A fund returned 12% with a market beta of 1.2 in a year when the market returned 8% and cash returned 2%. Compute its alpha, and say what a positive number there fails to establish.

A fund returned 12% with a market beta of 1.2 in a year when the market returned 8% and cash returned 2%. Compute its alpha, and say what a positive number there fails to establish.

Approach: Take the single factor expected return at the stated beta and subtract it. Then ask what any factor left out of the regression does to the intercept.

2.8%. The single factor expected return is rf + beta*(rm - rf) = 2 + 1.2*6 = 9.2%, so Jensen's alpha is 12 - 9.2 = 2.8 points. The number is a regression intercept, so it captures everything the market factor fails to explain, and every omitted factor exposure is paid straight into it. If small stocks beat large ones by 10% that year and the fund carries a 0.3 loading on size, 3 points of the 2.8 belong to that factor and the true alpha is negative. One year also gives a standard error on alpha of roughly the residual volatility, so 2.8% against a 6% residual volatility sits half a standard error from zero and establishes nothing about skill.

Follow-up: How many years of monthly data would make a 2.8% alpha with 6% residual volatility significant at t = 2?

Key concepts: jensen's alpha, market beta, factor exposure, residual volatility.