, and the average of the bottom half of the list is shown in the diagram. This number is
an indicator of down payment required: clearly many homeowners put down more than they had to, and that is why the top half is
dropped from the average. A 13% down payment in Q1 2000 corresponds to leverage of about 7.7, and 2.7% down payment in Q2
2006 corresponds to leverage of about 37. Note Subprime/AltA Issuance Stopped in Q1 2008. Source: Geanakoplos (2010).
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Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Leverage, Prices and Investment
The financial crisis was preceded by years in which leverage, prices
and investment increased dramatically.
Then all collapsed after the crisis. Leverage Cycle.
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0 CDS Jun-­‐10 Feb-­‐10 Oct-­‐09 Jun-­‐09 Feb-­‐09 Oct-­‐08 Jun-­‐08 Feb-­‐08 Over Investment
40 35 25 50000 20 40000 15 30000 5 CDS No'onal Amount in Billions U$S CDS
Oct-­‐07 Jun-­‐07 Feb-­‐07 Oct-­‐06 Jun-­‐06 Feb-­‐06 Oct-­‐05 Leverage
Jun-­‐05 Feb-­‐05 Oct-­‐04 Jun-­‐04 Feb-­‐04 Oct-­‐03 Jun-­‐03 Feb-­‐03 Leverage Model
Oct-­‐02 Jun-­‐02 Feb-­‐02 Oct-­‐01 Jun-­‐01 Feb-­‐01 Oct-­‐00 Jun-­‐00 Introduction
Conclusion
Two Financial Innovations: Credit Default Swaps and
Leverage
70000 60000 30 10 20000 10000 0 Source CDS: IBS OTC Derivatives Market Statistics
Avg Leverage for 50% Lowest Down Payment Subprime/AltA Borrowers 7 / 103
Jun-­‐00 0.00 CDS Jun-­‐10 Feb-­‐10 Oct-­‐09 Jun-­‐09 Feb-­‐09 Oct-­‐08 Jun-­‐08 Feb-­‐08 Oct-­‐07 Over Investment
200.00 140.00 50000 120.00 100.00 40000 80.00 30000 60.00 20000 CDS No'onal Amound in Billions U$S CDS
Jun-­‐07 Feb-­‐07 Oct-­‐06 Jun-­‐06 Feb-­‐06 Oct-­‐05 Leverage
Jun-­‐05 Feb-­‐05 Oct-­‐04 Jun-­‐04 Feb-­‐04 Oct-­‐03 Jun-­‐03 Feb-­‐03 Model
Oct-­‐02 Jun-­‐02 Feb-­‐02 Oct-­‐01 Jun-­‐01 Feb-­‐01 Oct-­‐00 Case-­‐Shiller Introduction
Conclusion
Credit Default Swaps, Prices and Investment
180.00 70000 160.00 60000 40.00 20.00 10000 0 Source CDS: IBS OTC Derivatives Market Statistics
Case Shiller NaAonal Home Price Index 8 / 103
Jun-­‐00 0 CDS Oct-­‐07 Jun-­‐10 Feb-­‐10 Oct-­‐09 Jun-­‐09 Feb-­‐09 Oct-­‐08 Jun-­‐08 Feb-­‐08 Over Investment
2000 1400 1200 1000 40000 800 30000 600 20000 CDS No'onal Amount in Billions of U$S CDS
Jun-­‐07 Feb-­‐07 Oct-­‐06 Jun-­‐06 Feb-­‐06 Oct-­‐05 Leverage
Jun-­‐05 Feb-­‐05 Oct-­‐04 Jun-­‐04 Feb-­‐04 Oct-­‐03 Jun-­‐03 Feb-­‐03 Model
Oct-­‐02 Jun-­‐02 Feb-­‐02 Oct-­‐01 Jun-­‐01 Feb-­‐01 Oct-­‐00 Investment in thousands Introduction
Conclusion
Credit Default Swaps, Prices and Investment
1800 70000 1600 60000 50000 400 200 10000 0 Source CDS: IBS OTC Derivatives Market Statistics.
Source Investment: Construction new privately owned housing units completed.
Department of Commerce.
Investment 8 / 103
Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Credit Default Swaps, Prices and Investment
Credit Default Swaps (CDS) was a financial innovation that was
introduced much later than leverage.
Peak in CDS coincides with lower prices and investment.
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Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Financial Innovation, Collateral, Prices and Investment
We show that financial innovations that change either:
-the set of assets that can be used as collateral
-or the types of promises that can be backed with the same
collateral
affect prices and investment.
We provide precise predictions.
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Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Results
I) The ability to leverage an asset generates higher prices and
over-investment compared to the Arrow-Debreu level.
II) The introduction of CDS generates lower prices and
under-investment with respect to the Arrow-Debreu level. It can
even destroy competitive equilibrium.
III)The ability to leverage an asset never generates marginal
under-investment in collateral general equilibrium models.
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Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Literature
-To collateral in a GE framework we follow the techniques
developed by Geanakoplos (1997, 2003,2010), Fostel-Geanakoplos
(2008, 2011, 2012, 2012, 2013)
-Related to a literature on Leverage as in Araujo et al (2012),
Acharya and Viswanathan (2011), Adrian and Shin (2010),
Adrian-Boyarchenko (2012), Brunnermeier and Pedersen (2009,
Brunnermeier and Sannikov (2011), Gromb and Vayanos (2002),
Simsek (2013).
-Financial innovations and asset pricing as in Fostel-Geanakoplos
(2012b) and Che and Sethi (2011).
