A three-period extension of the CAPM

Abstract

Purpose

Our result of this paper aims to indicate that the beta pricing formula could be applied in a long-term model setting as well.

Design/methodology/approach

In this paper, we show that the capital asset pricing model can be derived from a three-period general equilibrium model.

Findings

We show that our extended model yields a Pareto efficient outcome.

Practical implications

The capital asset pricing model (CAPM) model can be used for pricing long-lived assets.

Social implications

Long-term modelling and sustainability can be modelled in our setting.

Originality/value

Our results were only known for two periods. The extension to 3 periods opens up a large scope of applicational possibilities in asset pricing, behavioural analysis and long-term efficiency.

Citation

Habis, H. (2024), "A three-period extension of the CAPM", Journal of Economic Studies, Vol. 51 No. 9, pp. 200-211. https://doi.org/10.1108/JES-11-2023-0640

1. Introduction

The capital asset pricing model, routinely referred to as CAPM in the literature, accurately estimates the relationship between the risk and the expected return of an asset. Its foundations were established by Sharpe (1964), Lintner (1965) and Mossin (1966). The CAPM model is used for the estimation of expected returns of risky assets in equilibrium.

The CAPM can be derived from a two-period general equilibrium model which provides a sound theoretical basis for one of the essential tools of modern portfolio management: the return-beta relationship.

This paper extends the consumption-based capital asset pricing model to a three-period economy. This extension can potentially have remarkable effects on several other fields of application. For example, a minimum of three periods is both necessary for handling long term financial assets and adding time-inconsistent behaviour into the context of financial-economic modelling. We introduce the three-period intertemporal general equilibrium model with one asset and the consumption-based version of the popular CAPM model, the consumption capital asset pricing model (CCAPM).

In Section 2 we introduce the three-period general equilibrium model and show that the resulting consumption plan is efficient if markets are complete and that the first theorem of welfare economics remains fulfilled in the three-period model as well. Section 3 defines the CCAPM, which is followed by the derivation of the three-period CAPM in Section 4. As a foundation of our model, we use the well-known, two-period pricing equations described in the book by LeRoy and Werner (2001) which we frequently use as building blocks in this study.

2. The three-period finance economy

This section is dedicated to introducing the definitions and notations that are necessary elements for the dynamics of the model. The described structure is based on the one in the article by Habis and Herings (2011) where the competitive equilibrium is compared to cooperative outcomes.

Let t ∈ {0, 1, 2} = T denote the time periods. In periods t = 1, 2 one event out of a finite set occurs. At every state s∈S we denote the date-event at period t by st∈St, where the cardinality of St is S_t and S=⋃tSt for all t ∈ T. For t = 0 we define s₀ = 0, which is the current state with no uncertainty. Let st+ be the set of successors of s_t for all t = 0, 1 and st− the set of predecessors of s_t for all t = 1, 2. Note, that s1− then becomes simply state 0. In each period there is a single, non-durable consumption good.

There are a finite number of agents h ∈ H participating in the economy. Each agent h has initial endowments (esth)st∈{0}∪S1∪S2∈R(S1+S2+1). Agents have preferences over consumption bundles csth∈R(S1+S2+1) where st∈S. Each agent’s preferences are represented by a von Neumann-Morgenstern utility function that is additively separable over time and at period 0 it is defined by

(1)uh(ch)=v0h(c0h)+δ1∑s1∈S1ρs1vs1h(cs1h)+δ1δ2∑s1∈S1ρs1∑s2∈s1+ρs2vs2h(cs2h)

where ρs1 denotes the probability of occurrence of event s₁ and ρs2 denotes the probability of occurrence of event s₂ given event s₁ has occurred, δ_t is a one-period discount factor and vsth is a Bernoulli function.

We apply the following assumptions throughout the paper.

Assumption 2.1.

We assume that ρst>0 for all st∈St and ∑s1∈S1ρs1=1, ∑s2∈S2ρs2=1, δ₁, δ₂ > 0, the probabilities and discount factors are identical across agents, and that the Bernoulli utility function is strictly increasing. Furthermore c^h ∈ X^h where Xh⊂R1+S1+S2 and X^h is the vector of consumption bundles for agent h.

The constraint of ρst>0 means that the agents only take into account the future outcomes for which the objective probability of occurrence is positive, i.e. unlikely events do not affect their utility. A further simplifying assumption is that all agents apply the same discount factors and have no satiation point.

