Mathematical Behavioural...
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Mathematical Behavioural Finance
Mathematical Behavioural Finance
Xunyu Zhou
University of Oxford
May 2015 @ IPAM
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Mathematical Behavioural Finance
Part 3:
Market Equilibrium and Asset Pricing
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Mathematical Behavioural Finance
1 An Arrow-Debreu Economy
2 Individual Optimality
3 Representative RDUT Agent
4 Asset Pricing
5 CCAPM and Interest Rate
6 Equity Premium and Risk-Free Rate Puzzles
7 Summary and Further Readings
8 Final Words
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Mathematical Behavioural Finance
An Arrow-Debreu Economy
Section 1
An Arrow-Debreu Economy
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Mathematical Behavioural Finance
An Arrow-Debreu Economy
The Economy
Present date t = 0 (today) and a future date t = 1(tomorrow)
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Mathematical Behavioural Finance
An Arrow-Debreu Economy
The Economy
Present date t = 0 (today) and a future date t = 1(tomorrow)
(Ω,F ,P) at t = 1
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Mathematical Behavioural Finance
An Arrow-Debreu Economy
The Economy
Present date t = 0 (today) and a future date t = 1(tomorrow)
(Ω,F ,P) at t = 1
A single consumption good
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Mathematical Behavioural Finance
An Arrow-Debreu Economy
The Economy
Present date t = 0 (today) and a future date t = 1(tomorrow)
(Ω,F ,P) at t = 1
A single consumption good
A finite number of agents indexed by i = 1, . . . , I
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Mathematical Behavioural Finance
An Arrow-Debreu Economy
The Economy
Present date t = 0 (today) and a future date t = 1(tomorrow)
(Ω,F ,P) at t = 1
A single consumption good
A finite number of agents indexed by i = 1, . . . , I
Agent i has an endowment (e0i, e1i), where e0i is wealthtoday and F-measurable random variable e1i is randomendowment tomorrow
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Mathematical Behavioural Finance
An Arrow-Debreu Economy
The Economy
Present date t = 0 (today) and a future date t = 1(tomorrow)
(Ω,F ,P) at t = 1
A single consumption good
A finite number of agents indexed by i = 1, . . . , I
Agent i has an endowment (e0i, e1i), where e0i is wealthtoday and F-measurable random variable e1i is randomendowment tomorrow
Aggregate endowment is (e0, e1) :=(
∑Ii=1 e0i,
∑Ii=1 e1i
)
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Mathematical Behavioural Finance
An Arrow-Debreu Economy
Consumption Plans
Agents choose consumption for t = 0, and claims onconsumption for t = 1
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Mathematical Behavioural Finance
An Arrow-Debreu Economy
Consumption Plans
Agents choose consumption for t = 0, and claims onconsumption for t = 1
A feasible consumption plan of agent i is a pair (c0i, c1i),where c0i ≥ 0 is wealth consumed today and F-measurablerandom variable c1i ≥ 0 that consumed tomorrow
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Mathematical Behavioural Finance
An Arrow-Debreu Economy
Consumption Plans
Agents choose consumption for t = 0, and claims onconsumption for t = 1
A feasible consumption plan of agent i is a pair (c0i, c1i),where c0i ≥ 0 is wealth consumed today and F-measurablerandom variable c1i ≥ 0 that consumed tomorrow
The preference of agent i over (c0i, c0i) is represented by
Vi(c0i, c1i) = u0i(c0i) + βi
∫
u1i(c1i)d(wi P),
whereu0i is utility function for t = 0;(u1i, wi) is the RDUT pair for t = 1;βi ∈ (0, 1] is time discount factor
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Mathematical Behavioural Finance
An Arrow-Debreu Economy
Consumption Plans
Agents choose consumption for t = 0, and claims onconsumption for t = 1
A feasible consumption plan of agent i is a pair (c0i, c1i),where c0i ≥ 0 is wealth consumed today and F-measurablerandom variable c1i ≥ 0 that consumed tomorrow
The preference of agent i over (c0i, c0i) is represented by
Vi(c0i, c1i) = u0i(c0i) + βi
∫
u1i(c1i)d(wi P),
whereu0i is utility function for t = 0;(u1i, wi) is the RDUT pair for t = 1;βi ∈ (0, 1] is time discount factor
The set of all feasible consumption plans is denoted by C
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Mathematical Behavioural Finance
An Arrow-Debreu Economy
Pricing Kernel
The above economy is denoted by
E :=
(Ω,F ,P), (e0i, e1i)Ii=1, C ,
(
Vi(c0i, c1i))I
i=1
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Mathematical Behavioural Finance
An Arrow-Debreu Economy
Pricing Kernel
The above economy is denoted by
E :=
(Ω,F ,P), (e0i, e1i)Ii=1, C ,
(
Vi(c0i, c1i))I
i=1
A pricing kernel (or state-price density, stochastic discountfactor) is an F-measurable random variable ρ, withP(ρ > 0) = 1, E[ρ] < ∞ and E[ρe1] < ∞, such that anyclaim x tomorrow is priced at E[ρx] today
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Mathematical Behavioural Finance
An Arrow-Debreu Economy
Arrow-Debreu Equilibrium
An Arrow–Debreu equilibrium of E is a collection
ρ, (c∗0i, c∗1i)
Ii=1
consisting of a pricing kernel ρ and a collection (c∗0i, c∗1i)
Ii=1 of
feasible consumption plans, that satisfies the following conditions:
Individual optimality : For every i, (c∗0i, c∗1i) maximises the
preference of agent i subject to the budgetconstraint, that is,
Vi(c∗0i, c
∗1i) = max
(c0i,c1i)∈C
Vi(c0i, c1i)
subject to c0i + E[ρc1i] ≤ e0i + E[ρe1i]
Market clearing :∑I
i=1 c∗0i = e0 and
∑Ii=1 c
∗1i = e1
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Mathematical Behavioural Finance
An Arrow-Debreu Economy
Literature
Mainly on CPT economies, and on existence of equilibria
Qualitative structures of pricing kernel for both CPT and SP/Aeconomies, assuming existence of equilibrium: Shefrin (2008)Non-existence: De Giorgi, Hens and Riegers (2009), Azevedoand Gottlieb (2010)Under specific asset return distribution: Barberis and Huang(2008)One risky asset: He and Zhou (2011)
RDUT economy with convex weighting function: Carlier andDana (2008), Dana (2011) – existence
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Mathematical Behavioural Finance
An Arrow-Debreu Economy
Standing Assumptions
Agents have homogeneous beliefs P; (Ω,F ,P) admits noatom.
For every i, e0i ≥ 0, P(e1i ≥ 0) = 1, ande0i + P(e1i > 0) > 0. Moreover, e1 is atomless,P(e1 > 0) = 1, and e0 > 0.
For every i, u0i, u1i : [0,∞) → R are strictly increasing,strictly concave, continuously differentiable on (0,∞), andsatisfy the Inada condition: u′0i(0+) = u′1i(0+) = ∞,u′0i(∞) = u′1i(∞) = 0. Moreover, u1i(0) = 0.
For every i, wi : [0, 1] → [0, 1] is strictly increasing andcontinuously differentiable, and satisfies wi(0) = 0, wi(1) = 1.
