systemic stability of housing and mortgage market a state-dependent four-phase model qin xiao...

Systemic Stability of Housing and Mortgage MarketA state-dependent four-phase model

Qin Xiao

University of Aberdeen

Property

Introduction

Objective of the study: To construct a theoretical framework explicitly

modeling the dynamic interactions between housing and mortgage markets;

To establish conditions under which this system is dynamically stable.

Introduction

Justification:1. Housing is an important market:

the single most important asset for many households; exerts important influences on national economies

2. Existing consumption approach to housing market is largely based on the standard theory of consumer utility maximization But houses are not food, clothes or cars: It delivers

consumables; It also stores values

Introduction

Justification3. Existing asset-approach treats housing like

stocks and bond But houses are heterogeneous, lumpy, and require

mortgage financing

4. Supply are usually assumed to be fixed not a sensible assumption

US Home Sales

0

200

400

600

800

1000

1200

1400Ja

n-99

Jan-

00

Jan-

01

Jan-

02

Jan-

03

Jan-

04

Jan-

05

Jan-

06

Jan-

07

Jan-

08

Jan-

09

Jan-

10

Unit t

hous

ands

New 1 family house forsale

Exising home sales

Total home sales

US Home Sales as % of Stock

0.00%

0.20%

0.40%

0.60%

0.80%

1.00%

1.20%

Jan-

99

Jan-

00

Jan-

01

Jan-

02

Jan-

03

Jan-

04

Jan-

05

Jan-

06

Jan-

07

Jan-

08

Jan-

09

Jan-

10

New home sales as % ofstock

existing sales as % ofstock

total sales as % of stock

Introduction

5. Lack of an unified theory explaining the observed co-movement between Housing and mortgage market

Exception: Robert M. Buckley (19820 takes a simple graphical approach cannot capture the complexity of the

transmission mechanism; and stability is assumed rather than contested; no feedback on price)

Stylized facts

Housing market cycles are irregular and unpredictable;

Some facts about the cycle are predictable though:

1. When confidence , postpone planned purchases; When optimism , front-running

the expectation effect accelerates price changes in both directions

But the effect is asymmetric (loss-aversion, Genesove, D. and C. Mayer (2001)):

housing supply plays a key role which has so far largely been ignored by the literature (except for a few, e.g. Glaeser, Gyourko et al. 2008; Poterba 1984).

Stylized facts

2. The above two market phases are associated with high volatilities uncertainty, anxiety and controversial beliefs:

2. Has the price risen or fallen enough? 3. Is the market due for correction? 4. Is an observed market correction a temporary

random move, or a signal that the tide is now turning?

3. 有人辞官归故里，有人漏夜赶科场

Stylized facts

3. The other two phases of a market cycle. Stagnation in trading volume, sometimes with continued

price hike, on approaching a peak; Stagnation in both volume and price on approaching a

trough

4. In these two phases, low volatility is characteristic of the market ： more unanimous beliefs:

Has the highest bidder appeared? Has the seller with the lowest reservation price gone? Delay buying, until patience/resource runs out, and the

tide turns around.

Stylized fact

Phase II IP, hP small &< 0 Phase III

IP, hP large &> 0

Phase I IP, hP large & > 0

Phase IV IP, hP small & > 0

Transaction volume

Price

Recovering

Over-heating

Collapsing

Booming

Stylized fact: evidences?

England and Wales

55,000

75,000

95,000

115,000

135,000

155,000

175,000

195,000

21,000 41,000 61,000 81,000 101,000 121,000 141,000

Sales volume (unit)

Ave

rage

pri

ce (G

BP

)

(Jan 95 – Feb 10)

Stylized fact: evidences?

US Northeast

130000

150000

170000

190000

210000

230000

250000

270000

290000

310000

35 55 75 95 115 135

Total sales (unit th)

Med

ian

pri

ce (

US

D)

(Jan 99 – Apr 10)

Stylized fact: evidences?

