Equilibrium models with beliefs
heterogeneity
Elyes Jouini
Introduction
Starting point : Calvet, Grandmont and Lemaire (2002) construct
a representative agent in a static heterogeneous beliefs setting.
1. We extend this approach to an intertemporal framework,
2 We explore the implications of this aggregation procedure on the
risk premium and on the risk sharing rule.
The representative agent approach (Negishi (60), Rubinstein (74),
Breeden and Litzenberger (78), Constantinides (82)) is a cornerstone
of theoretical and applied macroeconomics and basis for
1. CAPM, Sharpe 1964 and Lintner 1965
2. CCAPM, Ingersoll, 1987, Huang and Litzenberger, 1988,
Duffie, 1996.
However ”difficulties remain, significant among which is the
restrictive assumption of homogeneous expectations” Williams
(1977).
Cragg and Malkiel (1982) studied the relationship between ex post
returns and various measures of risk and the measure that performed
best was the measure of divergence of opinion about the asset returns.
Our aim : to analyze the consequences of individual subjective
heterogeneous probabilities in an otherwise standard competitive
complete markets economy
The methdological approach
1) Is it possible to define consensus beliefs i.e. ”those beliefs which,
if held by all individuals (...) would generate the same equilibrium
prices as in the actual heterogeneous economy.” Rubinstein (1975)
2) Is it still possible to define a representative agent (or consensus
consumer).
3) Is it possible ”to construct sharing rules which indicate how
consumption or portfolio choices of a particular consumer deviate
from the per capita choices of the consensus consumer” (Rubinstein,
1976)
4) Is it possible to relate diversity of individual portfolios to
heterogeneity of individual beliefs.
5) To which extent our framework may contribute to a better
understanding of the equity premium puzzle (Mehra and Prescott,
1985) and of the risk free rate puzzle (Weil, 1989).
Derivative products
1) How to characterize and compare individual beliefs
2) How these characteristics are related to other individual characteristics.
3) How these characteristics may evolve through learning and
interactions
4) Is there a systematic cognitive bias in individual beliefs
Related papers
Heterogeneous agents
Lintner (1969), Rubinstein (1975,1976), Gonedes (1976),
Miller (1977), Williams (1977), Jarrow (1980), Mayshar (1981,
1983),Varian (1985, 1989),
Other imperfections related to our problem
Basak and Cuoco (1998), Detemple and Murthy (1994), Basak
(2000)
Distorded beliefs
Epstein and Wang (1994), Chauveau and Nalpas (1998) and
Hansen, Sargent and Tallarini (1999), Abel (2002), Wakker (2001,
2005), Chateauneuf (2005)
Empirical results and calibrations
Cragg and Malkiel (1982), Abel (1989), Ghysels (2002), Fried
and Givoly (1982), O’Brien (1988), Francis and Philbrick (1993),
Kang et al. (1994), Dreman and Berry (1995), Giordani and
Söderlind (2005).
Lectures based on a series of papers with Clotilde Napp
(Dauphine) :
• Pessimism, Riskiness, Risk Aversion and the Market price of
Risk, Submitted,
• Consensus consumer and intertemporal asset pricing with
heterogeneous beliefs, Submitted,
• Heterogeneous beliefs and asset pricing in discrete time: an
analysis of pessimism and doubt, to appear, Journal of Economic
Dynamics and Control
• Aggregation of heterogeneous beliefs, Submitted,
• Conditional comonotonicity, Decision in Economics and Finance,
2004, 27(2), 153 - 166
• Is there a pessimistic bias in individual beliefs ? Evidence from
survey data, with Napp and Ben Mansour, Working Paper,
The classical CCAPM model
• Time horizon T
´
³
Filtered probability space Ω, F, (Ft)Tt=0 , P
Investors indexed by i = 1, ..., N .
i
• Current income at date t denoted by e∗t
hP
i
T
• VNM utility function of the form E
t=0 ui (t, ct )
1. For all t, ui (t, ·) is of class C 1, strictly increasing and strictly
concave
2. For all t, ui (t, ·) satisfies Inada conditions,
PN ∗i
∗
3. The total wealth e ≡ i=1 e satisfies e≤ e∗t ≤ e uniformly in
(t, ω) for some positive constants e and e.
An equilibrium price process is a positive process q ∗ ≡ (qt∗)Tt=0
such that
N
X
¡ ∗ i¢
∗
i
e =
y q , e0
i=1
i
y (q, e0) = arg
E
PTmaxi
i
E [ t=0 qt yt ]≤e0
y i ≥0
" T
X
t=0
#
¢
i
¡
ui t, yt
Proposition (Negishi) For x = e, the solutions of
N
X
1
u (t, x) = Pmax
ui (t, xi)
N
λi
i=1 xi ≤x i=1
characterize Pareto optima and correspond to equilibria with transfer
payments.
An Arrow-Devreu equilibrium is a zero-tranfer equilibrium →
existence and
qt∗ = u0 (t, e∗t ) (representative agent)
We suppose the existence
¢
Qt ¡
0
0
f
• of a riskless asset with S0 = 1 and St = s=1 1 + rs for some
predictable risk free rate process rf .
• of a risky asset with rate of return between t and (t + 1) denoted
by Rt+1 ≡ SSt+1t − 1.