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Mathematical Behavioural Finance
Individual Optimality
Section 2
Individual Optimality
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Mathematical Behavioural Finance
Individual Optimality
Individual Consumptions
Consider
Max(c0,c1)∈C
V (c0, c1) := u0(c0) + β
∫ ∞
0w(
P(
u1(c1) > x))
dx
subject to c0 + E[ρc1] ≤ ε0 + E[ρε1]
(1)
where ρ is exogenously given, atomless, and ε0 and ε1 areendowments at t = 0 and t = 1 respectively
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Mathematical Behavioural Finance
Individual Optimality
Quantile Formulation
Recall the set of quantile functions of nonnegative randomvariables
G = G : [0, 1) → [0,∞] non-decreasing and right-continuous,
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Mathematical Behavioural Finance
Individual Optimality
Quantile Formulation
Recall the set of quantile functions of nonnegative randomvariables
G = G : [0, 1) → [0,∞] non-decreasing and right-continuous,
Problem (1) can be reformulated as
Maxc0≥0, G∈G
U(c0, G) := u0(c0) + β
∫ 1
0u1(G(p))dw(p)
subject to c0 +
∫ 1
0F−1ρ (1− p)G(p)dp ≤ ε0 + E[ρε1],
(2)
where w(p) = 1− w(1 − p)
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Mathematical Behavioural Finance
Individual Optimality
Quantile Formulation
Recall the set of quantile functions of nonnegative randomvariables
G = G : [0, 1) → [0,∞] non-decreasing and right-continuous,
Problem (1) can be reformulated as
Maxc0≥0, G∈G
U(c0, G) := u0(c0) + β
∫ 1
0u1(G(p))dw(p)
subject to c0 +
∫ 1
0F−1ρ (1− p)G(p)dp ≤ ε0 + E[ρε1],
(2)
where w(p) = 1− w(1 − p)
If (c∗0, G∗) ∈ [0,∞) ×G solves (2), then (c∗0, c
∗1), where
c∗1 = G∗(1− Fρ(ρ)), solves (1)
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Mathematical Behavioural Finance
Individual Optimality
Lagrange
Step 1. For a fixed Lagrange multiplier λ > 0, solve
Maxc0≥0, G∈G
u0(c0) + β
∫
1
0
u1(G(p))dw(p)
− λ
(
c0 +
∫ 1
0
F−1
ρ (1− p)G(p)dp− ε0 − E[ρε1]
)
.
The solution (c∗0, G∗) implicitly depends on λ
Step 2. Determine λ by
c∗0 +
∫ 1−
0F−1ρ (1− p)G∗(p)dp = ε0 + E[ρε1]
Step 3. c∗1 := G∗(1− Fρ(ρ))
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Mathematical Behavioural Finance
Individual Optimality
Finding Quantile
Obviously c∗0 = (u′0)−1(λ)
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Mathematical Behavioural Finance
Individual Optimality
Finding Quantile
Obviously c∗0 = (u′0)−1(λ)
So ultimately we need to solve
MaxG∈G
U(G;λ) :=
∫
1
0
u1(G(p))dw(p)−λ
β
∫
1
0
F−1
ρ (1− p)G(p)dp
=
∫ 1
0
[
u1(G(p))w′(1− p)−λ
βF−1
ρ (1− p)G(p)
]
dp
(3)
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Mathematical Behavioural Finance
Individual Optimality
Finding Quantile
Obviously c∗0 = (u′0)−1(λ)
So ultimately we need to solve
MaxG∈G
U(G;λ) :=
∫
1
0
u1(G(p))dw(p)−λ
β
∫
1
0
F−1
ρ (1− p)G(p)dp
=
∫ 1
0
[
u1(G(p))w′(1− p)−λ
βF−1
ρ (1− p)G(p)
]
dp
(3)
We have solved this problem ... provided that
M(z) = w′(1−z)
F−1
ρ (1−z)satisfies some monotone condition!
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Mathematical Behavioural Finance
Individual Optimality
Finding Quantile
Obviously c∗0 = (u′0)−1(λ)
So ultimately we need to solve
MaxG∈G
U(G;λ) :=
∫
1
0
u1(G(p))dw(p)−λ
β
∫
1
0
F−1
ρ (1− p)G(p)dp
=
∫ 1
0
[
u1(G(p))w′(1− p)−λ
βF−1
ρ (1− p)G(p)
]
dp
(3)
We have solved this problem ... provided that
M(z) = w′(1−z)
F−1
ρ (1−z)satisfies some monotone condition!
Difficulty: Such a condition (or literally any condition) is notpermitted in our equilibrium problem!
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Mathematical Behavioural Finance
Individual Optimality
Calculus of Variation
Set
G0 = G : [0, 1) → [0,∞] |G ∈ G and G(p) > 0 for all p ∈ (0, 1)
Calculus of variation shows that solving (3) is equivalent tofinding G ∈ G0 satisfying
∫ 1
q
u′1(G(p))dw(p)−λ
β
∫ 1
q
F−1ρ (1− p)dp ≤ 0 ∀q ∈ [0, 1),
∫ 1
0
(∫ 1−
q
u′1(G(p))dw(p)−λ
β
∫ 1
q
F−1ρ (1− p)dp
)
dG(q) = 0
(4)
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Mathematical Behavioural Finance
Individual Optimality
Equivalent Condition
Previous condition is equivalent to
K(q) ≥λ
βN(q) for all q ∈ (0, 1),
K is affine on
q ∈ (0, 1) : K(q) >λ
βN(q)
,
K(0) =λ
βN(0), K(1−) = N(1−)
(5)
where
K(q) = −
∫ 1
q
u′1(G(w−1(p)))dp
N(q) = −
∫ 1
q
F−1ρ (1− w−1(p))dw−1(p)
(6)
for all q ∈ [0, 1)
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Mathematical Behavioural Finance
Individual Optimality
How Concave Envelope is Formed
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Mathematical Behavioural Finance
Individual Optimality
How Concave Envelope is Formed
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Mathematical Behavioural Finance
Individual Optimality
How Concave Envelope is Formed
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Mathematical Behavioural Finance
Individual Optimality
How Concave Envelope is Formed
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Mathematical Behavioural Finance
Individual Optimality
Concave Envelope
K = λβN where N is concave envelope of N
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Mathematical Behavioural Finance
Individual Optimality
Concave Envelope
K = λβN where N is concave envelope of N
Recall K(q) = −∫ 1qu′1(G(w−1(p)))dp
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Mathematical Behavioural Finance
Individual Optimality
Concave Envelope
K = λβN where N is concave envelope of N
Recall K(q) = −∫ 1qu′1(G(w−1(p)))dp
We have u′1(G∗(1− w−1(1− q))) = K ′(q) = λ
βN ′(q) where
N ′ is right derivative of N
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Mathematical Behavioural Finance
Individual Optimality
Concave Envelope
K = λβN where N is concave envelope of N
Recall K(q) = −∫ 1qu′1(G(w−1(p)))dp
We have u′1(G∗(1− w−1(1− q))) = K ′(q) = λ
βN ′(q) where
N ′ is right derivative of N
G∗(q) = (u′1)−1
(
λβN ′(1− w(1− q))
)
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Mathematical Behavioural Finance
Individual Optimality
Concave Envelope
K = λβN where N is concave envelope of N
Recall K(q) = −∫ 1qu′1(G(w−1(p)))dp
We have u′1(G∗(1− w−1(1− q))) = K ′(q) = λ
βN ′(q) where
N ′ is right derivative of N
G∗(q) = (u′1)−1
(
λβN ′(1− w(1− q))
)
c∗1 = G∗(1− Fρ(ρ)) = (u′1)−1
(
λβN ′
(
1− w(Fρ(ρ))))
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Mathematical Behavioural Finance
Individual Optimality
Complete/Explicit Solution to Individual Consumption
Theorem
(Xia and Zhou 2012) Assume that ρ > 0 a.s., atomless, withE[ρ] < +∞. Then the optimal consumption plan is given by
c∗0 = (u′0)−1(λ)
c∗1 = (u′1)−1
(
λ
βN ′
(
1− w(Fρ(ρ)))
)
,
where λ is determined by
(u′0)−1(λ) + E
[
ρ(u′1)−1
(
λ
βN ′
(
1− w(Fρ(ρ)))
)]
= ε0 + E[ρε].
