optimal housing, consumption, and investment decisions ... · investment decisions over the...
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Introduction Model Full flexibility Comparative statics Limited flexibility
Optimal Housing, Consumption, and
Investment Decisions over the Life-Cycle
Holger Kraft1 Claus Munk2
1Goethe University Frankfurt, Germany
2Aarhus University, Denmark
Management Science, vol. 57(6), pp. 1025–1041, 2011
AARHUS UNIVERSITY AU
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Introduction Model Full flexibility Comparative statics Limited flexibility
Outline
1 Introduction
2 Model
3 Full flexibility
4 Comparative statics
5 Limited flexibility
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Introduction Model Full flexibility Comparative statics Limited flexibility
Overview and motivation
Important features in life-cycle decisions of individuals:
interest rate risk(Sørensen JFQA99; Campbell/Viceira AER01; Munk/Sørensen JBF04)
risky labor income(Bodie/Merton/Samuelson JEDC92; Cocco/Gomes/Maenhout RFS05; Munk/Sørensen
JFE-forthcoming)
housing decisions(Cocco RFS05; Yao/Zhang RFS05; Van Hemert wp08)
Existing papers with income and housing:
coarse and computationally intensive numerical solutiontechniques with unknown precisionprovide limited understanding of economic forces at play
This paper : explicit, “Excel-ready” solutions
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Introduction Model Full flexibility Comparative statics Limited flexibility
Financial assets
Available assets: cash, bond(s), stock
Short-term interest rate (= return on cash ):
drt = κ (r − rt ) dt − σr dWrt
Price Bt = B(rt , t) of an arbitrary bond :
dBt = Bt [(rt + λBσB(rt , t)) dt + σB(rt , t) dWrt ]
Zero-coupon bond: But = exp{−a(u − t)− Bκ(u − t)rt}, where
Bκ(τ) = 1κ (1− e−κτ ).
Stock price:
dSt = St
[(rt + λSσS) dt + σS
(ρSB dWrt +
√1− ρ2
SB dWSt
)]
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Introduction Model Full flexibility Comparative statics Limited flexibility
Housing
“Unit” house price Ht :
dHt
Ht=(
rt + λHσH − r imp)
dt + σH (ρHB dWrt + ρHS dWSt + ρH dWHt )
Housing positions:
owning ϕot housing units
renting ϕrt units at continuous rental rate per unit is νHt
investing in REITs , ϕRt units, with total return dHtHt
+ ν dt
Housing consumption: ϕCt = ϕot + ϕrt
Housing investment: ϕIt = ϕot + ϕRt
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Introduction Model Full flexibility Comparative statics Limited flexibility
Labor incomeIncome rate Yt until retirement at T :
dYt
Yt= (µY (t) + brt ) dt + σY (t) (ρYB dWrt + ρYS dWSt + ρY dWHt )
In retirement: Yt = ΥYT , t ∈ (T ,T ].
Human wealth/capital:The human capital is
Lt ≡ EQt
[∫ T
te−
∫ st ru duYs ds
]=
YtF (t , rt ), t < T ,
YT F (t , rt ), t ∈ [T ,T ],
where
F (t , r) =
∫ T
t e−A(t,s)−(1−b)Bκ(s−t)r ds + Υ∫ T
T e−A(t,s)−(Bκ(s−t)−bBκ(T−t))r ds, t < T ,
Υ∫ T
t e−a(s−t)−Bκ(s−t)r ds, t ≥ T
Here A(t , s) and A(t ,S) are deterministic functions stated in Appendix A.
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Introduction Model Full flexibility Comparative statics Limited flexibility
Wealth
Financial/tangible wealth: Xt
Total wealth: Xt + Lt
dXt = πStXtdSt
St+ πBtXt
dBt
Bt+ [Xt (1− πSt − πBt )− (ϕot + ϕRt )Ht ] rt dt
+ ϕot dHt + ϕRt (dHt + νHt dt)− ϕrtνHt dt − ct dt + Yt dt
= [Xt (rt + πStλSσS + πBtλBσBt ) + ϕItλ′HσHHt − ϕCtνHt − ct + Yt ] dt
+ (πStXtρSBσS + πBtXtσBt + ϕItHtρHBσH) dWrt
+
(πStXtσS
√1− ρ2
SB + ϕItHt ρHSσH
)dWSt + ϕItHt ρHσH dWHt ,
whereϕCt ≡ ϕot + ϕrt , ϕIt ≡ ϕot + ϕRt .
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Introduction Model Full flexibility Comparative statics Limited flexibility
The individual’s utility maximization problem
J(t , x , r ,h, y) = sup(c,ϕC ,ϕI ,πB ,πS)∈At
Et
[ ∫ T
te−δ(u−t) 1
1− γ
(cβu ϕ
1−βCu
)1−γds
+ εe−δ(T−t) 11− γ
X 1−γT
]
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Introduction Model Full flexibility Comparative statics Limited flexibility
Parameter values used in illustrations
Preferences and wealth: X0 = USD 20,000; δ = 0.03; γ = 4;β = 0.8; ε = 0; T = 30; T = 50
Interest rate and bond: κ = 0.2; r0 = r = 0.02; σr = 0.015;λB = 0.1; Tbond = 20 (running); σB = 0.0736
Stock: σS = 0.2, λS = 0.25, ρSB = 0
Housing: σH = 0.12; λH = 0.325; r imp = ν = 0.05 (soµH = 0.9%); ρHB = 0.65; ρHS = 0.5; H0 = USD 250 per “averagestandard” sq. footREITS drift 0.9%+5%=5.9%, low vol attractive investment
Labor income: Y0 = USD 20,000; b = 0.5; µY (t) = 0.01;σY (t) = 0.075; ρYB = −0.3; ρYS = 0; ρHY = 0.3509 (ensuresspanning)
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Introduction Model Full flexibility Comparative statics Limited flexibility
Solution to the HJB-equation...
