Loss Aversion and Asset Prices
Marianne Andries
Toulouse School of Economics
June 24, 2014
1
Preferences
In laboratory settings, systematic violations of expected utility theory
I
I
Allais Paradox
M. Rabin (2000)
D. Kahneman and A. Tversky: Prospect Theory: An analysis of Decision
under Risk, Econometrica 1979
2
Allais Paradox
Experiment 1
Gamble 1A
Winnings Chance
$1 million 100%
Experiment 2
Gamble 1B
Gamble 2A
Gamble 2B
Winnings Chance Winnings Chance Winnings Chance
$1 million
89%
Nothing
89%
Nothing
1%
$1 million
11%
$5 million
10%
Nothing
90%
$5 million
10%
!
In lab, most prefer 1A to 1B
3
e 1B
Allais Paradox
Experiment 2
Gamble 2A
Gamble 2B
Chance Winnings Chance Winnings Chance
89%
Nothing
89%
1%
$1 million
11%
10%
Nothing
90%
$5 million
10%
In lab, most prefer 2B to 2A
4
Allais Paradox
Experiment 1
Gamble 1A
Winnings Chance
$1 million 100%
Experiment 2
Gamble 1B
Gamble 2A
Gamble 2B
Winnings Chance Winnings Chance Winnings Chance
$1 million
89%
Nothing
89%
Nothing
1%
$1 million
11%
$5 million
10%
Nothing
90%
$5 million
10%
!
In lab, most prefer 1A to 1B
In lab, most prefer 2B to 2A
Violation of the independence axiom
5
Rabin: Small Gambles versus Large Gambles
M. Rabin: Risk Aversion and Expected-Utility Theory: A Calibration Theorem,
Econometrica
2000
All entries are rounded down to an even dollar amount.
in the table
4
$400
$600
$800
$1,000
$2,000
$4,000
$6,000
$8,000
$10,000
$20,000
$101
400
600
800
1,010
2,320
5,750
11,810
34,940
$105
420
730
1,050
1,570
$110
550
990
2,090
$125
1,250
If averse to 50-50 lose $100/gain bets for all wealth levels,
will turn down 50-50 lose /gain bets; ’s entered in table.
So, for instance, if a person always turns down a 50-50 lose
always turn down a 50-50 lose $800/gain $2,090 gamble. Entries of
/gain
gamble, she will
are literal: Somebody who
EU, concave and increasing utility function
always turns down 50-50 lose $100/gain $125 gambles will turn down any gamble with a 50%
Similar results if the small gamble is rejected only up to given wealth
chance of losing $600. This is because the fact that the bound on risk aversion holds everywhere
I
if (−$100/ + $125) is rejected up to wealth $300, 000 → reject
is bounded above.
(−$1000/ + $160 bn)
implies that
The theorem and corollary are homogenous of degree 1: If we know that turning down 50-50
lose /gain
gambles implies you will turn down 50-50 lose /gain
down 50-50 lose
/gain
, then for all
gambles implies you will turn down 50-50 lose
/gain
, turning
. Hence
6
Prospect Theory
D. Kahneman and A. Tversky: Prospect Theory: An analysis of Decision under
Risk, Econometrica 1979
Consider a (x, p; y, q) gamble
Under EU, its value is
pU (W + x) + qU (W + y)
Under Prospect Theory, its value is
π (p) v (x) + π (q) v (y)
”Simplest” functional form to represent lab decisions
7
176
Journal of Economic Perspectives
Prospect Theory Value Function v
Figure 1
The Prospect Theory Value Function
30
20
v(x)
10
0
–10
–20
–30
–100
1
2
–80
–60
–40
–20
0
x
20
40
60
80
100
Notes: The graph plots the value function proposed by Tversky and Kahneman (1992) as part of
cumulative prospect theory, namely v(x) = x α for x ≥ 0 and v(x) = – λ(– x)α for x < 0, where x is a dollar
Valuation on
andestimate
losses,
of current
wealth
(also, narrow
gain gains
or loss. The authors
α = 0.88 irrespective
and λ = 2.25 from experimental
data. The plot uses
α = 0.5
and λ = 2.5 so as to make loss aversion and diminishing sensitivity easier to see.
framing)
The more
fourth anddisutility
final component
of prospect
theory is probability
weighting.