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Mathematical Behavioural Finance
Individual Optimality
Concavity of N
N(q) = −∫ 1q
F−1
ρ(w−1(1−p))
w′(w−1(1−p))dp
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Mathematical Behavioural Finance
Individual Optimality
Concavity of N
N(q) = −∫ 1q
F−1
ρ(w−1(1−p))
w′(w−1(1−p))dp
N being concave iffF−1
ρ(p)
w′(p) being non-decreasing, or
M(z) = w′(1−z)
F−1
ρ(1−z)
being non-decreasing!
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Mathematical Behavioural Finance
Individual Optimality
Concavity of N
N(q) = −∫ 1q
F−1
ρ(w−1(1−p))
w′(w−1(1−p))dp
N being concave iffF−1
ρ(p)
w′(p) being non-decreasing, or
M(z) = w′(1−z)
F−1
ρ(1−z)
being non-decreasing!
When N is concave:
c∗1 = (u′1)−1
(
λ
β
ρ
w′(Fρ(ρ))
)
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Mathematical Behavioural Finance
Individual Optimality
Concavity of N
N(q) = −∫ 1q
F−1
ρ(w−1(1−p))
w′(w−1(1−p))dp
N being concave iffF−1
ρ(p)
w′(p) being non-decreasing, or
M(z) = w′(1−z)
F−1
ρ(1−z)
being non-decreasing!
When N is concave:
c∗1 = (u′1)−1
(
λ
β
ρ
w′(Fρ(ρ))
)
It recovers one of the results in Part 2!
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Mathematical Behavioural Finance
Representative RDUT Agent
Section 3
Representative RDUT Agent
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Mathematical Behavioural Finance
Representative RDUT Agent
Return to Economy E : Aggregate Consumption
Assumption. Agents have homogeneous probabilityweighting function w
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Mathematical Behavioural Finance
Representative RDUT Agent
Return to Economy E : Aggregate Consumption
Assumption. Agents have homogeneous probabilityweighting function w
Optimal consumption plan of agent i is
c∗0i = (u′0i)−1(λ∗
i ), c∗1i = (u′1i)
−1
(
λ∗i
βiN ′
(
1− w(Fρ(ρ)))
)
,
where λ∗i satisfies
(u′0i)
−1(λ∗i ) + E
[
ρ(u′1i)
−1
(
λ∗i
βi
N ′(
1− w(Fρ(ρ)))
)]
= e0i + E[ρe1i]
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Mathematical Behavioural Finance
Representative RDUT Agent
Return to Economy E : Aggregate Consumption
Assumption. Agents have homogeneous probabilityweighting function w
Optimal consumption plan of agent i is
c∗0i = (u′0i)−1(λ∗
i ), c∗1i = (u′1i)
−1
(
λ∗i
βiN ′
(
1− w(Fρ(ρ)))
)
,
where λ∗i satisfies
(u′0i)
−1(λ∗i ) + E
[
ρ(u′1i)
−1
(
λ∗i
βi
N ′(
1− w(Fρ(ρ)))
)]
= e0i + E[ρe1i]
Aggregate consumption is
c∗0 =
I∑
i=1
(u′0i)−1(λ∗
i ), c∗1 =
I∑
i=1
(u′1i)−1
(
λ∗i
βiN ′
(
1− w(Fρ(ρ)))
)
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Mathematical Behavioural Finance
Representative RDUT Agent
A Representative Agent
For λ1 > 0, . . . , λI > 0, set λ = (λ1, . . . , λI) and
h0λ(y) :=
I∑
i=1
(u′0i)
−1 (λiy) , h1λ(y) :=
I∑
i=1
(u′1i)
−1
(
λiy
βi
)
Define utλ(x) =∫ x
0h−1
tλ (z)dz, t = 0, 1
Then
c∗0= (u′
0λ∗)−1(1), c∗1= (u′
1λ∗)−1
(
N ′(
1− w(Fρ(ρ))))
Consider an RDUT agent, indexed by λ∗, whose preference is
Vλ∗(c0, c1) := u0λ∗(c0) +
∫
u1λ∗(c1)d(w P) (7)
and whose endowment is the aggregate endowment (e0, e1)
This representative agent’s optimal consumption plan is theaggregate consumption plan
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Mathematical Behavioural Finance
Representative RDUT Agent
What’s Next – Idea
Work with the representative agent
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Mathematical Behavioural Finance
Representative RDUT Agent
What’s Next – Idea
Work with the representative agent
Derive explicit expression of pricing kernel assumingequilibrium exists
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Mathematical Behavioural Finance
Representative RDUT Agent
What’s Next – Idea
Work with the representative agent
Derive explicit expression of pricing kernel assumingequilibrium exists
Turn an RDUT economy into an EUT one by a measurechange
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Mathematical Behavioural Finance
Representative RDUT Agent
What’s Next – Idea
Work with the representative agent
Derive explicit expression of pricing kernel assumingequilibrium exists
Turn an RDUT economy into an EUT one by a measurechange
Use existing results for EUT economy
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Mathematical Behavioural Finance
Asset Pricing
Section 4
Asset Pricing
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Mathematical Behavioural Finance
Asset Pricing
Explicit Expression of Pricing Kernel
Theorem
(Xia and Zhou 2012) If there exists an equilibrium of economy E
where the pricing kernel ρ is atomless and λ∗ is the correspondingLagrange vector, then
ρ = w′(1− Fe1(e1))u′1λ∗(e1)
u′0λ∗(e0)a.s.. (8)
Idea of proof. Market clearing –
e1 = c∗1 = (u′1λ∗)−1(
N ′(
1− w(Fρ(ρ))))
– manipulate quantiles
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Mathematical Behavioural Finance
Asset Pricing
Interpretations
ρ = w′(1− Fe1(e1))u′
1λ∗(e1)
u′
0λ∗(e0)
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Mathematical Behavioural Finance
Asset Pricing
Interpretations
ρ = w′(1− Fe1(e1))u′
1λ∗(e1)
u′
0λ∗(e0)
Pricing kernel is a weighted marginal rate of substitutionbetween initial and end-of-period consumption
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Mathematical Behavioural Finance
Asset Pricing
Interpretations
ρ = w′(1− Fe1(e1))u′
1λ∗(e1)
u′
0λ∗(e0)
Pricing kernel is a weighted marginal rate of substitutionbetween initial and end-of-period consumption
The weight is w′(1− Fe1(e1))
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Mathematical Behavioural Finance
Asset Pricing
Interpretations
ρ = w′(1− Fe1(e1))u′
1λ∗(e1)
u′
0λ∗(e0)
Pricing kernel is a weighted marginal rate of substitutionbetween initial and end-of-period consumption
The weight is w′(1− Fe1(e1))
An inverse-S shaped weighting w leads to a premium whenevaluating assets in both very high and very low futureconsumption states
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Mathematical Behavioural Finance
Asset Pricing
Implied Utility Function
Define uw by
u′w(x) = w′(1− Fe1(x))u′1λ∗(x)
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Mathematical Behavioural Finance
Asset Pricing
Implied Utility Function
Define uw by
u′w(x) = w′(1− Fe1(x))u′1λ∗(x)
Pricing formula rewritten
ρ =u′w(e1)
u′0λ∗(e0)
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Mathematical Behavioural Finance
Asset Pricing
Implied Utility Function
Define uw by
u′w(x) = w′(1− Fe1(x))u′1λ∗(x)
Pricing formula rewritten
ρ =u′w(e1)
u′0λ∗(e0)
A fictitious EUT economy (under P without weighting),where uw is outcome utility function of a “weighted”representative agent
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Mathematical Behavioural Finance
Asset Pricing
Implied Utility Function
Define uw by
u′w(x) = w′(1− Fe1(x))u′1λ∗(x)
Pricing formula rewritten
ρ =u′w(e1)
u′0λ∗(e0)
A fictitious EUT economy (under P without weighting),where uw is outcome utility function of a “weighted”representative agent
uw: implied utility function
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Mathematical Behavioural Finance
Asset Pricing
Implied Relative Risk Aversion
Implied relative index of risk aversion
Rw(x) := −xu′′w(x)
u′w(x)= −
xu′′1λ∗(x)
u′1λ∗(x)+
xw′′(1− Fe1(x))
w′(1− Fe1(x))fe1(x)
(9)
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Mathematical Behavioural Finance
Asset Pricing
Implied Relative Risk Aversion
Implied relative index of risk aversion
Rw(x) := −xu′′w(x)
u′w(x)= −
xu′′1λ∗(x)
u′1λ∗(x)+
xw′′(1− Fe1(x))
w′(1− Fe1(x))fe1(x)
(9)
It represents overall degree of risk-aversion (or risk-loving) ofRDUT agent, combining outcome utility and probabilityweighting
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Mathematical Behavioural Finance
Asset Pricing
Existence of Equilibria
Theorem
(Xia and Zhou 2012) If Ψλ(p) ≡ w′(p)u′1λ(
F−1e1
(1− p))
isstrictly increasing for any λ, and
E[w′(1− Fe1(e1))u1i(e1)] < ∞
E
[
w′(1− Fe1(e1))u′1i
(
e1
I
)]
< ∞
for all i = 1, . . . , I, then there exists an Arrow-Debreu equilibriumof economy E where the pricing kernel is atomless. If in addition
−cu′′1i(c)
u′1i(c)≤ 1 for all i = 1, . . . , I and c > 0,
then the equilibrium is unique.
