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In this paper, we consider an equilibrium insurance premium and risk exchange in a pure exchange economy with ambiguity or Knightian uncertainty. Each agent’s preference is represented by the expected utility with uncertainty probability (EUUP) theory. The Bühlmann’s economic premium principle is generalized under EUUP. Contrary to the existing models, our principle under uncertainty is given unanimously and can be calculated more easily and explicitly. Through comparative statics, we show that insurance transactions occur and demand for insurance is not always comonotonic due to the difference in the degree of ambiguity aversion even if all of the agents in the economy are ambiguity averse or ambiguity loving.

Equilibrium asset pricing models of financial and insurance markets have been extensively studied in the literature. Following the pioneering work of [

This paper further develops a model that can be used in empirical and behavioral studies by refining the separations between tastes and beliefs, and between risk and ambiguity. To this end, we adopt a new decision-making model called expected utility with uncertain probability (EUUP) theory of [

We consider a pure exchange economy where insurance is traded. The economy is identical with [

These results are our main contribution to existing literature. [

This paper is organized as follows. We formally state our model in Section 2. In Section 3, we derive our economic premium principle. Based on the equilibrium insurance premium, we perform some comparative statics in Section 4. Finally, we make concluding remarks in Section 5.

We consider a single period pure exchange economy that consists of n agents (each agent is denoted as agent i , i = 1 , 2 , ⋯ , n ). The commodities to be traded are quantities of money, conditional on the state s ∈ S ⊂ ℝ with a Borel measurable space ( S , B ( S ) ) 5. Let X i ( s ) denote the loss agent i faces, at the terminal time of the period if the state s ∈ S occurs. At the start of the period, to hedge the loss, each agent buys or sells an insurance policy that pays Y i ( s ) at the terminal time of the period if the state s occurs. Let ℙ = { P θ ( s ) ; s ∈ S } θ ∈ Θ be a set of probability measures on ( S , B ( S ) ) , where Θ ⊂ ℝ is an interval in the real line that stands for an arbitrary index set. Ambiguity is represented by a probability measure Q on ( Θ , B ( Θ ) ) , which is assumed to be the same among all agents6.

Agent i is characterized by her/his utility function u i : ℝ → ℝ . The utility function u i is assumed to be strictly increasing, strictly concave and twice continuously differentiable with the properties lim x → ∞ u ′ i ( x ) = 0 and lim x → − ∞ u ′ i ( x ) = ∞ .

For each s ∈ S , assuming that P θ ( s ) is B ( Θ ) -measurable, we define a probability measure on ( S , B ( S ) ) by P ¯ ( s ) = ∫ θ ∈ Θ P θ ( s ) d Q ( θ ) . P ¯ is the probability measure when agents are ambiguity neutral. We call the probability measure P ¯ the reference measure7 because this probability measure is known to every agent. Let E be the expectation under P ¯ . We also assume that insurance traded in the market is priced through P ¯ , that is, Y i = { Y i ( s ) : s ∈ S } can be bought or sold by agent i at a price;

p ( Y i ) = E [ Y i ϕ ] = ∫ s ∈ S Y i ( s ) ϕ ( s ) d P ¯ ( s ) ,

where ϕ : S → ( 0, ∞ ) is the state price density (SPD) which satisfies E [ ϕ ] = 1 . We note that although the expectation is taken under the reference measure, agents’ ambiguity attitudes (ambiguity averse, ambiguity neutral, or ambiguity seeking) are reflected in the SPD as shown later.

Let w i be the initial wealth of agent i, then, for a given ( w i , X i , Y i ) , the terminal wealth W i = { W i ( s ) : s ∈ S } of agent i is given by

W i ( s ) = w i − X i ( s ) + Y i ( s ) − p ( Y i ) , s ∈ S . ^{8} (1)

Then, assuming that each agent evaluates her terminal wealth by the EUUP without distortion of perceived probabilities9, the welfare of agent i is given by

V i ( W i ) = ∫ z ≤ 0 ( G u i ( W i ) ( i ) ( z ) − 1 ) d z + ∫ z ≥ 0 G u i ( W i ) ( i ) ( z ) d z , (2)

where G u i ( W i ) ( i ) denotes a capacity on ( S , B ( S ) ) 1^{0}, that is defined by

