_{1}

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This paper considers an equilibrium asset pricing model in a static pure exchange economy under ambiguity. Ambiguity preference is represented by the dual theory of the smooth ambiguity model [ 6]. We show the existence and the uniqueness of the equilibrium in the economy and derive the state price density (SPD). The equilibrium excess return, which can be seen as an extension of the capital asset pricing model (CAPM) under risk to ambiguity, is derived from the SPD. We also determine the effects of ambiguity preference on the excess returns of ambiguous securities through comparative statics of the SPD.

The state price density (SPD) is a central concept in modern asset pricing theory.^{1} Given the SPD and its probability distribution, we can price every asset. Thus, it is essential to derive the SPD for asset pricing. A lot of studies have attempted to derive the SPD from both a theoretical and an empirical viewpoint.

Most asset pricing models suppose that an agent can assign a unique probability distribution over a state space. However, it is commonly observed that such a unique probability is not available in the economy. It is known that expected utility, which is a dominant tool in asset pricing theory, cannot describe choice under ambiguity.^{2} Many ambiguity models, such as those of [

This paper unites the above two lines of research. The purpose of this paper is to characterize the SPD in the presence of ambiguity. More precisely, we examine equilibrium in a single-period pure exchange economy under ambiguity and derive the SPD by using the dual theory of the smooth ambiguity model [

The original theory of the smooth ambiguity model by [

A series of studies on equilibrium analysis in securities markets precede this paper. This paper derives the SPD in an economy with a representative agent. For the construction of the representative agent, we use the idea of assigning proper weights to each agent, which goes back to [

Using the SPD, we extend the classical CAPM to an economy under ambiguity. Thus, this paper is related to the recent literature on the CAPM under ambiguity. In particular, [

Because we adopt the dual theory of the smooth ambiguity model by [

The organization of this paper is as follows. The next section introduces some settings and notions about the economy and agents, and provides some preliminary analysis. In Section 3, we derive the SPD and show its existence and uniqueness in the economy. In Section 4, we apply the SPD to obtain an equilibrium asset return which can be rewritten as the CAPM in the presence of ambiguity. Furthermore, we show how an agent’s portfolio is decomposed and obtain the classical two-fund separation from [

We consider a single-period pure exchange economy. All of the uncertainty is described by a finite discrete state space Ω = { ω 1 , ⋯ , ω S } . Agents in the economy can trade a safe security and S ambiguous securities. The rate of return of the safe security is a constant r f . The rate of return of the i-th ambiguous security is given by a random variable r ˜ i . The rate of return r ˜ i , i = 1 , ⋯ , S , can be decomposed into a constant term μ i and a random term z ˜ i such that

r ˜ i = μ i + z ˜ i ,

where z ˜ i = z i s if ω s ∈ Ω occurs for s = 1 , ⋯ , S .

We define the matrix R by

R = ( r 11 ⋯ r S 1 ⋮ ⋱ ⋮ r 1 S ⋯ r S S ) ,

where r i s = μ i + z i s , i , s = 1 , ⋯ , S . We assume that the rank of R is equal to S, that is, the security market is assumed to be complete.^{3} Ambiguity is represented by a finite set of probability distributions on Ω :

p ( l ) = ( p 1 ( l ) , ⋯ , p S ( l ) ) , l ∈ { 1, ⋯ , L } .

Given a unique probability distribution ρ = ( ρ ( 1 ) , ⋯ , ρ ( L ) ) over the index set { 1, ⋯ , L } , a compound probability distribution P on Ω is defined by

P = ( P 1 , ⋯ , P S ) = ( ∑ l = 1 L p 1 ( l ) ρ ( l ) , ⋯ , ∑ l = 1 L p S ( l ) ρ ( l ) ) .

Each agent is assumed to consider this to be the reference probability distribution in the economy.^{4} Without loss of generality, we assume that E P [ z ˜ i ] = ∑ s = 1 S P s z i s = 0 , i = 1 , ⋯ , S .

We define the SPD as a non-negative random variable H = ( H 1 , ⋯ , H S ) on Ω satisfying

E P [ ( 1 + r f ) H ] = ( 1 + r f ) ∑ s = 1 S P s H s = 1 , (1)

E P [ ( 1 + r ˜ i ) H ] = ∑ s = 1 S P s ( 1 + r i s ) H s = 1 , i = 1 , ⋯ , S . (2)

There are K agents in the economy. Let ϵ k s > 0 , k = 1 , ⋯ , K , denote agent k ’s terminal income if the state ω s , i = 1 , ⋯ , S , occurs, let w k > 0 denote agent k ’s initial wealth, and let π k i denote the amount that agent k invests in the i-th ambiguous security. For a given state ω s , the terminal wealth W k s , i = 1 , ⋯ , S , of agent k is given by

W k = ( W k 1 ⋮ W k S ) = ( 1 + r f ) w k 1 S + ( R − r f 1 S × S ) π k + ϵ ˜ k = ( 1 + r f ) w k ( 1 ⋮ 1 ) + ∑ i = 1 S π k i ( r i 1 − r f ⋮ r i S − r f ) + ( ϵ k 1 ⋮ ϵ k S ) , (3)

where π k = ( π k 1 , ⋯ , π k S ) Τ and ϵ ˜ k = ( ϵ k 1 , ⋯ , ϵ k S ) Τ , and where 1 S = ( 1 , ⋯ , 1 ) Τ is an S-dimensional vector of 1s and 1 S × S is an S × S -matrix of 1s.^{5} The portfolio π k is called admissible for initial wealth w k if