Losses generate
than
comparable
gains,
noIn matter how
prospect theory, people do not weight outcomes by their objective probabilities p
small they are
→ bykink
at the
value
zero weights π . The decision weights
but rather
transformed
probabilities
or decision
computed with the help of a weighting function w(·) whose argument is an objecconcave forare
gains,
convex
forin losses
tive
probability.
The solid line
Figure 2 shows the weighting function proposed by
i
i
3
Tversky and Kahneman (1992). As is visible in comparison with the dotted line—a
45 degree line, which corresponds to the expected utility benchmark—the weighting
function overweights low probabilities and underweights high probabilities.
8
Thirty Years of Prospect Theory in Economics: A Review and Assessment
177
Prospect Theory Probability Weighting π
Figure 2
The Probability Weighting Function
1
0.9
0.8
0.7
w(P)
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
0.1
0.2
0.3
0.4
0.5
P
0.6
0.7
0.8
0.9
1
Notes: The graph plots the probability weighting function proposed by Tversky and Kahneman (1992)
as part of cumulative prospect theory, namely w(P ) = P δ/(P δ + (1 − P )δ)1/δ, where P is an objective
probability, for two values of δ. The solid line corresponds to δ = 0.65, the value estimated by the authors
from experimental data. The dotted line corresponds to δ = 1, in other words, to linear probability
weighting.
Over-weighting of tail events
Justifies both attraction to gambles and purchases of insurance
part, from the fact that people like both lotteries and insurance —they prefer a
0.001 chance
of $5,000
to a decision
certain gain ofweights
$5, but also prefer a certain loss of $5
Not erroneous
beliefs,
but
to a 0.001 chance of losing $5,000—a combination of behaviors that is difficult
to explain with expected utility. Under cumulative prospect theory, the unlikely
state of the world in which the individual gains or loses $5,000 is overweighted in
his mind, thereby explaining these choices. More broadly, the weighting function
9
Prospect Theory in Finance
1
Loss Aversion
I
I
I
2
< γσc
Over-weighting of low probability events
I
I
I
3
E(Rm )−Rf
Equity premium puzzle in C-CAPM with CCRA EU:
σ(Rm )
Locally infinite risk aversion at the kink in the value function
Participation puzzle
Over-pricing of deep out of the money options
Low or negative returns for right-skewed assets (IPO firms, single-segment
firms, OTC traded assets)
Under-diversified portfolios
Risk-aversion for gains, risk-seeking for losses
I
Disposition effect
N. Barberis: Thirty Years of Prospect Theory in Economics: A Review and
Assessment, JEP 2013
10
Loss Aversion - Some intuitions
Valuation of 50:50 gamble A − σ, A + σ
Utility
!
!
"# !
!
+" !
A
!
!
!
"
#
$
with EU,
#
%
U (payof f# ) ≈ v (A) −
&
with Loss Aversion,
1 2 00
σ v (A)
2
'
#!
U (payof f ) ≈ v (A) −
0
1 0
σ v− (A) − v+ (A)
2
11
Loss Aversion - Some intuitions
1
1st order pricing of risk relative to 2d order pricing of risk
I
I
2
equity premium puzzle
cross-sectional implications
M. Andries: Consumption-based Asset Pricing with Loss Aversion, 2012
(CAPLA)
Role of frequency and information
I
I
I
iid process with instantaneous growth rate µ and standard deviation σ
expected
√ growth increases with time interval T , standard deviation increases
with T
the pricing of risk becomes proportionally larger and larger the smaller the
time interval T
M. Andries and V. Haddad: Information Aversion, 2014 (IA)
12
Outline
1 Consumption-based Asset Pricing with Loss Aversion
2 Information Aversion
13
Plan
1 Consumption-based Asset Pricing with Loss Aversion
2 Information Aversion
14
CAPLA- Model of Preferences
Agents are loss averse:
I
I
consumption outcomes are valued relative to a reference point
losses relative to the reference create more disutility than comparable gains
Agents value the consumption stream recursively:
Vt = f (Ct , Et (g (Vt+1 )))
Loss aversion on the uncertain Vt+1
Reference point as an endogenous expectation
15
CAPLA- Main Results
A tractable consumption-based asset pricing model
Impact of loss aversion on expected excess returns:
I
I
Level effect
Cross sectional effect
Empirical implications:
I
I
Negative Premium for skewness
Security Market Line flatter than the CAPM
Dynamic implications for the pricing of risk
16
CAPLA- One-Period Model
At t = 1, the agent receives uncertain consumption C
Standard CRRA model:
U0 = E
C 1−γ
| I0
1−γ
γ > 1: risk aversion
Loss Aversion model:
1
2
3
homogeneous CRRA model above and below a reference point
continuous at the reference point
kink at the reference point (ratio of slopes) determined by a loss aversion
coefficient α ∈ [0, 1)
17
CAPLA- One-Period Model
Utility
!