J(t , x , r , h, y) =1
1− γ g(t , r , h)γ(x + yF (t , r))1−γ ,
g(t , r , h) = ε1γ e−Dγ (T−t)− γ−1
γBκ(T−t)r +
ην
1− β hk∫ T
te−d1(u−t)−β γ−1
γBκ(u−t)r du.
Here k = (1− β)(1− 1/γ), η = β1/γ(βν
1−β
)k−1, and Dγ(τ) and d1(τ) are
stated in the appendix.
πS ≡πSx
x + yF=
1γ
ξS
σS− σY (t)ζS
σS
yFx + yF
,
πB ≡πBx
x + yF=
1γ
ξB
σB−(σY (t)ζB
σB
yFx + yF
− σr
σB
yFx + yF
Fr
F
)− σr
σB
gr
g,
πI ≡hϕI
x + yF=
1γ
ξI
σH− σY (t)ζI
σH
yFx + yF
+hgh
g
c = ηβν
1− β hk x + yFg
, ϕC = ηhk−1 x + yFg
.
(when t ∈ [T ,T ]: σY (t) = 0 and y is to be replaced by YT )
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Introduction Model Full flexibility Comparative statics Limited flexibility
Expected consumption over the life-cycle
ct = ηβν
1− βHkt
Xt + YtF (t , rt )
g(t , rt ,Ht )νHt ϕCt =
1− ββ
ct
0
100
200
300
400
500
600
700
800
900
0
10
20
30
40
50
60
0 10 20 30 40 50
Exp
ect
ed
ho
usi
ng
un
its
Exp
ect
ed
co
nsu
mp
tio
n (
in t
ho
usa
nd
s)
Time, years
Perishable
House expend
House units
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Introduction Model Full flexibility Comparative statics Limited flexibility
Optimal investments – fractions of total wealth
Stocks πS =1γ
ξS
σS−σY ζS
σS
yFx + yF
,
5.5% 0↔ 31%
Bonds πB =1γ
ξB
σB−(σY ζB
σB− σr
σB
Fr
F
)yF
x + yF− σr
σB
gr
g,
−57% 0↔ 141% 0↔ −43% 49%
House πI =1γ
ξI
σH−σY ζI
σH
yFx + yF
+hgh
g86% 0↔ −104% 15%
speculative adjust for human wealth hedge
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Introduction Model Full flexibility Comparative statics Limited flexibility
Optimal investments and the composition of wealth
-60%
-40%
-20%
0%
20%
40%
60%
80%
100%
120%
0.0 0.2 0.4 0.6 0.8 1.0
Inve
stm
en
t /
Tota
l we
alth
Human wealth / Total wealth
Bond
Stock
House
Bond (retired)
Stock (retired)
House (retired)
Assuming fixed r , h and fixed FrF ,
grg (vary little over life anyway)
Main life-cycle effect is change in ratio of human-to-total wealth
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Introduction Model Full flexibility Comparative statics Limited flexibility
Expected wealth over the life-cycle
0
100
200
300
400
500
600
700
800
0 10 20 30 40 50
Exp
ect
ed
we
alth
(in
th
ou
san
ds)
Time, years
Total wealth
Financial wealth
Human wealth
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Introduction Model Full flexibility Comparative statics Limited flexibility
Expected investments over the life-cycle
-400
-200
0
200
400
600
800
0 10 20 30 40 50
Exp
ect
ed
inve
stm
en
t (i
n t
ho
usa
nd
s)
Time, years
Bond
Stock
House
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Introduction Model Full flexibility Comparative statics Limited flexibility
Housing consumption and over the life-cycle
0
200
400
600
800
1000
1200
1400
1600
1800
2000
2200
2400
0 10 20 30 40 50
Exp
ect
ed
ho
usi
ng
un
its
Time, years
Housing cons
Housing inv
Housing hedge
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Introduction Model Full flexibility Comparative statics Limited flexibility
Robustness of results
effects of γ, ε, H0, σH , Y0, σY , Υ, ρHY : see paper and appendix
here: effects of empirically estimated life-cycle income profiles,cf. Cocco, Gomes & Maenhout (RFS 2005)
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Introduction Model Full flexibility Comparative statics Limited flexibility
Expected income over the life-cycle for three
educational groups
0
5
10
15
20
25
30
35
40
45
50
25 35 45 55 65 75 85
Exp
ect
ed
an
nu
al in
com
e (
tho
usa
nd
s)
Age, in years
no high
high
college
constant
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Introduction Model Full flexibility Comparative statics Limited flexibility
Expected wealth over the life-cycle for three
educational groups
0
200
400
600
800
1000
1200
25 35 45 55 65 75 85
Exp
ect
ed
we
alth
(th
ou
san
ds)
Age, in years
Human, no high
Total, no high
Financial, no high
Human, high
Total, high
Financial, high
Human, college
Total, college
Financial,college
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Introduction Model Full flexibility Comparative statics Limited flexibility
Expected investments over the life-cycle for three
educational groups
-600
-400
-200
0
200
400
600
800
1000
1200
25 35 45 55 65 75 85
Exp
ect
ed
inve
stm
en
t (i
n t
ho
usa
nd
s)
Age, in years
Stock, no high
House, no high
Bond, no high
Stock, high
House, high
Bond, high
Stock, college
House, college
Bond, college
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Introduction Model Full flexibility Comparative statics Limited flexibility
Consumption of and investment in housing over the
life-cycle for three educational groups
-500
0
500
1000
1500
2000
2500
3000
3500
25 35 45 55 65 75 85
Exp
ect
ed
ho
usi
ng
un
its
Age, in years
Cons, no high
Inv, no high
Cons, high
Inv, high
Cons, college
Inv, college
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Introduction Model Full flexibility Comparative statics Limited flexibility
Limited flexibility in housing decisions
Scenarios considered:
1 constant number of units of housing consumed (closed-formsolution)
2 infrequent adjustments of housing investment and housingconsumption (MC results)
Percentage wealth-equivalent utility loss L due to limited flexibility:
J (t , x [1− L], r ,h, y [1− L]) = Jlimited(t , x , r ,h, y)
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Introduction Model Full flexibility Comparative statics Limited flexibility
Utility loss of constant housing consumption
8%
10%
12%
14%
16%
18%
20%
We
alt
h lo
ss c
om
pa
red
to
fu
ll f
lex
ibil
ity
0%
2%
4%
6%
0 100 200 300 400 500 600 700 800 900 1000
We
alt
h lo
ss c
om
pa
red
to
fu
ll f
lex
ibil
ity
Fixed level of housing units
Note: minimum loss is 0.26% for a constant housing consumption (USD 1,550 out of total wealth USD 596,400).
