p u d a 1-a q 1 d a k 1 1 a u d h 1-h u d h y h

Why Does Bad News Increase Volatility and Interest Rate, and Decrease

Optimism, Asset Prices and Leverage?

F. Albert Wang

University of Dayton

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

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

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

Asset Prices, Margin, Leverage, and Interest Rate

Asset prices: house (p); mortgage (q)

Margin:

Leverage: 1/m

Interest Rate:

1q

mp

11r

q

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

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

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

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

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?

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

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

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

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

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

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

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

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

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

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

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

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

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

Equilibrium Results: EBV Project vs. EGV Project

EBV project gives higher initial price and leverage

Variable EBV EGV

Marginal belief 0.69 0.57



House price 0.77 0.47



Mortgage price 0.46 0.18



0( )a

( )Ua( )Da

0( )p

( )Up( )Dp

0( )q

( )Uq

( )Dq

Variable EBV EGV

Interest rate 0.0500

0.0500


0.0500


0.0500

Leverage 2.53 1.63

Leverage 1.36

Leverage 1.64

Volatility 0.22 0.26



0( )r( )Ur

( )Dr

0( )l

( )Ul( )Dl

0( )( )U( )D

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)

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

Equilibrium Results: BV Project vs. GV Project

BV project gives higher initial leverage, but lower initial price

Variable BV GV










0( )a

( )Ua( )Da

0( )p

( )Up( )Dp

0( )q

( )Uq

( )Dq

Variable BV GV


0.0711


0.0555


0.0661

Leverage 17.83 15.06

Leverage 20.50 19.03

Leverage 8.34 16.14




0( )r( )Ur

( )Dr

0( )l

( )Ul( )Dl

0( )( )U( )D

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

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

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

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

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



0( )r

( )Ur

( )Dr

0( )l

( )Ul( )Dl

0( )( )U( )D

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

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

p u d a 1-a q 1 d a k 1 1 a u d h 1-h u d h y h

Documents

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