A simplified Capital Asset Pricing Model

We consider a Black-Scholes market in which a number of stocks and an index are traded. The simplified Capital Asset Pricing Model is the conjunction of the usual Capital Asset Pricing Model, or CAPM, and the statement that the appreciation rate of the index is equal to its squared volatility plus the interest rate. (The mathematical statement of the conjunction is simpler than that of the usual CAPM.) Our main result is that either we can outperform the index or the simplified CAPM holds.