Positive Prices in CAPM

Abstract

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. THE TWO-PERIOD MEAN-VARIANCE capital asset pricing model (CAPM) has the peculiar property that preferences may not be monotone. 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 the induced preferences for assets are not monotone. Nonmonotonicity of preferences in the mean-variance model is analyzed in Nielsen (1987). Because portfolio preferences are not necessarily monotone, equilibrium asset prices may be negative or zero. In fact, it is possible that the value of the market portfolio of risky assets is negative. This is demonstrated in Nielsen (1985, 1990a) and in two examples in the present paper. When the CAPM is used as a model of the prices of stocks, which in reality have limited liability, it is of course disturbing that it may predict negative prices. It can be seen as a symptom of the fact that mean-variance preferences are not always a good approximation to reality. This paper analyzes various conditions on the exogenous parameters which ensure that the preferences are sufficiently realistic to yield positive prices. The exogenous parameters in question are the expectations, variances and covariances of total returns, the initial allocations of the assets (endowments), and the investors' utility functions for mean and variance of return. The analysis applies to the classical version of the CAPM with a riskless asset, as developed by Sharpe (1964), Lintner (1965) and Mossin (1966), as well as the CAPM with risky assets only, which was introduced by Black (1972), and has been much used in empirical work under the name the zero-beta or two-factor CAPM.