∗
P ∗
We must
have
q
S
=
E
[q
t
t
t
t+1 St+1] or equivalently,
· 0
¸
∗
p u (t + 1, et+1 )
St = Et
St+1
∗
0
· 0 u (t, et )∗ ¸
µ 0
¶
∗
¢
u (t + 1, et+1)
p u (t + 1, et+1 ) ¡
P
P
1 + Et [Rt+1] + covt
, Rt+1
1 = Et
u0 (t, e∗t )
u0 (t, e∗t )
The same equation for· the riskless asset
¸
0
∗
(t
+
1,
e
)
u
1
t+1
=
Etp
f
u0 (t, e∗t )
1 + rt+1
hence
"
#
u0 (t + 1, e∗t+1)
f
P
P
£ ¡
¢¤ , Rt+1
Et [Rt+1] − rt+1 = − − covt
P
∗
0
Et u t + 1, et+1
Heterogeneous beliefs and representative
agent
Investors indexed by i = 1, ..., N .
i
• Current income at date t denoted by e∗t
hP
i
T
i
i
• VNM utility function of the form E
M
u
(t,
c
)
(
M
i
t
t
t
t=0
density of Qi ∼ P )
1. For all t, ui (t, ·) is of class C 1, strictly increasing and strictly
concave
2. For all t, ui (t, ·) satisfies Inada conditions,
PN ∗i
∗
The total wealth e ≡ i=1 e satisfies e≤ e∗t ≤ e uniformly in
(t, ω) for some positive constants e and e.
• The different subjective beliefs are considered as given.
• They reflect difference of opinion among the agents rather than
difference of information; “we assume that investors receive
common information, but differ in the way they interpret this
information” (Harris and Raviv, 1993).
• The different subjective beliefs might come from a Bayesian
updating of the investors predictive distribution over the uncertain
returns on risky securities (Williams (1977), Detemple and
Murthy (1994), Zapatero (1998), Gallmeyer (2000), Basak
(2000), Gallmeyer and Hollifield (2002))
• We do not make such a specific assumption; we only impose that
the subjective probabilities be equivalent to the initial one.
• The models with learning are not ”more endogenous”, since the
investors’ updating rule and the corresponding probabilities are
determined separately from the optimization problem (Genotte,
1986).
An equilibrium price process is a positive process q ∗ ≡ (qt∗)Tt=0
such that
N
X
¡ ∗ i i¢
∗
i
e =
y q , M , e0
i=1
i
y (q, M, e0) = arg
E
PTmaxi
i
E [ t=0 qt yt ]≤e0
yi ≥0
" T
X
t=0
#
¢
i
¡
Mtui t, yt
Is there a representative agent and a consensus belief?
Negishi’s method
u (t, x) = Pmax
N
N
X
Mi
ui (t, xi)
λ
i
i=1 xi ≤x i=1
Proposition (Cuoco-He) For x = e, the solutions characterize
Pareto optima and correspond to equilibria with transfer payments.
An Arrow-Devreu equilibrium is a zero-tranfer equilibrium →
existence and
qt∗ = u0 (t, e∗t ) (representative agent)
Problem : In fact u(t, x) = u(t, x, ω)! In other words u is a
stochastic combination of the ui’s and we have to solve (n − 1)
stochastic backward "differential" equations in order to determine
these weights.
Tractable if n = 2 (Basak).
Adjusted wealth method
Our aim is (without modifying the utility functions)
• to find an “equivalent equilibrium” in which the heterogeneous
subjective beliefs would be aggregated into a common belief M ,
• such that, like the M i’s, M is a positive martingale process
satisfying E [MT ] = 1, i.e. the positive density of a probability
measure Q equivalent to P
• and we want the “equivalent equilibrium” to generate the same
equilibrium price process q ∗ (every asset gets the same valuation
in both equilibria).
If we succeed : classical representative agent + qualitative
properties.
In particular, we need
³
´
¡
¢
i
Mtu0i t, yti = αiMtiu0i t, yt∗
Example for u0i(t, x) = exp −(x/θ), we obtain
!1/N
à N !1/N à N
Y
Y
Mt =
αi
Mti
i=1
i=1
and M is a martingale only if all the Mi’s are equal.
In general no solution to the aggregation problem.
Calvet, Grandmont and Lemaire (2002) (static) propose
• a scalar adjustment of the total wealth
• to replace the invariance principle on the aggregate wealth by an
invariance principle on the individual marginal valuation of assets.
³ ³ i ´´
Proposition (Calvet et al.) Consider an equilibrium q ∗, y ∗
¡ i¢
relative to the beliefs M , and the income processes ei with
PN i
e ≡ e∗. There exist a unique positive and adapted process
i=1 ¡
¢T
¤
£
M = M t t=0 with E M T = 1, a unique coefficient of adjustment
¡ i¢
PN i
∗
r ∈ R+, a unique family of income processes e ¡with
i=1 e =
¢
re∗ and a unique family of consumption processes y i such that
¡ ∗ ¡ i¢¢
is an equilibrium relative to the common belief
1. q , y
¡ i¢
¡ ¢T
M = M t t=0 and the income processes e
2. Individual trading volumes and individual marginal valuation of
assets remain the same before and after the aggregation procedure,
i.e. for all i = 1, ..., N
³
´
¡
¢
∗i
i
i
i
i 0
∗i
0
i
y − e = y − e and Mt ui t, yt = M tui t, yt .
¤
£
The condition E M T = 1 is sufficient to interpret M as a
belief (the density of an equivalent probability measure Q) only if
T = 1 (Calvet et al.).
In general, M should be a martingale. Is this the case?
A couter-example
Let Ω ≡ {ω 1, ω 2}, P = (1/2, 1/2) , Q1 ≡ P and Q2 = ( 13 , 23 )
α+1
with α ∈ [−1, 0[ .
u1 (t, x) = u2 (t, x) = xα+1
³
´
i
i
Take e∗ such that M iu0 e∗ does not depend upon i and we
³ i´
fix q ∗ ≡ M iu0 e∗ .
³ ³ i ´´
¡ i¢
∗
∗
is an equilibrium relative to the beliefs M and
q , e
i
the income processes e∗ .
We look for M, y 1³, y 2´and r such that for i = 1, 2, we have
X
X i
¡ i¢
0
i 0
∗i
i
i = 1, 2 and
Mu y = M u e
y =r
e∗
It appears that
£ ¤
E M1
M0
i

i
1/α
2/3
1/3


´α + ³
´α 
= 2 ³
1 + (2/3)1/α
1 + (4/3)1/α
which is equal to one if and only if α = −1.