There are Jst short-lived assets at each st∈{0}∪S1. The set of assets at event s_t is Jst. Each asset j pays (random) dividends dst+1,j at date-events st+1∈st+ and then it expires. We denote the vector of dividends by dst=(dst,1,…,dst,Jst−) where st∈S1∪S2, and the pay-off matrices by Ast=(d1,…,dJst)∈R|st+|×Jst where st∈{0}∪S1.

The price of asset j at date-events st∈{0}∪S1 is qst,j∈R. We denote the vector of asset prices by qst=(qst,1,…,qst,Jst), and the collection of prices over date-events by q=(qst)st∈{0}∪S1. We assume that assets are in zero net supply. At date-event st∈{0}∪S1 agent h chooses a portfolio-holding θsth=(θst,1h,θst,2h,…,θst,Jsth)∈RJst.

Definition 2.2.

The finance economy E=((uh,eh)h=1,…,H;(Ast)st∈{0}∪S1) is defined by the agents’ utility functions and endowments, and the pay-off matrices.

Definition 2.3.

A competitive equilibrium for an economy E is a collection of portfolio-holdings θ*=(θ1*,θ2*,…,θH*)∈RH×J×(S1+1), consumption c*=(c1*,c2*,…,cH*)∈RH×(S1+S2+1) and asset prices q*∈RJ×(S1+1) that satisfy the following conditions:

• (a) For h = 1, …, H,

(2) (ch*,θh*)∈argmaxch∈Xh,θh∈RJ×(S1+1)uh(ch)

s. t. c0h+q0θ0h=e0h,

cs1h+qs1θs1h=es1h+ds1θ0h,  for s1∈S1,

cs2h=es2h+ds2θs2−h,  for s2∈S2,

• (b)

(3) ∑h=1Hθh*=0,

• (c)

(4) ∑h=1Hch*=∑h=1Heh.

Note that the market clearing equations (b) for the financial contracts imply that the consumption c* is feasible; i.e. condition (c) is always satisfied when conditions (a) and (b) are. The feasibility of c* means that in a finance economy (a single commodity model) when the financial markets clear, the demand for the commodity induced by the portfolio-holdings equals its supply in each state; thus the commodity market clears in each state (Magill and Quinzii, 1996).

If Assumption 2.1 is met (i.e. agents have strictly increasing utility functions) equilibrium prices exclude arbitrage opportunities in the following sense.

Definition 2.4.

Asset prices q of short-lived assets are arbitrage-free if there is no θh=(θsth)st∈{0}∪S1 such that

(5) q0θ0h≤0,

(6) ∀st∈S1∪S2:qstθsth≤Ast−θst−h,

with at least one strict inequality.

Definition 2.5.

Markets are complete if for every income stream y∈RS1+S2 there exists a portfolio plan (θsth)st∈{0}∪S1 such that

∀s1∈S1:ds1θ0h−qs1θs1h=ys1;∀s2∈S2:ds2θs2−h=ys2.

That is, for each date-event st∈{0}∪S1 and arbitrary payoffs in immediate successors of s_t, there exists a portfolio that generates those payoffs. Such a portfolio exists if and only if Ast has rank |st+|, which is stated in the following proposition:

Proposition 2.6.

Markets are complete if and only if for every st∈{0}∪S1 the following condition is met

(7) rank(Ast)=|st+|.

Proof. The proof is given in (Habis and Herings, 2011). □

Proposition 2.7.

If there are no arbitrage opportunities in the financial markets and the markets are complete, then there exists a unique, strictly positive state price vector (πst)st∈{0}∪S1∈RS1+1 such that

(8) qst=πst⊤⋅Ast.

Proof. The proof is given in (Magill and Quinzii, 1996).

The following additional assumptions will be made throughout this section:

Assumption 2.8.

We assume that

1. asset 1 is risk free, so dst,1=1∀st∈S1∪S2, and its return is Rf=1/qst,1,

2. and {c^h ∈ X^h|u^h(c^h) ≥ u^h(e^h)} ⊂ int(X^h), which prevents the solution of the agent’s maximization problem from occurring at the boundary of the consumption set.

We use Est(cst+) to denote the expectation of cst+ conditional on date-event s_t, so Est(cst+)=∑st+1∈st+ρstcst.