W k ≥ 0. ^{6}

We assume that the axioms of the dual theory of the smooth ambiguity model [

E Q k [ u k ( W k ) ] = ∑ s = 1 S Q k s u k ( W k s ) . (4)

Here, Q k = ( Q k 1 , ⋯ , Q k S ) is a probability distribution on Ω defined by

Q k s = ∑ l = 1 L p s ( l ) ρ k ( l ) ,

where the probability distribution ( ρ k ( 1 ) , ⋯ , ρ k ( L ) ) is a distortion of

ρ = ( ρ ( 1 ) , ⋯ , ρ ( L ) ) that reflects agent k ’s attitude towards ambiguity (see Corollary 1 of [

continuously differentiable, with u ′ k ( ∞ ) = l i m x → ∞ d d x u k ( x ) = 0 and

u ′ k ( 0 + ) = l i m x ↓ 0 d d x u k ( x ) = ∞ .

For a given initial wealth w k , agent k chooses an admissible portfolio so as to maximize his welfare represented by (4) over the class of portfolios

A ( w k ) = { π k : E P [ H ( W k − ϵ ˜ k ) ] ≤ w k , E Q k [ u k − ( W k ) ] < ∞ } . ^{7} (5)

In other words, each agent computes the value function

V k ( w k ) = s u p π k ∈ A ( w k ) E k Q [ u k ( W k ) ] .

To solve this problem, we define a function X k by

X k ( λ ) = E Q k [ H L k − 1 ( I k ( λ L k − 1 H ) − ϵ ˜ k ) ] , λ ∈ ( 0, ∞ ) ,

where L k is the likelihood ratio defined by L k = ( L k 1 , ⋯ , L k S ) = ( Q k 1 P 1 , ⋯ , Q k S P S ) , and I k is the inverse function of the marginal utility u ′ k . We note that I k is a map from ( 0, ∞ ) onto itself with I k ( 0 + ) = u ′ k ( 0 + ) = ∞ and I k ( ∞ ) = u ′ k ( ∞ ) = 0 .

Under the settings above, the agent’s optimal wealth and the optimal portfolio are given by the following proposition.

Proposition 1. Suppose that

X k ( λ ) < ∞ , λ ∈ ( 0 , ∞ )

and that

V k ( w k ) < ∞ , w k ∈ ( 0 , ∞ ) .

Then agent k ’s optimal wealth W k * = ( W k 1 * , ⋯ , W k S * ) Τ and optimal portfolio π k * are given by

W k s * = I k ( λ k L k s − 1 H s ) , s = 1 , ⋯ , S , (6)

π k * = ( R − r f 1 S × S ) − 1 ( W k 1 * − ( 1 + r f ) w k − ϵ k 1 ⋮ W k S * − ( 1 + r f ) w k − ϵ k S ) , (7)

where λ k is a solution to the equation of the budget constraint:

X k ( λ k ) = ∑ s = 1 S P s H s ( I k ( λ k L k s − 1 H s ) − ϵ k s ) = w k , (8)

and ( R − r f 1 S × S ) − 1 is the inverse of ( R − r f 1 S × S ) .^{8}

In this section, we show the existence and the uniqueness of the equilibrium and derive the SPD. Before proceeding with the analysis, let us start by defining equilibria in the economy.

Definition 1. An equilibrium is defined as a set of pairs ( π k * , W k * ) , k = 1 , ⋯ , K , of an optimal portfolio and an optimal terminal wealth satisfying the following equations:

∑ k = 1 K π k * = 0 , (9)

∑ k = 1 K W k * = ( 1 + r f ) w 0 1 S + ε ˜ , (10)

where w 0 and ε ˜ = ( ε 1 , ⋯ , ε S ) Τ are the aggregate initial wealth and the aggregate terminal income defined by w 0 = ∑ k = 1 K w k and ε s = ∑ k = 1 K ε k s , s = 1 , ⋯ , S , respectively.

From (6), it follows that (10) is equivalent to

∑ k = 1 K I k ( λ k L k s − 1 H s ) = w 0 ( 1 + r f ) + ε s , s = 1 , ⋯ , S . (11)

For arbitrary Γ = ( γ 1 , ⋯ , γ K ) ∈ ( 0 , ∞ ) K and for each s = 1 , ⋯ , S , let the function I s ( ⋅ ; Γ ) be defined by

I s ( x ; Γ ) : = ∑ k = 1 K I k ( x γ k L k s ) .

Then (11) is equivalent to

I s ( H s ; Γ ) = w 0 ( 1 + r f ) + ε s , s = 1, ⋯ , S ,

with γ k = 1 λ k . If the inverse function H s ( ⋅ ; Γ ) of I s ( ⋅ ; Γ ) is defined by

I s ( H s ( x ; Γ ) ; Γ ) = x , s = 1, ⋯ , S , (12)

then the SPD in equilibrium is given by

H s = H s ( w 0 ( 1 + r f ) + ε s ; Γ ) .

The budget constraint (8) in equilibrium can be rewritten as

∑ s = 1 S P s H s ( w 0 ( 1 + r f ) + ε s ; Γ ) [ I k ( H s ( w 0 ( 1 + r f ) + ε s ; Γ ) γ k L k s ) − ϵ k s ] = w k . (13)

This means that the equilibrium can be characterized by Γ to satisfy (13).

