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?
1
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.
4
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
6
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
•
•
•
•
•
•
7
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
8
Dynamic portfolio and mortgage choice for homeowners
(h,t) for  =3
9
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
10
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

11
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
12
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
•
•
•
•
15
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
16
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)