Except for the log utility function, all the utility functions in our
class lead to processes M that are not martingales.
³ ³ i ´´
Proposition (CGL-JN) Consider an equilibrium q ∗, y ∗
¡ i¢
relative to the beliefs Q and the income processes ei with
PN i
∗
i=1 e ≡ e . There exist a unique probability Q, a unique positive
and bounded predictable
r, a unique family of
¡ i¢ adjustment
PN i process
∗
income processes e with
e
=
re
and a unique family of
¡ i¢ i=1
consumption processes y such that
¡ ∗ ¡ i¢¢
is an equilibrium
1. q , y
¡ i¢ relative to the common belief Q and
the income processes e
2. Individual trading volumes and individual marginal valuation of
assets remain the same before and after the aggregation procedure,
i.e. for all i = 1, ..., N
∗i
y − ei = y i − ei
and
" T
#
" T
#
³
´
X
X ¡
¢
i
Qi
0
∗i
Q
0
E
ui t, yt Dt = E
ui t, yt Dt
t=0
t=0
for every asset generating the dividends (Dt).
• Prices and volumes in heterogeneous beliefs economy are
the same as in an homogeneous setting modulo a predictable
adjustment of the total wealth
• Stochastic wealth adjustment and deterministic weights in the
representative utility function construction instead of stochastic
weights and no adjustment (Basak-Cuoco, 1998, restricted market
participation setting)
• The representative agent has the same utility function as in the
classical setting
• In particular, if all utility functions are state independent then so
is the representative agent utility function
• Generalization of Calvet-Grandmont-Lemaire (2000) where
t = 0, 1 and where consumption takes place at the final date only.
In that case the adjustment is a scalar adjustment.
• We have r = 1 for the logarithmic utility functions case
(Rubinstein 1975, 1976, Detemple and Murthy 1994)
The HARA case
u0i (t,x)
− ui”(t,x)
= θi + ηx > 0
Proposition If
0
PN
uλ (t,x)
1. − uλ”(t,x) = θ + ηx, where θ = i=1 θi.
2. the consensus probabilityQ (with density M ) and the adjustment
process r are given
– when η 6= 0, by
!1/η " N
#1/η
Ã
∗
X ¡ ¢η
θ + ηrtet
i
M
Mt =
T
i
t
θ + ηe∗t
i=1
– when η = 0, by
N
Y
¡ i¢θi/θ
(rt − 1) e∗t
Mt
Mt =
exp
θ
i=1
4. The adjustment process r satisfies
r (t, ω) ≥ 1 if η < 1
r (t, ω) ≤ 1 if η > 1
r (t, ω) = 1 if η = 1
PN
5. If θ ≡
i=1 θ i = 0, then we obtain a simple construction
algorithm
" N
#
X ¡ ¢η
¡ ¢η
Mt = rt
γ i Mti
i=1
and the predictable adjustment process r satisfies
r % if η < 1
r & if η > 1
• In the standard setting, when there is more risk involved and
when the investor is cautious or not (η < 1) the investor increases
current consumption acting as if future wealth was increased
• In our context, a possible interpretation consists in considering
the dispersion of beliefs as a source of risk, thereby leading for
the cautious representative agent to an upward adjustment of
aggregate wealth and then to an increase of current consumption
• We have r = 1 for the logarithmic utility functions case
(Rubinstein 1975, 1976, Detemple and Murthy 1994)
• The consensus probability is somehow a risk-tolerance weighted
average of individual probabilities
• Abel proved that "pessimism" increases the risk premium, we are
then interested by the average level of "pessimism" and by the
"correlation" between risk tolerance and "pessimism"
Other properties of this aggregation procedure
In an homogeneous beliefs setting all the individual allocations
are comonotonic (risk sharing rule).
¡ i
¢
∗i
i
i
In our setting yt = y t + ϕt Mt , M t where the y it’s are
comonotonic and ϕit (x, y) is monotone with respect to x with
ϕit (x, x) = 0.
• Beliefs heterogeneity induces then a distortion of the risk sharing
rule and, for each agent, this distortion is monotone in individual
beliefs deviations from the aggregate probability.
• Each investor’s observed demand is larger than his demand in the
“equivalent equilibrium” if and only if he attaches a subjective
probability that is larger than the aggregate common probability,
i.e.
i
yt∗ ≥ y it if and only if Mti ≥ M t
• This property (with the equilibrium price invariance) characterizes
this aggregation procedure (and might replace then the individual
marginal utility invariance requirement).
The adjusted discount factor procedure
• In the previous procedure the adjustment factor interfers too much
with the consensus probability
• Not easy to disentangle the adjustment effect and the change of
probability effect in risk premium analysis
• Even if this factor enlights the impact of beliefs heterogeneity, it
is not completely satisfactory
VNM utility functions
the form
" of
#
T
X
E
MtiBtiui (t, ct)
t=0
(Mti
density of Q ∼ P, Bti predictable discount factor)
Example
 :

i
T
X

u
(t,
c
)
i 
i
t 
EQ 
,
t−1
 t=0 Q

(1 + rsi )
i
EQ
·Z
T
0
¸
i
e− 0 rsdsui (t, ct) dt
Rt
i=0
We want to aggregate
• individual beliefs into a consensus belief,
• individual utility functions into a representative agent utility
function,
• individual discount factors into a common discount factor
Proposition (Jouini-Napp) Consider an equilibrium price process
q ∗ relatively to the beliefs (M i), the discount factors (B i) and the
income processes (ei).
There exists a positive martingale process M̄ with M̄0 = 1, and a
predictable positive process B with B0 = 1 such that
M̄tBtu0(t, e∗t ) = q ∗.