For each s = 1 , ⋯ , S , and ( γ 1 , ⋯ , γ K ) ∈ ( 0, ∞ ) K , we define the utility function of the representative agent by

U ( x ; L s ) = max x = ∑ k = 1 K x k , x k ≥ 0 ∑ k = 1 K γ k L k s u k ( x k ) ,

where L s denotes L s = ( γ 1 L 1 s , ⋯ , γ K L K s ) ∈ ( 0, ∞ ) K , s = 1 , ⋯ , S .

The SPD in equilibrium is characterized in the following proposition.

Proposition 2.

U ′ ( w 0 ( 1 + r f ) + ε s , L s ) = H s ( w 0 ( 1 + r f ) + ε s ; Γ ) .

We note that the utility function of the representative agent has positive homogeneity with respect to L s by the definition, that is, for any positive constant c , U ( x ; c L s ) = c U ( x ; L s ) . Hence, this proposition implies that H s ( w 0 ( 1 + r f ) + ε s ; Γ ) also has this property with respect to Γ . That is, for each s = 1 , ⋯ , S , and each positive constant c, the following equation holds:

H s ( w 0 ( 1 + r f ) + ε s ; c Γ ) = c H s ( w 0 ( 1 + r f ) + ε s ; Γ ) ∀ Γ ∈ ( 0, ∞ ) K . (14)

From this fact, we can also confirm that the budget constraint (13) does not change for any positive homogeneity with respect to Γ . That is, we can write

0 = ∑ s = 1 S P s H s ( w 0 ( 1 + r f ) + ε s ; Γ ) [ I k ( H s ( w 0 ( 1 + r f ) + ε s ; Γ ) γ k L k s ) − ϵ k s ] − w k = ∑ s = 1 S P s H s ( w 0 ( 1 + r f ) + ε s ; Γ ) [ I k ( H s ( w 0 ( 1 + r f ) + ε s ; Γ ) γ k L k s ) − ϵ k s − w k ( 1 + r f ) ] = ∑ s = 1 S P s H s ( w 0 ( 1 + r f ) + ε s ; c Γ ) [ I k ( H s ( w 0 ( 1 + r f ) + ε s ; c Γ ) c γ k L k s ) − ϵ k s − w k ( 1 + r f ) ] = ∑ s = 1 S P s H s ( w 0 ( 1 + r f ) + ε s ; c Γ ) [ I k ( H s ( w 0 ( 1 + r f ) + ε s ; c Γ ) c γ k L k s ) − ϵ k s ] − w k (15)

for every positive constant c .

Let κ be the index of absolute risk aversion for the representative agent, defined by

κ ( x ; L s ) = − U ″ ( x ; L s ) U ′ ( x ; L s ) , s = 1, ⋯ , S . (16)

We immediately obtain the following corollary to the above proposition.

Corollary 1.

H s ( w 0 ( 1 + r f ) + ε s ; Γ ) = e x p ( − ∫ 0 w 0 ( 1 + r f ) + ε s κ ( x ; L s ) d x ) ( 1 + r f ) E P [ e x p ( − ∫ 0 w 0 ( 1 + r f ) + ε ˜ κ ( x ; L ) d x ) ] .

We note that Corollary 1 is an extension of the general economic premium principle of Bühlmann under risk [

The following proposition states the existence and the uniqueness of the equilibrium.

Proposition 3. There exists a Γ ∈ ( 0, ∞ ) K satisfying (13). Furthermore, suppose that the marginal utilities u ′ k , k = 1 , ⋯ , K , satisfy the condition

x u ′ k ( x ) is increasing with respect to x ∈ ( 0, ∞ ) . (17)

Then Γ is unique up to positive constant multiples.

The following proposition characterizes the excess return in equilibrium.

Proposition 4.

μ i − r f = − E P [ U ′ ( w 0 ( 1 + r f ) + ε ˜ ; L ) z ˜ i ] E P [ U ′ ( w 0 ( 1 + r f ) + ε ˜ ; L ) ] . (18)

Let π M = ( π M 1 , ⋯ , π M S ) Τ be a portfolio satisfying

( U ′ ( w 0 ( 1 + r f ) + ε 1 , L 1 ) ⋮ U ′ ( w 0 ( 1 + r f ) + ε S , L S ) ) = ( R − r f 1 S × S ) π M . (19)

We note that there exists a unique π M because the market is complete. We refer to π M as the market portfolio. The terminology of the market portfolio is used as the common portfolio that every agent in the economy holds. At the end of this section, we show that the market portfolio degenerates to the classical market portfolio under certain conditions. The following corollary is a natural extension of CAPM [

Corollary 2. Let r ˜ M = ∑ i = 1 S π M i r ˜ i ∑ i = 1 S π M i be the rate of return of the market portfolio and let μ M = ∑ i = 1 S π M i μ i ∑ i = 1 S π M i be its expected return.

Then

μ i − r f = β i ( μ M − r f ) ,

where

β i = Cov ( r ˜ i , r ˜ M ) Var ( r ˜ M ) ,

and Cov and Var denote the covariance and the variance under P , respectively.

