Bolton Gerzensee (August 7, 2013)
Liquidity Transformation and Crises
1
Financial Intermediation
and
Liquidity Transformation
Main Ideas
Banks insure consumers against liquidity shocks.
Banks also monitor and supply liquidity to …rms.
Lending and Deposit-taking are complementary forms of
liquidity creation
How do Financial Intermediaries create Liquidity?
On the liability side: by issuing securities that can be easily
sold
On the asset side: through relationship lending
Bolton Gerzensee (August 7, 2013)
Liquidity Transformation and Crises
2
1) Liquidity transformation with no aggregate risk
Diamond and Dybvig (1983)
Main Idea:
(1) investors may be subject to liquidity shocks which force
them to liquidate their investments early,
(2) liquidation is ine¢ cient and the market structure is incomplete: no claims contingent on an individual’s liquidity
shock can be traded.
(3) bank deposit contracts can provide insurance against liquidity shocks, but
(4) bank deposit taking exposes banks to the risk of a bank
run
Bolton Gerzensee (August 7, 2013)
Liquidity Transformation and Crises
3
The Model:
Three periods:
Date 0:
- consumers have an endowment of one unit of consumptiongood (wheat) they can save
- two investment projects:
a storage technology which allows them to transfer the
good to future periods at no cost.
a long-term illiquid investment which delivers C > 1
units of good at date 2 for one unit invested at date 0.
Date 1:
- Liquidity shocks are realized. impatient consumers (type
1) consume at date 1.
liquidating the long term investment at date 1 generates
a return of C < 1 for any unit invested at date 0.
Date 2: patient consumers (type 2) consume at date 2.
Bolton Gerzensee (August 7, 2013)
Liquidity Transformation and Crises
Liquidity shocks are i:i:d: : (1
being a type 2 consumer
There is continuum of agents
4
) is the probability of
=)
(1
) is also the fraction of patient consumers
< 1 is the discount factor between periods 1 and 2.
Investors are risk-averse and maximize their expected V N M
utility function:
max
x1 ;x2
u(x1) + (1
)u(x2)
Benchmarks: I] Autarky: Suppose trade is not possible
at any date.
Agents can allocate of their initial endowment in the long
term asset and store 1
.
Bolton Gerzensee (August 7, 2013)
Liquidity Transformation and Crises
5
If at date 1 investor turns out to be patient he stores 1
for one more period and does not liquidate the long-term
investment
=)
x2
1
+C
If investor is impatient: he liquidates all his holdings in date
1:
=)
x1
max
u(1
1
+C
+ C ) + (1
)u (1
=)
Autarky solution requires:
u0(x?1) =
1
C 1 0 ?
u (x2)
1 C
)+C
Bolton Gerzensee (August 7, 2013)
Liquidity Transformation and Crises
6
The Pareto E¢ cient Allocation
Suppose that there is a “social planner” in this economy
who can observe liquidity shocks at date 1
The planner can o¤er an insurance contract: against one
unit of investment, the social planner promises x1 if the
investor is of type 1 and x2 otherwise
Expected costs of this policy for the social planner?
With a continuum of agents, each with the same probability
of facing liquidity shocks, the planner needs such that
1
= x1
(law of large numbers), and
C = (1
=)
)x2
Bolton Gerzensee (August 7, 2013)
The optimal
Liquidity Transformation and Crises
maximizes ex ante expected utility:
max
u(
1
) + (1
)u(
C
1
=)
PE
is such that:
u0(xP1 E ) = Cu0(xP2 E )
Compare with autarky solution:
u0(x?1) =
1
C 1 0 ?
u (x2)
1 C
)
7
Bolton Gerzensee (August 7, 2013)
Liquidity Transformation and Crises
8
Deposit Contracts and Banks
Suppose that contracts conditional on the identity of the
agents hit by a liquidity shock are not feasible.
Can we still implement the optimal allocation?
