I use alpha as defined by Jensen in the late 60's. That is, as the premium return above market averages. Often referred to as some added portfolio management skills.

A portfolio's expected return can have for expression: E[F(t)] = F(0)∙(1 + r_m)^t which represents some initial capital compounded over time.

In the stock market, r_m is given away, almost free. Buy low-cost index funds, and the job is done. No gurus, no academic papers needed, no programming of any kind, no AI required whatsoever. Simply keep on adding to your retirement fund as you go along, and beyond. It is what could be called an active trader industry killer.

If you look at the Capital Market Line definition found on Wikipedia you have: “This abnormal extra return over the market's return at a given level of risk is what is called the alpha.” This is expressed as: E[F(t)] = F(0)∙(1 + r_m + α)^t. Simply using what has been defined over half a century ago.

Jensen, in his seminal paper, wanted to show portfolio managers' skills as the reason for higher performance. The proposition was simple: if on average there were any management skills, they would show. And then, the following proposition would hold:

F(0)∙(1 + r_m + α)^t / F(0)∙(1 + r_m)^t > 1

However, what he found was a negative alpha of -1.7%. In fact, stating that the average portfolio manager was detrimental to his portfolio's health. You can imagine the uproar at the time. So, to appease the pundits, he later attributed the -1.7% negative alpha to commissions and fees. Nonetheless, it still said the same thing: average alpha contribution, technically zero, not even enough to cover commissions. This is as expected by Modern Portfolio Theory and efficient markets which do put expected alpha at zero. But, markets might not be that efficient!

The capital market line still holds. It is tangent to the efficient portfolio frontier at r_m and has a beta of one. The following chart might illustrate this better:

Capital Market Line

You have the usual CML (upward dark blue) with its risk-free intercept (r_f) and its tangential efficient frontier contact point at r_m. What is above the CML is considered as alpha, as a higher return than what was available from the CML. That you mitigate your return by having some of the assets at the risk-free rate, or leverage the portfolio to reach higher returns than r_m, Modern Portfolio Theory still does not see any alpha generation as long as you stay on that line.

To achieve more, we will have to do more. Alpha is just not given away, it has to be taken, extracted from what is there. It does not need to be a delicate process. It can be as rough as we want or the result of something we did that just turned out that way. Meaning, we could get some alpha and not really know why. But, I expect we would still take it anyway.

We can also get negative alpha, and this is simply underperforming the averages (r_m), the CML or even the efficient frontier. For sure, if you get negative alpha, meaning operating below the CML line, your better choice was, in fact, the index fund thingy. No hassle, no work and higher returns.

Now, if you go for zero-beta hedging by having half of your portfolio long and the other half short, you should get the red line with the Quantopian alpha definition which is, in reality, the return intercept that is very close to the risk-free rate (r_f). If Quantopian wants to define that as alpha, it is their choice. I for one am not changing definition. In the Quantopian alpha, I only see a subdued return that is very close to the risk-free rate.

That the regression line be: r = α + βx, or CML = r_f + β(r_m – r_f), it is the same line.

In clustered hedge trading, what I should expect to get is the differential drift since the hedged portfolio would have for equation: r_h = r_f + β(r_m – r_f) – β(r_m – r_f) + Σϵ, with Σϵ being the sum of residuals tending to zero. There can be a profit only if I make this binary directional bet in the right direction making it a low volatility bet with low returns, but still taking up my time. This is exemplified by the red line on the chart. I can not call that alpha generation.

To have stocks in a cluster, they need to be highly correlated, hence the differential drift is small. But this would still require making a directional bet. All that was done was reduce volatility to such an extent that you are forcing your portfolio to get by below its efficient frontier. You sacrifice return for stability. It is a choice.