In other words, there exists
• a consensus probability
• a common discount factor
• a "classical" representative agent
e∗t+1
e∗t
¢
¡
− θx
2
0
N µi, σ and ui(t, x) = e i then
• If under Qi, log
∼Ft
under Q,
θ
¢
¡ θ
e∗t+1
V
ar
B
(µ)
t+1
2
= exp −
log ∗ ∼Ft N E1 (µ) , σ and
et
Bt
2σ 2
¢
¡
e∗t+1
− θx
2
0
• If under Qi, log e∗ ∼Ft N µ, σ i and ui(t, x) = e i then under
t
Q,
¡ 2¢
θ
∗
θ
¡
¢¢
¡
σ
E
et+1
V
ar
B(t
+
1)
(σ)
−1
θ
2
log ∗ ∼Ft N µ, E−1 σ
and
=£
¤2 ∼ 1− θ 2
θ
et
B(t)
E1 (σ)
E1 (σ)
• More generally, with HARA utility functions,
– the consensus probability is, at the first order, a risk-tolerance
weighted arithmetic average of the M i
– the growth rate of B is, at the second order, proportional to the
dispersion of the M i
Continuous time example
We suppose that
de∗t = αte∗t dt + β te∗t dWt,
β ≥ 0,
dMti = δ itMtidWt,
M0i = 1
β it = β t
αit = αt + δ itβ t,
Then B and M̄ satisfy
dBt = µM (t)Btdt
dM̄t = δ M (t) M̄tdWt
The price q ∗ under heterogeneous beliefs = price in an economy
where
• all investors share the same belief Q with dQ
dP = M̄ and
ᾱt = αt + δ M (t)β t
• all investors
have the same ”discounted” utility function
Rt
exp 0 µM (s)ds × u(t, c) (possibly negative discount rate).
For linear risk tolerance utility functions, we have
δM =
N
X
i=1
i i
T
T δ = E [δ] ,
µM
1
= (η − 1)V arT [δ]
2
Two distinct effects of the introduction of some beliefs heterogeneity on the equilibrium price .
• A change of probability effect from P to the new common
probability Q,
– Weighted average of the individual subjective probabilities.
– The weights are given by the individual risk tolerances
– Q can be explicitly computed for l.r.t. utility functions
• An “aggregation bias”
– Of finite variation ↔ discount factor (possibly negative
discount rate)
– Can be computed for l.r.t. utility functions, associated to a
positive or negative discount rate depending on η
– Can be seen (at least in classical cases) as a measure of
dispersion of individual beliefs.
No such adjustment effect, i.e. B = 1 :
• if all investors share the same belief. The consensus belief M̄ is
equal to that common belief (homogeneity)
• if all the utility functions are logarithmic
In general, the discount factor makes the setting with heterogeneous beliefs fundamentally different from the homogeneous beliefs
setting
The impact of subjective beliefs
Risk-premium and risk-free rate
The market price of risk under heterogeneous beliefs is given by
MP R [heterogeneous] = MP R [standard] − δ M
= MP R [homogeneous under Q]
MPR (heter.) ≥ MPR (standard) ⇔ δ M ≤ 0 ⇔ Q pessimistic
Interpretation : under pessimism the representative agent
underestimates the rate of return of the risky asset ⇒ the equilibrium
risk premium%.
Abel (2000), (discrete time, uniform pessimism), Cechetti et al
(2000). (distorded belief)
Remark For linear risk tolerance utility functions, the consensus
probability belief Q is pessimistic if, for instance
• systematic pessimism at the individual level
• the optimistic (resp. pessimistic) investors have a risk tolerance
i
u0i (t,yt∗ )
Ti (t) ≡ − 00 ∗i lower (resp. higher) than the average
ui (t,yt )
– in the exponential case : risk averse agents are more optimistic
– in the power case : wealthier agents are more optimistic
Risk free rate
We obtain
rf = −µq∗ .
The risk free rate under heterogeneous beliefs is given by
rf [heterogeneous] = −µq − µM − δ M σ q
µ 00 ∗
¶
u (e )
= rf [standard] − µM + δ M − 0 ∗ βe∗
u (e )
= rf [homogeneous under Q] − µM
• The impact of the aggregation bias↔ µM .
µM ≥ 0 ⇒ risk free rate &.
Interpretation : (−µM ) ↔discount rate, a nonnegative µM ↔
future consumption is more important for the representative agent
⇒ lower equilibrium interest rate (Weil (1989) : ”a negative
discount...is a computer solution”).
• The impact of the change of probability effect from P to the
consensus probability Q
the risk free rate & ⇔ Q is pessimistic, i.e. δ M ≤ 0.
Interpretation : Under pessimism, the representative agent
underestimates the consumption growth rate ⇒ the equilibrium
risk free rate &
• Global effect : for small beliefs dispersion, the change of
probability effect is more important.
A two agents parametrization
Two agents and
σ R (t) = 16, 8%
µe(t) = 1.8%
σ e(t) = 3, 6%
(Mehra and Prescott, 1985).
δ 1 and δ 2 are measures of the return forecast errors
µi − µR
δi =
σR
2
ε = δ1−δ
2 measures beliefs dispersion
We have
w1 M1η +w2 M2η
η
• M ∼ w1+w2
2δ2
• E(δ M ) ∼ E( w1wδ11+w
+w2 )
The aggregate beliefs reflects the optimism/pessimism of the
wealthier agent.
If wealth is positively correlated with optimism we have
lower expected risk premium for assets with higher dispersion
(Consumer Sentiment Index of the University of Michigan reports
more optimistic forecasts for households with income>50 000$ than
for households with income<50 000$)
”Stocks with higher dispersion in analysts earnings forecast
earn lower future returns than otherwise similar stocks” (DMS) if
wealthier agents are more optimistic.