The individual can almost completely compensate for inflexibility in housing consumption by adjusting perishable consumption and
investments
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Introduction Model Full flexibility Comparative statics Limited flexibility
Loss due to infrequent housing adjustmentsSimulate BM’s and thus wealth, income (10,000 paths, 250 steps/year)When adjusted, use policies optimal with continuous adjustments
Initial income 10, 000 Initial income 20, 000
Adjustment frequency 2 years 5 years 2 years 5 years
Infrequent ϕC , frequent ϕI 0.03% 0.08% 0.03% 0.08%
Infrequent ϕI , frequent ϕC 0.21% 1.35% 0.23% 1.43%
Infrequent ϕC and ϕI 0.24% 1.52% 0.26% 1.61%
suggests moderate welfare effects of a well-functioning market forREITs or CSI housing contracts
suggests moderate effects of housing transactions costs
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Introduction General market setting Mean reversion in stock prices Stochastic interest rates Conclusion
Portfolio and Consumption Choice withStochastic Investment Opportunities and
Habit Formation in Preferences(Journal of Economic Dynamics and Control, 2008)
Claus Munk
August 2012
AARHUS UNIVERSITY AU
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Introduction General market setting Mean reversion in stock prices Stochastic interest rates Conclusion
Outline
1 Introduction
2 General market setting
3 Mean reversion in stock prices
4 Stochastic interest rates
5 Conclusion
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Introduction General market setting Mean reversion in stock prices Stochastic interest rates Conclusion
Motivation
Investment opportunities vary stochastically over timeI Many papers on optimal portfolio and consumption choice with
stochastic investment opportunitiesI Almost all assume utility of terminal wealth, E[u(WT )], or
time-separable utility of consumption, U(c) = E[∫ T
0 e−δtu(ct ) dt].
Many economists argue that individuals develop habits forconsumption.Preferences with habit formation help explain asset pricingpuzzles (Sundaresan 1989, Constantinides 1990, Campbell andCochrane 1999).Earlier studies of individuals’ optimal portfolio and consumptionchoice under habit formation assume constant investmentopportunities.
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Introduction General market setting Mean reversion in stock prices Stochastic interest rates Conclusion
Objectives
Derive optimal portfolio and consumption choice for individualswith habit formation in markets with stochastic investmentopportunities.Consider both general dynamics of investment opportunities andconcrete, special cases.Do stochastic investment opportunities affect the decisions ofindividuals with habit formation differently than individuals withtime-additive preferences?Are the effects of habit formation in preferences on optimaldecisions different in markets with stochastic investmentopportunities than in markets with constant investmentopportunities?
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Introduction General market setting Mean reversion in stock prices Stochastic interest rates Conclusion
Literature review
time-additive utility habit formation utilityconstant Samuelson (1969) Sundaresan (1989)inv.opp. Merton (1969, 1971) Constantinides (1990)
Ingersoll (1992)
general Merton (1971, 1973) Detemple and Zapatero (1992)stochastic Karatzas-Lehoczky-Shreve (1987) Detemple and Karatzas (2003)inv.opp. Cox and Huang (1989, 1991) Schroder and Skiadas (2002)
Liu (2007) THIS PAPERMunk and Sørensen (2004)
concrete Sørensen (1999) THIS PAPERstochastic Brennan and Xia (2000)inv.opp. Kim and Omberg (1996)
Wachter (2002)
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Introduction General market setting Mean reversion in stock prices Stochastic interest rates Conclusion
The financial market
Basic uncertainty modeled by n-dimensional SBM z = (zt )
“Bank account”: rate of return rt
n risky assets: dPt = diag(Pt ) [(rt1 + σtλt ) dt + σt dzt ]
Assume σt non-singularComplete market – unique state-price deflator:
ξt = exp
{−∫ t
0rs ds −
∫ t
0λ>
s dzs −12
∫ t
0‖λs‖2 ds
}
(and unique risk-neutral probability measure)
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Introduction General market setting Mean reversion in stock prices Stochastic interest rates Conclusion
The individual...
... wants to maximize
U(c) = E
[∫ T
0e−δt 1
1− γ(ct − ht )
1−γ dt
],
where
ht = h0e−βt + α
∫ t
0e−β(t−s)cs ds.
Note: requires ct ≥ ht for all t [not necessarily very restrictive].... chooses a consumption strategy c = (ct ) and a portfoliostrategy π = (πt ) under budget constraint Et
[∫ Ttξsξt
cs ds]≤Wt .
... receives no labor income.Wealth dynamics: dWt = [Wt (rt + π>
t σtλt )− ct ] dt + Wtπ>t σt dzt .
Indirect utility:Jt = sup(c,π)∈A(t) Et
[∫ Tt e−δ(s−t) 1
1−γ (cs − hs)1−γ ds].
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Introduction General market setting Mean reversion in stock prices Stochastic interest rates Conclusion
Auxiliary processesDefine the processes F = (Ft ) and G = (Gt ) by
Ft = Et
[∫ T
te−(β−α)(s−t) ξs
ξtds
]=
∫ T
te−(β−α)(s−t)Bs
t ds,
Gt = Et
[∫ T
te−
δγ (s−t)
(ξs
ξt
)1− 1γ
(1 + αFs)1− 1γ ds
].
