Asset Market Equilibrium with Short-Selling

Abstract

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 non-satiation 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. This paper analyses the existence of equilibrium in exchange economy models where choice sets may be unbounded below and satiation is likely. There are two main examples of such models. One is an asset market model where investors maximize expected utility and the total return to individual assets may be negative with positive probability. The other is the mean-variance Capital Asset Pricing Model (CAPM), where investors do not necessarily maximize expected utility but instead maximize a function of only the mean and variance of return to their portfolios. In both of these models, the investors' choice sets will be unbounded if short-selling is allowed. To short-sell a share of an asset means to borrow it and sell it, promising to buy it back and return it to the lender at a later date. Formally, short-selling corresponds to holding a negative number of shares. If unlimited short-selling is allowed, then there is no limit to how large negative numbers of shares can be held, and therefore the choice sets are unbounded. Satiation can also occur both in the mean-variance model and in the expected-utility model (unless there is a riskless asset). In the expected-utility model, there may be satiation if the individual assets have negative returns with positive probability. As the investor gets more and more shares, the potential positive returns get larger and larger, but so do the potential negative returns. If the investor is very averse to large negative returns, then he may eventually be satiated and not want any more shares. In the mean-variance model, the expected return to an investor's portfolio increases as he holds more and more shares of the assets, but so does the variance of return. It may be that at some point, the additional expected return gained from adding more shares to the portfolio is not sufficient to compensate for the increase in variance. If so, then there will be satiation. Satiation in the mean-variance model iie analysed in Nielsen (1987, 1988). The expected-utility model with short-selling has been used in many specific analyses of portfolio selection and asset pricing. To name but one, Connor (1984) studies arbitrage pricing in a general equilibrium model where there are no short sales restrictions. The mean-variance CAPM is of course very prominent in the finance literature. Because the