-Literature on existence: Polemarchakis and Ku (1990), Duffie and
Shaffer (86), Geanakoplos and Zame (1997).
-Macro /corportate finance literature: Bernanke and Gertler
(1989), Kiyotaki and Moore (1997), Holmstrom and Tirole (1997).
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Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Outline
1
Introduction
2
Model
3
Leverage
4
CDS
5
Over Investment
6
Conclusion
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Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Model Outline
Set-up.
Arrow Debreu Equilibrium.
Financial Innovation and Collateral.
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Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Set Up
We present a simple GE model with incomplete markets, collateral
and production, that we call the C-Model (C*-Model.)
In the paper we present a completely general GE model with
collateral.
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Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Time and Assets
Y s=U Time t = 0, 1.
X dYU=1 dXU=1 Two states of nature s = U, D at
time 1.
Two assets: risky, Y , and riskless, X .
Dividends in consumption good.
X can be thought as a durable
consumption good or cash,
numeraire. Price of Y at t = 0 is p.
s=0 ddYD < dYU dXD=1 s=D D
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Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Time and Assets
Y s=U Time t = 0, 1.
X dYU=1 dXU=1 Two states of nature s = U, D at
time 1.
Two assets: risky, Y , and riskless, X .
Dividends in consumption good.
X can be thought as a durable
consumption good or cash,
numeraire. Price of Y at t = 0 is p.
s=0 ddYD < dYU dXD=1 s=D D
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Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Time and Assets
Y s=U Time t = 0, 1.
X dYU=1 dXU=1 Two states of nature s = U, D at
time 1.
Two assets: risky, Y , and riskless, X .
Dividends in consumption good.
X can be thought as a durable
consumption good or cash,
numeraire. Price of Y at t = 0 is p.
s=0 ddYD < dYU dXD=1 s=D D
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Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Production
Agents have access to an intra-period production technology at
t = 0 that allows them to invest the riskless asset X and produce
the risky asset Y .
Z0h ⊂ R2 is the set of feasible intra-period production for agent
h ∈ H in state 0 (Z0h is convex and compact, (0, 0) ∈ Z0h and
Z0h = Z0 , ∀h.)
Inputs appear as negative components, zx < 0 of z ∈ Z h , and
outputs as positive components, zy > 0 of z ∈ Z0h .
Investment: zx .
Denote by Π = zx + pzy the profits from production plan (zx , zy ).
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Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Investors
Continuum of investors h ∈ H = [0, 1].
Risk neutral. No discounting. Consumption only at the end.
Expected utility to agent h is
h
h
U h (cU , cD ) = γU
cU + γD
cD
Each agent h ∈ H has an endowment x0∗ of asset X at time 0.
h .
The only source of heterogeneity is in subjective probabilities, γU
h
The higher the h, the more optimistic the investor (γU are
increasing and continuous).
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Introduction
Model
Leverage
CDS
Over Investment
Conclusion
C and C*-Models
C-Models are very tractable models. In particular, we can represent
the equilibrium in an Edgeworth Box even though we have a
continuum of agents.
We define a C*-Model as a C-model where:
-the space of agents H can be finite or a continuum.
h u h (c ) + γh u h (c ) can allow for
-the agents’ preferences U h = γU
U
D
D
risk aversion.
-initial endowments of X at time 0 x0h∗ can be arbitrary.
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Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Model Outline
Set-up.
Arrow Debreu Equilibrium.
Financial Innovation and Collateral.
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Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Arrow-Debreu Benchmark
Before focusing on financial innovation, let us consider the
Arrow-Debreu economy with production, without any type of
collateral considerations.
This will be an important benchmark throughout the paper.
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Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Arrow-Debreu Equilibrium
Since Z0h = Z0 , ∀h, then Πh = Π. Because of convexity, wlog we
may assume that production plans are the same across agents.
Then (zx , zy ) is also the aggregate production.
Arrow Debreu equilibrium is easy to solve.