Consider a bank o¤ering deposits to agents:
If an agent deposits 1 unit in t = 0, this agent can withdraw
R1 in t = 1 and R2 in t = 2.
deposit withdrawals are served sequentially until the bank
runs out of cash
suppose agent j wants to withdraw R1 in date 1. She then
obtains:
R1
0
if R1
Aj
otherwise
where Aj are the total cash reserves of the bank, after all
depositors before j have withdrawn their deposits
Bolton Gerzensee (August 7, 2013)
Liquidity Transformation and Crises
9
Candidate equilibrium:
the bank invests xP1 E in the storage technology and (1
)xP2 E in the long term asset.
R1 = xP1 E , R2 = xP2 E
Only type 1 depositors withdraw at date 1
a type 2 investor who withdraws at t = 1 gets xP1 E , which
can be stored until period 2
a type 2 investor who waits until period 2 gets xP2 E at t = 2.
=)
Bolton Gerzensee (August 7, 2013)
Liquidity Transformation and Crises
a type 2 investor prefers waiting if xP2 E
10
xP1 E ,
from u0(xP1 E ) = Cu0(xP2 E ) and (u is concave).
we observe that xP2 E
C
xP1 E provided that
1.
) If C < 1, everybody withdraws at date 1
when C
1 only type 1 investors withdraw at date 1,
assuming type 2 investors trust the bank
the proportion of withdrawals at t = 1 is then exactly
and the bank is solvent with probability 1
BUT.....
Bolton Gerzensee (August 7, 2013)
Liquidity Transformation and Crises
11
even if C
1, there is a second bank-run equilibrium
where the bank goes bust at t = 1 :
suppose the type 2 investor anticipates that all the other
type 2 investors withdraw their deposits
then bank is forced to liquidate its long-term investment,
generating cash of only
A = xP1 E + (1
)xP2 E C
=)
A is too small to cover all the withdrawals xP1 E
Bolton Gerzensee (August 7, 2013)
Liquidity Transformation and Crises
12
sequential service
=)
best response is to withdraw as soon as possible
)
Bank run as an (ine¢ cient) equilibrium.
Barring a bank-run (through forward induction argument?)
banks provide an e¢ cient ‘liquidity transformation service’,
which allows consumers to improve on their autarky payo¤s.
But, are …nancial intermediaries really needed? Can’t consumers achieve e¢ cient liquidity transformation by trading
assets or claims on their future output in a competitive secondary market?
Bolton Gerzensee (August 7, 2013)
Liquidity Transformation and Crises
13
Market Allocation:
Suppose now that there is no bank but there is a bond
market at date 1: Agents can trade the consumption good
against the promise to receive some quantity of goods at date
2.
Denote by r the return on the bond:
Suppose an impatient (type 1) investor borrows in the bond
market: he will receive C from the long-term investment
=) he can borrow
C
r
so that
x1 = 1
+
C
r
Patient (type 2) investors lend in the bond market and get
x2 = (1
)r + C
In equilibrium, must have rx1 = x2:
Why?
No arbitrage condition!
Bolton Gerzensee (August 7, 2013)
Liquidity Transformation and Crises
Utility maximization requires:
u0(x?1) + (1
) ru0(rx?1)
dx1
=0
d
=)
must have
=)
dx1
=0,r=C
d
A competitive equilibrium exists such that
=1
14
Bolton Gerzensee (August 7, 2013)
Liquidity Transformation and Crises
15
for then excess supply of consumption at date 1 by type 2
investors:
(1
)(1
)
equals excess demand for consumption by type 1 investors:
C
r
when
r = C.
and,
x?1 = 1
x?2 = (1
C
=1
r
)r + C = C
+
Note that the market allocation is better
(with autarky must have x1
1 and x2
C)
E¢ ciency gain comes from the fact that no ine¢ cient liquidation takes place at t = 1 when a secondary market is
operating.
