Abstract: The capital market for CAPM is incomplete and is a Hilbert space, we find out the analytic expression for the SDF mimicking payoff in this market. The CAPM formula holds under the partial equilibrium of purely risky assets, which is equivalent to the condition that the market portfolio is the tangency portfolio. Since the general solution to asset prices in CAPM has only one dimension, given all individual investor’s endowments and mean-variance preferences, the condition of CAPM equilibrium turns out to be an equation of only one variable. With the closed-form solution to CAPM equilibrium, we see more clearly that the risk-return characteristics is a false impression from the partial equilibrium of purely risky assets. Thus it is illusory, for the assets are priced as a whole and the prices are endogenous, the return of market portfolio is not exogenous but endogenous. By way of numerical examples, we show that CAPM equilibrium may coexist with arbitrage opportunities, and that the crisis of impossible frontier is due to market disequilibrium. We point out the incorrect practice of beta pricing by presenting negative prices of European call options.
Keywords: CAPM, Mean-Variance Rule, Beta, Arbitrage Opportunity, Risk Premium
We know that absolute pricing determines the prices of primitive securities in the equilibrium process of fund allocation. In absolute pricing, price is a function of the payoff of primitive security and every investor’s characteristics (endowment and preference). The pricing function is usually non-linear with respect to payoff. In contrast, relative pricing is the pricing of attainable payoffs based on the given prices of primitive securities in the process of fair trading. In relative pricing, price is a function of payoff. Due to the law of linear combination, the pricing function must be linear and only applicable to the payoff space spanned by given primitive securities. The CAPM formula is not an equilibrium pricing formula (absolute pricing), because it is only a result of partial equilibrium, and it has a one-dimensional general solution. Only when market return is given in advance, the CAPM formula could at best be considered a relative pricing formula. In this case it can only be used to price the portfolio of primitive securities. As relative pricing, it does not make sense to discuss risk in the CAPM formula. Because the relative pricing is based on the equilibrium prices of primitive securities and is realized through an arbitrage mechanism (replication), whereas arbitrage is not affected by risk preference. When the CAPM equilibrium prices are free of arbitrage, the CAPM formula must be a risk-neutral pricing formula.