Next, we show that the classical two-fund separation theorem [

Proposition 5. For agent k , the optimal portfolio π k ∗ can be decomposed as a sum of the market portfolio π M and his specific portfolio π k e as follows:

π k ∗ = π M + π k e , (20)

where

π k e = ( R − r f 1 S × S ) − 1 ( W k 1 ∗ − ( 1 + r f ) w k − ϵ k 1 − U ′ ( w 0 ( 1 + r f ) + ε 1 ; L 1 ) ⋮ W k S ∗ − ( 1 + r f ) w k − ϵ k S − U ′ ( w 0 ( 1 + r f ) + ε S ; L S ) ) .

The following is the two-fund separation theorem under ambiguity.

Proposition 6. Assume that all agents have quadratic utility functions, that they are all ambiguity neutral, and that all their terminal incomes are proportional to the aggregate income ε ˜ . Then the optimal portfolio for each agent consists of the market portfolio and the safe security.

We examine how ambiguity preference influences the SPD in equilibrium. We also determine the effects of ambiguity preference on equilibrium excess returns based on the CAPM derived in the previous section.

To keep the analysis simple, we consider a specific case with L = 2 , S = 2 , and K = 2 .^{9} We refer to this economy as a two-state economy. In the two-state economy, the probability distributions over the index set { 1,2 } are given as follows:

ρ ( 1 ) = ρ , ρ ( 2 ) = 1 − ρ ,

ρ k ( 1 ) = ρ k , ρ k ( 2 ) = 1 − ρ k , ρ , ρ k ∈ ( 0 , 1 ) , k = 1 , 2.

Then the reference probability distribution ( P 1 , P 2 ) , and agent k ’s probability distribution ( Q k 1 , Q k 2 ) , k = 1 , 2 , are given by

P 1 = P = p ( 1 ) ρ + p ( 2 ) ( 1 − ρ ) ,

P 2 = 1 − P = ( 1 − p ( 1 ) ) ρ + ( 1 − p ( 2 ) ) ( 1 − ρ ) ,

Q k 1 = Q k = p ( 1 ) ρ k + p ( 2 ) ( 1 − ρ k ) ,

Q k 2 = 1 − Q k = ( 1 − p ( 1 ) ) ρ k + ( 1 − p ( 2 ) ) ( 1 − ρ k ) , k = 1 , 2.

By Definition 1, the SPDs H s , s = 1 , 2 , in equilibrium are given by a solution of the following simultaneous equations:

( P H 1 ( I 1 ( H 1 γ 1 P Q 1 ) − ϵ 11 ) + ( 1 − P ) H 2 ( I 1 ( H 2 γ 1 1 − P 1 − Q 1 ) − ϵ 12 ) = w 1 , P H 1 ( I 2 ( H 1 γ 2 P Q 2 ) − ϵ 21 ) + ( 1 − P ) H 2 ( I 2 ( H 2 γ 2 1 − P 1 − Q 2 ) − ϵ 22 ) = w 2 , I 1 ( H 1 γ 1 P Q 1 ) + I 2 ( H 1 γ 2 P Q 2 ) = ( w 1 + w 2 ) ( 1 + r f ) + ( ϵ 11 + ϵ 21 ) , I 1 ( H 2 γ 1 1 − P 1 − Q 1 ) + I 2 ( H 2 γ 2 1 − P 1 − Q 2 ) = ( w 1 + w 2 ) ( 1 + r f ) + ( ϵ 12 + ϵ 22 ) . (21)

We impose the following assumption on the economy to get explicit results.

Assumption 1. a) Each agent gains strictly higher expected utility under the first-order probabilities with l = 1 than with l = 2 ; that is, for k = 1 , 2 ,

p ( 1 ) u k ( W k 1 ) + ( 1 − p ( 1 ) ) u k ( W k 2 ) < p ( 2 ) u k ( W k 1 ) + ( 1 − p ( 2 ) ) u k ( W k 2 ) .

b) Each agent has log utility; that is, u k ( x ) = log ( x ) , k = 1 , 2 .

Remark 1. We note that if Assumption 1 (a) holds and agent k, k = 1 , 2 , is strictly ambiguity averse (loving) in the two-state economy, then ρ k > ( < ) ρ holds from Corollary 1 and Proposition 1 of [

First, we consider the effects of ambiguity aversion and loving on the SPD in equilibrium.

Proposition 7. In the two-state economy, suppose that

p ( 1 ) < ( > ) p (2)

is satisfied under Assumption 1. Then the following statements hold.

1) If each agent is strictly ambiguity averse, then the SPD H 1 with ambiguity is strictly lower (higher) than that without ambiguity, and the SPD H 2 with ambiguity is strictly higher (lower) than that without ambiguity.

2) If each agent is strictly ambiguity loving, then the SPD H 1 with ambiguity is strictly higher (lower) than that without ambiguity, and the SPD H 2 with ambiguity is strictly lower (higher) than that without ambiguity.

As stated in Remark 1, if all agents are strictly ambiguity averse, then they uniformly increase the weights ρ k , k = 1 , 2 , of the index l = 1 as ρ k > ρ under Assumption 1 (a). This leads to Q k < ( > ) P for both k = 1 , 2 under the condition p ( 1 ) < ( > ) p ( 2 ) . As a result, ambiguity aversion increases (decreases) H 1 and decreases (increases) H 2 . The same reasoning can be applied to the case of ambiguity loving.