Bolton Gerzensee (August 7, 2013)
Liquidity Transformation and Crises
16
Comments:
Diamond and Dybvig:
Explains Financial Intermediation: F I useful as it provides insurance to agents against liquidity shocks
Explains speculative bank runs as a multiple equilibria
phenomenon =)
explains banking crises
Potential Solutions to Bank Run Risk:
Suspension of Convertibility: suppose the bank announces
that it will not serve more than xP1 E withdrawals at date
1; type 2 investors then know that the bank will be able
to meet obligations at date 2. =) they don’t withdraw
and no bank runs
Deposit Insurance
Lender of Last Resort
Bolton Gerzensee (August 7, 2013)
Liquidity Transformation and Crises
17
Criticisms
Role of Banks:
Other …nancial claims also provide liquidity: Jacklin
1987: agents could hold equity in a …rm:
–the …rm collects initial endowments against one share
to individuals, invests d in the storage technology
and (1 d) in the long-term asset
–Suppose equity pays a dividend d in t = 1; after
dividend is paid, agents can sell or buy equity at a
price p (but agents cannot take short positions in
the stock)
=)
x1 = d + p
type 2 agents buy x new equity at date 1 with their dividend:
x=
d
p
=)
d
x2 = (1 + )C(1
p
d)
Bolton Gerzensee (August 7, 2013)
Liquidity Transformation and Crises
18
=)
type 2 agents are willing buy x new equity at price p if and
only if:
d
(1 + )C(1
p
d)
C(1
d) + d
or,
p
C(1
d)
The price p is such that supply equals demand:
) dp .
=)
p=
1
= (1
d
=)
(1
)dP E
C(1
dP E )
where,
dP E = 1
PE
xP1 E and xP2 E can be implemented and there is no risk of a
Bank Run!
Bolton Gerzensee (August 7, 2013)
Liquidity Transformation and Crises
19
This only works, however, if agents cannot take short positions in the stock. If they can take short positions then again
we have a no arbitrage constraint:
x1 = px2
Then, as before, ex-ante utility maximization subject to
this constraint is achieved for
x1 = 1
and
x2 = C
Bolton Gerzensee (August 7, 2013)
Liquidity Transformation and Crises
20
Bank Runs:
Explains only speculative bank runs. Investment returns are certain, so no bad news trigger these runs.
Empirically, bank runs are also related to bad fundamentals, like poor performance of the loan portfolio.
Allen and Gale (1998) make C a random variable (aggregate
shock)
Cannot explain coexistence of Banks with a bond market
Townsend 1982 and Jacklin 1987
no arbitrage condition:
optimal bank deposit contract: R1 = xP1 E , R2 = xP2 E
must also satisfy:
rR1 = R2
but this implies that R1 = 1 and R2 = C!
Related Literature:
1) Gorton and Pennacchi (1990)
2) Allen and Gale (1997)
3) Diamond and Rajan (2002a,b)
4) Holmström and Tirole (1998)
Bolton Gerzensee (August 7, 2013)
Liquidity Transformation and Crises
21
2) Liquidity transformation with aggregate risk
Bolton, Santos and Scheinkman (2011) “Outside and Inside
Liquidity
Main Ideas
Model of liquidity demand from possible maturity mismatch between asset revenues and consumption
liquidity demand can be met with either cash reserves (inside liquidity) or with asset sales (outside liquidity)
what determines the mix of inside and outside liquidity in
equilibrium?
asymmetric information about asset values increasing over
time
existence of multiple equilibria: immediate-trading equilibrium –> asset trading in anticipation of a liquidity shock
delayed-trading equilibrium –> assets traded in response
to a liquidity shock
delayed-trading equilibrium is Pareto superior to the immediatetrading equilibrium, when it exists (despite adverse selection problems)
MOTIVATION
• Financial intermediaries engage in maturity transformation and demand liquidity whenever there
is a maturity missmatch.
• Financial intermediaries can meet this liquidity demand with
– Inside liquidity: Cash carried by financial intermediary.
– Outside liquidity: Cash carried by other investors who are willing to exchange this cash for
assets carried by the intermediary.
