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Transcript of p u d a 1-a q 1 d a k 1 1 a u d h 1-h u d h Y h.
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Why Does Bad News Increase Volatility and Interest Rate, and Decrease
Optimism, Asset Prices and Leverage?
F. Albert Wang
University of Dayton
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The Great Recession A joint collapse of the mortgage and the
housing markets during 2007-2009 A close interlock among mortgage,
housing, and credit markets A recurrent “leverage cycle” phenomenon
in American financial history (Geanakoplos 2010)
Bad news increases volatility and interest rate, and decreases optimism, asset prices and leverage
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Theoretical Framework
Collateral constraints on asset pricing (Geanakoplos 2003, 2010)
Mortgage (risky derivative asset) vs. house collateral (underlying asset)
Asset prices derived from a risk-neutral probability under no arbitrage
Subjective house values based on natural buyers’ heterogeneous beliefs about the housing market
The risk-neutral probability is the marginal belief
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Asset Prices under No Arbitrage
Three assets: house, mortgage, and risk-free bond with prices (p, q, k) and risk-neutral probability (a)
House (underlying asset) Mortgage (derivative asset)
p
u
d
a
1-aq
1
d
a
1-ak
1
1
a
1-a
( (1 ) ) / (1 )fp a u a d r ( 1 (1 ) ) / (1 )fq a a d r
0 1d u
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Asset Prices, Margin, Leverage, and Interest Rate
Asset prices: house (p); mortgage (q)
Margin:
Leverage: 1/m
Interest Rate:
1q
mp
11r
q
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Agents with Heterogeneous Beliefs
0 a 1
h
pessimists optimists
h = a = Marginal agent belief
0 h a 1a h
u
d
h
1-h
hp
( (1 ) ) / (1 )hfp h u h d r
h a h a h ap p p p
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Maximizing Expected Utility
Endowment:1 house (Y) and 1 consumption good (e)
Pessimists think house (Y) is overpriced Optimists think house (Y) is underpriced
u
d
h
1-h
0 0( , , ) ( (1 ) ) / (1 )h h h h h h hU D U D fU x x x x h x h x r
Yh
1-h0hx
hUx
hDx
( , , ) ( , , )h a h a h a h aU e u d e p e p U e u d e p
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Market Clearing Condition and Equilibrium
0 a 1
Pessimists sell all their houses, consume all endowment, and lend mortgage
Optimists buy houses using their endowment and borrowing to the max from mortgage
Combing no arbitrage asset pricing with equilibrium natural buyers to obtain a unique equilibrium: (a, p, q)
(1 )a p a e q
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Optimal Investment and Consumption
pessimists optimists consumption consumption
Optimists use mortgage to maximize their housing investment and consume none now
Pessimists lend mortgage, shun housing investment, and smooth consumption
e
1
d0
u-1
0
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The Dynamic Model
Extend the one-shot model into a dynamic model incorporating a possible crash in the interim period
Trading takes place at time 0 and subsequently at time 1 in either good state U or bad state D
The fundamental house value is realized at time 2
Mortgage principal links to the underlying house collateral price in each state of each time
Why bad news increases volatility and interest rate, and decreases optimism, asset prices and leverage?
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The Contingent House Prices:
Risk-neutral probabilities = marginal beliefs
0p
Up
Dp
UUp u
0aUDp v
DDp d01 a
1 Da
1 Ua
Ua
Da
0, ,( )U Da a a
0, ,( )U Dp p p
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The Contingent Mortgage Prices:
Mortgage principal is the current house price
Maturity misalignment between short-term mortgage and long-term house
0q
0p
DpDp
Up
Uq
Dq
v
d
0a
Ua
Da01 a
1 Da
1 Ua
0, ,( )U Dq q q
( )sp
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Marginal Agent Beliefs:
pessimists optimists
new new pessimists optimists new
new pessimists
optimists
0( 1)a h
0a
0, ,( )U Da a a
0(0 )h a
Da
(0 )Dh a 0( )Da h a
Ua
0( )Ua h a ( 1)Ua h
0a
1
10
0
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Housing Market in Bad State D
Optimists default and leave the market Pessimists (lenders) seize the house collateral New aggregate endowment: consumption
good (e) Existing agents trade once again among
themselves to maximize expected utility New pessimists think the house is overpriced
New optimists think the house is underpriced
0( )D Dh a h aD D Dp p p
0( 1)a h
0(0 )h a
0(0 )h a
0( )DD a h ah aD D Dp p p