Next, we consider the effect of more ambiguity aversion on the excess returns of the ambiguous securities in equilibrium. We compare two economies consisting of the same two-state economy except for the ambiguity preferences of each agent. To distinguish between the two economies, we call them Economy A and Economy B. We assume that all of the agents in Economy A are more ambiguity averse than those in Economy B in the sense of [

Corollary 3. Assume that the conditions of Proposition 7 hold. If the random terms of the rate of return for the i-th ambiguous security are arranged in the order

z i 1 < z i 2 , (22)

then the excess return μ i − r f in equilibrium in Economy A is lower (higher) than that in Economy B.

If the order of the random terms is reversed, that is, z i 1 > z i 2 , then the excess return μ i − r f in equilibrium in Economy A is higher (lower) than that in Economy B.

We note that, from Theorem 2 of [

This paper studies an equilibrium asset pricing model for a static pure exchange economy with ambiguity. The preference of an agent in the economy is represented by the dual theory of the smooth ambiguity model from [

This work was supported by JSPS KAKENHI Grant Number JP17K03825.

Iwaki, H. (2018) An Equilibrium Asset Pricing Model under the Dual Theory of the Smooth Ambiguity Model. Journal of Mathematical Finance, 8, 497-515. https://doi.org/10.4236/jmf.2018.82031

π k ∗ is trivially an admissible portfolio because W k s * ≥ 0 from (6). We first show that π k ∗ ∈ A ( w k ) . It is obvious that E P [ H ( W k * − ϵ ˜ k ) ] ≤ w k holds by (8).

Following [

u ^ k ( x ) = m a x y ∈ ( 0, ∞ ) [ u k ( y ) − x y ] , x ∈ ( 0, ∞ ) ,

and is a decreasing, convex and continuously differentiable function on ( 0, ∞ ) , satisfying

u ^ k ( x ) = u k ( I k ( x ) ) − x I k ( x ) ,

u ^ ′ k ( x ) = − I k ( x ) , x ∈ ( 0, ∞ ) .

Using the convex dual, we have

u k ( I k ( λ k L k s − 1 H s ) ) − λ k L k s − 1 H s I k ( λ k L k s − 1 H s ) ≥ u k ( 1 ) − λ k L k s − 1 H s

⇔ u k ( W k s * ) ≥ u k ( 1 ) + λ k L k s − 1 H s ( W k * − 1 ) , s = 1, ⋯ , S .

Because W k s * ≥ 0 , we have λ k L k s − 1 H s W k s * ≥ 0 , and so

u k ( W k s * ) ≥ u k ( 1 ) − λ k L k s − 1 H s , s = 1 , ⋯ , S . (23)

From (23), we have

E Q k [ u k − ( W k s * ) ] ≤ | u k ( 1 ) | + λ k E Q k [ L k − 1 H ] = | u k ( 1 ) | + λ k E P [ H ] = | u k ( 1 ) | + λ k 1 + r f < ∞ .

Thus π k ∗ ∈ A ( w k ) .

Next, we show that π k ∗ is optimal. For all λ > 0 and π k ∈ A ( w k ) ,

E Q k [ u k ( W k ) ] ≤ E Q k [ u k ( W k ) ] + λ { w k − E Q [ L k − 1 H ( W k − ϵ ˜ k ) ] } ≤ E Q k [ u ^ k ( λ L k − 1 H ) ] + λ ( w k + E Q [ L k − 1 H ϵ ˜ k ] ) = E Q k [ u k ( I k ( λ L k − 1 H ) ) ] + λ { w k − E Q [ L k − 1 H ( I k ( λ L k − 1 H ) − ϵ ˜ k ) ] } . (24)

The first inequality is due to the budget constraint, E Q [ L k H ( W k − ϵ ˜ k ) ] ≤ w k . The second inequality and the equality are due to the definition of u ^ k . From (6) and (8), the expression (24) holds with equality if and only if W k = W k ∗ and λ = λ k . This means that π k = π k * is optimal from (7).

A.2. Proof of Proposition 2Let W k s ⋄ be Defined by

W k s ⋄ = I k ( H s ( w 0 ( 1 + r f ) + ε s ; Γ ) γ k L k s ) . (25)

Then

∑ k = 1 K W k s ⋄ = ∑ k = 1 K I k ( H s ( w 0 ( 1 + r f ) + ε s ; Γ ) γ k L k s ) = I s H s ( ( w 0 ( 1 + r f ) + ε s ; Γ ) ; Γ ) = w 0 ( 1 + r f ) + ε s

by the definitions of I and H . This means that W k s ⋄ satisfies (10).

Since I k is the inverse of u ′ k , (25) can be rewritten as

γ k L k s u ′ k ( W k s ⋄ ) = H s ( w 0 ( 1 + r f ) + ε s ; Γ ) .

Since u k is strictly concave,

∑ k = 1 K γ k L k s u k ( x k ) ≤ ∑ k = 1 K γ k L k s { u k ( W k s ⋄ ) + ( x k − W k s ⋄ ) u ′ k ( W k s ⋄ ) } = ∑ k = 1 K γ k L k s u k ( W k s ⋄ ) + H s ( w 0 ( 1 + r f ) + ε s ; Γ ) ∑ k = 1 K ( x k − W k s ⋄ )

for all x k ∈ ( 0, ∞ ) . This holds with equality if and only if x k = W k s ⋄ . Hence, we have

U ( w 0 ( 1 + r f ) + ε s ; L s ) = ∑ k = 1 K γ k L k s u k ( W k s ⋄ ) = ∑ k = 1 K γ k L k s u k ( I k ( H s ( w 0 ( 1 + r f ) + ε s ; Γ ) γ k L k s ) ) .

d d x ∑ k = 1 K I k ( H s ( x ; Γ ) γ k L k s ) = 1.

