Two-fund separation in dynamic general equilibrium

Schmedders, Karl Article Two-fund separation in dynamic general equilibrium Theoretical Economics Provided in Cooperation with: The Econometric Society Suggested Citation: Schmedders, Karl (2007) : Two-fund separation in dynamic general equilibrium, Theoretical Economics, ISSN 1555-7561, The Econometric Society, New York, NY, Vol. 2, Iss. 2, pp. 135-161 This Version is available at: https://hdl.handle.net/10419/150094 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/2.5 Theoretical Economics 2 (2007), 135–161 1555-7561/20070135 Two-fund separation in dynamic general equilibrium KARL SCHMEDDERS Kellogg School of Management, Northwestern University This paper examines the two-fund separation paradigm in the context of an infinite-horizon general equilibrium model with dynamically complete markets and heterogeneous consumers with timeand state-separable utility functions. With the exception of the dynamic structure, we maintain the assumptions of the classical static models that exhibit two-fund separation with a riskless security. Agents have equi-cautious HARA utility functions. In addition to a security with state-independent payoffs, agents can trade a collection of assets with dividends following a time-homogeneous Markov process. We make no further assumptions about the distribution of asset dividends, returns, or prices. If the riskless security in the economy is a consol then agents’ portfolios exhibit two-fund separation. However, if agents can trade only a one-period bond, this result no longer holds. The underlying intuition is that general equilibrium restrictions lead to interest rate fluctuations that destroy the optimality of two-fund separation in economies with a one-period bond and result in different equilibrium portfolios. KEYWORDS. Portfolio separation, dynamically complete markets, consol, oneperiod bond, interest rate fluctuation, reinvestment risk. JEL CLASSIFICATION. D53, G11, G12. 1. INTRODUCTION The two-fund separation theorem, which is among the most remarkable results of classical finance theory, states that investors who must allocate their wealth between a number of risky assets and a riskless security should all hold the same mutual fund of risky assets. An investor’s risk aversion affects only the proportions of wealth that he invests in the risky mutual fund and the riskless security. The allocation of wealth across the different risky assets does not depend on the investor’s preferences. Cass and Stiglitz (1970) and Merton (1973) are perhaps the most prominent works on this fundamental result. Cass and Stiglitz (1970) derive necessary and sufficient conditions on investors’ utility functions for the optimal portfolio in investors’ static asset Karl Schmedders: [email protected] I thank Ken Judd, Felix Kubler, Christoph Kuzmics, and Costis Skiadas for helpful discussions and audiences at SUNY at Stony Brook, the 2004 NBER-NSF general equilibrium conference at UC Davis, the University of Frankfurt, and the ‘Weihnachtstreffen’ in Bielefeld for comments. I am indebted to Michael Magill and Martine Quinzii for making their unpublished book chapters on ‘Elements of Differential Topology’ available to me. I am grateful to the Co-Editor Ed Green and two anonymous referees for very useful suggestions on an earlier draft. Copyright c2007 Karl Schmedders. Licensed under the Creative Commons Attribution-NonCommercial License 2.5. Available at http://econtheory.org. 136 Karl Schmedders Theoretical Economics 2 (2007) demand problems to satisfy the separation property. They use the phrase “monetary separation” for the notion of portfolio separation that most people now have in mind when they talk about two-fund separation, specifically for the separation of investors’ portfolios into the riskless asset and a common mutual fund of risky assets. For examples, see Canner et al. (1997) and Elton and Gruber (2000). Merton (1973) establishes two-fund monetary separation for an intertemporal capital asset pricing model in a continuous-time setting without assumptions on utility functions but under the conditions that the returns of the risky assets are log-normally distributed and that the interest rate is constant through time. Although these two seminal works along with many other papers on the subject have established the two-fund separation paradigm at the heart of the foundations of modern finance theory, the paradigm has not been examined in the context of the celebrated Lucas asset pricing model (Lucas 1978). This paper aims to fill this gap in the literature. We prove that in a Lucas-style discrete-time infinite-horizon general equilibrium model with heterogeneous agents, two-fund monetary separation holds only if a consol, a perpetual bond with safe coupon payments and no maturity date, is available for trade on financial markets. Two-fund separation typically fails if only risky assets and a one-period bond (cash) can be traded on financial markets. In the model, markets are dynamically complete and utility functions are timeand state-separable. All asset dividends or payoffs follow a time-homogeneous Markov process. Efficient equilibria in this model have time-homogeneous consumption and asset price processes. After one round of initial trading, portfolios are constant over time. We emphasize that we do not impose this buy-and-hold nature of agents’ portfolios ex ante, but that this feature is a result of equilibrium efficiency in the general equilibrium model (Judd et al. 2003). We maintain the classical assumptions on utility functions (Cass and Stiglitz 1970). All agents have HARA utilities with linear absolute risk tolerances having identical slopes. We assume that in addition to stocks with Markovian dividend processes there is also a security with state-independent payoffs (consol or cash). The general equilibrium nature of our model prohibits us from making any further