G u i ( W i ) ( i ) ( z ) = φ i − 1 ( ∫ θ ∈ Θ φ i ( P θ { s ∈ S : u i ( W i ( s ) ) > z } ) d Q ( θ ) ) = φ i − 1 ( ∫ θ ∈ Θ φ i ( 1 − F W i ( u i − 1 ( z ) ; θ ) ) d Q ( θ ) ) , z ∈ ℝ ,

where φ i : [ 0,1 ] → [ 0,1 ] is a strictly increasing continuous function that represents agent i’s attitude toward ambiguity and that is referred to as the probability outlook function, and where, for each θ ∈ Θ , F W i ( ⋅ ; θ ) denotes the cumulative distribution function (cdf) of W i under the probability measure P θ . We note that the valuation function (2) coincides with the cumulative prospect theory (CPT) of [

We consider a problem in which each agent i decides the amount Y i to maximize the welfare V i ( W i ) . More precisely, agent i faces the following maximization problem;

| Maximize V i ( W i ) s . to w i = E [ ϕ ( W i + X i ) ] . (3)

Once an optimal terminal wealth W i * that solves (3) is given, agent i’s optimal insurance Y * is given by

Y i * − p ( Y i * ) = W i * + X i − w i . (4)

Here, we note that the optimal insurance Y i * is only determined up to an additive constant because, for any constant c, Y i * − c − E [ ϕ ( Y i * − c ) ] = Y i * − E [ ϕ Y i * ] . Hence, we normalize the optimal insurance Y i * so as to satisfy E [ ϕ Y i * ] = 0 hereafter.

To solve the maximization problem (3), we first represent the terminal welfare V i ( W i ) as the expected utility under agent i’s perceive probability measure as shown in the following lemma.

Lemma 1 Let F ^ i , W i : ℝ → [ 0,1 ] be agent i’s perceived-cumulative distribution function (perceived-cdf) of W i , defined by

F ^ i , W i ( x ) : = φ ¯ i − 1 ( ∫ θ ∈ Θ φ ¯ i ( F W i ( x ; θ ) d Q ( θ ) ) , x ∈ ℝ , (5)

where φ ¯ i be the dual of the outlook function φ i , that is defined by

φ ¯ i ( p ) : = 1 − φ i ( 1 − p ) , p ∈ [ 0,1 ] .

Then, the value function V i can be rewritten as

V i ( W i ) = ∫ x ∈ ℝ u i ( x ) d F ^ i , W i ( x ) . (6)

In other words, let P ^ i be the induced probability measure induced by the perceived-cdf F ^ i , W i . Then the value function V i is given by the expected utility under P ^ i as

V i ( W i ) = E ^ i [ u i ( W i ) ] ,

where E ^ i denotes expectation under P ^ i .

Proof. We first note that using the dual of the outlook function, we can rewrite G u i ( W i ) ( i ) as

G u i ( W i ) ( i ) ( x ) = 1 − φ ¯ i − 1 ( ∫ θ ∈ Θ φ ¯ i ( F i , W i ( u − 1 ( x ) ; θ ) ) d Q ( θ ) ) = 1 − F ^ i , W i ( u − 1 ( x ) ) , x ∈ ℝ .

By applying the change of variable

p = F ^ i , W i ( u − 1 ( z ) ) ⇔ z = u i ( F ^ i , W i − 1 ( p ) ) (7)

to (2), and performing integration by parts, we obtain

V i ( W i ) = − ∫ 0 F ^ i , W i ( u i − 1 ( 0 ) ) p d u i ( F ^ i , W i − 1 ( p ) ) + ∫ F ^ i , W i ( u i − 1 ( 0 ) ) 1 ( 1 − p ) d u i ( F ^ i , W i − 1 ( p ) ) = ∫ 0 1 u i ( F ^ i , W i − 1 ( p ) ) d p .

Applying the change of variable

x = F ^ i , W i − 1 ( p )

to the above equation leads to

V i ( W i ) = ∫ x ∈ ℝ u i ( x ) d F ^ i , W i ( x ) .

Hence, we obtain the result.