• Standard argument:
– Outside liquidity has difficulty flowing to financial intermediaries during liquidity crises, because
the latter have superior information about the quality of their assets:
∗ Effectively adverse selection acts as a barrier to outside liquidity.
• Here a different view on adverse selection:
– We emphasize the timing of adverse selection during a liquidity crisis.
– In particular we show that the anticipation of adverse selection at some future date may lead
to an inefficient acceleration of trade:
∗ Parties are going to liquidate before the onset of adverse selection problems.
– We show that this acceleration of trade is inefficient because it is associated with:
∗ low levels of origination and
∗ low levels of outside liquidity and high levels of inside liquidity.
• Liquidity has efficiency implications that are ex-ante in nature (origination); no efficiency implications ex-post.
• The model has policy implications, in particular regarding the timing of the provision of public
liquidity.
THE MODEL
I. Basic structure
• Four period economy.
• 2 types of agents, short and long run investors.
• The risky asset is the only source of risk.
II. Agents
• Short Run Investors (SRs):
u (C1, C2, C3) = C1 + C2 + δC3
with 0 < δ < 1
• Long Run Investors (LRs):
u (C1, C2, C3) = C1 + C2 + C3
III. Financial markets and investment opportunity sets
• Assets: Cash, a “long asset,” and a “risky asset.”
• Investment opportunities: We assume that
– LRs can invest in cash and in the long asset.
– SRs can invest in cash and in the risky asset.
Agents and their investment opportunities - 1
©©
Long run inv. κ
-
LR
*
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j
H
*
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-
SR
ϕ (κ − M )
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HH
HH
H
HH
Short run inv. $1
-
κ−M
M
-
M
-
M
-
M
1−m
-
?
-
?
-
?
m
-
m
-
m
-
m
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HH
HH
H
HH
j
H
t
t
t
t
t=0
t=1
t=2
t=3
Agents and their investment opportunities - 2
©©
Long run inv. κ
-
LR
*
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j
H
*
©
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-
SR
ϕ (κ − M )
©
©
©
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HH
HH
H
HH
Short run inv. $1
-
κ−M
M
-
M
-
M
-
M
1−m
-
?
-
?
-
?
m
-
m
-
m
-
m
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j
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t
t
t
t
t=0
t=1
t=2
t=3
The risky asset - 1
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1−λ
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1−θ
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HH
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ω
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1 − η HHHH
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j
w
t=0
w
t=1
aggregate
ω30
w
w
-
t=2
idiosyncratic
t=3
idiosyncratic
The risky asset - 2
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1−λ
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1−η
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1−θ
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ω3ρ
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2L H
HH
H
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H
ω
HH
1 − η HHHH
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j
w
t=0
w
t=1
aggregate
ω30
w
w
-
t=2
idiosyncratic
t=3
idiosyncratic
The risky asset - 3
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*
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ω1ρ
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λ©©©
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0 H
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1−λ
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θ
HH
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ω2ρ
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1−η
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1−θ
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HH
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ω
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1 − η HHHH
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j
w
t=0
w
t=1
aggregate
ω30
w
w
-
t=2
idiosyncratic
t=3
idiosyncratic
Two key parameters
I. The θ parameter
(A) It determines the probability that the risky asset pays early, when the short run investors value
it the most.
(B) It is also the parameter that determines the severity of the adverse selection problem:
• The higher it is the more likely the LR is to find “lemons” in period t = 2.
(C) It also determines the supply of the assets at t = 2:
• The higher the θ the lower the supply of the assets at t = 2 as more agents, θη, obtain
the realization ρ and don’t need to liquidate:
II. The δ parameter
• δ determines the value of the SR of retaining the asset and carrying it to date t = 3 and it
introduces existence and commitment issues.
Assumptions
• LRs carry cash only if they can deploy it to acquire the risky assets at very advantageous prices.
