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Financial market equilibria with heterogeneous agents: CAPM and market segmentation

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Abstract

We consider a single-period financial market model with normally distributed returns and heterogeneous agents. Specifically, some investors are classical expected utility maximizers whereas some others follow cumulative prospect theory. Using well-known functional forms for the preferences, we analytically prove that a Security Market Line Theorem holds. This implies that capital asset pricing model is a necessary (though not sufficient) requirement in equilibria with positive prices. We prove that equilibria may not exist and we give explicit sufficient conditions for an equilibrium to exist. To circumvent the complexity arising from the interaction of heterogeneous agents, we propose a segmented-market equilibrium model where segmentation is endogenously determined.

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Notes

  1. I wish to thank the Editor for pointing this out during the revision process.

  2. \(\mathbb{R }^N_{++}\) is short notation for \((0,+\infty )^N\). In this case, we mean that every asset is in strictly positive supply.

  3. For a set \(A,\,\# A\) denotes the cardinality of \(A\).

  4. We do not explicitly give a framing of loss aversion. The main reason is that LA is easy to explain using familiar language (“losses loom larger than gains”) but at the same time there is not a widespread accepted mathematical formulation. Common assumptions are \(u^h(y)+u^h(-y) < u^h(x)+u^h(-x)\) for \(y>x>0\) and \(\lim _{x \rightarrow 0} {u^{h}}^\prime (|x|)/\lim _{x \rightarrow 0} {u^h}^\prime (-|x|)>1\), which express LA for large and small stakes respectively. For an overview on LA and its implications, we refer the reader to [21, 33].

  5. Tangency occurs between the MV frontiers obtained with and without the risk-free asset. In the first case, it is formed by two straight lines on the mean/standard deviation plane. On the contrary, it is an hyperbola when only risky assets are available (see [16]).

  6. Depending on the preferences’ parameters, an optimal demand \(\Theta _M^{j \star }\) may not exist. In this case, we set \(\Theta _M^{j \star } = + \infty \).

  7. For CPT investors, strict versions of the inequalities can not be obtained since specific choices of \(u^h\) and \(w^h_\pm \) lead to indifference between values of \(\Theta _M^h\). This happens for example in the Linear LA and in the CRRA case, as it will be shown later.

  8. The Sharpe ratio of a security with expected return \(\mu \) and standard deviation \(\sigma \) is given by \(SR:=\tfrac{\mu - R_f}{\sigma }\).

  9. Given the utility function \(u\), the Absolute Risk Aversion coefficient is defined as \(A(x):=- \frac{u^{\prime \prime }(x)}{ u^{\prime } (x)}\).

  10. If \(q_M=0\), the return \(R_M=M/q_M\) is no more well defined. However, we can manipulate (9) to recover the CAPM using prices instead of returns.

  11. We drop the superscripts when unnecessary.

  12. Compare our Eq. (25) with the expression of \(g^i(q,\sigma )\) in the proof of Lemma 3 in [6]. In their notation, \(\sigma \) represents the standard deviation of the terminal wealth, equal to our \(\Theta _M^h \sigma _M\). The authors use condition (24) to prove the existence of an equilibrium in their Proposition 7. However, they do not check whether the resulting equilibrium price vector has strictly positive entries.

  13. In the case of indifference with respect to her risky position, an ill-behaved trader could clear the market by acquiring the risky securities that the other participants are not willing to purchase. For example, this would make sense in a stock-exchange placing where unsold securities are bought by a single institution. On the other hand, it is hard to think of a portfolio manager or a household who is indifferent in bearing undefined risks. We will not deepen on this type of equilibria.

  14. We could use a more general notion of segmented equilibrium where the group of participants coincide with that of well-behaved traders and the group of non-participants contains the ill-behaved traders. Unfortunately, it would prevent from obtaining explicit expressions in our results.

  15. With the exception of the Swedish case in July 2009, when the Riksbank (the Swedish central bank) used negative interest rate on deposits at \(-25\) bp, nominal interest rates never become negative. However, real interest rates often go below zero, as it happened for Italian BoT (Italian treasury bonds) in September 2008 and for U.S. Tips (Treasury inflation protected securities) in October 2010. Interpreting \({R}_0\) and \({R}_1\) as real expected returns, we can incorporate inflation in our model.

  16. For convenience, we do not explicitly write the arguments in the implicit functions. Clearly, each derivative has to be evaluated at equilibrium quantities.

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Acknowledgments

The author would like to thank the Editor and an anonymous referee for many valuable suggestions that helped to improve the quality of the paper considerably. He is also grateful to Luciano Campi for expository help on the manuscript and Maria Elvira Mancino for helpful comments.

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  1. Dipartimento di Matematica per le Decisioni Economiche, Università degli Studi di Firenze, Firenze, Italy

    Matteo Del Vigna

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Del Vigna, M. Financial market equilibria with heterogeneous agents: CAPM and market segmentation. Math Finan Econ 7, 405–429 (2013). https://doi.org/10.1007/s11579-013-0102-0

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