η between 0.5 and 1.1
w1 between 0.5 (same wealth) and 0.9 (optimistic agents are
much more wealthy)
δ̄ between −0.25 (growth is estimated in average at 2.7% instead
of 3.6%) and 0 (no aggregate bias)
ε between 0.30 (optimistic are 1.2% over the average) and 0 (no
dispersion)
(MP R%; r%)
η w1
0.5 0.5
0.9
0.9 0.5
0.9
1.1 0.5
0.9
δ̄ = 0
ε=0
ε = 0.10
(2.1;3.2) (2.1;2.9)
(2.1;3.2) (-0.2;3.7)
(1.2;1.8) (1.2;1.8)
(1.2;1.8) (-1.2;2.1)
(0.9;1.5) (0.9;1.5)
(0.9;1.5) (-1.4;1.8)
ε=0
(9.6;1.4)
(9.6;1.4)
(8.7;0.8)
(8.7;0.8)
(8.4,0.7)
(8.4,0.7)
δ̄ = −0.25
ε = 0.10 ε = 0.20
(9.6;1.1) (9.6;0.4)
(7.2;1.9) (4.8;2.2)
(8.7;0.8) (8.7;0.6)
(6.3;1.1) (3.9;1.4)
(8.4;0.7) (8.4;0.9)
(6.0;0.9) (3.6;1.3)
ε = 0.30
(9.6;-0.8)
(2.4;2.3)
(8.7;0.4)
(1.5;1.6)
(8.4;1.1)
(1.2;1.6)
Remarks on the aggregation procedure
• An aggregation procedure which permits to
– rewrite in a simple way the equilibrium characteristics (state
price density, market price of risk, risk premium, risk free rate)
– to compare them with an otherwise similar standard setting.
• In many cases, the impact can be easily estimated with a good
precision by considering the wealth weighted average belief.
• We explain Diether et al. (2002) findings.
• It is possible to construct specific parametrizations for which
– globally higher risk premia,
– lower risk free rates
– risk premia that are lower for assets with higher beliefs
dispersion.
• This aggregation procedure can also be used for the study of
models with constraints.
– Shortsales constraints : same allocations as in a model without
short sale constraint, where the initially constrained investors’
beliefs are replaced by more optimistic ones.⇒ optimistic
representative agent ⇒ MPR& and risk free rate%
– Borrowing constraints : oppposite results. Beliefs of constrained agents are replaced by more pessimistic ones ⇒
MPR% and risk free rate&
– Restricted market participation (Basak-Cuoco, 1998) : pessimistic aggregate probability, no aggregation bias (log
utility)⇒ MPR% and risk free rate&
• Pessimism and optimism play an important role
• Specific to continuous time models?
• Easy to define,
dQi
= Mi
dP
dMi = δ iMidW
β≥0
de = αdt + βdW P ,
de = (α + βδ i)dt + βdW P
• Characterizes Mi (at least locally)
Optimism and pessimism in a static setting
• We focus on the change of probability effect
• CCAPM like model with a subjective probability Q for the
representative agent
• standard 2 dates Lucas-fruit tree economy
• subjective belief/opinion for the representative agent Q << P,
dQ
dP = M
• standard utility maximization problem max u0(c0) + E Q [u(c)]
under the budget constraint
• characterization of the set of subjective probabilities leading to an
increase of the market price of risk for all utility functions in a
given class
• adjusted CCAPM formula
E P [R] − rf = −cov P
• classical CCAPM formula
·
E P [R] − rf = −cov P
0
¸
Mu (e)
,R .
P
0
E [Mu (e)]
·
¸
u0 (e)
,R .
E P [u0 (e)]
Measuring the subjective belief impact
the returns R obtained in equilibrium are not the same in both
settings
we consider an asset with terminal payoff = e (the aggregate wealth)
exists by market completeness but 6= price and return in both settings.
we propose to compare the market price of risk: ratio between the
risk premium and the “level of risk”
the subjective belief setting leads to a higher MPR iff
E Q [u0 (e) e]
E P [u0 (e) e]
− Q 0
≥− P 0
.
E [u (e)]
E [u (e)]
equivalent to rpQ(e) ≤ rpP (e)
Abel’s pessimism and doubt
Power utility functions
Impact of subjective beliefs on the RP (defined by
E P,Q [R]
1+rf )
RP Abel %⇐⇒ MP R %⇐⇒ rp(e) &
Pessimism : Q is pessimistic if P ºF SD Q (i.e. Q (e ≤ t) ≥
P (e ≤ t))
Uniform pessimism
Q (e ≤ t) = P (e ≤ t exp ∆) ,
∆>0
Proposition (Abel) Uniform pessimism⇒ MP R % (for power
utility functions).
FSD alone is not the right concept
Distribution functions of Q (Red) and P (Green). We have
P ºF SD Q.
1
0.75
0.5
0.25
0
1
1.5
2
2.5
3
Density of Pu = λu0 (Magenta) and dQ/dP (Blue). We have
Q|support(Pu) ºF SD P |support(Pu)
2
1
1.5
0.75
1
0.5
0.5
0.25
0
1
1.5
2
2.5
3
0
1
1.25
1.5
1.75
2
Even for power utility functions
Doubt : there is doubt if P ºSSD Q
Proposition (Abel) Doubt ⇒ MP R % (approximation for power
utility functions)
Our definitions
Converse approach
U1 : all nondecreasing functions s.t. lim supx→∞ u0(x) < ∞
U2 : all nondecreasing concave functions
Definition e ≥ 0 random variable on Ω, F = σ(e), Q1 and Q2
probability measures on (Ω, F ) such that E Qi [e] < ∞, i = 1, 2. We
say that Q1 <MP Ri Q2 when for all utility functions u in Ui,
E Q1 [u0 (e) e] E Q2 [u0 (e) e]
≤ Q2 0
Q
0
1
E [u (e)]
E [u (e)]
Remark There is no objective probability in the definition and there
is no a.c. conditions
Link with portfolio comparative statics
Equilibrium approach
X1 is cheaper than X2 if its relative equilibrium price is lower.
