Asset pricing in a pure exchange economy with heterogeneous investors

In the literature aggregation results for CRRA utility with different initial wealth levels but identical risk aversion and beliefs are well understood, see the recent analysis in [16]. They fully characterize the class of utility functions of heterogeneous households and study the existence of a representative consumer.

This paper considers a more general set of risk-aversion parameters, compared with [25]. In a different perspective, we propose a simple and intuitive closed-form solution which indeed enhances the understanding on the economics of financial markets. For example, the implications for the price-dividend ratio are very interesting. Compared to [25], our model newly implies that the price-divided ratio should be larger for firms with large size and low-risk aversion shareholders (e.g., institutional investors). This is consistent with the empirical evidence in [12]. Our framework can be easily applied to analyse different markets, e.g., bonds, stocks, options and funds. Further discussing a property of a particular market with an equilibrium model is beyond the scope of the current article.

The “complete” solution in this paper is relative to the partial solutions in the literature, e.g., lacking the closed-form solution for asset prices in [9] and missing the solution for the price of the stock yielding multiple or continuous dividends in [4].

Other literature studies on heterogeneous investors with margin constraints. Gârleanu and Pedersen [10] study a model with heterogeneous-risk-aversion agents facing margin constraints and discuss how margin constraints affect capital market equilibrium. Rytchkov [23] develop constant margin constraints in [10] into time-varying case. Recently, a two-tree [18] heterogeneous-belief model is studied by Han et al. [11].

In a constrained economy, finding the equilibrium reduces to solving a system of partial differential equations (PDEs) in [7] or ordinary differential equations (ODEs) in [8].

Perturbation method is a method of finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middle step that breaks the problem into “solvable” and “perturbation” parts. Historically this method was invented in the area of engineering science in solving some nonlinear problems in which it is difficult to obtain exact solutions. The validity of the approximate solutions obtained is often justified numerically with a reasonable range of parameters. A proof of convergence of this method is an open question. In finance, this method has been used by Kogan and Uppal [15] to analyse a heterogenous-agent economy in the presence of portfolio constraints.

We thank a reviewer for suggesting us to pay attention to the advantage of analytical simplicity and transparency.

With the standard logarithmic preferences (\(\gamma _H=\gamma _L=1\)), Yang and Zhang [27] propose a dynamic asset pricing model with heterogeneous sentiments and [19] develop a model with two institutional investors.

Kaniel and Kondor [14] propose a heterogeneous-agents Lucas tree model and distinguish investors (including fund managers, clients, direct traders and newborn investors) that trade on their own account from investors that trade through managed portfolios as opposed to low and high risk-aversion investors as in this paper. Kaniel and Kondor [14] therefore obtain an inverse U-shaped Sharpe ratio rather than our monotonic Sharpe ratio.

Unlike the incomplete information model [2] in which \(\lambda \) is stochastic as investors face the different state price densities, in our complete information model, \(\lambda \) should be constant (e.g., [3, 25]).

If we substitute Bhamra and Uppal’s [4] solutions of our model in “Appendix A” into the definition of the marginal utility of the representative investor in (11) and then solve the marginal utility, we will find that it is very difficult to explicitly solve the equilibrium price of long-lived stock (13). This is because the marginal utility is with respect to the \(\gamma _H\)th (or \(\gamma _L\)th) power of a hypergeometric function. Without a loss of generality, we only compare our solution with Bhamra and Uppal’s [4] as the closed solutions of [7, 8] have the same accuracy.

The perturbation method can be applied in solving the equilibrium in a more general settings, such as the case of heterogeneity in time discount factors and heterogeneity in assessment of the dividend mean growth rates. The details involving a lot of algebra will be presented in a subsequent research.

Following Wang [25], we focus on the consumption rate rather than the consumption over dividend (income) ratio. Based on the solutions in Eqs. (16)–(17), it does not make any difference using either one.

The proportion of institutional investors in the U.S. public equity market in 2010 is 67 %.

The optimal consumption for investor H in [25] isand Bhamra and Uppal’s [4] solution is given in “Appendix A”.

The only difference between the first-order and second-order perturbation solutions is that the risk-free rate and the price-dividend ratio are no longer constant. More details see “Appendix C”.

Corresponding to Wang [25], the trading strategies are expressed by the unique state variable \(D_t\) rather than \(W_{i,t}\) or \(S_t\). However, our solution are more explicit than [25] as \(S_t\) is solved in (19).

Given \(\beta , \gamma \) and \(\varepsilon \), we can solve a from a quadratic equation in (28) so that b can be determined by \(\beta , \gamma \) and \(\varepsilon \).

The risk-free interest rate in the second-order approximation is stochastic. More details see “Appendix C”.

The volatility in the second-order approximation is not constant. More details see “Appendix C”.

The second equality is proved by using the restricted equation of the weight ratio b in (28). By using Eq. (21) and the fact that the volatility of the stock is \(\sigma _D\), the equity premium in our model is \(\left( 1-\varepsilon \frac{a-1}{a+1} \right) \gamma \sigma _D^2+{\mathcal {O}}(\varepsilon ^2)= \gamma _A \sigma _D^2+{\mathcal {O}}(\varepsilon ^2)\) which is equivalent to the results solved in the classical equilibrium model with one representative investor (e.g., [20, 26]). It shows that our model can not fully explain the equity premium puzzle. However, the main target of the paper is to explore how the size of type L investors and risk-averse heterogeneity influence the prices of the assets and investments.

Here we set \(E[D_t]=D_0 e^{\mu t} =1.43\) as the value of \(D_t\).

The proof is available upon request.

Similarly to [3], the non-zero volatility of the risk-free interest rate or the stochastic risk-free interest rate is due to the more volatile consumptions in (16)-(17) in the second-order. In addition, it leads to that the volatility of the stock price is no longer \(\sigma _D\).

Similar to Wang [25], the wealth process can be expressed by the stock price and the divided. As there is only one state variable \(D_t\), the wealth process \(W_{H,t}\) is one of our solutions.

By using the restricted equation of the weight ratio b in (38), we can get the second equality in (39). As \(\gamma _H=\gamma (1+\varepsilon )\) and \(\gamma _H-\gamma (1-\varepsilon )\), only constant and the first order terms of \({\widehat{\phi }}_{H,0}\) contribute to the finally results.

Even though we set \(\beta <0.5\) (e.g., \(\beta =0.2\) or \(b=0.35\)) which means the initial size type L investors is quite small , the evolutions of Fig. 9 are not changed. It tells us that with the economy developing better and better, the type L investors become wealthier and wealthier. This is consistent to the fact that the size of type L investors jumps sharply from 7-8% in 1950 to 67% in 2010. Our model does explain the phenomenon that type L investors as risk takers become larger and larger in the financial market.