assumptions about the distribution of asset returns or prices. These quantities, along with interest rates, are determined in equilibrium. The underlying intuition for our results is that general equilibrium restrictions create interest rate fluctuation. This fluctuation affects agents holding a consol differently than agents holding a one-period bond. Portfolios are constant over time in the dynamic model equilibrium. When a consol is present, an agent establishes a desired position at time zero and then keeps this consol holding forever. Changes in the price (interest rate) of the consol do not affect the agent since she does not trade the consol. In an economy without a consol but with a one-period bond, equilibrium portfolios are also constant over time. Now, however, an agent must reestablish the constant bond position in every period. In such an environment the agent faces reinvestment risk due to bond price (interest rate) fluctuation in equilibrium. This fluctuation destroys the optimality of two-fund separation and leads to a different equilibrium portfolio. The first two-fund separation results were obtained by Tobin (1958) and Markowitz (1959), who analyze portfolio demand in the mean-variance framework. Subsequent Theoretical Economics 2 (2007) Two-fund separation 137 work on two-fund separation revealed that either assumptions on utility functions or on asset return distributions (or both) are necessary to establish further results. Cass and Stiglitz (1970) derive necessary and sufficient conditions on investors’ utility functions that ensure two-fund separation in investors’ static asset demand problems. Ross (1978) presents conditions on asset return distributions under which two-fund separation holds for static demand problems. Russell (1980) presents a unified approach of Cass and Stiglitz and Ross. Ingersoll (1987) provides a detailed overview of various separation results, and highlights the distinction between restrictions on utility functions and restrictions on asset return distributions. Rubinstein (1974) shows that if all investors have equi-cautious HARA utility, then the two-fund monetary separation theorem holds in an equilibrium model. Essentially he extends the Cass-Stiglitz result to equilibrium analysis. (We use the same assumption of equi-cautious HARA utilities in a dynamic general equilibrium model.) Gollier (2001) also states the separation result of Cass and Stiglitz in the context of a static equilibrium model. We cannot possibly do justice to the huge literature on portfolio separation and mutual fund theorems in static models and, therefore, just refer to textbook overviews such as Ingersoll (1987) or Huang and Litzenberger (1988), and to the summary in Ross (1978). The standard reference for two-fund separation in dynamic economies is Merton (1973), even though some of the results are already present in Merton (1971). Merton (1973) shows for his continuous-time capital asset pricing model that two-fund monetary separation holds independently of preferences, wealth distribution, and time horizon, if returns of risky assets are log-normally distributed and the investment opportunity set is constant. The latter assumption requires, among other conditions, the interest rate of the riskless asset to be constant over time. Merton also shows that two-fund separation no longer holds as soon as the (instantaneously riskless) interest rate changes stochastically over time. He establishes a “three fund” theorem, but both the composition of the fund of risky assets and agents’ holdings of the three funds change continuously over time. For a further generalization of the three-fund theorem to an m-fund theorem, see Merton (1990). We relate our results in this paper to Merton’s theorems in our discussion of a detailed example in Section 3. The paper is organized as follows. In Section 2 we describe the general equilibrium model and characterize efficient equilibria. Section 3 presents an illustrative example exposing the basic intuition underlying our results. In Section 4 we state some helpful equilibrium properties. Section 5 develops the two-fund separation theory for our dynamic model, proving the generalization of the classical static result when a consol is available for trade and showing that two-fund separation fails generically when there is only a one-period bond. Section 6 points to the relevance of our results for the asset allocation puzzle. The Appendix contains all technical proofs. 2. THE ASSET MARKET ECONOMY The purpose of this section is to introduce the economic framework for all the analysis in this paper. We first describe the general equilibrium model with dynamically complete markets. Next we explain how we can easily characterize efficient equilibria in 138 Karl Schmedders Theoretical Economics 2 (2007) the model. And finally we review the notion of two-fund monetary separation for the general equilibrium model. 2.1 General equilibrium model We examine a standard Lucas asset pricing model (Lucas 1978) with heterogeneous agents and dynamically complete asset markets. Time is indexed by t∈N0≡ {0,1,2,...