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Mathematical Behavioural Finance
Asset Pricing
Monotonicity of Ψλ
It is defined through model primitives:Ψλ(p) = w′(p)u′1λ
(
F−1e1
(1− p))
Monotonicity of Ψλ for any λ requires a concave impliedutility function for any initial distribution of the wealth.
Automatically satisfied when w is convex
Possibly satisfied when w is concave or inverse-S shaped
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Mathematical Behavioural Finance
Asset Pricing
Monotonicity of Ψλ: An Example
Example. Take w(p) = p1−α where α ∈ (0, 1), u1λ(c) =c1−β
1−β
where β ∈ (0, 1), and e1 follows the Parato distribution
Fe1(x) =
1−(
xm
x
)γx ≥ xm
0 x < xm.
In this case
Ψλ(p) = w′(p)u′1λ(
F−1e1
(1− p))
= (1− α)x−βm p
βγ−α
.
This is a strictly increasing function if and only if α < βγ.
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Mathematical Behavioural Finance
CCAPM and Interest Rate
Section 5
CCAPM and Interest Rate
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Mathematical Behavioural Finance
CCAPM and Interest Rate
Consumption-Based CAPM
r: rate of return of a security, and r = E[r]
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Mathematical Behavioural Finance
CCAPM and Interest Rate
Consumption-Based CAPM
r: rate of return of a security, and r = E[r]
rf : risk free rate
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Mathematical Behavioural Finance
CCAPM and Interest Rate
Consumption-Based CAPM
r: rate of return of a security, and r = E[r]
rf : risk free rate
g := e1e0
− 1: growth rate of aggregate endowment (assumedto be small)
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Mathematical Behavioural Finance
CCAPM and Interest Rate
Consumption-Based CAPM
r: rate of return of a security, and r = E[r]
rf : risk free rate
g := e1e0
− 1: growth rate of aggregate endowment (assumedto be small)
A rank-dependent consumption-based CAPM (CCAPM):
r − rf ≈
[
α+w′′(1− Fe1(e0))
w′(1− Fe1(e0))fe1(e0)e0
]
Cov(g, r)
where α := −e0u
′′
1λ∗(e0)
u′
1λ∗(e0)
and fe1 is density function of e1
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Mathematical Behavioural Finance
CCAPM and Interest Rate
Consumption-Based CAPM
r: rate of return of a security, and r = E[r]
rf : risk free rate
g := e1e0
− 1: growth rate of aggregate endowment (assumedto be small)
A rank-dependent consumption-based CAPM (CCAPM):
r − rf ≈
[
α+w′′(1− Fe1(e0))
w′(1− Fe1(e0))fe1(e0)e0
]
Cov(g, r)
where α := −e0u
′′
1λ∗(e0)
u′
1λ∗(e0)
and fe1 is density function of e1
Classical EUT based CCAPM: r − rf ≈ αCov(g, r)
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Mathematical Behavioural Finance
CCAPM and Interest Rate
Prices and Expected Consumption Growth
Again r − rf ≈[
α+w′′(1−Fe1
(e0))
w′(1−Fe1(e0))
fe1(e0)e0
]
Cov(g, r)
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Mathematical Behavioural Finance
CCAPM and Interest Rate
Prices and Expected Consumption Growth
Again r − rf ≈[
α+w′′(1−Fe1
(e0))
w′(1−Fe1(e0))
fe1(e0)e0
]
Cov(g, r)
Recall 1− Fe1(e0) = P (e1 > e0)
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Mathematical Behavioural Finance
CCAPM and Interest Rate
Prices and Expected Consumption Growth
Again r − rf ≈[
α+w′′(1−Fe1
(e0))
w′(1−Fe1(e0))
fe1(e0)e0
]
Cov(g, r)
Recall 1− Fe1(e0) = P (e1 > e0)
The subjective expectation (or belief) on general consumptiongrowth should be priced in for individual assets
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Mathematical Behavioural Finance
CCAPM and Interest Rate
Consumption-Based Real Interest
A rank-dependent consumption-based real interest rateformula:
1 + rf ≈1
βw′(1− Fe1(e0))
[
1 + αg +w′′(1 − Fe1(e0))
w′(1− Fe1(e0))fe1(e0)e0g
]
![Page 80: Mathematical Behavioural Financehelper.ipam.ucla.edu/publications/fm2015/fm2015_12823.pdfMathematical Behavioural Finance An Arrow-Debreu Economy ConsumptionPlans Agents choose consumption](https://reader033.fdocuments.in/reader033/viewer/2022060705/60714ecffc6a964b3f38e804/html5/thumbnails/80.jpg)
Mathematical Behavioural Finance
CCAPM and Interest Rate
Consumption-Based Real Interest
A rank-dependent consumption-based real interest rateformula:
1 + rf ≈1
βw′(1− Fe1(e0))
[
1 + αg +w′′(1 − Fe1(e0))
w′(1− Fe1(e0))fe1(e0)e0g
]
Classical EUT based real interest rate theory: 1 + rf ≈ 1+αgβ
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Mathematical Behavioural Finance
Equity Premium and Risk-Free Rate Puzzles
Section 6
Equity Premium and Risk-Free Rate Puzzles
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Mathematical Behavioural Finance
Equity Premium and Risk-Free Rate Puzzles
Equity Premium and Risk-Free Rate Puzzles
Equity premium puzzle (Mehra and Prescott 1985): observedequity premium is too high to be explainable by classicalCCAPM
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Mathematical Behavioural Finance
Equity Premium and Risk-Free Rate Puzzles
Equity Premium and Risk-Free Rate Puzzles
Equity premium puzzle (Mehra and Prescott 1985): observedequity premium is too high to be explainable by classicalCCAPM
Risk-free rate puzzle (Weil 1989): observed risk-free rate istoo low to be explainable by classical CCAPM
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Mathematical Behavioural Finance
Equity Premium and Risk-Free Rate Puzzles
Economic Data 1889–1978 (Mehra and Prescott 1985)
Consumption growth riskless return equity premium S&P 500 return
Periods Mean S.D. Mean S.D. Mean S.D. Mean S.D.
