# Difference between alpha and ROI ratios?

Given the intuitive definition of "alpha" as the percentage advantage of a portfolio over its benchmark market, what is the difference between ratio (%) of ROIs and alpha?

What I mean by "ratio of ROIs" is:

$$(\frac{(\frac{price(portfolio,end_t)}{price(portfolio,start_t) })}{(\frac{price(benchmark,end_t)}{price(benchmark,start_t)})}-1)*100$$

This ratio seems an intuitive match for the intuitive definition of alpha yet if we plug in the numbers for the S&P 500 over the last few weeks as against a portfolio that rises in value at \$1 per minute (starting at$1000), the ratio intuition comes out on the order of 300% while the linear regression method between minute-to-minute fractional gains yields an alpha of 0.03%.

Here is the scatter for that comparison, with a y-intercept ("alpha") of 0.0003 and slope ("beta") of -0.004: • The answer is, at least in part, alpha scales with the time interval of its constituent data points. The ROI ratio definition is one data point with a long time interval containing all the many time intervals of the datapoints going into the y-intercept definition. Presumably, this means multiplying the y-intercept alpha by the number of data points will bring it into the same order of magnitude (if not exactly the same value) as the as the ROI ratio. awgmain.morningstar.com/webhelp/glossary_definitions/… Nov 10 '20 at 1:26