From the above lemma, agent i’s welfare maximization problem can be rewritten as follows.

| Maximize E ^ i [ u i ( W i ) ] s . to w i = E [ ϕ ( W i + X i ) ] . (8)

That is, we can formulate the problem as a parallel problem to the classical expected utility maximization problem as can be seen in (8). We can solve this problem by an orthodox method. The results are shown in the following proposition.

Proposition 1 Let I i be the inverse function of the marginal utility u ′ i , and let L i be the Radon-Nikodym derivative of P ^ i with respect to P ¯ , i.e.,

L i = d P ^ i d P ¯ . (9)

Then, agent i’s optimal terminal wealth W i * is given by

W i * = I i ( λ i L i − 1 ϕ ) , (10)

where λ i is a positive constant defined by

w i − E [ ϕ X i ] = E [ ϕ I i ( λ i L i − 1 ϕ ) ] . (11)

Furthermore, agent i’s optimal insurance Y i * is given by

Y i * = I i ( λ i L i − 1 ϕ ) + X i ( s ) − w i . (12)

Proof. From (9), we can rewrite (8) as

| Maximize E ^ i [ u i ( W i ) ] s . to w i = E ^ [ ϕ ( W i + X i ) L i ] .

This is an orthodox concave maximization problem with a linear constraint. Hence we can immediately obtain the results of (10) and (11) with the Lagrange multiplier λ i (See e.g., Theorem 6.3 of [

Hence, (12) in Proposition 1, which gives the optimal insurance, is rewritten as

Y i * = I i ( λ i L i − 1 ϕ ) + X i − w i . (13)

We define an equilibrium in the economy described in the previous section as follows.

Definition 1 ( ϕ , Y 1 * , ⋯ , Y n * ) is an equilibrium if the following two conditions are satisfied.

For each i = 1 , ⋯ , n , Y i * is the Agent i’s optimal insurance policy given by Proposition 1;

∑ i = 1 n I i ( λ i L i − 1 ( s ) ϕ ( s ) ) = w − X ( s ) , s ∈ S , ^{11} (14)

where w : = ∑ i = 1 n w i and X ( s ) : = ∑ i = 1 n X i ( s ) are the aggregate initial wealth and the aggregate loss, respectively.

For each s ∈ S , let I ( x ; s , λ ) be defined by

I ( x ; s , λ ) : = ∑ i = 1 n I i ( λ i L i ( s ) x ) .

Then, (14) is rewritten as

I ( ϕ ( s ) ; s , λ ) = w − X ( s ) , s ∈ S .

Because I ( x ; s , λ ) is trivially a strictly decreasing function, it has its inverse function. Hence, if the inverse function H ( ⋅ ; s , λ ) of I ( ⋅ ; s , λ ) is defined by

I ( H ( x ; s , λ ) ; s , λ ) = x , s ∈ S , (15)

then the SPD in equilibrium is given by

ϕ ( s ) = H ( w − X ( s ) ; s , λ ) , s ∈ S .

Because in the equilibrium the budget constraint (11) can be rewritten as

E [ H ( w − X ; λ ) ( I i ( λ i L i H ( w − X ; λ ) ) + X i ) ] = w i , (16)

the equilibrium can be characterized by λ = ( λ 1 , ⋯ , λ n ) ∈ ( 0, ∞ ) n satisfying (16).

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

Proposition 2 There exists a λ ∈ ( 0, ∞ ) n satisfying (16). Furthermore, suppose that, for each agent i , i = 1 , ⋯ , n , the Arrow-Pratt relative risk aversion satisfies the condition:

− x u ″ i ( x ) u ′ i ( x ) ≤ 1, x ∈ ( 0, ∞ ) . (17)

Then, λ is unique up to positive constant multiples.

Proof. By the same arguments as the proof of Theorem 4.6.1 of [

We note that the restriction (17) on relative risk aversion is important in the consumption-saving problem to sign the comparative statics of (stochastic) changes in the interest rate, see [

When we specify the form of utility function as log or exponential functions, we can derive the SPD in equilibrium analytically as a closed-form function. In this subsection, assuming that each agent has an exponential utility function, we derive the SPD and optimal demands for insurance explicitly.

