Separate slope from average excess return
Beta equals sample covariance(strategy returns, benchmark returns) divided by sample benchmark variance. Let mean strategy return be μp, mean benchmark return be μb, and periodic risk-free return be rf. Arithmetic alpha per period is α = μp − rf − β(μb − rf). The displayed annualized arithmetic alpha is α multiplied by the entered periods per year. This scaling is a linear convention and does not compound alpha into an annual investment return.
A paired three-period example
Take benchmark returns of −1%, 1%, and 3%, and strategy returns of −1%, 2%, and 5%. Benchmark mean is 1% and strategy mean is 2%. Sample benchmark variance is 4 and covariance is 6, producing beta of 1.5. With a zero periodic risk-free input, alpha is 2% − 1.5 × 1% = 0.5 percentage points per period. For monthly observations, simple annualization gives six percentage points.
Choose a benchmark that answers the question
A zero-variance benchmark cannot identify beta. A highly volatile strategy can also have low beta if its movements differ from the chosen benchmark, so beta alone is not total risk. The model does not establish statistical significance, adjust for multiple strategy trials, or control additional factors. Match timestamps and fee treatment, and convert an annual cash rate to the intended periodic convention before entering it. Changing the benchmark changes the interpretation of alpha as well as beta.