1889–1978 1.83 3.57 0.80 5.67 6.18 16.67 6.98 16.541889–1898 2.30 4.90 5.80 3.23 1.78 11.57 7.58 10.02
1899–1908 2.55 5.31 2.62 2.59 5.08 16.86 7.71 17.21
1909–1918 0.44 3.07 -1.63 9.02 1.49 9.18 -0.14 12.81
1919–1928 3.00 3.97 4.30 6.61 14.64 15.94 18.94 16.18
1929–1938 -0.25 5.28 2.39 6.50 0.18 31.63 2.56 27.90
1939–1948 2.19 2.52 -5.82 4.05 8.89 14.23 3.07 14.67
1949–1958 1.48 1.00 -0.81 1.89 18.30 13.20 17.49 13.08
1959–1968 2.37 1.00 1.07 0.64 4.50 10.17 5.58 10.59
1969–1978 2.41 1.40 -0.72 2.06 0.75 11.64 0.03 13.11
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Mathematical Behavioural Finance
Equity Premium and Risk-Free Rate Puzzles
Equity Premium Puzzle
The observed equity premium of 6.18% corresponds to arelative index of risk aversion over 30 (Mankiw and Zeldes1991)
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Mathematical Behavioural Finance
Equity Premium and Risk-Free Rate Puzzles
Equity Premium Puzzle
The observed equity premium of 6.18% corresponds to arelative index of risk aversion over 30 (Mankiw and Zeldes1991)
A measure of 30 means indifference between a gamble equallylikely to pay $50,000 or $100,000 and a certain payoff of$51,209
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Mathematical Behavioural Finance
Equity Premium and Risk-Free Rate Puzzles
Equity Premium Puzzle
The observed equity premium of 6.18% corresponds to arelative index of risk aversion over 30 (Mankiw and Zeldes1991)
A measure of 30 means indifference between a gamble equallylikely to pay $50,000 or $100,000 and a certain payoff of$51,209
No human is that risk averse
![Page 88: Mathematical Behavioural Financehelper.ipam.ucla.edu/publications/fm2015/fm2015_12823.pdfMathematical Behavioural Finance An Arrow-Debreu Economy ConsumptionPlans Agents choose consumption](https://reader033.fdocuments.in/reader033/viewer/2022060705/60714ecffc6a964b3f38e804/html5/thumbnails/88.jpg)
Mathematical Behavioural Finance
Equity Premium and Risk-Free Rate Puzzles
Our Explanation
Probability weighting, in addition to outcome utility, alsocontributes to this total measure of 30
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Mathematical Behavioural Finance
Equity Premium and Risk-Free Rate Puzzles
Our Explanation
Probability weighting, in addition to outcome utility, alsocontributes to this total measure of 30
Recall r − rf ≈[
α+w′′(1−Fe1
(e0))
w′(1−Fe1(e0))
fe1(e0)e0
]
Cov(g, r)
![Page 90: Mathematical Behavioural Financehelper.ipam.ucla.edu/publications/fm2015/fm2015_12823.pdfMathematical Behavioural Finance An Arrow-Debreu Economy ConsumptionPlans Agents choose consumption](https://reader033.fdocuments.in/reader033/viewer/2022060705/60714ecffc6a964b3f38e804/html5/thumbnails/90.jpg)
Mathematical Behavioural Finance
Equity Premium and Risk-Free Rate Puzzles
Our Explanation
Probability weighting, in addition to outcome utility, alsocontributes to this total measure of 30
Recall r − rf ≈[
α+w′′(1−Fe1
(e0))
w′(1−Fe1(e0))
fe1(e0)e0
]
Cov(g, r)
w is typically inverse-S shaped
![Page 91: Mathematical Behavioural Financehelper.ipam.ucla.edu/publications/fm2015/fm2015_12823.pdfMathematical Behavioural Finance An Arrow-Debreu Economy ConsumptionPlans Agents choose consumption](https://reader033.fdocuments.in/reader033/viewer/2022060705/60714ecffc6a964b3f38e804/html5/thumbnails/91.jpg)
Mathematical Behavioural Finance
Equity Premium and Risk-Free Rate Puzzles
Our Explanation
Probability weighting, in addition to outcome utility, alsocontributes to this total measure of 30
Recall r − rf ≈[
α+w′′(1−Fe1
(e0))
w′(1−Fe1(e0))
fe1(e0)e0
]
Cov(g, r)
w is typically inverse-S shaped
It is plausible to assume P(e1 > e0) is large (close to 1)
![Page 92: Mathematical Behavioural Financehelper.ipam.ucla.edu/publications/fm2015/fm2015_12823.pdfMathematical Behavioural Finance An Arrow-Debreu Economy ConsumptionPlans Agents choose consumption](https://reader033.fdocuments.in/reader033/viewer/2022060705/60714ecffc6a964b3f38e804/html5/thumbnails/92.jpg)
Mathematical Behavioural Finance
Equity Premium and Risk-Free Rate Puzzles
Our Explanation
Probability weighting, in addition to outcome utility, alsocontributes to this total measure of 30
Recall r − rf ≈[
α+w′′(1−Fe1
(e0))
w′(1−Fe1(e0))
fe1(e0)e0
]
Cov(g, r)
w is typically inverse-S shaped
It is plausible to assume P(e1 > e0) is large (close to 1)
Hence 1− Fe1(e0) = P(e1 > e0) lies in the convex domain ofw
![Page 93: Mathematical Behavioural Financehelper.ipam.ucla.edu/publications/fm2015/fm2015_12823.pdfMathematical Behavioural Finance An Arrow-Debreu Economy ConsumptionPlans Agents choose consumption](https://reader033.fdocuments.in/reader033/viewer/2022060705/60714ecffc6a964b3f38e804/html5/thumbnails/93.jpg)
Mathematical Behavioural Finance
Equity Premium and Risk-Free Rate Puzzles
Our Explanation
Probability weighting, in addition to outcome utility, alsocontributes to this total measure of 30
Recall r − rf ≈[
α+w′′(1−Fe1
(e0))
w′(1−Fe1(e0))
fe1(e0)e0
]
Cov(g, r)
w is typically inverse-S shaped
It is plausible to assume P(e1 > e0) is large (close to 1)
Hence 1− Fe1(e0) = P(e1 > e0) lies in the convex domain ofw
Expected rate of return provided by our model is larger thanthat by EUT
![Page 94: Mathematical Behavioural Financehelper.ipam.ucla.edu/publications/fm2015/fm2015_12823.pdfMathematical Behavioural Finance An Arrow-Debreu Economy ConsumptionPlans Agents choose consumption](https://reader033.fdocuments.in/reader033/viewer/2022060705/60714ecffc6a964b3f38e804/html5/thumbnails/94.jpg)
Mathematical Behavioural Finance
Equity Premium and Risk-Free Rate Puzzles
Our Explanation (Cont’d)
Recall1 + rf ≈ 1
βw′(1−Fe1(e0))
[
1 + αg +w′′(1−Fe1
(e0))
w′(1−Fe1(e0))
fe1(e0)e0g]
![Page 95: Mathematical Behavioural Financehelper.ipam.ucla.edu/publications/fm2015/fm2015_12823.pdfMathematical Behavioural Finance An Arrow-Debreu Economy ConsumptionPlans Agents choose consumption](https://reader033.fdocuments.in/reader033/viewer/2022060705/60714ecffc6a964b3f38e804/html5/thumbnails/95.jpg)
Mathematical Behavioural Finance
Equity Premium and Risk-Free Rate Puzzles
Our Explanation (Cont’d)
Recall1 + rf ≈ 1
βw′(1−Fe1(e0))
[
1 + αg +w′′(1−Fe1
(e0))
w′(1−Fe1(e0))
fe1(e0)e0g]
We have argued 1− Fe1(e0) is normally close to 1
![Page 96: Mathematical Behavioural Financehelper.ipam.ucla.edu/publications/fm2015/fm2015_12823.pdfMathematical Behavioural Finance An Arrow-Debreu Economy ConsumptionPlans Agents choose consumption](https://reader033.fdocuments.in/reader033/viewer/2022060705/60714ecffc6a964b3f38e804/html5/thumbnails/96.jpg)
Mathematical Behavioural Finance
Equity Premium and Risk-Free Rate Puzzles
Our Explanation (Cont’d)
Recall1 + rf ≈ 1
βw′(1−Fe1(e0))
[
1 + αg +w′′(1−Fe1
(e0))
w′(1−Fe1(e0))
fe1(e0)e0g]
We have argued 1− Fe1(e0) is normally close to 1