Note: htFt is the cost of ensuring that future consumption equals thehabit level (cs = hs for all s ≥ t implies hs = hte−(β−α)(s−t)).Must assume W0 ≥ h0F0.
Dynamics:
dFt = −1 dt+Ft[(
rt + σ>Ftλt
)dt + σ>
Ft dzt], where σFt ≡
∫ Tt e−(β−α)(s−t)Bs
t σst ds∫ T
t e−(β−α)(s−t)Bst ds
.
dGt = Gt[µGt dt + σ>
Gt dzt]
for some µG, σG.
Note: σF and σG depend on the precise asset price dynamics.
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Introduction General market setting Mean reversion in stock prices Stochastic interest rates Conclusion
Theorem 1
Under two regularity conditions, the solution is as follows.Optimal consumption: c∗t = h∗t + (1 + αFt )
− 1γ
W∗t −h∗
t FtGt
.
Indirect utility: Jt = 11−γGγ
t (W ∗t − h∗t Ft )
1−γ.
Optimal portfolio:
π∗t =W ∗
t − h∗t Ft
W ∗t
1γ
(σ>t )−1λt︸ ︷︷ ︸
speculative
+W ∗
t − h∗t Ft
W ∗t
(σ>t )−1σGt︸ ︷︷ ︸
hedge
+h∗t Ft
W ∗t
(σ>t )−1σFt︸ ︷︷ ︸
habit “insurance”
.
Note: Both the speculative term and the hedge term are dampeneddue to habit formation. Additional effect on hedge term via G andhence σG.Jointly allowing for habit and stochastic IO’s has effects beyond theirseparate effects! Quantitatively important?
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Introduction General market setting Mean reversion in stock prices Stochastic interest rates Conclusion
Markovian asset pricesAssume rt = r(xt ) and λt = λ(xt ), where
dxt = m(xt ) dt + v(xt ) dzt .
Then Bst = Bs(xt , t), where Bs(x , t) solves
12
tr(∂2Bs
∂x2 v(x)v(x)>)
+ (m(x)− v(x)λ(x))>∂Bs
∂x+∂Bs
∂t− r(x)Bs = 0
with Bs(x , s) = 1, and σFt becomes
σFt =
∫ Tt e−(β−α)(s−t) ∂Bs
∂x (xt , t) ds∫ Tt e−(β−α)(s−t)Bs(xt , t) ds
.
Moreover, Gt = G(xt , t) and σGt = v(xt )> ∂G∂x (xt , t)/G(xt , t). G(x , t) will satisfy
∂G∂t
(x , t)+
(m(x)−
(1− 1
γ
)v(x)λ(x)
)>∂G∂x
(x , t)+12
tr(∂2G∂x2 (x , t)v(x)v(x)>
)+ (1 + αF (x , t))1− 1
γ =
(δ
γ+
(1− 1
γ
)r(x) +
‖λ(x)‖2
2γ
(1− 1
γ
))G(x , t)
with G(x ,T ) = 0.
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Introduction General market setting Mean reversion in stock prices Stochastic interest rates Conclusion
Constant investment opportunities: solution
If both r and λ are constant, then
F (t) =
∫ T
te−(r+β−α)(s−t) ds =
1r + β − α
(1− e−(r+β−α)(T−t)
)G(t) =
∫ T
te−( δ
γ +[1− 1γ ]r+ 1
2γ [1− 1γ ]‖λ‖2)(s−t) (1 + αF (s))1−1/γ ds
and σF = σG = 0.
Optimal consumption: c∗t = h∗t + (1 + αF (t))−1/γ W∗t −h∗
t F (t)G(t) .
Optimal portfolio: π∗t = 1γ (σ>
t )−1λ
W∗t −h∗
t F (t)W∗
t.
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Introduction General market setting Mean reversion in stock prices Stochastic interest rates Conclusion
Constant investment opportunities: example
Parameters: r = 0.03, σ = 0.2, λ = 0.3; T = 30, γ = 2, δ = 0.02, W = 100,h = 4.
habit formation dampens risky investment substantially
optimal stock weight decreases with length of investment horizon (forgiven W , h...): long horizon⇒ put more money aside to cover habit
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Introduction General market setting Mean reversion in stock prices Stochastic interest rates Conclusion
The financial market
constant interest rate rstock index: dPt = Pt [(r + σλt ) dt + σ dzt ], constant σmarket price of risk: dλt = κ
[λ− λt
]dt − σλ dzt
Note: perfect negative correlation – empirically not thatunreasonableWachter (2002) derives an explicit solution for time-additivepower utility of consumptionKim & Omberg (1996) derives an explicit solution for power utilityof terminal wealth, allowing for non-perfect correlation(incomplete market)
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Introduction General market setting Mean reversion in stock prices Stochastic interest rates Conclusion
Theorem 2Suppose that κ2 >
(1− 1
γ
)(σ2λ
(1− 1
γ
)+ 2κσλ
).
Indirect utility: J(W , h, λ, t) = 11−γG(λ, t)γ (W − hF (t))1−γ , where
F (t) =1
r + β − α
(1− e−(r+β−α)(T−t)
)G(λ, t) =
∫ T
t(1 + αF (s))1− 1
γ eg0(s−t)+g1(s−t)λ+ 12 g2(s−t)λ2
ds
and g0, g1, g2 are known in closed-form.Optimal strategies:
C(W , h, λ, t) = h + (1 + αF (t))−1γ
W − hF (t)G(λ, t)
,
Π(W , h, λ, t) =λ
γσ
W − hF (t)W
− σλσ
D(λ, t)W − hF (t)
W,
D(λ, t) =
∫ Tt (g1(s − t) + g2(s − t)λ) (1 + αF (s))1− 1
γ eg0(s−t)+g1(s−t)λ+ 12 g2(s−t)λ2
ds∫ Tt (1 + αF (s))1− 1
γ eg0(s−t)+g1(s−t)λ+ 12 g2(s−t)λ2 ds
.
