## Beta

The alpha and beta of a manager vs. a benchmark are obtained by fitting a straight line to the points in a scatter plot of the market returns vs. the manager’s returns. Alpha is the intercept of this straight line, while beta is the slope. Hence, if the market returns change by some amount x, then the manager returns can be expected to change by Beta * x.

Beta is defined as:

(covariance of manager and benchmark)
(variance of benchmark)

More explicitly, this is:

where:

n = number of returns
mi = i-th manager return
= average manager return
bi = i - th benchmark return
= average benchmark return

Beta is a measure of systematic risk, or the sensitivity of a manager to movements in the benchmark. A beta of 1 implies that you can expect the movement of a manager's return series to match that of the benchmark used to measure beta.

StyleADVISOR includes two Alpha and Beta statistics: Alpha and Beta and Cash-Adjusted Alpha and Beta. The Cash-Adjusted Alpha and Beta subtract the risk-free rate from the returns of both the manager and benchmark. Otherwise, the formulas are the same. It is the Cash-Adjusted versions that appear in StyleADVISOR's "Manager vs. Benchmark" chart and table.