}. A time-homogeneous recurrent Markov process of exogenous states (yt)t∈N0takes values in a discrete set Y={1,2,...,S},S≥2. The Markov transition matrix is denoted by Π. A date-event σtis the history of shocks up to time t, i.e. σt= (y0y1...yt). Let Σtdenote the possible histories σtup to time tand let Σ = ∪tΣtdenote all possible histories of exogenous states. We denote the predecessor of a date-event σ∈Σby σ∗. The starting node σ0=y0has a predecessor σ∗ 0=σ−1. There is a finite number of types H={1,2,...,H}of infinitely-lived agents. There is a single perishable consumption good, which is produced by firms. The agents have no individual endowment of the consumption good. The firms distribute their output each period to its owners through dividends. Investors trade shares of the firms and other securities in order to transfer wealth across time and states. There are J=Sassets traded on financial markets. An asset is characterized by its state-dependent dividends or payoffs. We denote asset j’s dividend or payoff by dj:Y → R+,j=1,...,S, which solely depends on the current state y∈ Y . Each security is either an infinitely-lived (longlived) asset or a single-period asset. There are Jl≥1 long-lived assets in the economy. The remaining S−Jlsecurities are short-lived assets that are issued in each period. A short-lived asset jissued in period tpays dj(y)in period t+1 if state yoccurs and then expires. For ease of exposition we collect the infinitely lived assets in a set L≡ {1,...,Jl} and the one-period assets in a set O≡ {Jl+1,...,S}. We denote the portfolio of agent hat date-event σ∈Σby θh(σ)≡(θhL(σ),θhO(σ)) = (θh1(σ),...,θhS(σ)) ∈RS. His initial endowment of asset jprior to time 0 is denoted by θhj −1,j∈L. Each agent has zero initial endowment of the short-lived assets and so these assets are in zero net supply. The infinitely-lived assets that represent firm dividends are in unit net supply. Other financial assets, such as a consol, are in zero net supply. We write θL −1≡(θhL −1)h∈H . The aggregate endowment of the economy in state yis e(y) = Pj∈LPh∈H θhj −1dj(y). Agent h’s initial endowment of dividends before time 0 is given by ωh(y) = Pj∈Lθhj −1dj(y)>0. To avoid unnecessary complications we assume that all agents have nonnegative initial holdings of each asset and a positive initial holding of at least one asset. Let q(σ)≡(q1(σ),...,qS(σ)) be the prices of all assets at date-event σafter dividends or coupon payoffs have been paid. At each date-event σ= (σ∗y)agent hfaces the budget constraint ch(σ) = X j∈L θhj (σ∗)(qj(σ) + dj(y)) + X j∈O θhj (σ∗)dj(y)− S X j=1 θhj (σ)qj(σ). Theoretical Economics 2 (2007) Two-fund separation 139 Each agent hhas a time-separable utility function Uh(c) = E¨∞ X t=0 βtuh(ct)«, where c= (c0,c1,c2,...)is a consumption process. All agents have the same discount factor β∈(0,1). We assume that the Bernoulli functions uh:X→Rare strictly monotone, twice differentiable, and strictly concave on some interval X⊂R. Below we discuss conditions that ensure equilibrium consumption at every date-event to always lie in the interior of an appropriately chosen consumption set X. Let the matrix d= (d1,...,dS) =     d1(1)··· dS(1) . . ..... . . d1(S)··· dS(S)     represent security dividends or payoffs.The vector of utility functions isU= (U1,...,UH). We denote the primitives of the economy by the expression E= (d,X,β,U;θL −1,Π). We define a standard notion of financial market equilibrium. DEFINITION 1. A financial market equilibrium for an economy Eis a process of portfolio holdings {(¯ θ1(σ),..., ¯ θH(σ))}and asset prices {(¯ q1(σ),..., ¯ qS(σ))}for all σ∈Σsatisfying the following conditions. 1. H X h=1 ¯ θh(σ) = H X h=1 θh −1for all σ∈Σ. 2. For each agent h∈ H , (¯ θh(σ))σ∈Σ∈argmax θ Uh(c)s.t. ch(σ) = X j∈L ¯ θhj (σ∗)( ¯ qj(σ) + dj(y)) + X j∈O ¯ θhj (σ∗)dj(y)− S X j=1 ¯ θhj (σ)¯ qj(σ) and sup σ∈Σ S X j=1 ¯ θhj (σ)¯ qj(σ)<∞. 2.2 Equilibrium in dynamically complete markets Judd et al. (2003) characterize efficient financial market equilibria in our model through a simple system of equations. (See also the comments by Bossaerts and Zame 2006 and the reply by Judd et al. 2006b.) Here we summarize their results and defer a more technical discussion of the underlying assumptions until Section 4. Two results of Judd et al. (2003) greatly simplify the equilibrium analysis. First, efficient equilibria exhibit time-homogeneous consumption processes and asset prices. That is, consumption allocations and asset prices in date-event σ= (σ∗y)depend only 140 Karl Schmedders Theoretical Economics 2 (2007) on the last shock y. Second, after one round of initial trading in period 0, each agent’s portfolio is constant across states and time. So equilibrium portfolios do not even depend on the last shock y. These results imply that we do not need to express equilibrium values as functions of the date-event σor through policy or value functions on some large state space. Instead we can index consumption and asset prices with the current exogenous shock ythrough a subscript. For example, ch ydenotes the consumption of agent hin state y. The simple structure of efficient equilibria means that computing an equilibrium reduces to finding finitely many numbers. We first can compute equilibrium consumption allocations using the Negishi approach (Negishi 1960) of Judd et al. (2003). For this purpose we define py=u0 1(c1 y)to be the price of consumption in state yand p= (py)y∈Y ∈ RS ++ to be the (column) vector of prices. We denote theS×Sidentity matrix by IS, Negishi weights by λh,h=2,...,H, and use ⊗to denote element-wise multiplication of vectors. If the economy starts in the state y0∈ Y at time t=0, then the Negishi weights and consumption vectors must satisfy the following equations. u0 1(c1 y)−λhu0 h(ch y) = 0, h=2,...,H,y∈ Y , (1) ([IS−βΠ]−1(p⊗(ch−ωh)))y0=0, h=2,...,H, (2) H X h=1 ch y− H X h=1 ωh y=0, y∈ Y . (3) Equations (1) require that marginal utility vectors are collinear. Equations (2) are the (infinite-horizon) budget equations for agents h=2,3,...,H, given that the economy starts in state y0. Walras’ Law allows us to omit the budget equation for the first agent in the presence of the market-clearing conditions (3). The equations (1)–(3) constitute a nonlinear system with HS + (H−1)unknowns and equations. There are HS unknown state-contingent consumption levels ch y,h∈ H ,y∈ Y , and H−1 Negishi weights λh, h=2,3,...,H. We can easily solve such a system of equations on a personal computer using Newton’s method. Once we know the consumption levels and thus the state price vector pwe determine asset prices from the Euler equations. The price vector qj= (qj y)y∈Y of a long-lived asset jis given by the linear expressions qj⊗p= [IS−βΠ]−1βΠ(p⊗dj). (4) The price of a short-lived