Definition X1 ºrpi X2 ⇐⇒ rp(X1) ≤ rp(X2), ∀u ∈ Ui
E[u0 (X1 )X1 ]
E[u0 (X2 )X2 ]
⇐⇒ − E[u0(X1)] ≥ − E[u0(X2)]
Let Q1 = PX1 and Q2 = PX2
Lemma The two following conditions are equivalent
a. X1 ºrpi X2
b. Q1 <MP Ri Q2 (for e(x) = x)
For X and ρ given we define
αX (ρ) = arg max u(wρ + α(X − ρ))
Lemma The following conditions are equivalent
a. X1 ºrpi X2
b. Q1 <MP Ri Q2 (for e(x) = x)
and imply αX1 (ρ) ≥ αX2 (ρ) , ∀u ∈ Ui, ∀ρ (more desirable)
Characterization of MPR1
Let Q1 and Q2 be given.
There exist X1 and X2 on (R, B, λ) s.t.
Qi(e ∈ A) = λ(Xi ∈ A) for all A ∈ B
Proposition (Landsberger and Meilijson) The following conditions
are equivalent
a. Q1 <MP R1 Q2
b. X1 ºMLR X2
c. Q2(e < e1) = 0, Q1(e > e2) = 0,
dQ2
dQ1
= h(e) % on [e1, e2]
and imply αX1 (ρ) ≥ αX2 (ρ) , ∀u ∈ U1, ∀ρ
Characterization of MPR2
Corollary The following conditions are equivalent
a. Q1 <MP R2 Q2
b. αX1 (ρ) ≥ αX2 (ρ) , ∀u ∈ U2, ∀ρ (portfolio dominance)
c. X1 º∩CR(ρ) X2 (Gollier)
d.
∀(x, y) ∈ R2+×R2+,
P2
P2
Q1
Q2
i=1 yi E [e1e≤xi ]
i=1 yi E [e1e≤xi ]
≤ P2
P2
Q
Q2
1
i=1 yi E [1e≤xi ]
i=1 yi E [1e≤xi ]
Simple properties and examples (MPR1)
1. If Q1 <MP R1 Q2, then for all x, Q1 [e ≤ x] ≤ Q1 [e ≤ x]
(stronger notion than Abel)
2. If e ∼Q1 N (µ1, σ 1) and e ∼Q2 N (µ2, σ 2) then Q1 <MP R1 Q2 ⇔
µ1 ≤ µ2 and σ 1 = σ 2
3. If ln e ∼Q1 N (µ1, σ 1) and ln e ∼Q2 N (µ2, σ 2) then Q1 <MP R1
Q2 ⇔ µ1 ≤ µ2 and σ 1 = σ 2
Notion of pessimism.
2
More generally dQ
dQ1 = h(e) % (pessimism).
Simple properties and examples (MPR2)
1. If Q1 <MP R2 Q2 and if E Q1 [e] = E Q2 [e] then V arQ1 [e] ≥
V arQ2 [e] (doubt)
2
2. If e under Q2 is symmetric (w.r. to ē) and if dQ
dQ1 = h(e) symmetric
(w. r.) to ē, &before ē and % after ē then Q1 <MP R2 Q2 (doubt)
y
2
1.5
1
0.5
0
-2.5
0
2.5
5
x
Simple properties and examples (MPR2)
1. If Q1 <MP R2 Q2 and if E Q1 [e] = E Q2 [e] then V arQ1 [e] ≥
V arQ2 [e] (doubt)
2
2. If e under Q2 is symmetric (w.r. to ē) and if dQ
dQ1 = h(e) symmetric
(w. r.) to ē, &before ē and % after ē then Q1 <MP R2 Q2 (doubt)
3. If for all x, E Q1 [e1e≤x] ≤ E Q2 [e1e≤x] and Q1 {e ≤ x} ≥
Q2 {e ≤ x} , then Q1 <MP R2 Q2 (pessimism)
4. If e ∼Q1 N (µ1, σ 1) and e ∼Q2 N (µ2, σ 2) then Q1 <MP R1 Q2 ⇔
µ1 ≤ µ2 and σ 1 ≥ σ 2 (pessimism and doubt)
Remark a. 3. is weaker than MP R1 (or MLR)
b. MP R2 ; F SD
FSD+...; MLR : e = (0, 1, 2), Q1 = ( 13 , 13 , 13 ) in Red and
Q2 = ( 18 , 48 , 38 ) in Green. Solid line = distribution. Dashed line =
E Qi [e1e≤x] (resp. E Q2 [e1e≤x]).Density = Blue.
2.5
2
1.5
1
0.5
0
0
0.5
1
1.5
2
MPR2; F SD : e = (0, 1, 2), Q1 = ( 13 , 13 , 13 ) in Red and
Q2 = ( 18 , 58 , 28 ) in Green. Solid line = distribution. Dashed line =
E Qi [e1e≤x] (resp. E Q2 [e1e≤x]). In the second figure, Green×λ with
λ = 16
18 .
1
1
0.75
0.75
0.5
0.5
0.25
0.25
0
0
0.5
1
1.5
2
0
0
0.5
1
1.5
2
Risk aversion
Equilibrium approach
v is more risk averse than u in the sense of the MPR when the risk
premium for v is higher than for u
Definition v <MP R u when for all (Ω, F, P ) and all e,
E[u0 (e)e]
E[u0 (e)]
E[v0 (e)e]
E[v 0(e)]
Proposition For utility functions u and v in U2, the following
conditions are equivalent
a. v <MP R u
b. h ≡
v0
u0
is nonincreasing
c. v = T ◦ u for some concave function T
d. if we further assume that u0 and v 0 are continuous then the
00
00
previous conditions are also equivalent to − vv0 ≥ − uu0 .