Specifically, ϕ0 (κ) > 1, and thus “cash-in-the-market pricing” obtains:
Price of risky asset =
Outside Liquidity
< Expected discounted payoff
Amount of risky assets supplied
• SRs do not want to invest in the risky asset in autarchy: They only invest in it if when it does not
pay off they can liquidate at attractive prices.
λρ + (1 − λ) [θ + (1 − θ) δ] ηρ < 1
• But investing in the risky asset is socially beneficial in that the expected return on the asset is
greater than cash.
ρ [λ + (1 − λ)η] > 1
The problem of the SRs and the LRs
I. The SRs decide
• how much inside liquidity to carry, m, and how much to invest in the risky technology, 1 − m
• and a liquidation policy in the lower branch of the tree.
– Essentially, whether to liquidate at date t = 1, q1, and/or t = 2, q2
II. The LRs decide
• how much outside liquidity to carry, M , and how much to invest in the long asset, κ − M
• when to step in to acquire assets at firesale prices.
– Essentially, whether to jump in the market at date t = 1, Q1, and/or t = 2, Q2.
The SR optimization problem
π [m, q1, q2] = m + λ (1 − m) ρ
+ (1 − λ) q1P1
+ (1 − λ) θη [(1 − m) − q1] ρ
+ (1 − λ) θ (1 − η) [1 − m − q1] P2
+ (1 − λ) (1 − θ)q2P2
+ δ (1 − λ) (1 − θ) η [(1 − m) − q1 − q2] ρ
max π [m, q1, q2]
m,q1 ,q2
subject to
m ∈ [0, 1]
with
q1 + q2 ≤ 1 − m
and
q1, q2 ∈ {0, 1 − m}
The LR optimization problem
Π [M, Q1, Q2] = M + ϕ (κ − M )
+ (1 − λ) [ηρ − P1] Q1
+ (1 − λ)E [ρ̃3 − P2| F ]Q2
max Π [M, Q1, Q2]
M,Q1 ,Q2
subject to
0≤M ≤κ
with
Q1P1 + Q2P2 ≤ M
and
Q1 ≥ 0, Q2 ≥ 0
Definition of equilibrium
I. Definition
• A vector of portfolio policies, [m∗, M ∗],
• prices, [P1∗, P2∗] and
• liquidation, [q1∗, q2∗], and acquisition policies, [Q∗1 , Q∗2 ], such that agents maximize and markets
clear.
• An equilibrium must also specify the price S1∗ that would obtain in event ω1L for payoffs in
period 3 and the price S2∗ for these payoffs that would prevail in period 2.
II. Two types of equilibria
• Immediate trading equilibrium: Trading occurs at date t = 1 - No adverse selection
• Delayed trading equilibrium: Trading occurs at date t = 2 - Adverse selection
EQUILIBRIUM UNDER FULL INFORMATION
• Under full information the only equilibrium is a delayed trading equilibrium.
• Why can’t we support an immediate-trading equilibrium?
– The conditions for a putative immediate-trading equilibrium are
P1i∗ ≥ θηρ + (1 − θ) P2i∗
and
P2i∗ ≥ P1i∗
– This implies
P1i∗ ≥ ηρ
⇒
P2i∗ ≥ ηρ
e
– Given that E[ρ|F
1 ] = ηρ under symmetric information, it follows that the expected return of
carrying cash for LRs cannot be greater than one and thus
Mi∗ = 0
which implies
m∗i = 0
• Under full information a delayed trading equilibrium always exists.
EQUILIBRIA UNDER ASYMMETRIC INFORMATION
I. Existence and efficiency
(A) Existence:
• Inefficient acceleration of trade: The immediate trading equilibrium always exists
• The delayed trading equilibrium exists as long the adverse selection problem is not too
severe (market breakdown).