asset jin state yis qj y=βΠy·(p⊗dj) py , (5) where Πy·denotes row yof the transition matrix Π. In the last step we can compute agents’ portfolios from their budget equations. After one initial round of trading at time 0, all agents hold a state-independent portfolio vector Θh≡θh yfor all y∈ Y . Define the matrix D= (d1,...,dJl,dJl+1−qJl+1,...,dS−qS). The Theoretical Economics 2 (2007) Two-fund separation 141 portfolio of agent h,h∈ H , is now the solution to the linear system of budget equations across all states y∈ Y , ch=DΘh. (6) In summary, the consumption allocations, asset prices, and portfolio holdings in every efficient financial market equilibrium solve the system of equations (1)–(6) (and vice versa). All our analysis in this paper is based on this characterization of equilibria. In Section 3 we solve these equations numerically to compute an equilibrium for an illustrative example. In Sections 4and 5we use the equations to prove general results for our model. 2.3 Two-fund separation Classical two-fund monetary separation (see, for example, Cass and Stiglitz 1970,Merton 1973,Ingersoll 1987,Huang and Litzenberger 1988) states that investors who must allocate their wealth between a number of risky assets and a riskless security should all hold the same mutual fund of risky assets. An investor’s risk aversion affects only the proportions of wealth that (s)he invests in the risky mutual fund and the riskless security. The allocation of wealth across the different risky assets does not depend on the investor’s preferences. In the context of our general equilibrium model with several heterogeneous agents this property states that the proportions of wealth invested in any two risky assets are the same for all agents in the economy. DEFINITION 2. Consider an economy Ewith an asset that has a riskless payoff vector, dS y=1 for all y∈ Y . The remaining S−1 assets are risky and in unit net supply. We say that portfolios exhibit two-fund monetary separation if qj yΘhj qk yΘhk =qj yΘh0j qk yΘh0k for all assets j,k6=Sand all agents h,h0∈ H in all states y∈ Y . All risky assets are in unit net supply and so market clearing and the requirement from the definition immediately imply that all agents’ portfolios exhibit two-fund separation if and only if Θhj = Θhk for all assets j,k6=Sand all agents h∈ H . That is, in equilibrium each agent must have a constant share of every risky asset in the economy. The ratio of wealth invested in any two risky assets j,k6=Sequals the ratio qj y/qk yof their prices and thus depends on the state y∈ Y . 3. ILLUSTRATIVE EXAMPLE The purpose of this section is to illustrate the basic theme of the paper in the context of an example. Two-fund separation holds in an economy with a consol but it typically fails in an economy with a one-period bond (under some standard assumptions on utilities). The following simple example has essentially the minimal structure to capture the main issues. Each period there are S=3 states, which are all equally likely. That is, all 142 Karl Schmedders Theoretical Economics 2 (2007) elements of the Markov transition matrix are 1 3. There are two stocks with the following dividend vectors. d1= (1.1,1.0,0.9)> d2= (1.4,0.8,0.8)>. The aggregate endowment in our economy is then e=d1+d2= (2.5,1.8,1.7)>. There are H=2 agents who have CARA (Bernoulli) utility functions uh(c) = −1 ahe−ahc with coefficients of absolute risk-aversion a1=1 and a2=4, respectively. The agents’ discount factor is β=0.95. Initially, the agents both hold half of each stock. That is, their initial holdings are θhj −1=1 2for h=1,2, j=1,2. The economy starts in state y0=1. Equations (1)–(3) determine the equilibrium consumption allocation in this economy. We can easily solve this system of 7 nonlinear equations in 7 unknowns with Newton’s method. Denoting the vector (1,1,1)>by 13we obtain the following consumption vectors (rounded to three digits). c1= (1.421,0.861,0.781)>=0.8 ·(d1+d2)−0.579 ·13 c2= (1.079,0.939,0.919)>=0.2 ·(d1+d2) + 0.579 ·13. Consumption allocations are determined by linear sharing rules. This feature of the consumption allocations is the key property underlying two-fund separation in an economy with a consol. Whenever agents’ consumption follows a linear sharing rule, asset portfolios exhibit two-fund separation in the presence of a consol. Next we can determine the stock prices. We know the consumption allocations of the first agent and thus the vector of state prices py=u0 1(c1 y)for y=1,2,3. Now the expressions (4) immediately determine the state-contingent stock prices. q1= (28.864,16.487,15.220)> q2= (27.346,15.620,14.419)>. For asset markets to be dynamically complete we need three assets in the economy. Suppose that in addition to the two stocks there is also a consol. The linear equation (4) for the consol allows us to trivially calculate the price vector qcof the consol, qc= (29.432,16.812,15.519)>. Finally, we can solve the linear system of equations (6) to determine agents’ equilibrium portfolios. This system is independent of the prices of the two stocks and the consol. Agents’ portfolio holdings depend solely on the stocks’ dividend vectors, the consol payoffs, and agents’ consumption vectors. Equations (6) for agent 1 are Θ11 ·d1+ Θ12 ·d2+ Θ1c·13=0.8 ·(d1+d2)−0.579 ·13. (7) Theoretical Economics 2 (2007) Two-fund separation 149 REMARK 1. The necessity of genericity with respect to transition probabilities in Lemma 1is now apparent. If transition probabilities are i.i.d. and all agents have HARA utility with γ=1 (but possibly Ah6=0), then the linear sharing rule leads to the bond price being a linear function of the endowment for any set of dividends and initial portfolios. If ch y=mhey+bhfor all y∈ Y then qy=β X s∈Y Πy s 1 es+Ph∈H Ah! ey+X h∈H Ah!. Making genericity arguments with respect to transition probabilities saves us from having to distinguish this case from the general case. We cannot allow for the popular perturbations of utility functions as, for example, in Cass and Citanna (1998) and Citanna et al. (2006), since we want to examine two-fund separation for specific classical families of utility functions. Two-fund separation in models with a one-period bond (see Section 5.2) depends crucially on whether the intercept of the sharing rules is zero. We prove the following lemma in Section B of the Appendix. LEMMA 3. Suppose all agents have equi-cautious HARA utility functions of the type [CARA] or the type [EC]with Ph∈H Ah6=0and [A3]holds. Then, for a generic set T ⊂ ∆H−1 ++ of initial holdings of the first asset, each agent’s sharing rule is linear with nonzero intercept; that is, bh6=0for all h ∈ H . For standard equi-cautious CRRA utility functions with Ah=0 for all h∈ H we have bh=0, ∀h∈ H . 