Introduction
Model
Solution
Empirical income
Robustness
Summary
Optimal Housing, Consumption, and
Investment Decisions over the
Life-Cycle
Holger Kraft1
1 Goethe
Claus Munk2
University Frankfurt, Germany
2 Aarhus
University, Denmark
Bachelier Finance Society
Toronto, June 2010
1 / 27
Introduction
Model
Solution
Empirical income
Robustness
Summary
Outline
1
Introduction
2
Model
3
Solution
4
Empirical income
5
Robustness
6
Summary
2 / 27
Introduction
Model
Solution
Empirical income
Robustness
Summary
Motivation
Labor income and housing decisions important for
most individuals
Some papers include labor income, some papers
housing decisions
The few papers including both aspects are restrictive
[Campbell/Cocco (QJE03), Cocco (RFS05), Yao/Zhang
(RFS05), Van Hemert (WP09)]
Difficult optimization problem – typically solved by
highly complex numerical methods
3 / 27
Introduction
Model
Solution
Empirical income
Robustness
Summary
This paper
Rich model: stochastic labor income, house price,
interest rate, stock price
Disconnect housing consumption and housing
investment
Closed-form “Excel-ready” solution
Model generates life-cycle behavior with many
realistic features
Non-negligible welfare gains from “perfect” house
price-linked financial contracts
4 / 27
Introduction
Model
Solution
Empirical income
Robustness
Summary
Outline
1
Introduction
2
Model
3
Solution
4
Empirical income
5
Robustness
6
Summary
5 / 27
Introduction
Model
Solution
Empirical income
Robustness
Summary
Financial assets
Short-term interest rate (= return on cash ):
drt = κ (r̄ − rt ) dt − σr dWrt
Price Bt = B(rt , t) of bond (20Y used later):
dBt
= (rt + λB σB (rt , t)) dt + σB (rt , t) dWrt
Bt
Stock price:
!
q
dSt
dWrt
= (rt + λS σS ) dt + σS ρSB , 1 − ρ2SB
dWSt
St
6 / 27
Introduction
Model
Solution
Empirical income
Robustness
Summary
Housing
“Unit” house price Ht : (unit ≈ 1 “average” sq. foot)


dWrt
dHt


= rt + λH σH − r imp dt + σH (ρHB , ρ̂HS , ρ̂H ) dWSt 
Ht
dWHt
Housing positions:
owning ϕot housing units
renting ϕrt units at rental rate νHt per unit
investing in REITs , ϕRt units, total return
dHt
Ht
+ ν dt
Housing consumption: ϕCt = ϕot + ϕrt
Housing investment: ϕIt = ϕot + ϕRt
7 / 27
Introduction
Model
Solution
Empirical income
Robustness
Summary
Labor income and wealth
Income rate Yt until retirement at T̃ :


