Mean-Variance Theory in Complete Markets

Two paradigms in the pricing of risky assets are the model of Arrow and Debreu and mean-variance pricing as embodied in the capital asset pricing model (CAPM). The former is primarily of theoretical interest. In fact, prior to the recent work of Banz and Miller (1978) and Breeden and Litzenberger (1978) there was little thought or hope of practical applications. On the other hand, the hegemony of the CAPM is due mostly to its apparent ease of applicability and, to a lesser extent, its empirical justification. In a competitive market with no taxes, transactions costs, or institutional constraints on portfolio holdings, the allocation of the available assets will be Pareto optimal. The end-of-period distribution of wealth (or the consumption goods), however, will typically be Pareto optimal only when there is available an Arrow-Debreu security for each contingency. Since, by definition, a Pareto superior reallocation is weakly preferred by all investors, a complete set of Arrow-Debreu markets should naturally arise provided they are not too costly to establish. The complete-markets model of Arrow and Debreu and the meanvariance capital asset pricing model (CAPM) are two paradigms of risky asset markets. This paper provides an integration of these two theories. We prove that the standard meanvariance separation theorem obtains in a complete market only if all investors have quadratic utility. In addition, the familiar CAPM pricing relation can hold for all assets in a complete market only if arbitrage opportunities exist. On the positive side, we show that the validity of this relation for the primary assets in a complete market is unaffected by the distribution of returns on the created financial assets.