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Introduction General market setting Mean reversion in stock prices Stochastic interest rates Conclusion
Properties of the solution
Theorem 3: Suppose γ > 1 and λt > 0. Then1 the hedge demand for the stock is positive,2 both the average and the marginal consumption/wealth ratio
increase with λ,3 the optimal fraction of free wealth invested in the stock,π ≡ πW
W−hF , increases with the horizon T .
Results as in Wachter (with π = π in (iii)...).Since F is increasing in T , π = π(1− hF/W ) may be decreasingin T – contrary to common advice.Typically, middle-aged investors will start to dissave (reducewealth) and increase consumption growth and habit hF/Wwill tend to increase and π decrease over time. Younger investorstend to build up wealth π may increase over time in thebeginning of life.
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Introduction General market setting Mean reversion in stock prices Stochastic interest rates Conclusion
Numerical experiment
Parameters: t = 0, T = 30, δ = 0.02, γ = 2, W = 100, r = 0.03,σ = 0.2, κ = 0.3, σλ = 0.1, λ = 0.3.State variables: λ = λ (expected excess return 6%), habit level h = 4.
α β F (t) D(λ, t) G(λ, t) πmyo πhedπhedπmyo
no habit — -0.2568 16.1 75.0% 12.8% 17.1%0.1 0.2 7.54 -0.2546 20.6 52.4% 8.9% 17.0%0.1 0.3 4.34 -0.2559 19.0 62.0% 10.6% 17.1%0.2 0.3 7.54 -0.2536 24.2 52.4% 8.9% 16.9%0.1 0.4 3.03 -0.2564 18.3 65.9% 11.3% 17.1%0.2 0.4 4.34 -0.2554 21.6 62.0% 10.6% 17.0%0.3 0.4 7.54 -0.2530 27.4 52.4% 8.8% 16.9%
NOTE: Hedge demand for stocks is dampened slightly more thanmyopic demand.
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Introduction General market setting Mean reversion in stock prices Stochastic interest rates Conclusion
NOTE: π is decreasing in the horizon (at least up to T ≈ 35 yrs)⇒ Horizon-effect due to habit beats horizon-effect due to mean-reversion
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Introduction General market setting Mean reversion in stock prices Stochastic interest rates Conclusion
The financial marketCIR model: drt = κ (r − rt ) dt − σr
√rt dz1t with λ1t = λ1
√rt/σr .
Zero-coupon bonds: Bst = Bs(rt , t) ≡ e−a(s−t)−b(s−t)rt
with κ = κ− λ1, ν =√κ2 + 2σ2
r , and
b(τ) =2(eντ − 1)
(ν + κ)(eντ − 1) + 2ν,
a(τ) = −2κrσ2
r
(12
(κ+ ν)τ + ln2ν
(ν + κ)(eντ − 1) + 2ν
),
WLOG assume trade in bank and T -zero (weight πB) withdynamics:
dBTt = BT
t[(rt + b(T − t)λ1rt ) dt + b(T − t)σr
√rt dz1t
].
Single stock (weight πS):
dSt = St
[(rt + σSψ(rt )) dt + σS
{ρdz1t +
√1− ρ2 dz2t
}]with ψ(r) = ρλ1
σr
√r +
√1− ρ2λ2.
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Introduction General market setting Mean reversion in stock prices Stochastic interest rates Conclusion
Theorem 4Indirect utility: J(W , h, r , t) = 1
1−γG(r , t)γ (W − hF (r , t))1−γ with
F (r , t) =∫ T
t e−(β−α)(s−t)−a(s−t)−b(s−t)r ds and G(r , t) solves the PDE
∂G∂t
+
(κr −
[κ−
(1− 1
γ
)λ1
]r)∂G∂r
+12σ2
r r∂2G∂r 2
+ (1 + αF (r , t))1− 1γ =
(δ
γ+
12γ
(1− 1
γ
)(λ2
1
σ2r
r + λ22
)+
(1− 1
γ
)r)
G
with the terminal condition G(r ,T ) = 0. Optimal strategies:
C(W , h, r , t) = h + (1 + αF (r , t))−1γ
W − hF (r , t)G(r , t)
,
ΠB(W , h, r , t) =W − hF (r , t)
W1
b(T − t)
[1γσr
(λ1
σr− ρλ2√
1− ρ2√
r
)−
∂G∂r (r , t)G(r , t)
]
+hW
1b(T − t)
∫ T
tb(s − t)e−(β−α)(s−t)−a(s−t)−b(s−t)r ds
ΠS(W , h, r , t) =W − hF (r , t)
Wλ2
γσS
√1− ρ2
.
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Introduction General market setting Mean reversion in stock prices Stochastic interest rates Conclusion
Properties
Hedge bond demand positive for γ > 1.Both myopic and hedge bond demand dampened due to habitformation.Additional habit effect on hedge demand through G.“Habit insurance” bond demand is positive – ensure c ≥ h by adynamic portfolio in the bond and the bank account.Bond/stock ratio different with habit formation than without.
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Introduction General market setting Mean reversion in stock prices Stochastic interest rates Conclusion
Explicit solution with time-additive power utility
G(r , t) =
∫ T
te−a(s−t)−
(1− 1
γ
)b(s−t)r ds, b(τ) =
2(
1 +λ2
12γσ2
r
) (eντ − 1
)(ν + κ) (eντ − 1) + 2ν
,
a(τ) =δ
γτ +
12γ
(1− 1
γ
)λ2
2τ −2κrσ2
r
(12
(ν + κ) τ + ln2ν
(ν + κ) (eντ − 1) + 2ν
),
where κ = κ+ 1−γγλ1, ν =
√κ2 + 2σ2
r
(1− 1
γ
)(1 +
λ21
2γσ2r
).