Differentiating the above equation completes the proof.^{10}

From (16), there is a constant A for which

U ′ ( x ; L s ) = A e x p ( − ∫ 0 x κ ( t ; L s ) d t ) .

The result then follows from Proposition 2 and the fact that E P [ U ′ ( w 0 ( 1 + r f ) + ε ˜ ; L ) ] = E P [ H ( w 0 ( 1 + r f ) + ε ˜ ; Γ ) ] = 1 1 + r f .

A.4. Proof of Proposition 3We first show the existence. Let K = { 1, ⋯ , K } be an index set of agents and let e 1 , ⋯ , e K ∈ ( 0, ∞ ) K be the K -dimensional unit coordinate vectors. For any B ⊂ K , we denote the convex hull of { e k : k ∈ B } by

S B = { ∑ k ∈ B γ k e k : ∑ k ∈ B γ k = 1, γ k ≥ 0, k ∈ B } .

Let S B + ⊆ S B be the set

S B + = { ∑ k ∈ B γ k e k : ∑ k ∈ B γ k = 1, γ k > 0, k ∈ B } .

For each k ∈ K , we define a function C k : S K → ℝ by

C k ( Γ ) = { ∑ s = 1 S P s H s ( w 0 ( 1 + r f ) + ε s ; Γ ) ( I k ( H s ( w 0 ( 1 + r f ) + ε s ; Γ ) γ k L k s ) − ϵ k s ) − w k if γ k > 0 , − ∑ s = 1 S P s H s ( w 0 ( 1 + r f ) + ε s ; Γ ) ϵ k s − w k if γ k = 0.

To prove that the proposition holds, we have to show that there exists a Γ ∈ S K satisfying C k ( Γ ) = 0 for each k ∈ K .

Because the function C k is continuous, the set

F k = { Γ ∈ S K ; C k ( Γ ) ≥ 0 }

is closed. On the other hand, from (11) and (12),

∑ k ∈ K C k ( Γ ) = 0 ∀ Γ ∈ S K . (26)

Now, suppose that there exists a Γ ^ ∈ S K such that Γ ^ ∈ ∪ k ∈ K F k . Then C k ( Γ ^ ) < 0 for all k ∈ K , which contradicts (26). Therefore,

S K = ∪ k ∈ K F k .

Furthermore, suppose that there exists a Γ ^ ∈ S B such that Γ ^ ∈ ∪ k ∈ B F k . Then C k ( Γ ^ ) < 0 for all k ∈ B . In this case, let γ ^ j = 0 for j ∈ K \ B . Then Γ ^ ∈ S K and ∑ k ∈ K C k ( Γ ^ ) < 0 , which again contradicts (26). Therefore,

S B ⊂ ∪ k ∈ B F k ∀ B ⊂ K . (27)

From (27) and the Knaster-Kratowski-Mazurkiewicz Theorem (cf. p. 26 of [

C k ( Γ * ) = 0 ∀ k ∈ K . (28)

Otherwise we would have ∑ k ∈ K C k ( Γ * ) > 0 , which contradicts (26). We also have Γ * ∈ S K + , because if there exists a k ∈ K such that γ k * = 0 , then C k ( Γ * ) < 0 , which contradicts (28). Therefore, we can conclude that there exists a Γ * ∈ ∩ k ∈ K F k that belongs to ( 0, ∞ ) K .

Next, we show the uniqueness. For any pair of vectors ( Γ ( i ) , Γ ( j ) ) ∈ ( 0, ∞ ) K × ( 0, ∞ ) K , we consider the usual order:

Γ ( i ) ≤ Γ ( j ) ⇔ γ k ( i ) ≤ γ k ( j ) ∀ k = 1 , ⋯ , K ,

Γ ( i ) < Γ ( j ) ⇔ Γ ( i ) ≤ Γ ( j ) , Γ ( i ) ≠ Γ ( j ) .

Let Γ ( 1 ) and Γ ^ be vectors in ( 0, ∞ ) , both of which satisfy (13). We define another vector Γ ( 2 ) by Γ ( 2 ) = c Γ ^ for a positive constant c = m a x k ∈ K γ k ( 1 ) / γ ^ k . Then, from (15), Γ ( 2 ) also satisfies (13), and Γ ( 1 ) ≤ Γ ( 2 ) . If Γ ( 1 ) = Γ ( 2 ) , then Γ ( 1 ) is a positive constant multiple of Γ ^ . Hence, we have only to show that Γ ( 1 ) < Γ ( 2 ) does not hold.