≥
Optimism and pessimism in a dynamic
setting
We say that a probability Q on (Ω, F, P ) equivalent to P, with
density process (Mt) is pessimistic (with respect to e∗) if for all t,
Mt+1 and −e∗t+1 are comonotonic conditionally to Ft.
A pessimistic probability is such that
• Mt+1 decreases with e∗t+1 conditionally to Ft for all t
• conditionally to date t information, it puts more weight on states
of nature where e∗t+1 is low
• (in a tree information structure) for each t and each date−t node
, the transition density between and its successors decreases
with e∗t+1
e∗t+1
e∗t
• If
∼ N (µ, σ) (resp. N (b
µ, σ
b)) under P (resp. Q) then Q is
pessimistic if and only if µ ≥ µ
b and σ = σ
b.
y
1
0.75
0.5
0.25
0
-2.5
0
2.5
5
x
e∗t+1
e∗t
• If
∼ N (µ, σ) (resp. N (b
µ, σ
b)) under P (resp. Q) then Q is
b and σ = σ
b.
pessimistic if and only if µ ≥ µ
• The same result holds for lognormal distributions and remains
valid if µ and σ are Ft measurable random variables and if the
distributions are replaced by conditional distributions
y
1
0.75
0.5
0.25
0
5
10
15
20
25
30
35
x
e∗t+1
e∗t
• If
∼ N (µ, σ) (resp. N (b
µ, σ
b)) under P (resp. Q) then Q is
pessimistic if and only if µ ≥ µ
b and σ = σ
b.
• The same result holds for lognormal distributions and remains
valid if µ and σ are Ft measurable random variables and if the
distributions are replaced by conditional distributions
• If (e∗t ) follows a CRR binomial process with ”returns” u and d
and associated transition probabilities π u and π d = 1 − π u (resp.
π
bu and π
bd = 1 − π
bu) under P (resp. Q), then Q is pessimistic if
and only if π u ≥ π
bu .
• The result remains valid if we introduce a Ft−measurable.time
and state dependence for u, d, π u and π
bu
Suppose that for all t, the Ft−conditional distribution of e∗t+1
under P is symmetric with respect to Et [e∗t+1] .
We say that a probability Q on (Ω, F, P ) equivalent to P, with
density process (Mt) exhibits doubt (resp. overconfidence) between
date t and (t + 1) (with respect to e∗) if for all t,
0
Mt+1
(ω) = ft(ω, e∗t+1 (ω) − Et [e∗t+1] (ω))
where ft is Ft−measurable with respect to its first variable, even and
nondecreasing (resp. nonincreasing) on R+ with respect to its second
variable.
• A probability measure, equivalent to P , exhibits doubt (resp.
overconfidence) between date t and (t + 1) if conditionally to date
t information, its density puts more (resp. less) weight on the tails
and less (resp. more) weight on the center of the distribution.
e∗t+1
e∗t
• If
∼Ft N (µ, σ) (resp. N (µ̃, σ̃)) under P (resp. Q), then Q
exhibits doubt (with respect to e∗) between t and (t + 1) if and
only if µ = µ̃ and σ ≤ σ̃.
y
2
1.5
1
0.5
0
-2.5
0
2.5
5
x
• A probability measure, equivalent to P , exhibits doubt (resp.
overconfidence) between date t and (t + 1) if conditionally to date
t information, its density puts more (resp. less) weight on the tails
and less (resp. more) weight on the center of the distribution.
e∗t+1
e∗t
• If
∼Ft N (µ, σ) (resp. N (µ̃, σ̃)) under P (resp. Q), then Q
exhibits doubt (with respect to e∗) between t and (t + 1) if and
only if µ = µ̃ and σ ≤ σ̃.
• If (e∗t ) follows a trinomial process with ”returns” at each period
u, m and d and associated transition probabilities π, 1 − 2π, and π
(resp. π̃, 1 − 2π̃ and π̃) under P (resp. Q), then Q exhibits doubt
if and only if π ≤ π̃.
Impact of pessimism and doubt on the market price of risk
Proposition
1. If the probability Q is pessimistic or exhibits doubt in the sense of
the previous definitions, then
MP Rsubj(Q) ≥ MP Robj
2. If Q exhibits pessimism and if the representative agent’s utility
function is concave and nondecreasing, then
subj(Q)
rf
≤ rf obj
3. If the subjective probability belief Q exhibits doubt and if the
representative agent’s utility function is nondecreasing, concave,
with a convex derivative, then
subj(Q)
rf
≤ rf obj
Application to the heterogeneous beliefs model
Proposition
1. If the consensus probability belief Q exhibits pessimism and/or
doubt then the market price of risk in the heterogeneous beliefs
setting is lower than in the standard objective belief setting.
2. If the representative agent utility function is in the HARA class
u0 (t,x)
(i.e. − u00(t,x) = θ + ηx) with a cautiousness parameter η > 1
and if Q exhibits pessimism and/or doubt then the equilibrium
interest rate in the heterogeneous beliefs setting is lower than in
the standard objective belief setting.
From individual to aggregate pessimism/doubt : an illustration
QN ¡ i¢θi/θ̄
With exponential utility functions, we have M = i=1 M
.
e∗t+1
e∗t+1
If e∗ ∼Ft N (µ, σ) under P , and e∗ ∼Ft N (µi, σ) under Qi.
t
t
³P
´
∗
e
N
θi
We have et+1
µ
∗ ∼Ft N
i=1 i θ , σ under Q.
t
If all θi are equal, the impact on the market price of risk is
given by the pessimism/optimism of the “equal-weighted average”
investor.
PN
1
If³the θi are different,
the expected
return under Q is N i=1 µi +
´
³
´
PN
PN
θi
1
1
µ
−
µ
−
i
i
i=1
i=1
N
N .