2.1 Efficiency

According to the First Welfare Theorem, the complete-markets equilibria provide Pareto-efficient consumption allocations. An allocation is Pareto-optimal if it is impossible to reallocate the total endowment so as to make some agents better off without making any agent worse off. Specifically, an allocation c^h is Pareto-optimal if there does not exist an alternative allocation c¯h which is feasible,

(9)∑h=1Hc¯h=∑h=1Heh,

weakly preferred by every agent,

(10)uh(c¯h)≥uh(ch),

and strictly preferred by at least one agent, so that (10) holds with strict inequality for at least one agent.

Proposition 2.9.

(First Welfare Theorem) Let (θ*, c*, q*) be a competitive equilibrium for E. If asset markets are complete, then c* is Pareto-optimal.

Proof. The proof can be obtained by contradiction. Suppose that c*^h is the complete-market equilibrium consumption allocation, and that there is a feasible allocation c̃h such that uh(c̃h)≥uh(c*h) for every h, with strict inequality for some h.

Using the framework of Definition 2.3, the consumption plan c*^h maximizes utility u^h(c^h) subject to the budget constraints

(11)c0*h=e0h−π0ds1θ0h

(12)cs1*h=es1h+ds1θ0h−πs1ds2θs1h

(13)cs2*h=es2h+ds2θs1h,

where πst is the unique state price vector associated with qst*. Note that πst is strictly positive.

Multiplying equation (13) by πs1 and adding the result to equation (12), we obtain

(14)cs1*h+πs1cs2*h=es1h+πs1es2h+ds1θ0h.

Multiplying equation (14) by π₀ and adding the result to equation ((11)), we obtain

(15)c0*h+π0cs1*h+π0πs1cs2*h=e0h+π0es1h+π0πs1es2h,

thus the budget constraints of the original utility-maximization problem in (2) are equivalent to equation (15). Consequently, the optimal consumption plan c*^h maximizes u^h(c^h) subject to equation (15).

Since u^h(c^h) is strictly increasing, we have

(16)c̃0h+π0c̃s1h+π0πs1c̃s2h≥c0*h+π0cs1*h+π0πs1cs2*h

for every h, with strict inequality for some h, who are strictly better off with c̃h than with c*^h. Summing over all agents and applying equation (15), we obtain

(17)∑h=1Hc̃0h+∑h=1h=Hπ0c̃s1h+∑h=1Hπ0πs1c̃s2h>e0+π0es1+π0πs1es2,

which contradicts the assumption that consumption allocation c̃h is feasible. □

Proving this proposition is a new development, and it is a crucial requirement for deriving the three-period model and finding a Pareto-efficient result at the same time.

When markets are incomplete, equilibrium consumption allocations are in general not Pareto-optimal and the first welfare theorem typically fails, since agents may not be able to implement the trades required to attain the optimal allocation. Equilibrium consumption allocations, however, can be optimal in a restricted sense. We turn now to a less ambitious notion of efficiency: are markets performing well in the sense that it is impossible to improve social welfare by using the asset market?

If we consider efficiency as a program carried out by a social planner with certain objectives we can distinguish myopic and forward-looking planners. Based on the results above, we can assume that the mentioned theorems can be proved in such constrained cases as well but that is the subject of future research.

In this section, we got familiarized with the model’s formalized general equilibrium framework. Now we move on to the asset pricing model.

3. The consumption capital asset pricing model

First, we shortly run through the most relevant aspects of the CAPM based on the relevant section of Bodie et al. (2011). Next, we move on to introduce the CCAPM.

The CAPM estimates the relationship between the risk and the expected return of an asset. The model assumes that the utility of an asset is dependent exclusively on the expected return, and the covariance of returns of the asset. The risk premium on the market portfolio can be given as a function of its risk and the risk aversion of the representative investor:

(18)E(rM)−Rf=A¯σM2

where σM2 is the variance of the market portfolio, A¯ is the coefficient of the average risk-aversion, and R_f is the risk-free rate.

The risk premium of the individual assets is proportional to the risk premium of the market portfolio and its beta coefficient. The beta describes the relationship between the individual asset’s return and the market portfolio’s return:

(19)βj=Cov(rj,rM)σM2,

Thus the risk premium in the case of individual assets is:

(20)E(rj)−Rf=Cov(rj,rM)σM2[E(rM−Rf]=βj[E(rM)−Rf].

which is the most popular expression of the CAPM: the expected return beta relationship.