Ref
!
!
C !
!
!
"
#
$
#
%
#
&
'
#!
C 1"#
1"#
!
!
18
CAPLA- One-Period Model
Utility
!
Ref
!
!
C !
!
!
"
#
$
#
%
#
&
'
#!
C 1"#
1"#
!
!
18
CAPLA- One-Period Model with Loss Aversion
1−γ̄
U0 = E
C
1−γ̄
C
| I0
1 − γ̄
 1−γ̄

C
= C 1−γ × (Ref )γ−γ̄

| {z }

!
for C ≤ Ref
for C ≥ Ref
scaling factor
γ̄ > γ determined by the ratio of slopes 1 − α =
1−γ
1−γ̄
Kahneman and Tversky (1979): α = 0.55
19
CAPLA- Multi-Period Model
Standard Epstein-Zin (1989) preferences:
Vt = (1 − β) Ct1−ρ + β (h (Vt+1 ))1−ρ
1−γ
h (Vt+1 ) = Et Vt+1
γ > 1: the risk aversion,β: the discount factor,
1
1−ρ
1
1−γ
1
:
ρ
the EIS
Add loss aversion on the CRRA model with reference point as an expectation
20
CAPLA- Properties of Multi-Period Model
1
1−γ̄
1−γ̄
h (Vt+1 ) = Et Vt+1
Vt+1
1−γ̄
 1−γ̄

Vt+1
1−γ
= Vt+1
× exp [(γ − γ̄) Et (vt+1 )]

|
{z
}

for vt+1 ≤ Et (vt+1 )
for vt+1 ≥ Et (vt+1 )
scaling factor
1
if the outcome Vt+1 is certain, then h (Vt+1 ) = Vt+1
2
h is increasing (first order stochastic dominance)
3
h is concave (second order stochastic dominance)
4
h is homogeneous of degree one (Vt homogeneous of degree one in
(Ct , Vt+1 ))
21
CAPLA- Representative Agent
Uniqueness of the solution to the optimization problem
Time consistency
h is concave (second order stochastic dominance)
Assume agents differ in their wealth only =⇒ with homothetic preferences,
the representative agent assumption is justified
22
CAPLA- Stochastic Discount Factor
+
St,t+1
−
St,t+1
=1−α
exp [Et (vt+1 )]
≤1
1−γ̄
exp [Et (vt+1 )] + αEt 1vt+1 ≥Et (vt+1 ) Vt+1
Discontinuity in the stochastic discount factor when α > 0
Discontinuity increases with loss aversion coefficient α
23
CAPLA- Main Results
Discontinuity in the SDF generates both a level and a cross-sectional effect
Annual Expected Excess Returns
0.08
Risk Price Elasticities
0.035
standard model
model with loss aversion
0.06
0.03
0.04
0.025
0.02
0.02
0
0.015
standard model
model with loss aversion
−0.02
0.01
−0.04
0.005
−0.06
−0.08
−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
Loadings on the consumption shocks
0.2
0.25
0.3
0
−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
Loadings on the consumption shocks
0.2
0.25
0.3
I use the parameters from Hansen, Heaton and Li (2008) for the consumption
process and β = 0.999, γ = 10, α = 0.55.
24
CAPLA- Equity Premium Calibration
CAPLA- Equity Premium Calibration
model with loss aversion
standard model
α = 0.10
α = 0.25
α = 0.55
risk aversion γ
γ = 10
0.94%
1.29%
2.14%
0.72%
γ = 15
1.34%
1.72%
2.74%
1.11%
γ = 20
1.74%
2.16%
3.39%
1.50%
Equity Premium from CRSP (1947-2010) = 6.09%
1
I use the quarterly parameters from Hansen, Heaton, and Li (2008) and β = (0.999) 4
Marianne Andries (TSE)
Loss Aversion and Asset Prices
June 2014
19 / 23
25
CAPLA- Value Premium Calibration
CAPLA- Value Premium Calibration
model with loss aversion
standard model
α = 0.10
α = 0.25
α = 0.55
risk aversion γ
γ=3
1.45%
2.68%
5.20%
0.65%
γ=5
2.04%
3.29%
5.98%
1.23%
γ = 10
3.53%
4.85%
8.05%
2.70%
Value Premium from Fama-French (1947-2010) = 4.22%
1
I use the quarterly parameters from Hansen, Heaton, and Li (2008) and β = (0.999) 4
Marianne Andries (TSE)
Loss Aversion and Asset Prices
June 2014
20 / 23
26
CAPLA- Prediction for CAPM
The model with loss aversion qualitatively predicts a security market line flatter
than the CAPM
Annual Expected Excess Returns
16
14
12
Ri−Rf in %
10
8
6
Positive Intercept
4
2
0
0
0.5
0.8
1
1.5
CAPM !