5. TWO-FUND SEPARATION: CONSOL VS.ONE-PERIOD BOND This section formalizes the intuition that we gained from our illustrative example in Section 3. A trivial proof shows that in an economy with a consol, portfolios exhibit twofund separation. We then prove that this property generically fails to hold in economies with a one-period bond. A discussion tying together our illustrative example and mathematical proofs concludes our analysis. 5.1 Economies with a consol In this subsection we assume that there are no short-lived assets; that is, Jl=S. Then equation (6) immediately yields that the consumption vector of every agent his a linear combination of the asset dividends (or payoffs), ch= (d1,...,dS)Θh. (9) Under the assumptions that all assets are infinitely lived and that there is a safe asset, we recover the classical two-fund monetary separation result for static demands of Cass and Stiglitz (1970) in our dynamic equilibrium context. 150 Karl Schmedders Theoretical Economics 2 (2007) THEOREM 1 (Two-Fund Separation Theorem). Suppose the economy Ehas S infinitely lived assets with linearly independent payoff vectors and satisfies Assumptions [A1]and [A2]. The first S −1assets are in unit net supply and asset S is a consol in zero net supply. If the agents have equi-cautious HARA utilities then in an efficient equilibrium their portfolios exhibit two-fund monetary separation. PROOF.Proposition 2 ensures that an efficient equilibrium exists. Lemma 2 implies that sharing rules are linear and ch y=mhey+bh∀h∈ H ,y∈ Y . Under the assumptions of the theorem, equation (9) has the unique solution ΘhS =bhand Θhj =mh∀j= 1,...,S−1.  We can easily extend Theorem 1 to economies with J<Slong-lived assets. Markets are dynamically complete with fewer assets than states and so portfolios exhibit twofund separation for J<Swhen a consol is present. Kang (2003, Chapter 4) observes that the results of Judd et al. (2003) can be generalized to economies with time-varying positive transition probabilities. In addition, he notices that the results also hold for finite-horizon economies with only long-lived assets. We can use these observations to extend the result of Theorem 1 to economies with a finite time horizon or time-varying (positive) transition matrices. Either change to our model affects the Negishi weights λh,h∈ H , and sharing rules mh,bh,h∈ H , but two-fund monetary separation continues to hold. 5.2 One-period riskless bond We now assume that the riskless asset is not a consol but instead a one-period bond. In addition the economy has S−1 infinitely lived assets in unit net supply. In such an economy two-fund monetary separation generically fails even when sharing rules are linear with nonzero intercepts. THEOREM 2. Consider an economy Ethat satisfies the following conditions. (i) There are J =S≥3assets. (ii) There are S −1infinitely lived securities in unit net supply. The last asset is a oneperiod riskless bond. (iii) Assumptions [A1]–[A3]hold. (iv) All agents have equi-cautious HARA utility functions of the type [CARA]or the type [EC]with Ph∈H Ah6=0. Then there are generic subsets T ⊂ ∆H−1 ++ of initial portfolios of the first asset and P ⊂ ∆S×(S−1) ++ of transition matrices such that each agent’s equilibrium portfolio does not exhibit two-fund monetary separation. Theoretical Economics 2 (2007) Two-fund separation 151 PROOF. All agents’ consumption allocations follow a linear sharing rule. Now suppose that equilibrium portfolios exhibit two-fund monetary separation, so agent hholds a portion ϑh≡Θhj of all infinitely lived assets j=1,...,S−1, and ΘhS of the one-period bond. Then equation (6) implies that the portfolio shares satisfy mh·e+bh1S=ϑh·e+ ΘhS(1S−qS)for all h∈ H , (10) where qSdenotes the bond price and 1Sthe vector consisting solely of ones. If bh=0 for all h∈ H then ΘhS =0 and mh=ϑhis a solution to this equation. Thus, two-fund monetary separation holds. But Lemma 3 states that under conditions (iii) and (iv) we have bh6=0 for all h∈ H for a generic set of initial portfolio holdings. Now suppose bh6=0 for all h. Then any solution to equation (10) must have ΘhS 6=0. Thus we can rewrite the equation as qS=ϑh−mh ΘhS ·e+ΘhS −bh ΘhS ·1S. But now the price of the one-period bond is a linear function of the aggregate endowment. Lemma 1, part (ii), states that for a generic set of initial portfolios and transition matrices there are no (endogenous) coefficients a,f∈Rsuch that qS=a·e+f. Hence, equation (10) generically does not have a solution. The intersection of generic sets is generic. The statement of the theorem now follows.  