As it holds true for individual assets, the equation holds for any linear combinations of these assets. This relationship can be understood as a risk-reward equation. The beta of the asset accurately describes the risk because it is proportional to the risk the asset contributes to the risk of the optimal portfolio. The graphical representation of this expected return beta relationship is the security-market line (SML).

Let us now move on to the CCAPM, where the CAPM is centred around consumption. The CCAPM was first introduced by Rubinstein (1976), Lucas (1978), and Breeden (1979).

We examine a life-long consumption plan, where the agents, in each period, need to decide about the division of their wealth between today’s consumption and the investments and savings that ensure the consumption of future periods. They reach the optimum if the marginal utility coming from spending an additional unit of wealth today equals the marginal utility coming from the expected future consumption that is financed using this same unit of wealth.

The future wealth can increase as a result of wage income and the return of the units of wealth invested in the optimal complete portfolio.

A financial asset is more risky in terms of consumption if it has a positive covariance with the increase in consumption. In other words, its payoff is higher when the consumption is already high, and lower when the consumption is relatively constrained [1]. As a result, the optimal risk premium is higher for those assets that show higher positive covariance with the increase in consumption. Based on this observation, we can describe the risk premium of an asset as a function of the risk of consumption:

(21)E(Rj)=βjC(E(rc)−Rf),

where the portfolio C can be translated as a consumption-tracking portfolio, which is the portfolio which correlates positively to the greatest extent with the increase in consumption.

The β_jC can be interpreted as the coefficient of the regression line where we explain R_j return premium of asset j using the return premium of the consumption-tracking portfolio as the explanatory variable.

With the previously defined risk-free rate R_f, we define the risk premium that is independent of the uncertainty of consumption as (E(rc) − R_f) which is also determined using the return premium of the consumption-tracking portfolio.

This is very similar to the traditional CAPM: the consumption-tracking portfolio plays the role of the market portfolio in the CAPM. However, opposing the original CAPM theory, the beta of the consumption capital asset pricing model is not necessarily 1, in fact, it is entirely realistic and empirically observed that this beta can be greater than 1. This means that the linear relationship between the market risk premium and the consumption portfolio can be written as

(22)E(RM)=αM+βMCE(RC)+ϵM

where α_M and ϵ_M ensure the possibility of empirical deviations from the exact model defined by equation (21), and that β_MC is not necessarily 1.

The CCAPM is attractive, as it compactly expresses the idea of consumption hedging and the potential changes in the investment opportunities. Furthermore, it integrates this in the parameter of the distribution of returns in a one-factor model setup.

As a summary, we define the CCAPM below in a format that fits the purposes of this study.

Definition 3.1.

The CCAPM is a version of the CAPM where the expected return premium of the market portfolio is replaced by the return premium of the consumption-tracking portfolio. This model establishes a relationship between the investors’ sensitivity to the changes in consumption and the risk of the assets.

4. The three-period CAPM

In this section, we prove that the β pricing formula, that relates the return of a risky asset to the return of the market portfolio can also be derived in the introduced three-period finance economy general equilibrium model.

Though many publications have tackled the possibility of deriving the CAPM in different environments (such as missing conditions or differing model environments) this perspective is a unique one as the capital asset pricing equation has not been derived in a three-period model previously. Though it is a topic of future research this result also means that the CAPM could be used for asset pricing in long term models with long-lived assets as well.

First, we define the utility function of the rational agents (h) as follows:

(23)uh(ch)=v0h(c0h)+δ1∑s1∈S1ρs1vs1h(cs1h)+δ1δ2∑s1∈S1ρs1∑s2∈S1+ρs2vs2h(cs2h).

Agent h maximizes this utility subject to her constraints on endowments, income and even costs which were formalized in Definition 2.3. Since markets are complete, it follows from Proposition 2.7, that there exists a unique and strictly positive state price vector πst. The asset price vector qst=πstT⋅Ast then follows from the agents’ optimization problem:

(24)Lh=uh(ch)−λ0h(c0h−e0h+q0θ0h)−λs1h(cs1h+qs1θs1h−es1h−ds1θ0h)−λs2h(cs2h−es2h−ds2θs2−h),

where λsth denote the Lagrange-multipliers. The first-order conditions, which are necessary and sufficient for (c^h*, θ^h*) to be a solution, are that there exist λh*∈R++1+S1+S2 such that

(25)∇Lh(ch*,θh*,λh*)=0,

which is equivalent to

(26)∇uh(ch*)=λh*,  and

(27)−qstλst+dst+λst+=0,∀st∈{0}∪S1.