The immediate trading equilibrium
λ
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1−λ
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1−θ
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H
H
1 − η HHHH
H
w
w
w
t=0
Q∗1,i
t=1
∗
= q1,i
= 1 − m∗i
∗
P1,i
=
Mi∗
1−m∗i
ω3ρ
t=2
∗
Q∗2,i = q2,i
=0
∗
P2,i
< δηρ
H
j
ω30
w
t=3
-
The delayed trading equilibrium
λ
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*
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H
0©
HH
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ω
*
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1−λ
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1−η
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1−θ
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1 − η HHHH
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w
t=0
w
w
t=1
∗
Q∗1,d = q1,d
=0
∗
P1,d
ω3ρ
Q∗2,d
t=2
∗
= q2,d
= (1 − m∗d) (1 − θη)
∗
P2,d
=
Md∗
(1−m∗d )(1−θη)
H
j
ω30
w
t=3
-
The immediate and the delayed trading equilibrium: θ = .35
Immediate versus delayed trading equilibrium: θ=.35
1
(M*,m*)
i
0.9
i
0.8
0.7
m
0.6
0.5
0.4
*
*
(Md,md)
0.3
0.2
0.1
0
0
0.02
0.04
0.06
M
0.08
0.1
The immediate and the delayed trading equilibrium: θ = .45
Immediate versus delayed trading equilibrium: θ=.45
(M*i,m*i)
0.9
0.8
0.7
0.6
m
0.5
0.4
0.3
(M*d,m*d)
0.2
IPSR
0.1
0
−0.1
0
0.02
0.04
0.06
M
0.08
0.1
(B) Efficiency
π*: The expected profit of the SRs in the delayed trading equilibrium
1.003
π*(θ)
1.002
m* >0
m* =0
d
d
1.001
1
0.35
0.4
0.45
θ
Π∗: The expected profit of the LRs in the delayed trading equilibrium
0.54
Π*(θ)
0.535
0.53
m* >0
d
m*d=0
Π*
i
0.525
0.35
0.4
θ
0.45
(C) Efficiency and the distribution of inside versus outside liquidity
• Efficiency gains occur in our set up whenever
1. more risky projects are implemented and
2. the lower the amount of cash carried by all parties.
• But, recall, the SRs do not want to implement the risky project in autarchy.
– They do so only if enough outside liquidity is brought to absorb the firesale.
– Less inside liquidity (more risky projects) can only occur if more outside liquidity is
brought in to absorb the firesales
• In the delayed trading equilibrium
– the amount of risky projects the SRs undertake is maximized and the overall amount of
cash carried by all parties is lower
– though outside liquidity is higher than in the immediate trading equilibrium.
• The efficiency gain associated with increased investment in risky projects more than compensates for the efficiency loss associated with the increase in outside liquidity.
• Intuitively the additional amount of outside liquidity that the LRs carry in the delayed
trading equilibrium is not very large relative to the outside liquidity carried in the immediate
trading equilibrium.
– The reason is that they only need to acquire the assets of SRs in states ω2L and ω20.
– SRs retain the “upside” of the risky asset: state ω2ρ.
– In the immediate trading equilibrium the LRs have to bring much more cash to absorb
the full measure of risky projects that now effectively include the upside associated with
state ω2ρ and thus the loss of efficiency.
(D) Comparative statics
1. Outside and inside liquidity as a function of θ
Cash position of the SRs in the delayed trading equilibrium as a function of θ
m*d(θ)
0.6
0.4
0.2
0
0.35
0.45
θ
Cash position of the LRs in the delayed trading equilibrium as a function of θ
0.08
M*d(θ)
0.4
0.07
0.06
0.05
0.35
0.4
θ
0.45
2. Prices and expected returns as a function of θ
Expected return of the risky asset at t=2 in the delayed trading equilibrium, R* (ω )
d
R*d(ωd)
4
3.5
m* >0
3
d
2.5
0.35
0.4
m*d=0
0.45
θ
Price of the risky asset at t=2 in the delayed trading equilibrium, P* (ω )
d
0.12
P*d(ωd)
0.11
0.1
0.09
0.08
0.35
*
md>0
0.4
θ
*
md=0
0.45
d
d
(B) Adverse selection and the existence of the delayed trading equilibrium
Existence of the delayed trading equilibrium
0.1
0.098
C
B
A
0.096
0.094
P∗ (ω )
P*2,d
0.092
d
d
0.09
0.088
0.086
0.084
0.082
0.08
0.4
PC
(ω )
d d
δηρ
0.41
0.42
0.43
0.44
θ
0.45
0.46
0.47
0.48
C
• Region C: The delayed trading equilibrium fails to exist as the candidate price, P2,d
, which
is unique, is such that
C
P2,d
< δηρ
• In this case the SRs in state ω2L prefer to carry the asset to t = 3 rather than liquidating,
destroying the pooling that sustains the delayed trading equilibrium.