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Introduction
Model
Leverage
CDS
Over Investment
Conclusion
The Arrow Debreu Equilibrium
h=1 Op(mists: buy Arrow U h1
Marginal buyer Pessimists: buy Arrow D h=0 23 / 103
Introduction
Model
Leverage
CDS
Over Investment
Conclusion
The Arrow Debreu Equilibrium: Edgeworth Box
cU
Y(dYU,dYD) Intra-­‐Period Produc:on Possibility Fron:er x0*(1,1) 45o
O cD
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Introduction
Model
Leverage
CDS
Over Investment
Conclusion
The Arrow Debreu Equilibrium: Edgeworth Box
cU
Y(dYU,dYD) Intra-­‐Period Produc;on Possibility Fron;er Q Economy Total Final Output x0*(1,1) 45o
O cD
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Introduction
Model
Leverage
CDS
Over Investment
Conclusion
The Arrow Debreu Equilibrium: Edgeworth Box
cU
Y(dYU,dYD) Intra-­‐Period Produc>on Possibility Fron>er Q Economy Total Final Output x0*(1,1) x0*+zX zY(dYU,dYD) 45o
O cD
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Introduction
Model
Leverage
CDS
Over Investment
Conclusion
The Arrow Debreu Equilibrium: Edgeworth Box
cU
Y(dYU,dYD) Q x0*(1,1) (1-­‐h1)Q 45o
O cD
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Introduction
Model
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CDS
Over Investment
Conclusion
The Arrow Debreu Equilibrium: Edgeworth Box
cU
Y(dYU,dYD) Slope –qh1D /qh1U Q x0*(1,1) (1-­‐h1)Q Price line equal to Indifference curve of h1 45o
O cD
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Introduction
Model
Leverage
CDS
Over Investment
Conclusion
The Arrow Debreu Equilibrium: Edgeworth Box
cU
Y(dYU,dYD) Slope –qh1D /qh1U Q x0*(1,1) (1-­‐h1)Q Price line equal to Indifference curve of h1 45o
O cD
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Introduction
Model
Leverage
CDS
Over Investment
Conclusion
The Arrow Debreu Equilibrium: Edgeworth Box
cU
Y(dYU,dYD) Slope –qh1D /qh1U Q x0*(1,1) (1-­‐h1)Q Price line equal to Indifference curve of h1 45o
O cD
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Introduction
Model
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CDS
Over Investment
Conclusion
The Arrow Debreu Equilibrium: Edgeworth Box
cU
Y(dYU,dYD) Produc>on Possibility Fron>er Slope –qh1D /qh1U Q x0*(1,1) (1-­‐h1)Q Price line equal to Indifference curve of h1 45o
O cD
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Introduction
Model
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CDS
Over Investment
Conclusion
The Arrow Debreu Equilibrium: Edgeworth Box
cU
Y(dYU,dYD) Produc>on Possibility Fron>er Slope –qh1D /qh1U C Q zydYU+x0*+zx x0*(1,1) (1-­‐h1)Q Price line equal to Indifference curve of h1 45o
O cD
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Introduction
Model
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CDS
Over Investment
Conclusion
Arrow Debreu Equilibrium Summary
Optimists consume only in the U state: (zy dUY + x0∗ + zx , 0)
Pessimists consume only in the D state: (0, zy dDY + x0∗ + zx )
The marginal buyer determines state prices.
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Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Model Outline
Set-up.
Arrow Debreu Equilibrium.
Financial Innovation and Collateral.
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Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Financial Contracts and Collateral
The heart of our analysis involves contracts and collateral.
In Arrow Debreu the question of why agents honor their promises
is ignored.
We explicitely incorporate in our model repayment enforceability
problems.
Collateral is the only enforcement mechanism: agents cannot be
coerced into honoring their promises except by seizing collateral
aggreed upon by contract in advance.
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Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Financial Contracts and Collateral
A financial contract j is an ordered pair
j = ((jU , jD ), cj )
Promise: j = (jU , jD ) denotes the promise in units of consumption
good in each final state.
Collateral: cj ∈ {X , Y } asset used as collateral.
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Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Financial Contracts and Collateral
We shall suppose every contract is collateralized either by one unit
of X or by one unit of Y .
Let J = J X ∪ J Y be the total set of contracts.
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Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Financial Contract Delivery
Actual delivery of contract j in each state is (no-recourse):
c
c
(min(jU , dUj ), min(jD , dDj ))
We are explicitely assuming repayment enforceability problems.
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Introduction
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CDS
Over Investment
Conclusion
No Cash Flow Problems
But crucially, we are assuming away cash flow problems:
c
c
The value of the collateral in the future, (dUj , dDj ):
-does not depend on the size of the promise
-or on who owns the asset at the end.
And every agent knows exactly how the future cash flow depends
on the exogenous state of nature.
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Introduction
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CDS
Over Investment
Conclusion
No Cash Flow Problems
This eliminates any issues associated with managerial hidden effort
or unobserved firm quality.
Promises will not be artificially limited. Agents can potentially
promise all their future cash flows coming from assets (or firm).
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Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Financial Contracts and Borrowing
Price of contract j ∈ J is πj .
An investor can borrow πj today by selling the contract j in
exchange for a promise tomorrow.
Let ϕj > 0 (< 0) be the number of contracts j sold (bought) at
time 0.
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Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Budget Set
B h (p, π ) =
{(x , y , zx , zy , ϕ, cU , cD ) ∈ R+2 × R− × R+ × R J × R+2 :
(x − zx − x0∗ ) + p (y − zy ) ≤
∑ ϕj πj
j ∈J
∑
max (0, ϕj ) ≤ x ,
j ∈J X
∑
max (0, ϕj ) ≤ y
j ∈J Y
z = (zx , zy ) ∈ Z0
cs = dsX x + dsY y −
∑
j ∈J X
ϕj min(js , dsX ) −
∑
ϕj min(js , dsY ), s = U, D }
j ∈J Y
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Introduction
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CDS
Over Investment
Conclusion
Collateral Equilibrium
((p, π ), (x h , y h , z h , ϕh , cUh , cDh )h∈H ) such that
R1
0
R1
0
R1
0
x h dh =
R1
y h dh =
R1
0
0
(x0h∗ + zxh )dh
zyh dh
ϕhj dh = 0, ∀j ∈ J
(x h , y h , z h , ϕh , cUh , cDh ) ∈ B h (p, π ), ∀h
(x , y , z, ϕ, cU , cD ) ∈ B h (p, π ) ⇒ U h (cU , cD ) ≤ U h (cUh , cDh ), ∀h.
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Introduction
Model
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CDS
Over Investment
Conclusion
Financial Innovation and Collateral
We regard the use of new kinds of collateral, or new kinds of
promises that can be backed by collateral, as financial innovation.
Hence, financial innovation in our model is a different set J.
We will show how different financial innovations, such as leverage,
and CDS can be cast within our model with collateral.
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Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Outline
1
Introduction
2
Model
3
Leverage
4
CDS
5
Over Investment
6
Conclusion
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Introduction
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Leverage-Economy
In this case J = J Y , and each j = (j, j ) for all j ∈ J = J Y .