Optimal investment strategy:
ΠB(r , t) =1γ
1σr b(T − t)
(λ1
σr− ρλ2√
1− ρ2√
r
)
+
(1− 1
γ
) ∫ Tt b(s − t)e−a(s−t)−(1− 1
γ)b(s−t)r ds
b(T − t)∫ T
t e−a(s−t)−(1− 1γ
)b(s−t)r ds,
ΠS =λ2
γσS
√1− ρ2
,
cf. Liu (1999,2007); Grasselli (2003) with utility of terminal wealth only.
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Introduction General market setting Mean reversion in stock prices Stochastic interest rates Conclusion
Numerical experiment(applies a Crank-Nicholson finite difference scheme)Parameters: t = 0, T = 30, δ = 0.02, γ = 2, W = 100, σ = 0.2,κ = 0.3, r = 0.05, σr = 0.1, ρ = 0.25, λ1 = 0.05, λ2 = 0.3.State variables: r = r , h = 4.
α β F G πS πBmyo πB
hed πBins πB
no habit — 15.7 77.5% 20.6% 40.7% 0.0% 61.4%0.1 0.2 6.42 19.6 57.6% 15.3% 31.9% 16.4% 63.6%0.1 0.3 3.94 22.8 65.2% 17.4% 36.7% 8.2% 62.3%0.2 0.3 6.42 28.1 57.6% 15.3% 33.5% 16.4% 65.2%0.1 0.4 2.83 24.1 68.7% 18.3% 38.9% 4.9% 62.1%0.2 0.4 3.94 26.2 65.2% 17.4% 37.3% 8.2% 62.9%0.3 0.4 6.42 31.6 57.6% 15.3% 33.8% 16.4% 65.5%
Hedge demand for bond not dampened quite as much as myopicdemand.
Significant “habit insurance” bond demand.
Bond/stock ratio is very different with habit formation.
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Introduction General market setting Mean reversion in stock prices Stochastic interest rates Conclusion
NOTE: Total bond demand relatively insensitive to T .
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Introduction General market setting Mean reversion in stock prices Stochastic interest rates Conclusion
Summary
Exact characterization of optimal strategies in general setting.Detailed analysis in two important special settings (the paperalso considers a combination of the two).No quantitatively surprising and/or dramatic effects of combininghabit formation in preferences and stochastic investmentopportunities.Main effect of habits: some assets (bonds, cash) are better thanothers (stocks) at ensuring that future consumption will not fallbelow the habit level
Possible extensions:Other habit-type utility functions.Different habits for different consumption goods.
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The set-up The case EIS equals 1 The case EIS different from 1 An example
Dynamic Asset AllocationEpstein-Zin recursive utility
Claus Munk
Aarhus University
August 2012
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The set-up The case EIS equals 1 The case EIS different from 1 An example
Motivation
Standard time-additive power utility imposes a close relationbetween the aversion against consumption gambles over statesand the aversion against substituting consumption over timeNo reason to believe in that relation?With Epstein-Zin preferences, the risk aversion and the elasticityof intertemporal substitution can be disentangledDiscrete-time portfolio application: Campbell & Viceira (QJE1999), Campbell et al. (EFR 2001)Continuous-time portfolio application: Chacko & Viceira (RFS2005)Often used for an infinite time horizonPlays an important role in the “long-run risk explanation” of theequity premium puzzle, cf. Bansal & Yaron (JF 2004)
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The set-up The case EIS equals 1 The case EIS different from 1 An example
Outline
1 The set-up
2 The case EIS equals 1
3 The case EIS different from 1
4 An example
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The set-up The case EIS equals 1 The case EIS different from 1 An example
Epstein-Zin utility in continuous timeThe utility index V c,π
t associated at time t with a given consumptionprocess c and portfolio process π over the remaining lifetime [t ,T ] isrecursively given by
V c,πt = Et
[∫ T
tf(cu,V
c,πu)
du + V c,πT
]. (*)
The so-called normalized aggregator f is defined by (γ > 1)
f (c,V ) =
{δ
1−1/ψc1−1/ψ([1− γ]V )1−1/θ − δθV , for ψ 6= 1δ(1− γ)V ln c − δV ln ([1− γ]V ) , for ψ = 1
where θ = (1− γ)/(1− 1ψ ).
δ: subjective time preference rate,γ: the degree of relative risk aversion towards atemporal bets,ψ > 0: the elasticity of intertemporal substitution (EIS) towardsdeterministic consumption plans.
Note: assume terminal utility V c,πT = α
1−γ (W c,πT )1−γ , where α ≥ 0 and
W c,πT is the terminal wealth induced by the strategies c,π.
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The set-up The case EIS equals 1 The case EIS different from 1 An example
Time-additive utility as special case
The special case ψ = 1/γ (so that θ = 1) corresponds to the classictime-additive power utility since the recursion (*) is then satisfied by
V c,πt = δ
(Et
[∫ T
te−δ(u−t) 1
1− γc1−γ
u du +1δ
e−δ(T−t) α
1− γ(W c,π
T
)1−γ])
,
which is a positive multiple of the traditional time-additive power utilityspecification.
Note that α = δ corresponds to the case where utility of a terminalwealth of W will count roughly as much as the utility of consuming Wover the final year.
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The set-up The case EIS equals 1 The case EIS different from 1 An example
Dynamic programming with recursive utilityIndirect utility is defined as Jt = sup(c,π)∈At
V c,πt .
Duffie and Epstein (1992) have demonstrated the validity of the dynamicprogramming solution technique with recursive utility.With a state variable xt and no income, wealth dynamics is
dWt =(
Wt
[r(xt ) + π>
t σ (xt , t)λ(xt )]− ct
)dt + Wtπ
>t σ (xt , t) dz t .
Assuming diffusion
dxt = m(xt ) dt + v(xt )> dz t + v(xt ) dzt ,
the indirect utility is of the form Jt = J(Wt , xt , t), and the HJB equation is
0 = LπJ(W , x , t) + supc≥0
{f(c, J(W , x , t)
)− cJW (W , x , t)
}+∂J∂t
(W , x , t)
+ JW (W , x , t)Wr(x) + Jx (W , x , t)m(x) +12
Jxx (W , x , t)(v(x)>v(x) + v(x)2),
where
LπJ = supπ∈Rd
{JW Wπ>σ (x , t)λ(x)+
12
JWW W 2π>σ (x , t)σ (x , t)>π+JWx Wπ>σ (x , t)v(x)}.