5.3 Discussion Recall that our illustrative example already provides us with much intuition for the main results of this paper. Now that we have seen the proofs of our two theorems we can continue the discussion from Section 3. Equation (10) in the proof of Theorem 2 is very instructive in providing intuition for the lack of two-fund monetary separation when the bond is short-lived. Recall that for an economy with a consol the corresponding equation would be mh·e+bh1S=ϑh·e+ ΘhS1Sfor all h∈ H . So the only difference that the short-lived bond introduces into the portfolio equation is that the bond position ΘhS is multiplied by the coupon payment minus the price instead of being multiplied only by the coupon payment. The economic reason for this difference is that the agent does not trade the consol after time zero but must reestablish the position in the short-lived bond in every period. This change has no impact on the portfolio weights if agents’ sharing rules have zero intercept and so the riskless security is not traded. But if sharing rules have nonzero intercept, then the bond price affects the portfolio weights. The appearance of the bond price still does not destroy two-fund monetary separation if this price is a linear function of the social endowment. But if that relationship does not hold, then the fluctuations of the bond price lead to a change of the portfolio weights that implement equilibrium consumption. In summary, fluctuations in the equilibrium interest rates (bond prices) of the shortterm bond lead to the breakdown of two-fund monetary separation. These fluctuations 152 Karl Schmedders Theoretical Economics 2 (2007) expose an agent holding a nonzero bond position in equilibrium to reinvestment risk because he must rebuild that position in every period. This reinvestment risk affects agents’ bond and thus stock portfolios and leads to a change of the equilibrium portfolio weights. In contrast, in an economy with a consol, the agent establishes a position in the consol at time zero once and for all. Fluctuations in the price of the consol therefore do not affect the agent, just as he is unaffected by stock price fluctuations. The agent does not have to reinvest the proceeds from an expiring security at fluctuating prices and so does not face reinvestment risk. This fact allows him to hold a portfolio exhibiting two-fund monetary separation. In our dynamic model only the consol is a riskless asset. The fact that interest rate variability has significant economic consequences in a dynamic equilibrium model has also been noted by Magill and Quinzii (2000). They examine an infinite-horizon CAPM economy with stochastic endowments and observe that with fewer assets than states an Arrow–Debreu allocation can be achieved only if a constant consumption stream can be spanned by the payoff matrix. But such a spanning condition may not hold if the interest rate fluctuates in equilibrium. As a consequence markets will be incomplete. 6. ASSET ALLOCATION PUZZLE In a paper that received a lot of attention in the finance literature, Canner et al. (1997) document recommendations from different investment advisers who all encourage conservative investors to hold a higher ratio of bonds to stocks than aggressive investors.1They point out that this financial planning advice violates the two-fund monetary separation theorem and call this observation the “asset allocation puzzle.” Similarly, Bossaerts et al. (forthcoming) state that the separation result cannot be reconciled with casual empirical observations and conclude that “many tests of asset pricing models address only the pricing predictions—but these pricing predictions rest on portfolio choice predictions which seem obviously wrong.” Both critiques assume that the classical notion of two-fund monetary separation is applicable to the dynamic world of modern financial markets. This supposition is a far-reaching generalization of the two-fund separation paradigm beyond the classical results (Cass and Stiglitz 1970,Merton 1973, and many others), which make strong assumptions on utility functions or return distributions. Moreover, both critiques clearly assume the existence of a riskless asset in actual financial markets. Canner et al. (1997) explicitly regard cash as the “safe” asset. In light of the results in this paper, it should come as no surprise that observed investors’ portfolios do not satisfy two-fund separation. Cash is a safe investment only in the very short term (see also, for example, Campbell and Viceira 2002). Because 1Canner et al. consider portfolios consisting of stocks, bonds, and cash, with cash being treated as the risk-free asset. They document that investment advisors recommend conservative (and even moderately risk averse) investors to hold a significant fraction of their wealth—beyond what liquidity needs would require—in cash assets. Advisors treat bonds as being somewhat risky relative to cash, so that the riskyasset portfolio consists of both stocks and bonds. The fact that the recommended ratio of these assets depends on the investor’s risk aversion violates portfolio separation. Theoretical Economics 2 (2007) Two-fund separation 153 investors must continually reinvest cash in the future at unknown and fluctuating interest rates, cash is not a safe asset for the long term. Instead cash positions expose investors to reinvestment risk. Therefore, we should not consider cash as a riskless asset in dynamic financial markets. Arguably investors do not have access to a truly safe asset. On modern financial markets, investors can certainly trade a huge number of financial assets including many finite-maturity bonds. But consols (with the exception of some perpetual bonds issued by the British Treasury in the 19th century) are not available for trade. And although (inflation-indexed) bonds with many different maturities do exist, investors cannot trade such bonds with a maturity matching just any desired (very long) investment horizon. In summary, the critiques of Canner et al. (1997) and Bossaerts et al. (forthcoming) are based on the incorrect assumption that investors have access to a safe asset. Their criticism of two-fund monetary separation is, therefore, not justified. For further discussion of the asset allocation puzzle, see Brennan and Xia (2000). Our results naturally lead us to question whether modern financial markets may enable investors to synthesize a consol through a variety of other assets, thereby leading to a