Proposition 3 Suppose that each agent have an exponential utility function with an index of constant risk aversion ρ i > 0 , that is,

u i ( x ) = 1 ρ i ( 1 − e − ρ i x ) , i = 1, ⋯ , n . (18)

Then, the SPD is given in equilibrium as

ϕ = e ρ ¯ X ∏ i = 1 n L i ρ ¯ ρ i E [ e ρ ¯ X ∏ i = 1 n L i ρ ¯ ρ i ] . (19)

where ρ ¯ is a constant defined by 1 ρ ¯ = ∑ i = 1 n 1 ρ i .

Proof. Because u ′ i ( x ) = e − ρ i x and I i ( y ) = − 1 ρ i log y in this case, the market clearing condition (14) is given as

w − X = ∑ i = 1 n I i ( λ i L i ϕ ) = − ∑ i = 1 n 1 ρ i log ( λ i L i ϕ ) = − ∑ i = 1 n 1 ρ i log ( λ i L i ) − 1 ρ ¯ log ϕ .

From this, we have

log ϕ = ρ ¯ X + log ∏ i = 1 n L i ρ ¯ ρ i − ρ ¯ w − log ∏ i = 1 n λ i ρ ¯ ρ i

or ϕ = e ρ ¯ X ∏ i = 1 n L i ρ ¯ ρ i κ ,

where κ is a constant defined by κ = e − ρ ¯ w ∏ i = 1 n λ i − ρ ¯ ρ i . On the other hand, from the definition of the SPD, because E [ ϕ ] = 1 , κ must coincide with

1 E [ e ρ ¯ X ∏ i = 1 n L i ρ ¯ ρ i ] .

Hence, we obtain the result.

We note that if L i = 1 , i = 1 , ⋯ , n , the SPD (19) coincides with that of [

Next, we derive the optimal insurance for each agent.

Proposition 4 Under the same assumption of Proposition 3, the agent i’s optimal insurance Y * is given as

Y i * = − ρ ¯ ρ i ( X − E [ ϕ X ] ) − ρ ¯ ρ i ∑ j ≠ i 1 ρ j ( log L j L i − E [ ϕ log L j L i ] ) + X i − E [ ϕ X i ] , s ∈ S . (20)

Proof. Because I i ( y ) = − 1 ρ i log y by (18), from (13), we have

Y i * = I i ( λ i ϕ L i ) + X i − w i = − 1 ρ i log λ i − 1 ρ i log ϕ + 1 ρ i log L i + X i − w i . (21)

Multiplying by ϕ on both sides of the above equations, and taking the expectation E leads to

0 = − 1 ρ i log λ i − 1 ρ i E [ ϕ log ϕ ] + 1 ρ i E [ ϕ log L i ] + E [ ϕ X i ] − w i . (22)

Here we note that we have used the fact that E [ ϕ Y * ] = 0 and E [ ϕ ] = 1 . Cancelling out − 1 ρ i log λ i from (21) and (22), we have

Y i * = − 1 ρ i ( log ϕ − E [ ϕ log ϕ ] ) + 1 ρ i ( log L i − E [ ϕ log L i ] ) + X i − E [ ϕ X i ] .

Finally, substituting (19) into the above equation, we obtain (20).

We also note that if L i ≡ 1 , i = 1 , ⋯ , n , optimal insurance Y * in (20) coincides with that of [

− ρ ¯ ρ i ( X − E [ ϕ X ] ) , and difference of attitude toward ambiguity − ρ ¯ ρ i ∑ j ≠ i 1 ρ j ( log L j L i − E [ ϕ log L j L i ] ) .

In this section, we make some comparative statics to examine the effects of ambiguity on insurance demand and premium under the assumption that all agents have exponential utility functions treated in the previous section.

We consider the economy consisting of two agents i = 1 , 2 . To eliminate other effects and focus only on the effect of ambiguity, we make the following assumptions. First, each agent faces the same amount of loss X i ( s ) = − e − s at the terminal time of the period, that is, the amounts of loss are strictly decreasing w.r.t. state s ∈ ℝ 1^{2}. Second, we assume that each agent shares the same type

exponential utility function such that u i ( x ) = 1 − e − ρ x ρ , i = 1 , 2 . That is, each

agent possesses the same constant index of risk aversion ρ . In other words, each agent shares the same attitude toward risk. In the sequel, we specify the value of the index of risk aversion as ρ = 0.00064 1^{3}. As to the ambiguity, the probability distribution of the state s ∈ ℝ follows the normal distribution with mean θ ∈ { − 1,0,1 } , and standard deviation 1. That is, the probability density function of the state is given by

f ( x ; θ ) = ( f ( x ; − 1 ) = 1 2 π e − ( x + 1 ) 2 2 w .p . 1 3 f ( x ; 0 ) = 1 2 π e − x 2 2 w .p . 1 3 f ( x ; 1 ) = 1 2 π e − ( x − 1 ) 2 2 w .p . 1 3