*
π : The expected profit of the SRs with and without commitment
1.004
π*(θ)
1.003
A
1.002
C
B
1.001
1
0.36
0.38
0.4
0.42
θ
0.44
0.46
0.48
Π∗: The expected profit of the LRs with and without commitment
0.54
Π*(θ)
0.535
C
B
A
0.53
0.525
0.36
0.38
0.4
0.42
θ
0.44
0.46
0.48
• Pareto improvement in the case of (state contingent) commitment.
• Can a monopolist do better?
Π: The expected profit of the competitive and the monopolist LR
0.54
Π*(θ)
0.535
0.53
B
A
C
competitive
profits
monopoly
profits
0.525
0.4
0.41
0.42
0.43
0.44
0.45
0.46
0.47
θ
Prices in t=2 for the monopolist and the competitive case
0.48
0.1
A
B
competitive
prices
P∗d(ωd)
0.095
C
monopoly
prices
0.09
0.085
0.4
0.41
0.42
0.43
0.44
θ
0.45
0.46
0.47
0.48
ROBUSTNESS
I. The instantaneous-trading equilibrium
• Would it be optimal for the SRs to originate assets and distribute them immediately (at t = 0)?
• No: An instantaneous-trading equilibrium does not exist either with full or asymmetric information.
·
c
c
c
– Assume prices that support such an instantaneous-trading equilibrium: P0, P1, P2
¸
– Then it has to be that
Pc0 ≥ λρ + (1 − λ) Pc1
and
[λ + (1 − λ) η] ρ ηρ
≥ c.
Pc0
P1
– This implies that
Pc0 ≥ [λ + (1 − λ) η] ρ,
– Given that ϕ0 (κ) > 1 LRs do not bring any capital to the putative instantaneous-trading
equilibrium.
II. General investment opportunity sets
• Are the results affected if we allow SRs to invest in the long run project and LRs to invest in
risky projects?
• No.
– SRs
∗ would not invest in long-run assets if δϕ0 (0) < 1.
∗ If they do, analysis goes through for the proportion of initial funds not invested in the
long-run asset (and SRs might sell the long asset to LRs at, for example, t = 2.)
– LRs
∗ Cash is a constant returns to scale technology. If SRs are partially invested in risky
assets then LRs are invested in cash and long-run assets.
∗ LRs may be holding risky assets when SRs are fully invested in risky assets but provide
an inadequate supply of risky assets.
III. Cash-in-the-market pricing and Arbitrage Contagion
• Cash-in-the-market pricing: Key to support an equilibrium where M ∗ > 0.
• Cash-in-the-market pricing and the absence of arbitrage implies that the price of one unit of
the long run project payable at date 3 has to be such that LRs are indifferent between holding
the long run project and cash at t = 1 and t = 2:
– In the immediate-trading equilibrium:
∗
S1i
P1i∗
<1
=
ηρ
and
∗
S2i
=1
– In the delayed trading equilibrium (when it exists)
∗
S1d
∗
P1d
≤1
=
ηρ
and
∗
S2d
• This occurs even when:
– There are no news about the long run project and
– The asset is not subject to distressed sales
∗
P2d
(1 − θη)
=
<1
(1 − θ)ηρ