US Midwest

110000

120000

130000

140000

150000

160000

170000

180000

190000

50 70 90 110 130 150 170 190

Total sales (unit th)

Med

ian

pri

ce (

US

D)

(Jan 99 – Apr 10)

Stylized fact: evidences?

US South

110000

120000

130000

140000

150000

160000

170000

180000

190000

200000

210000

100 150 200 250 300 350

Total sales (unit th)

Me

dia

n p

ric

e (

US

D)

(Jan 99 – Apr 10)

Stylized fact: evidences?

US West

170000

190000

210000

230000

250000

270000

290000

310000

330000

350000

60 80 100 120 140 160 180 200

Total sales (unit th)

Med

ian

pri

ce (

US

D)

(Jan 99 – Apr 10)

Stylized fact: disequilibrium?

Demand

House priceSupply

Units of housing asset

Phase I&IV

Phase II

Phase ???

Stylized fact: Continuous equilibrium?

D0: fundamental demand

D1: with speculation

D2: with speculation

Unites of housing asset

S

House Price

Phase II: market reversal when all foreseeable price appreciations have been fully exploited

Phase I: forward purchase in anticipation of future price rises

Phase III: over-pessimistic market in the face of uncertainty

Phase IV: recovery

Stylized fact: implication for time pathPrice

Volume

Phase I Phase II Phase IVPhase III

Volume leads the decline

Volume leads the recovery

Stylized fact: implication for time pathPrice

Volume

Phase I Phase II Phase IVPhase III

Volume leads the decline

Volume leads the recovery

House Price, Sales Volume and Mortgage Outstanding(YOY Jan 96 to Feb 10)

-80.00%

-60.00%

-40.00%

-20.00%

0.00%

20.00%

40.00%

60.00%

80.00%

100.00%

120.00%

Pri

ce a

nd

vo

lum

e

0.00%

2.00%

4.00%

6.00%

8.00%

10.00%

12.00%

14.00%

16.00%

18.00%

Mo

rtg

age

ou

tsta

nd

ing

England&Wales housesales (YOY)

England andWales averaehouse price(YOY)

UK net lendingsecued byhome (YOY)

Stylized fact: evidences?

Houses Price, Sales Volume and Mortgage Outstanding(YOY Jan 00 to Mar 10)

-30.00%

-25.00%

-20.00%

-15.00%

-10.00%

-5.00%

0.00%

5.00%

10.00%

15.00%

20.00%

Pri

ce a

nd

vo

lum

e

-15.00%

-10.00%

-5.00%

0.00%

5.00%

10.00%

15.00%

20.00%

Mo

rtg

ag

e o

uts

tan

din

g

US housesales (YOY)

US medianhouse price(YOY)

Loan securedby singlefamily homes(YOY)

Stylized fact: evidences?

Other stylized facts

5. The movement along the four-phase cycle is cyclical itself and again irregular;

6. Mortgage debt moves more in tandem with house prices than with transaction volumes (see previous two slides)

PH

IH HS

Equilibrium in both markets

PH=R/iH

MS’ more sensitive to iH

Arbitrage condition Housing market

Mortgage marketMS&MD

MS

LRHS

LRHS’

0

11’

LRHS’’

Model: Approach taken

Partial equilibrium Endogenous variables: price P, volume Q and

mortgage debt M; Exogenous variables: everything else.

Model: Demographic and Housing Tenure Distribution Housing consumption is the only consumption Housing production is the only production T generations of N(=nT) households (stationary

state) Exogenously given tenure type and fraction:

Single absentee landlord; Owner: n in each generation will own a house once in its

lifetime; Renter: never want never will own.

Renters are homogenous Owners are homogeneous only within a generation

the time pattern of house price and volume.