generalized form of two-fund separation. Judd et al. (2006a) study this question by examining families of bonds with variable but finite maturity structures. They argue that a finite number of bonds can span a consol for some non-generic transition matrices and dividend structures. In such situations all agents hold the same fund of risky stocks. Their computational exercises show that this result does not hold in general but that such portfolios can be approximately optimal in general if bonds with a sufficiently rich maturity structure are available for trade. APPENDIX A. PARAMETRIC SYSTEMS OF EQUATIONS We state the theorem on a parametric system of equations that we use in the genericity proofs below. THEOREM 3 (Parametric Systems of Equations). Let Ω⊂Rk, X ⊂Rnbe open sets and let h:Ω×X→Rmbe a smooth function. If n <m and for all (¯ ω,¯ x)∈Ω×X such that h(¯ ω,¯ x) = 0we have rank[Dω,xh(¯ ω,¯ x)] = m, then there exists a set Ω∗⊂Ωwith Ω−Ω∗a set of Lebesgue measure zero, such that {x∈X:h(ω,x) = 0}=;for all ω∈Ω∗. For a detailed discussion of this theorem see Magill and Quinzii (1996b, Paragraph 11; 1996a). This theorem is a specialized version of the parametric transversality theorem (see Guillemin and Pollack 1974, Chapter 2, Paragraph 3, and Mas-Colell 1985, Chapter 8). Billingsley (1986, Section 12) provides a detailed exposition on the kdimensional Lebesgue measure in Euclidean space. For an exposition of sets of measure zero see Guillemin and Pollack (1974, Chapter 1, Paragraph 7). A set is said to have full measure if its complement is a set of Lebesgue measure zero. An open set of full Lebesgue measure is called generic. 154 Karl Schmedders Theoretical Economics 2 (2007) B. PROOFS This section contains all proofs that are omitted in the main body of the paper. PROOF OF PROPOSITION 1. Market-clearing and collinearity of marginal utilities imply that in an efficient equilibrium all agents have state-independent consumption allocations. Define ˆ ch≡¯ ch yfor all y∈ Y . Equations (2) imply that the agents’ consumption allocations are ˆ ch=([IS−βΠ]−1)y0·ωh Ps∈Y ([IS−βΠ]−1)y0s for all h∈ H . The resulting asset prices are qj= [IS−βΠ]−1βΠdj,j∈Lfor infinitely lived assets and qj=βΠdj,j∈Ofor one-period assets. If the matrix dhas full column rank then the solution to equations (6) is unique and gives the agents’ holdings of infinitely-lived assets j∈Lin unit net supply, Θhj =ˆ ch ˆ e=1 ˆ e ([IS−βΠ]−1)y0·ωh Ps∈Y ([IS−βΠ]−1)y0s . (If the matrix ddoes not have full column rank then this solution is only one in a continuum of optimal portfolios.) The agents do not trade any of the other assets (including infinitely-lived assets in zero net supply). Note that all expressions in this proof are independent of the agents’ utility functions.  PROOF OF PROPOSITION 2. The existence result of Mas-Colell and Zame (1991) implies that there exist equilibrium state-contingent consumption values ch y,h∈ H ,y∈ Y , that solve the system of equations (1)–(3). The critical remaining issue for the existence of an efficient financial market equilibrium is now whether the matrix Dhas full rank. In that case equations (6) yield the agents’ equilibrium portfolios. (Actually, a careful reading of Judd et al. 2003 shows that we also need the matrix (d1+q1,...,dJl+qJl,dJl+1,...,dS) to have full rank. But that condition is equivalent to Dhaving full rank.) If all assets are long-lived then D=dand so Dhas full rank. We now show that for economies with a one-period riskless bond the matrix Dgenerically has full rank. If D does not have full rank then the following set of equations must have a solution. u0 1(c1 y)−λhu0 h(ch y) = 0, h=2,...,H,y∈ Y (11) [IS−βΠ]−1(p⊗(ch−X j∈L θhj −1dj))y0 =0, h=2,...,H(12) H X h=1 ch y−ey=0, y∈ Y (13) qS ypy−βΠy·p=0, y∈ Y (14) X j∈L dj(y)aj+ (1−qS(y)) = 0, y∈ Y . (15) Theoretical Economics 2 (2007) Two-fund separation 155 We denote the system of equations (11)–(15) by F((ch)h∈H ,(λh)h≥2,qS,a;(θh1 −1)h≥2,Π·1) =0. The expression F(i)=0 denotes equations (i). We now show that this system has no solutions for generic sets of individual holdings of the first asset and transition probabilities. The system (11)–(15) has HS +(H−1)+S+(S−1)endogenous unknowns ch,h∈ H , λh,h=2,...,H,qS, and aj,j=1,...,S−1, in (H−1)S+ (H−1) +S+S+Sequations. In addition, the function Fdepends on the (H−1) + Sexogenous parameters (θh1 −1)h≥2∈ ∆H−1 ++ and Π·1where Π−S∈∆S×(S−1) ++ denotes the first S−1 columns of Π. Assumption [A2] allows us to assume without loss of generality that e16=eS. We now prove that the Jacobian of Ftaken with respect to ch,qS,θh1 −1, and Π·1has full row rank (H−1)S+(H−1)+S+S+S. Denote by ΛS(x)∈RS×Sthe diagonal matrix whose diagonal elements are the elements of the vector x∈RS. We denote the derivative of the budget constraints (12) with respect to the agent’s initial holding in the first infinitely lived asset, −([IS−βΠ]−1(p⊗d1))y0, by η1. Note that η1<0. In order to keep the display tractable, we show the Jacobian of Ffor the special case of H=3 and write DΠ·1F(14)for −βΛ((p1−pS)·1S). c1c2c3qSθ21 −1θ31 −1Π·1 F(11)h=2ΛS(u00 1(c1)) ΛS(−λ2u00 2(c2)) 0 0 0 0 0 S F(11)h=3ΛS(u00 1(c1)) 0ΛS(−λ3u00 3(c3)) 0 0 0 0 S F(12)h=20 0 η10 1 F(12)h=30 0 0 η11 F(13)ISISIS0 0 0 0 S F(14)0 0 0 0 DΠ·1F(14)S F(15)0 0 0 −IS0 0 0 S S S S S 1 1 S The variables above the matrix indicate the variables with respect to which derivatives have been taken in the column underneath. The numbers to the right and below the matrix indicate the number of rows and columns, respectively. The terms to the left indicate the equations. Missing entries are not needed for the proof. We now perform column operations to obtain zero matrices in the first set of columns of the Jacobian. The sets of columns for ch,h∈ H , of the Jacobian then appear as follows. c1c2c3 F(11)h=20ΛS(−λ2u00 2(c2)) 0S F(11)h=30 0 ΛS(−λ3u00 3(c3)) S F(12)h=20 1 F(12)h=30 1 F(13)IS+ Λu00 1(c1) λ2u00 2(c2)+ Λu00 1(c1) λ3u00 3(c3)ISISS F(14)0 0 S