Traded instruments:
-risky asset Y and cash X
-non-contingent promises j (debt contracts or loans) using the
asset Y as collateral.
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Introduction
Model
Leverage
CDS
Over Investment
Conclusion
What does it mean to leverage Y?
U
Asset Y
Payoff
Family of debt contracts
dYU Residual
Debt contract promise j< j*
45o
dYD D
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Introduction
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Over Investment
Conclusion
What does it mean to leverage Y?
U
Asset Y
Payoff
Family of debt contracts
dYU Residual
dYU-j
Arrow U
Debt contract j>j*=dYD
45o
dYD D
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Introduction
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CDS
Over Investment
Conclusion
What does it mean to leverage Y?
U
Asset Y
Payoff
Family of debt contracts
d
Residual
dYU-dYD
Arrow U
Max min bond j=j*=dYD
45o
dYD
D
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Introduction
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CDS
Over Investment
Conclusion
L-Economy: Endogenous Leverage
But which contract is actively traded in equilibrium?
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Introduction
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CDS
Over Investment
Conclusion
L-Economy: Endogenous Leverage
The only contract traded in equilibrium is j ∗ = dDY and the
risk-less interest rate is equal to zero, so πj ∗ = j ∗ = dDY .
Geanakoplos (2003) and Fostel-Geanakoplos (JET, 2011).
Fostel-Geanakoplos (ECMA, forth) provide a complete
characterization showing that in all binomial economies with
financial assets we can always assume that the max min contract is
the only contract traded.
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Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Credit Surface
G (97) introduced the concept of
Credit Surface: the equilibrium
relashionship between LTVj and
1 + rj .
Borrowers can choose any contract
on the Credit Surface provided they
put up the corresponding required
collateral.
In the Arrow-Debreu budget set,
borrowers face in equilibrium a flat
Credit surface.
1+rj
B
A
1+r
LTVj*
100%
LTV
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Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Credit Surface
G (97) introduced the concept of
Credit Surface: the equilibrium
relashionship between LTVj and
1 + rj .
Borrowers can choose any contract
on the Credit Surface provided they
put up the corresponding required
collateral.
In the Arrow-Debreu budget set,
borrowers face in equilibrium a flat
Credit surface.
1+rj
B
A
1+r
LTVj*
100%
LTV
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Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Credit Surface
G (97) introduced the concept of
Credit Surface: the equilibrium
relashionship between LTVj and
1 + rj .
Borrowers can choose any contract
on the Credit Surface provided they
put up the corresponding required
collateral.
In the Arrow-Debreu budget set,
borrowers face in equilibrium a flat
Credit surface.
1+rj
B
A
1+r
LTVj*
100%
LTV
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Introduction
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CDS
Over Investment
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L-Economy: Equilibrium
h=1 Op(mists leverage Y using max min bond. They buy Arrow U. h1
Marginal buyer Pessimists lenders buy max min bond h=0 43 / 103
Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Numerical Example
We solve for equilibrium the Arrow Debreu and Leverage
economies just described for the following:
Production: Z0 = {z = (zx , zy ) ∈ R− × R+ : zy = −kzx }, k ≥ 0
h = 1 − (1 − h )2
Beliefs: γU
Parameter values: x0∗ = 1, dUY = 1, dDY = .2.
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Introduction
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CDS
Over Investment
Conclusion
Numerical Example
Equilibrium for k = 1.5.
Arrow Debreu Economy
L-economy
qY
0.6667
p
0.6667
qU
0.5833
h1
0.3545
qD
0.4167
−zx
0.92
h1
0.3545
zy
1.38
− zx
0.2131
zy
0.3197
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Numerical Example: Investment
Investment in Y: -­‐zx 1.2 1 0.8 Invesment L-­‐economy 0.6 Investment AD 0.4 0.2 0 1 1.1 1.2 1.3 1.4 1.45 1.5 1.55 1.6 1.65 1.7 k 46 / 103
Introduction
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Over Investment
Conclusion
Numerical Example: Welfare
Welfare 3 2.5 2 L economy 1.5 AD economy 1 0.5 0 h=0 h^LT_2=.348 h^AD=h^L=.3545 h^LT_1=.388 h=1 h 47 / 103
Introduction
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CDS
Over Investment
Conclusion
Theoretical Results: Over Valuation and Investment
Proposition: Over-Valuation and Investment compared to
Arrow Debreu in C-Models.
In C-Models p L ≥ p A , and zyL ≥ zyA .
48 / 103
Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Theoretical Results: Over Valuation and Investment
Proposition: Over-Valuation and Investment compared to
Arrow Debreu in C*-Models.
In C*-Models, p L ≥ p A , and zyL ≥ zyA .
49 / 103
Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Theoretical Results: Welfare
Proposition: Welfare in C*-Models
In C*-Models under constant return to scale, Arrow Debreu
equilibrium Pareto-dominates Leverage equilibrium.
50 / 103
Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Theoretical Results: Intuition
Y can be used as collateral to issue debt. Cash flows from Y can
be split into an Arrow U and a riskless part. X cannot be used as
collateral.
This gives Y an additional collateral value compared to the riskless
asset.
This gives agents more incentive to produce Y .
Agents are worse off over-investing.
51 / 103
Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Theoretical Results: Intuition
Marginal Utility of Money for h = .9 in equilibrium at time 0:
µh=.9 =
γU (.9)(dUY − dDY )
.99(1 − .2)
=
= 1.70.