dWrt
dYt


= (µ̄Y (t) + brt ) dt + σY (t) (ρYB , ρ̂YS , ρ̂Y ) dWSt 
Yt
dWHt
In retirement: Yt = ΥYT̃ , t ∈ [T̃ , T ].
Human wealth/capital:
Lt = EQ
t
Z
t
T
e−
Rs
t
(
Yt F (t, rt ), t < T̃ ,
ru du
Ys ds =
YT̃ F (t, rt ), t ∈ (T̃ , T ],
where F is known in closed form.
Financial/tangible wealth: Xt . Total wealth: Xt + Lt .
8 / 27
Introduction
Model
Solution
Empirical income
Robustness
Summary
The individual’s optimization problem
Z
J(t, X , r , H, Y ) = sup Et
T
e
−δ(u−t)
t
1 β 1−β 1−γ
c ϕ
ds
1 − γ u Cu
Choose:
ct perishable consumption rate
ϕCt housing units consumed
π̂It fraction of total wealth invested in house, π̂It =
Ht ϕIt
Xt +Lt
π̂Bt fraction of total wealth invested in bond
π̂St fraction of total wealth invested in stock
9 / 27
Introduction
Model
Solution
Empirical income
Robustness
Summary
Selected parameter values
Individual
Wealth
Risk aversion
Work life
Retirement
House
Exp. return
Volatility
Imputed rent
Rent
Unit price
20,000
4
30 Y
20 Y
1%
12%
5%
5%
250
Excess stock return
Income
Initial
Avg. growth
Volatility
Retirement
Correlations
income/stock,bond
house/stock
house/bond
income/house
5%
20,000
2%
7.5%
60%
0
0.5
0.65
0.57
10 / 27
Introduction
Model
Solution
Empirical income
Robustness
Summary
Outline
1
Introduction
2
Model
3
Solution
4
Empirical income
5
Robustness
6
Summary
11 / 27
Introduction
Model
Solution
Empirical income
Robustness
Summary
Solution to the HJB-equation...
J(t, X , r , H, Y ) =
1
g(t, r , H)γ (X + YF (t, r ))1−γ ,
1−γ
ην
Hk
g(t, r , H) =
1−β
Z
T
e−d1 (u−t)−β
γ−1
Bκ (u−t)r
γ
du,
t
βν
X + YF
Hk
,
1−β
g
X + YF
ϕC = ηH k−1
,
g
c=η
π̂S , π̂S , π̂S :
see below
12 / 27
Introduction
Model
Solution
Empirical income
Robustness
Summary
Investments – fractions of total wealth
σ Y ζS L
,
σS X + L
0 ↔ 33%
σY ζB
σr Fr
L
1 ξB
−
−
=
γ σB
σB
σB F X + L
−63% 0 ↔ 116% 0 ↔ −42%
Stocks
π̂S =
Bonds
π̂B
House
π̂I
=
1 ξS
γ σS
4%
1 ξI
γ σH
91%
speculative
−
σY ζI L
σH X + L
0 ↔ −109%
−
adjust for human wealth
Note: σY drops to zero at retirement, but L/(X + L) > 0
−
σr gr
,
σB g
49%
HgH
g
15%
+
hedge
jump
13 / 27
Introduction
Model
Solution
Empirical income
Robustness
Summary
Expected wealth over the life-cycle
800
Expected wealth (in thousands)
700
600
500
Total wealth
Financial wealth
400
Human wealth
300
200
100
0
0
5
10
15
20
25
30
35
40
45
50
Time, years
14 / 27
Introduction
Model
Solution
Empirical income
Robustness
Summary
Expected investments over the life-cycle
800
Expected investment (in thousands)
600
400
200
Bond
Stock
0
House
0
5
10
15
20
25
30
35
40
45
50
-200
-400
-600
Time, years
15 / 27
Introduction
Model
Solution
Empirical income
Robustness
Summary
... with age-dependent income volatility
1000
800
Expected investments (in thousands)
600
400
Stock, constant
200
House, constant
Bond, constant
Stock, decreasing
0
0
5
10
15
20
25
30
35
40
45
50
House, decreasing
Bond, decreasing
-200
-400
-600
-800
Time, years
16 / 27
Introduction
Model
Solution
Empirical income
Robustness
Summary
Housing consumption and investments
2500
2250
2000
Expected housing units
1750
1500
1250
Housing cons
1000
Housing inv
750
500
250
0
0
5
10
15
20
25
30
35
40
45
50
Time, years
17 / 27
Introduction
Model
Solution
Empirical income
Robustness
Summary
Outline
1
Introduction
2
Model
3
Solution
4
Empirical income
5
Robustness
6
Summary
18 / 27
Introduction
Model
Solution
Empirical income
Robustness
Summary
Empirical income profiles
Expected annual income (thousands)
50
45
40
35
30
no high
25
high
20
college
15
constant
10
5
0
25
35
45
55
65
75
85
Age, in years
19 / 27
Introduction
Model
Solution
Empirical income
Robustness
Summary
Expected investments again
1200
1000
Expected investment (in thousands)
800
Stock, no high
600
House, no high
Bond, no high
400
Stock, high
House, high
Bond, high
200
Stock, college
House, college
0
25
35
45
55
65
75
85
Bond, college
-200
-400
-600
Time (years)
20 / 27
Introduction
Model
Solution
Empirical income
Robustness
Summary
Housing consumption and investments again
3500
3000
Expected housing units
2500
Housing cons, no high
2000
Housing inv, no high
Housing cons, high
Housing inv, high
1500
Housing cons, college
Housing inv, college
1000
500
0
25
35
45
55
Time (years)
65
75
85
21 / 27
Introduction
Model
Solution
Empirical income
Robustness
Summary
Outline
1
Introduction
2
Model
3
Solution
4
Empirical income
5
Robustness
6
Summary
22 / 27
Introduction
Model
Solution
Empirical income
Robustness
Summary
Unspanned labor income
Our solution requires market completeness, i.e.,
spanned labor income
Labor income is much closer to being spanned when
housing assets are included – high income-house
price correlation
If labor income is unspanned, the implementation of
our consumption/investment strategy is sub-optimal
Bick, Kraft & Munk (presented Thursday): the welfare
loss is relatively small (magnitude ≤ 3%)
23 / 27
Introduction
Model
Solution
Empirical income
Robustness
Summary
Constant housing consumption
20%
18%
Wealth loss compared to full flexibility
16%
14%
12%
10%
8%
6%
4%
2%
0%
0
100
200
300
400
500
600
700
800
900
1000
Fixed level of housing units
Note: minimum certainty-equivalent wealth loss is only 0.24%
24 / 27
Introduction
Model
Solution
Empirical income
Robustness
Summary
Infrequent rebalancing of housing positions
Welfare loss
Adjustment frequency
2 years
5 years
Infrequent ϕC , frequent ϕI
0.03%
0.07%
Infrequent ϕI , frequent ϕC
0.43%
1.85%
Infrequent ϕC and ϕI
0.46%
1.96%
suggests moderate welfare gains from market for
REITs or CSI housing contracts
suggests moderate effects of housing transactions
costs
25 / 27
Introduction
Model
Solution
Empirical income
Robustness
Summary
Outline
1
Introduction
2
Model
3
Solution
4
Empirical income
5
Robustness
6
Summary
26 / 27
Introduction
Model
Solution
Empirical income
Robustness
Summary
Summary
Framework for consumption, housing, and investment
decisions over the life-cycle
High income/house correlation
life-cycle patterns
in optimal decisions, in particular housing investment
Calibrated model has many realistic features
Lots of comparative statics in the paper
Need to know more about typical life-cycle pattern in
income volatility and income/house price correlation
Our model is a benchmark for numerical solutions
with portfolio constraints and transaction costs
27 / 27