2
2.5
3
I use the parameters from Hansen, Heaton and Li (2008) for the consumption
process and β = 0.999, γ = 10, α = 0.55.
27
CAPLA- Conclusion
Tractable consumption-based asset pricing model with loss aversion and
recursive utility
Level effect on risk prices allows to match or improve on calibration
exercises that use moments in asset returns
Cross-sectional effect is a testable implication of my model
Empirical evidence on the fit of the CAPM model provides strong support
for my model with loss aversion
28
Plan
1 Consumption-based Asset Pricing with Loss Aversion
2 Information Aversion
29
IA- This Paper
Why don’t agents pay attention to information?
Micro founded models of risk → attitude towards information
Information aversion
Preference-based explanation of the cost of information
Characterize risk and information decisions when information costs are
endogenous:
Properties of optimal attention to savings:
I
I
Consumer Expenditure Survey (Dynan and Maki 2000): through a 15% rise
in the market, 1/3 of stockholders report no change to their portfolio value.
Alvarez, Guiso and Lippi (2012): household surveys in Italy, observe
portfolios 4 times a year.
Portfolio choice: home bias, underdiversification
30
Information Aversion Model
Disappointment aversion
31
Information Aversion Model
Disappointment aversion
Ability to close your eyes
31
Information Aversion Model
Disappointment aversion
Recursive dynamic implementation of piecewise linear case of Gul (1991)
Partial releases of information have a utility cost (Dillenberger 2010)
Micro evidence and successful macro applications (Ang et al. 2005,2006,
Routledge and Zin 2010, Bonomo et al. 2011, Lettau et al. 2013)
Ability to close your eyes
31
Information Aversion Model
Disappointment aversion
Recursive dynamic implementation of piecewise linear case of Gul (1991)
Partial releases of information have a utility cost (Dillenberger 2010)
Micro evidence and successful macro applications (Ang et al. 2005,2006,
Routledge and Zin 2010, Bonomo et al. 2011, Lettau et al. 2013)
Ability to close your eyes
No monetary or time cost of information
No limited cognition
Bayesian updating
31
IA- Results
Natural theory of the cost side of information acquisition
Which information flows are more costly?
I
I
I
Higher frequency
Higher risk
Infinite aversion to continuous Brownian flow, not to jumps
32
IA- Results
Natural theory of the cost side of information acquisition
Which information flows are more costly?
I
I
I
Higher frequency
Higher risk
Infinite aversion to continuous Brownian flow, not to jumps
Information choice in a consumption-saving problem
I
I
Infrequent observation of portfolio position
Tradeoff for optimal frequency of information. At lower frequency:
Misallocation of savings
Less “stressful” flow of information
I
More inattention in risky environments
32
IA- Results
Natural theory of the cost side of information acquisition
Which information flows are more costly?