∗
p − πj
.67 − .2
Payoff value of Y for h = .9 in equilibrium:
PVYh=.9 =
.99(1) + .01(.2)
= .58 < p
µh=.9
Hence the Collateral Value of Y for h = .9 in equilibrium:
CVYh=.9 = p − PVYh=.9 = .67 − .58 = .09.
52 / 103
Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Theoretical Results: Intuition
The utility from holding Y for its dividends alone is less than the
utility that could be derived from p dollars; the difference is the
utility derived from holding Y as collateral, measured in dollar
equivalents.
X cannot be used as collateral, so PVX = 1 and hence CVX = 0.
Agents have more incentive to produce goods that are better
collateral as measured by their collateral values. Investment
migrates to better collateral
53 / 103
Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Outline
1
Introduction
2
Model
3
Leverage
4
CDS
5
Over Investment
6
Conclusion
54 / 103
Introduction
Model
Leverage
CDS
Over Investment
Conclusion
What is a CDS?
Y Payoff
dYU 0
U CDS Payoff
dyD dYU -­‐ dYD 1 D 55 / 103
Introduction
Model
Leverage
CDS
Over Investment
Conclusion
CDS and Collateral
A seller of a CDS must post collateral typically in the form of
money that is worth dUY − dDY when Y pays only dDY in the down
state.
We can therefore incorporate CDS into our economy by taking J X
to consist of one contract called c promising c = (0, 1).
56 / 103
Introduction
Model
Leverage
CDS
Over Investment
Conclusion
The CDS-Economy
In this case J = J X J Y where:
-J X consists of the single contract called c promising c = (0, 1)
S
-J Y consists of contracts j = (j, j ) as described in the leverage
economy.
Agents can leverage Y and also can tranche X into Arrow
securities.
57 / 103
Introduction
Model
Leverage
CDS
Over Investment
Conclusion
What does it mean to tranche X?
U
Selling a CDS on Y
collateralized by X is like
selling an Arrow D promise:
Asset X Payoff
1
Residual
Arrow U
Sellers of promise c = (0, 1)
get the residual which is like
the Arrow U which pays 1.
45o
We call it Tranche X because
X is perfectly split into Arrow
securities.
1
D
Sell Promise Arrow D
58 / 103
Introduction
Model
Leverage
CDS
Over Investment
Conclusion
What does it mean to tranche X?
U
Selling a CDS on Y
collateralized by X is like
selling an Arrow D promise:
Asset X Payoff
1
Residual
Arrow U
Sellers of promise c = (0, 1)
get the residual which is like
the Arrow U which pays 1.
45o
We call it Tranche X because
X is perfectly split into Arrow
securities.
1
D
Sell Promise Arrow D
58 / 103
Introduction
Model
Leverage
CDS
Over Investment
Conclusion
What does it mean to tranche X?
U
Selling a CDS on Y
collateralized by X is like
selling an Arrow D promise:
Asset X Payoff
1
Residual
Arrow U
Sellers of promise c = (0, 1)
get the residual which is like
the Arrow U which pays 1.
45o
We call it Tranche X because
X is perfectly split into Arrow
securities.
1
D
Sell Promise Arrow D
58 / 103
Introduction
Model
Leverage
CDS
Over Investment
Conclusion
The CDS-Economy
Traded instruments:
-risky asset Y and cash X .
-non-contingent promises (debt contracts) using the asset Y as
collateral.
-contingent promises (CDS) using the asset X as collateral.
The equilibrium regime is as follows:
59 / 103
Introduction
Model
Leverage
CDS
Over Investment
Conclusion
CDS-Economy: Equilibrium
h=1 Op(mists: buy all remaining X and Y. Issue bond and CDS (holding the Arrow U) Marginal buyer h1
Moderates: hold the bond h2
Marginal buyer Pessimists: buy the CDS 60 / 103
Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Numerical Example
We solve for equilibrium in the Arrow Debreu, Leverage and CDS
economies just described for the following:
Production: Z0 = {z = (zx , zy ) ∈ R− × R+ : zy = −kzx }, k ≥ 0
h = 1 − (1 − h )2
Beliefs: γU
Parameter values: x0∗ = 1, dUY = 1, dDY = .2.
61 / 103
Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Numerical Example
Equilibrium for k = 1.5.
Arrow Debreu Economy
L-economy
CDS-economy
qY
0.6667
p
0.6667
p
0.6667
qU
0.5833
h1
0.3545 πj ∗
0.1904
qD
0.4167
−zx
0.92
πC
0.4046
h1
0.3545
zy
1.38
h1
0.3880
− zx
0.2131
h2
0.3480
zy
0.3197
−zx
0.14
zy
0.2
62 / 103
Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Numerical Example: Investment
Investment in Y: -­‐zx 1.2 1 0.8 Invesment L-­‐economy 0.6 Investment AD Investment CDS-­‐ economy 0.4 0.2 0 1 1.1 1.2 1.3 1.4 1.45 1.5 1.55 1.6 1.65 1.7 k 63 / 103
Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Numerical Example: Welfare
Welfare 3 2.5 2 L economy 1.5 AD economy CDS economy 1 0.5 0 h=0 h^LT_2=.348 h^AD=h^L=.3545 h^LT_1=.388 h=1 h 64 / 103
Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Under Valuation and Investment
Proposition: Under-Investment compared to First Best in
C-Models.
h is
In C-Models p A ≥ p CDS , and zyA ≥ zyCDS provided that γU
concave in h.
65 / 103
Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Under Valuation and Investment
Using X as collateral to sell a CDS splits its cash flows into Arrow
securities.
Using Y as collateral splits its cash flows into Arrow U and a
riskless bond.