Therefore, for an inverse-S shaped w, w′(1− Fe1(e0)) will belarger than one
![Page 97: Mathematical Behavioural Financehelper.ipam.ucla.edu/publications/fm2015/fm2015_12823.pdfMathematical Behavioural Finance An Arrow-Debreu Economy ConsumptionPlans Agents choose consumption](https://reader033.fdocuments.in/reader033/viewer/2022060705/60714ecffc6a964b3f38e804/html5/thumbnails/97.jpg)
Mathematical Behavioural Finance
Equity Premium and Risk-Free Rate Puzzles
Our Explanation (Cont’d)
Recall1 + rf ≈ 1
βw′(1−Fe1(e0))
[
1 + αg +w′′(1−Fe1
(e0))
w′(1−Fe1(e0))
fe1(e0)e0g]
We have argued 1− Fe1(e0) is normally close to 1
Therefore, for an inverse-S shaped w, w′(1− Fe1(e0)) will belarger than one
Our interest rate model indicates that an appropriate w canrender a lower risk-free rate than EUT model
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Mathematical Behavioural Finance
Equity Premium and Risk-Free Rate Puzzles
Our Explanation (Cont’d)
Recall1 + rf ≈ 1
βw′(1−Fe1(e0))
[
1 + αg +w′′(1−Fe1
(e0))
w′(1−Fe1(e0))
fe1(e0)e0g]
We have argued 1− Fe1(e0) is normally close to 1
Therefore, for an inverse-S shaped w, w′(1− Fe1(e0)) will belarger than one
Our interest rate model indicates that an appropriate w canrender a lower risk-free rate than EUT model
The presence of a suitable probability weighting function willsimultaneously increase equity premium and decrease risk-freerate under RDUT, diminishing the gap seen under EUT
![Page 99: Mathematical Behavioural Financehelper.ipam.ucla.edu/publications/fm2015/fm2015_12823.pdfMathematical Behavioural Finance An Arrow-Debreu Economy ConsumptionPlans Agents choose consumption](https://reader033.fdocuments.in/reader033/viewer/2022060705/60714ecffc6a964b3f38e804/html5/thumbnails/99.jpg)
Mathematical Behavioural Finance
Equity Premium and Risk-Free Rate Puzzles
Economic Data 1889–1978 (Mehra and Prescott 1985)
Consumption growth riskless return equity premium S&P 500 return
Periods Mean S.D. Mean S.D. Mean S.D. Mean S.D.
1889–1978 1.83 3.57 0.80 5.67 6.18 16.67 6.98 16.541889–1898 2.30 4.90 5.80 3.23 1.78 11.57 7.58 10.02
1899–1908 2.55 5.31 2.62 2.59 5.08 16.86 7.71 17.21
1909–1918 0.44 3.07 -1.63 9.02 1.49 9.18 -0.14 12.81
1919–1928 3.00 3.97 4.30 6.61 14.64 15.94 18.94 16.18
1929–1938 -0.25 5.28 2.39 6.50 0.18 31.63 2.56 27.90
1939–1948 2.19 2.52 -5.82 4.05 8.89 14.23 3.07 14.67
1949–1958 1.48 1.00 -0.81 1.89 18.30 13.20 17.49 13.08
1959–1968 2.37 1.00 1.07 0.64 4.50 10.17 5.58 10.59
1969–1978 2.41 1.40 -0.72 2.06 0.75 11.64 0.03 13.11
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Mathematical Behavioural Finance
Equity Premium and Risk-Free Rate Puzzles
Negative Real Interest Rates
Four periods, 1909–1918, 1939–1948, 1949–1958, and1969–1978, during which g > 0 but rf < 0
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Mathematical Behavioural Finance
Equity Premium and Risk-Free Rate Puzzles
Negative Real Interest Rates
Four periods, 1909–1918, 1939–1948, 1949–1958, and1969–1978, during which g > 0 but rf < 0
Not possible under EUT, since rf ≥ αg > 0 if g > 0
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Mathematical Behavioural Finance
Equity Premium and Risk-Free Rate Puzzles
Negative Real Interest Rates
Four periods, 1909–1918, 1939–1948, 1949–1958, and1969–1978, during which g > 0 but rf < 0
Not possible under EUT, since rf ≥ αg > 0 if g > 0
It can be accounted for by rank-dependent CCAPM
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Mathematical Behavioural Finance
Equity Premium and Risk-Free Rate Puzzles
Negative Real Interest Rates
Four periods, 1909–1918, 1939–1948, 1949–1958, and1969–1978, during which g > 0 but rf < 0
Not possible under EUT, since rf ≥ αg > 0 if g > 0
It can be accounted for by rank-dependent CCAPM
Recall1 + rf ≈ 1
βw′(1−Fe1(e0))
[
1 + αg +w′′(1−Fe1
(e0))
w′(1−Fe1(e0))
fe1(e0)e0g]
![Page 104: Mathematical Behavioural Financehelper.ipam.ucla.edu/publications/fm2015/fm2015_12823.pdfMathematical Behavioural Finance An Arrow-Debreu Economy ConsumptionPlans Agents choose consumption](https://reader033.fdocuments.in/reader033/viewer/2022060705/60714ecffc6a964b3f38e804/html5/thumbnails/104.jpg)
Mathematical Behavioural Finance
Equity Premium and Risk-Free Rate Puzzles
Negative Real Interest Rates
Four periods, 1909–1918, 1939–1948, 1949–1958, and1969–1978, during which g > 0 but rf < 0
Not possible under EUT, since rf ≥ αg > 0 if g > 0
It can be accounted for by rank-dependent CCAPM
Recall1 + rf ≈ 1
βw′(1−Fe1(e0))
[
1 + αg +w′′(1−Fe1
(e0))
w′(1−Fe1(e0))
fe1(e0)e0g]
It requires only a sufficiently large value of βw′(1− Fe1(e0)) –explainable by a proper inverse-S shaped w
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Mathematical Behavioural Finance
Equity Premium and Risk-Free Rate Puzzles
Great Depression
Great Depression (1929–1938) is the only 10-year periodduring which g < 0
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Mathematical Behavioural Finance
Equity Premium and Risk-Free Rate Puzzles
Great Depression
Great Depression (1929–1938) is the only 10-year periodduring which g < 01− Fe1(e0) = P(e1 > e0) would have lain in the concavedomain of w due to the overwhelmingly negative outlook ofeconomy
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Mathematical Behavioural Finance
Equity Premium and Risk-Free Rate Puzzles
Great Depression
Great Depression (1929–1938) is the only 10-year periodduring which g < 01− Fe1(e0) = P(e1 > e0) would have lain in the concavedomain of w due to the overwhelmingly negative outlook ofeconomy
Recall r − rf ≈[
α+w′′(1−Fe1
(e0))
w′(1−Fe1(e0))
fe1(e0)e0
]
Cov(g, r)
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Mathematical Behavioural Finance
Equity Premium and Risk-Free Rate Puzzles
Great Depression
Great Depression (1929–1938) is the only 10-year periodduring which g < 01− Fe1(e0) = P(e1 > e0) would have lain in the concavedomain of w due to the overwhelmingly negative outlook ofeconomy
Recall r − rf ≈[
α+w′′(1−Fe1
(e0))
w′(1−Fe1(e0))
fe1(e0)e0
]
Cov(g, r)
w′′(1−Fe1(e0))
w′(1−Fe1(e0))
fe1(e0)e0 should be negative
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Mathematical Behavioural Finance
Equity Premium and Risk-Free Rate Puzzles
Great Depression
Great Depression (1929–1938) is the only 10-year periodduring which g < 01− Fe1(e0) = P(e1 > e0) would have lain in the concavedomain of w due to the overwhelmingly negative outlook ofeconomy
Recall r − rf ≈[
α+w′′(1−Fe1
(e0))
w′(1−Fe1(e0))
fe1(e0)e0
]
Cov(g, r)
w′′(1−Fe1(e0))
w′(1−Fe1(e0))
fe1(e0)e0 should be negative
Our model would have predicted a lower equity premium
![Page 110: Mathematical Behavioural Financehelper.ipam.ucla.edu/publications/fm2015/fm2015_12823.pdfMathematical Behavioural Finance An Arrow-Debreu Economy ConsumptionPlans Agents choose consumption](https://reader033.fdocuments.in/reader033/viewer/2022060705/60714ecffc6a964b3f38e804/html5/thumbnails/110.jpg)