I
I
I
Higher frequency
Higher risk
Infinite aversion to continuous Brownian flow, not to jumps
Information choice in a consumption-saving problem
I
I
Infrequent observation of portfolio position
Tradeoff for optimal frequency of information. At lower frequency:
Misallocation of savings
Less “stressful” flow of information
I
More inattention in risky environments
Other features of portfolio allocation
I
I
I
I
Diversification
Background risk
Information delegation
Asymmetry between good and bad news
32
IA Preferences: Disappointment Aversion
Piecewise linear case of Gul (1991)
Lottery over final outcome X
Certainty equivalent:
µ(X) =
E
1 + θ1X≤µ(X) X
E 1 + θ1X≤µ(X)
Overweight “disappointing outcome”
I
I
θ > 0, coefficient of disappointment aversion
only source of aversion to risk comes from disappointment aversion
Certainty equivalent µ(X) is unique solution to a fixed-point problem
33
IA- Disappointment Aversion, dynamic implementation
Dynamic implementation: recursion on certainty equivalents
Value at time t, Vt , of lottery over continuation value Vt+1 : Vt = µ (Vt+1 )
Vt =
E
1 + θ1Vt+1 ≤Vt Vt+1 |It
E 1 + θ1Vt+1 ≤Vt |It
If no news is revealed: Vt = Vt+1
In continuous time: take the limit of discrete time sampling of information
34
Two-stage Lottery
P
α1
A Signal 3 F3
F2
F1
B F
α3
α2
Signal 2 Signal 1 αi Fi = F
C D A B C D 35
Two-stage Lottery
P
αi Fi = F
μ(F) α1
Signal 2 Signal 1 A Signal 3 F3
F2
F1
B F
α3
α2
C D A B C D 35
Two-stage Lottery
P
αi Fi = F
μ(F) α1
Signal 2 μ(F1) A Signal 3 F3
F2
F1
B F
α3
α2
C D A B C D 35
Two-stage Lottery
P
αi Fi = F
μ(F) α1
μ(F2) μ(F1) A μ(F3) F3
F2
F1
B F
α3
α2
C D A B C D 35
Two-stage Lottery
P
αi Fi = F
μ(F) μ({Fi, αi}) α1
μ(F2) μ(F1) A μ(F3) F3
F2
F1
B F
α3
α2
C D A B C D 35
Information Aversion
Disappointment aversion ⇒ information aversion
Agent prefers not to observe the signal
µ({Fi , αi }) ≤ µ(F )
I
Dillenberger (2010): Negative Certainty Independence ⇔ Preference for
One-Shot Resolution of Uncertainty
36
Information Aversion
Disappointment aversion ⇒ information aversion
Agent prefers not to observe the signal
µ({Fi , αi }) ≤ µ(F )
I
Dillenberger (2010): Negative Certainty Independence ⇔ Preference for
One-Shot Resolution of Uncertainty
Agent fears possibility of repeated changes in certainty equivalent
(
µ (Fi ) = µ (F ) or
µ({Fi , αi }) = µ(F ) ⇔ ∀i,
Fi is degenerate
36
IA- Endogenous Information Costs
Information aversion versus exogenous costs models
I
I
Endogenous information cost is zero if all or no information is revealed
Not monotonic increasing in quantity of information
Information aversion versus cognitive constraints
I
I
Endogenous information cost is zero for either fully informative or fully
uninformative signals
For any level of mutual information, we can construct signals with zero
endogenous cost: reveal the final value of the lottery with some probability
37
IA- Role of Frequency
Process Xt with i.i.d. growth
Observe its value at intervals of length T
Receive value of the process Xτ at time τ
38
IA- Role of Frequency
Process Xt with i.i.d. growth
Observe its value at intervals of length T
Receive value of the process Xτ at time τ
Information aversion
Prefer never to observe the intermediate values
Gneezy and Potters (1997), ...
How is the valuation of the lottery affected by
the observation interval?
the distribution of the process?
Input for consumption-savings problem
38
IA- Role of Frequency
Process Xt with i.i.d. growth
Observe its value at intervals of length T
Receive value of the process Xτ at time τ
Because growth is i.i.d
X(k+1)T
XT
X2T
µ
=µ
= ... = µ
X0
XT
XkT
Define instantaneous certainty equivalent rate v(T ):
XT
µ
= exp(v(T )T )
X0
Value at time 0 for payoff at time τ :
V0,τ (T ) = exp(v(T )τ )
38
IA- Role of Frequency
Process Xt with i.i.d. growth
Observe its value at intervals of length T
