General equilibrium in the classical two-period mean-variance capital asset pricing model is not unique. Corresponding to one single set of expectations, utility functions, and an initial wealth distribution, there may be several equilibria, and an asset may have different prices, expected rates of return, and betas in different equilibria. However, any equilibrium portfolio is sustained by a unique price system, and if investors have decreasing risk aversion, then any equilibrium allocation of the risky assets is sustained by a unique price system.

In the mean-variance CAPM without a riskless asset, the possibility of satiation sometimes leads to non-existence of general equilibrium. Moreover, because portfolio preferences are not necessarily monotone, equilibrium asset prices, when they exist, may be negative or zero. To demonstrate the possibility of non-existence, and to develop an intuitive understanding of when and why equilibrium does or does not exist, this paper fully investigates the special case of utility functions linear in mean and variance and partially extends the results to the general case.

This paper presents simple conditions and a simple proof of the existence of equilibrium in asset markets where short-selling is allowed and satiation is possible. Unlike standard nonsatiation assumptions, the one used here is weak enough to be reasonable in the mean-variance capital asset pricing model and in asset market models where investors maximize expected utility and where total returns to individual assets may be negative.

Some equilibrium prices in CAPM may be negative because of nonmonotonicity of preferences. We identify several sets of sufficient conditions for prices to be positive. The central conditions impose bounds on the investors' risk aversion. These bounds do not need to hold globally but only in a relevant range of portfolios or combinations of mean and standard deviation. The relevant range is specified on the basis of exogenous parameters and variables, and it must contain any endogenously determined equilibrium. The bounds on risk aversion ensure that the preferences for assets are sufficiently well‐behaved within the relevant range.

Some equilibrium prices in the capital asset pricing model may be negative because of nonmonotonicity of preferences. The authors identify several sets of sufficient conditions for prices to be positive. The central conditions impose bounds on the investors' risk aversion. These bounds do not need to hold globally but only in a relevant range of portfolios or combinations of mean and standard deviation. The relevant range is specified on the basis of exogenous parameters and variables, and it must contain any endogenously determined equilibrium. The bounds on risk aversion ensure that the preferences for assets are sufficiently well-behaved within the relevant range.