Terminal condition J(W , x ,T ) = α1−γW 1−γ .
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The set-up The case EIS equals 1 The case EIS different from 1 An example
Optimal portfolio
The maximization with respect to π is exactly as for the case withgeneral time-additive expected utility. The maximizer is
π∗ = − JW (W , x , t)WJWW (W , x , t)
(σ (x , t)>
)−1λ(x)− JWx (W , x , t)
WJWW (W , x , t)(σ (x , t)>
)−1 v(x),
which implies that
LπJ(W , x , t) = −12
JW (W , x , t)2
JWW (W , x , t)‖λ(x)‖2 − 1
2JWx (W , x , t)2
JWW (W , x , t)‖v(x)‖2
− JW (W , x , t)JWx (W , x , t)JWW (W , x , t)
v(x)>λ(x).
(Of course, J will be different with Epstein-Zin utility than withtime-additive utility.)
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The set-up The case EIS equals 1 The case EIS different from 1 An example
Optimal portfolio
We will study ψ = 1 and ψ 6= 1 separately, but in both cases theindirect utility will be of the form
J(W , x , t) =1
1− γG(x , t)γW 1−γ .
Hence, the optimal portfolio is
π∗t = − JW (W , x , t)WJWW (W , x , t)
(σ (x , t)>
)−1λ(x)− JWx (W , x , t)
WJWW (W , x , t)
(σ (x , t)>
)−1v(x)
=1γ
(σ (xt , t)>
)−1λ(xt ) +
Gx (xt , t)G(xt , t)
(σ (xt , t)>
)−1v(xt ).
speculative component the same as for time-additive power utilityhedge component may be differentwe have to determine G...
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The set-up The case EIS equals 1 The case EIS different from 1 An example
HJB with ψ = 1Recall that for ψ = 1,
f (c, J) = δ(1− γ)J ln c − δJ ln ([1− γ]J) .
Therefore HJB equation becomes
0 = LπJ(W , x , t) + LcJ(W , x , t)− δJ(W , x , t) ln ([1− γ]J(W , x , t)) +∂J∂t
(W , x , t)
+ JW (W , x , t)Wr(x) + Jx (W , x , t)m(x) +12
Jxx (W , x , t)(v(x)>v(x) + v(x)2),
whereLcJ = sup
c≥0{δ(1− γ)J ln c − cJW} .
The first-order condition for the consumption choice is
δ(1− γ)J1c
= JW ⇔ c = δ(1− γ)JJ−1W ,
which implies that
LcJ = δ(1− γ)J (ln δ + ln ([1− γ]J)− ln JW )− δ(1− γ)J
= δ(1− γ)J {ln δ + ln ([1− γ]J)− ln JW − 1} .
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The set-up The case EIS equals 1 The case EIS different from 1 An example
Solving the HJBSubstituting LπJ and LcJ into HJB:
0 = −12
J2W
JWW‖λ(x)‖2 − 1
2J2
Wx
JWW‖v(x)‖2 − JW JWx
JWWv(x)>λ(x)
+ δ(1− γ)J {ln δ + ln ([1− γ]J)− ln JW − 1}
− δJ ln ([1− γ]J) +∂J∂t
+ JW Wr(x) + Jx m(x) +12
Jxx (v(x)>v(x) + v(x)2),
Conjecture J(W , x , t) = 11−γG(x , t)γW 1−γ .
Then optimal consumption is c∗t = δWt .
We need G(x ,T ) = α1/γ and
0 =12
(‖v(x)‖2 + v(x)2
)Gxx +
(m(x)− γ − 1
γλ(x)>v(x)
)Gx +
γ − 12
v(x)2 G2x
G
+∂G∂t−(δ ln G +
γ − 1γ
δ[ln δ − 1] +γ − 1γ
r(x) +γ − 12γ2 ‖λ(x)‖2
)G.
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The set-up The case EIS equals 1 The case EIS different from 1 An example
Solving for GIf ‖v(x)‖2, v(x)2, m(x), λ(x)>v(x), r(x), and ‖λ(x)‖2 are all affine functionsof x , the PDE for G has the solution
G(x , t) = α1/γe−γ−1γ
A0(T−t)− γ−1γ
A1(T−t)x,
where A0 and A1 solve ODE’s with A0(0) = A1(0) = 0.
A′1(τ) = r1 +Λ1
2γ+
(m1 −
γ − 1γ
K1 + δ
)A1(τ)− γ − 1
2γ(V1 + γv1) A1(τ)2.
Note: closed-form solution with utility of intermediate consumption andincomplete markets – contrasts the results for time-additive power utility.
Then Gx/G = − γ−1γ
A1(T − t) so the optimal portfolio becomes
π∗t =1γ
(σ (xt , t)>
)−1λ(xt )−
γ − 1γ
(σ (xt , t)>
)−1v(xt )A1(T − t).
Only difference to time-additive power utility is the δ-term in the ODE!
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The set-up The case EIS equals 1 The case EIS different from 1 An example
HJB with ψ 6= 1Recall that for ψ 6= 1,
f (c, J) =δ
1− 1/ψc1−1/ψ([1− γ]J)1−1/θ − δθJ.
Therefore HJB equation becomes
0 = LπJ(W , x , t) + LcJ(W , x , t)− δθJ(W , x , t) +∂J∂t
(W , x , t)
+ JW (W , x , t)Wr(x) + Jx (W , x , t)m(x) +12
Jxx (W , x , t)(v(x)>v(x) + v(x)2),
where
LcJ = supc≥0
{δ
1− 1/ψc1−1/ψ([1− γ]J)1−1/θ − cJW
}.