Solving this system of equations for qst:

(28)qst=Astλst+hλsth, s.t. λsth≠0

Then we substitute with the respective values of the λ^h multipliers and get

(29)qst=Astδt+1∑st+∈St+ρst+∂vst+h(cst+h)/∂cst+h∂vsth(csth)/∂csth.

It becomes apparent that what we get is the marginal rate of substitution (MRS) between the consumption levels of the different periods. Equation (29) means that for each st∈{0}∪S1 date-event, an agent h invests in j assets, such that the marginal cost of each additional qst,j unit equals its marginal utility, which is the present value of the future dividends of agent h [2].

By the definition of the expected value described in Section 2, we substitute the respective part of equation (29) and we get [3]

(30)qst=δt+1Est[∂cst+vst+h(ch*)Ast]∂cstvsth(ch*)=E(MRSsthAst),   for all st∈{0}∪S1,

Where vst+=(vst+1)st+1∈st+ and we can see the MRS between the consumption levels of period t and of all states belonging to the period t⁺.

Equation (30) asserts that each agent h invests in each asset j at each date-event st∈{0}∪S1 in such a way that the marginal cost of an additional unit of the security qst,j is equal to its marginal benefit, the present value for agent h of its future stream of dividends. Although the MRSsth of each agent can be different as a result of the shape of the utility function (e.g. based on their attitude towards risk), they cannot disagree on asset prices in equilibrium. If one projects the individual MRSsths onto the marketed subspace 〈Ast〉 one obtains a unique pricing vector, given that qst=πst⊤⋅Ast which is the one defined in (30). For asset prices qst we define the one-period return rst+,θst for a portfolio θst, with qstθst≠0, by

(31)rst+,θst=Astθsthqstθsth.

This reflects the general definition of returns: we divide the pay-offs of the securities in the portfolio by their price. Using this formula for return, we can rewrite equation (30) in the following manner:

(32)1=δt+1Est[∂cst+vst+h(ch*)rst+,θst]∂cstvsth(ch*).

The expectation of the product of any two random variables can be written as their covariance plus the product of their expectations:

(33)E(yz)=cov(y,z)+E(y)E(z).

Now, using covst(xst+,yst+) to denote the conditional covariance between two variables we get

(34)1=δt+1Est[rst+,θst]Est[∂cst+vst+h(ch*)]∂cstvsth(ch*)+δt+1covst(∂cst+vst+h(ch*),rst+,θst)∂cstvsth(ch*).

Rearranging this yields the equation of the one-period expected return

(35)Est[rst+,θst]=∂cstvsth(ch*)δt+1Est[∂cst+vst+h(ch*)]−covst(∂cst+vst+h(ch*),rst+,θst)Est[∂cst+vst+h(ch*)]

where the expression

(36)Rstf=∂cstvsth(ch*)δt+1Est[∂cst+vst+h(ch*)]

is the return of the one-period risk-free asset [4]. Plugging this into equation (35) we retrieve the consumption-based capital asset pricing formula

(37)Est[rst+,θst]=Rstf−δt+1Rstfcovst(∂cst+vst+h(ch*),rst+,θst)∂cstvsth(ch*).

This equation shows that for each asset the risk premium (which is the difference between the expected return of the risky assets and the risk-free rate) is proportional to the covariance between its return rate and the MRS between the date-events of s_t and st+ (with a negative proportionality constant).

To be precise, ∂cst+vst+h(ch*)/∂cstvsth(ch*) in equation (37), is not the MRS between the state-dependent consumptions of date-events st+ and s_t, as the probabilities are missing. Similarly, we will refer to the marginal utility of consumption by the notion ∂cst+vst+h(ch*), although the probabilities are missing here as well. There is no reason to be held up by this terminological imprecision, as we are not diverting from the conventional methodology of the literature, see LeRoy and Werner (2001). For a strictly risk-averting decision maker, ∂cst+vst+h(ch*) is a negative function of the consumption in st+. Thus, the security, that has a high pay-off when the consumption is high, and has a low pay-off when the consumption is low as well, has a greater expected return than the risk-free security. Let us now, in contrast, consider a security, that has a high pay-off when the consumption is low, and has a low pay-off when the consumption is high. Following the above concept, such a security would have an expected return which is less than that of the risk-free asset. Such securities can then be used to decrease the risk of consumption for the decision makers. If the covariance of an asset’s return and the MRS is zero, the asset has the same expected return as the risk-free asset.