F(15)0 0 0 S 156 Karl Schmedders Theoretical Economics 2 (2007) The transformed matrix has submatrices of the following ranks. c1c2c3qSθ21 −1θ31 −1Π·1 F(11)h=20S0 0 0 0 0 S F(11)h=30 0 S0 0 0 0 S F(12)h=20 0 1 0 1 F(12)h=30 0 0 1 1 F(13)S S S 0 0 0 0 S F(14)0 0 0 0 S S F(15)0 0 0 S0 0 0 S S S S S 1 1 S The term DΠ·1F(14)=−βΛ((p1−pS)·1S)has rank Ssince p16=pSbecause e16=eS. This matrix has full row rank (H−1)S+ (H−1) + 3Swhich exceeds the number of endogenous variables by 1. The function Fis defined on open sets with ch∈int(X)for all h∈ H ,λh∈RS ++ for h≥2, a∈RS−1,qS∈RS ++,(θh1 −1)h≥2∈∆H−1 ++ , and Π−S∈∆S×(S−1) ++ . Hence Fsatisfies the hypotheses of the theorem on parametric systems of equations, Theorem 3. We conclude that there exist subsets T ⊂ ∆h−1 ++ and P ⊂ ∆S×(S−1) ++ of full Lebesgue measure such that the solution set of the system (11)–(15) is empty. The sets Tand Pare open. The solutions to (11)–(15) change smoothly with the exogenous parameters. A small variation in initial portfolios and probabilities does not lead to a solvable system if there is no solution for the original parameters. We conclude that the matrix Dhas full rank Sand so an equilibrium exists for initial holdings (θh1 −1)h≥2∈ T of the first asset and transition matrices such that Π−S∈ P . PROOF OF LEMMA 1. Part (i). The price of the one-period bond isqb y=βΠy·u0 1(c)/u0 1(cy), where u0 1(c)denotes the column vector of utilities u0 1(cy),y∈ Y . If the social endowment eis not constant, every agent has nonconstant consumption. Choose y1∈ argmin{c1 y:y∈ Y } such that Πy1s>0 for some s/∈argmin{c1 y:y∈ Y }. Similarly, choose y2∈argmax{c1 y:y∈ Y } such that Πy2s>0 for some s/∈argmax{c1 y:y∈ Y }. Obviously, y16=y2. Then Πy1·u0 1(c)<u0 1(cy1)and Πy2·u0 1(c)>u0 1(cy2)and so qb y2> β > qb y1. If there is no aggregate risk in the economy, then the bond price equation immediately yields qb=β. Part (ii). The price of the one-period bond in state y∈ Y satisfies qypy=βΠy·p. If in equilibrium the price is a linear function of the social endowment e, then the following set of equations must have a solution. u0 1(c1 y)−λhu0 h(ch y) = 0, h=2,...,H,y∈ Y (16) [IS−βΠ]−1(p⊗(ch−X j∈L θhj −1dj))y0 =0, h=2,...,H(17) H X h=1 ch y−ey=0, y∈ Y (18) (a ey+f)py−βΠy·p=0, y∈ Y . (19) Theoretical Economics 2 (2007) Two-fund separation 157 We denote the system of equations (16)–(19) by F((ch)h∈H ,(λh)h≥2,a,f;(θh1 −1)h≥2,Π·1) = 0. The expression F(i)=0 denotes equations (i). The system has HS + (H−1) + 2 endogenous unknowns ch,h∈ H ,λh,h=2,...,H,a, and fand (H−1)S+ (H−1) +S+S equations. (Note that the coefficients aand fof the linear price function are endogenous variables.) In addition, Fdepends on the (H−1)+Sexogenous parameters θh1 −1,h= 2,...,H, and Π·1. The aggregate endowment is not constant and so we can assume without loss of generality that p16=pS. The Jacobian of Ftaken with respect to ch,θh1 −1, and Π·1is identical to the respective columns of the corresponding submatrix in the proof of Proposition 2. After performing the same column operations as in that proof we obtain a transformed matrix with submatrices of the following ranks. c1c2c3θ21 −1θ31 −1Π·1 F(16)h=20S0 0 0 0 S F(16)h=30 0 S0 0 0 S F(17)h=20 1 0 1 F(17)h=30 0 1 1 F(18)S S S 0 0 0 S F(19)0 0 0 0 S S S S S 1 1 S This matrix has full row rank (H−1)S+ (H−1) +S+Swhich exceeds the number of endogenous variables by S−2≥1. The function Fis defined on open sets with ch∈int(X) for all h∈ H ,λh∈RS ++ for h≥2, a,f∈R,(θh1 −1)h≥2∈∆H−1 ++ and Π−S∈∆S×(S−1) ++ . Hence, Fsatisfies the hypotheses of Theorem 3 and the proof proceeds like that of Proposition 2. PROOF OF LEMMA 3. We first consider an economy where all agents have equi-cautious HARA utility functions of the type [EC]. Then if bh=0 for some agent hin equilibrium the following equations must hold. u0 1(c1 y)−λhu0 h(ch y) = 0, h=2,...,H,y∈ Y (20) [IS−βΠ]−1(p⊗(ch−X j∈L θhj −1dj))y0 =0, h=2,...,H(21) H X h=1 ch y−ey=0, y∈ Y (22) −A¯ h+(λ¯ h) 1 γ Pi∈H (λi) 1 γX i∈H Ai=0, for one ¯ h∈ H . (23) We denote the system of equations (20)–(23) by F((ch)h∈H ,(λh)h≥2;(θh1 −1)h≥2) = 0. The system has HS + (H−1)endogenous unknowns ch,h∈ H and λh,h=2,...,H, and (H−1)S+(H−1)+S+1 equations. In addition, the function Fdepends on the H−1 exogenous parameters θh1 −1,h=2,...,H. We show that the Jacobian of Ftaken with respect to ch,λh, and θh1 −1has full row rank (H−1)S+ (H−1) +S+1. 158 Karl Schmedders Theoretical Economics 2 (2007) For ¯ h≥2 denote the derivative in equation (23) with respect to λ¯ hby η2 ¯ h= 1 γ(λ¯ h) 1 γ−1·PH i=1(λi) 1 γ−(λ¯ h) 1 γ PH i=1(λi) 1 γ2 H X i=1 Ai. For ¯ h=1 we cannot take the derivative in (20) with respect to λ1, since it does not appear (it is normalized to one). Instead we differentiate with respect to λ2and obtain η2 1=− 1 γ(λ2) 1 γ−1 PH i=1(λi) 1 γ2 H X i=1 Ai. Note that under the condition from the lemma, Pi∈H Ai6=0, we have η2 ¯ h6=0. For the special case of H=3 the Jacobian Dch,θh1 −1,λ¯ hFappears as follows. c1c2c3θ21 −1θ31 −1λ¯ h F(20)h=2ΛS(u00 1(c1)) ΛS(−λ2u00 2(c2)) 0 0 0 S F(20)h=3ΛS(u00 1(c1)) 0ΛS(−λ3u00 3(c3)) 0 0 S F(21)h=20η10 0 1 F(21)h=30 0 η10 1 F(22)ISISIS0 0 0 S F(23)0 0 0 0 0 η2 ¯ h1 S S S 1 1 S After the same column operations as in the proof of Proposition 2 we obtain the following ranks for the various submatrices of the transformed matrix. c1c2c3θ21 −1θ31 −1λ¯ h F(20)h=20S0 0 0 S F(20)h=30 0 S0 0 S F(21)h=20 1 0 0 1 F(21)h=30 0 1 0 1 F(22)S S S 0 0 0 S F(23)0 0 0 0 0 1 1 S S S 1 1 S This matrix has full row rank HS +(H−1)+1 which exceeds the number of endogenous variables by 1. The function Fis defined on open sets with ch∈int(X)for all h∈ H , λh∈RS ++ for h≥2, and (θh1 −1)h≥2∈∆H−1 ++ . Hence, Fsatisfies the hypotheses of Theorem 3 and the proof proceeds like that of Proposition 2. We can perform the proof for each agent and then take the intersection of generic sets which in turn yields a generic set for which no agent has a linear sharing rule with zero intercept.