The collateral value of X is higher than the collateral value of Y .
This gives agents less incentive to use X to produce Y .
There is no welfare domination: moderate agents in the CDS
economy are better off than in the Arrow Debreu economy.
66 / 103
Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Theoretical Results: Intuition
Marginal Utility of Money for h = .9 in equilibrium at time 0:
µh=.9 =
γU (.9)(dUY − dDY )
.99(1 − .2)
=
= 1.66.
∗
p − πj
.67 − .1904
Payoff value of Y for h = .9 in equilibrium:
PVYh=.9 =
.99(1) + .01(.2)
= .60 < p
µh=.9
Hence the Collateral Value of Y for h = .9 in equilibrium:
CVYh=.9 = p − PVYh=.9 = .67 − .6 = .07.
67 / 103
Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Theoretical Results: Intuition
Payoff value of X for h = .9 in equilibrium:
PVXh=.9 =
.99(1) + .01(1)
= .60
µh=.9
Hence the Collateral Value of X for h = .9 in equilibrium:
CVXh=.9 = 1 − PVXh=.9 = 1 − .60 = .40.
So whereas the collateral value of Y accounts for 10.5% of its
price, the collateral value of X accounts for 40% of its price.
Agents have more incentive to produce goods that are better
collateral as measured by their collateral values. Investment
migrates to better collateral
68 / 103
Introduction
Model
Leverage
CDS
Over Investment
Conclusion
CDS and Robust Non-Existence
We saw that selling a CDS on Y using X as collateral is like selling
an Arrow D using X as collateral.
The only difference between a CDS and an Arrow D is that when
Y is not produced the CDS is no longer well-defined.
It is precisely this difference that can bring about interesting
existence problems: introducing CDS can robustly destroy
collateral equilibrium in economies with production. Non-Existence
69 / 103
Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Outline
1
Introduction
2
Model
3
Leverage
4
CDS
5
Over Investment
6
Conclusion
70 / 103
Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Outline
Geometrical Proof of the Over-Investment Result.
Discussion: Over Investment without Cash Flow Problems.
Marginal Over-Investment and Collateral Value.
71 / 103
Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Over-Investment in C-Model
First we show a geometrical argument in the case of C-Models.
72 / 103
Introduction
Model
Leverage
CDS
Over Investment
Conclusion
L-Economy: Equilibrium
h=1 Op(mists leverage Y using max min bond. They buy Arrow U. h1
Marginal buyer Pessimists lenders buy max min bond h=0 73 / 103
Introduction
Model
Leverage
CDS
Over Investment
Conclusion
L-Economy: Edgeworth Box
Y(dYU,dYD) cU
Intra-­‐Period Produc1on Possibility Fron1er Q (1-­‐h1)Q 45o
Slope –qh1D /qh1U x0*(1,1) Price line equal to indifference curve of h1 45o
O cD
74 / 103
Introduction
Model
Leverage
CDS
Over Investment
Conclusion
L-Economy: Edgeworth Box
Y(dYU,dYD) cU
Intra-­‐Period Produc1on Possibility Fron1er Q (1-­‐h1)Q 45o
Slope –qh1D /qh1U x0*(1,1) Price line equal to indifference curve of h1 45o
O cD
74 / 103
Introduction
Model
Leverage
CDS
Over Investment
Conclusion
L-Economy: Edgeworth Box
Y(dYU,dYD) cU
Produc9on Possibility Fron9er Q zYdYD Slope –qh1D /qh1U x0*+zX x0*+zX (1-­‐h1)Q 45o
zYdYD x0*(1,1) C Price line equal to indifference curve of h1 zY(dYU-­‐dYD) 45o
O cD
74 / 103
Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Over Valuation and Investment Geometrical Proof
cU
ARROW DEBREU Y(dYU,dYD) Y(dYU,dYD) cU
Slope –qh1D /qh1U C Q zYdYD zydYU+x0*+zx LEVERAGE ECONOMY Q Slope –qh1D /qh1U x0*+zX x0*+zX x0*(1,1) (1-­‐h1)Q 45o
zY
(1-­‐h1)Q dY
D x0*(1,1) C Price line equal to Indifference curve of h1 Price line equal to indifference curve of h1 zY(dYU-­‐dYD) 45o
O 45o
cD
O cD
Proof
75 / 103
Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Over Valuation and Investment Geometrical Proof
cU
ARROW DEBREU Y(dYU,dYD) Y(dYU,dYD) cU
Slope –qh1D /qh1U C Q zYdYD zydYU+x0*+zx LEVERAGE ECONOMY Q Slope –qh1D /qh1U x0*+zX x0*+zX x0*(1,1) (1-­‐h1)Q 45o
zY
(1-­‐h1)Q dY
D x0*(1,1) C Price line equal to Indifference curve of h1 Price line equal to indifference curve of h1 zY(dYU-­‐dYD) 45o
O 45o
cD
O cD
Proof
75 / 103
Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Over Valuation and Investment Geometrical Proof
cU
ARROW DEBREU Y(dYU,dYD) Y(dYU,dYD) cU
Slope –qh1D /qh1U C Q zYdYD zydYU+x0*+zx LEVERAGE ECONOMY Q Slope –qh1D /qh1U x0*+zX x0*+zX x0*(1,1) (1-­‐h1)Q 45o
zY
(1-­‐h1)Q dY
D x0*(1,1) C Price line equal to Indifference curve of h1 Price line equal to indifference curve of h1 zY(dYU-­‐dYD) 45o
O 45o
cD
O cD
Proof
75 / 103
Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Over-Investment in C*-Model
The geometrical argument in the case of C*-Models is as follows
76 / 103
Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Over-Investment in C*-Model
xU
Y(dYU,dYD) L L AD e N AD AD 45o
O x
77 / 103
Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Over-Investment in C*-Model
E(pU)
EN(pU)
EL(pLU)
pU
EAD(pU)
78 / 103
Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Outline
Geometrical Proof of the Over-Investment Result.