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Market Clearing Condition in Bad State D
0
New pessimists sell all their houses, consume all endowment, and lend mortgage
New optimists buy houses using their endowment and borrowing to the max from mortgage
(0 )Dh a
0( )Da h a
Da 0a
0
0 0
DDD D
a aap e q
a a
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Housing Market in Good State U Optimists pay off mortgage principal and
keep the house Pessimists get payment and leave the market Net aggregate endowment after debt
payment: Existing agents trade once again among
themselves to maximize expected utility New pessimists think the house is overpriced
New optimists think the house is underpriced
0( )U Ua h a h aU U Up p p
0( 1)a h
0(0 )h a
1( )U Uh a a hU U Up p p
0( )e p
0( )p
0( 1)a h
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Market Clearing Condition in Good State U
1
New pessimists sell all their houses, consume all endowment, and lend mortgage
New optimists buy houses using their endowment and borrowing to the max from mortgage
0( )Ua h a
( 1)Ua h
Ua
00
0 0
1( )
1 1U U
U U
a a ap e p q
a a
0a
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Maximizing Expected Utility in Initial State 0
Endowment:1 house (Y) and 1 consumption good (e)
Pessimists think house (Y) is overpriced Optimists think house (Y) is underpriced
h
1-h
0 0( , , ) ( (1 ) ) / (1 )h h h h h h hU D U D fU x x x x h x h x r
Yh
1-h0hx
hUx
hDx
0 0 0 00 0 0( , , ) ( , , )h a h a h a h a
U D U DU e p p e p e p U e p p e p
Up
Dp
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Market Clearing Condition in Initial State 0
0 1
Pessimists sell all their houses, consume all endowment, and lend mortgage
Optimists buy houses using their endowment and borrowing to the max from mortgage
0 0 0 0(1 )a p a e q
0a
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The Equilibrium of the Dynamic Model
There exists a unique equilibrium of the model:
Extend Geanakoplos (2003, 2010) under risk-free mortgage to a general model under risky mortgage
Yield pro-cyclical mortgage credit, consistent with Schularick and Taylor (2012)
Provide endogenous leverage cycle and interest rate dynamics
0 0 0( , , , , , , , , )U D U D U Da a a p p p q q q
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A Special Case under Risk-free Mortgage(Foster and Geanakoplos 2012)
Agents choose between Extreme Bad Volatility (EBV) and Extreme Good Volatility (EGV) projects
EBV or EGV: payoffs only volatile in bad or good times
Agents prefer EBV projects because they offer higher initial price and leverage
So, bad news increases volatility and decreases leverage
Re-examine this issue with two restricted assumptions: (1) extreme payoff structure and (2) risk-free mortgage
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Extreme Payoff Structure:
E. bad volatility (EBV) project: (u,v,d)=(1,1,0.2)
E. good volatility (EGV) project: (u,v,d)=(1,0.2,0.2)
0p
Up
Dp
u
0av
d01 a
1 Da
1 Ua
Ua
Da
( , , )u v d
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Risk-free Mortgage:
Mortgage principal is the low value of house next period, i.e., the recovery value
0q
Dp
Dpd
v
Uq
Dq
v
d
0a
Ua
Da01 a
1 Da
1 Ua
0, ,( )U Dq q q
( )sDp
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Equilibrium Results: EBV Project vs. EGV Project
EBV project gives higher initial price and leverage
Variable EBV EGV
Marginal belief 0.69 0.57
Marginal belief 1.00 0.69
Marginal belief 0.39 0.57
House price 0.77 0.47
House price 0.95 0.72
House price 0.49 0.19
Mortgage price 0.46 0.18
Mortgage price 0.95 0.19
Mortgage price 0.19 0.19
0( )a
( )Ua( )Da
0( )p
( )Up( )Dp
0( )q
( )Uq
( )Dq
Variable EBV EGV
Interest rate 0.0500
0.0500
Interest rate 0.0500
0.0500
Interest rate 0.0500
0.0500
Leverage 2.53 1.63
Leverage 1.36
Leverage 1.64
Volatility 0.22 0.26
Volatility 0.00 0.37
Volatility 0.39 0.00
0( )r( )Ur
( )Dr
0( )l
( )Ul( )Dl
0( )( )U( )D
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Summary of Findings: EBV vs. EGV
Agents prefer EBV projects because they offer higher initial price and leverage
Flat interest rates dynamics EBV project: extreme optimism and infinite
leverage in good times EGV project: extreme optimism and infinite
leverage in bad times Replicate results of Foster and Geanakoplos
(2012)
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The Dynamic Model under Risky Mortgage
Relax the two restricted assumptions: (1) extreme payoff structure and (2) risk-free mortgage
Agents choose between bad volatility (BV) and good volatility (GV) projects under a general payoff structure
BV project: GV project: Examine the general properties of the
