Existence Theorems in the Capital Asset Pricing Model

Uniqueness of Equilibria in the CAPM

“…And indeed in this case, as Allingham (1991) has demonstrated, equilibria are unique even when there is no riskless asset.…”

“…by Nielsen (1990), Allingham (1991), Dana (1999) and Hara (1998). For the case without a riskless asset under some additional assumptions Nielsen (1990), Allingham (1991) and Laitenberger (1998) give existence results. In any case, if there is an equilibrium, the next proposition shows that it features a lot of powerful properties.…”

“…Indeed, we will show in section 4 that even in the case when all investors exhibit constant absolute risk aversion, examples of economies with multiple equilibria can be constructed for the CAPM without a riskless asset. The only condition we know which guarantees uniqueness of equilibria even for the CAPM without a riskless asset has been given in Allingham (1991). When the CAPM utility functions are derived from expected utility maximization and when asset returns are normally distributed, then agents do not only exhibit constant absolute risk aversion but represented in a mean-variance diagram indifference curves are parallel straight lines.…”

A note on the existence of CAPM equilibria with homogeneous cumulative prospect theory preferences

“…In the CAPM world with only EU decision makers, the problem of existence of equilibria with positive prices is well-known (see Nielsen 1988;Allingham 1991;Nielsen 1992, andLevy 2007, among others). Exactly the same question arises in homogeneous CPT agents models, and the reason for negative prices is similar to that in the aforementioned papers.…”

Capital market equilibrium without riskless assets: heterogeneous expectations

“…The traditional approaches to equilibrium theory of commodity markets are not applicable to the CAPM with satiation portfolios either, because they exclude the case that satiation occurs only inside the set of feasible and individually rational allocations. Nielsen (1990) and Allingham (1991) investigate the existence of equilibrium in the classical capital asset pricing model without riskless assets. They assume that agents have homogeneous expectations on the return distribution.…”