Model: Demographic and Housing Tenure Distribution (time t, no speculation)

Owners N

Renters (1-)N

Generation 0

Generation 1,2,…T-2

Generation T-1

Buyer n

Holder (T-2)n

Seller n

Renters (1-)n

Renters (T-2)(1-)n

Renters (1-)n

Model: Demographic and Housing Tenure Distribution (time path without speculation)

tborne

t+1t+2

t+3 exit

Buy

Do not buy

Hold

Do not buy

Sell

No action

Owner path

Renter path

Model: Demographic and Housing Tenure Distribution (time path with speculation)

Buy

Do not buy

Hold

Sell

Buy

Do not buy

Sell

Sell

No action

No action

tborne

t+1t+2

t+3 exit

Speculative owner paths

Model: Ownership financing

Endowment: wealth (debt) of exiting generation new-borne generation

When inheriting a debt: borrow from a single absentee financier at rate r;

When inheriting a wealth: wi,t < Pt deposit with the financier at rate r, or use as a down-payment for the purchase of a

housing asset or invest in another risky asset.

Model: Ownership financing

The real return of the other risky asset ,

t

t

tt

ds

E

rr

..

0

~

Model: Ownership financing

Assume, mortgage rate

M

Mt

Mt

Mt

Mt

Mt

ds

E

rr

..

0

~

Model: Ownership financing

Factors determining the volume of mortgage debt (continuous equilibrium MDt = MSt= Mt) The price and volume of housing asset

transaction The borrowing constraint The opportunity costs of buying a housing asset Return volatilities.

Model: Ownership financing

dtdXX

0 QMMQ

0 PMM P.0 MM

.0 MMM M

0 LTVMM LTV

?M

rrMM M

tMMfttt LTVrrrQPMM ,,,,,,,

0

0

f

r

r

rMM

rMM

f

Model: Ownership financing

###:

0

0

tQtPt

t

MMf

QMPMMHence

VTL

rrr

tLTVM

rM

r

f

rtQtP

t

VTLMMM

rMrMrMQMPM

M

M

Mf

Model: Ownership financing

Implication: other things equal Mortgage debt moves in tandem with house price

and transaction volume Lower return and greater uncertainty in stock

market direct speculative money to housing market

Credit crunch is typically associated with a sudden drop in house price.

Model: The Investment Market for Housing Assets Assumption: constant rate of housing ownership, Bench marking case: perfect information, hence

absence of speculation, the new-born households purchase at the point of entry

or never all house owners sell at the point of exit. the transaction volume of housing asset, Qt, will be

constant over time, the house price, Pt, will fully reflect the value of the

housing services and grow in line with the rental price, Rt.

nQt

ttt RP

Model: The Investment Market for Housing Assets Case with information uncertainty, price

expectation, and speculation the new-born to-be-owners may delay buying the existing owners may sell before exit If a large enough number of owners expect a

reduction in future prices, the current price will fall with uncertain impact on the volume

If a large enough number of future owners believe buying now is more profitable than in the future, the current price will rise, again with uncertain impact on the volume

Model: The Investment Market for Housing Assets House price dynamics

Households optimization and continuous equilibrium imply

where te the time-t expected owner cost associated

each pound worth of housing capital

ttet RP

et

et

et

p

pr

Model: The Investment Market for Housing Assets

tTtttet WWHRPP ,1,0 ,...,,

Ptt

ett PP

Ptt ttt 3322

10 62

0,0,0 321

0PtE P

tPtds ..

PHPL

Pt ,

PH

PL

Model: The Investment Market for Housing Assets Probability of state unknown but inferable

using Markov chain: tttttktttt PP ;;Pr;,...,,Pr 121

HHHH

LLLL

HHHL

LHLL

1

1

Model: The Investment Market for Housing Assets

02

122

22

2321

22321

22321

JifttRP

JiftttRP

JtttP

P

tt

tt

et

t

Model: The Investment Market for Housing Assets Table 1 Price Growth and Price Volatility

Model: The Investment Market for Housing Assets One man’s gain is another man’s loss Trading in the housing asset market is a

zero-sum game!