Suppose that Γ ( 1 ) < Γ ( 2 ) . Then, from the definition of H s ,

H s ( w 0 ( 1 + r f ) + ε s ; Γ ( 1 ) ) < H s ( w 0 ( 1 + r f ) + ε s ; Γ (2) )

holds. Hence, for k ∈ K such that γ k ( 1 ) = c γ ^ k = γ k ( 2 ) , we have

1 γ k ( 1 ) ∑ s = 1 S P s H s ( w 0 ( 1 + r f ) + ε s ; Γ ( 1 ) ) ϵ k s < 1 γ k ( 2 ) ∑ s = 1 S P s H s ( w 0 ( 1 + r f ) + ε s ; Γ ( 2 ) ) ϵ k s . (29)

On the other hand, because I k is the inverse of u k , (17) is equivalent to x I k ( x ) decreasing with respect to x ∈ ( 0, ∞ ) . Hence, noting that L k s > 0 , s = 1 , ⋯ , S , k ∈ K , we have

H s ( w 0 ( 1 + r f ) + ε s ; Γ ( 1 ) ) γ k ( 1 ) I k ( H s ( w 0 ( 1 + r f ) + ε s ; Γ ( 1 ) ) γ k ( 1 ) L k s ) ≥ H s ( w 0 ( 1 + r f ) + ε s ; Γ ( 2 ) ) γ k ( 2 ) I k ( H s ( w 0 ( 1 + r f ) + ε s ; Γ ( 2 ) ) γ k ( 2 ) L k s ) .

This inequality leads to

1 γ k ( 1 ) ∑ s = 1 S P s H s ( w 0 ( 1 + r f ) + ε s ; Γ ( 1 ) ) I k ( H s ( w 0 ( 1 + r f ) + ε s ; Γ ( 1 ) ) γ k ( 1 ) L k s ) ≥ 1 γ k ( 2 ) ∑ s = 1 S P s H s ( w 0 ( 1 + r f ) + ε s ; Γ ( 2 ) ) I k ( H s ( w 0 ( 1 + r f ) + ε s ; Γ ( 2 ) ) γ k ( 2 ) L k s ) . (30)

Combining (29) and (30), we have

1 γ k ( 1 ) C k ( Γ ( 1 ) ) > 1 γ k ( 2 ) C k ( Γ ( 2 ) ) .

However, this contradicts (28). That is, Γ ( 1 ) < Γ ( 2 ) never holds.

A.5. Proof of Proposition 4From (2), we have

E P [ H ( w 0 ( 1 + r f ) + ε ˜ ; Γ ) ( 1 + r ˜ i ) ] = E P [ H ( w 0 ( 1 + r f ) + ε ˜ ; Γ ) ( 1 + r f + μ i − r f + z ˜ i ) ] = 1

⇔ μ i − r f = − E P [ H ( w 0 ( 1 + r f ) + ε ˜ ; Γ ) z ˜ i ] E P [ H ( w 0 ( 1 + r f ) + ε ˜ ; Γ ) ] ,

where we use E P [ H ( w 0 ( 1 + r f ) + ε ˜ ; Γ ) ] = 1 / ( 1 + r f ) by (1).

A.6. Proof of Corollary 2From (18) and (19),

μ i − r f = − E P [ U ′ ( w 0 ( 1 + r f ) + ε ˜ ; L ) z ˜ i ] E P [ U ′ ( w 0 ( 1 + r f ) + ε ˜ ; L ) ] = − E P [ ( ∑ j = 1 S z ˜ j π M j ) z ˜ i ] E P [ U ′ ( w 0 ( 1 + r f ) + ε ˜ ; L ) ] , (31)

where we have used the fact: E P [ z ˜ j ] = 0 in the second equality. From (31) and the definition of μ M , we have

μ M − r f = ∑ i = 1 S π M i ∑ j = 1 S π M j ( μ i − r f ) = − E P [ ( ∑ j = 1 S π M j z ˜ j ) 2 ] ∑ j = 1 S π M j E P [ U ′ ( w 0 ( 1 + r f ) + ε ˜ ; L ) ] . (32)

Canceling out E P [ U ′ ( w 0 ( 1 + r f ) + ε ˜ ; L ) ] from (31) and (32), we obtain

μ i − r f = ∑ j = 1 S π M j E P [ z ˜ i ∑ j = 1 S z ˜ j π M j ] E P [ ( ∑ j = 1 S π M j z ˜ j ) 2 ] ( μ M − r f ) .

Here, recalling

Cov ( r ˜ i , r ˜ M ) = 1 ∑ j = 1 S π M j E P [ z ˜ i ∑ j = 1 S π M j z ˜ j ]

and

Var ( r ˜ M ) = 1 ( ∑ j = 1 S π M j ) 2 E P [ ( ∑ j = 1 S π M j z ˜ j ) 2 ] ,

we obtain the result.

A.7. Proof of Proposition 6Because all agents have quadratic utility functions, we can assume that the marginal utility for each agent k, k = 1 , ⋯ , K , is given by

u ′ k ( x ) = − x + α k , x < α k ,

for some constant α k . Noting that ambiguity neutrality implies that L k s = 1 , s = 1 , ⋯ , S , (see, [

W k s * = α k − λ k H s . (33)

Because the terminal income is proportional to the aggregate income ε ˜ , there exists a constant η k satisfying ϵ ˜ k = η k ε ˜ and ∑ k = 1 K η k = 1 . Hence, from (8),

λ k = α k 1 + r f − w k − η k E P [ H ε ˜ ] E P [ H 2 ] . (34)

Substituting (34) into (33), we have from (3) that

W k s * = α k − α k 1 + r f − w k − η k E P [ H ε ˜ ] E P [ H 2 ] H s = ( 1 + r f ) w k + ( r 1 s − r f , ⋯ , r S s − r f ) π k * + η k ε s