The first-order condition for the consumption choice is
c = δψJ−ψW ([1− γ]J)ψ(1− 1θ) ,
which implies that
LcJ =1
ψ − 1δψJ1−ψ
W ([1− γ]J)ψ(1− 1θ) .
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The set-up The case EIS equals 1 The case EIS different from 1 An example
Solving the HJB
With conjecture J(W , x , t) = 11−γG(x , t)γW 1−γ we get the optimal
consumptionc∗t = δψG(xt , t)−ψγ/θWt
and G must solve
0 =12
(‖v(x)‖2 + v(x)2
)Gxx +
(m(x)− γ − 1
γλ(x)>v(x)
)Gx +
γ − 12
v(x)2 G2x
G
+∂G∂t
+θ
γψδψG
γψ−1γ−1 −
(δθ
γ+γ − 1γ
r(x) +γ − 12γ2 ‖λ(x)‖2
)G
with terminal condition G(x ,T ) = α1/γ .
Term with Gγψ−1γ−1 prevents explicit solution – unless ψ = 1/γ, then
time-additive CRRA utility as in earlier chapters.
Approximate closed-form solution (next!) or numerical solution (maybe set upthe problem in discrete time from the beginning)
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The set-up The case EIS equals 1 The case EIS different from 1 An example
Approximate closed-form solutionWith constant investment opportunities:
0 = G′(t) +θ
γψδψG(t)
γψ−1γ−1 − AG(t), A =
δθ
γ+γ − 1γ
r +γ − 12γ2 ‖λ‖
2.
A Taylor approximation of z 7→ ez around z gives ez ≈ ez(1 + z − z), so
G(t)γψ−1γ−1 = G(t)G(t)
γ(ψ−1)γ−1 = G(t)e
γ(ψ−1)γ−1 ln G(t)
≈ G(t)eγ(ψ−1)γ−1 ln G(t)
(1 +
γ(ψ − 1)
γ − 1[ln G(t)− ln G(t)]
)= G(t)G(t)
γ(ψ−1)γ−1
(1 +
γ(ψ − 1)
γ − 1[ln G(t)− ln G(t)]
).
Using that approximation in the ODE, we get
0 = G′(t)− a(t)G(t)− b(t)G(t) ln G(t),
a(t) = A− δψG(t)γ(ψ−1)γ−1
(θ
γψ+ ln G(t)
), b(t) = δψG(t)
γ(ψ−1)γ−1 .
Solution with G(T ) = α1/γ is
G(t) = α1/γe−D(t), D(t) =
∫ T
te−
∫ st b(u) du
(a(s) + b(s)
1γ
lnα)
ds.
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The set-up The case EIS equals 1 The case EIS different from 1 An example
Approximate closed-form solution, cont’dConsumption: c∗t = δψG(xt , t)−ψγ/θWt = δψα−
ψθ e
ψγθ
D(t)Wt
How to choose G(t)?
One possibility: presume that the optimal consumption/wealth ratioc∗t /Wt = δψG(xt , t)−ψγ/θ is close to δ which is the optimalconsumption/wealth ratio for ψ = 1:
δψG(t)−ψγ/θ ≈ δ ⇒ G(t) ≈ δ−γ−1γ ≡ G(t).
In that case, the functions a and b are simply constants,
b = δ, a = A− δ(θ
γψ− γ − 1
γln δ),
so that D(t) reduces to
D(t) =
(Aδ− θ
γψ+γ − 1γ
ln δ +1γ
lnα)(
1− e−δ(T−t)).
Precision of the approximation is not clear!
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The set-up The case EIS equals 1 The case EIS different from 1 An example
Approximate closed-form solution, cont’d
The approximation can be generalized to stochastic investmentopportunities, but will then involve recursive procedure
Not clear how the hedge portfolio with Epstein-Zin utility differs from thehedge portfolio with time-additive power utility – study on a case-by-casebasis?
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The set-up The case EIS equals 1 The case EIS different from 1 An example
The modelChacko & Viceira: Dynamic Consumption and Portfolio Choice withStochastic Volatility in Incomplete Markets, Review of FinancialStudies, 2005.
Constant interest rate rSingle risky asset with price dynamics
dPt
Pt= µdt +
√1yt
dzSt ,
with
dyt = κ[y − yt ] dt + σ√
yt
[ρdzSt +
√1− ρ2 dzyt
].
Note:
µ = r + σtλt = r +
√1ytλt ⇒ λt = (µ− r)
√yt
High y ∼ low volatility ∼ good state (in contrast to Liu-Pan model)ρ > 0 implies that high yt ∼ low volatility AND high stock priceEpstein-Zin utility with infinite time horizon
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The set-up The case EIS equals 1 The case EIS different from 1 An example
Exact solution for ψ = 1
J(W , y) =1
1− γeAy+BW 1−γ ,
c∗t = δWt ,
π∗t =1γ
(µ− r)yt︸ ︷︷ ︸λtσt
=λt√
yt
+γ − 1γ
(−ρ)σAyt︸ ︷︷ ︸negative hedge for ρ > 0
,
where A,B are constants, and A = A/(1− γ) > 0.
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The set-up The case EIS equals 1 The case EIS different from 1 An example
Approximate solution for ψ 6= 1
J(W , y) =1
1− γe−
1−γ1−ψ (A1y+B1)W 1−γ ,
c∗t = δψe−A1yt−B1Wt ,
π∗t =1γ
(µ− r)yt +γ − 1γ
(−ρ)σA1yt︸ ︷︷ ︸again negative for ρ > 0
,
where A1,B1 are constants, and A1 = A1/(1− γ) > 0.
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The set-up The case EIS equals 1 The case EIS different from 1 An example
Some quantitative results: portfolios
Table 2 from Chacko and Viceira (2005).Little action across EIS.
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The set-up The case EIS equals 1 The case EIS different from 1 An example
Some quantitative results: consumption
Table 4 from Chacko and Viceira (2005).Lots of action across EIS.