Discussion: Over Investment without Cash Flow Problems.
Marginal Over-Investment and Collateral Value.
79 / 103
Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Discussion: Over Investment without Cash Flow Problems
Over-valuation and over-investment due to leverage may seem
surprising.
Many macro models (like Kiyotaki-Moore (97), Bernanke-Gertler
(89), Mendoza (10)) with financial frictions get the opposite result:
lower price and investment with respect the first best allocation.
Intuitive: one would expect that the need for collateral would
restrict borrowing and hence investment.
Why do we get different results?
80 / 103
Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Discussion: Over Investment without Cash Flow Problems
The reason for the discrepancy is that in the macro-corporate
finance literature it is assumed that there are cash flow problems:
The value of the collateral depends on the size of the promise or
on who owns the asset at the end. Hence agents cannot pledge the
whole future value of the assets they produce.
This naturally imposes a limit on borrowing and hence depresses
investment.
We can clearly see this looking at the Credit Surface implied by
models with cash flow problems.
81 / 103
Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Credit Surface
All contracts j ∈ J = J Y with j = (j, j ) have a price in
equilibrium, πj . Hence:
All contracts define a gross interest rate 1 + rj = j/πj .
All contracts have a well defined LTV j =
πj
p .
82 / 103
Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Credit Surface
G (97) introduced the concept of
Credit Surface: the equilibrium
relashionship between LTVj and
1 + rj .
Borrowers can choose any contract
on the Credit Surface provided they
put up the corresponding required
collateral.
In the Arrow-Debreu budget set,
borrowers face in equilibrium a flat
Credit surface.
1+rj
B
A
1+r
LTVj*
100%
LTV
83 / 103
Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Credit Surface
G (97) introduced the concept of
Credit Surface: the equilibrium
relashionship between LTVj and
1 + rj .
Borrowers can choose any contract
on the Credit Surface provided they
put up the corresponding required
collateral.
In the Arrow-Debreu budget set,
borrowers face in equilibrium a flat
Credit surface.
1+rj
B
A
1+r
LTVj*
100%
LTV
83 / 103
Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Credit Surface
G (97) introduced the concept of
Credit Surface: the equilibrium
relashionship between LTVj and
1 + rj .
Borrowers can choose any contract
on the Credit Surface provided they
put up the corresponding required
collateral.
In the Arrow-Debreu budget set,
borrowers face in equilibrium a flat
Credit surface.
1+rj
B
A
1+r
LTVj*
100%
LTV
83 / 103
Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Discussion: Over Investment without Cash Flow Problems
1+rj
B
A
1+r
πj*
p
Borrowing πj
84 / 103
Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Discussion: Over Investment without Cash Flow Problems
1+rj
B
A
1+r
p
Borrowing πj
p is fixed at the value of the firm without external financing 84 / 103
Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Discussion: Over Investment without Cash Flow Problems
1+rj
B
A
1+r
p
Borrowing πj
p is fixed at the value of the firm without external financing 84 / 103
Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Discussion: Over Investment without Cash Flow Problems
In a family of models (C and C*) we show that when we
disentangle cash flow problems from repayment enforcement
problems we always get over valuation and over investment
compared to the Arrow Debreu level.
85 / 103
Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Outline
Geometrical Proof of the Over-Investment Result.
Discussion: Over Investment without Cash Flow Problems.
Marginal Over-Investment and Collateral Value.
86 / 103
Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Marginal Over Investment and Collateral Value
Investment and prices can be above or below Arrow Debreu levels
in GE collateral models. As we saw in C and C*-models they are
above. But in general we don’t know.
We show that in GE collateral models there is never marginal under
investment in equilibrium due to the presence of collateral value.
87 / 103
Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Marginal Over Investment
Proposition: No Marginal Under-Investment.
There is never marginal-under investment on assets that serve as
collateral in collateral general equilibrium models due to
non-negative collateral values.
88 / 103
Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Marginal Over Investment and Collateral Value
Concept of marginal over-investment is a “local” measure of
inefficiency.
Given all spot prices, no agent would prefer to invest an extra unit
of money in raising production over the equilibrium level, even if he
had access to the best technology available in the economy.
89 / 103
Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Marginal Over Investment and Collateral Value
Need to post collateral may constrain borrowers in equilibrium.
But when one considers in the same model many durable goods
than can be produced with different collateral values, investment
migrates to “good” collateral.
Hence, we expose a countervailing force in the incentives to
produce:
-when only some assets can be used as collateral, they become
relatively more valuable, and are therefore produced more. Example
90 / 103
Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Outline
1
Introduction
2
Model
3
Leverage
4
CDS
5
Over Investment
6
Conclusion
91 / 103
Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Conclusion
We show that financial innovation affect prices and investment.
Leverage can generate higher prices and over-investment compared
to the Arrow-Debreu first best level. In C and C*-models it always
does.
Leverage never generates marginal under-investment in assets that
can be used as collateral due to the presence of collateral value.
CDS can generate lower prices and under-investment with respect
to the Arrow-Debreu first best level. In C-Models always does.
And their introduction can even destroy equilibrium.
