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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!
Comments?