What it is:
The fundamental quandary for investors of how get the highest return possible for the least amount of risk can be measured by alpha. It is a measurable way to determine whether a manager's skill has added value to a fund on a risk-adjusted basis.
The very existence of alpha is controversial, however, because those who believe in the efficient market hypothesis (which says, among other things, that it is impossible to beat the market) attribute alpha to luck instead of skill, and base this belief on the fact that most managers fail to beat the market over the long-term.
How it works/Example:
Let's assume you are a portfolio manager who expects your client's portfolio to return 15% next year. The year goes by and the portfolio actually returns 16%. In its most basic sense, the alpha of the portfolio = 16% - 15% = 1%.
Mathematically speaking, alpha is the rate of return that exceeds what was expected or predicted by models like the capital asset pricing model (CAPM). To understand how it works, consider the CAPM formula:
r = Rf + beta * (Rm - Rf ) + alpha
r = the security's or portfolio's return
Rf = the risk-free rate of return
beta = the security's or portfolio's price volatility relative to the overall market
Rm = the market return
The main part of the CAPM formula (except the excess-return factor) calculates what the rate of return on a certain security or portfolio ought to be under certain market conditions. Note that two similar portfolios might carry the same amount of risk (same beta) but because of different alphas, it's possible for one to generate higher returns than the other. This is a fundamental quandary for investors, who always want the highest return for the least amount of risk.