Mathematical Behavioural Finance
Equity Premium and Risk-Free Rate Puzzles
Great Depression
Great Depression (1929–1938) is the only 10-year periodduring which g < 01− Fe1(e0) = P(e1 > e0) would have lain in the concavedomain of w due to the overwhelmingly negative outlook ofeconomy
Recall r − rf ≈[
α+w′′(1−Fe1
(e0))
w′(1−Fe1(e0))
fe1(e0)e0
]
Cov(g, r)
w′′(1−Fe1(e0))
w′(1−Fe1(e0))
fe1(e0)e0 should be negative
Our model would have predicted a lower equity premium
Corresponding premium, 0.18%, is lowest in Table 1
![Page 111: Mathematical Behavioural Financehelper.ipam.ucla.edu/publications/fm2015/fm2015_12823.pdfMathematical Behavioural Finance An Arrow-Debreu Economy ConsumptionPlans Agents choose consumption](https://reader033.fdocuments.in/reader033/viewer/2022060705/60714ecffc6a964b3f38e804/html5/thumbnails/111.jpg)
Mathematical Behavioural Finance
Equity Premium and Risk-Free Rate Puzzles
Great Depression
Great Depression (1929–1938) is the only 10-year periodduring which g < 01− Fe1(e0) = P(e1 > e0) would have lain in the concavedomain of w due to the overwhelmingly negative outlook ofeconomy
Recall r − rf ≈[
α+w′′(1−Fe1
(e0))
w′(1−Fe1(e0))
fe1(e0)e0
]
Cov(g, r)
w′′(1−Fe1(e0))
w′(1−Fe1(e0))
fe1(e0)e0 should be negative
Our model would have predicted a lower equity premium
Corresponding premium, 0.18%, is lowest in Table 1
In general, at times when most people believe that economy isin a downturn, expected rate of return provided by RDUT issmaller than that provided by EUT model
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Mathematical Behavioural Finance
Equity Premium and Risk-Free Rate Puzzles
Great Depression
Great Depression (1929–1938) is the only 10-year periodduring which g < 01− Fe1(e0) = P(e1 > e0) would have lain in the concavedomain of w due to the overwhelmingly negative outlook ofeconomy
Recall r − rf ≈[
α+w′′(1−Fe1
(e0))
w′(1−Fe1(e0))
fe1(e0)e0
]
Cov(g, r)
w′′(1−Fe1(e0))
w′(1−Fe1(e0))
fe1(e0)e0 should be negative
Our model would have predicted a lower equity premium
Corresponding premium, 0.18%, is lowest in Table 1
In general, at times when most people believe that economy isin a downturn, expected rate of return provided by RDUT issmaller than that provided by EUT modelHence we should investigate asset pricing by differentiatingperiods of economic growth from those of economic depression
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Mathematical Behavioural Finance
Summary and Further Readings
Section 7
Summary and Further Readings
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Mathematical Behavioural Finance
Summary and Further Readings
Summary
Conditions on an RDUT economy provided under which theArrow-Debreu equilibrium exists uniquely
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Mathematical Behavioural Finance
Summary and Further Readings
Summary
Conditions on an RDUT economy provided under which theArrow-Debreu equilibrium exists uniquely
Motivated to re-study RDUT portfolio choice problemwithout any monotonicity condition
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Mathematical Behavioural Finance
Summary and Further Readings
Summary
Conditions on an RDUT economy provided under which theArrow-Debreu equilibrium exists uniquely
Motivated to re-study RDUT portfolio choice problemwithout any monotonicity condition
At equilibrium one cannot distinguish between RDUT andEUT economies; however, representative risk aversion level is(possibly substantially) altered
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Mathematical Behavioural Finance
Summary and Further Readings
Summary
Conditions on an RDUT economy provided under which theArrow-Debreu equilibrium exists uniquely
Motivated to re-study RDUT portfolio choice problemwithout any monotonicity condition
At equilibrium one cannot distinguish between RDUT andEUT economies; however, representative risk aversion level is(possibly substantially) altered
Asset prices not only depend upon level of risk aversion andbeta, but also upon agents’ belief on economic growth
![Page 118: Mathematical Behavioural Financehelper.ipam.ucla.edu/publications/fm2015/fm2015_12823.pdfMathematical Behavioural Finance An Arrow-Debreu Economy ConsumptionPlans Agents choose consumption](https://reader033.fdocuments.in/reader033/viewer/2022060705/60714ecffc6a964b3f38e804/html5/thumbnails/118.jpg)
Mathematical Behavioural Finance
Summary and Further Readings
Summary
Conditions on an RDUT economy provided under which theArrow-Debreu equilibrium exists uniquely
Motivated to re-study RDUT portfolio choice problemwithout any monotonicity condition
At equilibrium one cannot distinguish between RDUT andEUT economies; however, representative risk aversion level is(possibly substantially) altered
Asset prices not only depend upon level of risk aversion andbeta, but also upon agents’ belief on economic growth
Probability weighting may offer a new way of thinking inexplaining many economic phenomena
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Mathematical Behavioural Finance
Summary and Further Readings
Essential Readings
H. Shefrin. A Behavioral Approach to Asset Pricing (2ndEdition), Elsevier, Amsterdam, 2008.
J. Xia and X. Zhou. Arrow-Debreu equilibria forrank-dependent utilities, Mathematical Finance, to appear;available at http://people.maths.ox.ac.uk/˜zhouxy/download/AB.pdf
R.A. Dana. Existence and uniqueness of equilibria whenpreferences are additively separable, Econometrica, 61:953–957, 1993.
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Mathematical Behavioural Finance
Summary and Further Readings
Other Readings
M. Abdellaoui. A genuine rank-dependent generalization of the von Neumann-Morgenstern expected utility
theorem, Econometrica, 70:717–736, 2002.
E.M. Azevedo and D. Gottlieb. Risk-neutral firms can extract unbounded profits from consumers with
prospect theory preferences, Journal of Economic Theory, 147:1291–1299, 2012.
N. Barberis and M. Huang. Stocks as lotteries: The implications of probability weighting for security
prices, American Economic Review, 98:2066–2100, 2008.
R.A. Dana. Existence, uniqueness and determinacy of Arrow-Debreu equilibria in finance models, Journal
of Mathematical Economics, 22:563–579, 1993.
R.A. Dana. Comonotonicity, efficient risk-sharing and equilibria in markets with short-selling for concave
law-invariant utilities, Journal of Mathematical Economics, 47:328–335, 2011.
E. De Giorgi, T. Hens and M. O. Rieger. Financial market equilibria with cumulative prospect theory,
Journal of Mathematical Economics, 46:633–651, 2010.
H. Follmer and A. Schied. Stochastic Finance: An Introduction in Discrete Time (3rd edition), Walter de
Gruyter, Berlin, 2011.