We assume that the function that represents agent i’s attitude toward ambiguity follows CAAA (Constant Absolute Ambiguity Aversion) type; such that

φ i ( x ) = 1 − e − τ i x τ i ,

where τ i = − φ ″ i ( x ) φ ′ i ( x ) is a positive constant that represents her/his degree of

ambiguity aversion. We note that if τ i > 0 then agent i is ambiguity averse, and otherwise if τ i < 0 then she/he is ambiguity loving, and that if τ 1 > ( < ) τ 2 then agent 1 is more ambiguity averse (loving) than agent 2 (see [

First, we show the behavior of the likelihood ratio L i ( s ) . ^{4}. Here and hereafter, in all the graphs, the horizontal axis denotes the state s ∈ ℝ . Because it is assumed that the value of the state s ∈ ℝ is larger, the terminal wealth W i ( s ) of each agent is larger, the case where the value of the state is large is referred to as a “good state,” and the case where the value of the state is low is referred to as a “bad state.” From

On the other hand, if an agent is ambiguity loving, i.e., τ i = − 0.7 or τ i = − 1.4 , the curve seems to be symmetric w.r.t. the horizontal line such that L i ( s ) = 1 . That is, the value of the capacity against the corresponding reference probability measure is lower in the bad states and it is higher in the good states. The more ambiguity loving the agent is, the larger the difference between the capacity and the reference probability measure.

From these results, we can see that if an agent is more ambiguity averse (loving), she perceives the likelihood of a bad state to be higher (lower) and that of a good state to be lower (higher).

Next, we compare the behavior of the SPD ϕ under the EUUP with that of ϕ 0 under the Bühlmann model. First,

Finally, we investigate insurance demand in the equilibrium.

From

In summary, from the above numerical examples, we can see more ambiguity averse agents overestimate (underestimate) the bad (good) states compared with the reference probability measure and receive (pay) the insurance amount in the good (bad) states.

In this paper, we derive an equilibrium insurance premium and risk exchange in a pure exchange economy with ambiguity, where agents follow EUUP theory of [

Contrary to the economic premium principle under the max-min EU by [

Assuming each agent have an exponential utility function, we also show the optimal insurance or risk exchange decomposes into individual risk, market risk, and difference of attitude toward ambiguity.

Finally, we conduct some comparative statics on insurance premium and risk exchange numerically and examine the influence of attitude for ambiguity aversion on the insurance premium and insurance demand.

First we show the state price density (SPD) or pricing kernel under EUUP is not monotonically deceasing. This is in contrast to the SPD under the Bühlmann model, which monotonically decreases with respect to the state. According to the emprical studeis, the actual SPD takes the bumped shape ( [^{5}. Hence our model might give a one of explanation of this puzzle.

Next, unlike the result of existing research, we show that each agent’s optimal demand for insurance is not always comonotonic even if all the agents in the economy are ambiguity averse or ambiguity loving. That is, even if all the agents in the economy are ambiguity averse or ambiguity loving, insurance trade or risk exchange occurs depending on the difference of attitude fowards ambiguity. For example, [

In the comparative static analysis, we only considered the influence of attitude toward ambiguity, but we also need to consider the influence of the magnitude of ambiguity. This remains for future studies. Furthermore, we need to specify the functional form of the probability outlook function to get more realtic implications. However, we do not know what is a real one to the best of our knowledge. To solve this problem, we must wait for the results of more empirical studies.

This study is supported by a JSPS KAKEN JP20K01761. The author is grateful for valuable comments of anonymous referees.

The author declares no conflicts of interest regarding the publication of this paper.

Iwaki, H. (2021) Risk Exchange under EUUP. Journal of Mathematical Finance, 11, 512-527. https://doi.org/10.4236/jmf.2021.113029