92 / 103
Introduction
Model
Leverage
CDS
Over Investment
Conclusion
CDS and Robust Non-Existence
The only difference between CDS and Arrow D is that when Y
ceases to be produced the CDS is no longer well-defined.
We show how introducing CDS can robustly destroy collateral
equilibrium in economies with production.
93 / 103
Introduction
Model
Leverage
CDS
Over Investment
Conclusion
CDS and Robust Non-Existence
Suppose we introduce into the L-economy a CDS. We call this the
LC -economy.
Equilibrium in the LC -economy equals:
-equilibrium in the LT -economy if Y is produced.
-equilibrium in the L-economy if Y is not produced.
Thus, if all LT -equilibria involve no production of Y and all
L-equilibria involve production of Y, then there cannot exist a
LC -equilibrium.
94 / 103
Introduction
Model
Leverage
CDS
Over Investment
Conclusion
CDS and Robust Non-Existence
Constant return to scale production:
Z0 = {z = (zx , zy ) ∈ R− × R+ : zy = −kzx }, k ≥ 0.
Consider any k ∈ (1, 1.4). Rest of parameters and beliefs as before.
Then LC -equilibrium does not exist.
95 / 103
Introduction
Model
Leverage
CDS
Over Investment
Conclusion
CDS and Robust Non-Existence
Y Volume CDS volume 1.8 High CDS volume with low underlying Y volume 1.6 1.4 1.2 Y Volume L-­‐economy 1 Y Volume AD Y Volume LT-­‐ economy 0.8 CDS volume 0.6 0.4 0.2 0 1 L=LT=AD No produc?on 1.1 1.2 1.3 1.4 Non-­‐existence region for CDS 1.45 1.5 1.55 1.6 1.65 LC=LT with produc?on 1.7 k 96 / 103
Introduction
Model
Leverage
CDS
Over Investment
Conclusion
CDS and Robust Non-Existence
The equilibrium in the LC economy does not exist for a robust set
of parameters. Back
97 / 103
Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Over Valuation and Investment Geometrical Proof
In the L-economy, optimists collectively consume zyL (dUY − dDY ) in
state U while in the Arrow Debreu economy they consume
zyA dUY + (x0∗ + zxA ). The latter is evidently much bigger, at least as
long as zyA ≥ zyL .
So suppose, contrary to what we want to prove, that
Arrow-Debreu output were at least as high, zyA ≥ zyL and p A ≥ p L .
98 / 103
Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Over Valuation and Investment Geometrical Proof
cU
Y(dYU,dYD) QA QL x0*(1,1) 45o
O cD
99 / 103
Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Over Valuation and Investment Geometrical Proof
cU
Y(dYU,dYD) QA Slope –qhL1D /qhL1U QL zLy(dYU-­‐dYD) x0*(1,1) (1-­‐h1L)QL 45o
O cD
99 / 103
Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Over Valuation and Investment Geometrical Proof
cU
Y(dYU,dYD) QA Slope –qhL1D /qhL1U QL zLy(dYU-­‐dYD) x0*(1,1) (1-­‐h1L)QL (1-­‐h1L)QA 45o
O cD
99 / 103
Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Over Valuation and Investment Geometrical Proof
cU
Y(dYU,dYD) QA Slope –qhL1D /qhL1U QL zLy(dYU-­‐dYD) x0*(1,1) (1-­‐h1L)QL (1-­‐h1L)QA (1-­‐h1A)QA 45o
O cD
99 / 103
Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Over Valuation and Investment Geometrical Proof
cU
Y(dYU,dYD) QA Slope –qhL1D /qhL1U QL zLy(dYU-­‐dYD) x0*(1,1) (1-­‐h1L)QL (1-­‐h1L)QA (1-­‐h1A)QA 45o
O cD
99 / 103
Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Over Valuation and Investment Geometrical Proof
cU
Y(dYU,dYD) QA Slope –qhL1D /qhL1U QL zLy(dYU-­‐dYD) x0*(1,1) (1-­‐h1L)QL zAydYU+x0*+zAx (1-­‐h1L)QA (1-­‐h1A)QA 45o
O cD
99 / 103
Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Over Valuation and Investment Geometrical Proof
Back
100 / 103
Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Marginal Over Investment and Collateral Value
Will illustrate the concept with our previous numerical example
that also has zero consumption at time 0.
Consider our numerical example with production
Z0 = {z = (zx , zy ) ∈ R− × R+ : zy = −kzx }, k = 1.5, beliefs:
h = 1 − (1 − h )2 and x ∗ = 1, d Y = 1, d Y = .2.
γU
0
U
D
In the L-economy equilibrium is given by
h1 = .35, p = .67, zx = −.92 and zy = 1.38.
101 / 103
Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Marginal Over Investment and Collateral Value
To fix ideas let’s consider one of the optimists h = .9.
Marginal Utility of Money for h = .9 in equilibrium at time 0:
µh=.9 =
.99(1 − .2)
= 1.70
.67 − .2
Marginal Expected Utility of a dollar invested on Y for h = .9 in
equilibrium:
.99(1.5) + .01(.2).1.5 = 1.48.
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Introduction
Model
Leverage
CDS
Over Investment
Conclusion
Marginal Over Investment and Collateral Value
There is marginal over-investment in equilibrium. No agent would
use an extra unit of cash in producing the asset if he could not also
borrow to do it. In fact, the agents do not borrow to buy the
asset, they buy the asset because it allows them to borrow (and
hence consume only in the up state).
KM(97) despite cash flow problems also had marginal
over-investment in equilibrium.
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