Hence

and

001

0

1

0

T

ii

T

iit WthenHif

JttHRPP tHttt 32

###3 JPIRPP ltPHtt

3

Model: The Investment Market for Housing Assets The dynamics of transaction volumes

Continuous equilibrium: QDt = QSt = Qt

Housing supply Qt Housing stock Ht

Model: The Investment Market for Housing Assets

0

0

,,

,...,2,1,,

1

1

tH

ltltC

ttltltlt

tfttttltltlt

t

Hhh

CII

tcoefficienfixedthe

HPhPCI

TfHPEPhPECI

Q

Model: The Investment Market for Housing Assets

Model: The Investment Market for Housing Assets

Phase II IP, hP small &< 0 Phase III

IP, hP large &> 0

Phase I IP, hP large & > 0

Phase IV IP, hP small & > 0

Transaction volume

Price

Booming

Over-heating

Collapsing

Recovering

IPII, h P

II <0

IPI, hP

I>0

IPL & h P

L

IPIII, hP

III>0

IPIV, hP

IV>0

Model: The Investment Market for Housing Assets

###:

:

0:

:

1

1

11

1

ltPHtPltPt

ltHtPltPt

t

ltt

tHtPltPltCt

PIhPhPIQand

IhPhPIQhence

tCassume

IHnow

HhPhPICIQ

Model: Analysis of the Dynamic System

JQMPMM tQtPt

01 ltPHtPltPt PIhPhPIQ

JPIRPP ltPHtt 3 3

Model: Analysis of the Dynamic System Solution:

A column vector of particular integrals: the intertemporal equilibrium

A vector of complementary functions: the deviation of these variables from the vector of equilibrium values

PPP QPM

CCC QPM

Model: Analysis of the Dynamic System General solution:

thenIRif PH 042

MtAtAeM tt sincos 43

PtBtBeP tt sincos 43

QtCtCeQ tt sincos 43

Model: Analysis of the Dynamic System Intertemporal equilibrium are state-dependant due

to Ip & hp (each has four values) and

2

3

22 tIRt

JP

PH

2

32

22

2

tIRt

JtIhthIQ

PH

PHPP

2

32

22

2

tIRt

JttIhhIMMM

PH

PHPPQP

Model: Analysis of the Dynamic System

Phase II IP, hP small &< 0 Phase III

IP, hP large &> 0

Phase I IP, hP large & > 0

Phase IV IP, hP small & > 0

Transaction volume

Price

Booming

Over-heating

Collapsing

Recovering

IIIIII MQP ,,

III MQP ,,IIIIIIIII MQP ,,

IVIVIV MQP ,,

Model: Analysis of the Dynamic System Off-equilibrium cyclical movements are state-

dependant tAtAeM t

C sincos 43

tBtBeP tC sincos 43

tCtCeQ tC sincos 43

2

1 24

2

1 PH IR

Model: Analysis of the Dynamic System Stability?

Stable if the real root a < 0, i.e.

intuitively, a high general inflation rate would reduce the rate of real house price inflation hence the real growth rate of mortgage debt, making the system more sustainable.

Puzzle: only in Phase II can this condition be satisfied

r

042 PH IR

042 PH IR

Conclusion

The aim of the model is to capture The non-linearity and cyclicality of growths in price, volume

and mortgage debt The dynamic interactions among house price, housing

transaction volume, and mortgage debt; The non-linear relationship is modeled with state-

dependent coefficients; The solution to the model produce the desired

cyclical pattern, provided certain parameter restrictions are met;

Multiple channels of interactions established with sound economic reasoning.

Some puzzle though!

Future extension

Adding stochastic term to the differential equations

The value of which may not be great in terms of generating additional insight

Will be more valuable for studying REITS (traded daily and second by second).

Thank you!

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