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To Alpha, or not to Alpha

Q Alpha

@Guy: I understand for illustration purpose the assumption of ± 1% alpha factor.. are there general "scale" for alpha?
For instance, is 0.01 a threshold level compared to, say 0.05 or 0.10 being good, and 0.50+ to be exceptional?

@Karl, the question is almost irrelevant having no real predictability at that level. What would be the alpha on a particular stock in 40 years? I certainly can not answer that. Well, yes I can: I do not know. It might even be nonexistent to outright negative pushing a stock to extinction.
Evidently, it should be the highest possible alpha, even higher than what was presented. This would apply to a portfolio of strategies as well; as I illustrated in the Alpha Vertex thread:

The maximum alpha on the above chart was 0.10 making it a maximum 20% CAGR. Close to Mr. Buffett's long term CAGR. The range is from 20% down to 1% CAGR in steps of 1%. In real life, these lines would be jagged, but the end points would still hold. There is no lost in generalities by having smoother lines.

Traders don't seem to consider such a chart, but they should. Even in trading the same phenomenon occurs. Traders could compensate, even accentuate this kind of development. Notice that the chart does not say continuous rebalancing to equal weights. On the contrary, it says let your high fliers shine, let them take in all the weight they can.

However, I hardly see any presented code on Q even trying to account for it. They will trade stocks like AMZN, AAPL,..., make a few bucks out of them, but have no provision for keeping them in their portfolios even though it was easy to do so. What I often see in strategies is throwing those few best performers away as if they had no use or need of them. Sure, trading strategies can not do everything, but... throwing away the best makes it that much harder to outshine with the rest.

That chart is something Mr. Buffett understood a long time ago. It is a side effect of having a portfolio of stocks each growing at their respective CAGR rates. The highest of the group will take the lion's share of the returns. We should therefore facilitate its growth and not curtail it.

5 responses

It sounds like what you're describing is momentum. I agree I agree that it's typically not smart to rebalance from your winners to your losers... but sometimes it is. Sometimes that kind of rebalancing is advantageous. Case in point, when a mean reversion force kicks in. If it were simply a matter of allocating your portfolio in proportion to recent returns, then you can very quickly replicate that strategy with pipeline and see for yourself that this premise isolated fails to beat the market average. Returns, alpha, and beta are never consistent through time or easily predictable. Past performance is not a reliable indicator of future yada yada yada...

Warren Buffet's strategy is also a bit more nuanced. He buys companies at extreme discounts that he's checked out and that are in industries he understands really well. So basically he knows it's going to mean-revert. As you say Karl, he knows the real value is higher than the price.

Obviously in a stationary portfolio the best performing assets will grow in relative weight. So it's not clear to me how this applies to algorithmic trading in general, except in specific cases, like for example with momentum strategies where it makes sense. Each case is different and worth investigating. But as per the example, the positions are larger because they have outperformed so far, but not necessarily because they'll continue to do so.

Are you guys referring to alpha as a shorthand for returns, or as it is used on Quantopian -- idiosyncratic returns?

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.

Understood the Jensen's Alpha in E[F(t)] = F(0)∙(1 + r_m + α)^t

Suppose that +α can be captured by Q, would the Zero-Beta line not assume the new normal at E[rm] from the Rf intercept?

@Karl, I would say no. This would be like extracting some alpha out of thin air.

The problem is in the two definitions of alpha. Quantopian uses the intercept of a regression line on the return axis while Jensen's alpha is what is above the average market return, an expression of portfolio management skills. The intercept will not change because you change its name.

Maybe I should have expressed this using:
E[r_q] = r_f + β(r_m – r_f) – β(r_m – r_f) + Σϵ = r_f + Σϵ where r_q is the Quantopian expected return.

Take it like this: if you go long r_f and short the equivalent of r_f in some manner, how much do you think you are going to get? I think you would get about the same as: β(r_m – r_f) – β(r_m – r_f).

What has not been shown to my satisfaction in the zero-beta simulations I have seen is: E[r_q] > E[r_m] when all expenses are included: commissions (including minimums), slippage, margin and leveraging fees. This does not even touch the problem of shorts availability when your program wants them.