X. He and X. Zhou. Portfolio choice under cumulative prospect theory: An analytical treatment,
Management Science, 57:315–331, 2011.
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Mathematical Behavioural Finance
Final Words
Section 8
Final Words
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Mathematical Behavioural Finance
Final Words
Two Revolutions in Finance
Finance ultimately deals with interplay between market riskand human judgement
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Mathematical Behavioural Finance
Final Words
Two Revolutions in Finance
Finance ultimately deals with interplay between market riskand human judgement
History of financial theory over the last 50 years characterisedby two revolutions
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Mathematical Behavioural Finance
Final Words
Two Revolutions in Finance
Finance ultimately deals with interplay between market riskand human judgement
History of financial theory over the last 50 years characterisedby two revolutions
Neoclassical (maximising) finance starting 1960s: Expectedutility maximisation, CAPM, efficient market theory, optionpricing
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Mathematical Behavioural Finance
Final Words
Two Revolutions in Finance
Finance ultimately deals with interplay between market riskand human judgement
History of financial theory over the last 50 years characterisedby two revolutions
Neoclassical (maximising) finance starting 1960s: Expectedutility maximisation, CAPM, efficient market theory, optionpricingBehavioural finance starting 1980s: Cumulative prospecttheory, SP/A theory, regret and self-control, heuristics andbiases
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Mathematical Behavioural Finance
Final Words
Neoclassical vs Behavioural
Neoclassical: the world and its participants are rational“wealth maximisers”
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Mathematical Behavioural Finance
Final Words
Neoclassical vs Behavioural
Neoclassical: the world and its participants are rational“wealth maximisers”
Behavioural: emotion and psychology influence our decisionswhen faced with uncertainties, causing us to behave inunpredictable, inconsistent, incompetent, and most of all,irrational ways
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Mathematical Behavioural Finance
Final Words
Neoclassical vs Behavioural
Neoclassical: the world and its participants are rational“wealth maximisers”
Behavioural: emotion and psychology influence our decisionswhen faced with uncertainties, causing us to behave inunpredictable, inconsistent, incompetent, and most of all,irrational ways
A relatively new field that attempts to explain how and whyemotions and cognitive errors influence investors and createstock market anomalies such as bubbles and crashes
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Mathematical Behavioural Finance
Final Words
Neoclassical vs Behavioural
Neoclassical: the world and its participants are rational“wealth maximisers”
Behavioural: emotion and psychology influence our decisionswhen faced with uncertainties, causing us to behave inunpredictable, inconsistent, incompetent, and most of all,irrational ways
A relatively new field that attempts to explain how and whyemotions and cognitive errors influence investors and createstock market anomalies such as bubbles and crashesIt seeks to explore the consistency and predictability in humanflaws so that such flaws can be avoided or even exploited forprofit
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Mathematical Behavioural Finance
Final Words
Do We Need Both?
Foundations of the two
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Mathematical Behavioural Finance
Final Words
Do We Need Both?
Foundations of the two
Neoclassical finance: Rationality (correct beliefs oninformation, risk aversion) – A normative theory
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Mathematical Behavioural Finance
Final Words
Do We Need Both?
Foundations of the two
Neoclassical finance: Rationality (correct beliefs oninformation, risk aversion) – A normative theoryBehavioural finance: The lack thereof (experimental evidence,cognitive psychology) – A descriptive theory
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Mathematical Behavioural Finance
Final Words
Do We Need Both?
Foundations of the two
Neoclassical finance: Rationality (correct beliefs oninformation, risk aversion) – A normative theoryBehavioural finance: The lack thereof (experimental evidence,cognitive psychology) – A descriptive theory
Do we need both?
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Mathematical Behavioural Finance
Final Words
Do We Need Both?
Foundations of the two
Neoclassical finance: Rationality (correct beliefs oninformation, risk aversion) – A normative theoryBehavioural finance: The lack thereof (experimental evidence,cognitive psychology) – A descriptive theory
Do we need both? Absolutely yes!
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Mathematical Behavioural Finance
Final Words
Do We Need Both?
Foundations of the two
Neoclassical finance: Rationality (correct beliefs oninformation, risk aversion) – A normative theoryBehavioural finance: The lack thereof (experimental evidence,cognitive psychology) – A descriptive theory
Do we need both? Absolutely yes!
Neoclassical finance tells what people ought to do
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Mathematical Behavioural Finance
Final Words
Do We Need Both?
Foundations of the two
Neoclassical finance: Rationality (correct beliefs oninformation, risk aversion) – A normative theoryBehavioural finance: The lack thereof (experimental evidence,cognitive psychology) – A descriptive theory
Do we need both? Absolutely yes!
Neoclassical finance tells what people ought to doBehavioural finance tells what people actually do
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Mathematical Behavioural Finance
Final Words
Do We Need Both?
Foundations of the two
Neoclassical finance: Rationality (correct beliefs oninformation, risk aversion) – A normative theoryBehavioural finance: The lack thereof (experimental evidence,cognitive psychology) – A descriptive theory
Do we need both? Absolutely yes!
Neoclassical finance tells what people ought to doBehavioural finance tells what people actually doRobert Shiller (2006), “the two ... have always been interwind,and some of the most important applications of their insightswill require the use of both approaches”
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Mathematical Behavioural Finance
Final Words
Mathematical Behavioural Finance
“Mathematical behavioural finance” leads to new problems inmathematics and finance
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Mathematical Behavioural Finance
Final Words
Mathematical Behavioural Finance
“Mathematical behavioural finance” leads to new problems inmathematics and finance
But ... is it justified: to rationally and mathematicallyaccount for irrationalities?
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Mathematical Behavioural Finance
Final Words
Mathematical Behavioural Finance
“Mathematical behavioural finance” leads to new problems inmathematics and finance
But ... is it justified: to rationally and mathematicallyaccount for irrationalities?
Irrational behaviours are by no means random or arbitrary
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Mathematical Behavioural Finance
Final Words
Mathematical Behavioural Finance
“Mathematical behavioural finance” leads to new problems inmathematics and finance
But ... is it justified: to rationally and mathematicallyaccount for irrationalities?
Irrational behaviours are by no means random or arbitrary
“misguided behaviors ... are systamtic and predictable –making us predictably irrational” (Dan Ariely, PredictablyIrrational, Ariely 2008)
![Page 142: Mathematical Behavioural Financehelper.ipam.ucla.edu/publications/fm2015/fm2015_12823.pdfMathematical Behavioural Finance An Arrow-Debreu Economy ConsumptionPlans Agents choose consumption](https://reader033.fdocuments.in/reader033/viewer/2022060705/60714ecffc6a964b3f38e804/html5/thumbnails/142.jpg)
Mathematical Behavioural Finance
Final Words
Mathematical Behavioural Finance
“Mathematical behavioural finance” leads to new problems inmathematics and finance
But ... is it justified: to rationally and mathematicallyaccount for irrationalities?
Irrational behaviours are by no means random or arbitrary
“misguided behaviors ... are systamtic and predictable –making us predictably irrational” (Dan Ariely, PredictablyIrrational, Ariely 2008)
We use CPT/RDUT/SPA and specific value functions as thecarrier for exploring the “predictable irrationalities”
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Mathematical Behavioural Finance
Final Words
Mathematical Behavioural Finance
“Mathematical behavioural finance” leads to new problems inmathematics and finance
But ... is it justified: to rationally and mathematicallyaccount for irrationalities?
Irrational behaviours are by no means random or arbitrary
“misguided behaviors ... are systamtic and predictable –making us predictably irrational” (Dan Ariely, PredictablyIrrational, Ariely 2008)
We use CPT/RDUT/SPA and specific value functions as thecarrier for exploring the “predictable irrationalities”
Mathematical behavioural finance: research is in its infancy,yet potential is unlimited – or so we believe