Receive value of the process Xτ at time τ
Because growth is i.i.d
X(k+1)T
XT
X2T
µ
=µ
= ... = µ
X0
XT
XkT
Define instantaneous certainty equivalent rate v(T ):
XT
µ
= exp(v(T )T )
X0
Value at time 0 for payoff at time τ :
V0,τ (T ) = exp(v(T )τ )
With drift g and martingale component Y :
vX (T ) = g + vY (T )
38
IA- Frequency and Geometric Brownian Motion
dXt
= σdWt
Xt
Certainty Equivalent Rate,
θ=1, σ=1
Certainty equivalent rate v(T)
0
−0.5
−1
−1.5
−2
−2.5
0
0.5
1
1.5
Observation interval T
Equivalent rate as a function of
observation interval
39
IA- Frequency and Geometric Brownian Motion
dXt
= σdWt
Xt
Certainty Equivalent Rate,
θ=1, σ=1
Distaste for frequent partial
information:
I
I
equivalent rate increasing in
observation interval
optimally choose never to look
at any information
Certainty equivalent rate v(T)
0
−0.5
−1
−1.5
−2
−2.5
0
0.5
1
1.5
Observation interval T
Equivalent rate as a function of
observation interval
39
IA- Frequency and Geometric Brownian Motion
dXt
= σdWt
Xt
Certainty Equivalent Rate,
θ=1, σ=1
0
I
I
equivalent rate decreasing in risk
σ
equivalent rate decreasing in risk
aversion θ
Certainty equivalent rate v(T)
Risk aversion:
−0.5
−1
−1.5
−2
−2.5
0
0.5
1
1.5
Observation interval T
Equivalent rate as a function of
observation interval
39
IA- Frequency and Geometric Brownian Motion
dXt
= σdWt
Xt
Certainty Equivalent Rate,
θ=1, σ=1
Infinite risk aversion at high
frequency:
I
I
Value for t = τ payoff equals
lowest possible outcome in the
continuous information limit
expansion around 0:
κ(θ)σ
v(T ) ≈0 − √
T
I
first-order risk aversion:
√
−σ T
×
τ /T
| {z }
|{z}
observation discount
# observations
Certainty equivalent rate v(T)
0
−0.5
−1
−1.5
−2
−2.5
0
0.5
1
1.5
Observation interval T
Equivalent rate as a function of
observation interval
39
IA- Frequency and Jump process
Distaste for frequent partial
information
Risk aversion
Certainty Equivalent Rate,
θ=1, σ=.2, λ=5
0
Certainty equivalent rate v(T)
dXt
= λσdt − σdNt
Xt
Nt : Poisson counting process, intensity
λ
−0.2
−0.4
−0.6
−0.8
−1
0
0.5
1
1.5
Observation interval T
Equivalent rate as a function of
observation interval
40
IA- Frequency and Jump process
I
limiting behavior
v(T ) −−−→ −θσλ
T →0
I
no first order risk aversion:
infrequent large risks vs.
frequent small risks
Certainty Equivalent Rate,
θ=1, σ=.2, λ=5
0
Certainty equivalent rate v(T)
dXt
= λσdt − σdNt
Xt
Nt : Poisson counting process, intensity
λ
Finite limit at high frequency:
−0.2
−0.4
−0.6
−0.8
−1
0
0.5
1
1.5
Observation interval T
Equivalent rate as a function of
observation interval
40
IA - Portfolio problem
How does the disappointment averse agent decide to consume, save, and
observe information?
Choice between risk-free and risky savings
Setup of the fixed cost of information/transaction literature:
I
I
Duffie and Sun (1990), Gabaix and Laibson (2001), Abel et al. (2007,
2013), Alvarez et al. (2013).
Baumol-Tobin model (1952, 1956)
No exogenous cost of information/transaction, but agent free to close her
eyes
41
Setup
Preferences:
(µθ [Vt+dt |Ft ])1−α
Vt1−α
C 1−α
= t
dt + (1 − ρdt)
.
1−α
1−α
1−α
θ: coefficient of disappointment aversion
1/α: intertemporal elasticity of substitution
ρ: rate of time discount
42
Setup
Preferences:
(µθ [Vt+dt |Ft ])1−α
Vt1−α
C 1−α
= t
dt + (1 − ρdt)
.
1−α
1−α
1−α
θ: coefficient of disappointment aversion
1/α: intertemporal elasticity of substitution
ρ: rate of time discount
Opportunity sets:
Information: choose time until next observation T
Investment:
I
I
I
Instantaneous consumption Ct
Buy St shares of the risky asset, price Xt , instantaneous certainty
equivalent rate v(T )
Remainder in risk-free asset, rate of return r
Budget constraint:
dWt = −Ct dt + St dXt + r(Wt − St Xt )dt
42
Setup
Preferences:
Vt1−α
=
1−α
T
Z
0
e−ρτ
1−α
Ct+τ
(µθ [Vt+T |Ft ])1−α
dτ + e−ρT
.
