Alpha is a measure of the active return on an investment, the performance of that investment compared with a suitable market index. An alpha of 1% means the investment's return on investment over a selected period of time was 1% better than the market during that same period; a negative alpha means the investment underperformed the market. Alpha, along with beta, is one of two key coefficients in the capital asset pricing model used in modern portfolio theory and is closely related to other important quantities such as standard deviation, R-squared and the Sharpe ratio. In modern financial markets, where index funds are widely available for purchase, alpha is commonly used to judge the performance of mutual funds and similar investments. As these funds include various fees normally expressed in percent terms, the fund has to maintain an alpha greater than its fees in order to provide positive gains compared with an index fund. Historically, the vast majority of traditional funds have had negative alphas, which has led to a flight of capital to index funds and non-traditional hedge funds. It is also possible to analyze a portfolio of investments and calculate a theoretical performance, most commonly using the capital asset pricing model (CAPM). Returns on that portfolio can be compared with the theoretical returns, in which case the measure is known as Jensen's alpha. This is useful for non-traditional or highly focused funds, where a single stock index might not be representative of the investment's holdings. Definition in capital asset pricing model The alpha coefficient () is a parameter in the single-index model (SIM). It is the intercept of the security characteristic line (SCL), that is, the coefficient of the constant in a market model regression. where the following inputs are: : the realized return (on the portfolio), : the market return, : the risk-free rate of return, and : the beta of the portfolio. It can be shown that in an efficient market, the expected value of the alpha coefficient is zero. Therefore, the alpha coefficient indicates how an investment has performed after accounting for the risk it involved: : the investment has earned too little for its risk (or, was too risky for the return) : the investment has earned a return adequate for the risk taken : the investment has a return in excess of the reward for the assumed risk For instance, although a return