dynamic model under risky mortgage, assuming
( , , ) ( , , 3 ) (1.2,1,0.4)UU UD DDp p p v v v
( , , ) ( 3 , , ) (1.6,1,0.8)UU UD DDp p p v v v
( , ) (0.2,0.05)fe r
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Equilibrium Results: BV Project vs. GV Project
BV project gives higher initial leverage, but lower initial price
Variable BV GV
Marginal belief 0.95 0.94
Marginal belief 0.99 0.98
Marginal belief 0.86 0.89
House price 1.07 1.41
House price 1.14 1.52
House price 0.87 0.93
Mortgage price 1.01 1.32
Mortgage price 1.09 1.44
Mortgage price 0.77 0.87
0( )a
( )Ua( )Da
0( )p
( )Up( )Dp
0( )q
( )Uq
( )Dq
Variable BV GV
Interest rate 0.0594
0.0711
Interest rate 0.0513
0.0555
Interest rate 0.1363
0.0661
Leverage 17.83 15.06
Leverage 20.50 19.03
Leverage 8.34 16.14
Volatility 0.06 0.14
Volatility 0.02 0.07
Volatility 0.21 0.06
0( )r( )Ur
( )Dr
0( )l
( )Ul( )Dl
0( )( )U( )D
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General Properties of the Dynamic Model
Agents still prefer BV projects because they offer higher initial leverage, though not higher price
Pro-cyclical optimism and asset prices BV project: pro-cyclical leverage; counter-
cyclical volatility and interest rate Give a unified explanation: why bad news
increases volatility and interest rate, and decreases optimism, asset prices and leverage
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Double Leverage Cycle (Geanakoplos 2010)
The Great Recession is particularly bad because it suffers from a “double leverage cycle” problem
One primary cycle in the housing market and one secondary cycle in the mortgage securities market
The same collateral (house) backs the mortgage payment first and the mortgage securities again
The two cycles reinforce each other in a positive feedback loop, resulting in greater volatility, more severe leverage cycle, and worse financial crises
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The Extended Model with Double Leverage Cycle
Add a secondary cycle in the mortgage securities
Let mortgage principal be a weighted average of the current house price and the recovery value
The “funding margin” (n) of the secondary cycle
(1 ) , 0 1s s sDF n p n p n sUp
sDpsp
( ) / ( ) /s s s s sD sp F p n p p p n
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The Extended Model (Cont.)
Mortgage prices under double leverage cycle
The market clearing condition in good state U
BV (u,v,d) = (1.2,1,0.4) vs. GV (u,v,d) = (1.6,1,0.8)
Loose funding (n=0) vs. tight funding (n=0.2)
The marginal effect of tightening the funding margin in the secondary cycle on the primary cycle
( ((1 ) ) (1 ) ) / (1 )s s s sD s sD fq a n p n p a p r
00
0 0
1( ((1 ) ))
1 1U U
U D U
a a ap e n p n p q
a a
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Equilibrium Results
BV Project vs. GV ProjectLoose Funding (n=0) vs. Tight Funding (n=0.2)
Variable BVn=0
BVn=0.
2
GVn=0.
2
M. belief 0.95 0.91 0.88
M. belief 0.99 0.98 0.95
M. belief 0.86 0.73 0.81
H. price 1.07 1.05 1.36
H. price 1.14 1.14 1.49
H. price 0.87 0.80 0.92
M. price 1.01 0.94 1.17
M. price 1.09 1.06 1.31
M. price 0.77 0.60 0.83
0( )a
( )Ua
( )Da
0( )p
( )Up
( )Dp
0( )q
( )Uq
( )Dq
Variable BVn=0
BVn=0.
2
GVn=0.2
Int. rate 0.0594
0.0705
0.0865
Int. rate 0.0513
0.0524
0.0659
Int. rate 0.1363
0.1935
0.0709
Leverage 17.83 8.98 7.18
Leverage 20.50 13.71 8.07
Leverage 8.34 4.07 11.11
Volatility 0.06 0.10 0.19
Volatility 0.02 0.03 0.13
Volatility 0.21 0.27 0.08
0( )r
( )Ur
( )Dr
0( )l
( )Ul( )Dl
0( )( )U( )D
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Main Findings of the Extended Model
Agents still prefer BV projects because they offer higher initial leverage, though not higher price
BV project: pro-cyclical optimism, asset prices and leverage; counter-cyclical volatility and interest rate
Bad news increases volatility and interest rate, and decreases optimism, asset prices and leverage
Tightening funding margin magnifies the leverage cycle and volatility
Double leverage cycle leads to more severe leverage cycle, thus resulting in worse financial crises
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Conclusion Combine no arbitrage asset pricing with
equilibrium natural buyers to obtain a dynamic model of leverage cycle and interest rate
Extend Geanakoplos (2003, 2010) under risk-free mortgage to a general model under risky mortgage
Explain why bad news raises volatility and interest rate, and reduces optimism, asset prices and leverage
Yield new testable implications: the marginal effect of funding margin on the leverage cycle
Double leverage cycle leads to more severe leverage cycle and worse financial crises