1−α
1−α
θ: coefficient of disappointment aversion
1/α: intertemporal elasticity of substitution
ρ: rate of time discount
Opportunity sets:
Information: choose time until next observation T
Investment:
I
I
I
Instantaneous consumption Ct
Buy St shares of the risky asset, price Xt , instantaneous certainty
equivalent rate v(T )
Remainder in risk-free asset, rate of return r
Budget constraint:
dWt = −Ct dt + St dXt + r(Wt − St Xt )dt
42
Basic Properties
Homothetic preferences
Linear opportunity set for consumption
i.i.d. dynamics
⇒
Constant observation interval T
Consumption-wealth ratio and asset allocation functions of wealth at last
observation and time since last observation
43
Basic Properties
Homothetic preferences
Linear opportunity set for consumption
i.i.d. dynamics
⇒
Constant observation interval T
Consumption-wealth ratio and asset allocation functions of wealth at last
observation and time since last observation
Remark: Fixed cost models lose homotheticity or use ad hoc assumptions
on the scaling of the cost
43
IA- Consumption and Investment Decisions
Given observation interval T :
Consumption between observations deterministic, financed at the risk-free
rate r
Inter-observation savings:
I
I
all risk-free if r > v(T )
all risky if r < v(T )
44
IA- Consumption and Investment Decisions
Given observation interval T :
Consumption between observations deterministic, financed at the risk-free
rate r
Inter-observation savings:
I
I
all risk-free if r > v(T )
all risky if r < v(T )
Fraction of wealth allocated to consumption:
ρ
1−α
C (T ) = 1 − exp − +
max(v(T ), r) T
α
α
Consumption path, for τ ∈ [0, T ]:
Ct+τ ∝ C(T )e
−ρ+r
τ
α
44
IA- Role of Observation Interval
Geometric brownian motion: dX/X = gdt + σdWt
Value Function
Consumption allocation
2.3
0.7
2.2
0.6
2.1
0.5
2
0
C
V0
0.4
1.9
0.3
1.8
0.2
1.7
0.1
1.6
1.5
0
0.5
1
1.5
2
Observation interval T
2.5
3
0
0
0.5
1
1.5
2
2.5
3
Observation interval T
Parameters values: θ = 1, α = 0.5, σ = 1, g − r = 1, ρ = 0.1.
45
IA- Role of Observation Interval
Geometric brownian motion: dX/X = gdt + σdWt
Value Function
Consumption allocation
2.3
0.7
2.2
0.6
2.1
0.5
2
0
C
V0
0.4
1.9
0.3
1.8
0.2
1.7
0.1
1.6
1.5
0
0.5
1
1.5
2
Observation interval T
2.5
3
0
0
0.5
1
1.5
2
2.5
3
Observation interval T
Parameters values: θ = 1, α = 0.5, σ = 1, g − r = 1, ρ = 0.1.
→ Infrequent observation and investment in risky asset iff g > r
45
IA- Role of Observation Interval
Geometric brownian motion: dX/X = gdt + σdWt
Value Function
Consumption allocation
2.3
0.7
2.2
0.6
2.1
0.5
2
0
C
V0
0.4
1.9
0.3
1.8
0.2
1.7
0.1
1.6
1.5
0
0.5
1
1.5
2
Observation interval T
2.5
3
0
0
0.5
1
1.5
2
2.5
3
Observation interval T
Parameters values: θ = 1, α = 0.5, σ = 1, g − r = 1, ρ = 0.1.
→ Infrequent observation and investment in risky asset iff g > r
More generally, need v(0) < r < v(∞)
45
IA- Optimal Information Choice
Key result: Optimal observation interval exists and is such that:
∂v(T )
f
v (T ) −
ρ
!
=
1−α
∂ log(T )
where f (x) = − exp
1−α xT
α
v (T ) −
ρ
!
f
1−α
v (T ) −
ρ
!
−
1−α
r−
ρ
!
f
1−α
ρ
r−
!!
1−α
/ 1 − exp 1−α xT
α
Marginal cost of infrequent observation (RHS)
I
I
lost consumption through financing risk-free rather than risky between
observations
increasing in equivalent rate differential between v(T ) and r
Marginal benefit of infrequent observation (LHS)
I
I
I
higher certainty equivalent for higher observation interval
increasing in certainty equivalent elasticity ∂v(T )/∂ log(T )
missing elasticity of fixed cost models
46
IA- Optimal Information Choice
Key result: Optimal observation interval exists and is such that:
∂v(T )
f
v (T ) −
ρ
!
=
1−α
∂ log(T )
where f (x) = − exp
1−α xT
α
v (T ) −
ρ
!
f
1−α
v (T ) −
ρ
!
−
1−α
r−
ρ
!
f
1−α
ρ
r−
!!