⇔ ( ( 1 + r f ) − H s E P [ H 2 ] ) ( α k 1 + r f − w k ) + η k ( E P [ H ε ˜ ] H s E P [ H 2 ] − ε s ) = ( r 1 s − r f , ⋯ , r S s − r f ) π k * . (35)

Summing the above equations for k = 1 , ⋯ , K , and applying (9), we have

E P [ H ε ˜ ] H s E P [ H 2 ] − ε s = − ( ( 1 + r f ) − H s E P [ H 2 ] ) ζ ,

where ζ = ∑ k = 1 K ( α k 1 + r f − w k ) . Substituting this into (35), we have

( ( 1 + r f ) − H s E P [ H 2 ] ) ( α k 1 + r f − w k − ζ η k ) = ( r 1 s − r f , ⋯ , r S s − r f ) π k * .

Hence, from Proposition 2 and by the definition of the market portfolio π M , the optimal portfolio π k * satisfies

( r 1 s − r f , ⋯ , r S s − r f ) π k * = ι k ( ( 1 + r f ) − 1 E P [ H 2 ] ( r 1 s − r f , ⋯ , r S s − r f ) π M ) ,

where ι k = α k 1 + r f − w k − ζ η k . This means that the optimal portfolio consists of the safe security and the market portfolio.

A.8. Proof of Proposition 7Under the condition: p ( 1 ) < ( > ) p ( 2 ) , if each agent is strictly ambiguity averse, from the definition of the probability distributions ( Q k 1 , Q k 2 ) , k = 1 , 2 , and Remark 1,

Q k = p ( 1 ) ρ k + p ( 2 ) ( 1 − ρ k ) < ( > ) p ( 1 ) ρ + p ( 2 ) ( 1 − ρ ) = P .

Similarly, under the same condition, if each agent is strictly ambiguity loving, Q k > ( < ) P . Hence, to prove the proposition it is sufficient to show that

∂ H 1 ∂ Q k > 0 , ∂ H 2 ∂ Q k < 0 , k = 1 , 2. (36)

From Assumption 1 (b), (21) can be explicitly rewritten as

( γ 1 = w 1 + P H 1 ϵ 11 + ( 1 − P ) H 2 ϵ 12 , γ 2 = w 2 + P H 1 ϵ 21 + ( 1 − P ) H 2 ϵ 22 , γ 1 Q 1 + γ 2 Q 2 H 1 P = ( w 1 + w 2 ) ( 1 + r f ) + ϵ 11 + ϵ 21 , γ 1 ( 1 − Q 1 ) + γ 2 ( 1 − Q 2 ) H 2 ( 1 − P ) = ( w 1 + w 2 ) ( 1 + r f ) + ϵ 12 + ϵ 22 . (37)

Solving the above equations with respect to ( H 1 , H 2 ) , we obtain

( H 1 H 2 ) = 1 W 0 + a + b ( 1 P ( Q 1 w 1 + Q 2 w 2 + b 1 + r f ) 1 1 − P ( ( 1 − Q 1 ) w 1 + ( 1 − Q 2 ) w 2 + a 1 + r f ) ) ,

where we put W 0 = ( w 1 + w 2 ) ( 1 + r f ) , a = ( 1 − Q 1 ) ϵ 11 + ( 1 − Q 2 ) ϵ 21 and b = Q 1 ϵ 12 + Q 2 ϵ 22 . From this, we obtain for k = 1 , 2 :

∂ H 1 ∂ Q k = w k ( W 0 + a + b ) + ϵ ¯ k ( 1 ) w 1 + ϵ ¯ k ( 2 ) w 2 + ϵ k 1 b + ϵ k 2 a 1 + r f P ( W 0 + a + b ) 2 ,

∂ H 2 ∂ Q k = − w k ( W 0 + a + b ) + ϵ ¯ k ( 1 ) w 1 + ϵ ¯ k ( 2 ) w 2 + ϵ k 1 b + ϵ k 2 a 1 + r f ( 1 − P ) ( W 0 + a + b ) 2 ,

where we put ϵ ¯ s ( k ) = Q k ϵ s 1 + ( 1 − Q k ) ϵ s 2 , s , k = 1 , 2 . Hence we obtain (36).

A.9. Proof of Corollary 3Let O = ( O 1 , O 2 ) be a probability distribution over the states defined by

O s = H s E P [ H ] P s , s = 1,2. (38)

We first note that, from (18) and Proposition 2, the excess returns in equilibrium are given by

μ i − r f = − E O [ z ˜ i ] (39)

where E O denotes the expectation under O . For each state s = 1 , 2 , let H s A and H s B denote the SPD in equilibrium for Economies A and B, respectively. Similarly, let O A and O B denote the probability distribution O s for Economies A and B, respectively. From Theorem 2 of [

p ( 1 ) < ( > ) p ( 2 ) ⇒ H 1 A < ( > ) H 1 B and H 2 A > ( < ) H 2 B .

From (38), this implies that if p ( 1 ) < p ( 2 ) , then O 1 A < O 1 B and O 2 A = 1 − O 1 A > 1 − O 1 B = O 2 B , while if p ( 1 ) > p ( 2 ) , then O 1 A > O 1 B and O 2 A < O 2 B . Hence, from (39), we obtain the result.