θ
• first effect : average level of optimism/pessimism
• second effect : “correlation” between risk tolerance/aversion and
optimism/pessimism
A third effect : the cautiousness effect
³P
´ η1
1
η
N
i
u0i(x) = (θi + ηx)− η and M =
.
i=1 γ i M
Assume that
• all the agents are equally weighted in the definition of M (no
weights effect)
• there is no systematic
bias in the average returns individual
P
estimations (i.e. N1 N
i=1 µi = µ).
The equally weighted geometric average belief is then equal to
the objective belief and the consensus belief is an equally weighted
average of the individual beliefs.
However, this last average is an η−average and not a geometric
average and, contrarily to the exponential case, the consensus belief
is not equal to the objective one.
In fact, Q exhibits doubt and the market price of risk is higher
than in the standard setting. It is easy to check that this doubt effect
increases with the cautiousness parameter η.
From individual to aggregate pessimism/doubt : general result
u0i (t,x)
− u00(t,x)
i
= θi + ηx for η 6= 0, and that the
Proposition Assume that
equally-weighted η 0−average of the individual beliefs corresponds
to the objective probability P.
1. If η = η 0 and if the pessimistic (resp. optimistic) agents have a
risk tolerance higher (resp. lower) than the average, then
MP Rsubj(Q) ≥ MP Robj
2. If η = η 0 and if the agents that exhibit doubt (resp. overconfidence)
have a risk tolerance higher (resp. lower) than the average, then
MP Rsubj(Q) ≥ MP Robj
3. η > η 0 generates additional doubt
Behavioral issues
Are individuals pessimistic on average?
Difficult question and many different approaches!
Psychology
Notion based on the concept of pessimism as a negative
conception of life. Two different measures
1. in relation to events which have a direct impact on the well being
of the individual
2. in relation to events which have a direct impact on the well being
of the society.
Intuitively, our pessimism (beliefs about investment opportunities) should be close to the notion of personal pessimism.
To measure personal pessimism: questions aim at evaluating the
way they perceive their future.
Takes into account how individuals might influence future events
(self esteem or overconfidence)!
Our aim : direct personal impact but exogeneous events
Empirical studies
Analysis of analysts, economists from industry, government,
banking forecasts on earnings, dividends or on GDP, consumption,
etc..
Fried and Givoly (1982), O’Brien (1988), Francis and Philbrick
(1993), Kang et al. (1994) and Dreman and Berry (1995) provide
evidence that analysts’ forecasts on earnings are overly optimistic.
Converse result about professional forecasts on GDP in Giordani
and Söderlind (2005).
Schipper (1991), Mc Nichols and O’Brien (1997), Abarbanell
and Lehavy (2001), Darrough and Russell (2002) : professionals’
forecasts may be biased by environmental factors.
• many analysts are employed by brokerage firms : incentives to
promote purchase of stock or maintain acccess to top executives
• because of their close contact with company management,
analysts take on characteristics of insiders and tend to overweight
good news and underweight bad news (Darrough and Russell,
2000) : Kahneman and Lovallo’s (1993) insider bias
Decision theory : 1st set of experimental studies
Prospect Theory, Rank Dependent Expected Utility, Cumulative
Prospect Theory,... (Quiggin, Yaari,...)
Aim : to calibrate models of Prospect Theory in order to
determine the shape of the probability weighting function.
Parametric (Tversky-Kahneman, 1992, Camerer-Ho, 1994,
Wu-Gonzalez, 1996, Gonzalez-Wu, 1999) as well as non parametric
approaches (Prelec, 1998, Bleichrodt-Pinto, 2000, Abdellaoui, 2000)
They all agree on an inverse S-shaped probability weighting
function, which means that it overweights unlikely (extreme)
outcomes and underweights outcomes with a medium or large
probability relative to the objective probability.
Distortion
function
w
1
.51
1/3
2/3
Events ranked from the best (0) to the worst (1)
The slope = relative probability
Convex shape = pessimism
Concave shape = optimism
Conclusion : overweight extreme events, insensitivity in the middle, dose of pessimism
0
1
p
Learning : 2nd set of experimetal studies
Aim : to investigate forecasting ability of human subjects in an
effort to identify possible sources of forecasts bias: Affleck-Graves
et al. (1990), Maines and Hand (1996), Calegari and Fargher (1997)
and Gillette et al. (1999).
Subjects are given data on the EPS or dividends of a given asset
and are asked to forecast the next EPS or the next dividends.
Authors find that the forecasts exhibit significant positive bias
and conclude that there is an optimism bias.
Data do not correspond to something owned by the participants!
High values are neither “good” nor “bad”
They only reflect the way individuals extrapolate future terms of
a partially observed series of numbers.
Requirements for a good measure instrument
Based on hypothetical scenarios : to avoid environmental effects
(analysts)
Sample large enough for cross sectional analysis
Consistent with pessimism as a transformation of an objective
distribution into a subjective one.
Setting simple enough so that the obtained measure is independent of the choice of a specific decision theory model.
Should lead to a direct measure of the level of optimism/pessimism
and should not involve other individual characteristics like risk aversion, or other feelings like overconfidence, loss aversion, regret,
doubt, etc.
States should be identified as unambiguously good (resp.
bad) and clearly correspond to good (resp. bad) outcomes for the
individual.
States should be equiprobable to avoid underweighting of high
probabilities.
A simple experiment
"The participant is offered the opportunity of entering a heads or
tails game in one draw. More precisely, a coin is being tossed once;
if heads occurs, the participant is supposed to get 10 Euros, and if
tails occurs, the participant is supposed to get nothing. Then the
participant is confronted with the opportunity to play ten times this
game. The question consists in asking for his/her own estimation,
according to his/her experience and his/her luck, of the number of
times heads will occur, i.e. how many times (out of ten) he/she thinks
he/she is going to win."
Is there a correlation between pessimism/doubt and risk
tolerance?
• direct measures : risk averse agents are more optimistic
• optimal illusions and disappointment aversion (Gul, 1991)
• differential information
• Bayesian learning process (model+experiment)