An Analytic Solution to the CAPM Equilibrium

Abstract

We determine the one-dimensional general solution to the semi-equilibrium prices of primitive securities in the equation for the capital asset pricing model (CAPM). Furthermore, considering the value clearing condition, we determine the analytical equilibrium solution to the CAPM market, which reveals the overall thinking in equilibrium pricing. We use a numerical example to illustrate that the CAPM equilibrium in an incomplete market does not exclude arbitrage opportunities. In addition, we show that the beta pricing formula can only be used to price marketable (within the market payoff space) assets, because the beta pricing formula is nothing more than a manifestation of the law of asset portfolio, that is, beta pricing is based on the equilibrium prices of primitive securities to compute the linear pricing of the asset portfolio in the market.