1−α
/ 1 − exp 1−α xT
α
Marginal cost of infrequent observation (RHS)
I
I
lost consumption through financing risk-free rather than risky between
observations
increasing in equivalent rate differential between v(T ) and r
Marginal benefit of infrequent observation (LHS)
I
I
I
higher certainty equivalent for higher observation interval
increasing in certainty equivalent elasticity ∂v(T )/∂ log(T )
missing elasticity of fixed cost models
Certainty equivalent elasticity
I
I
I
Independent of the drift
Typically decreasing in the observation interval
Non-trivial dependence to the shape of the return distribution
46
IA- Role of risk
Geometric brownian motion: dX/X = gdt + σdWt
→ Optimal observation interval increasing in risk σ:
Standard effect: equivalent rate of return v(T ) decreases. Can be
compensated by higher average rate g
Information aversion effect: less willingness to take on the information
flow, higher elasticity ∂v/∂ log(T )
Observation interval
Value
1.4
2.5
1.3
2.45
1.2
2.4
1.1
2.35
Consumption Allocation
0.5
0.45
1
C0
V0
T
0.4
2.3
0.9
2.25
0.8
2.2
0.7
2.15
0.35
0.3
0.25
0.5
1
Volatility m
1.5
2.1
0.5
1
Volatility m
1.5
0.2
0.5
1
1.5
Volatility m
47
Predictions
Observation interval decreasing in expected stock returns
Observation interval increasing in volatility
I
I
I
Even when compensated by higher expected returns
“Scary information flow”
“Ostrich effect” (Karlson et al. 2009), follow-up paper on VIX level and
inattention (Sicherman et al. 2014)
More disappointment averse agent observe their portfolios less frequently
I
Alvarez et al. (2013): more risk averse agents check their accounts less
often
All else equal, in response to exogenous decrease in observation interval,
increase in stock holdings
I
I
Driven by corner solution in asset holdings
Consistant with Beshears et al. (2012), also finding an increase in trading
activity
48
IA- Diversification
What if you can get
(1)
1
X
2 t
(2)
+ 12 Xt
(1)
rather than Xt ?
Standard benefit: less risk
49
IA- Diversification
What if you can get
(1)
1
X
2 t
(2)
+ 12 Xt
(1)
rather than Xt ?
Standard benefit: less risk
Limits of diversification: If asynchronous forced information arrival,
increase in scope for disappointment
49
IA- Diversification
What if you can get
(1)
1
X
2 t
(2)
+ 12 Xt
(1)
rather than Xt ?
Standard benefit: less risk
Limits of diversification: If asynchronous forced information arrival,
increase in scope for disappointment
Result:
With independent Brownian motions, diversification is still valuable with
non-instrumental information.
The gains to diversification go to 0 as observation becomes continuous.
49
IA- Diversification
What if you can get
(1)
1
X
2 t
(2)
+ 12 Xt
(1)
rather than Xt ?
Standard benefit: less risk
Limits of diversification: If asynchronous forced information arrival,
increase in scope for disappointment
Result:
With independent Brownian motions, diversification is still valuable with
non-instrumental information.
The gains to diversification go to 0 as observation becomes continuous.
Work in progress:
Role of background risk: risky portfolio can be decreasing in risk in
presence of background risk
Home/local bias: anchor on forced information flows vs diversification
benefits
49
IA- More General Information Choices
So far, limited to simple information structure: open or closed eyes
With the help of machines or people, can better taylor the information flow
Result: Simple ”alarm” when the risky asset reaches some thresholds
provides more utility
I
State-dependent trading rules do better than time-dependent rules, in
contrast to fixed information cost (Abel et al. 2013)
In practice:
I
I
Useful to have your broker send you an email following extreme
performance, good or bad
Media reporting large events
50
IA- Information Intermediaries
Other individuals can not only curate information, but also take actions for
the agent
I
I
Portfolio managers, investment funds, ...
Optimal opaqueness: complex or illiquid securities hard to mark-to-market,
...
Information sets differ ⇒ need to appropriately incentivize the informed
decision maker
51
IA- Conclusion
Information aversion: a novel foundation for inattention
Disappointment aversion creates information aversion, fear of repeated
disappointment
Without use of information agent always prefers to close her eyes
More averse to flows:
I
I
I
about more risky outcome
with frequent small news than infrequent large news
about likely bad news than likely good news
Simple way to summarize information aversion: certainty equivalent rate
v(T )
More questions:
Multi-asset decisions, background risk
Delegated management
Combined learning and frequency decisions
Multiple agents
52