I think the chart in the website linked has a pretty good one stop shop to see how PE funds have performed historically using direct alpha metric and the S&P 600 index. What is missing is that direct alpha has no concept of risk adjustment which is required for fiduciary work. We know that traded firms are levered in the 30% range and PE firms in the 70% range so just that would suggest a beta of around 2 if assets had similar risk. I think a more reasonable guess is 1.3-1.5 but it is up for debate.