Based on equation (37) the risk premium of a security is solely dependent on the covariance between its return and the MRS between the date-events s_t and st+. This covariance can be understood as the degree of risk of the security, which has two significant features. Firstly, it can only be used if the economy is in the state of equilibrium. Secondly, this covariance-measure provides not just a partial but a complete ordering of the risk of returns.

If the MRS is constant, the consumption-based asset pricing equation defined in equation (37) gives a fair price. The MRS can be deterministic in two cases: if the consumption of the agent is deterministic as well, and if the agent is risk-indifferent.

In order to illustrate further details of the optimization process of the agents, in the next assumption, we will show the vsth utility function which is quadratic with respect to the t + 1 period consumption.

Assumption 4.1.

Let Xh=R1+S1+S2 and vsth(csth)=ξtcsth−12αt(csth)2 be a quadratic utility-function.

Substituting this into Equation (37) we get

(38)Est[rst+,θst]=Rstf−δt+1Rstfcovst(ξt+1−αt+1cst+h,rst+,θst)ξt−αtcsth,

then it follows that the expected return of an arbitrary asset j is

(39)Est[rst+,j]=Rstf+δt+1αt+1Rstfξt−αtcsthcovst(cst+h,rst+,j).

In a securities market economy the aggregated endowment is in the asset span which means it can be attained from the pay-offs of the portfolio of some securities. This portfolio is the market portfolio with its return denoted by rst+M. Equation (39) holds for returns of portfolios as well. In particular it holds for the market return rst+M so that

(40)Est[rst+M]=Rstf+δt+1αt+1Rstfξt−αtcsthcovst(cst+h,rst+M).

Dividing Equation (39) by (40) after subtracting Rstf from both and thus eliminating the term δt+1αt+1Rstfξt−αtcsth one obtains

(41)Est[rst+,j]−RstfEst[rst+M]−Rstf=covst(cst+,rst+,j)covst(cst+,rst+M)

where, as we assume, the market risk premium is nonzero.

If equilibrium consumptions lie in the span of the market return and the risk-free return, then cst+h and rst+M are perfectly correlated. Accordingly cst+h can be replaced by φrst+M. Finally, for a portfolio θsth∈RJst we define βθst-t

(42)βθst=covst(rst+M,rst+,θ)var(rst+M).

This βθst will be the consumption beta of the CCAPM, mentioned in Section 3, which reflects how the risk of a security is related to the risk of the market portfolio.

Then the following CAPM-pricing formula holds for each θsth∈RJst thus

(43)Est[rst+,θ]−Rstf=βθst(Est[rst+M]−Rstf);

which is, in fact, the formula of the security market line:

(44)Est[rst+,θ]=Rstf+βθst(Est[rst+M]−Rstf).

As it is also stated in LeRoy and Werner (2001), the assumption, that the equilibrium consumption choice is in the span of the market return and risk-free return is trivial in a representative-agent economy. This is because the optimal consumption of each agent in the economy is equal to the per capita pay-off of the market portfolio. Since we assumed that all agents have the same quadratic utility function this holds true for the economy defined in this paper.

Hence, we have proven that the CCAPM formula can be derived from a three-period utility maximization model; in other words, we extended the results of the widely known two-period model to three periods. This is significant as a stand-alone result but it can also provide a basis for numerous future research topics which require a multi-period model. One such case is the analysis of long term securities or the long term efficiency of incomplete markets.

Notes

1. We also note this when we discuss the three-period model later.

2. Note, that when markets are incomplete, initial endowments and consumption plans need not lie in the asset span. Then agent h’s endowment can be decomposed into a marketed and a non-marketed part. There may no longer exist a unique state price vector since the MRS need not be the same among agents due to the non-traded part of the endowments.

3. For the sake of clearer notation, we will substitute the traditional notation (∂f(x)∂x) of the partial derivative of any function f(x) with respect to x variable by simply writing ∂_xf(x).

4. The definition of the risk-free asset is the one described in LeRoy and Werner (2001) as Rstf=1∑st∈{0}∪S1∪S2qst which, in equilibrium, is equivalent with Rstf in our equations.