Download Dynamic portfolio and mortgage choice for homeowners

Dynamic portfolio and mortgage choice for homeowners
Dynamic portfolio and mortgage choice for
homeowners
Otto van Hemert, Frank de Jong
Joost Driessen
January 23th, 2006
0
Dynamic portfolio and mortgage choice for homeowners
Research Agenda
• For many investors, house is largest asset, and
mortgage largest liability
• Research questions
– How does optimal financial portfolio depend on
housing tenure and size?
– What mortgage type to finance your house?
– How to hedge house price/future housing cost risk?
– When to own, when to rent?
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Dynamic portfolio and mortgage choice for homeowners
Main findings this paper
• Unconstrained investor: closed-form solution
–
–
–
–
Mean-variance tangency portfolio
Portfolio hedging real interest rate (/inflation) risk
Portfolio hedging house price risk
Leverage financial positions to get desired risk
exposure total (financial + housing) wealth
– Weights depend on effective housing wealth
• Constrained investor with mortgage choice
–
–
–
–
2
ARM alleviates short-sale constraint on cash
FRM alleviates short-sale constraint on 20y bond
Risk-tolerant homeowner chooses ARM
Risk-averse homeowner chooses FRM (or hybrid)
Dynamic portfolio and mortgage choice for homeowners
Two complementing papers
• Dynamic portfolio and
mortgage choice for
homeowners
–
–
–
–
Utility from terminal wealth
Capitalised labor income
No tenure choice
Fixed house size
• Implicit closed-form
solution unrestricted case
enhancing intuition
restricted case
• Interpretation: retired or
wealthy investor
3
• Life-cycle housing and
portfolio choice with bond
markets
–
–
–
–
Full-fledged life-cycle model
Stochastic labor income
Choice renting/owning
House size choice
• Brute force solution with
aid supercomputer
• Builds on intuition acquired
in other paper
Dynamic portfolio and mortgage choice for homeowners
Related literature
• Brennan-Xia (2002,JF), Campbell-Viceira (2001,AER)
– Portfolio choice with bonds. No house, no labor income.
• Flavin-Yamashita (2002,AER)
– Static mean-variance setting. Little role for bonds, no advice
on mortgage choice
• Cocco (2005,RFS), Yao-Zhang (2005,RFS)
– Life-cycle model with house. Only cash and stocks as
financial assets. No mortgage choice.
• Campbell-Cocco (2003,QJE)
– Mortgage choice in life-cycle setup. No housing or portfolio
choice. No persistent real interest rate shocks.
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Dynamic portfolio and mortgage choice for homeowners
Investor’s objective
• At
t  0, T  the
investor solves:

max Et u wTF  x   wTH , H
x   A,
t  T
where utility is given by:
w
u w , H  

T
with wt  wtF  wtH
5

~
1 1
T H
1  ~

wT1

H
1 
Dynamic portfolio and mortgage choice for homeowners
Housing ratio
• We define h  wH /( wTF  wTH )
• We interpret financial wealth as including
capitalised labor income and maintenance
costs
• We typically think of housing to total
wealth ratio in order of magnitude of 0.2 to
0.4
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Dynamic portfolio and mortgage choice for homeowners
Price dynamics
Stocks: dS / S  R f   S S dt   S dzS
Real riskless rate: dr   r  r dt   r dzr
Expected inflation rate: d      dt    dz
imp

dt  ' dz
dQ
/
Q

R


'


r
House price:
f
Price level: d /   dt   ' dz  u dzu
Model extends Brennan and Xia (2002,JF)
with house price process
• Market imp. rent: corr. for housing services
•
•
•
•
•
•
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Dynamic portfolio and mortgage choice for homeowners
Th. II: cf sol. when no constraints
1  h  h  1  S  S  1   S  h  S
xS 
 1    

1  h   S
    S  1 h  S
• [..] is blend of the two Brennan-Xia portfolios.
1) mean-variance tangency portfolio
2) portfolio hedging real interest rate (/inflation) risk
•  S /  S originates from
3) portfolio hedging house price risk
• 1 / 1  h is leverage factor to obtain desired
exposure for total portfolio
• Investor acts as if house is worth only h because
– Adjustment for PV(market imputed rent until T)
– Take into account unhedgeable idiosyncratic house risk
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Dynamic portfolio and mortgage choice for homeowners
(h,t) for  =3
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Dynamic portfolio and mortgage choice for homeowners
Calibration asset price parameters
• Step 1: estimate term structure model
– 1973Q1-2003Q4 data on nominal interest
rates and inflation
– Kalman filter technique
• Step 2: determine correlations
– residuals step 1
– 1983Q1-2003Q4 data on stock and house
prices
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Dynamic portfolio and mortgage choice for homeowners
Implication bond price dynamics
• Nominal bond price dynamics:


dP / P  R f  B  r r  C     dt  B  r dzr  C    dz
• Half life dzr shocks: 1.1 years
• Half life dz shocks: 12.6 years
  5 years
  20 years
B 
1.48
C 
4.37
1.54
12.15
• 20-year bond has slightly larger exposure to  r dzr shocks
and much larger exposure to   dz shocks
• $1 in 5-year bond, -$4.37/12.15 in 20-year bond
real interest rate hedge without exp. inflation exposure

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Dynamic portfolio and mortgage choice for homeowners
Unconstrained portfolio choice ( =3)
h
Stocks
5y bond
T=5
20y bond
Cash
Stocks
5y bond
T=20
20y bond
Cash
0
0.35
6.73
-2.06
-4.02
0.35
6.77
-2.08
-4.04
0.2
0.42
8.54
-2.60
-5.36
0.40
8.26
-2.52
-5.14
0.4
0.52
11.10
-3.35
-7.27
0.46
9.81
-2.98
-6.29
• Leverage & effective wealth effect clearly visible
– In stock allocation; in allocation to different bonds
• 5y bond allocation positive for it has relative large
negative exposure to real interest rate risk
– Hedge against real int. rate risk; exploit risk premium
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Dynamic portfolio and mortgage choice for homeowners
Constrained portfolio choice ( =3)
h
Stocks
5y bond
T=5
20y bond
Cash
Stocks
5y bond
T=20
20y bond
Cash
0
0.35
0.61
0.04
0.00
0.35
0.61
0.04
0.00
0.2
0.42
0.38
0.20
0.00
0.40
0.45
0.15
0.00
0.4
0.51
0.07
0.41
0.00
0.45
0.28
0.27
0.00
• Leverage & effective wealth effect clearly visible
– In stock allocation; in duration of bond portfolio
• 20-year bond has slightly larger exposure to  r dzr
shocks and much larger exposure to   dz shocks
13
Dynamic portfolio and mortgage choice for homeowners
Constrained Portfolio choice
(T=20,  =3)
100%
90%
80%
Portfolio choice
70%
60%
20y bond
50%
5y bond
Stocks
40%
30%
20%
10%
0%
0
0.1
0.2
0.3
h
14
0.4
0.5
Dynamic portfolio and mortgage choice for homeowners
Mortgage choice
• Mortgage modeled as negative position in bond
– Valued at market price
– Costless rebalancing size and type
•
•
•
•
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Up to market value of the house
Adjustable-rate mortgage (ARM): -cash
Fixed-rate mortgage (FRM): -20y bond
Hybrid mortgage (hybrid): -cash and -20y bond
Dynamic portfolio and mortgage choice for homeowners
Portfolio choice with a mortgage ( =3)
Mortgage
Stocks
5y bond
T=20
20y bond
Cash
weq gain
no
0,45
0,28
0,27
0,00
h=0.4
FRM ARM hybrid
0,45
0,48
0,48
0,28
1,18
1,18
0,27
0,00
0,00
0,00 -0,67 -0,67
0,00% 6,46% 6,46%
• Desire to leverage risk exposure
• Cash constraint binding
• ARM utility enhancing
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Dynamic portfolio and mortgage choice for homeowners
Portfolio choice with a mortgage ( =7)
Mortgage
Stocks
5y bond
T=20
20y bond
Cash
weq gain
•
•
•
•
17
no
0,23
0,71
0,00
0,06
h=0.4
FRM ARM hybrid
0,16
0,23
0,19
1,06
0,72
1,48
-0,22
0,00 -0,33
0,00
0,05 -0,34
3,26% 0,11% 5,73%
5y bond to hedge real interest rate
20y bond constraint binding
FRM utility enhancing
Hybrid mortgage even better!!!
Dynamic portfolio and mortgage choice for homeowners
Main findings
• Unconstrained investor: closed-form solution
–
–
–
–
Mean-variance tangency portfolio
Portfolio hedging real interest rate (/inflation) risk
Portfolio hedging house price risk
Leverage financial positions to get desired risk
exposure total (financial + housing) wealth
– Weights depend on effective housing wealth
• Constrained investor
–
–
–
–
18
ARM alleviates short-sale constraint on cash
FRM alleviates short-sale constraint on 20y bond
Risk-tolerant homeowner chooses ARM
Risk-averse homeowner chooses FRM (or hybrid)